Online poker - rigged or not? A case study: Pokerstars

2017

ONLINE POKER - RIGGED OR NOT? A CASE STUDY: POKERSTARS

IONUÈš APAHIDEANU, PhD 0

The story My relationship to poker has been a sinuous one for as far as I remember. In hindsight, it seems like that kind of first love that one, unable, or unwilling, to forget, keeps returning to repeatedly over a lifetime.

I remember first trying the game back when I was a kid, maybe eleven-twelve years old. I was playing with my cousin - the classic five cards draw. Back then, it was the only type of poker we knew. Nowadays there are so many versions of the game, one can hardly keep track of them. I also vividly remember how, in order to make every hand more spectacular, more adrenalin-laden, like the ones we saw in the movies, we used to eliminate from the deck all cards lower than seven. Thus, we got to enjoy plenty of "sensational" duels Aces full of Jacks vs. three Queens, Kings quads vs. the nut flush, etc. Nowadays, they seem to be called "coolers" and omnipresent on online poker sites. But let's not go there, yet.

Time passed, with me occasionally playing the game with friends, until I got to university. There, and then, I rediscovered poker. Or vice versa, can't say exactly. We engaged in the most passionate relationship possible. True love, what can I say!

It lasted for about two years.

Then time passed again, with me getting older, marrying, getting a job - you know, the usual stuff. Until the dawn of the Golden Age of poker: Chris Moneymaker winning the WSOP main event. All of a sudden, everywhere I looked around, it seemed the whole planet was playing Hold'em! Specialized sports' TV stations were broadcasting tournaments or high stakes cash games, there was talk everywhere about some hand Ivey had won against Antonius, Negreanu was becoming the most popular player around, the "Poker Brat" was having a problem with Northern European players to the amusement of TV viewers worldwide, and it seemed that everyday a new poker platform popped up on the Internet.

It was simply impossible for me to resist the temptation. I downloaded the Pokerstars platform, the first one I found on the Internet, and I started playing again. Only this time solely on play-money: I felt I had gotten older, or shall we say 'matured', and didn't need the adrenalin of real money anymore. In parallel, I met with friends on poker nights. We had all bought one or another version of a poker kit, and played whenever we could, as a hobby. It was fun!

Gradually, the popularity of the game decayed worldwide. One day, I remember replacing my desktop computer with a new one, when noticing the poker application on the old one. I asked myself: do I still need this? Nah, barely played it the last two years, and most likely won't need it anymore.

Almost another decade passed yet again, with me playing the game on maybe a handful of occasions, live, among friends, before I once again returned to poker, but this time in more serious manner. Feeling a bit rusty after the prolonged break, I re-installed the Pokerstars application and granted myself about two months of playing on virtual, playmoney, to get back in shape, before jumping to real-money games. Eight to ten, sometimes twelve or more hours a day. Simultaneously, every day I kept (re-) studying the game theoretically, devouring every book or article I could get my hands on: the psychology of the game, video analysis, the tactics, the mathematics.

Admittedly, the first time during this training period that I lost three times in a row holding pocket Kings, it felt a bit awkward, and even more so after I calculated the probability of this chain of "accidents" happening. But, still, I had barely restarted playing, right, and accidents do indeed happen in nature, right? The next day or so, at a six-players table, three players had been dealt hole cards of the same suit, all of them eventually making a flush. It did make me raise an eyebrow, but, again, this accident was also "possible", as 1

some online poker spokesperson always say. Another day, in another session, I bumped into a higher pocket pair three of the five times. Again, I jumped into calculating the probability.

The next day, I lost with Aces against Kings, on the river. A few minutes later, however, my pocket Queens defeated the opposition Aces. And so on, for the whole two months or so of training. Things seemed a little bit dubious, but all my theoretical and practical knowledge of probabilities, my very background of a researcher made me 'know better': the sample of hands played was still arguably small, randomness in nature implies even such "accidents", overall the figures could have been normalized, etc. Convincing enough for me to dismiss all of the online allegations that were claiming online poker in general and Pokerstars in particular would be "rigged", allegations that I had begun finding on the Internet, I didn't find any solid, scientific and therefore credible demonstration of such preposterous-seeming claims. Moreover, despite those awkward events that I had occasionally witnessed on the platform, I thought my training was going better than I hoped for: I had more than doubled my initial stack of play-money, gaining almost 20k big blinds in a couple of months.

All these elements considered, I threw in a couple hundred dollars (not more, given my remnant suspicions), then also bought online a poker data-analysis software, and started playing against the "grinders" at the lowest stakes levels. With no effort of readapting to everything a real money game supposes, things were going smoothly. True, already on my first day of playing (April the 9th), I flopped directly an Aces full, then I lost with pocket Queens against KJ off-suite, the opponent flopping a trip of Kings, but only eight minutes later, I was once again dealt pocket Queens, this time winning the pot, etc. In retrospect, maybe it was the fact that I had ended the session on a little profit that made me not pay much attention to some otherwise suspect events.

Things went on and after a good ten thousand hands played, I was on profit, and in full line with my elaborated plan of gradually moving up through the stakes in order to cash in enough to pay the entry fees at live tournaments. All the way to the WSOP series!

However, despite everything going according to plan, I couldn't stop noticing that I had already witnessed some truly horrendous "accidents" in terms of anything that a genuinely "random" generator of numbers would actually suppose. Some of them were so truly grotesque, that I instantly screen-shot them, posting of few on my social media accounts. I had for instance already been through Aces vs. Kings duels twice as often than normal at a 6players table. I had seen some of the most improbable flops imaginable, the kind of one in tens of millions of hands. Within pair vs. pair duels I seemed to win visibly more often as an underdog than expected to. And my pre-flop all-ins were completely off the charts, the majority of them having an outcome contrary to the mathematically normal and expected one.

I went back to what still seemed like nothing more than online hearsay. I revisited all the hundreds of allegations and complaints already read and explored other thousands of new ones. On the basis of my mere 10k hands played, plus other over 60k played on play-money, I could relate to many of them. What was being said there had also happened to me! And things on Pokerstars did indeed seem a little "fishy", to say the least.

Still, I was torn. My inner voice of a researcher kept telling me that at least some of the events witnessed were hard to classify as anything closely related to randomness, and that the situation definitely needed to be investigated scientifically, while the voice of the poker player in me pushed me to keep playing, to win more money and follow my initial plan.

Eventually, the researcher's voice prevailed. I threw in another USD 150 or so, speculating a deposit bonus offered by the platform, then won some profitable Spin & Go-s, gathering enough money to even start experimenting things, without the risk of going bankrupt. Thus, contrary to what any rational poker player would do, I started calling my opponents tens of times, despite all the signals telling me I was already beaten, only to reach showdown, so that I could check if another "accident", like AA vs. KK vs. some third pocket pair, or more generally some other form of "cooler" had once again, absolutely "randomly", happened on Pokerstars. Whenever I was losing too much money on these little experiments at the lowest 2

stake levels, I climbed a level or two, where I went back to playing seriously, meaning winoriented, in order to make again enough money to continue testing things. And the more I played, the more showdowns I got to see, and I discovered new anomalies. In fact, any frequency, any indicator, any parameter I was regularly checking did not seem right: from the frequency of triple pair situations to the one of flopping at least a set when holding a pocket pair; or from the conversion rate of my flopped flush draws into flushes up to the recorded rate of winning as a favorite on the turn calculated as a percentage of the expected rate indicated by my equity.

Somewhere along the way, probably half way through my series of 55k hands played on real money, I gave up entirely any intention of making money on the platform. My new goal was clear: produce a serious, solid, statistically-based investigation of how truly "random" Pokerstars' algorithm is dealing the cards. I once again returned to the thousands of complains spread all over the Internet, this time evaluating them in a more applied manner and trying to reformulate them in the form of statistically verifiable hypotheses. Gradually, the design of the research was outlined. Afterwards the methods and items to be tested were expanded and refined. As a sample volume, I targeted a number of 50k hands played. I ended up having played over 55k. As a final step taken during the last three days of playing, I even engaged into a full-scale experiment over more than 2.5 k hands: every time I had been dealt a pocket pair, regardless which one exactly, I simply open-limped, then check-called on each and every street, with the sole purpose of dragging my opponent(s) to showdown, where I could see his/her hole cards, which enabled me to expand and refine the data gathered.

In petty terms, let's put it like this: at the end of this enterprise, my net financial loss, all things considered, barely amounted to USD 110. It has been one of the most profitable investments I could have possibly made: it has led to a research that represents, to the best of my knowledge, the first statistical investigation of an online poker platform done by an external and truly independent source and simultaneously made freely available to anyone in the world.

Above anything else though, the truth is the truth. One cannot put a price tag on it. And if the game on a poker platform is rigged, then the truth has to be known. All the players still actively engaged in online poker deserve the truth. The game of poker - that kind of first love that one cannot or will not forget, deserves the truth.

The following presents this research.

3

Contents I. INTRODUCTION .................................................................................................................................... 6 I.1 The (more than) methodological challenge ................................................................................... 6 I.2 What and how this research investigates ...................................................................................... 9 I.3 Main hypotheses tested and limits of interpretation .................................................................. 14 I.4 General approach, structure, and nature of findings .................................................................. 18 II. HOLE CARDS' DISTRIBUTION ............................................................................................................. 21 II.1 General statistical analysis of the hole cards' distribution on the platform ............................... 21 II.2 Targeted analysis ......................................................................................................................... 24 II.2.1 Suited hole cards' distribution ............................................................................................. 24 II.2.2 Hierarchical groups of hands in relation to their dealing frequency ................................... 28 II.2.3 Pocket pairs' distribution ..................................................................................................... 29 II.3 Pokerstars' "coolers" ................................................................................................................... 32 II.3.1 Coolers.................................................................................................................................. 32 II.3.2 Pocket pair vs. pocket pair: a Pokerstars "classic" ............................................................... 38 II.4 An experiment: (open-) limp-check-calling with pocket pairs all the way to showdown ........... 50 II.5 Summary of the findings ............................................................................................................. 53 III. THE FLOPS: when you take the bait ................................................................................................. 56 III.1 An overall look at the flops on Pokerstars ................................................................................. 56 III.2 A case study: my pocket Queens ............................................................................................... 65 III.2.1 The data commented .......................................................................................................... 65 III.2.2 Main findings ....................................................................................................................... 72 III.3. A quadruple targeted analysis of Pokerstars' flops ................................................................... 82 III.3.1 Flopping a set or better when starting with pocket pairs ................................................... 83 III.3.2 Flopping a flush (draw) when starting with suited hole cards ............................................ 84 III.3.3 Flopping a straight (draw) when starting with middling connectors .................................. 86 III.3.4 Flopping at least one pair when holding un-paired hole cards ........................................... 88 III.3.5 A retesting of the leveling the field hypothesis .................................................................. 90 III.4 Summary of findings .................................................................................................................. 96 IV. AFTER THE FLOP: when things go south ........................................................................................ 101 IV.1 An analysis of six variables on the post-flop streets ................................................................ 104 IV.1.1 Open-ended / double inside straight draws turned into straights ................................... 106 IV.1.2 Flopped inside or semi-open straight draws turned into straights .................................. 108 IV.1.3 Flopped sets turned into full houses or quads ................................................................. 109 IV.1.4 Flopped two pairs turned into full houses ........................................................................ 110 IV.1.5 Flopped flush draws turned into flushes .......................................................................... 110 IV.1.6 Backdoor flush draws turned into flushes ........................................................................ 111 IV.2 Integrating the findings and re-contextualization ................................................................... 112 4

IV.3 The "Riverstars" (sub-)hypothesis............................................................................................ 122 V. A DEEPENED (RE-)TESTING OF THE LEVELING-THE-FIELD HYPOTHESIS .......................................... 130 V.1 A summary of the findings so far .............................................................................................. 130 V.2. Pair vs. pair duels ..................................................................................................................... 132 V.3 A methodological problem and a revisiting of my limp-check-call experiment ....................... 135 V.3.1 A methodological mini-case study: my pocket KK hands .................................................. 136 V.3.2 A revisiting of my limp-check-call experiment ................................................................... 143 V.4 Pre-flop heads-up all-ins ........................................................................................................... 145 V.5 Pre-flop equities in relation to hands' outcomes over a sample of 2k showdowns ................. 151 V.5.1 My perspective................................................................................................................... 155 V.5.2 The favorites' perspective .................................................................................................. 156 V.5.3 A parenthesis: tournaments .............................................................................................. 161 V.6 Integrating the findings ............................................................................................................. 167 VI. CONCLUSIONS ................................................................................................................................ 171 APPENDIX 1 ......................................................................................................................................... 179 APPENDIX 2 ......................................................................................................................................... 180

5

I. INTRODUCTION "Online poker - rigged or not?" Quite an interrogation, right? Most crucially: how does one actually establish whether it is rigged or not? And, on second thought: what would "rigged" mean, precisely?

It is exactly the nature and scope of such a line of questioning and its implicit methodological issues raised that may explain why, from this very beginning and all the way through its four chapters up to the Conclusions, this undertaking may look like something totally unusual. It is not, strictly speaking, an academic paper, since it does not fully comply with the rigors of academic writing1. But nor is it an investigative, non-academic, report, since all concepts, methods and instruments employed are entirely and directly transferred from the academic, scientific world. Let's put it like this for the moment: it's none of these entirely, but it's simultaneously both of them.

It is at least something for sure: a pioneering effort. Despite thousands, if not tens of thousands, of complaints spread all over the Internet about online poker in general, and the Pokerstars platform as a case in particular, being "rigged", so far, to the best of my knowledge, there has been no single attempt, by a truly independent source2, at providing a rigorous, extensive, and ideally credible demonstration of one or more of the accusations formulated in social media discussions. As a consequence, all that remained essentially within the realm of public debate has been something like a dialogue of the deaf: demoralized players, thousands of them deserting Pokerstars in recent years, kept accusing the platform of being rigged, while the latter's representatives kept delivering the same, already standardized, excuses as answers, such as: online, people play far more games than in real life, so that those terrible "accidents" invoked by so many critics would in fact be only natural to occur over so many hands; people tend to remember only, or to a higher rate at least, those situations when they were unlucky, and not also the ones where they have been lucky; the samples of hands gathered as evidence by one or another criticizing player have been too small to allow any reliable generalizations, etc. Such answers seemed to only further frustrate many critics, with the accusations escalating and expanding slowly, but surely, up to the point where pejorative terms such as "Riverstars" or "Jokerstars" actually became so widespread, that they turned into distinct hashtags on Twitter and even made their way into urban dictionaries 3. To this, Pokerstars retaliated by apparently hiring an entire legion of paid "drones" entrusted with the sole task of countering criticism and "cleaning" the company's reputation all over the Internet.

I.1 The (more than) methodological challenge The conflict continues to this day. No serious and credible statistical testing has been advanced to break the deadlock one way or the other. And it was only after concretely engaging in the enterprise detailed below that I have understood why. No wonder it hasn't been done before: aside from the tremendous amount of necessary work, the actual, coreproblem is one of a (more than) methodological nature: how does one reformulate online accusations, so differently expressed, respectively understood, by people all over the world wide web, in a form that allows a rigorous scientific testing, and would be approved as such by the scientific community, while nevertheless trying permanently to also make it fully understandable for poker players who are not familiarized with statistics? And therein lies the 1

Some expressions in the text may seem frivolous, there are bold-marked fragments in the middle of a text and other "unorthodox" editing instances, partially anecdotic analogies, numerous screen captures of hands played, meant to simplify the understanding of the reader, etc.

2 Meaning any other source than the one company who "audits" the world's biggest online poker platforms.

3 See for instance http://www.urbandictionary.com/define.php?term=RiverStars (retrieved July 16, 2017).

6

problem: in order to adequately tackle the problem addressed in this investigation, one would normally need two simultaneous sets of knowledge: of poker and of statistics. Reckoning that such a double requirement would inevitably and severely narrow down the audience, whereas the problem of online poker potentially being rigged seems a problem of a much bigger impact, and as such of interest to a considerably broader audience, I was left with a fundamental first challenge, the one of bridging, as much as possible4, a serious gap between two types of potential readers.

Thus, one the one hand, readers who know solely statistics, but not poker, may for instance not understand why in some instances, classic methods and associated instruments and terms of statistical hypothesis testing (T-test, p and Îą, "false positive", Z-test, etc.) could not, unfortunately, be applied in this investigation - it has to do with the specific way the game is played, and, more importantly, with how a data-storing software actually stores, processes, and displays all data (more about this below). For the same reason, the usual approaches such as testing a "null hypothesis" also encountered certain limits of their applicability. Or, to give just another example as a preface: this first category of readers, unless they know poker, might not fully understand why for instance an analysis of the series of the river cards (or the first cards on the flop, for that matter) across all my games played otherwise a correct randomness testing solution - cannot be applied, since, when looking how exactly the game is played, specifically what hole cards are usually kept and played by most players, it becomes clear that the distribution of the cards on the river street actually should not be expected to be random.

One of the biggest issues in this regard has been the otherwise classic problem of dealing with incomplete series of data, and consequently the nature, limits, and reliability risks of extrapolations.

Subsequently, but only in part overlapping, over the course of the research, I came up with methodological instruments that may seem difficult to understand at first glance, such as the matrix combining two plus one dichotomies: a) the one between a.1.) hands played effectively by me (meaning hands where I did remain actively engaged in the game to showdown), and a.2.) hands where I folded somewhere along the way before showdown, but at least two players have continued playing all the way till showdown. The latter allows the Hold'em Manager 2 software5, directly connected to the platform's software (within certain limits, obviously) to display a series of information such as: the known hole cards of both the two players having reached showdown, and mine, together with the equities, thus allowing me to somewhat counterfactually establish whether I would have actually won the hand, had I stayed in the game; b) the one between b.1.) hands with a certain "outcome" (a term employed either with reference to the winner or, and more often so, in relation to my hole cards and targeted combination6) vs. b.2) hands with an uncertain outcome, meaning for instance hands that have ended on the flop, or on the turn, before thus knowing for a fact whether I would have hit my targeted combination or not, respectively if I would have won.

The latter category raised a whole new category of problems in terms of how to correctly approach the factual/counterfactual outcome, etc.

4

Meaning that, even considering all my clarifications and simplifications of explanations and interpretations, the reader will still require a minimal knowledge of, especially, the ABC of poker (the hierarchy of combinations, the positions at the table, etc.). I can only hope I have let this amount to be indeed the minimum possible.

5 See below.

6 E.g. the binary outcome of me making, or not, a flush, starting from a flush draw on the flop.

7

As such, throughout my research, I had to often innovate, and sometimes even build from scratch, (new) approaches, methods and indicators, such as the above matrix, or the alternation of personal and favorite's perspectives in terms of equities and corresponding ways of accounting the results of the analysis, or the 13-lines of cards scale, etc.7 Aside from these methodological issues, for this first category of readers there is potentially the additional problem of some of the very main, most commonly used, terms used by this paper, many of them possibly unknown to a non-poker player, a problem which I already tried addressing preliminary in the table below, and more extensively, whenever I felt necessary, throughout the four chapters of this investigation.

On the other hand, readers who only know poker, without however a certain amount of knowledge and skills in statistics and more generally in methodological issues, might find it difficult to form an opinion on some of the significance of the statistical indicators used throughout the analysis: correlation coefficients, standard deviations, squared Rs, probability values and confidence intervals, sample representativeness, kurtosis and skewness8 of a particular distribution of hole cards or of the conversion rates of a draw depending on the type of the hole cards, average recorded frequencies as percentages of the corresponding expected one, etc. Nor may they know how many of the probabilities discussed throughout the analysis are actually calculated (although in many more complicated, or at least unusual, cases, I did display the exact method of calculation in order to also avoid any potential controversy).

It is in consideration of this latter category of audience that, without any intention of offending the former one, I have tried, sometimes maybe in a frivolous manner, to simplify the explanations and interpretations as much as possible without altering the very meaning and significance of the findings, by employing various accessible analogies and metaphors, such as mixing boiling and freezing water in order to get a normal temperature, or playing the roulette in a casino, or flipping a coin, or making a poll, etc.

Additionally, the same task of trying to make everything understandable to an audience as broad as possible, combined with the pioneering profile of the research and the peculiarities of the reality investigated, may also explain in part the considerable size of this document: a final table for instance, with data reprocessed in chain, which in itself occupies let's say - only a few lines, however required a considerably larger space in advance for me to explain, in an broadly accessible manner, what, why and how I measured something.

And now, before moving on to what, under any other circumstances, an Introduction should actually address, allow me to briefly clarify, to all readers, the most commonly used abbreviations or terms used in an adapted meaning9: 7

And it is also in this sense that the present Introduction is an atypical one: if, as customary, it would outline to the needed depth level the methodological approach employed, the Introduction would risk becoming - and this is no exaggeration - as voluminous as two chapters combined.

8 The very second I typed this term, the Microsoft Word dictionary immediately notified me, by underlining it in red, that it does not recognize the word. This illustrates my above point.

9 In addition to these common terms, sometimes, for reasons of simplicity, I have used slang terms maybe unfamiliar to persons not playing poker, such as "quads" (instead of "four of a kind"), "cooler", or certain expressions such as "to hit a two outer", "to draw dead", "to flop a full house", etc.

8

"Hold'em Manager 2" software open-ended straight draw gut-shot straight draw double belly buster straight draw (a.k.a. double-inside straight draw) Random Number Generator. According to Pokerstars' website, it is "a system, device or module that creates a sequence of apparently unrelated numbers" R squared = "coefficient of determination", essentially signifying what Rsq proportion of the dependent variable's recorded variation can be pinned on, or is predictable from, the independent variable (Pearson) (linear) correlation coefficient correl. "average"; unless differently specified, means "weighted arithmetic mean" "avg." suited combination(of hole cards) XY-s off-suite combination XY-o hole cards the starting hand (the two cards a player is dealt, as opposed to the "board" or "community" cards) "hand" used alternatively in two meanings: 1. as an individual game in itself (e.g. "I have played 51,989 hands in cash games on the HM2 OESD GSSD 2xSD RNG

platform")

2. as a signifier of the 169 versions of hole cards that can be dealt to a player in the Texas Hold'em game. In this second meaning, the term "hand" does not differentiate among the four possible suits, only indicating whether the hand is suited or off-suite. As such there are 169 "hands" corresponding to 1,326 "combinations" possible; e.g. an AA hand consists of 6 possible combinations, a AK-o hand of 12, an AK-s of 4, etc. "street" a term used not in its standard meaning, but as an adapted denominator of a game's four phases: pre-flop, flop, turn, and river (cf. Wiki:) the scaled commission fee taken by a card-room operating a poker "rake" game "equity" the probability of one player to win or tie the hand (as displayed by both Pokerstars and the HM2 software. Both indicate the combined probability of winning or splitting the pot) "favorite" the player best positioned in terms of equity at a certain moment of the game One last specification: unless differently mentioned, the chapters use the same general 4-color shading system in all tables: light green: recorded deviation of < +5% of the normal, mathematically expected, frequency; dark green: bigger than +5%; pink: smaller than -5%; respectively red: bigger than -5%.

I.2 What and how this research investigates With all of the above being necessarily (pre-) clarified, allow me to now properly "introduce" the research: it analyzes the series of 55,320 Texas Hold'em hands that I played between the 9th of April and the 28th of May, 2017, on real money10, on the Pokerstars platform11, under the username "Apasu76". Of these hands, 51,989 have been played in cash games of a zoom-format, all of them in 6-players rings, at various stake levels. The other 3,331 have been played in tournaments, within the same timeframe, in different competitions (Sit&Go, Spin&Go, and, mostly, proper multi-table tournaments (MTTs)) in various numberof-players-formats, from a minimum of three up to a maximum of nine.

10 11

As opposed to so-called "play money". http://www.pokerstars.com

9

Given certain problems of standardizing probabilities, many of which differ depending on the number of players at the table, most of the analysis covers only my 51,989 hands played in cash games. Whenever methodologically possible (and always specified as such), the analysis is expanded to cover all hands played in both cash games and tournaments.

10

In terms of the representativeness of this sample of hands, for the reader who is not familiarized with either statistics, or the game in itself, allow me to put it like this: there are 169 distinct hands, respectively 1,326 distinct combinations that a player can be dealt in a Texas Hold'em games. In relation to my combined number of 55,320 games, this means that, on the average, I have been through each starting hand 327.3 times12, and through each starting combination a number of 41.7 times.

As technological support of the research I have used the Hold'em Manager 2 software. Available for purchase online, this software stores a multitude of data, including video storage, pertaining to all hands one plays on a mutually agreed poker platform, such as Pokerstars. In addition to the storing function, it also allows certain processing of the data collected.

Specifically, for those not familiar with it, it separately stores data from tournaments and cash games, and displays among other things, in different selectable ways of arranging:  the hole cards of the user,  the time and date of each hand,  his actions on the four streets (e.g. "X" for check in the third column below, "F" for fold, etc.),  the board / community cards,  the stakes,  the user's net win or loss,  the pre-flop actions,  the frequency of dealing of all 169 hands (the lower-right smaller icon overlapped), etc.

Crucially, it also has video footage of the hands played. If a certain hand has not been played all the way till showdown, only the user's cards remain visible when double clicking it:

12

For more details, see the first table of the next chapter.

11

In the hand screen shot above, holding pocket 4s, I check-folded on the flop, so that all the other players' hole cards remain unknown.

If, however, a hand has reached showdown (regardless whether with the user still present at the table, or the hand continued by (at least) two of his/her opponents), HM2 will display all known hole cards and, for the first streets within the game, pot size, playing parameters of the opponents, each player's equity (the red figure below the hole cards), meaning the probability of him/her winning or splitting the pot at the given moment of the game (bottom left), etc:

12

This is how a hand looks when screen-shot directly on the Pokerstars platform (with me holding 54-s and having folded pre-flop):

And this is how the same hand is displayed in the HM2 database, captured at showdown, with my cards hidden, since I had already folded pre-flop, meaning my equity evolution in the left bottom icon is also not shown:

With - as far as I remember - no more than a maximum of seven exceptions, all screen captures presented throughout the paper are made from within the hands where I was personally involved in the game, and technically from within the HM2 database and not from the platform itself, for reasons of convenience related to the data that are displayed and can be correspondingly processed in an easy manner.

13

I.3 Main hypotheses tested and limits of interpretation With these technical support issues now also being clarified, let us move forward, and get to the actual point of this enterprise: to test whether online poker, in this case the Pokerstars platform, is "rigged" or not. As said, over the last years there have been thousands of complaints all over the Internet supporting this idea - discussions on online poker forums, mass media articles, innumerous screen and video captures of millions of hands, even a few lawsuits, etc.13 However, exactly because of this spectacular variety of statements, there has never been a clear, unified, precisely formulated, and universally accepted meaning, let alone demonstration, of what "rigged" actually means.

Hereby I propose an operational meaning of the term, defined by opposition to what the Pokerstars, and most online poker websites more generally, claim, i.e. that the software operates on the basis of a "Random number generator" (RNG), which, selfunderstood, deals the cards in a random manner. Conversely, in my reading, what the critics argue is exactly that Pokerstars' RNG distributes cards non-randomly.

13

See for instance, among so many: https://legitorscam.org/pokerstars-is-a-scam/, https://www.pokertube.com/article/is-pokerstars-rigged, http://www.pokerupdate.com/poker-opinion/1122-ispokerstars-rigged/ (see the comments, especially), http://www.pokerscout.com/AllReviews.aspx?id=1, http://forumserver.twoplustwo.com/25/probability/pokerstars-rigged-447349/, http://forumserver.twoplustwo.com/54/poker-beats-brags-variance/pokerstars-rigged-proof-tba-1559406/, https://www.youtube.com/watch?v=2Ty5NmT-4-4, https://www.youtube.com/watch?v=kuBvAWwRY_o, https://www.youtube.com/watch?v=EHQidtck9IY, https://www.youtube.com/watch?v=dt7cfdaTCbg, https://www.youtube.com/watch?v=vSnfaik92pc, https://www.youtube.com/watch?v=92tDwMyn0pI, http://pokerstars-rigged-scam.blogspot.ro/, https://www.complaintsboard.com/complaints/pokerstars-anaheimcalifornia-c366882.html, https://bonuscodepoker.com/aga-pokerstars-battle, or http://forumserver.twoplustwo.com/153/high-stakes-pl-omaha/massive-bot-ring-pokerstars-party-how-spotthem-1537778/

14

Now, under any normal circumstances, there actually should be no debate and nothing for me to research, since any, and I repeat, any IT specialist will tell you there simply is no such thing as a (truly, genuinely) "random" number generator. Any software operating with such an algorithm is conceived by a human mind, which makes it by default anything, but random. What players should expect at best is a pseudo-random distribution of cards. In this adapted meaning, what an online poker platform could do is at best a satisfactory enough mimicking, or simulation, of the randomness otherwise existent, and measurable as such, in nature. Depending on the degree to which their distribution of cards comes close to what a natural random distribution would look like, poker sites RNGs might subsequently be evaluated individually and hierarchically classified as more or less close to randomness.

In direct relation to this new adapted meaning and understanding, when further exploring the allegations made in the online realm, the all-so-frequent term "rigged" would implicitly define a deliberate, systematical, and (statistically) significant distancing from what randomness would imply. And this, as is probably obvious already, suddenly becomes considerably more difficult to verify and, if the case, demonstrate: whereas certain recorded variations and deviations from randomness pose no difficulty in terms of measurement and evaluation, given the existence of multiple exact, precise, evaluations of the normality (i.e. randomness) of a distribution, universally employed and accepted, a demonstration of intent, of deliberation, of premeditation, is a completely different matter. After all, accidents do indeed happen in nature. Similarly, an online poker RNG may suddenly be affected by an unintentional error, by a glitch, etc. Where does true intent begin, and how does one demonstrate it beyond any reasonable doubt? Obviously, there is no single, universally accepted, answer to such a question.

And herein lies the second fundamental challenge for the researcher of such a topic: how to interpret and evaluate whatever deviations from randomness may be found during the investigation in such terms that both the scientific community approves and the ordinary poker players understand. Should, for instance, a measured linear correlation coefficient of minus .81, covering 3k+ showdowns, between a player's recorded equity on a specific street of the game and the factual outcome of the hand, meaning him/her winning, losing, or splitting the pot, be enough? Of course, correlation does not equate causation, but what if such correlations are repeatedly discovered throughout the testing, and, additionally, and crucially, they all work the same way, meaning their effect is always the same, for instance making the underdog win more frequently than mathematically expected to? Or what if, for instance, when analyzing a player's pre-flop engaged heads-up all-ins, it is discovered that not only did he actually win at a rate of little below 3/4 of the expected win rate, but also, that this "deficit" has been recorded in seven of the eight methodologically-separated all-in categories? Does this indicate something like a deliberate and systematic non-random action of the software? Or, for fans of statistics: what does a p value of, say, 0.044 suggest and how should one relate to it when interpreting the results of a measurement? Interrogations such as these converge towards a fundamental idea: the heuristic limits of any methodological approach, and hence the need to take all figures for what they are and nothing more, nor less: statistical indicators, usually operating in a specific context pertaining to a certain confidence interval, a certain level of statistical significance, etc.

For both potential categories of audience, here is a quote from one of the best books on poker I have ever read - Barry Greenstein's Ace on the River: An Advanced Poker Guide.14 Amusing as it is, this hypothetical situation does seem to capture the very heuristic nature of the above-discussed problem, thus sparing me a considerable and most likely redundant amount of further explanation:

14

Last Knight: Fort Collins, CO (2005, p. 154).

15

Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and is comes up heads. What is the probability that it will come heads on the eleventh flip?

A novice gambler would tell you, "Tails is more likely than heads, since things have to even out and tails is due to come up."

A math student would tell you, "We can't predict the future from the past.

The odds are still even."

A professional gambler would say, "There must be something wrong with the coin or the way it is flipped. I wouldn't bet with the guy flipping it, but I'd bet someone else that heads will come up again."

On a more serious note, it is within the frontiers of the above-indicated limits of interpretation and adaptations of some common, but otherwise inoperative terms such as "rigged", that I formulated the main hypotheses of this investigation and subsequently tested them. A careful reading, over weeks, of the main allegations roaming on the Internet has led me to summarize the most frequent and serious ones as follows:  the hole cards on Pokerstars would be dealt in a non-random manner; most of these allegations further targeted a specific type of situation - the so-called "coolers" such as, notorious among the platform's clients, AA vs. KK vs. QQ three-way duels;  the pre-flop engaged all-in duels in a heads-up format (meaning 1 vs. 1) would show spectacularly abnormal deviations from anything mathematically expected;  the flop cards would further multiply the "cooler" situations, way beyond anything normal in terms of frequency, by simultaneously providing the players that remained at the table with promising, attractive combinations; Some critics specifically use the term "second shuffle", one supposedly occurring after the pre-flop round is finished.  after the flop, the turn and especially the river cards (hence the nickname "Riverstars") would also be dealt non-randomly, usually to the direct favor of the underdog, who, it is alleged, would win considerably more often than mathematically correct in probabilistic terms - hence frequent terms such as "leveling the field" or "balancing the odds";  roughly every - and here estimations differ - every fifth, tenth, or twentieth hand would be "manipulated" by a non-random acting software, meaning these hands would register an outcome contrary to the mathematically expected one;  many players at the cash games tables (but not only) would actually be bots, some of them acting to the direct benefit of Pokerstars, and with their tacit consent.

The general logic underpinning these allegations in terms of purpose, or motivation, is one and the same: these deviations from statistical normality implying randomness are deliberate and systematic and serve one purpose: the increase of the proportional rake the platform collects and is ultimately making a living of: "coolers" will make players bet, raise, and re-raise higher, which translates into a bigger rake to be collected; enticing flops act towards the same final outcome; a insidious assisting of the underdog prevents recreational players to lose too much money too fast, thus protecting the player base and the platform's source of income, etc.

With the last accusation, albeit it an extremely serious one, remaining obviously un-testable, at least from my position, I have rephrased and regrouped all the other ones in forms that will hopefully be approved by both the scientific community as being correctly formulated and verifiable and the poker players as accurately reproducing the content of the accusations. Oppositely to the general-umbrella equivalent of a "null hypothesis", which would state that anything pertaining to cards distribution on Pokerstars is entirely random (or 16

at least that the recorded deviations are not significant statistically), the hypotheses thus reformulated and correspondingly tested in this investigation are these:  Hypothesis H1: the "non-randomness" of the hole cards dealing; Pokerstars' RNG distributes the hole cards in a manner that is statistically significantly distant from what a normal, probabilistic, distribution should look like; This hypothesis is formulated solely in reference to a player's hole cards. o Sub-hypothesis H1*: the "coolers"; the recorded frequency of "cooler" situations is significantly higher than the normal one. This sub-hypothesis is formulated in direct reference to opponents and their hole cards.  Hypothesis H2: the "second shuffle" hypothesis; in similar terms, the flop cards are also being dealt non-randomly, in the sense that the factually recorded frequency of flopping useful combinations (for exactly the players that have remained at the table after pre-flop) are higher, category-by-category, than their mathematically expected, normal, counterparts; this includes any new addition of "coolers" or a development of new ones;  Hypothesis H3: the "leveling-the-field" (or "balancing the odds") hypothesis: throughout the entire game, the underdog is directly and significantly assisted by the non-random distribution of cards, in that he/she ends up winning significantly more often than mathematically expected to.

This hypothesis is tested solely across the series of cash games, and not also tournaments15, and assumes two meanings, or "dimensions", plus two sub-hypotheses.  a.) the first dimension is defined in relation to an opponent, in the sense that the "underdog", meaning the lower positioned player in terms of probabilities, wins more often than normal, to the detriment of the "favorite", meaning the player at the table with the mathematically highest probability to win or tie;  b.) the second dimension is formulated in relation to one's hole cards, stating that the better those cards, the less they actually end up winning eventually, respectively proportionally to their strength16. o Sub-hypothesis H3A: the "Riverstars" sub-hypothesis; the non-randomly dealt cards on the river are directly responsible for the underdog winning more frequently than probabilistically expected to; o Corollary H3B: "Every tenth hand is manipulated"; roughly every tenth hand that is effectively played (meaning not folded pre-flop by all but one player) is manipulated, meaning hijacked from its natural, normal, mathematically expectable, outcome. By its very formulation not a testable (sub-) hypothesis, but rather an orientative, indicative, statement, it admits at most circumstantial evidences, in the form of more instances in which the discovered deviations come close to the referential 10%.

15

The same online folklore claims that in tournaments the situation would actually be the opposite, meaning favorites would win more than expected. The underlying business-motivated logic is that, unlike the case of cash games, in tournaments, the only source of income for the platform consists of the entry fees paid by participants.

As such, the platform would have no interest whatsoever in prolonging the games by leveling the field, quite the contrary: the sooner a tournament ends, the sooner new ones can be opened, with more entry fees being paid.

16 This "strength" is made operational by employing a classification of the general, by default, probability to win or tie a hand for each of the game's 169 hands - a so-called "power ranking" of the hands. For more details, see the first chapter.

17

I.4 General approach, structure, and nature of findings The above-operated reformulation of online allegations in the form of testable hypotheses has obvious advantages in terms of the potential knowledge to be gained in a reliable, safe, manner, but it also has direct implications on the very structure of the research, one directly conditioned by the hypothesis' formulation and place of testing. Thus, in relation to the four phases of the game, some phenomena associated with the hypotheses are to be identified and verified on the (chronological) longitudinal, whereas others should manifest on the transversal of the phases, which allows neither of two possible ways of approach and subsequent structuring: on the longitudinal, meaning chronologically corresponding to the four streets of the game; on the transversal of the game's streets.

Given the peculiar nature of the phenomenon investigated and the corresponding way in which the general logical approach conditions any possible structuring, I have opted for a solution that once again may deem this investigation as one somewhat atypical: a mixture of longitudinal and transversal approaches: the first three chapters are delineated chronologically, corresponding roughly to the game's phases (pre-flop, flop, and after the flop), but chapters two and three are each supplemented with a transversal investigation grouping all findings related to the leveling-of-the-field hypothesis and its spin-offs, while the fourth, and last, chapter adopts a nominally transversal approach across the four streets by further expanding and deepening the testing of the leveling-of-the-field hypothesis.

In the framework of this general mixed approach, the first chapter addresses Hypothesis H1 and its derived sub-hypothesis. It starts with a general statistical testing of the hole cards' dealing in terms of the normality of the distribution and then continues with a targeted analysis of three specific distributions, or items: of suited hole cards; of hierarchically arranged groups of hands; of the infamous "coolers", with a particular emphasis on pocket pair vs. pair duels. Finally, it addresses a very revealing experiment that I tried on the poker platform: (open-)limp-check-calling each time I had been dealt pocket pairs with a twofold purpose in mind - testing the alleged randomness of the hole cards distribution and 18

alleviating the methodological problem of the unknown represented by hands that ended before showdown.

The findings not only confirm most of the online allegations, but also expose a spectacularly elaborated, sophisticated, way for the Pokerstars software to permanently keep the final, overall, average parameters absolutely "normal" at the end of the day, although the specific manner in which this is achieved has nothing to do with randomness, quite the contrary, it is rather "anti-random". It is a series of such findings throughout the investigation that strongly hint, and sometimes seem - no matter how much caution the researcher tries to keep - to bluntly indicate, beyond any reasonable doubt, evidences of intent, deliberation, premeditation.

The second chapter covers the flops in search of any traces of the alleged "second shuffle" that some claim would occur at the end of the pre-flop. In a first step, it statistically analyzes flops in a general manner. As a second step, all my hands where I have been dealt pocket Queens are instrumented as a case study of the flops. After the latter's summarizing and contextualization by expanding the same analytical approach over other pocket pairs, the third step is a quadruple targeted analysis of four flop situations: 1.) a set or better when starting with pocket pairs; 2.) a flush or flush draw when starting with suited hole cards; 3.) straight or draw when starting with middling connectors; 4.) at least one pair when starting with non-paired hands. These four variables address both the second shuffle hypothesis more generally, and the renewed or extended "cooler" situations in particular.

Multiple, statistically significant, mutually reconfirming and further reinforced by the discoveries within the other chapters, the findings of the second chapter cast a huge shadow of doubt over all claims stubbornly repeated by Pokerstars representatives of its software distributing cards in a "random" manner. Moreover, the one-way direction in which the quasi-totality of the recorded anomalies all manifest themselves in a systematical and synergic manner, i.e. towards an increase of the rake collected by the poker operator, considerably adds substance to the online allegations made by critics and further hints at premeditation, at pure intent, at a deliberate conception of the software to act in a certain direction that does not even try to mimic randomness.

The third chapter investigates the distribution of cards on Pokerstars in relation to what happens after the flop, meaning on the turn and on the river, in a both longitudinal and transversal approach. In a first step, it builds on the findings of the previous chapter, further measuring the post-flop recorded conversion or improvement rates of six draws or combinations in comparison to the normal, probabilistic, rates to be mathematically expected in a randomness-governed environment: 1.) of flopped OESD or 2xSDs into straights; 2.) of flopped GSSDs straights; 3.) of flopped sets into full houses or better; 4.) of flopped two pairs into full houses; 5.) of flopped flush draws into flushes; 6.) of flopped backdoor flush draws into flushes. The second step isolates and extensively tests, over no less than 3k showdowns, the "Riverstars" sub-hypothesis, one to be verified on the last street of the game.

Whereas, as shall be seen, the sub-hypothesis has been definitely disconfirmed in its strict sense, the collateral findings of the research however are some of the most shocking of the entire underlying investigation. In addition, the chapter's findings not only capture significant multiple differences to the events happening on the flop, in contrasts extremely hard to justify if presuming randomness, but also reconfirm both the non-randomness of Pokerstars' flops (via a different path) and the leveling the field hypothesis.

The last chapter's investigation is pursued on the transversal of the game's streets, specifically testing, in a deepened and extended approach the "leveling the field" hypothesis, right after summarizing the previously gathered findings. It targets four distinctively addressed items, by permanently comparing recorded win plus split rates with the 19

mathematically normal values to be expected in any environment supposed to be random: 1.) pure heads-up pair vs. pair duels; 2.) a revisiting of my (open-) limp-check-call experiment by focusing on the favorite vs. underdog comparative evaluation of probabilities and recorded results - supplemented by a methodological discussion on the limits of any extrapolation into hands folded pre-showdown; 3.) my entire series of pre-flop engaged heads-up all ins; 4.) a comparison, extended over a sample of over 2k showdowns reached among my played hands, between the pre-flop equities of the favorites at the tables and the final outcomes of the hands in terms of the winner.

The findings not only reconfirm the previous, partial, and disparate preliminary conclusions formulated, but also provide a considerably broader and deepened confirmation of the leveling-of-the-field hypothesis in ways and at values of the various coefficients and indicators employed that are in my opinion simply too many and too spectacular to allow any further talk about "accidents", "variance", or any such similar terms so often used by Pokerstars' representatives.

Additionally, although nominally outside the scope of the hypothesis tested in the chapter, tournaments that I played are, as a one-time exception, integrated into the analysis of the last chapter in a comparison to cash games operated from the same perspective of the favorite vs. underdog dichotomy. Even if only in light of the shocking nature of the findings, the reader may deem this exception as being justified.

20

II. HOLE CARDS' DISTRIBUTION When verifying the claimed randomness of a poker platform such as, in this case, Pokerstars, the first thing, in both chronological and importance terms, to be investigated, is the distribution of the hole cards. This chapter specifically addresses this issue. It starts with a general statistical analysis, and continues with a targeted analysis separately covering three items: suited hole cards; hierarchical groups of hole cards; respectively the pocket pairs' distribution. Separately, due to its significance and the shocking nature of the findings, the third step of the analysis addresses the so-called "coolers" situations occurring on Pokerstars, both generally and particularly in regard to pair vs. pair duels. A fourth step presents the results of an experiment I made in May 2017 on the platform, playing 2,573 hands over two consecutive days in a specific and deliberately conceived and applied manner, while the final part summarizes the findings of the chapter.

II.1 General statistical analysis of the hole cards' distribution on the platform In analyzing the hole cards' distribution on the poker platform, the table below highlights the general situation of the combined 55,320 games that I participated in (both the 51,989 cash games and the 3,331 tournament ones) in the timeframe April 9 - May 28, 2017.17 The results are integrated using the usual suited - off-suited - pairs diagonal matrix, with the suited cards occupying the upper-right triangle in relation to the pairs diagonal, and the off-suite hole cards placed in the opposite triangle. For each of the 169 hands / hole cards, the figure placed above in their cell indicates the hand's absolute frequency (i.e. how many times I was dealt that particular hand), whereas the figure underneath it displays its relative frequency, calculated as a percentage of its normal, expected, probabilistic, frequency. For example: within the total 55,320 hands played, I received pocket Aces a total 240 times, which is 95.9% of the normal, expected frequency (which is 0.4525% of 55,320, meaning I should have received Aces 250,32 times)18.

In terms of the shading, the colored are used in this manner: Shadings: recorded frequency as % of the expected frequency = below 89.9% = 90 - 94.9% = 95 - 99.9% = 100 - 104.9%

= 105 - 109.9%

= above 110%

At first glance, as displayed below, everything seems to be in order: all 6 colors seem relatively uniformly dispersed across the 26 columns and lines of the tables, while the aggregated figures on the last column and line are all within a reasonable Âą5% margin of variation in relation to their expected, normal, frequencies. True, no less than 20 of the 169 hands' recorded frequencies exceed that margin, which in itself is reason for some concern, but as said, at a superficial look at the table nothing particularly suspicious catches the attention.

17

All of the hands, as mentioned in the Introduction, being recorded and stored, including video footage, on the Hold'em Manager 2 software I have been using.

18 There are 6 possible pairs (i.e. distinct combinations) of Aces, comprising 0.4525% of the total 1,326 twocards combinations one can be dealt in Texas Hold'em.

21

Table: My hole cards distribution over 55,320 hands played:

Anyhow, not at peace with this superficial undertaking, I opted to specifically verify if there is any correlation between a hand's power ranking19 (i.e. their win + split rate in a 6-players game, meaning how strong a hand is, by default) and it's recorded frequency of being dealt to me calculated as a % of its normal, expected frequency20. Because power rank(ing)s differ among them depending on the number of players at the table, and the 3,331 hands I played in tournaments have been played in various formats (from 3 to 10 players at the table), I isolated the 51,989 hands played in cash games, all of them in 6-players rings.

Convincingly enough, for these hands, there is absolutely no mathematical relation between how strong a hand is and how often it has been dealt to me, the Pearson correlation coefficient being a mere +0.0079, which suggests independence between the two variables beyond any doubt.

19

For a detailed hierarchy of such power ranks in a 6-players game, see this excellent website: https://wizardofodds.com/games/texas-hold-em/6-player-game/ (retrieved June 27, 2017), the data being also copied in the Appendix to this document.

20 E.g. my AJs hands: normally, there are 4 AJs combinations among the total 1,326 possible in a 52 cards deck, meaning a relative frequency of 0.302%. Applied to the total 51,989 hands I played in cash games on Pokerstars, this would mean a normal, expected, absolute frequency of 0.00302*51989 = 156.83. In the recorded reality, I have been dealt AJs hands 146 times, which means that the recorded frequency represents 146*100/156.83 = 93.09% of the expected frequency. This percentage has been calculated for all the 169 hands, and then correlated to their power ranking.

22

Graph: hole cards' power ranking in relation to their dealing frequency: (N = 51,989 hands played in cash games, 6-players rings)

Furthermore, even when reframing the entire 169 hands sample analysis in terms of the normality of the dealing frequency, the recorded data convincingly meet the statistical requirements necessary to be considered a normal, Gaussian, probabilistic distribution: Graph: Dealing frequency of hole cards (169 hands) in terms of normal distribution: (N = 51,989 hands played in cash games, 6-players rings)

Thus, aside from the tiny triple mode peculiarity21 (attributable to the size of the statistical population analyzed - 169 individuals = hands), all other parameters are confined within normal intervals of variation. Thus, the mean and median are practically equal. The skewness 21

There are three groups of hands, each equally comprising six cases, which represent the mode. These clusters of 3 cards are overrepresented in relation to their normal, expected frequency, to degrees of 101.3%, 101.9%, and 109.7%.

23

is weakly positive, indicating a slight tilt of the distribution curve to the right, one however statistically not significant. The excess kurtosis is positive, which means (what is already visible on the graph) a leptokurtic distribution (an aspect already visible on the graph).

Finally, the standard deviation - delineated variation intervals of the distribution also look normal: 69.8% of the cases lie within 1σ distance from the mean (vs. 68.27% normally), 94.7% within 2σ (vs. 95.45% normally), the rest of the cases not exceeding a 3σ distance from the series' mean.

So, after two in-depth verified statistical indicators both suggesting everything has been normal on the Pokerstars platform, the case should be closed, right? Well, not really.

And this is actually a masterpiece of the poker platform's software: somehow, despite the sometimes grotesque statistical anomalies recorded in various instances, overall, at the end of the day, the general, average values of most of the parameters will be normal. But how exactly this is achieved, represents is a completely different story. To put it as simple as possible: it is like mixing boiling water with ice; the two have extreme and opposite temperatures, but, at the end of the day, after all the mixing is done, the recorded temperature is indeed normal, isn't it? Consequently, if some auditor at the end of the day shows up and verifies only this indicator - i.e. the average general temperature, things will be in order.

Without prematurely going into details, here is an example that may help clarify the explanation: in the Hold'em game, in a pair vs. pair heads-up encounter, the higher pair is, mathematically, a roughly 81.5% favorite to win the duel. However, as I shall detail later on in this analysis, on Pokerstars, due to a business-motivated strategy of "leveling the field" or "balancing the odds", the lower pair (and more generally, the underdog) clearly, constantly, and significantly wins more often than mathematically expected to. This works both ways, meaning both for you and for your opponent. Thus, when you bump into an opponent with a higher pair, you will win more than 18.5% of times, but so will your opponent in case it is you holding the higher pair! Well, at the end of the day, if you are curious enough to calculate your average equity for all pair vs. pair or any other type of encounter you have been through, you will find out that it is respected, meaning that, for instance, if you have been a 66.6% average favorite to win, you will have indeed won roughly 2/3 of the duels. However, this has been achieved by mixing in opposite statistical and mutually compensating exceptions, meaning you actually win more often than normal when being the underdog and less often than normal when being the top dog!

II.2 Targeted analysis In relation to the above-captured apparent "normality" of the hole cards distribution on Pokerstars, there are at least four clear examples that I can provide as arguments in support of my above statement: • the distribution of all suited hole cards specifically distinguished by the suite type; • the frequency of hole cards along power-ranked hierarchical groups of hands; • the distribution of pocket pairs; • the frequency of pocket pair duels, part of a phenomenon usually nicknamed "coolers". The following addresses these four points with the last one of them, as already mentioned, addressed separately for reasons previously explained.

II.2.1 Suited hole cards' distribution The following continues the above analysis in terms of the statistical normality of the hole cards' distribution, but this time specifically isolating the suited hands I have been dealt in the 51,989 cash games I went through, and, additionally, differentiating among the four possible suits: clubs (♣ - colored in green in the table below), diamonds (♦ - blue), hearts (♥ - red) and spades (♠ - black). Statistically, any of the 1326 two-cards combinations 24

corresponding to a 52 cards deck has an equal chance of being dealt to a player, one of 1/1326 = 0.0754%. This applies to any pocket pair combination (6 possible for each pocket pair hand), to any off-suite combination (12 possible for each off-suite hand), and to any suited combination (4 possible for each suited hand). For my 51,989 games recorded, this means it is expected for each of the 1,326 combinations to have been dealt to me 0.0754%*51,989 = 39.207 times. Here are the recorded absolute frequencies (abbreviated "F") for the 12,299 suited hands that I have been dealt, differentiated by the specific suit and arranged in a decreasing order for each suit, from the most frequent (52 of clubs) to the most rare (K2 of hearts) combination (abbreviated "C"): Table: suited hole cards' frequency of being dealt to me: (out of 51,989 hands analyzed, cash-games, 6-players rings)

Summarizing the findings, I have been dealt clubs-suited hands 3,039 times, hearts 3,174 times, diamonds 3,075 times, and spades 3,011 times. At first glance, the heartssuited recorded frequency stands out among the four, but the recorded surplus in relation to the expected frequency (of 3058) is of only 3.8%, which is indeed considerable, but not worrisome. Also, as detailed below, one should notice that I have been dealt suited hole cards

25

more often than expected, but, again, not to a significant degree22. Furthermore, even when dissecting and recomposing the aggregated data in terms of the normality of the distribution, the result is almost too good to be true, meaning the distribution is as normal as one could expect, if not more, actually!

Thus, the excess kurtosis is a staggering -0.002, the skewness, although clearly visible above, is half the threshold value beyond which one may conclude some abnormality, whereas mean, mode and median overlap almost perfectly. In fact, even after employing more advanced statistical methods and techniques, such as CDF or Z-value, the difference between the recorded distribution and a mathematically perfectly normal, Gaussian, random-specific, is, as shown below, almost invisible!

60

50

40

recorded

30

expected 20

10

1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199 208 217 226 235 244 253 262 271 280 289 298 307

0

22

Specifically, 12,299 times as recorded vs. 12,232.7 as expected.

26

And it is exactly this perfect overlapping that is highly suspicious. It is as if Pokerstars somehow periodically makes sure that the distribution of the hole cards on the overall is constantly Gaussian all the way to the third decimal place! The recorded figures are simply unbelievably... "perfect".

Simply amazed by these almost-too-good-to-be-true results obtained, I went back to the table listing the dealing frequency of the suited hands. One thing in particular immediately caught my attention, namely the suited connectors. For the readers not so familiar with the Hold'em game, there are probably two explanations required at this point: firstly, the term "connectors" signify any two consecutive-valued cards, such as QJ, 87, or 32; secondly, among all 78 suited hands, the suited connectors are actually the best ones, meaning they are the hands most likely to win a pot, since they can transform, using the board, or community, cards, in either a flush or a straight, aside from their individual values.

With this being clarified, a recalculation of the figures recorded above highlights a slight overrepresentation of the 12 suited connectors among all the hands I have been dealt (3.71% recorded vs. 3.62% expected), but a way more interesting finding is the spectacular, indisputable, correlation between the suited connector's power rankings and the frequency of them being dealt to me as hole cards: a negative 0.762, meaning the better the suited connectors, the less often they have been dealt to me23: Suited hand dealt

Hand's Absolute power frequency ranking of dealing (6-players games)

AK KQ QJ JT T9 98 87 76 65 54 43 32 Correl.:

141 159 137 149 153 155 185 174 162 161 178 175 -0.762

32.1 29.5 27.6 26.3 23.9 21.8 20.4 19.3 18.4 17.8 15.9 14.0

Not coincidentally in this light, it is exactly 5 of the best 6 suited connectors whose recorded frequency is significantly below their expected frequency. Anyhow, the coefficient's value is simply too high to attribute such an anomaly to randomness, suggesting instead human, nonrandom, intervention, in the process of dealing the hole cards.

Additionally, when looking closer at the very first table of this paper, the one listing all the 169 hands' frequencies, one can easily notice that, of all the total 26 lines and columns of the matrix, two of the four most underrepresented ones are the first line from top (97.8% of the expected frequency), respectively the first column from the left (98.5% of the expected frequency). These are exactly the line and the column corresponding to the Ace card, which needless to mention, is the best possible card in the game! This observation in 23

Readers not really fond of statistics should now the linear correlation coefficient's scale stretches from -1 (total negative, inverse, relation) to +1 (total positive, direct, relation).

27

turn determined me to take one step further and target the single-card dealing frequency registered on the platform along my combined cash plus tourney 55,320 hands, whose results, although not part of the present analysis of suited hands, I nevertheless display below: Graph: single cards distribution on the poker platform:

The findings speak for themselves: not only it is exactly the Ace, the best card in the game, that has been dealt to me the least often, but, more generally, the low cards have also been more frequent than the high ones. True, one may legitimately start thinking that the last two exposed anomalies (that is the suited connector's conditioning of frequency by their individual value and the single card distribution) should be themselves sufficient to begin thinking of Pokerstars' RNG as one not really" random". Let us however move on with the analysis and, at least for the moment, try to pretend that it still might be randomness involved, in the sense that the two bizarre results could still have been just an "accident". After all, why couldn't one simply be an unlucky guy, whose most rarely distributed single card happens to be exactly the best one in the game, right?

II.2.2 Hierarchical groups of hands in relation to their dealing frequency By keeping in mind the boiling vs. freezing water metaphor and hence ignoring the already established fact that, overall, no correlation has been identified between hole cards' values and their frequency of being dealt to me, the present analysis focuses, for reasons of necessary standardization, only on the 51,989 cash games I have participated and delineates hierarchical groups of hole cards according to their power ranking in 6-players games24. Specifically, the table below focuses on the best 30 of the total 169 hands that can be dealt in a 6-players Hold'em game:

24

The reader will allow me to remind that a hand's initial power ranking differs depending on the number of players at the table. Take for instance the AK suited hand: in a 6-players game, it is the fifth best hand in the game, with a 31.1% probability to win or split the pot; in a 10-players game, it is the fourth best (20.7%), whereas in a heads-up game, it is only the eighth (67.0%). Such discrepancies are the reason for me isolating only my 6-players cash games, where all figures are standardized, as the tournaments I have participated in were of various formats, from 3 to 9 players rings.

28

The figures indicate a slight, but still visible, relation between the top hands and the frequency of them being dealt to me; on the overall, the top 30 hands in the game have been dealt to me more frequently than normal; along the groups the five, the first two of them are also over-represented in terms of dealing them as hole cards; the situation normalizes relatively beneath the cluster of the best 25 hands in the game but, after the best 30 hands are counted, the overall cumulated frequency is still slightly higher than the expected one. This means that the overall lack of correlation I discussed above has been the result of some distribution anomalies (big enough to alter the measurable correlation) registered somewhere among the weaker rated hands.

II.2.3 Pocket pairs' distribution Pocket pairs represent one of the most striking issues noticeable on the Pokerstars platform in at least three regards: their dealing frequency - one clearly, constantly, and consistently higher than normal; their probability of encountering at least another pocket pair at the table - one also clearly above normal, as I shall detail at a later point in this analysis; their conversion rate along the next three streets in the game (flop, turn, river) into sets or better - an aspect that I shall also investigate in-depth at later.

For now, let us deal with the first aspect and focus on the overall recorded dealing frequency of pocket pairs along my entire 55,320 series of cash and tournament games played on the poker platform, by starting with two specifications: regardless of the number of players at the table, the probability of being dealt a specific pocket pair is 6/1326 = 0.4525%; in aggregated terms, the probability of being dealt any pocket pair is 78/1326 = 5.88%, meaning roughly once in every 17 hands dealt.

The discrepancies between the above-mentioned normal frequencies, to be expected in a probabilistic, random environment, and the ones actually recorded on Pokerstars are, as visible in the graph below, quite remarkable. Thus, in comparison to an expected absolute frequency of 250.3 times, the recorded average frequency of pocket pairs that have been dealt to me is 258. On the overall, I have been dealt pocket pairs 3,354 times within a total 55,320 games played, meaning a relative frequency of 6.06% in comparison to the normal 5.88%, this surplus having manifested itself in both cash and tournament games.

Clearly, and, as detailed below, continuously, pocket pairs on the Pokerstars platform are being dealt more frequently than normal.

29

Pocket pairs distribution (No.) (55,320 hands played, cash games + tournaments) 22s

278

KK

277

99s

264

TT

264

QQ

263

88s

261

44s

259

average

258.0

55s

258

JJ

255

normal frequency

250.3

33s

250

77s

66s AA

243

242 240

This recorded anomaly is by no means an "accident". It cannot, at least reasonably, be attributed to randomness, or variance, or whatever term Pokerstars usually employs in trying to justify some of the weird things happening on the platform, for at least two reasons. Firstly, as one can notice above, it is not one, not two, not three, but 9 of the total 13 pocket pairs that are situated above the normal frequency to be expected in a genuinely random environment, meaning the overall figures cannot be attributed to some single case outside of the third standard deviation interval, which would alter the recorded average.

30

Neither can these findings be attributed to some short-term incredible variance that might have happened to me. Specifically, because the frequency of pocket pairs and, partly overlapping, the frequency of pocket pair vs. pair situations both already caught my attention after a week or two of playing on the platform, back when I was doing it only on play-money, the moment I started playing at real money tables, I began monitoring the pairs situation and kept a detailed account of it. Here are the results recorded at three separate moments in time in relation to the 51,989 cash games I played: after the first 15,503 hands played (the lower segment of the columns in the graph below); after 29,313 hands played (the middle segment of each column); respectively after 44,312 hands played (the upper segments):

Frequency of pocket pairs dealt along three consecutive timeframes (after 15,503 games; after 29,313; after 44,312) (N = 51,989 cash games played)

250

200

75

64

83 64

67

62

64

68

66

73

76 68.3

64

62

67.9

150

65 80 100

61

70

69

50

62

68

AA

KK

78

74

63

67

72

JJ

TT

87

74

66 64

63

61

65

77s

66s

64

63

65

75

71

76

78

55s

44s

33s

22s

66.7

62.4

71.8

70.1

0

QQ

99s

88s

31

average normal

Using the same 4-colors shading as usual (dark green for over +5% variation in relation to the expected frequency, light green for less than +5%, pink for less than -5%, and dark red for over -5%), one can easily notice, looking at the two columns placed right in the graph, that, regardless which timeframe is chosen, the average recorded frequency of pocket pairs has constantly been higher than the normal, expected one. This suggests intentionality, premeditation, on the part of the people who conceived and approved Pokerstars' cards distribution algorithm, as well as the impossibility of attributing the findings to randomness.

The explanation for this anomaly is quite simple and reconfirmed by the findings throughout this analysis regardless what indicator is taken into account: strong, high powerranked hole cards are dealt more often. For the players involved, this simply makes the game more fun, more spectacular, more attractive, which may subsequently lead to an increase of the player-base, which consequently implies more profit for the poker platform. And in this explanatory light, it should be emphasized that, at least in 6-players games, pocket pairs comprise the best group of hands. Thus, not only are five of the best six hands25 in the game pocket pairs (AA, KK, QQ, JJ, and TT), but, statistically, and essentially, a pocket pair holding player will almost always be the favorite at the table unless the one exception-case in which he encounters a higher pair. Otherwise, even a 44 pocket pair for instance will still be favorite (albeit it, narrowly) against an AK-s! Hence, having a pocket pair will tend to make a player bet. In the elaborate scenario in which there is an opponent who also has a pocket pair (more on this in the next subchapter), there is an increased chance there will be some raising and maybe even re-raising. And, logically, bigger pots mean bigger rake for the poker platform to collect. It is, truly, that simple! And therefore it is also no coincidence that on Pokerstars, at least over the 55,320 games analyzed here, pocket pairs, the best group of hands, are dealt more frequently, followed by suits as the second best group of cards, whereas off-suite cards are being dealt below their normal frequency.

Before concluding the analysis here, there is another remark to be made: the rarest pocket pair that I have been dealt in the combined 55k games played has been AA, exactly the best pair and also the best hand possible in the game. And, when engaging into calculations, the results indicate that the overall underrepresentation of the Aces as single card cannot be attributed solely to the lower frequency of the AA pocket pairs. This sheds a different light on the finding above (when I invited the reader to give Pokerstars the benefit of the doubt), namely that the rarest single card distributed to me was the Ace; in any normal circumstances, such a coincidence would be highly suspicious; on Pokerstars, as this study demonstrates, it can no longer be considered just a "coincidence".

II.3 Pokerstars' "coolers" Aside from one individual's hole cards themselves, what has been at least equally suspicious to me on the investigated platform, way before engaging in this investigation, is the interactive distribution of hole cards. This refers to the hole cards that two, sometimes three, players are sometimes dealt simultaneously (well, truth be said, not sometimes, but quite often, actually), despite some extremely low probabilities of such events to happen). The following section firstly treats the Pokerstars "coolers" situations in a general manner, and secondly targets specifically the pair vs. pairs situations.

II.3.1 Coolers The slang term "cooler" signifies a situation in which a very strong hand loses, but not because of the player committing a mistake, but against an even stronger hand. As such, 25

Or 7, depending on the source and the type of ranking, i.e. by win/split rate or by the expected value.

32

the loss is due to a particular, and usually very improbable, exceptional, way in which the deck cards have been "shuffled". Needless to say it again, the one certain consequence of such a cooling distribution of the hole cards is the increase of the pot and therefore a bigger rake for the poker platform to collect.

Obviously, the most common, notorious, and simple, example of "coolers" are the AA vs. KK duels, meaning an encounter of the best two hands in the game. However, in the meaning in which it is used throughout this analysis, the term "cooler" does not refer strictly to double or triple high pair situations. It more broadly encompasses any situation in which, pre-flop and/or on the flop, two or more players each have extremely attractive, albeit equally improbable, hole cards or combinations using the flop cards. The hand screen-captured below, with me holding Q4-o, and folding pre-flop, is a perfect example of such a "cooler":

33

In the example above, two players each flop an enticing trip, while the third one gets directly a full house. The conditioned probability of such a flop? An amazing 0.013%.

What do you think happened on such a flop? Well, quite predictably, a first player 26 bet, the second one raised (53-o), the third one (holding 88) called the raise, then the fourth one, holding the A5-o, went all-in, which the third and fourth player called. This is how, in cash games, such "coolers" ensure that people will bet and raise massively, thus increasing the pot-size and subsequently increasing Pokerstars' collected rake and it's profit.

An image being worth a thousand words, allow me, before engaging into a detailed analysis, to first display five screen shots of such coolers recorded on Pokerstars. The first two, as an exception, are from games that I did not participate in, but only observed27, the other from games in which I was directly involved in.

The first example can be regarded as a cooler not so much because of the hole cards dealt to the three players, but because of the flop:

Let's be clear as possible: (at least) two of the six players at the table have pocket pairs and, additionally, the third has one card of a rank equal to the JJ pocket pair! Extraordinarily, since in the 46-cards remaining deck there are, at best28, one Jack, two 3s and three Kings left (meaning a total 6 cards), the flop brings one of each of these three values on the table. The probability of such a flop, which insidiously encourages all three players at the table to bet, thus automatically also increasing the collectible rake, is a ridiculous 6/15180 = 0.039%, meaning it'll happen once in every 2564 times that these three players will find themselves in the same starting position, which in itself is already a low probability!

The second example, captured below, is one that I find astonishing even by Pokerstars standards: it shows us how one can lose a $2,477 pot on this platform, despite having quads! I think a little math is required here: in a four players game, (at least) two players have been dealt pocket pairs, both of them flop a set, then both of them make the quads on turn, respectively river!

26

The one placed right below the 53-o holding player. He folded along the way, so his hole cards remained unknown.

27 The Pokerstars platform offers its users the "observe" option at all the stakes levels, allowing at least a (carefully chosen) selection of hands to be observed.

28 Meaning we assume that these cards have not been distributed to the other two players who had folded preflop!

34

Let's again explore the associated probabilities, all calculated from the perspective of the player holding the pocket 4s:  the probability of being dealt a pocket pair other than Aces: 72/1326 = 5.43%  the probability of any of the three opponents having Aces 1-[((C50,2)-6)/(C50,2))^3] = 1.47%  the probability of the flop offering both players a pure set: 2*2*44 (useful flops) / 17296 (total flops possible) = 1.02%  the probability of both of them making quads, one (any) of them on turn, the other on river = (2/45)*(1/44) = 0.1515% Thus, the conditioned probability of the entire chain of events happening is a dumbfounding 5.43% * 1.47% * 1.02% * 0.1515% = 0.00000123347%. Formulated alternatively, it means that something like the above will happen once in every 81,071,840.23 hands played! Or: if playing 81 million hands, an event like the one above could be considered as statistically "normal". Now, of course I don't know for sure, since I don't know the pocket 4s holding player, but I may rhetorically ask: would Pokerstars representatives try to convince us that he/she did indeed play eighty-one-point-zero-seven million hands on the platform?!

Since such "cooler" situations are seemingly a deliberate policy of Pokerstars, it is only logical that I personally also could not avoid them when playing on the platform. Here are three examples from the games that I was directly involved in.

In the first one selected, a hand screen-captured directly on the poker platform, I had been dealt 54s and was smart enough to fold pre-flop, when the raise - re-raise - re-reraise - re-re-re-raise spiral erupted (with two players eventually going all-in pre-flop), which, admittedly, only made me curious enough to check the fold-but-keep-watching game option available on the platform.

35

With this happening on May 18, after a good 5 weeks spent on the platform, I have to confess I was already convinced even pre-flop that this was another classic AA vs.

KK situation, all so frequent on Pokerstars (see below), the only surprise being the third player, who also held pocket Kings. With the mathematics detailed below, let me just specify the probability of such an event happening, as calculated for any of the two players holding Kings: 0.09% of all those situations in which he/she has been dealt pocket pairs!

Whereas in the previous example, with a weak hand such as 54s, I made the easy decision to fold, I cannot say the same about the next displayed hand, where I did remain in the game all the way to showdown, only to discover that three players at the table had hearts hole cards, all of them making a flush!

Without delving into boring mathematical formulas again, let me again specify the probability of such an event happening: 20.52% * 12.16% * 5.39% * 22.15% = 0.0298% of all those hands in which one has already been dealt suited hole cards in a 6-players game, or once in every 3,356 suited hands played!

As a third example, one of my luckiest hands on the platform, as captured by the HM2 2 software: 36

Spectacular, sensational, unbelievable? I have frankly run out of labels, or, better said, Pokerstars has consumed all the labels on my stock. Let's again get specific: the conditioned probability, when holding pocket Kings in a 6-players game, of bumping not only into the single higher pair (Aces), but simultaneously into a third pair at the table (any other than AA or KK), and, additionally, flopping the quads, is a staggering 0.00026% of all hands of pocket KK played. What this further means is that something like this will happen once in every... 384,705 KK hands played in a 6-players ring! It is probably redundant to specify, but, of all my 51,989 cash games played, the total number of KKs that have been dealt to me was 259 (see capture below), out of which a mere 139 hands have made it to at least the flop! And still, this ridiculously minuscule probability materialized. Or, from the general perspective, considering the frequency of being dealt Kings, an event like the above will happen every 85,111,726 hands dealt, whereas, in reality, the total number of hands that I played in 6-players cash games has been less than 52k!

37

The problem is not that such events, that, in order to be plausibly considered statistically "normal", require players to have played hundreds of thousand or even tens of millions of hands, do happen. After all, they are improbable, but, anyhow, possible; the core problem is that all of these "coolers" have one single thing in common, aside from their nanoscopic probabilities: they make sure that players will bet, raise, and re-raise, thus guaranteeing bigger rakes, which is exactly what Pokerstars makes a living of!

Or, to put it conversely: no "non-cooler", or say we shall "dull", "boring", nonspectacular, situation, has ever been found in this analysis, nor ever reported by any player on Pokerstars for that matter, to have occurred at a frequency lower than its expected one. Say for instance playing 100 KK hands and not encountering Aces at a 6-players table; or playing 1,000 suited hole cards hands and not encounter a situation where three players simultaneously make a flush; or of having 2,000 flopped sets and not encountering an opposition flopped set; or playing 20,000 hands in a 6-ring and not going through the classic AA vs. KK vs. QQ situation; etc. Such "boring" distributions, with people not betting or even folding, fail to achieve a crucial task: the increase of the rake29: it's that simple.

II.3.2 Pocket pair vs. pocket pair: a Pokerstars "classic" Within the broader category of "cooler situations", which comprises, as selectively captured above, various types of situations, such as triple pairs, triple flushes, flopped sets vs. sets, full house vs. flushes, or quads vs. quads, there is one subcategory that, given the particulars of both the Texas Hold'em game and the mechanics of the Pokerstars software, can and needs to be addressed in-depth: the pocket pair vs. pair dealing situations.

Mathematically, in a genuine randomness, probabilistic, context, when holding a certain pocket pair in a 6-players game, considering that within the whole 50 cards left deck (meaning 1225 combinations of two cards) there are 73 paired combinations possible30, and there are 5 opponents at the table, the probability that at least one of them also holds a pocket pair, regardless whether it is equal or non-equal to your own, is 26.45%, as can be easily calculated using the formula [1 - ( ((C50,2)-(2*12+1)) / (C50,2) )^5] * 100 = 26.45% Without going into further mathematical detail, let us now specify six other probabilities, associated with the situation of holding a pocket pair in a 6-players game, which will show the relevance of the below findings: ď&#x192;ź the probability of encountering, somewhere among the 5 opponents, a specific pair is 2.425%. For instance, if one holds KK in a 6-players game, the probability that (at least) one of the 5 opponents holds Aces (any of the 6 AA possible combinations) is 2.4251%, that he holds tens the same 2.4251%, etc. ď&#x192;ź the probability that one of the 5 opponents holds exactly the same pocket pair as you is 0.407%. That means if you for instance have been dealt pocket 9s, there is only a 0.407% probability that one, any, of your 5 opponents also holds a 99 pair; ď&#x192;ź the probability of one of the 5 opponents also holding a pocket pair, but one (any of the 12 pair hands left) different to your one is 26.13%. For instance, if you hold pocket

29

And more so, since, it should be specified, Pokerstars does not collect rake from hands that end pre-flop, with all but one player folding. This type of hands and situations does simply not serve the platform owners' business interest.

30 Six combinations each for the other twelve pocket pair hands possible (meaning 6 AA possible, 6 KK possible, etc.) + 1 left of the same value (e.g. if you have 22s, there is only one other possible combination of 22 left within the deck.

38

Deuces, the probability that one of your opponents hold a pocket pair any from AA to 33 included, but not deuces, is 26.13%; ď&#x192;ź the conditioned probability of two of the five opponents also having pocket pairs (regardless which) is 5.754%; ď&#x192;ź the probability of two other opponents also having pocket pairs, but with two of the three pocket pairs at the table being of the same rank, is 0.185% ď&#x192;ź the probability of (at least) three players having pocket pairs, and one of the two opponents having a same rank pair with yours is 0.09%.

These figures might put some of the situations occurring on Pokerstars in a quite different explanatory light. Specifically, as players usually bet high, and extremely often go all-in with the highest pocket pairs (Aces, Kings, and to a lesser degree, Queens), such pairs' simultaneity subsequently means bigger pots, and consequently more rake for online poker platforms to collect. This is why, on Pokerstars (but, truth be told, also on other platforms I have tried over the course of time), the frequency of such high pair vs. high pair encounters is way above the normal, probabilistic, frequency.

In my case, of the 51,989 hands played in cash games in a zoom format at 6players tables, I held for instance pocket Aces 232 times (see the HM2 capture displayed above). Applying the mentioned normal frequency, I should have met opposition Kings in 2.425% of the Aces hands I played, which would be 5.63 times. Well, I bumped into Kings AT LEAST 10 times, which equates a percentage of 4.3, which in turn means 1.8 times the normal frequency: 1.) April 17, at 17:14 EET; 2.) April 19, at 14:38; 3.) April 22, at 13:01; 4.)

April 22, at 13:0931; 5.) May 2, at 15:50; 6.) May 3, at 14:52; 7.) May 21, at 12:04; 8.) May 23, at 12:38; 9.) May 25, at 16:45; and 10.) May 28, at 16:46.

31

Yes, that's right, on the same day, only 8 minutes after the previous case!

39

I cannot stress enough the term "at least" which I have used above: these 10 cases are the ones in which the existence of Kings in one of my opponent's pocket can be proven, 40

since both I and my opponent went to showdown and saw each other's hole cards, hence the above screenshots. These cases are certain. However, after having played a combined 110,000+ hands on Pokerstars on both real and play money, and judging by some of the betting patterns of my opponents in a few of the other cases I held Aces, but we didn't reach showdown, I have extremely strong reasons to suspect there were at least 22 other instances in which my opponent held Kings (or - a slight possibility - Queens). Most likely, they had the wisdom to fold, either pre-flop against my 3- or 4-bet, or on the later streets, when it might have become clear to them that I had Aces.32 This would increase the frequency of AA vs. KK encounters to 5.1%, which is closer to the 2.2 (if not 2.5) higher than normal frequency I generally estimate for pair vs. pair situations in 6-players games on the overall on the Pokerstars platform.

And if that wouldn't in itself be enough in regard to Pokerstars RNG's supposed "random" dealing of the cards, it should be noted that two of those hands I held pocket Aces and encountered Kings were consecutive, and placed at a time distance of only 8 (eight) minutes (on May 22, at 13:01 and 13:09), as captured below by the HM2 software:

Well... the conditioned probability of bumping twice in row, when having a pair (AA in this case), into the same other pair (KK in this case) is a minuscule (0.2613*00.2425)*100 = 0.63%, meaning once in every 159 hands of that particular pocket pair that are being played, whereas the total number of my pocket Aces that have reached at least the flop has been only 131. And, as repeatedly specified throughout this research, neither can this anomaly be attributed to other pocket pair hands that I've played, that would supposedly compensate it33; take for instance my pocket 8s hands: an event such as the one above has happened not once, not twice, but three times. Specifically, I bumped consecutively into: Kings (May 27, at 19:04, and then on May 28, at 16:25); Queens (May 19, at 16:34 and then at 16:47, so 13 minutes away); respectively 9s (May 10, at 13:18, and then the very next hand on the same day, at 14:17). As a recurrent conclusion of this analysis, in order for some statistics on Pokerstars to be possibly considered statistically "normal", I should have played

32

See for instance my Aces hands played on: May 26; May 21, at 15:29; May 17, at 16:08 (called 3-bet, folded on turn); May 7, at 12:40 (called 3-bet, folded on flop); April 19, at 14:24 (same playing line); and especially: May 20, at 12:54 (3-bettor who folded in front of my massive 4-bet, April 18, at 10:38 (called 3-bet, folded on river), and April 15, at 14:38 (called my 4-bet, called on turn, folded on river).

33 Meaning that, in relation to the overall number of pocket pair hands that I have been dealt, the total frequency should potentially be normal.

41

thousands and thousands of hands more than I actually played, sometimes 25-30 times more hands than I actually did!

Returning to the item-example of my pocket Aces, overall, out of the total 232 times they have been dealt to me, I reached showdown in 75 cases - enough to discover pair in my opponents' pockets no less than 34 times (Kings, as mentioned, 10 times, Queens 6 times (also a little higher than the normal, expected frequency)34 - see the screenshots below, 7s and 4s three times each, etc.).

Additionally, I encountered another opponent with pocket Aces twice in 232 hands that I played at least on the flop35, which would place such an event on a 2/232 = 0.86% frequency, more than 2 times higher than the normal one of 0.407% (see below for a more detailed discussion of this type of situation).

For all pocket pairs, other than Aces36, that I held, it is impossible to irrefutably prove the real percentage of cases in which one of my opponents held another pocket pair, regardless if higher or lower than mine. The obvious reason for it is that both me and my opponents, being rational beings and playing on real money, obviously did not go every time all the way to showdown independent of how the betting line evolved, of what the board cards 34

Which is also above the normal, expected, frequency of such encounters: May 26, at 18:30; May 26, at 14:21; May 20, at 20:46; May 18, at 11:37; May 6, at 11:49; April 20, at 13:47.

35 I use this aggregated figure (flops both played and folded pre-flop) as reference since it's safe to presume that in none of the 101 hands not played beyond pre-flop could an opponent have folded with pocket Aces.

36 Which I folded just one time, for the specific purpose of testing how the RNG will "react" :)

42

were, etc. Numerous times, either me or my opponents folded along the way when I held pocket pairs (I personally folded two KK hands before showdown with an Ace among the board cards and my opponents betting aggressively both pre-flop and afterwards). As there are no showdowns that could have been screen-captured, I thus unfortunately cannot provide a solid, irrefutable, accounting. However, glimpses of Pokerstars overinflated percentage of pair vs. pair situations can be brought forward by some disparate cases, such as for instance: - in at least 7 of the 259 times I held pocket Kings, when reaching showdown (i.e. 94 times), I encountered Aces37 - meaning 2.7% of the KK hands played, which is already above the normal frequency of 2.42%; also with Kings in my pocket, I encountered Jacks also at least seven times38, and 9s an additional six (three times actually on the same day (May 26), within little over 4 hours of game-play!)39 - of the 249 hands of pocket Queens, 95 did not reach the flop; of the remaining ones, I reached showdown with 86 of them, enough to discover in my opponent's pockets Aces at least 6 times40, same for Kings41, respectively Jacks42; - having pocket Jacks (a total 235 times), I only reached showdown 77 times, enough yet to certify in my opponents pockets: Aces at least 5 times, Kings at least 6 times, 10s at least 5 times, 4s at least 6 times (which beat me 3 times) - a total staggering 37 times that at least one of my opponents also had a pocket pair, of, I repeat, only those cases in which the game reached showdown; - of the 249 hands of pocket 8s I was dealt, 90 of them reached showdown - in at least 36 of them, at least one another opponent also had a pocket pair; thus, I bumped into Tens 6 times, into Queens43 also 6 times (in two of those instances there was also (at least) a third opponent also with a pocket pair!), into Kings 7 times44, etc.

Surely, or at least very likely, Pokerstars' representatives might try counterarguing by saying that in the hands that did not reach the flop, or have been played, but folded before showdown, my opponents did not have pocket pairs, so that, overall, meaning hands ended pre-flop plus hands played further, the overall frequencies are actually normal.

However, cases such as those when I held AA or KK and did go all the way to showdown any hand I had received, cases in which the recorded frequencies of pair vs. pair encounters are significantly above the normal, expected one, cast a huge doubt over any such possible refuting tentative. These are the specific cases one has to focus on when trying to estimate the frequency of pair vs. pair encounters, since in these cases the percentage of hands taken all the way to showdown is the biggest, hence the images captured by them at showdowns will be the closest to the actual reality on the poker platform. For the lower pairs I held, the % of hands that reached showdown decreases visibly, and subsequently directly alters the reliability of the findings at showdown: 37

May 26, at 18:13; May 25, at 17:18; May 21, at 16:32; May 16, at 15:39; May 18, at 16:09; April 17, at 11:36; April 11, at 15:40. I also encountered Aces once within 18 KK hands that I have been dealt in tournaments (of which 11 made it to showdown).

38 May 26, at 18:13; May 14, at 16:54; May 14, at 15:20; May 10, at 14:54; April 29, at 10:12; April 13, at 17:27; and April 11, at 17:42.

39 I even managed to lose 50% of the times against 9s, despite being a 81.5% favorite. My KK hands when I encountered 9s were played on: May 26, at 17:23, at 12:27, and at 11:14; on May 19, at 13:20; on May 13, at 14:30; and on May 7, at 12:29.

40 May 26, at 15:10; May 25, at 13:31; May 23, at 13:14; May 20, at 12:27; May 19, at 19:10; and April 18, at 00:15.

41 May 22, at 16:31; May 19, at 16:55; May 14, at 16:20; April 21, at 13:58; April 14, at 17:11; and April 11, at 12:30.

42 May 22, at 12:33; May 16, at 12:31; May 16, at 11:45; May 11, at 11:19; April 13, at 15:19; and April 12, at 11:47.

43 May 19, at 16:47; May 19, at 16:34 (so less than 15 minutes distance from the previous encounter); May 13, at 15:52; May 10, at 13:18; May 2, at 11:31; and April 18, at 18:15.

44 May 28, at 16:25; May 27, at 19:04; May 20, at 19:53; May 2, at 11:31; April 26, at 11:22; April 20, at 12:53; and April 16, at 12:32.

43

My pocket pair

total hands dealt

Aces Kings Queens Jacks Tens 99s 88s 77s 66s 55s 44s 33s 22s total

232 259 249 235 240 252 249 230 222 243 239 237 259 3146

Showdowns reached No. 75 94 87 77 73 88 90 73 66 59 63 59 71 975

% of hands 32.3 36.3 34.9 32.8 30.4 34.9 36.1 31.7 29.7 24.3 26.4 24.9 27.4 31.0

The figures and the associated graph capture a crystal clear tendency: the higher the pocket pair I held, the bigger the chance of reaching showdown with it. This is only logical: despite occasional experiments such as the one separately discussed in this paper, when I open-limped and check-called each time I had a pocket pair, with the sole purpose of dragging my opponent to showdown, in order to test some statistics, in the overwhelming cases of all 51,989 hands that were dealt to me, I did, or at least tried to, play rationally, meaning the stronger the hole cards, the higher my likelihood of staying in the game till showdown.

Hence, as argued above, the best available indicators when trying to grasp the frequency of pair vs. pair encounters on the Pokerstars platform are those hands where I held strong pocket pairs, whose percentage of showdowns reached is considerably higher. This logical reframing of the analysis sheds some additional explanatory light on the findings in the next table:

44

My pocket pair

Showdowns reached

Aces Kings Queens Jacks Tens 99s 88s 77s 66s 55s 44s 33s 22s total

75 94 87 77 73 88 90 73 66 59 63 59 71 975

Showdowns where 1+ opponent had pocket pair No % 34 45.3 38 40.4 33 37.9 36 46.8 25 34.2 29 33.0 34 37.8 25 34.2 16 24.2 25 42.4 17 27.0 16 27.1 23 32.4 351 36.0

The tendency is as clear as possible: the higher my own pocket pair, the higher the percentage of cases with at least one of my opponents also having a pocket pair. True enough, the overall percentage of 36.0% of showdowns highlighting at least two pocket pairs at the table is higher than the normal 26.45% one, but, this general discrepancy could still be attributed to those hands folded before showdown, claiming that in those cases opponents would not have had pocket pairs, so that the overall frequencies would be eventually right. However, when isolating, for reasons explained above, the cases in which I held a strong pocket pair, say the best five pocket pairs (AA to TT), one discovers that in no least of 166 of the 406 showdowns reached, meaning 40.9%!, there was at least one other pocket pair at the table. That's 1.55 times more frequent than normal, in a random environment. And this is based only on those hands in which, simultaneously, both myself and my pocket-pair holding opponent have chosen to stay in the game all the way to showdown! If, however, Pokerstars' representatives would push the speculation, in trying to explain such anomalies, by claiming that the hands folded before showdown must have been compensating the overall statistics, I think the further analyses below shall deem any such tentative as futile.

45

ď&#x192;&#x2DC; Equal pair encounters' frequency If a general and superficial look at the overall frequency of pair vs. pair duels casts a substantial doubt on Pokerstars' claim of its RNG being random, but still cannot definitively prove non-randomness, other items verified shall definitely clarify the situation such as, for instance, the very frequency of equal pairs being dealt at a 6-players table on the Pokerstars platform (situations as the one screen-captured below), which is also significantly above the normal, expected one of 0.407%

Specifically, as partially highlighted in the table below, I encountered an equal rank pair twice when I held Aces, once for each of the KK, QQ, and JJ, and 44 series of hands I had been dealt, and again 2 times each for my TT and 77 series of hands. In an effort to approximate the frequency of this type of situation on the platform investigated, I isolated the total 1,215 hands of AA, KK, QQ, JJ, and TT that I have been dealt, excluding from the analysis other lower pocket pairs, since in the latter type of case the number of showdowns reached covers only a fraction of the total hands dealt (the weaker the pocket pair, naturally, the lower the chance of me reaching showdown). In turn, the higher pairs cases I selected ensure approximately one third of all hands having reached showdown45.

My pocket pairs:

Total hands dealt

Ended preflop

Played, but ended before showdown

Showdowns

Aces Kings Queens Jacks Tens Total

232 259 249 235 240 1215

101 120 95 82 79 477

56 45 67 76 88 332

75 94 87 77 73 406

Equal pair encounters % of hands No. dealt 2 0.86 1 0.39 1 0.40 1 0.43 2 0.83 7 0.58

Equal pair situations displayed at showdown, when I held: Aces - May 19, at 17:51, and May 22, at 11:22; Kings - May 24, at 15:48; Queens - April 14, at 14:52; Jacks - May 26, at 16:50; Tens - May 6, at 17:11, and April 27, at 11:17.

45

The simple correlation coefficient between my pocket pairs values and their associated % of hands that reached showdowns is +0.772 .

46

Thus, in 0.58% of the total pocket pairs hands selected that I was dealt, there has been an opponent with an equal pair, a figure already, and significantly so, above the normal, expected one of 0.407%. And, it needs to be stressed, this is by basing the inventory exclusively on the 406/1215 = roughly one third of the hands dealt that did reach showdown.

Of the total showdowns reached, no less than 1.7% displayed an equal pair vs. pair situation, which would be over four times the normal frequency! One can safely presume that, among the 2/3 hands that failed to get to showdown, there have been other instances when an opponent (or me, for that matter) also had an equal pair in his/her pocket but, considering how the board evolved, folded before showdown! So what would be the real percentage of equal pairs encounters? Only Pokerstars' employees can answer that. From the outside, even the frequency recorded (solely) at showdowns, but based on all the hands dealt, is already above the normal, expected one! ď&#x192;&#x2DC; Triple pair situations' frequency Aside from the above anomaly, there is another pre-flop recognizable problem with Pokerstars' claim of its algorithm being random: the frequency of triple pairs situations, such as the one captured here:

As mentioned above: the conditioned probability of two of the five opponents in a 6-players game also having pocket pairs (regardless which) is 5.754%; the probability of two other opponents also having pocket pairs, but, specifically, with two of the three pocket pairs at the table being of the same rank, is 0.185%, and the probability of (at least) three players having pocket pairs, and one of the two opponents having a same rank pair with yours is 0.09%.

As an example, I will pick the pocket 99s that I have been dealt over the entire 51,989 hands that I played on the Pokerstars platform. Of the total 252 cases of pocket 9s, 68 have ended pre-flop, 96 have been played at least to the flop, but were folded before showdown, while the number of hands taken all the way to showdowns was a mere 88. Well, these 88 were enough to discover a triple pairs situation in no less 4 times (meaning 4.5%!): May 14, at 14:32 (Hand No.1 in the table below); May 13, at 13:47 (Hand No.2); May 2, at 16:50 (Hand No. 4).

47

1.

2.

3.

4.

This has got to be Pokerstars at its best. Not only do these hands strikingly defy statistics in terms of the chance of dealing such hole cards, but, when looking closer at the hands and their unfolding (images on the right), all that has been so vehemently and frequently criticized online in regard to Pokerstars suddenly and vividly comes to light: minuscule probabilities materialized, the most improbable flops, the "coolers" meant to keep players in the game and bet massively, the sensational reversals of situations of turn and river, again defying statistics, etc. Take for instance the first hand captured: had I stayed in the 48

game, despite my pre-flop equity of only 20%, I would have won the hand by making the set on the turn (a roughly 8.4% probability of hitting it on either of the last two streets, respectively 4.65% of doing it specifically on the turn!). The second hand is similar: having started from a 15% initial equity, I would have won again the hand had I stayed in the game, by hitting a straight on the river (a 4-outer, to be more exact). Finally, the "winner" is probably the fourth hand: not only are 3 of the 6 players being dealt pocket pairs, with me smart enough to fold pre-flop, but the flop bears a set vs. set situation of Queens vs. Jacks!

And if you think that might have been an awkward, but statistically still possible, exception, think again! 59 showdowns that have been reached when I held pocket 3s were also enough to discover an equal number of 4 triple pair cases (meaning 6.78%), and this in relation only to those hands that did reach showdown.

The aggregated image covering all triple pair situations that I discovered at showdowns is captured in the table below.

Table. Triple pocket pair situations recorded at showdowns:

Now, obviously, the overall percentage of 1.74% is inferior enough to the normal, expected one, of 5.75%. This, however, does not actually mean that by some miraculous happening, the occurrence of triple pair situations on Pokerstars would be lower than normal.

Quite the contrary, in fact, when looking closer at the table.

Firstly, it can easily be noticed that no less than 9 of the total 17 triple pairdisplaying showdowns, meaning 52.9% of them, involved a player with pocket Aces. This in turn does also not mean that Aces' frequency of dealing would be higher than normal, but only, and as simply as possible, that it is only when holding an extremely strong pair, actually the best hand in the Hold'em game, that players will not fold and instead go all the way to showdowns. It is exactly because the players engaged had Aces in their pocket that these cards' frequency is so high among the selected showdowns. With lower pocket pairs dealt, 49

people will naturally venture into showdowns way more seldom, folding along the way, especially if they did not hit a set or better on the flop, to a much higher degree. After all, this analysis covers real cash games, played on real money, which makes it only reasonable to attribute rationality to the players involved. This means inclusively that, when starting with a weak pocket pair, and not hitting an advantageous board, and additionally confronted with the opponents betting, they will be folding.

Starting with this logic, and assuming, as if we were Pokerstars' attorney, that in no initial triple pocket pair situations did the players with AA fold before showdown 46, in relation to which 9 triple pair situations have been recorded, then, by extrapolation, we can estimate grosso modo that, corresponding to all 13 possible pocket pair hands existent, there should actually have been (13*9)*2 = 117 such instances of triple pairs among the showdowns, meaning 117*100/975 = exactly 12% of all hands taken to showdowns. This new figure is not only significantly higher than the mathematically normal frequency of 5.75% (meaning more than double), but also, judging in contextualization with all the dozens of items analyzed throughout this investigation47, also extremely plausible for the Pokerstars platform.

Secondly, a closer look at the table reveals two cases in which two of the three pocket pairs were of equal rank: 99 vs. 33 vs. 33 and 55 vs. TT vs. 55. Then again, the probability of one single such situation is, as already indicated, a mere 0.185% in relation to a player's series of being dealt pocket pair hands. Well, if this recorded case represents that percentage, than it means, by extrapolation, that the total number of showdowns needed for these two events to be normal should have been 2*100/0.185 = 1081.08, which, again, is absurd, since the total number of showdowns that I have reached when holding a pocket pair was only 975. And, again, no, one cannot take into consideration the total number of pocket pair hands that I have been dealt since it can easily be counter-argued that among the hands folded pre-showdown there have also been cases of triple pair situations.

The conclusion is as recurrent as redundant: the "cooler"-type situations on the Pokerstars platform occur significantly more often than in a genuinely random environment, and, no matter which technical method or argument or perspective is chosen, the recorded anomalies cannot be attributed to any randomness. They only represent the visible result of a deliberate, business-motivated, strategy of the poker platform.

II.4 An experiment: (open-) limp-check-calling with pocket pairs all the way to showdown Essentially when dealing with a situation such as the analyzed here, i.e. an online poker platform suspected to use a non-random software, there is, actually, one single definitive and irrefutable, but unfortunately impossible to apply method to prove that something is wrong48. This method would suppose that all the players should remain at the table every hand all the way to showdown, for a sufficient number of hands, in order to thus exhaustively and reliably test the randomness of the dealing algorithm. Surprisingly or not, this single usable option is one unavailable on Pokerstars, not even on play money: you cannot simply take 5 friends, open a table and just sit together till showdown, so you can test the statistics. There isn't, simply, such an option available, as one cannot open such a table.

46

Although, obviously, this also happens in reality depending on the board cards and the opposition's betting manner), and knowing AAs represent only one of the total 13 possible pair hands (or combinations).

47 From the frequency of heads up AA vs. KK situations, which can also offer an extrapolation starting point, all the way to the frequency of set vs. set situations on the flop.

48 Other than, obviously, having access to the cards-dealing software.

50

Given this insurmountable obstacle, I came up with the possibly second best method: for an entire 2,573 hands played in the timeframe May 26, 17:40 - May 27, 14:48 EET, at cash games, zoom format, 6-players rings, 1/2c stakes, each time I was dealt a pocket pair, I (open-) limped and simply check-called on all streets, with the sole purpose of dragging my opponent to showdown, in order to test two things: the recorded frequency of pair vs. pair encounters; the actual, measured, general win rate of pocket pairs in comparison to their mathematical, expected, win rate in the given game circumstances. With the second issue discussed later on in this study, the following addresses the first one.

Table. Hands' outcomes when I held pocket pairs during the experiment:

Opponent's hole cards 1 2 3 4 5 6 7 8 9 10 11 12 13 My PP QJo 54o T7s KJo KQs KTo 22/T6o AA QQ 88 66 106o ATs/KQs K5o A6o J6o A4s ATo QJo KK AA/JJ AJo KJo K4s A9o 84o 97o A5s KJs 43o QQ 44 55 82s A7s 95o 72o JJ AA AQo Q7s Q9s ATo 42s AKo J9s TT 33 66 AJs JTo A8o AKs A3o 65s A3s Q4o 98s/87s A3s J4o T7o 99 88 J9o AJo/76s QTs AQs AKo AJs J6o KTs/K3o A3s 62o 99/K10s 88 44 44 108o AKo A8s 83o AKo T8s KQo 77 66 AQs A8o ? (BB)* 86o K7o A8s Q2o 75s Q9o KQo Q9o T8s 66 88 42o A6o 95s T6o T3o 54s/84s A6s 76o A4s A9o J2o 55 44 KJo 87o AQo Q7o KQo 108s 74s K8s 87s/T8s 97o 44 AA 87s AJs A9o 97o 33!

AA 99 TT QQ 86o ATo 43s J7s 22!

JJ 99 77 * All my opponents folded pre-flop, with me in the big blind position.

Shadings: green = I won; pink = I lost; blue = split pot

14

AQo

AKs

Despite my (non-)betting trick meant to not dissuade any participation in the game, it is more than obvious I still haven't managed to persuade each player who held a pocket pair to walk along me till showdown. Players with, say, 22s or 33s might have folded pre-flop in front of me, not opening with such cards from early positions; others probably folded their pocket payers behind me, when confronted with an intermediary opponent's 3-bet, etc.

Yet, this method provides a slightly better approximation of the pair vs. pair situations' frequency on Pokerstars. True, on the overall, nothing seems suspicious: Out of the 131 times I held a pocket pair, at showdown I encountered an opposition non-equal pair only 24 times, meaning in 18.3% of the cases, which is still significantly below the normal, expected, frequency of 26.13%. Thus, the findings are at first glance inconclusive.

51

But, as said, my method still could not ensure each pocket pair holder coming along to showdown. And in this regard, going back to the 2s and 3s pocket pairs mentioned above, they are actually those pairs providing the best approximation of the frequency sought: when I held 2s or 3s, my opponents, if they had any pairs, would have had better ones, pairs which they wouldn't have folded that easily, thus increasing the chance of reaching showdown. And the corresponding numbers speak for themselves: in no less than 46.7% of the cases (7/15, that is), I was dealt pocket 2s or 3s, at least one of my opponents, the one who stayed till showdown, also had a (different rank) pocket pair. This percentage is significantly higher than the normal, expected, frequency of 26.13 associated with non-equal pair vs. pair encounters in 6-players games.

Extremely interesting in regard to this 46.7% figure, if we were to try another approximation in a completely different manner, and extrapolate on the basis of the cases in which, in all the 51,989 cash games played by me, I held Aces and encountered opposition Kings, knowing that the probability of such an event is 2.425%, and knowing that the normal pair vs. pair encounters' frequency is 26.45%, we would reach a necessary absolute frequency of Aces vs. any other pocket pair of 26.45*100/2.425 = 109.07 cases. Relating this to the total 232 hands of Aces that I have been dealt, it would mean that in 109.07*100/232 = 47.01% of all the cases I held Aces, there must have been at least another player at the table who had also been dealt a pocket pair. The difference between the two figures reached via separate methods, 46.7% in the first case vs. 47.01% in the second, is spectacularly small, both methods converging towards the conclusion that, on the Pokerstars platform, the frequency of pair vs. pocket pair situations is at a minimum 1.75 times higher, and deliberately so, than the normal, expected one in a genuinely random, probabilistic environment.49 However, still not at peace with being incapable of getting my opponents to showdown, it crossed my mind that in Pokerstars play money games people usually play significantly more loose, since no real money is involved, thus increasing the chances of reaching showdown. Thus, on the same evening of May the 27th, I ended the experiment and jumped onto play money tables. Since there is absolutely no reason to suspect that the platform uses one type of RNG in real money games and another one in play money games 50, it seemed like a feasible plan. So I picked up the lowest stakes level and played the same strategy of open-limping and just check-calling whenever I was being dealt a pocket pair. It worked; play money players are indeed looser, calling and opening with a significantly larger range of hands, so that I had the opportunity of taking all my pocket pairs to showdown. Well, after not even 150 hands played, I had already been dealt pocket pairs 11 times. In no less than 6 of these cases (that is a staggering 54.5%, but, remarkably, a figure not so distant from the overall 46.7-47.0% reached above!), at showdown I discovered one of my opponents also having a pocket pair: one time I was dealt Aces, and bumped directly into Kings, to my adversary's despair; one time I had Kings, and discovered an opponent with pocket 6s; my single time I was dealt Jacks was enough to bump into pocket 9s; of the two times I had tens, 49

No matter how you approach the findings within the experiment, they converge towards the same conclusion line, i.e. the significant over-representation of pocket pair duels. Let's return for a last time to the experiment, only this time from another angle, specifically focusing on those cases in which I encountered, distinctively, a pocket over-pair. Thus, I bumped for instance into Aces (never mind the simultaneous Jacks) when holding pocket Kings, despite the probability of such a scenario, in a 6-players ring, being a mere 2.425%. This means out of 50 KK hands played, I will / should encounter Aces 1.21 times - well, in my experiment, this happened, though I have played Kings only 9 times. If also taking into account the Jacks my second opponent had at the same hand, the probabilities go beyond ridiculous. And the same applies to Jacks: even though the probability of encountering an opposition over-pair is a mere 7.2% (meaning roughly one time for every 13.9 pair of Jacks played), it happened for me despite having played Jacks only 5 times!

50 If some Pokerstars representative would try claiming that there are two different RNGs used, depending on the real- or play- money nature of the games, it would, if you think about it, be the equivalent of shooting themselves in the foot.

52

one time I encountered Aces; three times I had 7s, and in one of them I hit an opponent with Queens; finally, having played pocket 4s one time was enough to bump, again, into Aces.

You get the general idea. I repeat: in my estimation, based on combined 110,000+ hands played on both real and play money on Pokerstars, and on other thousands I have not participated in, but only observed, and using multiple separate methods of measurement and eventual extrapolation, I feel it is safe enough to estimate that the real frequency of pair vs. pair encounters in 6-players game is at least (i.e. a measured) 1.75, but possibly (i.e. remeasured, plus estimated by various methods) even 2.1 times, higher51 than the normal 26.45% aggregated one and than the 2.425% specifically-targeted one. As a final remark, I think it is crucial to specify that, based on multiple measurements engaged, my conviction is that these inflated frequencies recorded on the poker platform are not the sole consequence of players being dealt pocket pairs more frequently, but the result of an additional software scheme that separately and increases the likelihood of such encounters.

II.5 Summary of the findings Covering all the 55,320 hands that were dealt to me in both cash and tournament games, the general statistical analysis of the hole cards' distribution on the Pokerstars platform, as the first step of this chapter's analysis, failed, aside from some minor exceptions, to discover any significant anomalies in relation to an expected, mathematically normal, Gaussian, distribution, which would prima facie suggest that the platform's RNG deals the cards in a truly random manner.

Even when starting with the second step of the analysis, namely a targeted analysis of the suited hole cards' distribution nothing suspicious appeared within my scope of observation, quite the contrary. Thus, all the general statistical indicators employed found recorded values that are almost perfectly overlapping the expected, normal, ones, with some of the differences even placed at the third decimal's position! Thus, overall, meaning at a general and superficial look, the distribution recorded on the poker platform is as normal as possible, and actually, and arguably, the most normal one could imagine / the most normal one has ever seen in nature!

However, my52 educated guess that the software somehow manages to always provide normal average values for the general parameters at the end of the day (despite brutally intervening not parallel to, but flagrantly against randomness) has been satisfactorily confirmed when narrowing and deepening the analysis in four exploratory directions. Firstly, at an intermediary level of segmenting the 169 hands to be possibly dealt in the game, one easily notices that, in their decreasing order of their value in the game: pocket pairs are dealt the most frequently in relation to both their expected frequency and other possible hole cards; the next ones, also above the normal frequency, are the suited hole cards, which, on the average, make up the best hands other than pairs; off-suite hole cards, the weakest among the three groups, are being dealt below their expected frequency.

Whereas, judged in itself, meaning isolated, this overlapping of power ranking and frequency of distribution might be still be a coincidence, things get a bit more difficult to justify, if walking in Pokerstars' shoes, when discovering narrowed down statistical anomalies, such as an indisputable correlation (-0.76), along the suited connectors series, between their power ranking in 6-players games and the frequency of the RNG dealing them 51

This is just a "guesstimate" based on the same 110k+ hands played, a speculation that I did not engage to test, but it's possible that the higher one's pocket pair is, the higher the chance of bumping into another high pocket pair, while the corresponding frequency for lower pairs might be smaller.

52 And not only mine, as one can easily notice when using online search engines to read out about Pokerstars' alleged irregularities.

53

to me. Additionally, there is also some discernible relation between the best hierarchical groups of hole cards (arranged according to their win plus split rate) and their dealing frequency, although, overall, meaning along all the 169 hands in the game, the general correlation is, again, almost inexistent.

The second confirmatory direction targeted separately the pocket pairs' distribution. In this case, all the statistical methods and indicators employed converge towards a definitive and categorical conclusion: the systematic over-representation of pocket pairs as hole cards cannot, by any means nor by any logical contortionism or "variance" invoking, be considered just an(other) "accident" and attributed to randomness; it substantially hints at a deliberate, premeditated, strategy of the poker platform, justified by the very way in which it makes a living of: rake. To this end, by dealing good, strong, hands, more frequently, it increases the "fun" for the players involved, which in turn may attract new players, which, subsequently and decisively, will only increase the rake to be collected by the platform.

The third explored and confirmatory direction of analysis, directly related to the previous one, focused on the so-called "coolers" on Pokerstars. And there have been thousands of people worldwide that have been complaining on the Internet about these "coolers", starting from the classic AA vs. KK duels, all of them significantly more frequent the normal (read: almost twice as frequent), and all the way up to quads vs. quads duels. And indeed some of the "accidents" happening on Pokerstars are truly grotesque judging by their minuscule probabilities, all clearly indicating anything but randomness. Among the extremely selective chosen examples from this chapter, one can find situations such as  me holding Aces and running into opposition Kings 1.8 more frequent than normal (and that is a minimum, assuming no KK holder ever folded a hand against my Aces before showdown!), two of the cases being consecutive AA hands played by me, which, in order to be normal, would require me to have played 159 hands, whereas the real, actual, number has only been 131;  three people at a 6-players table all having hearts-suited hole cards, with the five board cards giving them the hearts flush - a probability of 0.0298% of all the suited hands played;  an AA vs. KK vs. KK situation at a 6-players table - again, a infinitesimal probability;  a KK (me) vs. AA vs. JJ situation at the 6-players table, with me flopping a quads of Kings, an event that, in order to be considered statistically normal, would have required me to have played some 85 million hands, whereas my total combined cash + tourney games has barely reached 55.3k!;  a equally sensational 4s vs. As quads at a 4-players table, something which, from the perspective of the 4s-holding player, should happen to him once in every 81 million hands dealt!

One of Pokerstars' favorite excuses over time has been that anomalies such as the ones exemplified happen in online poker for the sole reason that one is playing significantly more hands than in any live form of poker. Well, I would say: go ahead and convince me that, in regard to the third example, I did indeed play 85 million hands! Or try explaining to the poor guy holding the pocket 4s in the last example why that "accident" happened to him, even though one could safely bet he did not play that necessary 81 million hands!

And, rhetoric aside, the crucial problem, as explained, is not that such things do happen, since they are (indeed, extremely) improbable, but nevertheless possible. After all, people get hit by meteorites, others win the big lottery, while others suffer from a 1 in 10 million rare disease, etc. The problem, and this is where Pokerstars' representatives have a tremendously difficult, or rather impossible, task in justifying themselves, is that all of these beyond-exceptional hands have one thing in common: all of them lead naturally, surely, implacably, to the same outcome - an increase of the pots by betting, raising and reraising, which in turns provides the platform with a higher rake to collect!

54

The fourth direction, a spin-off of the third, has extensively addressed the pocket pair vs. pair situations as another aspect all to frequent on the Pokerstars platform, and, as a subtype of the more broadly delineated "coolers" also ensuring a rake increase. Measured and re-estimated by various separate methods, and re-confirmed by a definitely unusual limpcheck-call experiment that I engaged in over 2.5k hands played, the frequency of pair vs. pair duels on the investigated poker platform is a measured minimum of 1.75 times, respectively an estimated possible maximum of 2.1 times higher than the normal frequency to be expected in a genuinely randomness-characterized environment. Finally, the same conclusion has only been reconfirmed when specifically analyzing the frequency two distinct phenomena, namely equal pair encounters and triple pair situations in 6-players games, with both types of events occurring, based on a set of extrapolations, twice more often than normal.

Correlation, regression, extrapolation, standards deviation and normality of distribution, empirical observation, individual and cumulated absolute and relative frequencies, reductio ad absurdum, timeframe segmenting, - you name it, I've tried it. All of these methods, indicators, parameters, and techniques employed in this chapter across the four mentioned exploratory analysis directions converge towards one single categorical conclusion: the way the hole cards are dealt by Pokerstars supposedly "random" numbers generator is anything, but random. Moreover, as the next chapters of this investigation shall also demonstrate and gradually reconfirm, what happens on the platform is not the visible effect of some unintended error, of some "glitch" within programming, but the expression of a deliberate, premeditated, and systematically applied policy of the platform's owners.

55

III. THE FLOPS: when you take the bait The present chapter investigates the alleged randomness of the flops I have encountered on the Pokerstars platform, by testing two fundamental hypotheses: a) the so-called "second shuffle" hypothesis, which states that the flops are not really random, but rather directly related to the exact hole cards of the players that have remained at the table beyond pre-flop, meaning that, after the initial hole cards dealing, there could be an additional, second, and severely non-random "shuffle", and b) the "leveling the field", a.k.a. "balancing the odds" hypothesis, which also suggests nonrandomness, but specifically looking at how the favorite vs. underdog duels on the platform illustrate a higher than mathematically expected win rate for the underdog.

To this end, after a general analysis as a first step, this chapter continues with a second step in the form of a case study based on all the hands (within the cash games) that I have been dealt pocket Queens. This case study shall re-target the two hypotheses, in a preliminary and partial confirmation, in order for the third part of this chapter to deepen the analysis of Pokerstars' flops across four analytic items: i.) flopping a set or better when starting with pocket pairs; ii.) flopping a flush draw or better when starting with suited hole cards; iii.) flopping at least one pair (a cumulative category, segmented at the right time) when starting with non-paired hole cards; iv.) flopping straight draws or straights when starting with middling connectors as hole cards, regardless whether they are suited or not.

III.1 An overall look at the flops on Pokerstars Much has been debated online over the years about the implausible frequency of situations in which either on the turn or, especially so, on the river, equities are reversed in spectacular turns of the tide, which has even earned Pokerstars one (other) nickname "Riverstars". It needs to be said that none of the dozens of items (each containing from tens up to thousands of cases) I have analyzed confirm this speculation; wherever I looked, from all-ins engaged on turn or river and up to hands that reached showdown step-by-step, probabilities on the two streets are generally and at least satisfactorily respected provided the sample of hands is representative enough.

Indeed, in cash games and tournaments as well, and regardless of the game format or stakes size, the Pokerstars platform does host some "accidents", some of them truly "horrendous", I'd say, such as, for instance, among the hands I personally played: ď&#x192;&#x2DC; the "classic" of hitting a 2-outer on the river (2 useful cards out of a deck of 44 cards, meaning a probability of 4.55%) exactly when you need it the most:

56

ď&#x192;&#x2DC; or reversing a 2% equity on the flop, against a flopped set, by hitting a backdoor straight on turn and river:

ď&#x192;&#x2DC; or winning the hand despite a... 1% equity on the flop, against a player who had flopped directly a full house(!):

ď&#x192;&#x2DC; or, finally, achieving the same performance in terms of equity reversal, but in a different manner:

In spite of such dramatic reversals of situations, easily etched in the memory of players, especially if they happened to their detriment, as said, over volumes of hands large enough, probabilities seem to be respected on Pokerstars at least to a satisfactory degree. For instance, within the total 51,989 hands that I played in cash games, I isolated those hands when, simultaneously, a.) I have been dealt pocket pairs; b.) the hands reached showdown; c.)

57

on the turn I was either a 95.45% favorite to win the hand, or the 4.55% underdog (meaning, simplified, I needed any of only two outs to win the hand). The results are these: ď&#x20AC;­ of the 112 hands in which, on the turn, I was a 95.45% favorite to win, I won 104, meaning 92.86%. The percentage is a little smaller than the expected one, but on the one hand not significantly so (minus 2.7% variation of the normal figure), and on the other hand it could indeed be argued that, whereas I personally played thousands of hands in which I deliberately stayed till showdown, despite strongly suspecting I was the underdog or even that I would not win, my opponents should have played more win-oriented. This would mean there could have been cases in which my 95% advantage on turn did hold, meaning I had mathematically won the hand(s), but my opponents folded on the river, just before showdown. If this was indeed the case, not the same argument could in turn be made in relation to ď&#x20AC;­ the 194 hands when I was the 4.55% underdog on the turn, out of which I won 9, meaning 4.64%, which is even slightly above the expected frequency.

Thus, as a general conclusion, probabilities at least on the turn, before the river, seem to be generally respected on the Pokerstars platform. The actual issue, or better said, this other issue, with Pokerstars has nothing (or, better said, not so much) to do with the turn or the river cards, but, and this might come as a surprise, with the flops! The reason why, to the best of my knowledge, no one has focused on them so far, has probably to do with two reasons. Firstly, sensational twists on the river such as hitting a two-outer (as mentioned, a probability of less than 5%), especially when to one's detriment and direct financial loss, will naturally be more easily remembered. Secondly, unlike the river, the flop contains not one card, but three, which makes the mathematics associated with it significantly more complex and correspondingly more difficult to grasp, let alone master.

Whereas in a heads-up situations, once the players have reached the turn, there are only 44 mathematically possible river cards53 and only one card to be dealt, a flop, even if considering the opponent's hole cards as known, may contain no less than 17.296 cards combinations54. Hence, "guesstimating" or actually calculating the probability of a specific river card is one thing, whereas doing the same thing for one particular flop is a completely, and significantly more difficult, matter. As an additional factor, only a fraction of the hands are played, by at least two players, till showdown, thus allowing us to see in what manner the flop cards related to the hole cards of the players at the table. Hence, even if something about the flop would be peculiar, or significantly improbable, for the average player who is not a mathematician and does not possess a software to store the played hands (and indicate equities) it would be extremely difficult to notice it.

And indeed there is something... "fishy" about the flops on Pokerstars, many of which cannot, by any overstretch of the argument, be attributed in a compensating-type explanation not even to those flops that weren't continued till showdown, flops which, in the speculative explanation, are claimed to have compensated any anomaly among the hands that did reach showdown. The underlying logic would be something like: "True, your flops have illustrated some exceptional probabilities, but all the other flops, among the hands that did not reach showdown, were 'normal', so that, on the overall, statistics are respected, and things are not uncommon!". Again, I categorically reject such a possible pseudo-argument.

Allow me to first remind the reader of two examples already given in the previous chapter, and then offer another set of five examples, plus a bonus, from among the hands that I played.

The two previously listed examples are the 0.039% probable J3K flop at a table with three opponents holding in their pockets KJo, JJ, respectively 33. The second, that one lucky hand of mine, was the triple pair situation where I held pocket Kings, suddenly woke up 53 54

Assuming, of course, both sets of hole cards are known.

Respectively 19,600 if the hole cards of the opponent are unknown.

58

with one opponent having Aces, the other Jacks, only for me to hit the jackpot of flopping Kings quads. Leaving aside the already tiny pre-flop probability of three players having exactly those pocket pairs, the probability of such a flop, in that given situation, was a mere 0.29%, which means that in a 6-players game, such an event will happen once in every 345 hands of KK that I am dealt (whereas, of course, redundantly specified) I did not actually get 345 KK hands.

Now let's get to another 5 + 1 new examples of flops that defy statistics and refute speculations, all of them selected from the subsample of hands in which I had been dealt pocket pairs, and all of them reinforcing the recurrent idea that Pokerstars will abundantly and deliberately provide players with statistically quasi-impossible flops, meant only to determine the players to stay in the game, bet high and thus increase the rake collected by the platform.

The first flop, captured below, is already a shocker, and strikingly resembles the first of the two examples recounted above:

One opponent has a pocket pair, while his two opponents somehow manage to simultaneously flop two pairs each, starting from non-paired hole cards. Well, for those not really into math, let's simplify the explanation. Considering the 6 known hole cards of the players, of the total C(46, 3) = 15,180 flops possible, there are only 18 providing the above situation: 2 (any of the two Kings left in the deck) * 3 (any of the three 10s in the deck) * 3 (any of the three Jacks in the deck). Thus, the probability of such a flop, considering the starting pre-flop hole cards, is 18*100/15180, meaning a minuscule 0.118%! To formulate it differently: out of 1,000 such starting situations, a flop such as the one above will be dealt a little over one time! Obviously, I did not start form such a pre-flop position 1,000 times; there was only a total 3,146 times that I have been dealt pocket pairs, of which only 2,132 times did I get to play or at least see the flop. Still, it seemingly was enough for me to encounter such a statistical improbability!

The second example is one equally spectacular, or actually even more so, considering it's probability to be dealt is slightly smaller, namely 0.104%!

59

Of the total - in this case - no less than 17,296 flops possible (C 48,2), only 2 (any of the two 6s in the deck) * 3 (any of the three 7s in the deck) * 3 (any of the three 8s in the deck), meaning only 18, could have generated a situation in which one player has a set, whereas the other has two pairs. And still, this exceptional case also happened!

I guess that, bearing a 0.237% probability, the third flop used here as an example should be considered at least in comparison to the previous two, as being extremely likely?!

60

Out of 15,180 flops possible, I was apparently lucky enough to get exactly one of the only 18 that could simultaneously guarantee my two opponents the same Jacks trip!

The fourth example is a hand when I was inspired enough to fold pre-flop, but my two opponents continued the hand all the way to showdown, so that the relation of the flop to their cards could be visualized:

The flop manages to simultaneously grant the players: a) a top-pair or at least what the AJs player might think is a top-pair; b) a set for the JJ-holding player, and c) another set, the lower one, for me, had I stayed in the game. Without going into further mathematical details, let's just state the probability of such a flop: 0.57%.

If these superficially selected four cases weren't enough, the fifth example should, normally, close the case. The two screenshots below capture two clusters of hands of pocket 4s that I played in cash games on Pokerstars, as stored by the HM2 software, with my pairs in the second column from the left, while the fourth one indicates the board cards for that respective hands:

So, in a matter of less than 24 hours, between April 14, 17:23, and April 15, 14:57, there were three consecutive instances among the pocket 4s hands that I had been dealt that I flopped a set! Well, the per se probability of flopping a set is 11.17% (or 11.76% for the cumulative category "set or better", which includes sets, full houses, and quads). The conditioned probability of flopping a set or better three times in a row is a mere (0.1176)^3 = 0.16%! But that's not it; less than a month later, I managed (well, correction: Pokerstars' RNG actually managed) to do the same thing four times in a row55 (out of which one was a quads case), a chain of events with a conditioned probability of... 0.0019%!

55

The intermediary hand of 44s that did not reach the flop can obviously not be taken in account, since anything could have happened in regard to it.

61

That would mean that, in a genuinely statistically normal, meaning authentically random, environment, such a series of flops would be expected to happen (almost) once in every 50,000 hands of pocket pairs played!56 To put it differently: if I had played 50k pocket pairs, then would such an event be possibly regarded as statistically normal, if there were any doubts left, allow me to specify: the total number of all pocket pairs I was dealt over all the 51,989 hands in cash games on Pokerstars has been 3,146, out of which a mere 2,132 made it (by me personally playing or my opponents continuing them) to the flop! And still, such a astonishing exceptionality once again made its way through Pokerstars' RNG mechanics!

As a parenthesis: similar things happen even when having non-paired cards; one can, for instance, flop two pairs 4 out of 5 consecutive times, which equals a probability of 2.02% * 2.02% * 97.98% * 2.02% * 2.02% = 0.0000163%, meaning an event expected to happen once in every 61.35 million non-paired hands played, where in reality the total number of flops I have played or seen when holding non-paired hole cards has been less than 20k! It is that simple: Pokerstars apparently has no hesitation whatsoever in providing you with the most extraordinarily improbable flops, all apparently in your favor, as long as you bet, raise, and re-raise, thus guaranteeing an increase of the rake it collects!

56

Or, by introducing the 5.88% probability of being dealt a pocket pair, it should happen once in every 1.12 million hands played (regardless whether I was dealt paired or nonpaired hole cards)! All this not even taking into account either a.) the probability of the first two flops in discussion (the inferior screen capture) containing the same value cards - K, 9, and 4, or b.) the probability that both of the first two hands ended up with me making a full house!

62

Finally, the promised bonus: my flopped royal flush, the best hand in poker!

For those not very familiar with probabilities, allow me to specify the one of an event such as the above: there are 1326 two-cards combinations possible in relation to a 52-cards deck that you can be dealt at the table. Out of those 1326, there are only 40 that will put you pre-flop in the necessary position to flop a royal flush draw, the made royal flush included: 4 (combinations) for each of these ten hands: AKs, AQs, AJs, ATs, KQs, KJs, KTs, QJs, QTs, JTs. That means that in only 100*40/1326 = 0.302% of all hands to be possibly dealt, one gets, in a first step, in the pre-flop position necessary to aim at flopping the royal flush.

However, even if you are dealt one of those 40 combinations, there is the second step, even more improbable than the first one: only one of 19,600 possible flops (C50,3) will contain the targeted combination, meaning those specific 3 cards that will give you the royal flush; for instance, if you were dealt AJ of diamonds, the three flop cards must be the K, the Q and the T of diamonds, and none else. In conditioned terms, that means a probability of (40/1326)*(1/19600)*100 = 0.0001539% of flopping the royal flush or, in an alternative formulation, such an event will happen once in every 649,772 hands played! Well, apparently not on the Pokerstars platform, where my 51,989 hands played in cash games have proven to be more than enough for such a sensationally improbable event to occur!

And no, that was not an "accident", although, under normal circumstances, such an event might, indeed, happen, in the sense that it is possible, though extremely improbable.

If further proof for such a statement was needed, then I should add that, among the exactly 999 flops that I have reached in total when holding any of the 40 above-listed necessary combinations, I have additionally flopped a royal flush draw no less than 6 (six) times57, 57

Out of which one even turned into a royal flush (see above) - an in itself probability of 4.25%.

63

which, excluding the made royal flush, equals a recorded frequency of 6/998 = 0.601%, which is 2.5 times higher than the normal, expected one, of 0.239%!58

It seems that everywhere you look, the flops are exceptional in terms of probabilities. In order to encounter such unbelievable "miracles", or, depending on the perspective and outcome, terrible "misfortunes", I (and to an equal degree, your average player on Pokerstars) should have played hundreds of thousands of games, which, of course, is not the case. And, as repeatedly argued and stressed throughout this paper, the problem is not that such events do occur, but that all of them tend to happen in the same direction, i.e. directly corresponding to the hole cards of players remaining at the table and to their apparent benefit!

More generally, such observed anomalies are simply the visible effect of an articulate and deliberate business strategy implemented by the poker platform, meant, among other things, to increase the fun, to deliver adrenalin and spectacle via all kinds of sensational hands and turnings of situations, in order to increase its player base and subsequently the rake it collects. To this end, on Pokerstars, as this entire study demonstrates, better cards will be dealt to you more often than normal, flops will contain enticing combinations more frequently, combinations such as sets, full houses or quads when starting with pocket pairs, or flush draws when starting holding two suited cards, or straight draws, trips and two pairs, when holding non-paired cards. Equally true, there will also be an abundance, again significantly above normal parameters, of the so-called "coolers" (e.g., pair vs. pair duels, straight draws vs. flush draws, flushes vs. sets, even quads vs. quads, etc.), in order to push players to bet and raise and thus increase the pot and the rake to be collected by the platform.

58

Or a recorded 0.700% vs. an expected 0.245% if the flopped royal flush is taken into account.

64

III.2 A case study: my pocket Queens In another distinct testing of this analysis' main hypotheses, the following isolates as a study case the sample of 249 pocket Queens that I have been dealt over the investigated timeframe in 51,989 cash-games. Of the total 249, - 95 I won without the hands being played (opponents folded pre-flop), - 67 have been played at least on flop, but were folded before showdown by either me (14) or my opponents (53), and - 87 have reached showdown.

III.2.1 The data commented The table below lists all the 87 hands that reached showdown, inventorying my hand, the opponents' hands, the flop cards along, when needed, a short comment, and the final outcome of the hand in terms of both me winning or losing, and the favorite (regardless whether it was me or the opponent) winning or losing. All probabilities listed in the table have been recalculated accordingly to the two set of hole cards known. In terms of the shadings in the extreme right column, green indicates that the pre-flop favorite won, red that he lost, while blue signifies a split pot, while "W", "L", and "S" indicate whether I won, lost or tied.

No.

Date

My cards

Opp. cards

Flop

28.5 27.5

Time (EET) 16:23 13:08

1.

2.

QQ QQ

QT 34

J6A 8Q8

3.

27.5

12:44

QQ

4.

5.

27.5 27.5

11:59 00:27

QQ QQ

6.

27.5

00:10

QQ

7.

26.5

23:33

QQ

8.

26.5

22:37

QQ

Situation on flop (probability of hitting that situation of flop (if relevant)):

Opponent: flush draw + gut-shot straight draw (2.20%) Me: full house (0.76%) Me: set (11.17%) KJ 10 Q 3 Opponent: open-ended straight draw (2.13%) Opponent: one pair (33.63%) + backdoor flush / straight draw A5 725 Opponent: one pair (33.63%) + backdoor straight draw 79 389 Comment: with a 25% equity on flop, my opponent hits his second pair (7s) on turn and wins 55

4K8

W W L W

Me: set (11.17%) 48 76Q Opponent: gut-shot straight draw (12.49%) Comment: with a 15% equity on flop, my opponent hits the 5 on turn, making a straight, and wins 44

Outcome (for me): W W

5AK

L W

65

9.

26.5

21:51

QQ

A9

387

10.

11.

26.5 26.5

19:02 18:36

QQ QQ

4K KJ

K37 KJ7

12.

26.5

18:17

QQ

AJ

752

13.

26.5

16:14

QQ

KA

922

14.

26.5

15:10

QQ

15.

16.

17.

25.5 25.5 23.5

18:25 13:31 13:14

QQ QQ QQ

18.

23.5

13:02

19.

23.5

20.

22.5

Comment: after the turn 7, with an equity of 14%, my opponents hits K on the river and wins Me: set (11.17%) AA JQT Opponent: gut-shot straight draw (28.10%) Comment: Despite a mere 18% pre-flop equity, I flop a set at the right time and win, with opponent missing both the straight and a set on turn or river (22% equity on flop) Opponent: gut-shot straight draw + backdoor flush draw T9 865 Opponent: pocket over-pair + gut-shot straight draw (1.48%) AA 342 AA

K23

QQ

QT

379

11:54

QQ

9T

K54

16:31

QQ

21.

22.5

14:43

QQ

22.

22.5

14:25

QQ

23.

22.5

14:19

QQ

24.

22.5

12:23

QQ

Me: pocket over-pair (61.63%) Opponent: backdoor straight draw + backdoor flush draw Opponent: top (over-) pair (17.96%) Opponent: two pairs (2.18%) Me: pocket over-pair Opponent: two over-cards + backdoor flush draw & straight draw

Me: pocket over-pair Opponent: 2 over-cards + backdoor straight + flush draw

88

W L

W W L L W

2TJ

472

W

W W W

Me: pocket over-pair (57.12%) Opponent: flush draw (2.54%) Comment: the 2 on turn places my opponent on a 25% equity, but the A on river gives him the flush JJ

L L

W

Me: backdoor straight draw Opponent: pocket over-pair + backdoor straight draw Comment: my 10% equity on flop increases to 14% after the 9 on turn. The 8 on river gives me the straight and I win Me: set (11.17%) J7 2QJ Opponent: one pair (33.63%) Me: pocket over-pair (61.63%) AJ 69J Opponent: top-pair (33.63%) KK

W

734

66

L

25.

22.5

10:33

QQ

KQ

9JJ

26.

21.5

14:27

QQ

88

46T

27.

21.5

11:40

QQ

28.

20.5

12:27

QQ

29.

30.

31.

32.

19.5 19:5 19.5 19.5

19:21 19:10 18:41 18:27

Q Q Q Q

Q Q Q Q

Me: pocket over-pair (61.63%) Opponent: gut-shot straight draw at higher end (4.25%)

W

Me: pocket over-pair (61.63%) Opponent: 2 over-cards + backdoor flush draw Me: backdoor straight draw AA 82T Opponent: pocket over-pair Comment: despite equities of only 18% pre-flop and 10% on the flop, I once again manage to beat Aces, making a straight with 9 on the turn and J on the river AJ

79 AA 22 A7

483

4A9 J7A K83 588

Opponent: top-set (11.17%)

Comment: placed on a 19% equity on flop, my opponent hits the A on turn and wins Me: pocket low-middle pair + closed-ended straight draw KK TJA Opponent: pocket high-middle pair + closed-ended straight draw

33.

19.5

16:55

QQ

34.

35.

19.5 19.5

14:26 13:30

QQ QQ

36.

19.5

11:16

QQ

37.

18.5

10:00

QQ

38.

16.5

16:25

QQ

39.

16.5

12:31

QQ

JJ

894

40.

16.5

11:45

QQ

JJ

4J5

41.

42.

14.5 14.5

17:05 16:29

QQ QQ

AJ KK

7K2 AA8

TJ 88

W

8K8 969

Opponent: pocket middle-pair + backdoor straight draw Me: pocket over-pair JA 557 Opponent: 2 over-cards + backdoor flush draw Me: pocket middle pair J8 8TA Opponent: one pair + backdoor straight draw (2.2%) Comment: despite equities of 17 and 19%, my opponent hits the J on turn and wins with two pairs Me: pocket over-pair AK 9J5 Opponent: two over-cards + backdoor straight draw

W

W W L W L L W W W L W W

Me: (what I thought was a) pocket over-pair Opponent: set (11.17%)

L W L

67

43.

14.5

15:35

QQ

44.

14.5

14:17

QQ

45.

13.5

17:49

QQ

46.

13.5

15:34

QQ

47.

13.5

14:18

QQ

48

13.5

13:34

QQ

49.

50.

51.

52.

12.5 12.5 11.5

10.5

14:29 11:09 11:19

16:28

QQ

QQ QQ

QQ

Me: pocket middle-pair + backdoor straight draw Opponent A: two top pairs (2.37%) Opponent B: open-ended straight draw (4.93%) Comment: A quite remarkable hand, but one not so unusual on Pokerstars. Even leaving aside both my loss, despite a 52% pre-flop equity, and the conditioned probability for the three players of flopping such a combination, the equally surprising fact is that the J on river gives opponent B the straight and makes him the winner, despite his equities of 17% pre-flop and 14% on flop (vs a flop equity of 65% for opponent A on the flop.

Me: (what I thought to be a ) pocket over-pair 78 838 Opponent: trip (1.46%) Me: pocket over-pair AJ 3J8 Opponent: top pair (33.63%)

KT

78

99

KT9

8JA

L

L W W

Me: pocket over-pair AK 578 Opponent: Two over-cards Comment: With a 30% equity on flop, opponent hits K on the turn and wins Me: (what I thought would be) pocket over-pair + board pair (31.66%) Opponent B: trip (1.58%) 85 43 545 Opponent C: two pairs (4.23%) Comment: Once again, an opponent with two non-paired off-suit cards flops a trip, enough for him to win, despite a 18% pre-flop equity (vs. 65% (me) vs. 17% (opponent B) Opponent: top (over-) pair (17.96%) KQ 52K Opponent: one pair (33.63%) 3Q 2A3 Opponent: set (11.17%) JJ 9AJ Comment: Another Pokerstars trademark hand; my opponent, having started as a 18% underdog, is lucky enough to flop a set, which places him as a 88% favorite to win the hand. However, after the blank 6 on the turn, when my equity had dropped to 4.55%, I managed to hit a two-outer on the river, making a set of queens and winning! In terms of the conditioned probability of this happening, it means that out of 100 such encounters, an event like the one above (with me hitting a set and turning the tide on either turn or river) will happen 1.06 times.

Me: pocket over-pair (61.63%) A8 85J

68

L

L

L W

W

W

Opponent: one pair (33.63%) 53.

9.5

16:35

QQ

JK

A62

54.

7.5

13:57

QQ

QK

10 J Q

55.

6.5

10:12

QQ

TT

956

56.

3.5

15:22

QQ

33

42K

57.

58.

59.

3.5

1.5

30.4

13:14

17:12

12:33

QQ

QQ

QQ

KJ

8A7

27.4

10:03

QQ

AJ

AT6

61.

22.4

12:51

QQ

55

K62

63.

22.4

21.4

12:50

13:58

QQ

QQ

Me: set (6.25%) Opponent: one pair + open-ended straight draw (0.18%) Me: pocket over-pair (57.12%) Opponent: (an apparent) pocket over-pair + backdoor straight door (8.88%)

Comment: My Queens hold, despite the 5 on the turn granting my opponent an open-ended straight draw and an equivalent 23% chance to win the hand Me: (what looked like a) pocket over-pair (57.12%) 99 J39 Opponent: set (11.17%) Comment: Opponent wins as the initial underdog (18% equity pre-flop), despite the 10 on the turn giving me an open-ended straight draw (corresponding to a 23% equity) Me: pocket over-pair + board pair 88 939 Opponent: pocket middle pair + board pair Comment: a Pokerstars classic example of how to lose two times in a row against a lower pair (i.e. a 3.4% conditioned probability); after the blank 2 on the turn, my opponent's equity had plummeted to 4.55%. Still, he manages to hit the 8 on the river and win the hand with a set!

60.

62.

W W W W

L

L

W Me: pocket high-middle pair + flush draw Opponent: top (over-) pair (17.96%)

L W

Me: full (0.76%) TK QJJ Opponent: board pair + open-ended straight draw (0.14%) Comment: my second pocket pair of Queens dealt in less than a minute of playing time and my second flopped full house in less than 100 QQ hands. In a classic example of Pokerstars so-called "coolers", meant to keep players in the game and stimulate them to bet and increase the pot size, not only do I flop a full house and my opponent an open-ended straight-draw, but after the 3 on the turn, he also gets a 9 on the river, thus making both a straight and a flush.

Me: set (11.17%) KK Q5T Opponent: (an apparent) pocket over-pair + backdoor straight draw 69

W

W

64.

65.

66.

67.

68.

19.4 18.4

18.4

18.4 17.4

12:39 17:22

12:24

00:15 15:30

QQ QQ

QQ

QQ QQ

AJ A5

65A 356

Opponent: top (over-) pair (17.96%) Opponent: one pair + flush draw + backdoor gut-shot straight draw (1.44%) Comment: second time in a row my Queens lose against an opponent with an over-card (a 3.22% conditioned probability per se, not taking into account his additional flopped draws). After 3 on the turn, on a 30% equity, my opponents gets the A on the river and wins.

Opponent: middle pair (17.96%) KQ KA6 Comment: third loss in a row on the flop against an over-card (i.e. a 0.58% conditioned probability.

This means out of 200 such situations, this chained series of events should happen roughly once. This of course borders the absurd, since my total all-time number of Queens played at least on flop has been 154). Additionally, this hand is also the fourth in a row that ends with the victory of the pre-flop underdog (which in itself bears a conditioned probability of 0.104%, meaning such a thing happens once in every 1,000 QQ hands played!)

AA KT

37J 67K

17.4

13:17

QQ

70.

16.4

13:41

QQ

66

A3K

71.

15.4

15:02

QQ

AQ

KTT

72.

73.

74.

15.4 14.4 14.4

13:50 17:11 16:52

QQ QQ QQ

8K KK QQ

A94 53A AT3

75.

14.4

16:19

QQ

9T

T52

76.

77.

13.4 13.4

15:34 15:19

QQ QQ

77 JJ

A23 24K

L

L

L

Opponent: top (over-) pair (17.96%) Comment: the probability of losing for the fourth time in a row, on the flop, with pocket Queens against one over-card is 0.1%, meaning in a genuinely random ambient, it should happen once in every 1,000 QQ hands played! Mind you, I only played 154 QQ hands on the flop in all of my cashgames experience on Pokerstars!

Me: pocket over-pair (57.12%) T7 T96 Opponent: top pair + gut-shot straight draw (1.85%)

69.

L

L

W W

Opponent: gut-shot straight draw + board pair + backdoor flush-draw + one over-card (0.14%)

W W L S

Me: pocket over-pair + flush draw (1.39%) Opponent: top-pair (30.18%)

W W W

70

78.

13.4

13:46

QQ

79.

13.4

11:11

QQ

80.

13.4

10:31

QQ

81.

82.

83.

12.4

12.4 11.4

13:50

11:49 15:11

QQ

QQ QQ

A2

AJ3

Opponent: top (over-) pair + flush draw (1.27%) Me: pocket over-pair (57.12%) 7T 398 Opponent: open-ended straight draw + backdoor flush draw (0.57%) Me: pocket over-pair (57.12%) JT T42 Opponent: top-pair (34.4% maximum) Comment: having started on a 18% pre-flop equity, which remained unchanged on the flop, my opponent hits the J on the turn and wins with two pairs Me: set (11.17%) 8T TKQ Opponent: bottom pair + flush draw (3.82%) Comment: situated on a 29% equity on the flop, my opponent hits the 4 on the turn, making a flush and winning the hand JJ 33 78

A68 24K

86.

87.

11.4 9.4

9.4

9.4

12:30 14:12

13:18

12:36

QQ QQ

QQ

QQ

KK AT

W

L

L W

Comment: Having started from a 62% equity pre-flop, I climb to 83% on the flop, but still lose, as the 5 on the turn and the 6 on the river provide a straight for my opponent A (with the pair of 3s, who started pre-flop with a 17% probability of winning, whereas on the turn his chance had been 14%)!

84.

85.

L

T65 AT6

L L

Opponent: top two pairs (1.87%) Comment: Another Pokerstars "classic"; having started as the underdog (33% equity pre-flop), my opponent is incredibly lucky to flop two pairs, which propels him to a 90% probability of winning.

However, the Q on the turn gives me a winning set.

Me: pocket over-pair 98 946 Opponent: top pair + backdoor straight draw Comment: the same "classic", only this time the other way around; despite my pre-flop equity of 80%, which remained almost unchanged on flop and even increased to 89% after the K on the turn, the 8 on the river offers my opponent a winning two pairs.

Me: two pairs + backdoor straight draw JK KTK Opponent: trip (1.46%) Comment: In hindsight, my very first hand of Queens played on real money on Pokerstars couldn't have resulted in any other outcome than my opponent starting from a 28% pre-flop equity and hitting a sensational 1.46% probability of flopping a trip, could it?!

71

W

L

L

III.2.2 Main findings A. "Leveling the field" hypothesis & "Every tenth hand..." corollary confirmed Focusing on the extreme right column's shadings in the table above, it can be noticed that, leaving aside the one singular hand in which two QQ pocket pairs ended up splitting the pot, in 31 of the remaining 86 hands that reached showdown, the initial, pre-flop, favorite to win actually lost the hand. This is a staggering 36% of the cases, so more than a third of those hands that reached showdown! And - it needs to be emphasized - this has nothing to do with any methodologically incorrect aggregation of the situations regardless of each one's specific equities ratio, an assertion that can be proved in two ways.

Firstly, when taking into account, for each hand, the favorite's pre-flop equity59, his/her resulting average equity over the total 86 hands that reached showdown turns out to be a substantial 75.92%. This means that the favorite should have won 75.92*86/100 = 65.3 of the hands. Which further means that his/her actual, recorded win rate of only 55 cases, meaning 63.9%, represents only 84.22% of the normal, mathematically correct and expected rate. That is a staggering 10 out of 65 expected victories that simply went "m.i.a." Or, to formulate it from another perspective: 10 of the total 86 hands, that is 11.63%, had their expected, outcome hijacked in favor of the underdog60.

Secondly, even when segmenting different types of situation with their corresponding equities, the recorded findings again suggest there is something "rotten" with Pokerstars' RNG. For instance, in the uniform subsample of the 32 (pure heads-up, unequal) pair vs. pair encounters61, the underdog (regardless if that was me or my opponents), despite starting with an average equity of 18.5%, actually won 8 times, which is equivalent to a 25.0% win rate! Or, as another example, out of the strict 24 encounters of my pocket Queens with an over-card of an opponent, I won only 15 times, meaning 62.5% of the cases, which is also significantly below the normal, mathematically correct win rate of 69Âą2%, etc.

This set of preliminary findings confirms, alongside the "every tenth hand is manipulated" corollary, the "leveling the field" or "balancing the odds" hypothesis, recurrently addressed throughout this study and essentially stating that Pokerstars' RNG mechanics is not at all random, but intentionally engineered to constantly and deliberately assist the underdog. The purpose of this premeditatedly altered and non- (to not say anti-) random algorithm, is two-fold. One the one hand, it keeps the weak players (who will find themselves positioned as underdogs more often than experienced players 62) attached to the game, by preventing their demoralization and ultimately abandonment of the game on the poker platform. On the other hand, it prevents what may be the true, ultimate, nightmare, for 59

That is for instance 84% in hand No. 1 (with me as a favorite), 83% in hand No. 2 (also with me as the favorite), 68% in hand No. 3, etc.

60 Here is a "teaser" - the figure is amazingly close to at least two others: a.) the 11.73% pre-flop heads-up all-ins whose natural, mathematically correct outcome we also discovered to have been hijacked in favor of the underdog; b.) the 10.19% of "anti-mathematical" outcomes discovered in the experiment undertaken on May 2667 and also described separately in this study (for details, see chapter five). These findings preliminary substantiate the corollary that on the Pokerstars platform, roughly every tenth hand has an outcome directly contrary to the expected mathematical outcome that should be found in a genuinely random environment!

Finally, if any further (re-)confirmation was needed, I applied the same method to the KK hands I took to showdown (except the one hand with split pot against KKs). Placed on a weighted average equity of 77.86%, the favorite (regardless whether that was me or an opponent) won only 63/93 cases = 67.7% of times, instead of the expected 72.4 times. This means a (72.4-63)*100/93 = 10.1% of hands had their outcome manipulated against the mathematically normal course. The aligning of these percentages is remarkable and strongly hints at premeditation.

61 Again leaving aside hand No. 74, in which two hands of pocket Queens split the pot, but also Hand No. 83, where three players reached showdown.

62 Since good players are good also, or especially, because they tend to know what hands when to play, and not only how.

72

any poker platform, namely the good players rapidly cleaning up the house to the detriment of weaker players and then withdraw all their money from the platform. Additionally, from a broader marketing perspective, such sensational hands and dramatic twists of situations will only make the game more spectacular and funnier and thus potentially attract more and more players on the platform.

Finally, contextualizing by returning to the extreme right column in the table containing all my played hands of pocket Queens, one also needs to notice that the spectacular and mathematically bizarre win rate of the (initial, pre-flop) underdog in the cases listed is not due to some equities reversal on turn or river. This would anyhow, and naturally so, attract (even more) attention from experienced players, at least. No, the actual twist happens (already) on the flop, and that is where things are anything but random! That's the street where, way too often on Pokerstars, the initial pre-flop top dog suddenly wakes up in the position of the underdog, and vice versa! Four example-categories are relevant in this regard.

B. Separately targeted items The following section reframes the analysis within the case study along four items, three from the opponents' perspective, one from my own: • flopping a trip when holding two unpaired cards against my pocket QQ; • flopping two pairs (using both cards) starting from the same opponent's situation; • flopping the needed over-card (meaning A or K) against my QQ; • flopping a set or better from my QQ-holding perspective Item No. 1: flopping a trip when starting with two non-paired cards Let's take a closer look to the hands listed in the table and offer a convincing first example; again, the reader may notice how often either my opponents or me hit a an unbelievably improbable flop.

Take for instance, as captured below, my very first hand of Queens that I played on real money on the platform - hand No. 87 in the table: although I start as a 72% favorite to win, on the flop my equity plummets to 7%, as my opponent hits a 1.46% probability of flopping a trip starting from two non-paired off-suit cards! Well, it appears he hit that jackpot exactly when playing against me, which is upsetting for me, but, let's say it again, statistically possible, isn't it? It could still be "random", couldn't it?

Well, I will say this as categorically as possible: no, it couldn't, and no, it is not random! For this given situation, when looking closer at the table, we find another two cases: hand No. 44, where my opponent flops a trip of 8s, and hand No. 48, where one of my opponents flops a trip of 5s. For these given, and known, situations and players' hole cards, the probability for my opponents to flop a trip were: 1.46%; 1.58%; and 1.46%, which would mean an average of exactly 1.5% probability. Then again, at least among the 87 hands that reached showdown, my opponents (or, better said, at least one of them) started only 54 times from the same situation, i.e. two non-paired off-suit cards63. This means that they flopped a trip 3/54 times = in 5.55% of the cases, which is no less than 3.7 times higher than the normal frequency of such an event happening! This has nothing to do with randomness, quite the contrary!

63

Meaning all hands that reached showdown in any other configuration than pocket pair vs. pocket pair.

73

74

Without any malice, I can even anticipate Pokerstars' possible excuse: the above cases are only the ones in which my opponents hit their trip on the flop, and it is exactly for this reason that they went all the way till showdown. In all the other cases, meaning hands that did not reach showdown, they didn't, and therefore they folded. I shall say it again: false!

Let's do some simple first grade arithmetic: of the total 249 pocket Queens that I have been dealt in 51,989 cash games on Pokerstars, 95 of them did not reach the flop, in all of these cases my opponents having folded pre-flop. We cannot speculate anything about these hands, neither what hole cards the opponents had, and even less so, what the boards would have looked like.

This leaves a remaining 154 hands that reached at least the flop. Of them, 87 reached showdown (and are listed in the table above), whereas 67 have been folded before showdown - 53 by my opponent(s), meaning I won, and 14 by me (meaning I lost). Now, let's try some grotesque contortionism of logic and mathematics and assume that, somehow, in all the 67 hands folded before showdown, my opponent(s) had two non-paired cards (meaning, ad absurdum, obviously, that in all these cases they had neither pocket pairs, nor two overcards, nor two undercards, nor a Queen and some other card (i.e. high- or low-blocked cards).

Well, even if so, that would leave only a total of 67 hands folded before showdown plus another 46 that reached showdown (as in the other 87-54 = 33 hands reaching showdown they certifiably had a combination other than non-paired cards). This would thus mean a theoretically - 67+54 = 121 hands. Now let's refer to this figure the 3 recorded, proven, cases of flopped trips: 3/121 = 2.48%. So, even in this fractured logic, the recorded frequency of flopping a trip is still significantly higher than the normal, mathematically correct, one of 1.5%, thus highlighting Pokerstars' flops as anything but random!

Thus, one cannot reasonably talk about "accidents" anymore. The fundamental thesis here is this: the flops on Pokerstars are not at all random, and deliberately so; following a probable so-called "second shuffle" after the pre-flop phase has ended, they will all too often, in a premeditated way, be exactly and directly related to the hole cards of those players that have remained at the table. The reason for this is subordinated to the fundamental business strategy of the platform: stimulate players to bet big, by granting them good, advantageous, useful and enticing flops, which translates into bigger pots, which in turn ensures higher rakes for the platform to collect. It's the very same fundamental reason that explains, inter alia, why pocket pairs are constantly being dealt more often than normal, why the frequency of pair vs. pairs encounters (and especially "coolers" such as AA vs. KK) is roughly twice the normal, why the percentage of flops containing a flush draw when starting with two suited hole cards is also higher than the normal, why the number of flopped straights is higher than the expected rate, etc.: all these statistical anomalies serve the same business oriented purpose of Pokerstars - make players bet bigger, which in turn ensures bigger rakes to collect.

And this is why, as we return to the table, one can notice incredibly improbable flops, with so many of them coming in a "delivered" manner, in direct relation to the players' hole cards and apparently to their direct benefit. To put it in a simplistic form with a didactical purpose: after the initial dealing of the hole cards, when players have bet/folded/raised/etc., there is a second shuffle of the cards, directly related to the hole cards.

Item No. 2: flopping two pairs when starting with two non-paired cards Arguing in favor of the above, let's take another look at the table, at the same type of opponents' hole cards situation, meaning two non-paired cards, only this time isolating those flops that provided them with two pairs (using both hole cards). Exactly as with the trips, this scenario also occurred three times: hand No. 11 (KJo - 2.18% probability in that given situation), hand No. 43 (KTs - 2.37%), and hand No. 85 (ATs). Three times per the same 54 total cases means the same 5.55% probability, which, again, is significantly higher (specifically three times higher) than the normal, expected, frequency of 2.18% associated 75

with flopping two pairs when knowing one opponent's hole cards. And again, practicing the same logical ballet as in the previous example and assuming that in all the 67 hands folded pre-flop when I held pocket Queens, my opponents had the same two non-paired hole cards version, it would still mean a frequency of flopping two pairs of 3/121 = 2.48%, which, again, is above the normal, mathematically normal, one of 2.02% (respectively 2.18% presuming one opponent's hole cards are known), which one should expect in a genuinely random environment.

Item No. 3: flopping the A- or K- top pair against QQ A third example: the flops that I played against opponents that had one over-card in hand in relation to my pocket QQ (so either an Ace or a King): 11 of the 30 flops played and continued till showdown provided my opponents exactly that one over-card they needed that is an amazing 36.67%, more than twice the normal probability of 17.96%! And staying with this same type of situation, as already mentioned in the table's section of comments, I even managed the astonishing performance of losing four times in a row against opponents with one over-card (hands No. 64, 65, 66, and 68), the conditioned probability of such a chain of events happening being a ridiculous (0.1796)^4 = 0.104%! That means that, if playing pocket Queens 1,000 times, the above accident will happen once! Needless to remind you, the total number of Queens I have taken to showdown has been 11.5 times less, or 87, specifically! Even if absurdly taking into account the total number of pocket Queens that saw at least the flop, the number, i.e. 154, would still be 6.5 times lower than the 1,000 required for the above accident to happen under statistically normal circumstances.

I will say it again: neither this third example of significant anomaly can be attributed to those hands that did not reach showdown, for at least two reasons. Firstly, in relation to the total 154 hands that made it at least to flop, the very frequency of flops containing at least one over-card (Ace or King) has been 66/154, meaning 42.86%, above the normal, expected frequency of 41.43%. In fact, the most frequent flop card when I held pocket Queens has been Kings64!

Secondly, when looking at those flops that did not reach showdown, one discovers a significant number of hands that contained A and/or K in the flop, which, combined with the betting patterns of the opponents both pre- and post-flop, strongly suggests there have been at least ten cases in which, again, the flop delivered exactly that one over-card needed by my opposition. In more precise terms, of the 53 hands my opponents folded before showdown, 8 contained A/K in the flop. These one could, I think, be safely ruled out as not belonging to the here discussed category of situations; if my opponents had that flopped over-card in hand, thus getting in a top-pair situation, they probably wouldn't have folded. However, of the subtotal 14 hands that I folded before showdown, 10 contained Ace(s) or/and King(s), with my opponents betting aggressively, which in fact ultimately determined me to fold.

Presuming that in only 2/3 of these cases, my opponents did indeed have the A or K making a top-pair on the flop, while in the other 1/3 of the cases, they either bluffed or did not have a non-paired hand combination (but something else, such as pocket pairs), the overall mathematics still makes no sense, unless the actual frequency of opponents flopping their A or K top-pair has been significantly higher than the normal, expected, one.

Item No. 4: flopping a set or better when starting with pocket QQ A fourth example, this time in regard to my perspective and position: of the total 154 times I played my pocket Queens beyond pre-flop, I amazingly flopped two full houses (Hands No. 2 and 62), one quads (see capture below) and, aggregated, sets or better 19 times (10 times among the 87 hands taken to showdown and another 9 times among the 67 hands ended before showdown) - all three recorded frequencies being above their normal, expected 64

44 entries, equally placed with 3s.

76

counterpart65, regardless if the latter is calculated by taking into account known opposition hole cards or not: - full houses: 1.30% recorded vs. 0.73% normal; - quads: 0.65% vs. 0.245% normal; - sets or better: 12.33% vs. 11.76% normal.

Just in case you (or, read: Pokerstars' representatives) think that a singular instance such as the Queens quads is not enough and that, perhaps, the situation normalized along other pocket pair that I played (meaning a balancing and normalization of probabilities materialized), think again! Moving one level up, from Queens to pocket Kings, of the latter's 139 instances that (at least) the flop was reached, two of the flops provided me quads, the first of them already screen-captured above in this paper66, the second being this one:

65

Since this includes all 154 flops, regardless if they were continued to showdown or not, the expected frequencies indicated are the usual ones, calculated obviously on the basis of not knowing opposition's hole cards.

66 The AA vs. KK vs. JJ hand where I flopped four of a kind.

77

So, even leaving aside the minuscule probability of the first instance's three players being dealt (exactly) those pocket pairs and the 19% underdog flopping quads exactly then, 2 flopped quads in 139 flops seen means a frequency of 1.44%, which is almost 6 times higher than the normal frequency of 0.245%. Or, adopting a combined Queens-Kings approach, the recorded frequency of quads reached 3/293 = 1.02%, meaning over 4 times more often than normally expected if things on Pokerstars were really random.

C. A contextualization Finally, for a probably useful contextualization, I should mention a similar segmented analysis undertaken for all the KK hands I played highlights essentially the same anomalies as was the case with Queens. Reminding you there have been 45 KK hands that didn't make it to showdown and 94 that did (out of which 1 was a KK vs. KK duel), the summarized findings are the following:  total showdowns including at least two pocket pairs: 37 (out of which, hereinafter excluded, one the KK vs. KK encounter, one included a triple pocket pair situation, and one included two pairs and a third opponent with non-paired hole cards);  starting from two non-paired hole cards, my opponents flopped exactly one pair 22/58 times = 37.9% (vs. 33.6% normal);  from the same starting situation, they flopped directly two pairs 3/58 times, meaning 5.17% (vs. 2.18% normal) - which could be normal only if absurdly assuming that in all the hands folded before showdown, my opponents had the same type of hole cards and none of them made 2 pairs on the flop;  from the same situation, they flopped trips 1/58 times = 1.72 times (vs. 1.46% normally, a figure theoretically attributable in a compensation logic to the hands folded);  with an over-card in hand, my opponents paired that card on the flop 8/27 times = 29.6% (vs. 17.96% normally);  out of 34 heads-up (unequal) pair vs. pair duels that made it to showdown, the lower paired underdog won 8 times = 23.5% (vs. 18.5 normally);  two times when starting with pocket pairs they flopped directly a full, meaning a frequency of 2/36 = 5.55% (vs. 0.76% normally, a figure that cannot be attributed even ad absurdum to folded hands, as the corresponding ratio of 2*100/(36+45) = 2.47% would still be over three times higher than the normal frequency);

78

ď&#x192;ź from the same starting situation, they flopped a pure set 6/36 times = 16.7% (vs.

11.17% normally); ď&#x192;ź two out of 36 pair vs. pair encounters bumped into a flop with a set vs. set situation, meaning a dumbfounding 5.05% (another figure that could not be compensated in relation to the normal 1.02% even if absurdly taking into account folded hands);

ď&#x192;ź as mentioned, I also flopped quads 2/139 times = 1.44% (vs. 0.245% normally) and a full house 1/139 (0.72%, probably the only relatively normal-value parameter in all this list); 79

ď&#x192;ź finally, of the total 26 times my opponents started with suited hole cards and reached showdown, they flopped a backdoor flush draw 14/26 times = 53.8% (vs.

41.6% normally), a strict 4-cards flush draw 5/26 times = 19.2% (vs. 10.9%), and even a served flush 1/26 = 3.85% (vs. 0.84% normally).

An expansion of the analysis over the entire series of my pocket pair hands, by pre-filtering out methodologically inadmissible hands (i.e. equal pair duels) and isolating all showdown cases in which at least one opponent at the table had non-paired hole cards, leads to these results: ď&#x20AC;­ the frequency of flopping two pairs: 23/653 = 3.52%, meaning 1.61 times higher than the normal frequency of 2.18% (considering my hole cards as being known); ď&#x20AC;­ the frequency of flopping a trip = 14/653 = 2.14%, meaning 1.47 times higher than the identically calculated normal frequency of 1.46%.

These aggregated figures are lower than the ones derived solely on the basis of the QQ case study, but, it needs to be stressed, they represent the minimum, recorded, proven, frequencies. Since the lower the pocket pair, the more folds on my part, hence the less showdowns seen, it is only reasonable to assume that the higher figures calculated when isolating the KK and QQ hands are significantly closer to the real frequencies occurring on Pokerstars.

Finally, after again filtering out all equal pair situations (regardless whether in a heads up format or with more than two players reaching showdown), the recorded frequency of set vs. set duels on the flop67 reached 8/333 = 2.40%, which is 2.35 times higher than the normal frequency of 1.02% (calculated by considering all known hole cards).

The main, fundamental, recurrently reinforced thesis remains the same: the flops on Pokerstars do not seem random at all, quite the contrary: in way too many cases, and statistically very significantly so, the flops comes as if they were "called" by the players that remained at the table, meaning they contain cards (no matter how amazingly exceptional the associated probabilities of flopping such combinations) that directly assist at least one of those players, everything being meant to stimulate betting, increase the pot and grant higher rakes for the poker platform to collect. Let's take for instance, before closing the argument, one last look at the table, this one analyzing the very last 5 hands of Queens that I took to showdown.

1. 28.5 16:23 Q Q Q T J6A In an utterly dominated position pre-flop (against my hi-blocking pair of Queens), with an equity of a mere 16%, my opponent somehow manages to flop a combination that provides him with both a flush draw and a gut-shot straight draw, enough, of course, to keep him in the game (47% equity on the flop) and possibly stimulate him to bet and thus ensure more rake for the poker platform. Considering my known hole cards, the probability of my opponent landing in that exact double-draw position on the flop was a mere.... 2.197%!

2. 27.5 13:08 Q Q 3 4 8Q8 Nothing spectacular this time for my opponent, but in turn I flop directly a full house, the probability of this event occurring in that given type of situation being a spectacular 0.763%!

3. 27.5 12:44 Q Q K J 10 Q 3 A classic: Not only do I flop a set (11.17% probability per se), but, simultaneously, my opponent flops an open-ended straight draw, despite me blocking two of four of his needed 67

These are the eight cases of flopped set vs. set that I encountered: when holding pocket 5s - May 22 (at 11:06), May 19 (at 16:02), and May 15 (at 13:24); with pocket Kings - May 26 (at 12:27) and May 3 (at 13:15); with pocket 7s - May 25 (at 14:13); with pocket 9s - April 17 (at 17:18); with pocket 6s - May 22 (at 12:59).

80

Queens. The conditioned probability of such a flop considering the two pocket hands is another sensational... 2.13% 4. 27.5 11:59 Q Q A 5 7 2 5 This time I somehow I manage to avoid the nightmare of another over-card in the flop (an otherwise considerable probability of 61.63% per se), but my opponent also manages to flop one pair plus an additional backdoor flush draw and a straight draw!

5. 27.5 00:27 Q Q 7 9 3 8 9 Whereas for me, as in the above case, the probability of not flopping an over-card is considerable, the probability of my opponent simultaneously a.) hitting a top-pair and b.) no over-card, and c.) a backdoor straight draw of 3 successive cards (without trips or two pairs or better, nor with me making a set) is another sensational 2.08%!

And now, as done in the previous section, let's write down some of the flopassociated probabilities materialized in the QQ hands table: 2.2%, 0.76%, 2.13%, 2.18%, 1.48%, 2.54%, 4.25%, 2.2%, 2.37%, 1.46%, 1.58%, 0.18%, 0.14%, 0.76%, 1.85%, 0.14%, 1.39%, 1.27%, 0.57%, 1.87%, 1.46%, and 2.08%. And these figures are derived from solely the QQ hands analyzed - a mere total of 249 dealt, of which only 87 have reached showdown!

It seems at every flop you look, something sensational, meaning something almost unbelievably improbable, happens. It is like most of the flops are actually mathematical exceptions, of minuscule associated probabilities.

Moreover: whereas sometimes it seems that the pre-flop favorite to win the hand flops something truly exceptional, upon closer examination, however, one can notice that by far more often, the flop is meant to (at least) keep the initial underdog in the game (alongside the favorite), by offering him at least something, a combination, a draw, to hang on to.

Usually, the flop actually produces a complete reversal of the pre-flop equities. It is this twist that happens all too often on Pokerstars' flops that indicates anything but randomness, and a clear tendency of mainly helping the underdog in a general business-motivated strategy of "leveling the field". Here is a summarizing view of this kind of twists that occur on the flop, all structured from the initial pre-flop equities of the favorite.

Table. Favorite's initial, pre-flop, equity after the flop (when I played pocket Queens to showdown): Cases No. Conserved/increased Diminished, but still favorite Completely reversed Total

55 13 18 86

% 63.9 15.1 20.9 100.0

End result Wins Losses No. % No. % 46 83.6 9 16.4 7 53.8 6 46.1 2 11.1 16 88.9 55 63.9 31 36.1

(hand with the split pot against pocket QQ not included) (Favorite = the best starting hand, regardless if it was mine or the opponent's)

The findings are astonishing: in no less than 36.0% of the cases analyzed, the flop produces either a diminishing of the pre-flop favorite's equity, or a complete reversal of it in favor of the underdog! Amazingly, in more than every 5th of all the hands analyzed, the initial top dog suddenly wakes up on the flop in the position of the underdog!

Furthermore, 7 out of 10 hands in which the favorite saw his equity on flop either diminished or completely reversed eventually end up with the favorite losing. And even when the flop has at least conserved his/her initial equity, the favorite will still lose roughly every sixth such 81

hand! This is Pokerstars' usage of flops as a leveling instrument in all its splendor. And this is achieved by a deliberate and massive alteration of any "randomness" the platform's RNG might, theoretically, have applied when dealing the flop cards. Or, as an alternative explanatory perspective, the RNG algorithm has actually never been conceived and utilized as a genuinely random cards distributor, but has deliberately been engineered in such a manner as to level the field, to balance the odds.

Applying the same analytic method to the pocket Kings I took to showdown (93 68 clear cases ), the findings are equally suspicious. In 59.1% of the cases (55), the initial favorite kept or even increased his equity on the flop. In no less than a staggering 40.9% of the cases (38), the favorite's pre-flop either diminished (22.6% of the cases, with 1/4 of them eventually ending up in a loss), or was completely reversed (18.3%, all of them having ended with the initial favorite's loss!). Such a figure is simply astonishing, since, in regard to the favorite, we are talking here about either Aces or Kings, the two best hands in the Hold'em game, with initial heads-up power rankings (expected win rates) of 85.3%, respectively 82.4%69! Still, an impressive 40.9% of the flops seen somehow managed to at least diminish their starting equities, if not completely reverse them to the direct benefit of the underdog!

The legitimate question is: what possible flops could have been delivered by the poker platform's RNG in order to produce such a figure. The answer is only one: non-random flops.

And, as already explained before, if a Pokerstars' employee would try and attribute such an anomaly to all the KK hands that did not reach showdown (arguing that in all those hands, players folded their hand before showdown exactly because Aces or Kings held their position and materialized their starting equity), then such a speculation could again be easily refuted by the reductio ad absurdum method. Specifically, let's adopt lower 17% as the normal ratio of cases in which Aces/Kings are indeed mathematically expected to lose.

Well, if the 38 cases of Aces' or King's equity decrease or reversal on the flop represent the normal 17%, that means the favorite should have played no less than 100*38/17 = 223.5 hands in order for the overall mathematics to be correct. And this is again absurd, since the total number of hands actually played here is only 93 (KK taken to showdown) + 45 (folded before showdown) = 139. This leaves a single possible explanation: flops on Pokerstars do not "land" randomly, but are deliberately manipulated and selected so as to specifically help the underdog to the detriment of the top dog (while nevertheless giving the latter the illusion of a strong position), within a systematically applied and business-motivated strategy of "leveling the field".

III.3. A quadruple targeted analysis of Pokerstars' flops The present analysis departs from the above-finished case study, returning to the higher level, general, approach of the flops on Pokerstars, but specifically targets, over 51,989 cash games played at 6-players tables, four flop situations: • a set or better when starting with pocket pairs; • a flush or flush draw when stating with suited hole cards; • straight or draw when starting with middling connectors; • at least one pair when starting with non-paired hands. Thus segmented, the four items actually cover the overwhelming majority of all the possible types of situations one can find him- or her-self on the flop in a Texas Hold'em game.

Additionally, the leveling the field hypothesis shall be retested based specifically on the findings associated with the four investigated items.

68 69

I.e. leaving aside 2 hands in which I encountered another KK pocket pair, both having ended with a split pot.

I adopted the heads-up figures since in the 90+% of the cases, the flop has been reached by only two players.

82

III.3.1 Flopping a set or better when starting with pocket pairs Along my entire series of 51,989 cash games on the investigated poker platform, the flops corresponding to situations in which I was dealt pocket pairs are summarized in the table below, with the color coding being the usual in this paper, i.e. red meaning below the normal, expected, frequency, and green above it.

Table. Flopping a set or better (i.e. full house or quads) when holding a pocket pair:

My starting pocket pair Aces Kings Queens Jacks TTs 99s 88s 77s 66s 55s 44s 33s 22s total

Flops providing me: Total flops seen 131 139 154 153 161 184 190 169 164 171 165 174 177 2132

Pure sets 18 13 16 20 12 21 20 17 22 26 20 17 23 245

Full houses

Quads

1 1 2 3 1 1 2 1 1 0 1 3 0 17

0 2 1 0 0 1 1 1 0 0 1 1 1 9

Sets / better (cumulated) 19 16 19 23 13 23 23 19 23 26 22 21 24 271

Recorded frequency (%)

11.49

0.80

0.422

12.71

Normal frequency (%):

11.17

0.73

0.245

11.76

Calculated as percentages of the normal, expected, values, the frequencies recorded on the platform thus reach: - pure sets: 102.9%; - full houses: 109.6% - quads: 172.2% - sets or better: 108.1%.

And in a graphical representation, the findings look like this:

83

As a conclusion: not only are three of four recorded frequencies significantly above the normal ones, but all of them exceed the normal values expected to be found in a truly random environment. Furthermore, only one of the 13 pairs (TT, specifically) has flopped a set or better to a significantly lower degree than expected, the other two red-marked cases (KK and 77) are placed within a reasonable 5% variation margin, thus leaving all of the remaining 10/13 pocket pairs having flopped a set or better of 10/13 pairs above the normal frequency line70! These findings can categorically not be attributed to any supposed randomness.

Instead, it is, as in the so many cases of other items analyzed in this research, the visible effect of a deliberate strategy of the poker platform investigated to grant players so-called â&#x20AC;?action flopsâ&#x20AC;? meant to stimulate them to bet and thus ensure the rake that the platform ultimately makes a living off.

III.3.2 Flopping a flush (draw) when starting with suited hole cards When holding suited hole cards, regardless of the number of players at the table, the associated probabilities on the flop are - flopping a flush : 0.84%; - flopping a flush draw: 10.84% - flopping a flush or flush draw: 11.765% - a flop of (entirely) another suit: 4.38%. This latter category covers cases such as having two hearts hole cards, with the flop being entirely mono-color, but of another suit than yours. An example is the below screenshot, capturing my QJs hands that I played in tournaments, with the checked instances marked by the red arrows at the left:

70

It may be also relevant that, in the case analyzed here, there is absolutely no mathematical relation between the pocket pairs' strength and the percentage of flops granting a set or better (Rsq = ascending 0.01), meaning that, at least in relation to this item, there are no recognizable traces of the "leveling the field" mechanism.

84

In my case, of all the suited hands that I have been dealt over the combined 55,320 hands played in cash games and tournaments, 6,351 have reached the flop71 (5,745 in cash games, 606 in tournaments). In relation to the four above-mentioned types of situation and associated probabilities, the recorded results are the following: Total methodologically Recorded frequency admissible flops (6,351) No. % 52 0.82 Flushes 729 11.48 Flush draws 781 12.30 Flush or flush draw 295 4.64 Another monocolor

Expected frequency (%) 0.84 10.94 11.76 4.38

Recorded as % of expected 97.6 104.9 104.6 105.9

Whereas in the case of flopped flushes, given the extremely low starting probability of reference, no safe conclusions can be made, in the cases of flopped flush draws and flops of another color, the figures capture a considerable discrepancy between recorded and expected frequencies, with the latter case's occurrences clearly above the statistical significance threshold, and the former almost overlapping it.

As another potentially relevant observation, it should be added that I have not found any significant difference between on the one hand cash games and on the other hand tournaments in terms of the frequencies recorded for each category: flopped flushes - 0.82 (tourneys) vs. 0.78% (cash); flush draws - 11.41 vs. 11.51%; another monocolor flop - 4.61 vs. 4.64%. 72 Conclusively, in relation to the item investigated here, the flops that have been verified display another deviation from the frequency to be expected under truly random circumstances. This deviation also works "in favor" of the player and is extremely close to being statistically significant.

71

The figures combines both the flops when I was effectively still in the game and the flops from hands that I did not play (having folded pre-flop), but were continued by at least two opponents. More on this distinction in the next subchapter addressing straights and straight draws.

72 Nor have I found any significant difference when measuring the frequency of flopped sets or better (when starting with a pocket pair), so that in the following, the cash vs. tourney comparative framework shall be abandoned.

85

III.3.3 Flopping a straight (draw) when starting with middling connectors Two methodological specifications are necessary at this point, before engaging into the analysis per se. Firstly, I need to specify that, as a methodological inconvenience, the probabilities of flopping a straight or a straight draw substantially differ depending on the relation between the two hole cards - connectors (e.g. 98), 1- (e.g. KJ), 2- (e.g. 96), or 3gappers (e.g. A5). Given this methodological constraint, I chose to select as a sample the series of middling connectors, meaning connectors twice triple-open: JT, T9, 98, 87, 76, 65, and 54. Such connectors can develop a straight in both value directions, both upwards (ex.

QJT when holding 98) and downwards (e.g. 765 for the same exemplified case), hence my label "twice triple open"). Conveniently enough, the sample of flops associated to hands in which I started with these connectors as hole cards is big enough - there have been 1,973 flops that I have played effectively or that have been played by my opponents at the table. For this category of hole cards, in mathematical terms, the probabilities on flop, are the following:  flop a pure open-ended straight draw (i.e. does not include made straights or "double belly" / "double inside" straight draws73): 9.80%;  flop a pure "gut-shot" (or "inside") straight draw: 18.94%;  flop a served, made, straight: 1.31%.

The second specification refers to a methodological distinction that I think is critical for this paper in relation to the "second shuffle" hypothesis, namely the one between hands A.) "played" vs. B.) "not played". Both have actually been played by me, in the sense that in all of these hands, I was at some table and have been dealt some hole cards. But, in some of the cases (meaning category A.)) I did indeed play the hand, meaning I limped / checked / called / raised and remained at the table along the board's unfolding for at least another street, whereas in other cases (meaning B.)) I folded my hole cards.

The latter category of situations admits two subcategories: B.1.) hands that have not been continued by other opponents and thus ended on the very same street that I folded, respectively B.2.) hands that, after my fold, have been continued by at least two opponents to at least the next street, which means that both the Pokerstars and my HM2 software have recorded and stored the continuation of the hand.

This mentioned software function is crucial to this analysis, since it allows us to somewhat counterfactually verify what would have happened had I remained at the table, which is impossible in cases of the subcategory B.1). For instance, suppose I was dealt T8 of hearts, and there were three players, me included, who had not folded pre-flop and were still at the table when the flop came: A of hearts, K of spades and 5 of hearts. Let's suppose further that, even though I had a flush draw of hearts, one opponent bet so aggressively that I didn't want to take the risk and decided to fold, but the third player at the table called, and the two of them went farther all the way to showdown. Well, if at least one of the turn or river cards was a heart - which we can check both on Pokerstars and on HM2, it means I would have made a flush. Or, as another example, let's say I folded 97 off-suite on the pre-flop street, but two opponents continued the game, so that a flop such as 8K6 would have provided me an openended straight draw.

The above exemplified dichotomy is no futile counterfactual reasoning, but a potentially crucial instrument to establish whether, in terms of the "second shuffle" hypothesis, Pokerstars' RNG  deals the board cards in a truly random manner, regardless both who the players at the table are what hole cards they have, or

73

E.g. hole cards JT on a flop AQ8: both a K and a 9 on the turn or river would make a straight, hence the terms "double belly gut-shot" or "double inside" draw.

86

ď&#x20AC;­ on the contrary, it deals them specifically and in direct relation to the players' hole cards (meaning there could be a second, significantly altered, shuffle of the deck after the pre-flop phase has ended).

For instance, in a truly random environment, when starting at the table with, say, connectors as hole cards, there should be no significant difference in terms of flopping an open-ended straight draw between those cases in which I did indeed play the hand and stayed on the flop, and those cases in which I folded pre-flop. Along the two types of situations, the flop should contain an equal 9.80% of times an open-ended straight draw? Why else should there be a difference on the Pokerstars platform, depending on whether I remained at the table or folded, right? Well, surprisingly enough, there actually is a difference! The table below lists, separately for played and not played hands in which I had been dealt connectors, all the played/seen flops in terms of straights draws (open-ended or gut-shot) and straights provided in relation to my hole cards: Table: flopping a straight or straight draw when holding connectors:

Played: Not played: Total:

Total flops (No) 648 1325 1973

a pure OESD No. % 65 10.03 105 7.92 170 8.62

Normal, expected %:

9.80

Flops providing: a pure GSSD No. % 123 18.98 233 17.58 356 18.04 18.94

a straight No. % 16 2.47 21 1.58 33 1.67 1.31

green: significantly above normal frequency; red: significantly below; blue: normal, expected

The recorded discrepancies between played and not played flops, especially along open-ended straight draws (OESDs) and served straights, are significant, to not say shocking: when I stayed at the table, Pokerstars' RNG provided me with an open-ended straight draw in 10.03% of the cases, which is slightly, but not significantly, above the normal, expected, frequency. However, once I fold, the recorded frequency drops dramatically, to 7.92%, which is 80.8% of the normal, expected, frequency, respectively 79% of the frequency recorded when playing the flop! Similarly, within all the cases when I remained at the table at least until the flop, I flopped a made straight in 2.47% of them, which is no less than 1.9 times more often than normal, whereas when folding pre-flop, the flops would have provided me with a straight also more frequently than normal, but to a significantly lower degree (120.6% of the normal frequency, respectively 64% of the frequency associated with played flops). The discrepancies highlighted by figures such as 80.8% or 120.6% of the normal frequencies, after almost two thousand cases counted, are, to put it non-academically, miles above any statistical significance threshold a human being could choose to employ and, no matter how one twists possible interpretations, they admit only two alternative explanations: a) either I am one of the best poker players ever to have walked on Earth, since I apparently have an uncanny and unparalleled ability to know whether the flop will be useful or not for me, b) or, and more plausibly so, RNG's algorithm distributes the flop cards depending directly to the hole cards of the players that remained in the game, to, at least apparently, their direct benefit, with the percentage of useful (or shall we say "enticing"?!) flops significantly more frequent than normal.

To put joking aside, the findings actually only make sense, as they reinforce the recurrent conclusion transparent throughout the paper, namely that the flops are being intentionally dealt in a seriously non-random manner, providing the players (often simultaneously so) with attractive, enticing, draws meant to not only keep them in the game, but actively stimulate them to bet and raise, thus increasing the pot size and augmenting the 87

collectible rake for the poker platform. In this contextual light, there should thus be no surprise in regard to these last findings, since they simply reconfirm what has already been repeatedly discovered in the present analysis, starting with the increased number of sets, full houses, and quads flopped when holding a pocket pair, and all the way to the spectacular frequencies of flopping a royal flush draw when starting with any suited hand in the A to T interval.

In the light of this new revelation, I think it has become critical to revisit the previously analyzed two items by using the methodological framework employed here. In this regard, the results of recounting the flops that provided me with a set or better when starting with a pocket pair in the cash games are rather inconclusive: of the total 2,132 flops, 2,001 have actually been played by me, with a recorded number of flopped sets or better of 253, meaning a frequency of 12.64%. Among the 131 flops actually only seen, and not played (meaning, I repeat, I folded pre-flop), 18 would have provided me with a set or better, meaning 13.74%. True, there is a visible difference among the two percentages, but the size of the second subsample74, in trying to measure a frequency whose normal value is anyhow a mere 11.76% (i.e. sets or better), remains unfortunately too small to allow any reliable conclusions.

As far as flushes and flush draws are concerned, the analytical reframing does capture some notable and interesting differences along the played. vs. not played delineation, but they seem rather contradictory: Type of hand

situations Flop directly a flush Flop a flush draw Another monocolor flop

Played (N1 = 2804)

Not played (N2 = 2941)

No.

No.

20 324 132

% 0.71 11.55 4.71

25 334 133

% 0.85 11.46 4.52

Normal, expected, % 0.84 10.94 4.38

(51,989 cash games analyzed, N = 5,745 flops)75

Thus, of the three parameters isolated, when staying in the game, two have worked to my detriment (meaning I flopped a flush less often than normal, whereas another monocolor flops, also working to my detriment, have been more frequent than normal), while the frequency of flopped flush draws has been higher than normal in both types of situations, with a slightly better percentage recorded in the cases where I did effectively play the hand.

As such, the results remain inconclusive.

III.3.4 Flopping at least one pair when holding un-paired hole cards Again selecting all my 51,989 cash games, there have been a total 19,921 flops that I have played or seen in situations where I started with non-paired hole cards (suited and off-suite combined). The results of the overall and of the binary counting are captured in the table below. On the overall, the recorded frequencies, regardless whether played or not played hands, are not significantly different from the expected ones: for the cumulative category "at least one pair", the recorded frequency is 99.6% of the expected one. Along the four subcategories, I have flopped strictly one pair to a frequency of 99.34% of the expected one, two pairs to a relative ratio of 102.0%, and trips, the only relatively significant case, to a frequency of 104.7% of the normal, expected one. As far as full houses and quads are 74

I.e. the 131 hands not played, where I would have flopped a set or better.

The subsample of not played hands in tournaments is unfortunately too small to allow any reliable conclusions along the played vs. not played delineation, so all tournament hands have been discarded from this analysis.

75

88

concerned, although both have been clearly rarer than expected, one might invoke the extremely small probability to begin with.

Flopped / would have flopped At least one pair Strictly one pair Two pairs* Trips Full houses Quads

Flops played (N1 = 5843) No. 1956 1743 123 89 3 0

% 33.48 29.83 2.11 1.52 0.05 0.00

Flops not played (N2 = 14078) No. % 4476 31.79 3988 28.33 287 2.04 191 1.36 10 0.07 0 0.00

Total flops (N = 19921) No. 6432 5731 410 280 13 0

% 32.29 28.77 2.06 1.41 0.07 0.00

Normal, expected, % 32.43 28.96 2.02 1.347 0.092 0.010

* using both hole cards Italics: the higher values between the first two main columns (flops played vs. not played)

Thus, along the six items verified, in 4 cases the values are higher (and all so in my favor) when staying in the game than when folding pre-flop, in only one case are things the other way around, whereas in one case the percentages are equal. So, reconfirming the conclusion drawn when verifying the flopped straight and straight draws, it again appears that a player, in this case, me, is advantaged if staying in the game at least on the flop, instead of folding pre-flop, thus partly reconfirming the second shuffle hypotheses.

Such discrepancies become even more striking when isolating specific lines of hole cards, such as for instance hole cards that contained an Ace:

In this subcategory of cases, when effectively playing the hand, I flopped at least one pair in 33.15% of the cases, but among the hands that I folded pre-flop, the counterfactual 89

corresponding figure is only 29.82%! That's a 3.33% absolute difference, which, as a deviation, represents no less than 10.27% of the expected general frequency of 32.43%!

Furthermore, if focusing on the first four items analyzed, with associated subsample of an adequate volume, the differences between played and not played hands become conspicuous when calculating the frequency recorded in played hands as a percentage of the one corresponding to hands that I didn't effectively play, with all four figures higher than 100%, and three of them even higher than 105%: - flopping at least one pair: 33.48/31.79 = 105.3%; - flopping exactly one pair: 29.83/28.33 = 105.3%; - flopping two pairs: 2.11/2.04 = 103.4%; - flopping trips: 1.52/1.36 = 111.8%.

Such discrepancies are simply too big to be attributable to randomness for at least three reasons: firstly, their effective value as a deviation from the expected results is higher than 5% in three of the four cases; secondly, all four of them manifest themselves in the same direction (i.e. rewarding me if I remain at the table); thirdly, in context, they are confirmed by the findings achieved when analyzing the variable of flopping a straight or straight draw.

III.3.5 A retesting of the leveling the field hypothesis The present investigation retests the "leveling the field" hypothesis based on the previous four sets of measurements regarding the flops on the Pokerstars platform, only this time in relation not to opponents and in terms of equities, but in relation to the very value of the hole cards. Specifically, it tests whether the frequency of flopping sets or better, a flush or a flush draw, a straight or a straight draw, respectively at least one pair, is in any way related to the value of the hole cards as indicated by the power ranking (the % win plus split rate) in 6-players games uniformly used throughout this paper. One could, theoretically, level the field, or balance the odds, inclusively by hindering the stronger hole cards' chance on the flop.

Logically speaking and methodologically corresponding to a "null hypothesis", if the platform's RNG would deal cards in a random manner, there should be no relation whatsoever between how strong a certain pocket hand is and it's recorded percentage of flopping a specifically targeted combination.

And indeed, confirming the null hypothesis, in three of the four cases reanalyzed along the 51,989 cash games played, there is absolutely no mathematical relation whatsoever: the correlation between the power rank of a pair and it's percentage of cases flopping a set or better is a negative 0.047, the correlation between a suited pocket hand's strength and it's frequency of flopping a flush draw or a flush is 0.01, whereas, for middling connectors, the relationship between their power rank and the percentage of cases having flopped a gut-shot straight draw76 is a negative 0.07. Thus, it is more than safe to concluded that, at least in relation to these three variables, the relation between a hand's strength and it's frequency of flopping a targeted combination is one of absolute mathematical independence.

The findings, however, change dramatically when investigating the fourth variable, namely the frequency of flopping at least one pair when starting with two non-paired cards. True, on the overall of the 156 non-paired hands series, there is no discernible correlation, with the regression line in the graph below satisfactorily horizontal, and a convincing Rsq value of only 0.018, which would again imply mathematical independence between the two variables.

76

Truth be told, there has been a correlation of -0.39 between the middling connectors' strength and the frequency of flopping an open-ended straight draw, one which I am however dismissing based on the small size of the flopped OESDs (n=170).

90

Graph: the relation between a non-paired hand's power rank and its recorded % of flopping at least one pair:

Hand's percentage of flopping at least one pair

50.00 45.00 40.00

35.00

y = 0.1353x + 29.789 R² = 0.0183

30.00

25.00 20.00

15.00 10.00 5.00 0.00 0

5

10

15

20

25

30

35

Hands' power rank (i.e. their % win + split rate) in 6-players games

However, after doing the above calculations and associated graphical representation, I remembered that, when counting the flop-indicated percentages for all, I started the listing in the standard decreasing order of how the hands are arranged in the HM2 database (meaning first AKs, then AKo, then AQs, then AQo, all the way downwards to 32o), which gave me the opportunity to notice how the cumulated percentages evolved in an extremely curious fashion, meaning a strikingly linear unfolding.

91

This recalling, combined with my metaphor of mixing boiling with frozen water, first determined me to regroup the 156 hands in 13 clusters: hands that contained an Ace, hands that contained a K, etc., thus delineating 13 "lines of cards". Then, in need of a instrument to hierarchically rank these lines in terms of strength, I came up with the most commonplace methodological artifice possible: I assigned values to the non-numerical cards in the direct succession to the latter group, meaning J (following 10) was to have a value of 11, Q - 12, K - 13, and A - 14. Consecutively, for all the 13 card lines in discussion I calculated their aggregated thus assigned values. For instance, the A line's aggregate value is: 14+2 (A+2)

+ 14+3 + 14+3 + .............. + 14+12 (A+3)

(A+4)

..................

(A+Q)

+

14+13 = 258, etc.

(A+K)

Subsequently rearranging the 13 lines of cards and recalculating correspondingly the percentages of having flopped at least one pair, I came to the spectacular discovery graphically represented below:

So, suddenly, and spectacularly so, one can see a clear-cut, irrefutable relation between a certain group of card's and their recorded % of having flopped at least one pair over a combined series of 19,921 flops played or seen! Let's get this clear: in order to confirm null hypothesis, meaning in order for these dealings of cards to be considered as random, the correlation coefficient should be significantly smaller, whereas, on the graph above, the regression line should basically be horizontal.

Thus, in the factual reality measured on the Pokerstars platform, the stronger a cards line is, the more frequent it flops at least one pair! This is anything, but randomness!

Now, admittedly, these findings are as surprising as possible, not because they (again) depict some non-randomness evolutions on the investigated platform (thus reconfirming the general thesis regarding the non-randomness of the flops). They are surprising because they indeed reject the null hypothesis, but in a direction contrary to what I had been expecting in terms of the leveling the field hypothesis, at least in regard to its second dimension (meaning in self92

reference to one's hole cards, not to an opponent). Specifically, strong cards are not "balanced", but, on the contrary, are "assisted" further on the flop in providing them with the targeted useful combination - in this case at least one pair!

Going back to the story, after this first idea that I tested, I went back to the data series, this time in order to target specifically how the aggregated cumulated % of having flopped at least one pair evolved step-by-step in terms of adding hands counted, in a decreasing order of the hole card's power rank, so in a more scientific and standardized approach.

Let's try to make the above method description a bit easier to understand: I started with the AKs hands, the strongest non-paired hand in a 6-players game, and counted how many flops provided me with at least one pair, when starting at the table with these two hole cards. 28 of 76 flops played / seen had / would have provided me with at least one pair, meaning a relative frequency of 36.84%. Then, I added the second best hand in a 6-players game - AQs. In this case, it was 36 of the total 101 flops that had given me at least one pair, meaning an individual relative frequency of 35.64%. After these two hands, the cumulated frequency thus came at 100*(28+36)/(76+101) = 36.16%. After counting the third best hand, meaning KQs, the cumulated frequency had thus become 100*(28+36+45)/(76+101+109) = 38.11%. And so on, I continued with this algorithm all the way down to the weakest hand in a 6-players game, that is 72-o, while simultaneously verifying how the cumulated correlation between hand strength and % of useful flops evolved:

Now let's take a look at the quite shocking results in a graphical manner:

93

Graph: Power rank-descending arranged cumulated frequency of flopping at least one pair depending on the hands' power ranking:

94

If the above graph is not explanatory enough for readers with less knowledge of statistics, let me try making it a bit more simple: of a total 19,921 flops seen or played, it is only after the best 10,142nd, meaning after more than (the best) half of all flops, that the cumulated frequency finally begins to border the expected, normal, frequency of 32.43% associated with flopping at least one pair when starting with non-paired cards. This is anything, but normal!

Normally, in regard to the blue line on the graph, there should be a few hands along which the line would be expected to oscillate, maybe even dramatically, across the 32.42 green threshold, but afterwards, once the sample of hands counted has become representative enough, it should be relatively flat. Correspondingly, after some initial oscillations, the regression line should become steadily horizontal, with Rsq tending towards zero. It is not, and its spectacular value clearly indicates a positive relation between the strength of a hand and its chance to flop at least one pair. It's similar to, say, polling: obviously you do not need to interview half the population of state in order to produce reliable measurements of opinions! If that were the case, nobody on this planet would do polls anymore! Suppose, for instance, it is a known, measured, fact, that 1/4, so 25%, of the US population has green eyes. As a non-believer, let's further suppose you are not convinced that things are indeed that way, so one day you begin checking people, starting in an organized way, with the people on your street, then continuing with people on a 1 mile radius, then 2 miles, etc. Well, equally obvious, you should not have already counted half of the state's population and still not have reached the given 25% of green-eyed people in your sample!

This is, of course, unless someone has tampered with the natural geographical situation of the population (presumed as random in regard to eyes color), such as for instance moving in a concentrated manner a massive number of non-green eyed people around you, on a hundreds of miles radius!; or, returning to Pokerstars, unless someone has tampered with the odds in a well-organized, systematic manner, by specifically "aiding" strong hands to flop at least one pair more frequently!

Finally, in terms of how the cumulated correlation evolved along the 156 hands series, the findings are equally conclusive:

Let me emphasize this: if the dealing of the flops would indeed be random, then after the natural initial oscillation (also existent in the measured reality and visible in the graph above), once the sample of hands becomes big enough, the blue line above should 95

indeed become and then remain horizontal, in this manner indicating no relation between hand strength and % of having flopped at least one pair, but by no means should it be situated that high above the 0-line! A linear correlation of +0.882 after almost 20k flops counted is simply dumbfounding, and it indicates a clear, systematic, and premeditated non-random way of dealing the flops.

III.4 Summary of findings Structured in three consecutive steps, this chapter's analysis has brought to light some extremely intriguing findings, which in their overwhelming majority reinforce themselves, converging towards the same conclusion: the flops on the Pokerstars platform are anything, but random. To probably an even higher degree than the distribution of hole cards pre-flop, it is the flop among the four streets of the game where, proportionally: on the one hand, the most improbable events happen, which only deepen the already improbable situations recorded pre-flop; on the other hand, but inter-related, the leveling-the-field mechanisms manifest themselves most pregnant, with significantly more often than normal cases where pre-flop equities are balanced or even completely reversed in favor of the initial underdog at the table. Not coincidentally, it is exactly the flop whose associated mathematics are, at least for the average player, the most complex and inaccessible among the four streets, thus making it the perfect spot for statistically grotesque improbabilities to happen.

In summarizing the findings of the first analytical step, the one addressing flops on the overall, it should be recalled that, of the total 51,989 hands that I have been dealt in 6players cash games on the platform, 22,053 have made it to the flop one way or another77 (with 2,131 flops seen or played when I held pocket pairs, respectively 19,921 in situations where I had been dealt non-paired hole cards). It turns out that these 22k flops were enough for me to have witnessed, as exemplified, some sensationally improbable situations such as: • set vs. set. vs. two pairs - a 0.039% probability from the given starting pre-flop situation, i.e. not taking into account the probability of being dealt that hole cards simultaneously; • a triple pair AA vs. KK vs. JJ situation where I flop the Kings quads - a 0.29% probability calculated solely in relation to the pre-flop given situation,!78; • two pairs vs. two pairs (one of them being common) vs. another initially dealt pocket pair - a 0.118% probability; • set vs. two pairs - 0.104% probability; pocket pair vs. twice the same trip - 0.237%; set vs. twice the same trip - 0.57%; flopping a set four times in a row for the same starting pocket pair - a 0.0019% probability, meaning such a chain of events happens once in every 50,000 pocket pair situations that a player has been through, respectively once in every 1.12 million hands played by him/her in total; • flopping two pairs (using both hole cards) 4 out of 5 consecutive times when starting with unpaired cards - a probability of 0.0000163%, meaning an event expected to happen once in every 61.35 million unpaired hole cards played at a poker table; flopping a royal flush draw - with a recorded frequency 2.5 times higher than the normal one to be expected under genuinely random condition; flopping directly the royal flush, an event which happens once in every roughly 650,000 hands that a player is dealt in a Hold'em game!

These general findings only reconfirm the particular conclusions of the QQ case study in terms of the "second shuffle" hypothesis, which highlighted recorded flops of minuscule probabilities associated: 2.2%, 0.76%, 2.13%, 2.18%, 1.48%, 2.54%, 4.25%, 2.2%, 2.37%, 1.46%, 1.58%, 0.18%, 0.14%, 0.76%, 1.85%, 0.14%, 1.39%, 1.27%, 0.57%, 1.87%, 77

I.e. either with me effectively remaining in the game, or with me folding, but at least two opponents continuing the hands at least to the next street, so that the flop cards became visible and also stored on and by both the Pokerstars platform and my installed HM2 software 78 But, considering all elements involved (KK running into the single higher pair, plus another pocket pair at the table, and the frequency of being dealt pairs), such a chain of events is actually expected to happen once in every, roughly, 85 million hands played. See the previous chapter for details.

96

1.46%, or 2.08%. As said, Pokerstars shows no restraint or hesitation in offering you the most sensational flops imaginable, no matter how nanoscopic their probability, as long as the result is you betting and re-raising and helping them collect a bigger rake.

The problem, as already emphasized in the analysis, is not the fact that such things happen, as they are extremely improbable, but still possible. The problem is that essentially all of them have, aside from their minuscule associated probabilities, one single thing in common: all too often, the flops are related exactly and directly to the hole cards of those players that have remained at the table, which strongly suggests a "second shuffle" after the pre-flop. They deliver a impossible-to-be-refused offer, keeping at least two the players in the game, by providing them with the most attractive combinations possible, which is only meant to stimulate them to bet and (re-) raise, thus increasing the pot and subsequently increasing the proportional rake that is to be collected by the poker platform. And it is this second commonality of examples such as the ones selected that explains why events that would require a player to have necessarily played hundreds of thousands, or even tens of millions of hands, when it is clearly not the case, can by no means be attributed to "randomness", "accidents", "variance", nor a software "glitch", etc., but instead clearly hint at a deliberate, systematic, and premeditatedly applied and business-motivated strategy of the poker platform's owners.

By analyzing hand-for-hand all my 249 pocket Queens that I have been dealt, out of which 154 made it at least to the flop, the second step of the analysis, conceived as a case study, confirmed the conclusion regarding the non-randomness of the flops, and has itself been reconfirmed by the re-contextualization brought by an extension of the same analytical framework to cover all the KK hands that I have been dealt. Specifically, this case study:  pre-confirmed the "leveling the field" hypothesis: the initial underdog is systematically assisted by a deliberately non-random manner in which the cards are dealt.

The initial, pre-flop underdog wins on the average 36% of encounters, a ratio proven as significantly higher than the normal one even when a.) judging by his weighted average pre-flop equity (24.1%, meaning the underdog wins 1.3 times more frequent than supposed to!), or b.) segmenting different situations and corresponding equities types of situations: in pure, heads-up, pair vs. pair encounters, the lower pair wins 25.0% of the duels (instead of the normal 18.5%); in pair vs. one over-card encounters, the underdog wins 37.5% of the duels (instead of the normal 31%), etc.;  pre-confirmed the "every tenth hand is manipulated" corollary: 11.63% of the Queens, respectively 10.1% of the Kings hands that reached showdown had an outcome "engineered" contrary to anything that would be mathematically correct and as such to be expected to happen;  showed that the leveling of the field is achieved by stridently non-random flops. Too many flops directly relate to exactly the hole cards of those players who chose to remain at the table, not folding pre-flop. Among such examples selected as analyzed items:  the frequency of flopping a trip when starting with two un-paired hole cards is 3.7 times higher than normal (5.55% recorded vs. 1.5% expected), and, starting from the same situation, of two pairs (using both hole cards) is 2.8 times higher than normal (5.55% vs. 2.18%);  the frequency of flopping one of the two winning over-cards (A or K, in relation to pocket Queens) is 2 times higher than normal (36.67% vs. 17.96%);  the frequency of flopping a set+ is 5% higher than normal (12.33% vs. 11.76%)79;  the frequency of flopping a full house (starting with pocket pair) is 1.8 times higher than normal (1.30% vs. 0.73%); 79

As all the other within the above list, this figure is also derived in relation to my pocket pair hands. In the larger context of the 328 showdowns reached by me with a pocket pair against another opposition pair, the opponents flopped a set or better 45 times, which is a considerable 13.72%.

97

 the frequency of flopping quads (starting from pocket pair) is 2.5 times higher than normal (0.65% vs. 0.245%)80;  Even by extrapolation, these significant anomalies cannot be attributed to all those hands that were folded before reaching showdown. Even if practicing logical contortionism and adopting logically absurd assumptions, the recorded frequencies still remain significantly higher than normal, sometimes necessarily requiring that I should have played thousands of pocket QQ hands!  For the entire series of hands where I have been dealt pocket pairs and that reached showdown, the recorded frequencies of opponents flopping two pairs or a trip have been 1.61, respectively 1.47 times higher than normal. The frequency of set vs. set situations on the flop has similarly been 2.35 times higher than normal.  As a directly subordinated instrument meant to achieve the "field-leveling", it is not, or rather not so much, the turns or the rivers (as popularly believed), but the flops which produce spectacular and statistically abnormal turnings of the tide and reversals of equities to the direct benefit of the underdog. In the case of pocket Queens, regardless who was the favorite to win, more than every third flop either diminishes the initial underdog's equity, or completely reverses it, whereas in the case of pocket Kings, the figure reached an astonishing (and absolutely unexplainable) 40.9%! It is in this manner that the favorite ends up winning significantly less than mathematically expected to on Pokerstars, with the opposite result for the underdog.

Finally, departing from the isolated case of pocket Queens and returning to the general analysis of the flops, the third step of the chapter separately targeted four flop-related recorded frequencies: a set or better when starting with pocket pairs; a flush or flush draw when stating with suited hole cards; straight or draw when starting with middling connectors; at least one pair when starting with non-paired hands. A correspondingly divided cluster of 13 methodologically reliable items investigated81 captured the following integrated image:

Hole cards

Pocket pairs (N1 = 2,132 flops)

Suited (N2 = 6,351 flops) Middling connectors (N3 = 1,973 flops) Non-paired (N4 = 19,921 flops)

Flopping: (pure) sets full houses quads sets of better (cumulated) flush (pure) flush draw another monocolor flush or flush draw (cumulated) open-ended straight draw gut-shot straight draw straight (strictly) one pair two pairs trips ≥ one pair (cumulated)

80

Recorded frequency as % of expected frequency 102.9 109.6 172.2 108.1 97.6 104.9 105.9 104.6 88.0 95.3 127.5 99.3 102.0 104.7 99.6

As shown, basically all items identically investigated for the sample of KK hands I played converge to the same conclusions as in the case of Queens.

81 Meaning that, considering their extremely small associated probabilities, flopped full houses and quads (when starting with non-paired hole cards) have been left out of the analysis.

98

Thus, in 61.5% of the cases (8 out of 13), the flop came directly "to my help", meaning the recorded frequency of the targeted combinations on the flop has been higher than the expected one, and in half of them by more than 5% so. Oppositely, in only 23.1% of the cases (3 out of 13) has the recorded frequency been lower than the expected one, and in only them by more than a negative 5%. In two cases the difference between the two frequencies has been less than 1%. Such a discrepancy can hardly be attributable to randomness. As a conclusion reinforcing the findings of the first two steps of the analysis, Pokerstars' flops tend to be directly related to the hole cards of the players at the table, apparently to their favor.

Anyhow, the dark red - marked case anomaly in the table made me reanalyze the whole situation by employing a methodological artifice: a delineation between hands that I did effectively play (meaning I was still at the table, in the game, when the flop came) and hands that I had folded pre-flop, but have been continued by at least two opponents at the table at least till the flop, which made the three cards visible and stored both on the platform and my data-processing software. When re-isolating the four items across the 51,989 cash games series and filtering out the subsamples not big enough to be admissible82, the results obtained display a general concluding image as clear as possible:

Flopped target combination:

Open-ended straight draw Gut-shot straight draw Straight Flush or flush draw Strictly one pair Two pairs Trips

Conversion % % recorded recorded in hands in hands not played played 10.03 7.92 18.98 17.58 2.47 1.58 12.27 12.31 29.83 28.33 2.11 2.04 1.52 1.36

Recorded % in played hands calculated as % of the recorded % in hands not played 126.6 108.0 156.3 99.67 105.3 103.4 111.8

Of the seven thus re-segmented items, in no less than six of them the variable of flopping the targeted combination records clearly and significantly higher values when having effectively participated in the game, whereas in the seventh case the differences are almost inexistent (0.33%). The finding is a tremendous revelation, re-confirming the "second shuffle" hypothesis: if cards on the platform would indeed be dealt in a random manner, there should make absolutely no difference whether a player remains or not at the table; the frequency of flopping the targeted combination should be the same, or at least the differences between the two value series should not be significant! Let's try an analogy: imagine a friend of yours repeatedly flipping a coin; whenever you are next to him in the room, the results are 60% heads vs. 40% tails; however, once you leave the room (and say you left it seven times, as in the table above), it's suddenly 40% heads vs. 60% tails. This is anything, but an expression of randomness!

Hence, on Pokerstars the recorded differences are not only significant, but, additionally, and thus casting a huge shadow of a doubt over all claims of randomness, they all manifest themselves in the same direction: when staying at the table, one is directly "rewarded", with the flops landing in a certain manner, directly related to the hole cards, which is not the case when folding pre-flop.

82

I.e. the sets or better flopped within tournament games.

99

Finally, as the last finding of this chapter, a retesting of the "leveling the field" hypothesis, in its second dimension, by investigating any possible relation between on the one hand hole cards' strength (operationalized by their win plus split rate in 6-players games) and on the other hand their recorded frequency of flopping the targeted combination led to a bizarre conclusion. Thus, for the three of the fourth items checked (sets or better, flushes or flush draws, respectively straight or straight draws), there is clearly no relation, the corresponding three correlation coefficients reaching, convincingly, absolute values of 0.05, 0.01, and 0.07. However, in the fourth case, meaning flopping at least one pair when starting with non-paired hole cards, the relation is as clear as possible, but, surprisingly so, not in the sense of the hypothesis investigated. On the contrary, as recorded on the platform, the stronger a pocket non-paired combination is, the higher its chance to flop at least one pair!

This conclusion, which, admittedly, remains unexplainable for the moment, is underlined not only by a re-segmenting into 13 card lines (with Rsq being 0.48), but also by the measured cumulative correlation coefficient (+0.882) when operating with the entire 156 hole cards series.

As an overarching conclusion, it can be stated that the roughly 22k flops analyzed across my games on the Pokerstars platform display multiple, intense and systematic elements of non-randomness or even, judging by both their direction of manifestation and their end, of what might be called "anti-randomness". As a general rule, consecutive to the significantly non-random distribution of hole cards, the flops dealt are, considerably more often than normal, directly related to exactly the hole cards of the players that remained at the table after the pre-flop street. At first glance "helping" both the two (or more) players that remained at the table, but the underdog significantly more so, the flops on Pokerstars will bring along, considerably often, combinations of cards of minuscule, infinitesimal probabilities, some of them happening once in tens, if not hundreds, of millions of hands played, as long as they achieve one business-motivated objective: players enticed by these flops betting, raising and re-raising, which translates into bigger pots, which in turn results in more and bigger rakes collected by the poker platform.

100

IV. AFTER THE FLOP: when things go south Whereas, as shown in the previous chapter, the flops on Pokerstars tend to "help" a player, meaning they will provide him/her with the targeted combination significantly more often than normal, and especially so if he/she effectively plays the hand, meaning he/she remains at the table, after the flop the situation seems to reverse completely. Once you took the bait cunningly delivered to you on the flop, things appear to suddenly go south.

Let's pick for instance situations where you flop two pairs using both hole cards (a remarkable 2.02% probability on the flop in itself), one of them being Aces. What better flop could you have hoped for, right? Unless something really, really terrible happens, of course.

For instance, your opponent, with an equity of only 16% on the flop, to manage to hit his gutshot needed card and defeat you:

Or you get to find out you have actually been baited, being doomed already on the flop, as your opponent had also flopped two pairs, but better ones (a 0.1% flop probability):

101

Or, in another classic "cooler" so frequent on Pokerstars' flops, while enjoying the relish of your two pairs, you suddenly notice your opponent actually flopped directly a straight!

Or a set:

Or, never minding your flopped two pairs, your opponent comes back with a vengeance, hitting a sensational backdoor flush!

102

Or, though probably never hoping for such a miraculous happening, your opponent makes a winning flush starting from two off-suite hole cards!

And this is basically the moment where you start asking yourself: are two flopped pairs actually able to win a pot on Pokerstars? Indeed, you already know of the dramatically implausible abundance of coolers on the platform's flops But, still, you also know for a fact that, on such a two-paired flop, you still have a 16.47% chance of making a full house on either turn or river, which would, or better yet, should normally still give you the victory. Is this 16.47% probability respected on Pokerstars? What about flopped flush draws? Did they convert to flushes on turn or river to their expected ratio of 34.97%? And how about the field, is it still being "leveled" after the flop?

103

IV.1 An analysis of six variables on the post-flop streets This sub-chapter tries to answer questions such as the ones above by specifically investigating what happens on Pokerstars after the flop, on the turn and river streets, in terms of recorded vs. expected frequencies. For reasons of minimal subsamples size necessary, the following analysis separately analyzes as variables six post-flop conversion rates, each with its own associated expected probability: 1.) of flopped open-ended or double inside straight draws into straights (a 31.45% probability on the flop to hit the targeted combination by the river); 2.) of flopped gut-shot or semi-closed straight draws into straights (16.47%); 3.) of flopped sets into full houses or better (33.33%); 4.) of flopped two pairs (using both hole cards) into full houses; 5.) of flopped flush draws (using both hole cards) into flushes (34.97%); 6.) of flopped backdoor flush draws into flushes (4.168%).

On the transversal of these items, the previous chapter's methodological delineation between played and not played hands will be kept and employed whenever possible, as it has proven extremely relevant, but two observations are required here. Firstly, in relation to the post-flop phase of the game, the term "not played" has a slightly different meaning, in the sense that it covers not only hands that I folded pre-flop, but also hands that I folded on either flop or turn, at a moment when my targeted combination had not yet been achieved. Say for instance I had KJ of hearts in my pocket, the flop came T and 2 of hearts plus 3 of diamonds; the turn brough a T of spades, so I folded; two opponents remained in the game, and the river came 5 of hearts, meaning I would have made a flush. However, I wasn't in the game anymore. Such hands will be included to the "not played" category.

Secondly, this dichotomy between played and not played hands shall be supplemented by an additional, refined, dichotomy between hands with a certain outcome vs. hands with an uncertain outcome. Thus, both played and not played hands, as well as both of them taken together, can be regrouped into two categories depending on whether their outcome in relation to my targeted combination (say for instance making a set starting from two flopped pairs) is certain or not: i.) hands with a certain outcome, meaning hands where - either I went all the way and know, for instance, if I did eventually make a set when starting with paired hole cards, or if I did hit a full house when having flopped two pairs; - or I folded somewhere along the way, but at least two opponents continued on later streets, meaning I saw what cards have entered the board, thus knowing if I would have or not hit my set had I remained at the table83; and ii.) hands with an uncertain outcome, meaning hands that have ended (i.e. only one player remained at the table) before the river, implicitly before I could see for sure whether I would or would not have obtained the desired outcome. These are hands that ended either on the flop (when my targeted combination had not yet been obtained) or on the turn (under the same circumstances).

Here are some clarifying screenshots, three for each of the two types of situation, using the data stored by my HM2 software, data of which I picked up as examples hands in which I aimed at straight starting from flopped straight draws:

83

Both cases obviously include all scenarios in which my targeted combination came on the turn, regardless whether the hand continued to river or not.

104

 Example of hand with a certain outcome - had I remained at the table beyond flop (when I folded) with my pocket KJs, I would have made a straight (A to 10) on the turn:

 Example of hand with a certain outcome: having KJs in my pocket, I folded on the flop, when the board had given me an inside straight draw; had I stayed in the game, I would not have made a straight:

 Example of hand with a certain outcome: having folded my ATo pre-flop, I would have made a straight on the river:

 Example of hand with an uncertain outcome: I raised pre-flop and bet on the flop (letter "R,B" in the second column in HM2 database), but, since my opponent folded, and the hand ended with me collecting the pot, we cannot know for sure if my flopped 105

10-to-7 open-ended straight draw would have eventually turned into a straight on turn or river:

 Example of hand with uncertain outcome: I folded my T7o pre-flop (letter "F" in the second column). The hand reached the flop, which wouldn't have granted me a straight draw (the only thing certain). Whether or not I would have hit a backdoor straight on turn and river remains uncertain, since of the two players left, one folded, so the hand ended prematurely, before its outcome in relation to my targeted combination could be certain:

 Example of hand with uncertain outcome: I folded my T6o pre-flop, I would have flopped an inside straight draw, but whether it would have become a straight or not remains uncertain:

This certain vs. uncertain delineation might prove to be a crucial instrument in testing the randomness of the cards dealing on the Pokerstars platform, although it might simultaneously stir an ardent (methodo-) logical debate: should hands with an uncertain outcome be taken into any consideration? The prima facie and most simple answer would be "No", since you never know what might have happened if the game had continued after it's recorded premature end. Logically speaking, anything could have happened. One can never know, obviously. Which begs the question: why then open this apparently futile discussion and not simply stick with hands whose outcome was certain? I shall try answering this in an applied and hopefully clarifying manner in the very first of the five targeted analyses below.

IV.1.1 Open-ended / double inside straight draws turned into straights Within my 51,989 cash games played on the platform, the underlying analysis isolates as a sample to be analyzed those flops that simultaneously met three requirements:  provided me with a either an open-ended84 or a double inside straight draw85 (both these types of draws having an equal 31.45% probability of turning into a straight on either turn or river);  granted me such draws under the condition that both hole cards have been necessary for making the draw86; but  regardless of the specific relation between my two hole cards, meaning connectors (e.g. 98, KQ, 54), 1-gappers (e.g. 97, KJ, 53), or 2-gappers (e.g. 96, KT, 52).

Of the total hands played, there have been 437 cases of flops fulfilling the above criteria, whose results in regard to both the played vs. not played and the certain vs. uncertain outcome dichotomies are these:

84

E.g. 98 in hand on a T7x board, or 53 in hand on a 46x flop, or 96 in hand on a 87x flop), etc.

E.g. QT hole cards on a 986 flop: both a J or a 7 on either turn or river would give me a straight, hence the interchangeable terms "double belly buster" or "double inside" or "double gut-shot" straight draws.

86 For instance: a KT pocket hand in relation to a QJ4 board is taken into account, as both hole cards need to be combined in order to form the open straight draw; however, a KT pocket hand on a 987 board is filtered out, since the 7-to-T draw does not use both my hole cards.

85

106

Flops providing me with a draw (No.) Played (144) Not played (296) Total (437)

Hands with a certain outcome Hands with an uncertain Flopped draws outcome, which have ended on: turned into No. straights No. % flop turn total 111 32 17 13 30 28.83 143 53 102 51 153 37.06 254 85 119 64 183 33.46 (51,989 cash games analyzed; N = 437 admissible flops)

The results are surprising in two regards. Firstly, among hands with a certain outcome, the aggregated played plus not played conversion rate of open-ended or doubleinside straight draws into straights is 33.46%, which is, significantly, 106.4% of the normal, expected, frequency of 31.45%. Secondly, the conversion rate differs significantly depending on whether I did effectively play the hand or not: in cases where I did remain in the game, only 28.83% of the draws eventually turned into straights (a mere 91.7% of the expected frequency). Oppositely, however, across the hands that I folded prematurely, no less than 37.06% of my flopped draws would have turned into straights (which is a spectacular 117.8% of the expected frequency). This is exactly the opposite of what happened on flops, where, if I did stay in the game, I was considerably rewarded in terms of flopping useful combinations.

And this is where I go back to the methodological debate on whether to take into account uncertain outcome hands or not: in the table above, one can easily notice there has been a considerable number of 183 hands (more than twice the number of certain outcome hands) which have not reached the river, so that it remains uncertain whether in those cases I would have eventually made a straight or not. As said, at first glance it only seems normal and methodologically rigorous to ignore all hands with an uncertain outcome and take into account exclusively those with a certain outcome. Upon closer examination however, there is a significant difference between those hands that ended on the flop and those that ended on the turn: whereas in the former case, "anything could" indeed "have happened", with my associated probability of hitting the straight being the same 31.45% as in the case of certain outcome hands, I shall argue that hands ended on the turn cannot and should not be ignored: true, their final outcome, which would have been clarified on the river, remains uncertain.

Nevertheless, we know for a fact that in all these cases, the turn did not bring a straight, which means that at that point, just before the river, the probability of making a straight had already decreased (from 31.45% on the flop) to 17.39%!87 And this, indeed, is a certainty. Adopting this logic, hands that ended on the turn could actually be considered "semi-certain": the uncertain half is the final outcome that would have been brought along by the river card; the certain half is that at least on the turn, the straight had not come yet, and my conversion probability had already decreased.

Considering this argument, for all six items analyzed here, I have chosen to calculate not only the targeted conversion rates for certain outcome hands, but, separately, also what might have happened in all the hands ended on the turn, by attributing them counterfactually the expected conversion rate of 17.39%, and then to recalculate a combined factual plus counterfactual conversion rate. So, we shall simply assume that across all hands ended on the turn, the conversion rate would have been the normal one. For the item analyzed here, that means that we could expect 17.39% of the 64 hands that ended on the turn to have brought me a straight on the river - meaning 11.13. Combining this figure with the results registered along certain outcome hands, the new combined factual plus counterfactual conversion rate of flopped open-ended plus double inside straight draws into draws 87

Meaning 8 outs needed / 46 cards left in the deck.

107

would be (85 + 11.13) * 100 / (254 + 64) = 30.23%. This new figure suddenly drops considerably below the rate measured exclusively for certain outcome hands and, simultaneously, is only 96.12% of the overall expected frequency of 31.45%! Moreover: for the first cluster, when counterfactually mixing in hands that have ended on the turn, the combined conversion rate further drops to an alarming 27.63%, representing 87.8% of the expected frequency! For the second, it also drops, in this case from 37.06% to 31.89%.

Whereas the latter figure is slightly above the expected frequency, the conversion rate measured across effectively played hands, the only ones that actually matter for a player on the platform, remains spectacularly small.

Thus, surprisingly, at least on the basis of this first item analyzed, the post-flop situation seems diametrically opposed to the one recorded on the flop: whereas, as shown in the previous chapter, the flops came to my favor, the post-flop situation seems to do the opposite, and considerably so. Whether or not this conclusion can be extended to cover all items analyzed in this chapter, will be verified in the following section. For now, as a final remark, let us also notice that (100 - 87.8)% = 12.2% "missing" straights along effectively played hands once again comes remarkably close to the figure of 10% pinned down by the "every tenth hand is manipulated" corollary.

IV.1.2 Flopped inside or semi-open straight draws turned into straights Across the same series of 51,989 cash games played and also under the necessary condition of both hole cards having been used, the present analysis verifies the conversion rate of flopped inside / gut-shot88 and semi-open straight draws89 into straights (both with an equal associated probability of 16.47%):

Flops providing me with a draw (No.)

Played (522) Not played (957) Total (1,479)

Hands with a certain outcome Hands with an uncertain Flopped draws outcome, which have ended on: turned into No. straights No. % flop turn total 350 66 96 76 172 18.86 488 94 282 187 469 19.26 838 160 378 263 641 19.09

This second set of findings is as spectacular as the previous one, but in a different sense: in this case, among hands with a certain outcome, both the played- and not played hands display overall frequencies significantly above the expected one, specifically 114.5% and 116.9% of it, such deviations being practically impossible to attribute to randomness. As a further difference, the difference between played and not played, again in favor of the latter, is however negligible, which in itself would argue in favor of randomness, if, however, in context, both recorded frequencies would not be that high.

The aggregated played plus not played figure of 19.09% in the table above is interesting. Redoing the math in terms of expected vs. recorded absolute frequencies, we can for instance establish that, in relation to the expected 138.02, so roughly 138, draws turned into straights, there has been an extra 21.98, let's round it to 22, that should have not have converted, but did. Or, judging in terms of the relative frequencies of 19.09% recorded vs.

16.47% expected, in means that, approximately, for every six draws correctly turned into straights, there appears to have been one bonus that the platform gifted me with.

88 89

E.g. a KJ hand in relation to a T94 flop, needing a Q to make a straight E.g. an AK hand in relation to a QJ2 flop, needing a T to make the straight.

108

Again, the question is: could these considerable deviations be somewhat compensated by the semi-certain hands that have ended on the turn, a point where the probability of a conversion had, as a fact, dropped to 8.696%?90 Redoing the math by applying the same algorithm as in the previous analysis, in a combined factual plus counterfactual reasoning, the new figure of (160 + 0.08696*263) * 100 / (838 + 263) = 16.61% comes extremely close to the expected frequency of 16.47%. All of a sudden, the dark green colors in the table above, suggesting an apparent "help" coming to my benefit, lose their relevance.

IV.1.3 Flopped sets turned into full houses or quads In regard to flopped sets that turned into full houses or quads across the series of 51,989 cash games played, the number of not-played hands is unfortunately too small to provide any reliable conclusions so that the following abandons this first methodological dichotomy, but keeps the second one, between certain and uncertain outcome hands. As an exception within this chapter, motivated by the need of a satisfying sample size, the tournament hands have also been taken into account, so that the results below cover the entire 55,320 cash + tournament games on the platform in which, starting with pocket pairs, I did or would have flopped a pure set.

Pocket pair flopping a pure set (No.) AA (19) KK (15) QQ (18) JJ (20) TT (12) 99 (21) 88 (20) 77 (18) 66 (24) 55 (29) 44 (23) 33 (17) 22 (25) total (261)

Hands with a certain outcome Hands with an uncertain outcome, which have ended Flopped sets on: turned into sets No. or quads No. % flop turn total 17 4 23.5 1 1 2 11 6 54.5 4 4 14 5 35.7 4 4 10 3 30.0 2 8 10 9 3 33.3 1 2 3 11 4 36.4 4 6 10 15 4 26.7 2 3 5 14 6 42.8 1 3 4 16 4 25.0 6 2 8 22 8 36.4 7 7 20 6 30.0 1 2 3 12 6 50.0 1 4 5 17 8 47.1 2 6 8 188 67 32 41 73 35.64

As was the case with the previously analyzed item, the recorded conversion rate of flopped sets into full houses or quads along the series of certain outcome hands is also significantly higher than the normal, expected one, of 33.39% (specifically, 106.7% of it)91.

Furthermore, it should be noticed that, of the total 13 possible pocket pair types held, 5 led to sets that turned into full houses or better below the normal frequency, whereas 7 displayed higher recorded frequencies, so that the overall recorded percentage cannot be 90

Four outs in a 46-cards deck.

When indicating the probability of a flopped set to turn into a full house or better, many people forget to take into account the scenarios in which the turn and river cards are of the same value, despite them being non-equal to the flop cards. For instance a KK pocket pair in relation to a K85QQ board also makes a full house.

91

109

attributed to any anomaly case(s) in terms of standard deviation. Additionally, the very value of the verified normal frequency of 33.39% is big enough to dismiss any discussions about some potentially weak representativeness of the 188-sized subsample of certain outcomes cases of the series analyzed92.

However, when once again taking into account the hands that have ended on the turn, where my probability of making a full house had for a fact dropped to 21.79%, the overall projected factual plus counterfactual frequency is (67 + 0.2174*41) * 100 / (188+41) = 30.22% again plummets below the expected frequency of 33.39%. Thus, it actually covers and this might not be a surprise anymore when keeping in mind the "every tenth hand" corollary - 90.51% of it, meaning that roughly every tenth flopped set (9.49% more exactly) that should, under genuinely random circumstances, have brought a full house, actually did not.

IV.1.4 Flopped two pairs turned into full houses In relation to the 16.47% probability of two flopped pairs (using both hole cards) to turn into a full house by either turn or river, the frequencies recorded within 51,989 cash games played, out of which 410 provided the targeted flop situation, are once again significantly higher, blatantly surpassing the threshold below which anomalies would theoretically be still attributable to randomness: 111.84% of the expected value for hands effectively played, 121.4% for hands not played, respectively 118.03% on the overall. Thus, approximately, for each 5.5 cases of mathematically correct, normal, expected conversions into full houses, Pokerstars gifted, or would have gifted me, with one extra.

Additionally, once again, the second frequency (hands not played - 20.00%) surpasses the first one (hand effectively played - 18.42), to a ratio of 1.08 to 1 and in a pattern completely opposite to the one discovered when studying the flops.

Flops providing me with two pairs (No.) Played (120) Not played (290) Total (410)

Hands with a certain outcome Hands with an uncertain outcome, which have ended Flopped two on: pairs turned into No. full houses No. % flop turn total 76 14 25 19 44 18.42 140 28 103 47 150 20.00 216 42 128 66 194 19.44

Finally, when applying the factual plus counterfactual algorithm, taking into account hands that ended on the turn, at a moment when, for a fact, I still haven't made my full house yet, and where the new probability of achieving it had dropped to 8.696%, the resulting projected frequency of 100* (42 + 0.08696*66) / (216 + 66) = 16.92% is reasonably close to the expected frequency's values of 16.47%, even slightly higher.

IV.1.5 Flopped flush draws turned into flushes When isolating flopped flush draws (using both hole cards) across the 51.989 games played and verifying their rate of conversion into flushes by the turn or the river, in relation to an associated probability of 34.97%, the results are surprising. Thus, in a vivid contrast to all of the four previously analyzed items, in this case all the three recorded frequencies measured 92

Furthermore, the percentage of flopped sets that turned exactly into quads is exactly the normal one of 4.25.

110

along the series of certain outcome hands are below the normal, expected one of 34.97%, and significantly so: 93.42% of it for played hands, 96.40% for hands not played, respectively 94.65% on the overall93.

Flops providing a flush draw (No.)

Played (321) Not played (339) Total (660)

Hands with a certain outcome Hands with an uncertain Flopped draws outcome, which have ended on: turned into No. flushes No. % flop turn total 251 82 48 22 70 32.67 175 59 108 56 164 33.71 426 141 156 78 234 33.10

Moreover, when taking into account the 78 hands that have ended on the turn, when I hadn't yet made the flush, and when the new associated probability had dropped to 19.56%, the resulting factual plus counterfactual new frequency is (141 + 0.1956*78) * 100 / (426 + 78) = 31.00%. This is a mere 88.65% of the expected frequency94, in a discrepancy which once again strongly relates to the "every tenth hand is manipulated" corollary.

IV.1.6 Backdoor flush draws turned into flushes As was the case with flopped sets turned into full house or better, the item investigated here also doesn't allow a reliable application of the played vs. not played dichotomy: the isolated sample revolves around a tiny 4.1628% probability on the flop95 and displays only 55 positive outcomes (meaning draws that did turn into flushes). Hence, the table below operates solely with the certain vs. outcome delineation. In this regard, as a specific difference to all the previous five items analyzed, as a backdoor flush needs both the turn and the river cards to be of the targeted suit, in this case the hands that have ended on the turn are hands in which the turn has brought a card of the targeted suit (otherwise, obviously, the hand would be reclassified as having had a certain, i.e. negative outcome).

Total flops (2367)

Hands with a certain outcome Hands with an uncertain Flopped draws outcome, which have ended on: turned into No. flushes No. % Flop turn total 1603 55 692 72 764 3.43

The table captures a severe discrepancy between the expected and recorded conversion frequencies, with the latter reaching only 82.42% of the former, which indicates non-randomness. Within the extended factual plus counterfactual reframing, the corresponding frequency expected on the turn is 19.56%96, which leads to a new combined 93

As a possibly interesting side note, in my played tournaments on Pokerstars, the overall conversion rate of flopped flush draws into flushes by turn or river has reached a spectacular 41.51%, but, it needs to be specified, the flops sample of only 53 may dismiss any comments as being imprudent.

94 Along the series of hands actually played, the corresponding factual plus counterfactual projection leads to an overall frequency of 31.61%, which is 90.4% of the expected frequency of conversion.

95 10/47 * 9/46, corresponding to the available number of targeted suit cards on turn plus river.

96 Nine cards of the targeted suit out of 46 cards in the deck.

111

figure of (55 + 0.1956*72) * 100 / (1603 + 72) = 4.12%. The new figure almost overlaps the expected 4.16% one.

IV.2 Integrating the findings and re-contextualization Integrating the played vs. not played, certain vs. uncertain outcome and certain vs. projected factual plus counterfactual methodological dichotomies for the six variables investigated, the table below captures some significant and simultaneously bizarre deviations from what a genuinely randomness-governed situation would look like:

Type of conversion (expected frequency)

OESD into straight1,3 (31.45%) GSSD into straight2 (16.47%) 2 pairs3 into full house+ (16.47%) Flush draws into flushes3 (34.97%) Sets into full houses+ (33.39%) Backdoor flush draws into flushes3 (4.16%)

Played Not played Total Played Not played Total Played Not played Total Played Not played Total

Projected conversion frequencies when adding into account hands that have ended on the turn4 Recorded Recorded Recorded Recorded frequency frequency frequency frequency as % of as % of out of out of expected expected % % N= N= frequency frequency 111 124 28.83 91.7 27.63 87.8 143 194 37.06 117.8 31.89 98.6 254 318 33.46 106.4 30.23 96.1 350 426 18.86 114.5 17.04 103.5 488 675 19.26 116.9 16.33 99.1 838 19.09 115.9 16.61 1,101 100.8 76 95 18.42 111.8 16.48 99.9 140 187 20.00 121.4 17.16 104.2 216 282 19.44 118.0 16.92 102.7 251 273 32.67 93.4 31.61 90.4 175 231 33.71 96.4 30.28 86.6 426 504 33.10 94.6 31.00 88.6

Total

35.64

188

106.7

30.22

229

90.5

Total

3.43

1603

82.4

4.12

1,675

99.0

Conversion frequencies for hands with a certain outcome Type of hand

1

includes double belly buster or double inside straight draws includes semi-open straight draws 3 using both hole cards.

4 with the turn having not provided me with the targeted combination (except backdoor flushes) 2

There are five major and critical conclusions that can be reached in the light of these findings.

1. After the enticing flop, things go south, to the detriment of the player The first one, formulated in terms of the played vs. not played hands distinction, is that across the hands with a certain outcome, in a pattern completely opposite to the one discovered 112

on the flop, the recorded conversion frequencies are lower, and considerably so, along the hands effectively played than along those where I folded, but were continued by some opponents at the table:  28.83% vs. 37.06% in the case of OESDs or 2xSDs (with the former being 77.8% of the latter);  18.86 vs. 19.26% in the case of inside or semi-open / gut-shot straight draws (97.9%);  18.42% vs. 20.00% in the case of two pairs (92.1%); respectively  32.67% vs. 33.71% in the case of flush draws (96.9%)97.

It's like the promising events on the flop, commented in the previous chapter, afterwards take a turn for the worse, and to my direct detriment, on the turn and river streets.

Thus, whereas flops clearly rewarded me when effectively remaining in the game, providing me with useful situations more frequently than expected, the turns and rivers do the exactly opposite thing, meaning had I stayed in the game, I would have obtained my desired combination considerably more often than in the cases where I actually remained at the table and effectively played the hand. This finding thus explains the meaning of the chapter's title: once a player on Pokerstars takes the bait on the flop, things suddenly go south.98 2. The recorded anomalies cannot be attributed to randomness The second conclusion departs from the played vs. not played delineation and addresses in a combined manner the total frequencies recorded across hands with a certain outcome, in order to state that the recorded statistical anomalies cannot be attributed to randomness; in relation to the expected frequencies, the recorded proportional deviations reach (in the descending order of the table above): +6.4%, +15.9%, +18.0%, -5.4%, +6.7%, respectively 17.6%. The sample of a combined 3,525 flops analyzed across only hands that had a certain outcome is also big enough to grant reliability to this conclusion.

3. Pokerstars' dilemma The third conclusion involves the certain vs. uncertain dichotomy. Two observations can be made here. Firstly, when taking into account the hands (both played and not played) that have ended on the turn, we notice that the resulting projected frequencies have been lower than the expected frequency in three of the six cases (and considerably so), essentially equal to it in two cases (meaning a deviation less than 1%), and higher in only one case 99 - thus reconfirming that, after the flop, things take a turn for the worse. As a second observation, the way I see it, Pokerstars representatives are confronted with a dilemma in the true meaning of the term, meaning having to choose between two unsatisfactory options:  either they reject the inclusion into calculations of hands that ended on the turn, a case in which they are left with the tremendously difficult task of explaining the above indicated statistical deviations recorded, solely across certain outcome hands, in relation to the expected conversion frequencies;  or (as a slightly less worse option when looking at the results in the table), they could accept the methodological solution, but then again they would still have to explain what seems to be another strong indicator of the leveling-the-field strategy, 97

Whereas, I repeat, the subsamples' sizes for the last two variables do not allow for a viable, reliable, played vs. not played analytic dichotomy.

98 I admit I have no explanation for this fact in terms of the poker platform owners' purpose of doing it. It might be another application of the leveling-the-field strategy: when staying in the game and flopping a combination with a high conversion probability, a player is dragged downwards, with the decreased recorded frequency of conversion working in favor of the underdog. Or, it might be some sort of teaser, meant to discourage players from folding: once they fold, they discover all so often that, had they remained in the game, they would have made their targeted combination. Or it might something else, or even a combination of other reasons.

99 I.e. two pairs on the flop turned into a full house or better.

113

specifically why, in the right half of the table, when combining hands played with those not played, it seems that the more likely a draw is to hit its targeted combination on turn or river, the less it actually does so.

Mathematically, there is a spectacular negative correlation of 0.79 between the probability on the flop and the actual, combined factual plus counterfactual frequency calculated in relation to the probability - a figure again hardly attributable to randomness, "variance", sample size, "accidents" or "bad luck", etc:

Conversion type Flush draw to flush Sets to full house or better OESD to straight GSSD to straight Two pairs to full or better Backdoor FD to flush

Projected frequency as % of expected frequency 88.6 90.5 96.1 100.8 102.7 99.0

Probability on the flop 34.97 33.39 31.45 16.47 16.47 4.16 Correlation:

- 0.79

Furthermore, in an interesting recontextualization of the findings, two of the six projected overall frequencies across the combined series of played and not played hands come surprisingly close to the referential percentage of 10 stated by the "every tenth hand" corrolary, with both deviations to the player's direct detriment, in what once again looks like a method of levelling the field. Specifically, 9.49% of the flopped sets that, under random conditions, should have turned into a full house or better, did not; and 11.35% of the flopped flush draws expected to turn into flushes also did not.

4. Leveling-the-field hypothesis reconfirmed in its second dimension In addition to the two partial confirmations discussed above, one can find more further reconfirmations of Pokerstars' post-flop continued strategy of leveling the field, respectively striking traces of the non-randomness of the flops, when looking at two of the six items investigated: flopped flush draws turned into flushes, respectively, closing the circle by returning to the beginning of this chapter, flopped two pairs analyzed in terms of winning.

In regard to the conversion rate of flopped flush draws into flushes, as graphically shown above, an application of the same algorithm as in the previous chapter, of separating 13 114

card lines100 and assigning them aggregated values, captures a considerable relation between the initial pocket hand's strength and it's likelihood of eventually making the flush by the river - specifically, a linear correlation of minus 0.662 for exclusively those hands that had a certain outcome in regard to my draw. This is a significant indicator of non-randomness: under random conditions, there should be a considerably lower correlation (preferably zero, as a signifier of complete mathematical independence), and the regression line should be horizontal.

Even if distinctively covering the 78 suited hole cards, the resulting aggregated correlation coefficient between the starting hole cards' power rank in 6-players games, arranged in a descending order, and their corresponding cumulated % of flopped flushes, is still unacceptably high: -0.72.

Conclusively, in relation to a correlation coefficient not spectacularly high, but anyhow not negligible, it can be stated that, to a certain degree, the stronger the starting suited hole cards are in a 6-game, the lower their likelihood to lead to a flush by the river.

5. Non-randomness of Pokerstars' flops reconfirmed As far as flopped two pairs are concerned, they do indeed seem to end up losing the pot suspiciously often despite their seemingly high equity estimated on the flop. True, a standardized indicator meant to evaluate their recorded win rate in comparison to their expected one is hard, if not impossible, to construct, as it would necessarily have to take into account various factors such as the entire pre-flop hole cards distribution at the table, the opponent's equity on the flop, both player's separated probabilities to hit a certain combination by the river, etc. However, let us put it this way: of 74 hands in which I flopped two pairs and the hands did reach showdown, I won only 52 of them, meaning 70%, lost 27%, and split the pot in 2 cases. I cannot come up with a mathematically precise evaluation indicator, but still, most poker players will probably agree that a recorded win rate of only 70% for two already flopped two pairs seems a tiny bit too small.

And this has nothing to do with either hands folded pre-flop, nor some specific playing style that I have practiced. True, there have been a few dozen hands that I won before showdown, following a fold of my opponents, meaning I would have probably won them anyway at showdown, but so have there also been cases in which I felt compelled to fold, in 100

Suited Aces, suited Kings, suited Queens, etc.

115

situations such as this one, with my opponent's betting on the turn either making him a good bluffer, or, rather, the holder of a straight or two pairs, one of them Aces:

Or this one hand, where I did eventually fold on the river, just before showdown, when my opponent once again bet high, strongly indicating he had the straight:

As such, for all hands that I won before showdown, which could mean my two pairs held, meaning they would have won anyway at showdown, there have been other cases in which, all things considered, I probably would have lost had I remained at the table by the showdown.

And, as said, this also has nothing to do with my playing style, i.e. me folding inadvertently or incorrectly. Taking a look at hands that I folded along the way (most of them pre-flop), but which have been continued by my opponents all the way to showdown, one discovers that, out of 70 such hands (with those where I would have made a full house filtered out), I would have won only 39 (meaning 55.7%), split 2, and actually lost no less than 29 (meaning 41.4%) in all kinds of highly improbable situations, such as for instance: ď&#x192;&#x2DC; emblematic for Pokerstars' "coolers", I would have flopped my two pairs (J6), despite one opponents occupying one of my needed cards (thus also making a pair on the flop), only 116

for both of us to discover that the third player at the table had already flopped a flush - a "cooler" flop of a 0.296% probability for the given pre-flop situation!:

ď&#x192;&#x2DC; or: had I stayed in the game with my pocket Q9s, I would have flopped two pairs, but so did the K9 holding player, and, for the "circus" to be complete, both of us would have lost on the turn against the AKo-holding player. In relation to the three known sets of hole cards, the probability of such a "cooler" flop that grants the players twice two pairs plus another pair for the third player is 0.079%:

ď&#x192;&#x2DC; or, I would have lost against a flopped set, despite one of the two Ks needed by my opponent being blocked by the third player - a flop of a 0.059% probability:

ď&#x192;&#x2DC; in another classic "cooler" situation, in spite of my counterfactually flopped two pairs, I would have anyway lost to one opponent who had also flopped two pairs, but better ones, while the third one was kept in the game by an enticing flush draw - a flop of a 0.046% probability!: 117

ď&#x192;&#x2DC; or: I would have flopped two pairs (JJ-44), again, despite one opponent blocking one of my needed cards (J), but, simultaneously, a third player at the table flopped a combined flush plus gut-shot and was set to hit the winning straight on the turn. In the starting circumstances of the three players' known hole cards, the probability of a flop providing them, in the given situation, with two pairs, one pair, respectively a flush plus straight draw is a mere 0.039%, meaning it happens once in every 2564 such starting positions.

And still, on Pokerstars it happened!

ď&#x192;&#x2DC; or, as a last example, just in case you may have thought there couldn't have been even more improbable flops for me to lose with two flopped pairs: how about a flop of a...

0.026% probability?! Take a look at the screen-shot hand below, where I folded on the flop, when I felt something was not quite right!

Let's get this clear: in relation to my T9s, one opponent is blocking one of the three tens that I need, but is himself blocked by the K of the third player. So, in the 46 cards deck there are only two Ts, three 9s, and two Ks left. Miraculously, the flop manages to provide: me with two pairs; the KTs-holding player also with two pairs, and even better ones; the K4s- holding 118

player with top pair plus flush draw. Now, out of 15,180 flops possible considering the known six hole cards (C46, 3), there are only four (4) flops that can simultaneously offer the players those simultaneous combinations (and this is regardless which of my two opponents may flop a flush draw!):  T9K  T9K  T9K  T9K In relation to the 15,180 flops possible, these four combinations, the only ones simultaneously putting the three players in that particular position on the flop, make up for a probability / expected frequency of 0.0263%, meaning such a flop is expected to happen once in every 3802 hands played from that particular starting position, which is in itself already associated with a small probability No matter how one forces interpretations of the results, a win rate of a mere 51.7% across all those hands that reached showdown, under the condition of having actually flopped two pairs, remains extremely suspicious and intensely suggests a deliberate levelingof-the-field method applied by the Pokerstars' platform. Moreover, even if one forcibly and erroneously adds into account all the 26 hands in which I would have made a full, regardless whether those hands reached showdown or not, by hastily assuming any full house is a winner, the correspondingly new counterfactual win rate of my flopped two pairs would still have been (39+26)*100/(70+26) = only 67.7%. This figure comes significantly close to my 70% win rate recorded exclusively across hands that I did effectively played and that also reached showdown! But, as a parenthesis, it needs to be stressed: on Pokerstars, full houses, not even flopped ones, should never be hastily considered a certain winner! If the reader may find such a statement outrageous, let us take a look at this hand, one I luckily wasn't involved in, screen-shot directly on the platform:

So: flopped full house vs. quads at a 6-players table - a flop of a 0.0116% probability in relation to the given pre-flop starting position. Taking into account the probability of being dealt a pocket pair and simultaneously at least one opponent also having been dealt a pocket pair results in a conditioned probability of 0.000178%, meaning such a chain of events is expected to happen once in every 561.798 hands played at a 6-player table. Did the players in the image play that number of hands? I seriously doubt it.

119

And if you still might think the above hand was just a singular accident on the platform, think again, while re-taking a look at another hand, this time with me involved directly in the game, a hand already presented in the second chapter of this paper:

All things considered101, from the perspective of the 5s-holding player, the conditioned probability of such a chain of events is a sensational 0.0000403%, meaning such an "accident" is expected to happen once in every 2.48 million hands played at a 6-players table! I can't wait for some Pokerstars' representative to show me that that particular player did indeed go through 2.48 million hands on the platform!

Actually, on Pokerstars, you'll never be sure. Not even if you flop four of a kind!

The probability, from the perspective of the 4s holding player, of playing Hold'em at a six players table and suffering such an "accident"? It's 5.43% (the probability of being dealt a pocket pair other than Aces) * 2.425% (of an opponent having Aces) * 0.254% (of flopping the quads) * 4.44% * 2.27% (the conditioned probability of Aces on both turn and river) = 0.000000334% 101

I.e. the probability of being dealt a pocket pair in a 6-players ring; the probability of at least another opponent also having a pocket pair; the probability of flopping a full house in a combination that does not help the other players; the probability of the non-paired opponent to hit a full house on turn plus river.

120

Or, formulated alternatively: normally, an event such as the above happens once in every 299,347,873.4 hands dealt, so roughly once in every 300 million hands played at a 6-ring table.

Closing the above parenthesis and returning to my flopped two pairs cases, I'd say it can be safely concluded that the leveling-the-field hypothesis has been re-confirmed. What, however, is equally interesting in the context of this entire paper is the fact that, among hands both played and not played, in most of the cases, in terms of equities, my flopped two pairs have, or would have lost, already on the flop, as a direct consequence of cooler-like dealings of the card. Thus, among hands not effectively played, I bumped twice into flopped sets, three times into better two pairs plus once into equal ones, once into a served, made, flush, once into a made straight, and over a dozen times into flush or straight draws, usually combined with a pair made on the flop, etc., less than eight of my counterfactual defeats having remained attributable to some turning of the tide on turn and/or river.

As already extensively analyzed in the previous chapter, it is the flop among the four streets of the game that brings along the most strikingly improbable twists of situation, usually by an overinflated number of "cooler"-type situations. Let's revisit the flop probabilities in the six examples screen-shot above: 0.296%; 0.079%; 0.059%; 0.046%; 0.039%; and 0.026%. And these six examples are selected solely from within the 70 hands that I folded, but have been taken to showdown by two of my opponents! Correspondingly, each of these six flops is expected to happen once in every 338, 1266, 1695, 2174, 2564, respectively 3846 hands, but, it needs to be specified, these 3846 are only the already-given pre-flop situations, which are in themselves significantly improbable in terms of the hole cards' dealing!102 When taking into account the probability of the players (at a 6-players table) of being dealt exactly those hole cards needed to put them simultaneously in that given preflop positions, one discovers that for such events to occur at their normal, expected, frequency, I should have been playing hundreds of thousands of hands, whereas: my total number of cash games has been less than 52k; in four of the 70 hands taken to showdown, I discovered another opponent that had also flopped two pairs, despite the maximum probability103 of such a situation being only 0.104%.

Finally, similarly to the examples selected above, the hands that I did effectively play, and lost at showdown, also display cooler-type flops in frequencies that cannot be attributed to randomness. Specifically, of the total twenty that have reached showdown, five could be safely considered to have been lost for me after the flop 104, while in the remaining 3/4 of them, despite having two pairs at the moment of the flop, I was actually, although unknowingly, already the underdog. Four times I bumped into a flopped set, another four times into a flush draw that did eventually turn into a winning flush, twice into a gut-shot straight (plus one pair on the flop) draw that also turned into a straight, twice I encountered another two pair at the table, but better ones, once my opponent flopped directly the straight, etc. Now consider in regard to the first case for instance, that, if one player has two nonpaired hole cards and an opponent a pocket pair, the probability of a flop that provides them simultaneously with two pairs, respectively a set, is a mere 0.104%, meaning such a flop is expected to come roughly once in every 1,000 hands of that given, albeit highly improbable, 102

Obviously, these are not the only anomalies registered in regard to flop probabilities. Inter alia, I would have bumped for instance into an opponent holding equal hole cards as me, both of us flopping the same two pairs (a 0.11% probability to flop such cards) (May 20, 11:10, with me holding J8o), into a flopped set by one opponent, with the other opponent flopping a flush draw, which has converted into a flush on the turn (a 0.12% flop probability) (May 13, at 14:05, with me holding 93s), etc.

103 Meaning excluding scenarios where two players flop exactly the same two pairs - a considerably smaller probability, one however materialized at least once (see the first example in the footnote above).

104 With my opponent for instance hitting a backdoor flush, a backdoor straight, or a trip on turn plus river.

121

starting situation. Well, the 70 hands that I did effectively get to showdown, I encountered opponents with a flopped set four times!

Let's put this in a more simple and simultaneously clarifying way, by using the situation-example in which both I and an opponent flop two pairs at a 6-players table, leaving aside the probability of us being dealt those specific hole cards. Say, for instance, I hold KJ and my opponent Kx (i.e. he needs to have at least one equal card with me in order for both of us to flop two pairs105): of 17,296 flops possible, considering the known hole cards, there are only a total 2*3*3 = 18 which provide simultaneously a King (there are two left in the deck), a Jack (three left in the deck) and a "x" (three left in the deck). Thus, the probability of such a flop is 18*100/17,296 = 0.104%. Relating this probability, meaning expected frequency, to the total number of 410 cases (both played and not played hands) in which, as a certain fact, I flopped (or would have flopped) two pairs, we would expect the event of two simultaneously flopped two pairs to have happened 0.104*410/100 = 4.264 times. Well, in the factual reality manifested on Pokerstars, it happened at least 6 times, four each for my played and my not played hands (out of which in one case, my opponent had hole cards equal to mine!). This means a recorded frequency 1.41 times higher than the normal one to be expected in a genuinely randomness-governed context! And, it needs to be stressed, this is the minimal, certain, and provable, figure, since it is based exclusively on the total 90 cases of double two pairs on the flop that did reach showdown, and it is simultaneously assuming that in all the 410 cases, my opponent at the table had non-paired hole cards!

How frequent this type of situation has actually been in reality on Pokerstars, along the entire series of 410 cases, one can only speculate.

Similarly, the recorded frequency of flopped two pairs vs. flopped sets (6 such cases), of also the same probability106 of 0.104%, is equivalently a minimum, certain, and provable, 1.41 times higher than the expected one, the frequency of two pairs vs. flush draw on flop has been 1.93 times higher, etc.

The conclusion is as clear as possible. In the larger context of this investigation, the findings above reconfirm, and categorically so, albeit via a different path, the main conclusion of the previous chapter, namely the blatant non-randomness of the flops on this poker platform, one especially recognizable in the spectacularly inflated frequency of "cooler"-like situations. Any flop, no matter how minuscule it's probability, is going to be provided by Pokerstars' "random" numbers generator, as long as it satisfies a single condition: grant at least two players at the table attractive combinations they can't refuse, so they bet, raise, and re-raise, with the implacable outcome that is an increase of the rake the poker platform collects.

IV.3 The "Riverstars" (sub-)hypothesis As prefaced in the Introduction, the so-called "Riverstars" sub-hypothesis is one placed chronologically in terms of the game's four streets, thus finding its place in this chapter, although in the logical approach of this study, it is directly derived from the transversally followed more general "leveling the field" hypothesis.

Essentially, the methodologically operational(ized) meaning of the sub-hypothesis is that the river cards all too often (i.e. more often than mathematically expected) disfavor the favorite, meaning he or she wins less often than normal in mathematical terms. So far, as prefaced in this chapter, when checking the showdown-recorded results from my perspective, regardless whether on the turn I had been the top - or the under-dog, and based solely on those hands where I had been dealt pocket pairs, the findings failed to capture any significant 105

Obviously, even if the opponent has Jx instead of Kx, the probability remains the same.

For a KJ vs. XX scenario, there are 3 Kings, three Jacks, respectively two Xs useful in relation to 17,296 flop combinations possible.

106

122

anomaly. The issue, however, has its very essence related not to a particular player, but to the favorite. So, the immediate question is: how to test this hypothesis from the methodologically separated perspective of the favorite on the turn?

Well, quite simply: say the favorite on the turn has a 95% equity (meaning his/her probability to win or split the pot). That means that, of all those cases with him/her on a 95% equity on the turn, there should be an expected 95% victories plus ties for him/her.

I expanded this logical approach to all possible equities registered on the turn by the favorite, meaning the probabilistically speaking best positioned player. As a sample within my 52k cash games, I selected all hands where I had been dealt pocket pairs. From within those, I isolated the ones that reached showdown (with or without me still at the table). Then I went back to the turn using the HM2 software, and, after filtering out all methodologically inadmissible hands107, I wrote down the favorite's / best positioned player's equity, meaning the cumulated probability of him/her winning or splitting the pot. As a last step, I then checked the showdown of the hand, to see if he/she won, lost, or tied. After doing this for all 744 methodologically admissible hands, I grouped all the favorites' equities on 5% large variation intervals (the left column in the table below), except for two: ď&#x201A;ˇ those of equities below 70%, which were too few numerically to be grouped in 5% intervals, so I aggregated them in one single category; ď&#x201A;ˇ the 95% equities (the classic "two outer" situations") - which, oppositely, have been many, 308 specifically, which allowed me to count in a separate, distinct, category.

Correspondent to all these intervals and values, I calculated the weighted average recorded equity of the favorite on the turn, as displayed by the HM2 software. For the first line of the table for instance, for the 33 cases where, on the turn, the equity of the best positioned player was 95% or more, the mean value of all factually recorded 33 equities has been 97.94%. I reported this value to the total number of cases, calculating how many of the total 33 hands, based on this average equity, the favorite was expected to win or tie: 32.3%.

Then, as a last step shown in the extreme right column of the table, I calculated his recorded win plus split rate as a percentage of the expected win + split rate. For the example here discussed, 33/32.3 = 102.17%. Replicating this simple algorithm for all 774 hands has led to the following results:

At first glance, the results, and especially the huge correlation discovered, by far the biggest one discovered so far in this analysis, which no sane person on this planet would dare defy by still talking of "randomness", are shocking. The image couldn't be more 107

Meaning hands with no favorite on the turn, i.e. either hand where on the turn there was either a 50-50% (or any other perfect parity) situation, or hands that had been already won mathematically by a player, with the other(s) drawing dead.

123

convincing: not only are 5 of the 8 registered deviations from the expected values significant, but the general trend is clear - the stronger favorite one is, the less he/she actually wins on the average. This conclusion overlaps to a satisfying degree the meaning of our hypothesis, stating that the river card tends to help the underdog far more frequent than normal.

However, allow me to once again try to come to the defense of Pokerstars and counter-argue with four objections: 1. firstly, such huge correlations might still not indicate causation or deliberation, given the small number of both the units in each series (eight) and some of the subsamples (33, 49, etc.) in relation to their corresponding probabilities checked; 2. secondly, of the total 774 cases in the table, 162 have been a part of my limp-check-call experiment already described, an experiment whose implications might have affected the reliability of the findings. Specifically, whereas, when having been dealt pocket pairs, I remained in the game all the way to showdown, regardless how the hand played out, my opponents might have played more rationally and folded for instance right before showdown (i.e. hands unaccounted for in the table) in many cases where I was a favorite and my probabilistic advantage had indeed held. Subsequently, the recorded proportion of cases where the favorite's equity on the turn did not hold might be inaccurate due to a selection bias; 3. thirdly, and expanded from the previous objection, a selection of only pocket pair hands might have put me in the position of the turn favorite more often, since, by default, pair are favorite against any hand other than a higher pair; 4. fourthly, when further processing the data, it turns out that, placed on a weighted average equity of 87.76% on the turn, the favorite actually ended up winning to a frequency of 111.69% of his indicated probability so that the favorite has clearly not been disadvantaged, quite the contrary in fact.

Such a threefold counterargument actually both a) fails to explain the extremely conspicuous mathematical trend in the right column of the table, since in my experiment I stayed at the table till showdown regardless what specific pocket pair I had been dealt, and b) leaves the derived problem of the fourth point unanswered, but, other than that, the objections do indeed at least seem legitimate.

Given these possible objections, I decided to adequately expand the sample of hands analyzed. Aiming at a threefold increase in volume, I threw in all the hands where, of my two hole cards dealt, one was either an Ace or a King (or both, obviously). Doing this, I ended up with a new sample volume almost four times the initial one, covering 3012 cases corresponding to 36 hands of the total 169 existent in the game. Obviously, exactly as in the previous case, I first filtered out all cases of equal parity on the turn, such as for instance

, 124

and all those hands where on the turn there was already a mathematical winner, meaning the opponent(s) was or were drawing dead, such as for instance:

Thus reconfigured, the sampling deems all previous potential objections futile, considering that:  the total number of showdowns within my experiment now represent only 5.38% of the expanded sample of 3,012 cases;  the entire subsample of my pocket pair hands cover only 25.7% of the whole sample;  in terms of representativeness of the sample in relation to all my 52k hands dealt in cash games, it covers 21.3% of the game's 169 hand in terms of hole cards possible dealt, respectively 23.6% of the entire number of showdowns;  I was a favorite on the turn in less than 50% of the total cases, which eliminates any objections pertaining to my playing style;  I actually was still at the table on the turn in only roughly 64% of the cases, meaning there is a significant proportion of hands in which the favorite has been any of my thousands of opponents that I encountered in my cash games108.

In regard to the last remark, maybe a specification is necessary: all my 51,989 cash games were played on the Pokerstars platform in the so-called "zoom format". For those unfamiliar with the term, it means that, once a player folds his hand, he is automatically and instantaneously moved to another table. Generally speaking, it is an extremely attractive tool, since it allows players to go through an incomparably higher number of games per time unit, which is especially useful for the so-called "grinders". As a consequence of this software, the number of opponents one faces over a session also increases exponentially. In my case, the HM2 software has recorded over 1,000 opponents of mine who played at least 50 hands each against me, with the others (with less than 50 hands) numbering at least as many (see the capture below). Thus, whenever I talk of the "favorite", it means any of those thousands of opponents, which guarantees, from a methodological point of view, that no potential unusual playing style of a certain player could affect the reliability of the data measured.

108

More on this in the next chapter.

125

Furthermore, this considerable expansion of the sample allowed me to operate with measurements even more precise than the previously used 5-points equity intervals grouped in 8 ordinal categories. Unfortunately it did not allow the usage of a full, perfect, 1point scale, given that the mathematical situations and associated probabilities in the Hold'em game on the turn are only so many, but it allowed at least a considerably more accurate classification and evaluation of equities. The following presents the manner in which I grouped all equities in new unequal intervals (encircled in the table below), by trying to segment, as far as possible, ordinal intervals of at least 90, ideally 100, cases each109:

109

Since all calculations are done in relation to the weighted average recorded values of the equities as displayed by the HM2 software, the unequal sizes of the intervals obviously don't matter in themselves.

126

Subsequently, I applied the same algorithm as in the case of the small sample for all 16 thus separated intervals and values of the turn favorites' equities. The integrated results are these:

Some of the most important discoveries in history have been made by mere accident. On a different scale, the same could be said about the findings above. At the beginning of this subchapter, I hit the road with the specific task of testing the so-called "Riverstars" sub-hypothesis, which states essentially that, on the Pokerstars platform, the turn's favorite, meaning the best positioned player in terms of his/her equity, actually ends up winning less often than normal. The table above categorically and definitively contradicts this thesis. As a matter of fact, over a sample of 3,012 showdowns analyzed, the favorite on the turn actually wins plus splits more, and not less, often than normally, mathematically expected: instead of the expected 84.42%, the favorite actually won plus split 86.52% of times, which equates 102.49% of the expected rate (2606 times vs. 2542.84). The results are safely placed beyond any doubt and leave no room for interpretation; out of 16 categories of equities that have been analyzed, the turn's favorite won more often than expected in no less than 13 of them, and in only one of the other three cases has the recorded deficit been higher than 5%. Thus, the "Riverstars" hypothesis can definitely, and methodologically most safely, be repudiated, the recorded reality actually indicating, albeit statistically not significant, the opposite as being the truth..

However, the research has led to a collateral by-product that represents undoubtedly the most compelling evidence so far in favor of the broader "leveling the field" hypothesis from which the here analyzed sub-hypothesis is actually derived: the stronger a favorite one is on the turn, the less he/she actually wins plus splits

127

proportionally to his/her expected rate. Thus, the preliminary conclusion drawn on the basis of the previous smaller sample of hands has actually proven to be correct110.

Specifically, measured over 3,102 cases grouped in 16 ordinal categories, the coefficient of the linear correlation between on the one hand the factually recorded weighted average equity of the favorite on the turn and on the other hand the recorded win plus split rate calculated as a percentage of the corresponding expected one reaches a most sensational minus 0.821.

110

A fact that actually had become clear already after an expansion of the sample to 1,4-1,5k hands, with the vast majority of the values stopping to oscillate significantly, meaning the sample had already reached a satisfactory representativeness.

128

Formulated in the most simple terms possible for the readers not familiar with statistics: if the cards would be indeed dealt randomly by Pokerstars' software, then: ď&#x201A;ˇ in the upper graph above, the red and blue lines should overlap - completely in an ideal world, respectively to variations not higher than 5% each when adopting more realistic expectations; ď&#x201A;ˇ in the lower graph, the red line should be, within the same analogous confidence interval of [95%-105%], horizontal, and overlapping the 100% referential line.

Not only are these conditions not met in the recorded reality on the poker platform, but the values reached by the Pearson correlation coefficient, the corresponding R2, and the probability p converge towards one single and categorical conclusion: statistically significant, the cards, or, specifically for this case, the river cards on Pokerstars are dealt non-randomly, but rather anti-randomly, in a premeditated and systematical manner, with the generalized effect that, the stronger a favorite one is mathematically on the turn, the less he or she actually ends up winning plus splitting in relation to the corresponding expected rate. For any sane person on this planet, let alone a student in statistics, the case is closed, the recorded values leaving no room whatsoever for doubt, relativism, methodological contortionism, or interpretative ballet: the "field" on Pokerstars is "leveled", significantly so in statistical terms, and deliberately so in intentional terms, to the direct benefit of the underdog.

In context, the findings offer additional substance to the already partial conclusions of the previous chapters regarding the investigated hypothesis. Whether or not they can be expanded to cover the entire streets and aspects of the game on the platform requires however a broadened and deepened analysis, which is exactly the purpose of the next chapter.

129

V. A DEEPENED (RE-)TESTING OF THE LEVELING-THE-FIELD HYPOTHESIS The following starts with a brief revisiting of the previously gathered findings related to the "leveling the field" / "balancing the odds" hypothesis and then proceeds with a deepened and expanded retesting of it, in both its dimensions and in the form of its corollary regarding every tenth hand effectively played at the tables.

I chose to dedicate it an entire separate chapter of this analysis for two reasons.

Firstly, as in part already shown previously, my more than 55k hands, plus other tens of thousands either observed or played on play money, all offer an abundance of strident instances, placed beyond any reasonable doubt, of field-leveling, making this general policy one of Pokerstars' biggest problems in terms of the fairness of the game provided to its clients.

Secondly, as a direct and unavoidable consequence, over the last years it has been by far one of the two most important and frequent accusations coming from thousands of players all over the Internet, alongside the issue of the "coolers" deliberately and blatantly dealt by the platform's alleged "random" numbers generator (an issue covered in the first chapter); of the hundreds and hundreds of easily accessible screen-shots and video footages of hands played on Pokerstars and posted online by various players, more than half of them are in my estimation directly related to the hypothesis investigated here.

V.1 A summary of the findings so far In the first meaning of the "leveling the field" hypothesis (i.e. in relation to an opponent), the first revealing proof so far seems to have been provided by the case study of my pocket Queens hands. Among other things, it showed that:  despite an initial, pre-flop, average equity of 75.92%, the favorite (regardless if that was me or an opponent), actually won (plus split) a mere 63.9% of the hands that reached showdown (meaning a recorded win + split rate of only 84.22% of the expected win rate in relation to the pre-flop situation). The separately addressed extension of the analysis to cover the pocket Kings I played highlighted the same issue, with the pre-flop favorite winning 67.7% of the times, making up for only 86.9% of the expected win rate of 77.86%.  Within the subsample of pure heads-up pair vs. pair duels, the underdog turned out to win at an average rate of 25%, instead of the expected 18.5%, in the same pattern that an underdog player holding an over-card in relation to the QQ also won more often than mathematically normal (37.5% vs. 31±2%). When returning to the case of pocket KK, the recorded anomalies remained equally significant and manifested in the same direction of favoring the underdog: 23.5% (vs. 18.5% expected), respectively 29.6% (vs. 17.96%).  Additionally, the analysis of both KK and QQ hands played captured across non-pair holding opponents abnormal recorded frequencies of flopping a trip or two pairs (using both hole cards) of 1.2-3.7, respectively 2.4-2.8 times higher than normal, with three of the four figures remaining absolutely impossible to be attributed, even by absurd extrapolations, to all hands that ended before showdown.  Relevant in the context of the second chapter, most of the field-leveling assisting the underdog is operated already on the flop (and not on the turn or the river 111): in 36%, 111

Which, in the context of the third chapter led to a rejection of the possible "Riverstars" hypothesis.

130

respectively 41%, of the KK and QQ hands played to showdown, the flop brought along either a decrease of the pre-flop favorite's equity, or even a complete equities reversal to the benefit of the underdog, with the recorded anomalies also hardy attributable to hands folded pre-showdown.

In addition to the expanded QQ-KK case study, the most shocking findings have been provided, as a collateral by-product, by the previous chapter's investigation of the "Riverstars" sub-hypothesis. Thus, measured over a distinct 3k+ showdowns, the analysis showed, as statistically significant as possible, that, the stronger a favorite the player is mathematically on the turn (meaning the higher his/her equity), the lower his/her factually recorded win plus split rate calculated as a percentage of the expected rate; measured across two series of values segmented on a 16-points scale, the correlation coefficient reached a staggering minus 0.82, in what has so far been the most compelling evidence of the levelingthe-field mechanism systematically applied by Pokerstars.

An exploration of the hypothesis' second dimension (i.e. in reference to one's own hole cards, and not to an opponent) led firstly to a surprising conclusion: as shown in the third chapter, a quadruple targeted analysis of Pokerstars' flops clearly concluded that in three of the four cases (i.e. flopping a set, a flush draw, or an OESD) there definitely was no relation whatsoever between the strength of specific hole cards and the likelihood of flopping the correspondingly targeted combination. As a surprise, the fourth item investigated highlighted a clear (with R2 absolute values of 0.23 and 0.78) between the strength of a non-paired pocket hand and it's frequency of flopping at least a pair, but not in the sense expected, but to the contrary: the stronger the hand, the more often it flops at least one pair. Thus, at least in regard to the flops, the second meaning of the hypothesis has been disconfirmed.

This, however, has not been the case when analyzing the post-flop unfolding of hands within the fourth chapter. Specifically, when adopting the methodological artifice of projecting a combined factual plus counterfactual conversion rate by taking into account hands that ended on the turn without the draws having been converted yet, one discovers an extremely interesting, and spectacularly intense negative correlation of minus 0.79 between on the one hand the likelihood of a certain flopped combination to convert into a better / final combination and the recorded frequency of it actually happening. To put it simply, the more likely a draw is to hit its targeted combination on turn or river, the less it actually does so in relation to its expected frequency.

Finally, in regard to the "every tenth hand..." corollary, in addition to the partial conclusions provided by the Queens' case study and it's Kings addition (i.e. 11.6%, respectively 10.1% of the favorite's expected win rate that did not materialize), along the same series of 51,989 cash games played, there have been other three confirmatory instances out of the six types of flop-related variables analyzed: ď&#x201A;ˇ 12.2% of the open-ended and double-inside flopped straight draws (among hands effectively played and with a certain outcome) that were mathematically expected to be turned into straight did not; ď&#x201A;ˇ similarly, when aggregating both types of hands and counterfactually projecting a conversion rate to draws that were ended right before the river, it turned out that 9.49% of the flopped sets that, under random conditions, should have turned into a full house or better, did not; ď&#x201A;ˇ similarly, when applying the analytical factual plus counterfactual methodological framework to hands both played and not played, it has been discovered that 11.35% of the flopped flush draws expected to turn into flushes also actually did not convert; across

131

draws within hands effectively played by me, the corresponding frequency makes up for only 90.4% of the expected, normal, conversion rate, meaning a deficit of 9.6%.112 The following builds, in a gradually expanded scope, on these disparate and preliminary findings by separately investigating four items analyzed by comparing the favorite vs. underdog pre-flop equities with the hands' outcomes: 1) pair vs. pair duels, extended beyond the QQ and KK hands in order to cover all thirteen pocket pairs, and analyzed in two directions; 2) a revisiting of my limp-check-call experiment starting from a broader issue of methodological nature; 3) my entire series of pre-flop heads-up all ins on the poker platform; 4) the sample of almost 1k hands where I had been dealt pocket pairs and that did reach showdown, 5) a parenthesis addressing tournaments, meant to provide better context to the findings pertaining to cash games.

V.2. Pair vs. pair duels Nowadays many online poker players seem to record and often even broadcast live on various social platforms tournaments or cash games they participate in. Recently, on my Twitter newsfeed, I bumped into such an example: engaging in an all-in on pre-flop with pocket AA, and losing, the player had an immediate reaction after the river card had been dealt, a reaction that I find symptomatic for what happens on the platform: "What? Aces lose again???". The keyword here is, obviously, "again".

Let's see, firstly, if there is indeed some problem with Aces or any other particular pocket pair, by isolating, across all my 51,989 cash games played, the series of hands when I was dealt pocket pairs and encountered an opposition pair in a pure heads-up format. The table below lists, corresponding to my pocket pair type, all pure113 heads-up duels of, obviously, unequal pairs, presented however not from my perspective, but from the one of the favorite, whoever that was - me or the thousands of my opponents114. For all such situations, the average pre-flop equity of the player holding the higher pair is 81.5%, with the underdog assuming the remaining 18.5% probability to win.

Well, let's look at the table below: judging by their recorded win rate in pair vs. pair duels of 69%, instead of the expected 81.5%, Aces are definitely not the weakest pair.

They are the second weakest after Tens! Joking aside, the overall percentage of 75.2% wins is an alarming one. It specifically makes up for only 92.27% of the expected win rate, which is not only statistically significant, but also, in its difference to 100, close enough to the 10% figure implied by the "every tenth hand..." corollary. Inexplicably and worryingly, it can moreover be noticed that of the total 13 cases the initial favorite won less often than expected in no less than 11 of them, which closes any possible discussion regarding representativeness or variance. Furthermore, the table begs the question: how is it that all for all seven high pairs (i.e. AA to 99 included) involved in heads-up pair vs. pair duels, the favorite always wins less often than expected, and (statistically) significantly so?!

112

Equally true, the other three conversion rates measured did not provide any evidence to substantiate the "every tenth..." corollary, in any of the four analytic cells implied by the played - no played + certain - uncertain outcome matrix.

113 Meaning engaged pre-flop and arrived at showdown in a one vs. one format.

114 Which, from a methodological point of view, dismisses any speculation that could suggest a possible measurement bias based on my unorthodox playing style when testing things, i.e. that I might have stayed in games longer than supposed to, refusing to fold, whereas my opponents, a contrario, folded whenever it was reasonable to do so, so that the list would capture incorrectly more of those cases in which equities had been reversed to my detriment.

132

Correlating the values within the first and fifth columns of the table (pair's power rank in 6-players games and the higher pair's recorded win rate against lower pairs) offers a crystal-clear image:

A linear correlation coefficient of minus 0.69 indicates a strong and negative (inverse) relation between the two variables and is attributable to anything but randomness115. Ideally, it should be zero-point-zero. Realistically, it should be anywhere reasonably close to zero. But under no circumstances, as long as randomness is claimed, should it have such a high value. It is simply just another strident example and proof as such of how Pokerstars deliberately and methodically implements a policy of leveling the field to the detriment of the top dog and to 115

In what is probably no surprise anymore, if attributing mathematical values to the non-number cards comprising pocket pairs (i.e. A = 14, K = 13, Q = 12, J = 11), and recalculating correspondingly the values of the 13 pocket pairs, the correlation of these values' series with the recorded win plus split rates of the pairs reaches minus .713, almost equal to the coefficient calculated above in a scientifically more rigorous manner.

133

the direct benefit of the underdog: the stronger one's pocket pair is, the less it will actually cover of its expected win rate against lower pairs.

Aside from this first discovery, which reconfirms the "leveling the field" hypothesis in its second dimension (i.e. by testing it in relation to one's own hole cards), the same sample of pair vs. pair encounters offers a relevant reconfirming of also the first dimension of the same hypothesis (i.e. in relation to the opposition) when isolating the cases of flopped sets or better (from my perspective) depending on three types of opposition hole cards: a higher pocket pair; a lower pocket pair; non-paired hole cards. Normally, assuming randomness, the percentage of flopped sets should be the relatively equal for all three categories of situations, placed somewhere between 6.25 and 12.5%, depending on various factors (how many opposition hole cards are known, whether the non-paired opposition cards are maybe half-blocked by my pair, etc). Well, in the reality recorded on Pokerstars, the frequencies differ dramatically depending exactly on the opposition's hole cards.

The table below lists the cases of me flopping sets or better, across hands that reached showdown, with the three categories of opposition hole cards placed on columns and delineated non-exclusively, meaning some cases being listed in both one and another category. For instance, a flopped set of mine when holding pocket 8s and encountering at showdown both pocket Jacks and another non-paired hand of a third player is registered simultaneously in the first and the third column; a situation with me facing both a higher pocket pair and a lower one will be listed in both the first and the second column, etc.

My pocket pair AA KK QQ JJ TT 99 88 77 66 55 44 33 22 total %

Cases of flopping a set or better when encountering (at least)...: a higher pair a lower pair non-paired hole cards 0/0 4/33 5/43 1/7 3/30 4/58 2/12 0/21 7/54 2/13 1/22 3/42 3/13 0/9 5/52 2/15 0/17 5/62 4/25 0/10 12/65 3/15 0/8 2/53 1/13 1/3 11/54 4/17 2/6 4/42 2/15 1/1 10/50 1/15 1/1 9/49 5/22 0 9/61 30/182 13/161 86/685 16.48 8.07 12.55

The results once again expose Pokerstars' leveling-the-field mechanism in all its "splendor": leaving aside the third column, with figures relatively normal statistically, the likelihood of flopping a set or better turns out to be directly conditioned by the opponents' hole cards. Thus, it is twice more likely to flop a set or better as the underdog, meaning when encountering a higher pocket pair, than as the top-dog, meaning when facing a lower pocket pair: 16.48% vs. 8.07%, or a ratio of 2.04 to 1! As repeatedly stated throughout this investigation, it is as if Pokerstars' software knows exactly when and how to intervene when dealing the flop cards so that the field is "leveled" to the favor of the underdog and the detriment of the top dog. You got the lower pocket pair and are the underdog? No worries, Pokerstars will gift you a flopped set, or better, so that the bad top dog doesn't win!

134

V.3 A methodological problem and a revisiting of my limpcheck-call experiment: How much deviation can one attribute to hands folded pre-showdown?

As an idea recurrently stated, explicitly or implicitly, throughout this study, if and when confronted with brutal anomalies in terms of (non-)randomness recorded at showdowns and significant deviations from the normal values to be expected in a truly random context, Pokerstars, and for this matter, all online poker platforms, could - and not rarely actually do so - invoke the hands folded before showdown, by pinning on them - by compensation - all recorded deviations that otherwise would remain unexplainable.

Say for instance a player checks his hands that reached showdown and notices with frustration how his higher pair actually won significantly less than expected against lower pairs (and this is actually no hypothetical example, but a reality of hundreds of complaints all over the Internet). The poker platform's representative suddenly becomes a methodological expert in extrapolations and attributes all the deviation recorded at showdowns to the hands that have ended before it. The underlying logic is quite simple: say for instance the discussion is about Aces, who would have won, as in the recorded reality of the previous sub-chapter, only 69% of times instead of the expected 81.5%, when confronting other pocket pairs. The representative will simply argue that, among those pocket Aces hands that did not reach showdown (as the player's opponents folded on flop, or turn, or river), there have been plenty of cases in which opponents had been dealt lower pairs, and in which the Aces' pre-flop equity held, meaning they were about to mathematically win, but the opponent realized he was beaten and folded, before the hand could reach showdown. So, if taking into account these pre-showdown folded hands, things, on the overall, must have been OK, right?

By compensation: among showdowns, Aces won only 69% of the times, but among hands ended pre-showdown, they would have won significantly more often, so that the overall win rate should be close to the expected one.

Such an excuse is an extremely convenient one, and all the more so because, as also explained in this document, aside from tens of thousands of allegations all over the Internet, in a rigorous methodological approach, there is actually one single exhaustive, definitive, irrefutable, way to prove the poker platform's representative is wrong, and that the RNG is anything but random: a group of players pick an empty table, play a certain number (big enough) of hands, all of them and all the way to showdown. In this scenario, placed comfortably above any methodological pseudo-debate, everything would become clear: the real frequency of pair vs. pair duels in the given ring-format, the real frequency of multiple pair situations, the real frequency of "cooler"-type situations, the real conversion rates of draws, the real win rates of the favorite over the underdog for all pre-flop equities types, etc.

Guess what? It is this very option, of players in a group picking an empty table and playing together that is not available to clients, neither on Pokerstars, nor on any other online poker platform that I know of, and neither on play-, nor on real-money. If provided, such an option might actually signal the end of the entire online poker industry, as it would put an end to all claims of randomness made by the poker sites.

However, this methodological deadlock is actually not as implacable as it may seem at first glance, which also means that poker representatives' convenient excuse cannot hold forever. Thus, as a first observation, let us recall that among the previous findings of this study, there have been multiple instances in which, probably being driven by Gordon Gekko's famous "Greed is good!", Pokerstars exaggerated to such a degree when tampering with the supposed random dealing of the cards, that the anomalies recorded at showdowns remain 135

unexplainable and unjustifiable even when trying to pin them on all those hands that did not reach showdown. For instance116:  me flopping a set four times in a row with the same pocket pair would have required me to have played, for such an event to be statistically "normal", 50k thousands of pocket pair hands played, whereas, in reality, I've been through such a situation a little over 3k times (meaning all cases of being dealt pocket pairs, leaving aside the fact that only a fraction of them made it to, at least!, the flop);  the recorded event of me flopping directly the royal flush is expected, under, of course, randomness-governed circumstances, to happen once in every 650k hands played at a poker table, whereas the total hands dealt to me in cash games has been less than 52k;  a flop situation of full house vs. quads would have required 562k hands to have been played;  an event such as one of my opponents flopping directly a full house and still lose to a better full made on the turn plus river, starting from non-paired hole cards, is expected to happen once in every 2.48 million hands played;  the event of me holding KK, and simultaneously bumping, at a 6-players table, into pocket Aces and Jacks and flopping quads would have required me to have played 85.1 million hands;  item-by-item, the frequency of my opponents flopping trips and two pairs when I held pocket KK or QQ also remains significantly higher than normal even if absurdly assuming that, simultaneously, in all those hands that ended pre-showdown, my opponents had non-paired hands, and they did not flop two pairs or trips, and they all folded before showdown;  or, as shown in the previous chapter, the frequency of flopping simultaneously two pairs or two pairs vs. a set also remains in both cases 1.41 times higher than normal even if absurdly taking into account all hands folded before showdown and assuming that in none of them did an opponent flop the targeted combination; etc.

In addition to this set of findings, the following further undermines the usability of the poker sites' frequent excuse in two directions: the first picks up my set of KK holding hands in order to expose the certain limits of the pseudo-extrapolation method; the second one is a revisiting, as a third distinct item verified in this chapter, of my already described limpcheck-call experiment undertaken for all hands where I have been dealt pocket pairs, an experiment conceived exactly to overcome the here discussed methodological obstacle.

V.3.1 A mini-case study: my pocket KK hands Having to pick one of my two most frequently dealt pocket pair as a sample, Deuces and Kings, I opted for the latter, considering the substantially higher fraction of hands that reached showdown, which allows more reliable observations.

Methodologically, of the total 259 KK hands, the 120 that didn't even make it to the flop, having ended pre-flop, obviously cannot be taken into account, since it remains completely uncertain what hole cards my opponents have been dealt, what the board would have looked like, and what the final outcomes of the hands might have been. This leaves us with 139 hands, out of which 45 have ended before showdown (folded by me two times, folded by my opponents 43 times), while 94 have reached showdown. For the latter category, the table below displays: my opponent's hole cards and, within the brackets underneath, my pre-flop equity; the hand's outcome for me: win (green shading), loss (red), or split pot (blue).

116

For more such examples, see the third chapter.

136

Table. A full inventory of my KK hands that made it to showdown: No. of cases KK situation: Vs. higher pair

Vs. lower pair

As overpair

vs. 1 over-card As Hi-blocking pair As Lo-blocking pair Vs. equal pair Other cases (>2 players)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

AA (17) AA AA AA AA AA (18) (18) (18) (18) (18) 99 77 QQ 77 77 99 QQ JJ JJ QQ 99 JJ 66 77 44 (81) (80) (82) (80) (80) (81) (82) (82) (81) (81) (80) (82) (80) (80) (81) 99 99 66 77 99 44 TT (81) (80) (80) (80) (81) (82) (80) J8o QJo J6o T6o JTs T8o J9o J5o Q5o JTs QJo Q9o Q3s QJo 76s (84) (86) (88) (86) (81) (83) (85) (87) (88) (80) (86) (86) (85) (86) (77) QJs T9s 98s T9s J9o 53o J7o (83) (78) (78) (79) (85) (84) (86) T2o (89) A4s AQs A5s A9s A9o AQo AJo AJo ATo A4s A9s AJs AQo AQs (67) (68) (66) (68) (72) (72) (71) (71) (71) (67) (68) (68) (72) (68) ATo A6o A9s A8s A8s A4s A5s AJs AQo AJo ATo (71) (71) (68) (68) (68) (67) (66) (68) (72) (71) (71) K5o KTo KQo KTs (93) (91) (92) (85) AKo (70) AKs (66) KK (50) KK (55) > ATs (31) & KQs (14) KK (57) > A5s (31) & QJs (13) KK (18) > AA (67) & JJ (15) KK (62) & JTs (19) < 55 (19)

137

16

17

18

19

20

21

JJ (82)

QQ (83)

88 (81)

JJ (82)

JJ (82)

TT (81)

87o (82)

The key-question in relation to these data is: what must have happened in the cluster of hands that were folded before showdown, in order for the overall mathematics of the showdown-proven outcomes to be correct in terms of pre-flop equities? As an example meant to facilitate the understanding, let's isolate for instance those showdowns in which my pocket Kings discovered they had bumped into a lower pair in the pocket of my opponents.

Of the total 28 cases that fall within this category, my Kings, despite being 81% favorite to win on the average117, have won only 21 times, meaning 75%. Or, to put it the other way around, my lower pair-holding opponent won 7 out of 28 times, meaning 25% of the duels, despite having an average equity of only 19%.

In order for the overall mathematics to be correct, this can only, or should only, mean one thing: somewhere along the way, among those hands that did not get to showdown, there must have been a certain number of cases in which my KK higher pair had its advantage materialized, meaning I would have won, but, as said, my opponent folded before showdown, so I was robbed of the opportunity to actually see my win confirmed.

In this logic, the math is quite simple. If, in the given situation category, my opponent's recorded 7 wins represents the normal, correct, 19% win-rate in the given situation, than this means that I should have played a total of 100*7/19 = 36.84 times with my Kings as a higher pair, in order to also achieve my normal win rate.

7 wins ............... 19% x ...................... 100% Out of the overall result of 36.84, it was 28 times that me and my opponent did reach showdown, so that, by difference, among the hands that were folded (let me remind you, there have been a total 45 such hands), there must have been 36.84 - 28 = 8.84 hands that I was in the same position, and my equity advantage materialized, and I would have won, but only for my opponent to be inspired enough to fold somewhere along the way before showdown.

The same goes for all situations listed in the table above. Another brief example before combining all the figures into an integrated image and drawing the necessary conclusions: the cases in which my Kings faced an over-card. If my opponents' recorded 11 wins (out of 25 instances) account for his average pre-flop equity of 30.8%, that means, following the same method, that the total number of such KK vs. one over-card instances within hands both having and having not reached showdown, must have been 100*11/30.8 = 35.71. Subtracting the 25 cases that reached showdown, the result is a 10.71 necessary hands in which my equity advantage materialized, but my opponent folded before reaching showdown.

As for the four situations in which three players, me included, reached showdown, let's clarify the math using as an example the case in which both I (with KK, 62% equity) and an opponent with JTs (19% equity) lost to a third player with pocket fives and 19% equity.

Well, if this victory accounts for 19%, then somewhere along the way there must have been a total 5.26 cases, out of which, subtracting the single recorded case at showdown, there must have been 4.26 cases in which the pocket five-holder (or any holder of a lower pair for that matter) did not win and also folded his hand before showdown.

A full inventory of these hands necessary to both a.) have been played and b.) have had a specific outcome is provided by the next table.

117

I.e. the mathematical average of all equities within brackets in the isolated category of cases. This recorded, actual, figure is satisfactorily close to the general 81.5% equity mathematically associated with a higher pair in general.

138

My KK situation:

W

L

S

Necessary additional number of hands to have been played 0.08 8.84 18.71 10.7 0.43 3.12 0.05* 4.55

With the necessary outcome (in regard to my Kings):

Vs. higher pair 1 5 0 W Vs. lower pair 21 7 0 W As an overpair** 16 7 1 W Vs. 1 overcard 14 11 0 W As hi-blocking pair 4 0 0 L As low-blocking pair 1 1 0 W Vs. equal pair 0 0 1 S Vs. a higher and a lower 1 0 0 L pair Vs. a lower pair vs. 2 0 1 0 4.26 3.26W, 1L undercards Vs. 1 over-card vs. hi1 0 0 0.82 L blocked cards Vs. 1 over-card vs. 2 1 0 0 0.75 L undercards 60 32 2 total 52.32 44.7 W, 7.6 L, 2 S W = win; L = loss; S = split pot * based on the normal frequency of encountering an equal pocket pair in a 6-players game ** based on the number of 7 losses Thus, in order for the overall mathematics of pre-flop equities to be respected, among the hands that didn't reach showdown, I should have played 52.32 hands (out of which I should have won 44.7, lost 7.6, respectively split 0.05). And this is absurd, since the total combined number of my KK hands that were played pre-flop, but did not reach showdown, has been only 45118. It means I should have necessarily played a number of hands that exceeds the number of hands that have actually been played! Hence the conclusion: the mathematics of Pokerstars' "Random" Numbers Generator has not been correct. Or, to formulate it differently: normal, mathematically correct, pre-flop equities are not respected by Pokerstars' algorithm, with the underdog being constantly and deliberately helped within a generalized logic, and subsequent business strategy, of "leveling the field" or "balancing the odds". In context, this conclusion only reinforces the same findings discovered when investigating the series of pre-flop heads-up all-ins I engaged into, with the underdog almost universally winning significantly above their normal, expected, win-rate.

Surely, one could argue that the pocket Kings sample of cases analyzed is too small and that, somehow, these statistical anomalies would have been compensated within the larger sample of all pocket pairs that I played. To put it more simply: the KK sample could have been the "exceptional" case within the series of pocket payers that I have been dealt. My answer to this objection is a blunt "False!" Naturally, an irrefutable demonstration based on the entire series of pocket pairs that I have been dealt is impossible to provide since, logically, the lower the pair dealt, the higher the number of folds and, correspondingly, the lower the number of hands that actually reached showdown, so that the overall number of pocket pair hands that didn't reach showdowns is high enough to argue that anything could have happened.

However, even within the subsample of pocket pairs' hands that reached showdown there are a some elements relevant enough and numerically sufficient to refute 118

As mentioned, 43 of them were folded by my opponents, and the remaining 2 by me.

139

such an objection and reinforce the general conclusion stating that, on the Pokerstars platform, the underdog is being constantly and systematically helped to win significantly above his normal, correct, win-rate. Let's take a look at two suggestive examples.

The first one draws back to the table above, specifically to the KK hand that, as an over-pair, encountered T2 off-suit, and ended with the pot being split! The hand was played on May 19, at 14:29:03 EET. Here are its pre-flop and showdown captures by the HM2 software:

Shocking, but, still, mathematically possible, right? Well, allow me two clarifications. Firstly: in the table above, within the lines of the over-pair situation, the calculations are based on the number of cases (7) in which the underdog won.

140

7...... 16.39% x ...... 100% x = 42.71. Minus 24 hands actually played => 18.7 hands that must have been played with me as a winner.

Secondly, in the average equity calculation I have used the 89% indicated by HM2 (see the upper photo above). However, this figure is actually an aggregated one, combining both the % of (pure) winning cases and the % of pot splitting cases. If one uses alternatively a more accurate pot odds calculator, that differentiates between wins and splits, such as the one provided by the "CardsChat" website119, it turns out that the probability, given the two player's hole cards, to split the pot in a tie, is... 0.55%!

Rest assured, it's not an error. Two other online calculators that I used alternatively (specifically CardsPlayer120 and Pokernews.com121) both indicate an even lower probability, namely 0.50%. Well, even if choosing the 0.55 percentage, if this single case of my KK splitting the pot with T2o accounts for that particular percentage, this means that, in order for the overall mathematics to be respected, I did or should have played a total of 1*100/0.55 = a staggering 181.82 hands of KK that encountered two undercards, out of which I won / should have won 83.78% = roughly 152 times, lost / should have lost about 28.8 times, plus the single, materialized case in which I split the pot. Of course, this is also absurd; even when relating the 181.82 figure to the total 259 hands of KK I was dealt (and not the mere 45 played but ended before showdown), the difference of 259 - 181.82 = 77.18 does not leave enough room for the normal percentage of cases in which my Kings should have encountered not 2 undercards, but lower pairs + Aces as higher pair, one over-card + one under-card, and high + low-blocked cards. I should have hit these opposition hands, in their aggregated number, way too seldom for the mathematics related to the T2o case to be correct.

Anyhow, if, as said, one would argue that the general KK-hands anomaly would have somehow been compensated within the aggregated cluster of all the other pocket pairs, allow me to provide another example, this time referring to the pocket Aces I have been dealt, specifically to the hand played on May 22, at 11:50 EET:

119

See https://www.cardschat.com/poker-odds-calculator.php (accessed June 10, 2017, at 10:13 EET) http://www.cardplayer.com/poker-tools/odds-calculator/texas-holdem (accessed June 10, 2017, at 10:33 EET) 121 https://ro.pokernews.com/poker-tools/poker-odds-calculator.htm (accessed June 10, 2017, at 10:57 EET) 120

141

So, basically the same tie result, this time, however, against a lower pair, starting from the following pre-flop equities122: me winning: 81.77 - 81.95%; my opponent winning: 17.64 17.51%; tie: 0.54 - 0.59%. Well, if this recorded tie accounts for the average 0.55% probability, that means a necessary total number of 1*100/0.55 = 181.82 cases in which my Aces did or should have encountered Deuces, with another result than a tie. Which, again, is absurd, since I was dealt Aces only 232 times, with 101 hands ended pre-flop (i.e. not played), 75 hands reaching showdown and the remaining 56 having been folded before showdown. A figure of 181.82 such cases would mean 74.8% of the total 232 AA hands I have been dealt, which, again, would be absurd, considering the normal cumulated probabilities of hitting some opposition hole cards other than a lower pair (i.e. two undercards or high-blocked cards). Statistically normal, the probability of encountering an(y) opposition pair when holding a pair is merely 26.45%, and not 74.8%.

And these, as a conclusion, are the certain, insurmountable limits to the type of speculative argument used generally by online poker platforms: those multiple cases, wherever one looks in which the spectacular deviations from anything normal that have been 122

Provided by the same online odds calculators.

142

recorded at showdowns cannot mathematically be explained even if trying to pin them on all those hands that have ended before reaching showdown.

V.3.2 A revisiting of my limp-check-call experiment Let us briefly recall my experiment on the poker platform, described in the second chapter of this study: for an series of successive 2,573 hands played in the timeframe May 26, 17:40 - May 27, 14:48 EET, at zoom cash games, 6-players rings, 1/2c stakes, each time I was dealt a pocket pair, I (open-) limped and simply check-called on all streets, with the sole purpose of dragging my opponent to showdown, in order to test two things: firstly, the recorded frequency of pair vs. pair encounters; secondly, the actual, measured, general win rate of pocket pairs in comparison to their mathematical, expected, win rate in the given game circumstances. Simply open-limping-check-calling was, as explained, meant to ensure that none of my opponents, whenever I held pocket pairs, was folding before showdown.

Whereas the first exploratory direction led to a satisfactorily reliable conclusion123, the second one is presented here, with the key question at this point being: do pairs at least win at their normal win rate, or is the software leveling the field by increasing the underdog's chances? Isolating all the 123 cases of pure and finalized heads-ups (i.e. from pre-flop to showdown in a one vs. one format), so that we can operate with standardized expected win rates124, the results, in light of the previous findings, are anything, but surprising:

My pocket pair in relation to opponent's hole cards Higher pair Overpair Pair vs. 1 overcard Pair vs. 2 overcards Low-blocking pair High-blocking pair total

Recorded wins No.

%

17 out of 23 73.9 16 out of 19 84.2 13 out of 28 46.4 (+ 1 split) (or 50.0) 21 out of 47 44.7 2 out of 4 50.0 1 out of 1 100% 70 out of 122 57.4 (or 71/123) (or 57.7)

123

Normal, expected win+split rate (%) 81.5 81.5 69.5 54.4 66.6 93.0 67.9

See chapter II.4 of this study.

For a detailed discussion of these expected win rates depending on various types of situations, see below the subchapter dealing with the pre-flop heads-up all ins.

124

143

Glimpses of the leveling-the-field mechanism were already recognizable in the first, hand-forhand arranged, table: notice how, for instance, I lose with pocket Aces against inferior pocket Queens, but, in turn, I manage to win with KK over opposition Aces? Or how I lose, despite being a 81.5% favorite, with pocket 9s against 8s, only to win later with 3s against Aces?! Or how, as a consequence of this turning upside down of equities, on the overall of pair vs. pair duels, the favorite manages to win only 73.9% of the times, instead of 81.5%, meaning only 90.67% of his/her expected, mathematically normal, win rate? This means in turn a 9.33% deficit, of "vanished" wins, a percentage remarkably close to the "every tenth hand..." corollary.

This is the meaning in which I used the metaphor of mixing boiling with freezing water and achieving a normal overall temperature, so that things may seem completely normal when looking at things superficially and from the outside in aggregated, average, terms. If one player checks his recorded win rate in a given type of duel situation, it should be expected to be reasonably close to the normal, mathematically, correct one. But this has actually been achieved, as the final overall result, by a premeditated and sophisticated mechanism of "mutually compensating exceptions", meaning one wins more often than normal when being the underdog and less often than normal when being the top dog, with the final average result looking absolutely normal. The key here in exposing the mechanism is to switch one's perspective when measuring outcomes, from the personal one to the favorite's one, regardless who that was.

Here is the overall, concluding, perspective on the basis of this experiment: judging by my (weighted, of course125) average equity of 67.91%, of the total 123 cases listed above in which I held pocket pairs and got to showdown, I should have won 67.91*123/100 = 83.53 of these hands. I actually won only 70 + 1 (split) of them. This means that no less than [(83.53-71)]*100/83.53 = 15.0% of my normal, mathematically correct, win rate has simply been denied by the poker platform's algorithm.126 I find the figure astonishing; it means that out of roughly every 6.6 hands that I should mathematically have won, one was simply stolen from me. Or, from an alternative perspective related to the "every tenth hand" corollary, it means that [(83.53-71)]*100/123 = 10.19% of the total hands played and listed above had their normal, expected, outcome "hijacked" by the mechanics of Pokerstars' "random" numbers generator, all of them to my detriment, meaning I should, mathematically, have won them, but I didn't.

For now, before continuing with the next steps of the analysis, let us offer a summarizing view: so far, within this chapter's reminders and newly engaged analyses, we ran into no less than seven figures situated spectacularly close to the "every tenth..." corollary's value, all of them manifested not only as significant deviations from the expected values, but also, in their entirety, to the direct benefit of the initial, pre-flop, underdog: 10.1%, 11.65%, 12.2%, 9.49%, 11.35%, 9.6%, and 10.19%.

125

That is, calculated as: [(81.5*23+81.5*19+69.5*29+54.4*47+66.6*4+93*1)]*100/(23+19+29+47+4+1).

Interestingly, but hardly a surprise, the leveling of the field mechanism works, obviously, both ways, meaning independently on whether one is the favorite or the underdog. Specifically, within the same experiment, there have been multiple instances where i held non-paired hole cards and bumped into opposition pairs. Well, despite being the underdog, I managed to for instance beat opposition's pocket 5s with T8o, 9s with AQo (twice), Queens with AKo, or 7s with A9s, and split a pot, when holding T8o, with an opponent having pocket 8s, etc.

126

144

V.4 Pre-flop heads-up all-ins One of the items invoked quite often in online discussions as an example of the leveling-the-field mechanisms practiced Pokerstars is the pre-flop heads-up all-in situation, the accusation being that, under the effect of the platform's RNG, the pre-flop underdog ends up winning significantly more often that mathematically expected to. Admittedly, I noticed the same apparent anomaly in numerous instances already when I was still playing on the platform on play money, which became an even bigger source of frustration when I made use of Pokerstars' option to have the equities displayed on the screen, thus knowing in real time not only that I was the favorite, but also my exact probability to win! Among such situations that I recall at this point, I remember for instance: - having lost with pocket Kings three times in a row against opponents placed pre-flop on, roughly, 18.5%, 19.0%, and 6.5% equities127, meaning a conditioned probability of such a chain of events to happen of 0.023%; - or losing with Aces another three times in a row; - or losing 9 consecutive all ins, regardless whether I was the favorite or not, etc.

But let's leave both (what might still be) urban legend and play-money games aside and focus on those heads-up all-ins that I engaged already pre-flop on the Pokerstars platform within the series of 51,989 cash games played. Specifically, there have been 108 such pre-flop, pure heads-up, all-ins, stretching temporally from April 11 to May 28, 2017, from within games played at cash tables, in a zoom format, at 6-players rings, at the lowest four stakes levels. Of the total 108 all-ins, 2 started on relatively equal equities and eventually ended in a split pot128. The following analyzes the other 106.

Not really a surprise anymore in light of the previous findings, of the 55 cases in which I started as the favorite, I won only 30 = 54.5% (and lost the remaining 45.5%), whereas of the 51 cases in which I started as the underdog, I won a bare 10, meaning 19.6% (losing the other 80.4%). Thus, to put it synthetically:  as a pre-flop favorite, I won 54.5% of the times; however, if my opponent, regardless who the person was, was the favorite, he/she/it129 won in 80.4% of the cases, meaning 1.47 times more often than me;  when being the underdog, I won 19.6% of the times; however, if my opponent was the underdog, he/she/it wins 45.5% of the times, meaning 2.32 times more often than me!

Since, however, the above aggregated perspective can, and, legitimately enough, should be criticized as forcefully combining the all-ins into a brute binomial distribution, by ignoring the specific equities of each hand played, the following differentiates among various hole cards categories and corresponding equity situations and restructures the 106 all-ins series accordingly: A. Pair vs. pair (top pair is mathematically 81.5±3% favorite to win); B. Pair vs. 2 over-cards (pair is 54±2.5% favorite); C. Pair vs. 1 over-card (pair is 69±2% favorite to win) D. High-blocking pair (e.g. AA vs. A7, QQ vs. QJ) (pair is 91±2.5% favorite) E. Low-blocking pair (e.g. QQ vs. AQ, 88 vs. J8) (pair is 66.6±2% favorite) F. Over-pair (ex. QQ vs. 96, KK vs. QJ) (pair is 81.5±1% favorite to win) G. a.) "adjacency" (e.g. AQ vs. Q8, KJ vs. J9) OR b.) equal low card (e.g. A8 vs. J8, Q9 vs. T9) (the hand containing the higher card is a 72±4% favorite to win) H. better kicker (e.g. K8 vs. K5, QT vs. Q9) - the better kicker hand is a 70±3.5% favorite to win) I. Two over-cards (e.g. AQ vs. J7, K8 vs. 76) (over-cards are 62±3% favorite to win) 127

I even wrote it down on paper, anxious to immediately calculate the probability of such an "accident" to happen.

128 19th of May, 17:51 EET (Aces vs. Aces) and 20th of May, 19:46 EET (AKo vs. AKs).

129 We cannot rule out bots opponents, can we?!

145

J. Other situations (residual category): a.) "zipper" (e.g. AJ vs. K8, K9 vs. J4) (the hand with the higher card is 62.5±2% favorite to win) b.) "pliers" (e.g. A8 vs. KT, Q9 vs. JT) (the higher card hand is 57.5±2.5% favorite to win) c.) equal value cards, suited vs. off-suit (suit is 52.5% favorite) Making use of this methodological instrument of classification, let us take an exhaustive look at all the 106 pre-flop heads-up all-ins in a descending chronological descending order, with the 4th column from the left to the right specifying the all-in type situation in accordance with the classification, and the sixth and eighth indicating the opponent's equities as displayed, for each hand and on all four streets, by the HM2 software.: (Legend: - green shading: favorite won; red shading: underdog won - [in "Type" column] .1 = favorite; .2 = underdog)

Crt. No. 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

Date

Time (EET)

Type

11.4 11.4 11.4 11.4 11.4 11.4 12.4 12.4 13.4 13.4 13.4 13.4 13.4 15.4 16.4 16.4 17.4 17.4 17.4 17.4 18.4 18.4 18.4 19.4 19.4 20.4 20.4 22.4 22.4 22.4

11:12 11:22 12:30 13:51 15:40 17:42 11:31 16:52 12:57 13:46 16:38 17:52 20:16 15:27 12:07 14:44 11:36 13:06 13:38 16:45 00:15 13:02 16:09 11:48 13:41 11:05 13:47 09:57 12:47 13:01

H.2.

H.2.

A.2.

C.2.

A.2.

A.1.

H.2.

A.2.

H.2.

C.1.

C.2.

A.1.

C.2.

C.1.

E.1.

F.1.

A.2.

C.1.

D.1.

B.1.

A.2.

J.c.2.

A.2.

G.b.2.

A.2.

D.1.

A.1.

A.2.

E.1.

A.1.

Me Hole Equity (%) cards Ad Jd 29.6 As 9d 25.0 Qs Qh 18.8 As 8s 31.6 Kh Kc 18.1 Kc Kh 82.2 Ah Jh 30.6 7h 7d 18.2 Ah Qh 30.3 Qh Qd 71.0 As Ts 32.3 Jh Js 80.9 Kh Qh 47.3 Kc Kh 72.0 Kd Kh 69.8 Kh Ks 77.0 Ks Kh 18.1 Kh Ks 68.1 As Ah 93.5 4s 4h 51.0 Qs Qc 18.5 Qd Jd 44.3 Kc Ks 18.1 Kh Qh 28.6 Ts Tc 19.3 Ah Ad 93.5 As Ah 81.5 4h 4s 18.0 9s 9d 69.3 As Ad 81.9 146

My opponent Hole Equity (%) cards As Qd Ac Qc Kc Kd Kd Ks Ac As Jh Jc Ac Ks 8h 8s As Kd Ah 2s Kc Ks 7h 7d 9h 9s Ac Qh Ah Kc 7d 6d Ad Ah As Js Ac Kh Qh Jc Ah As Ks Tc As Ad Ac Qc As Ad As Kh Qc Qs Ah As Ks 9h Ks Kh

70.4 75.0 81.2 68.4 81.9 17.8 69.4 81.8 69.7 29.0 67.7 19.1 52.7 28.0 30.2 23.0 81.9 31.9 6.5 49.0 81.5 55.7 81.9 71.4 80.7 6.5 18.5 82.0 30.7 18.1

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

22.4 22.4 26.4 27.4 29.4 2.5 2.5 2.5 3.5 3.5 6.5 6.5 7.5 8.5 8.5 9.5 9.5 10.5 12.5 13.5 13.5 13.5 13.5 13.5 13.5 14.5 14.5 14.5 14.5 15.5 16.5 16.5 16.5 16.5 16.5 18.5 18.5 19.5 19.5 19.5 19.5 19.5 20.5 20.5 20.5 20.5

13:09 15:43 11:38 09:28 09:29 14:21 14:46 15:50 13:14 14:52 10:49 11:49 12:19 13:29 14:45 16:48 16:58 13:47 12:59 14:11 14:18 15:04 16:43 16:45 19:13 12:57 13:54 17:36 17:43 11:19 11:19 11:35 11:45 11:56 12:31 09:23 11:37 14:02 16:47 19:10 19:12 20:20 10:06 11:25 11:58 12:07

A.1.

I.1.

A.1.

B.1.

C.1.

D.2.

A.2.

A.1.

A.1.

A.1.

B.2.

A.1.

B.1.

B.1.

B.2.

F.2.

D.2.

J.b.2.

F.1.

J.a.1.

B.1.

E.1.

A.2.

B.2.

G.a.1.

A.1.

E.2.

A.2.

B.1.

J.a.2.

D.1.

B.1.

A.1.

B.1.

A.1.

B.1.

A.1.

A.2.

A.2.

A.2.

D.2.

B.2.

C.1.

D.2.

H.2.

G.b.2.

Ah Ad Td 9d Kc Kd 5s 5c Jd Jh Ah Jh Jd Jh Ac As Qc Qd Ah Ac Ad Kh Ac Ad Jc Jd 4s 4d Ah Jh 7s 2s As Kh Kh Jh Kh Kd Ah Qc Qs Qh Kc Ks Js Jc Ac Ks Ah Kh Kc Ks Ad Kd Th Ts 4s 4d Jc 8c Ah Ac 8d 8c Qh Qs 2c 2d Qd Qs 8h 8c As Ah 9h 9c 8s 8d Qd Qs Ad 7h As 9s Kc Ks As Kc As Tc Qh Jh 147

81.9 70.1 82.6 56.3 70.3 12.9 18.5 81.9 81.9 82.6 43.1 81.5 56.9 54.7 47.1 15.2 6.5 43.5 87.1 58.6 53.8 65.9 18.5 43.1 77.2 81.2 34.1 18.9 51.8 41.8 93.6 55.2 82.0 52.6 82.0 55.3 81.5 19.8 19.8 17.8 7.2 45.9 68.3 6.5 26.0 30.2

Kc Kh 8h 4d Qc Qd Ac 6h Ad 5h Ac Ad Kc Kh Ks Kh 9d 9c Kh Kc Jd Jc Qs Qd Ad Ks Ah 7d 9s 9h Ts Td Ac Ah Ac 4c Jd 5c Ks Js Ad Kd Ad Kd Kh Kc Jc Jd Kd 9c Qh Qd Kc Kh Ks Kd Jc 6c Ad 9c As Kd Ad Ks Js Jc Ac Ks Jc Js Kh 9c Qd Qh Kd Ks Qc Qh As Ad As Ac 8c 8d Ac 9c Ac Ah Ac Kh Ah Js

18.1 29.9 17.4 43.7 29.7 87.1 81.5 18.1 18.1 17.4 56.9 18.5 43.1 45.3 52.9 84.8 93.5 56.5 12.9 41.4 46.2 34.1 81.5 56.9 22.8 18.8 65.9 81.1 48.2 58.2 6.4 44.8 18.0 47.4 18.0 44.7 18.5 80.2 80.2 82.2 92.8 54.1 31.7 93.5 74.0 69.8

77.

78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

105.

106.

20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 21.5 22.5 23.5 25.5 25.5 25.5 25.5 25.5 26.5 26.5 26.5 26.5 26.5 26.5 26.5 26.5 27.5 27.5 27.5 27.5 28.5

12:31 14:06 15:29 15:54 17:30 19:20 19:23 19:24 20:39 12:04 10:33 12:38 11:35 13:09 13:26 13:26 16:45 11:14 11:55 15:10 17:43 19:01 22:05 22:16 23:38 14:11 14:42 18:29 18:58 16:46

E.2.

C.2.

B.1.

B.1.

I.1.

D.2.

C.1.

A.2.

J.b.1.

A.1.

E.1.

A.1.

G.a.1.

D.2.

J.b.2.

J.b.2.

A.1.

A.1.

C.1.

A.2.

B.1.

H.2.

G.b.2.

I.1.

B.2.

B.1.

D.2.

A.2.

D.2.

A.1

As Ks Jc 9c 5d 5h 9d 9h Ac Kh Ah Jh 7d 7c 8h 8c Ad Ts Ad Ac Qc Qh As Ac Ad Jc Ah Jc Ks Jc Qh Td Ah Ad Kc Ks 4c 4d Qh Qd 6s 6h Ac Td 5h 3 h Ac Tc Ad Kc 6c 6s Ac Kd Jc Js Ah 8h Ah As

34.1 32.3 53.5 52.4 65.3 12.9 70.6 19.5 59.8 81.9 68.5 81.3 71.5 8.3 42.2 42.2 82.6 80.2 69.7 18.5 52.2 26.7 32.1 69.2 42.8 52.5 6.5 18.5 12.0 81.9

Kd Kh Td Tc Kc Qs Ac Kc Jd 8c As Ad Ad 6s Ad Ah Kc Qh Kc Kh Kh Qd Kh Kd Js Tc Ad As Ac 8d Ad 6d Kd Kh 9h 9d 6s 3h As Ah Ah Qh Ah Kc Ad 3c 8c 3c Qh Qs Ac Kc Ah Ad Kd Ks Ac As Kc Ks

65.9 67.7 46.5 47.6 34.7 87.1 29.4 80.5 40.2 18.1 31.5 18.7 28.5 91.7 57.8 57.8 17.4 19.8 30.3 81.5 47.8 73.3 67.9 30.8 57.2 47.5 93.5 81.5 88.0 18.1

The findings are staggering: o of the 19 cases in which I held a higher pair and encountered a lower pair, I won 13, meaning 68.4% of the times, instead of the normal, expected, 81.7% (calculated as the equity average of the category's cases)!; o the other way around, when bumping into a higher pair, I won a single time in 17 such duels, meaning 5.9%, instead of the normal 18.6%! o with a pocket pair vs. the over-cards of an opponent, I won 5 out of 13 times, meaning 38.5%, instead of the mathematically normal 53.7%, o on the opposite end, I won only 1 of the 5 cases, meaning 20% instead of 44.4%; o of the 7 cases in which I held a pair and confronted an opponent with one over-card, I won also just one time, meaning 14.3%, instead of the normal 70%; etc.

148

Even if some Pokerstars' representative would try some methodological contortionism by invoking the supposedly small size of such subsamples, the unavoidable question remains: how come that in the overwhelming majority of these subsamples, the outcome is to my direct detriment? If the problem would only lie with sample sizes, then, in any normal world (read: any genuinely randomness context), some variables would be in my favor, and some to my detriment! The above data don't point, even remotely, to such a distribution.

Here is a graphical representation (from the perspective of my hole cards, with the number of each category's cases within brackets) of the above data when filtering those pre-flop all-in categories that gather each at least 5 cases:

Pre-flop heads-up all-ins

90

left columns : % won

81.7

right columns: % should have won (mathematically)

80

70

70.0

68.4

60

53.7 50

44.4 38.5

40

30.0 30

25.0 18.6

20

28.0 25.0

20.0 16.7

14.3 10

9.1 5.9

0

higher pair (19) lower pair (17) pair vs 2 Ocs (13) 2 Ocs vs. pair (5) pair vs 1 OC (7) 1 OC vs pair (4)

149

vs hi-blocking pair (8)

vs better kicker (6)

Astonishingly, Pokerstars' RNG somehow managed to "randomly" develop such a series of scenarios that I have been favored in one single situation-type: when encountering a high-blocking pair. In all the other main categories, my actual win rate has been below, and significantly so, the normal, expected, probabilistic rate.

Or, to put it more accurately in mathematical terms as a overall conclusion: of the total 106 pre-flop heads-up all-ins I engaged in, at my weighted average equity of 49.46%, I should have won 52.43 of the 106 duels. Well, as it turns out, I only won 40. This means that [(52.43 - 40) / 52.43] * 100 = 23.7% of my normal, expected, mathematically correct, win rate somehow "evaporated" under Pokerstars' (truly) "random" numbers generator's action. Or, formulating it from another angle: [(52.4 - 40) / 106] * 100 = 11.73% of the preflop heads-up all-ins have had an outcome contrary to the normal, mathematically correct, one, that would be associated with a genuinely random environment. And, thus, we stumble again upon a figure strikingly close to the 10% stated by the "every tenth hand..." corollary130.

Thus, confirming our hypothesis and potentially adding fuel to thousands of complaints all over the Internet about Pokerstars being "rigged", this general flaw of the platform software's pre-flop heads-up all-in mechanics does in no way seem to be an accident, a programming error, but a deliberate, consistent, and systematically applied algorithm. Not only do the pre-flop all-ins outcomes blatantly defy any logic, any probability, or any discussion on statistical significance, but - and here comes the shocker! - Pokerstars' probabilistic regime of all-ins engaged on flop and also on turn is as accurate and precise as a Swiss clock! The 200+ cases in which I went all-in on either flop or turn do not indicate any statistical anomaly, quite the contrary: when for instance I was a roughly 95% favorite, of 20 cases I played, I won 19; if, say, I was a 66% favorite, I approximately won 20 out of 30 times; etc. The fact that on pre-flop, all probabilities are way off any "normality" line, while on flop and turn they all of a sudden apply rigorously to the second decimal, and all this over hundreds and hundreds of cases verified, cannot be attributed to any "glitch", "error" or "accident". It's simply a premeditated policy of leveling the field or balancing the odds, it's simply how Pokerstars deliberately chose to build its software, despite all their representatives' claims of an alleged "random" numbers generator being applied in the games.

130

As a parenthesis, things get all the more interesting when isolating within the above 106 all-ins series the cases in which I held pocket Kings or Queens. Mathematically, if things are genuinely random, then in any heads-up duel, Kings are supposed to win 82.4% of times, while Queens in 79.9% of the cases. Well, not on Pokerstars, or at least not if it's me playing: 14 times I went all-in in pre-flop with Kings; I won 7 times (cases No. 6, 14, 15, 16, 33, 49, and 56 in the table above) and lost an equal 7 times (and only 3 of these against Aces)130 - so, a win rate of only 50% (Cases No. 5, 18, 23, 48, 52, 73, and 94 in the table above). Another ten times when I had the unfortunate idea to go all-in in pre-flop with pocket Queens - out of which I somehow managed to win 3 times130, which equates a win rate of only 30.0%, not only half of the normal, expected, 79.9% win-rate (whereas in the 7 loss instances, I was an underdog just 3 times) (Cases No. 65, 87, and 96 in the table above. I lost in cases No. 3, 10, 21, 39, 51, 63, and 70 in the same table). As for my pocket Aces pre-flop all-ins, things look slightly better, or, let me rephrase it, less worse: of 14 all-ins, I won with Aces 11 times, that is 78.6% of times, which is still below the normal, mathematical winning rate of 85.3% in heads-ups. Interestingly enough, my Aces have beaten opposition Kings only 5 out of 8 times in pre-flop all-ins, so in 62.5% of times, despite being a 81.5Âą1% favorite!

150

V.5 Pre-flop equities in relation to hands' outcomes over a sample of 2k showdowns In trying to deepen the above investigations and offer a better context, as a last part of this chapter's analysis, I isolated all hands in which I have been dealt pocket pairs and that did reach showdown, regardless how many players came along all the way - two, three or even more. Over my entire series of cash games, there have been 977 such hands. For all these hands, I measured how exactly, or to what degree, the pre-flop equities, as stored and indicated by HM2, have been respected.

A few methodological observations are, I think, required here. Firstly, it needs to be said that I picked those hands when I held a pocket pair, and not some other off-suite and/or non-paired hole cards, as an extremely convenient sample in two regards: on the one hand, they combine into a cluster of a satisfactory 977, so almost 1k, hands that did make it to showdown131; on the other hand, as a uniform sample of situations, they allow a mathematically standardized approach in terms of analyzing the hand's outcomes in comparison to the pre-flop equities.

Secondly, considering that the HM2 software indicates, for each hand played beyond pre-flop in one way or another, the pre-flop equities (as well as the equities on flop, turn, and river), I was able to sample and analyze not only hands effectively played by me, meaning with me still at the table at showdown, but also hands where I folded pre-showdown, for which, consecutively, I could establish for a fact whether I would or not have won the pot had I stayed at the table. Take for instance my pocket 8s hand from April 16, at 12:32 EET. As indicated in the first, pre-flop, capture, right below the hole cards of each player, the KK holding player was a 56% favorite to win or split, the player holding A3o had a 25% chance, while I was positioned third, with a 19% probability of winning/splitting:

On the flop, I folded. It seemed like the right decision, although, at that moment, I obviously did not know my equity had further dropped to 13%: 131

Regardless whether with me still at the table or having folded before, but with my opponents continuing all the way.

151

However, the hand has been continued by my two opponents all the way to showdown, which not only allowed my HM2 software to indicate for each street each equity, but also offered me the opportunity, in hindsight, to know whether I would have won or lost the pot. And, surprisingly, I would have! My 8 of spades would have been enough to make the winning flush. With me gone, the opponents ended up splitting the pot, both having a straight:

Thirdly, returning to the methodological solution I employed in the previous chapter132, namely of constantly switching the perspectives, and corresponding measurements, from the personal one to the favorite's one, for all hands sampled, the pre-flop equities' 132

One which, I repeat, dismisses any excuses the poker platform might come up in relation to my playing style, meaning I stayed too long at the table when not supposed to, thus altering the representativeness of the hands that reached showdown, etc.

152

evolutions have been counted twofold. For the above example, from my perspective, the hand is registered as a win from the initial position of the underdog (19% equity); separately, from the favorite's perspective (the KK holding player, with a 56% equity), the hand is registered as a loss. This separate accounting has been done for all the 977 hands, while simultaneously calculating in a cumulated manner my / the favorite's average pre-flop equity, equivalent to the expected win rate, and the actual, recorded win rate.

Take for instance the last six hands played by me, when holding pocket 9s, with the winner of the hand marked with bold characters: Me No. of hands

Hole cards

1.

2.

3.

4.

5.

6. ...

99 99 99 99 99 99 ...

Opponents My preflop equity 41% 55% 89% 55% 70% 72% ...

Hole cards

Preflop equity

J2-s KJ-o 97-o AQ-o T7-o J4-o ...

17% 45% 11% 45% 30% 28% ...

Hole cards

Preflop equity

AKs ...

42% ...

So, for these selected hands, from my perspective, I have started from an average pre-flop equity of (41+55+80+55+70+72)/6 = 63.67%. Relating this expected win + split rate to the six hands played, it means I was expected to win or split 63.67*6/100 = 3.82 times (I won 4 times, but obviously for the given situation, the sample is too small to draw any conclusions yet).

Oppositely however, meaning from the pre-flop favorite's perspective, which was me in five of the six cases (i.e. in the first hand, which reached showdown, the favorite being the player holding the AKs hole cards), the average pre-flop equity was (42+55+89+55+70+72)/6 = 63.83%, meaning he/she was expected to win or split 63.83*6/100 = 3.83 times.

This double accounting algorithm has thus been extended to cover all the 977 hands sampled, while also keeping separately the thirteen situations differentiated by the specific pocket pair I have been dealt.

Finally, in regard to the difference in sample size - 952 cases analyzed from the favorite's perspective vs. 977 from my own perspective is covered by all those hands which started from a 50% vs. 50% equity balance, meaning they had no pre-flop favorite, so in the first accounting column they have been left out. They are, however, accounted for in the analysis from my perspective, in relation to the hand's outcome133. In cases of hands ended with a split pot, the result has been accounted as a half victory, meaning 0.5 has been added into the columns of recorded wins for each such case.

The table below displays the integrated results of this multiple accounting:

133

As, on a side note, it may be said that most of these parity situations didn't even end up in a split pot; specifically, out of 26 such hands where I had pocket pairs and that started on a perfect 50%-50% heads-up parity (meaning, for each player, a combined probability of winning or splitting the pot), I won or split the pot 22 times (12 wins and 10 splits), meaning 84.6%, and lost 4 times (15.4%).

153

Table. Pre-flop equities in relation to hand's outcomes from two perspectives (I):

Look at that color pattern in the right column! The data are so revealing, to not say shocking, one can hardly know where to start commenting them. Contextually I would say that aside from, arguably, the pre-flop heads-up all ins, they provide the most compelling evidence within this chapter of Pokerstars' systematic policy of leveling the field to the benefit of the underdog. Both methodological perspectives highlight spectacularly abnormal deviations from anything that should be expected under genuinely randomness-governed circumstances.

154

V.5.1 My perspective Let's start with how the results look from my perspective. Remember my boiling plus freezing water metaphor? Meaning that, overall, things will almost always look normal, with the recorded win rate extremely close to the expected one, this however being achieved by a mechanism of "mutually compensating exceptions"? The extreme right column captures this mechanism in its full manifestation: the stronger my pocket pair was, starting from Aces downwards, the less its recorded win plus split rate managed to cover its expected win plus split rate, although, on the overall, the recorded win rate calculated as a % of the expected rate is an almost perfect 100.57% (meaning I actually won plus would have won even slightly more than expected to)! On the average, mathematically expected to win or split 56.486% of the times, I did or would have done so in 56.806%.

Now take the Aces for instance: judging by their pre-flop equities, they were supposed to win or split 61.63 out of the total 75 duels; in reality, however, they did so only 60 times (57 victories and 3 split pots, more exactly), meaning 80% of times, which equals 97.36% of their expected, mathematically normal, win plus split rate. Fast forward to the weakest pair, Deuces: instead of an expected 25.59 times, they actually won 28 times, meaning 109.4% of their expected, mathematically normal, win rate. And indeed, when reinspecting my hands of pocket deuces hand-for-hand, I found out, suggestively, that of all their 28 victories recorded, in 12 of them this has happened when starting as the top dog, 12 when starting as the underdog, while the remaining four victories were recorded in 50-50 preflop equities starting situations.

Along the entire column, the pattern of the colors is simply astonishing: all the red and pink is situated in the upper half, while the lower half is colored entirely in dark green; this means the best eight pocket pairs in the game - let's put this in an analogy - as "underachievers" in reference to their expected win plus split rate, whereas the weakest seven pair are all significant "over-achievers". In the reality recorded on the Pokerstars platform, everything is turned upside down. Conclusively enough to dismiss any potential doubt, ten of the total deviations recorded are well above any statistical significance threshold Pokerstars might chose to employ. It is that clear: as a consequence of a premeditated mechanism of leveling the field, the better the hole cards are, the less they shall, proportionally, actually win, and vice versa. And if a unique and standardized mathematical expression of this policy would be still needed, then let us take a look at how the correlation between each of the thirteen pocket pair's power rating in 6-players games and its recorded win + split rate as a % of its expected one:

Pair's recorded win+split rate as % of its pre-flop expected rate

The relation between pocket pairs' power rank and their recorded win+split rate 160

140

120

100

80

correl: - 0.688 60

40

20

0 0

10

20

30

40

50

60

Pair's power rank in 6-players games (its % expected win + split rate)

155

A negative 0.688 linear correlation coefficient measured on a [-1, +1] scale, resulted over almost 1k hands, instead of a 0.00 one to be ideally expected if between the two variables there would indeed be mathematical independence, makes it essentially impossible for Pokerstars to continue claiming that its software deals the cards in a random manner. It instead indicates, intentionality, deliberation, blatant premeditation.

The way I see it, this is another dead end, or a dilemma, for any attempt by Pokerstars' representatives to explain the situation highlighted above. They could, as a first option, disagree for instance with attributing split pots the same 1 point as for pure wins, despite the equities displayed by HM2 clearly indicating the aggregated win plus split probability. Not only would this actually be similar to shooting themselves in the foot (as a recounting of all the split pots across the 1k hands above and the other 1k treated below indicates a recorded frequency of pure split pots around 2.6 times higher than normal134), but in relation to the above table, nothing would change essentially. If considering a split equivalent only to 0.5 victories, then: ď&#x192;ź the overall recorded win rate would become 99.3% of the expected one, so still spectacularly close to it; ď&#x192;ź the colors pattern in the extreme right column would however become even clearer, with seven of the upper eight cells colored in dark green, all the lower cells remaining unchanged; ď&#x192;ź the new correlation coefficient between hole cards' strength and their coverage of their own expected win plus split rate would even increase slightly to minus 0.694!

Or, alternatively, Pokerstars could try to argue that, among other factors pertaining to my playing style, my limp-check-call experiment might explain some of the variation, in the sense that maybe I remained too long at the table when not supposed to, unlike my opponents, who, when I had already won mathematically, folded before showdown. Such an argument would be wrong for at least three reasons: firstly, my entire experiment led to a mere 162 showdowns when I held pocket pairs, which account for only 16.58% of the total 977 showdowns listed in the table above; secondly, during the entire experiment, I uniformly attempted to get to showdown regardless how strong my pocket pair was - I applied the same method when holding Aces and when holding Deuces; thirdly, they would still have to explain why, as strikingly visible in the table, when holding any of the weakest seven pocket pairs, I won or would have won a spectacular 1.21 times more frequent than normal, meaning for every five deserved, i.e. mathematically correct and expected, victories, Pokerstars seemed to gift me with one bonus, as if it was manifesting its gratitude to me for playing the weakest pocket pairs!

Thus, no matter how they could try to twist it, the platform's representatives would still have to explain the brutal anomalies recorded, anomalies that indicate anything but randomness, specifically capturing the leveling-the-field mechanism in all its splendor.

V.5.2 The favorites' perspective Let us now switch the perspective and check how things played out looking from the position of the favorite, whoever that was in each case - me or one of the thousands of opponents at the tables I played at. What immediately catches the attention when looking back at the above table, only this time focusing on its left half, is the figure of 96.49% representing the actual recorded plus projected win + split rate calculated as a percentage of the expected, mathematically normal, rate. It means that on the overall, whenever players found themselves in the position of the pre-flop favorite, they actually ended up losing a bit less often than 134

I ran simulations on the three aforementioned online odds calculators introducing all data pertaining to the 52 split pots' cases (out of which 4 were / would have been split among three players)

156

expected to. And, in light of the methodological specification made above, it becomes clear that the registered anomaly this time cannot be attributed to any particular playing style (i.e. refusing to fold when believing one is favorite and playing like a "calling station"), since the data are aggregated for thousands of players! In absolute frequencies, the favorites, regardless who those were, should have won a projected 636.33 times, whereas in reality, the figure has been only 614. This represents a gross confirmation of the leveling-the-field hypothesis.

However, there are at least two reasons to believe that the 972 showdowns' sample is still not representative enough, meaning, specifically, that much of the variance could still be attributed to my personal playing style. Firstly, the fact that my average pre-flop equity has been 56.486%, whereas the favorite's one 66.842%, meaning that, in the majority of cases analyzed, it was actually still me being the favorite at the table 135. Secondly, and interrelated, even from the favorite's perspective there is a -0.646 linear correlation between (my) pocket pair power rank in 6-players game and the favorite's recorded coverage of his expected win plus split rate which in itself makes no sense, unless the correlation is also related to my hole cards. Acknowledging this bias in the construction of the sample, I opted in favor of expanding it by including other non-paired hands that I played136. In doing this, I opted for one of the most simple and simultaneously truly random sampling techniques - the every nth selection from within the data-base. So, I opened the HM2 software, arranged "all hands" by cards values, and picked up every tenth non-paired hand.

Thus, I started with AK-s, for which I inventoried all showdowns reached, then moved to the next tenth hand (A8-s), repeating the procedure, then jumped to A3-s, then KTo, and so on. Once I reached the bottom of the list, I went back to the beginning in a second step of the count, continuing the same algorithm, up until I surpassed 1k showdowns, which I then added to the pocket pair hands already analyzed. In order to reach the targeted 2k showdowns, I thus counted 37 of the total 169 hole cards types of the game, meaning the sample makes up for over a fifth of the total statistical population represented by "hands" in the meaning of hole card types.

The results of this corresponding recalculation, captured in the table below, may once again seem surprising, all the more so as they also foster contradictions. On the one hand, the favorite's recorded win + split rate has indeed come remarkably close to the normal 100% (specifically 99.68), while the distribution pattern of the colors in the extreme right column does tend to seem a bit more random, meaning a mixture of green and red in the lower half of the table. However, this single finding, one that would suggest a tendency towards normalization corresponding to the sample's increase in size, is directly contradicted by at least three observations.

Firstly, the overall recorded win plus split rate as % of the expected one has suddenly jumped to a considerable 104.95%, meaning that for every twenty hands I did or would have won/split correctly from a mathematical point of view, there has been one extra; this is all the more interesting when considering my average equity has been 45.05%, meaning I started the hands more often in the position of the underdog.

Secondly, the correlation coefficient between the power rank of the hole cards in 6-players games and my proportional win plus split rate has indeed decreased to minus 0.343, but this is still unacceptable considering the sample makes up for 21.8% of the statistical population analyzed137; in fact, the very existence of a correlation is unacceptable if claims of randomness were to be believed.

135

Which only makes sense considering that in the overwhelming majority of cases, pocket pairs are basically favorites to win against any hole cards of an opponent, except for higher pocket pairs.

136 Due to time constraints, I opted in favor of sampling, and not a complete count, which would have taken me over a week. However, in terms of representativeness, the key figures (see below) stopped changing significantly already after 25-28 hands.

137 Furthermore, the small sizes of some subsamples in comparison to other additionally and considerably alter the measured correlation.

157

Table. Pre-flop equities in relation to hand's outcomes from two perspectives (II):

* Two subsamples combined due to their small sizes (n = 12, respectively 21)

158

Thirdly, no less than 32 of the total 34 cases of hands display registered deviations of more than |5%|, corresponding to a standard deviation within the extreme right column's series of values of no less than 22.59 (in relation to a expected average percentage of 100!).

As far as the favorites' positions are concerned, 26 of the 36 variations measured also exceed the 5% interval, corresponding to a considerable 12.96 standard deviation, which only makes the overall 99.58% coverage of the expected win plus split rate all the more intriguing in terms of the boiling - freezing water analogy.

In fact, it seemed so bizarre when I processed and studied the data, that I decided to deepen the analysis by reanalyzing all 2,019 hands counted on the basis of a more detailed accounting in terms of the favorites' specific pre-flop equities, in the same manner done at the end of the previous chapter. These equities, as shown in the capture below, I regrouped in 5% variation intervals, noting for each corresponding case two elements: the exact pre-flop equity of the favorite as displayed by the HM2 software; the exact outcome of the hand, from the favorite's perspective: win (colored green), loss (red) or split pot (blue):

Take for instance the 85-89.99% interval: the first figure in the left column means the favorite of the hand was placed on, specifically in that case, a 86% equity, and the green color indicates that he ended up winning the pot; the red-colored 85 figure to the right of the first one indicates a loss, etc.

159

After collecting all data following the explained algorithm, I further measured the actual, recorded, pre-flop equity based on all cases within each interval, which, obviously, indicates the probability of the top dog winning or splitting the pot. Finally, after filtering out all intervals with not more than ten cases each138, the integrated results suddenly reframed the entire series of data in a crystal-clear image:

From where I sit, this table looks like yet another "smoking gun". The picture is as revealing as possible in terms of the leveling-the-field hypothesis. Thus, along no less than two thirds of the intervals analyzed (8 out of 12), the initial, pre-flop, favorite (regardless whether it was me or the thousands of my opponents, and regardless of how many players are taken into account at showdown - two, three, four, sometimes even five139) actually wins less often than expected under the assumption of randomness, and in 7 out of those 8 cases even by less than 5%! It does indeed seem hard to be a favorite when playing poker on Pokerstars!

Furthermore, the smaller his/her equity, the higher actually his/her coverage of the expected win plus split rate, with the highest rate per rate values recorded for equities below 60%! In terms of the correlation between the recorded average equity and its percentage of making up for the expected rate, the coefficient reaches a remarkable minus 0.5025, which indicates a strong, intense, negative relation, which in turn implies a deliberate, methodical, tampering with randomness. As said, presuming randomness, that coefficient should be somewhere close to zero, thus signaling mathematical independence. Ideally zero, realistically speaking, very close to it, but by no means 0.5! Graphically, this mathematically clear relation could be represented as this:

138

That is the interval [95-99.99] and all those below the 34.99% equity threshold.

Which, obviously, explains the less than 50% equities - those hands have been disputed among more than two players, with the equities indicating the probability of the best placed player to win or split the pot.

139

160

120.0

100.0

recorded as % of expected

80.0

expected

60.0

Linear (recorded as % of expected)

40.0

20.0

0.0

To put is as simply as possible: if things on Pokerstars would indeed be random, the dotted red trend line in the graph above should be horizontal. It actually being divergent from the lower blue one highlights how, the higher the favorite's pre-flop equity is, the lower, proportionally, his/her actual win plus split rate.

In context, these findings actually have the potential to underpin all major assumptions underlying this chapter and this study in its entirety:  they confirm what I had concluded when analyzing the 2k hands solely from my perspective, i.e. that the stronger one's hole cards, the more intense the field is going to be leveled;  they confirm all the findings collected when studying the previous four items within this chapter;  they show how one can mix boiling and freezing water (meaning in this case winning less when stronger pre-flop and vice-versa) and get at the end of the day a normal temperature (meaning a coverage rate of 99.79%);  they expose what I called a "mechanism of mutually compensating exceptions", with three quarters of the recorded deviations exceeding |5%|, while the overall figure still remains normal;  and, last but not least, six of the twelve deviations (in relation to the expected 100% coverage), and the standard deviation of 7.75 as well, all lie within a 2.5% distance from the 10% value implied by the "every tenth hand..." corollary.

V.5.3 A parenthesis: tournaments As clarified already in the Introduction, the "leveling the field" hypothesis investigated in this report refers exclusively to cash games. Urban online folklore specifically claims that in the tournaments played on the Pokerstars platform, the favorite (and, more generally, the bigger stacks) is actually helped, meaning he/she wins more often than supposed to, a claim which is by its very formulation contrary to our investigated hypothesis.

This claim does actually make sense from a business-anchored point of view. Thus, unlike to cash games, tournaments not only take a lot of time, but also do not provide rake for the poker 161

platform. The only profit the platform makes comes from the entry fee paid by the participants. As such, it would make no sense whatsoever for the platform to be interested in prolonging and balancing the games, quite the contrary: the faster a tournament ends, the more new tournaments can be opened, which more players can join, this implying more entry fees to be paid, etc. And, logically, any "unorthodox" help provided to underdogs would in fact only prolong the tournaments, whereas, by contrast, increasing the real odds of the favorite works the opposite way.

Subsequently, none of the multiple testing of the leveling-the-field hypothesis undertaken in this analysis has addressed tournaments held on Pokerstars. The following shall remain the only exception in this regard. There are three reasons for me doing this: a) the first one is to collaterally verify if urban folklore is (again) right; b) the second one is to provide a possible relevant context to the findings of the previous subchapters pertaining to pre-flop equities in comparison to hands' outcomes; c) the third one has to do more generally with the fairness of the game provided by Pokerstars, in direct relation to its official claim that its software is "random".

Thus motivated, the analysis below simply replicates the previously applied algorithm, by specifically placing on an ordinal 5-points scale the pre-flop equities of the favorite, then measuring the average recorded equity for all situations, and finally verifying to what degree those equities have or would have been140 mathematically respected by the hands' final outcomes as displayed at showdowns.

A few methodological clarifications might be useful at this point:  as already mentioned, my total number of hands that I have been dealt in tournaments has been 3,331. Of these, 1,559 have reached showdown141, with or without me still at the table at showdown; all of them have been counted, so the following is not based on a sample, but on the entire statistical population;  all of these tournament hands having been played in various number-of-players formats (from two to nine), "favorite" naturally means the player best positioned pre-flop in terms of the probability to win or split the pot; due to many games having been played in large rings, the average equity always indicated refers thus to the probability of the player bestpositioned pre-flop to win or tie;  the same combined factual plus counterfactual method of evaluation is applied, meaning the registered and classified outcomes are both certain ones and counterfactually ones, meaning what would have happened if I had remained at the table;  all hands where there has been no clear favorite (meaning two or more players sharing the same best equity at the table), regardless of the final outcome142, have been filtered out.

This being clarified, without further ado, here are the aggregated results: across 1,559 showdowns, placed pre-flop on an weighted average equity of 52.64%, the favorite ended up winning or splitting 846 times (810 victories and 39 ties), meaning 54.27% times, which equates an actual coverage of his expected rate of 103.09%. The figure, which is surprising at least in comparison with the way hands played out in cash games from the favorite's perspective, may have two, but not necessarily alternative, explanations.

The first one is, as simply as possible, that urban folklore once again had it right: in tournaments, unlike cash games, the pre-flop favorite wins more often than mathematically expected to, in order for the platform to shorten the duration of tournaments and increase their total number, thus also increasing the entry fees collected from players. It would, indeed, be at least time-effective.

140

Meaning the same combined factual plus counterfactual method of evaluation is applied.

This proportion strikingly higher than in cash games naturally has to do with the structure of tournaments, where the increasing big blinds formats plus antes force players to take considerably more risks and engage in hands all the way to showdowns.

142 A hint to the fact that not all of these situations have ended up in a split pot, many times a 50% player defeating his equally positioned 50% player (the same goes for any equity value).

141

162

The second possible explanation is of a pure methodological nature and has to do with the already mentioned problem of split pots, specifically the problem of how to count them. In the calculations summarized above, I counted them each a unit in the wins plus ties columns, since, as specified, the equities displayed by the HM2 software indicate the cumulated, combined, probability, of the player to win or tie, so victories and ties should, obviously, be counted together.

Of the 846 total wins plus ties, no less than 39 have been ties: 34 between two contenders, 4 among three players, and 1 even among four players! Overall, no less than a intriguing 2.5% of all hands reaching showdown actually ended with a split pot - a figure that might raise an eyebrow of any poker player accustomed to data-collecting and processing software, considering that I pre-excluded from counting all hands in which the two players best positioned pre-flop actually shared an equal equity (e.g. 50 vs. 50, 39 vs. 39, etc.).

Thus, the first possible explanation is that Pokerstars' algorithm artificially and significantly increased the occurrence of split pot scenarios. Admittedly, for the moment I have no explanation for why they would do this, but, anyhow, taking another look at all hands counted, some of the tied ones are indeed the kind of extremely exceptional cases one can probably find only on Pokerstars. Take a look at this one for instance (from May 16, at 21:51 EET), with me holding pocket Aces:

There is no possible better pre-flop situation one can imagine in the Texas Hold'em game: not only do I have the best hole cards in the game, but I also start from the best possible position in relation to my opponent - the high-blocking pair (i.e. I am blocking my opponent's Ace), a type of situation in which the player holding the pocket pair is a 90Âą3% favorite.

In my particular case, as displayed above, the HM2 software indicated a probability of 92% for me to win or split the pot. In search of more rigor, I introduced the data into the three online odds calculators aforementioned, with the following results: Online odds calculator

Probability of me winning (%)

CardsChat CardsPlayer PokerNews

91.59 91.5 91.5

Probability of opponent winning (%) 7.14 7.19 7.19

Probability of a tie (%) 1.27 1.31 1.31

So, on the three values' average, the probability of a pure tie was a mere 1.297%. This figure means that, out of 100 such given pre-flop situations, where I have pocket Aces and my opponent AT-o, 1.3 is expected to end up in a split pot. And, as usual, an event such as this 163

one again happened to me on Pokerstars, in spite of the fact that in all tournaments combined, I have been dealt Aces a mere 8 times, of which only 5 got to showdown!

Or take this hand, with me holding pocket nines:

Despite a pre-flop win plus split equity of a dumbfounding 4%, the player holding A9-o still manages to split the pot with me. The average probability of a pure tie calculated by the three websites? 2.82% Or, as a last example, the 4-way split pot:

Whereas a split among the three A-holding players (me included) is indeed, or at least, a reasonable 5% likely, from the position of the deucing holding player, the probability of splitting the pot together with all of his opponents is 0.52%!

Conclusively, this second explanation suggests that the recorded frequency of pure ties, one higher than normal, might be responsible for the pre-flop favorite winning to a ratio of 1.03 to 1 of the expected win rate. And indeed, if assigning each split not 1, but only 0.5 points, the overall % recorded as % of % expected drops to a more reasonable 100.7%.

But this, however, would be incorrect, as said, since the equities displayed by HM2 and used by me in the calculations indicate the aggregated win plus tie probability.

Returning to the results obtained, a segmenting along 5-point intervals of pre-flop equities of the player best positioned pre-flop and a filtering out of those intervals with less than ten cases143 leads to an image completely opposite to the one discovered when analyzing cash games: the bigger favorite one is, the higher his recorded coverage of his/her expected win plus split rate!

143

Specifically, the [90-94.99] interval, with only three cases - equities of 93% (win), 92% (tie), and 91% (win).

164

Pre-flop equity interval (%)

total showdowns (No.)

85-89.99 80-84.99 75-79.99 70-74.99 65-69.99 60-64.99 55-59.99 50-54.99 45-49.99 40-44.99 35-39.99 30-34.99 25-29.99 Total/avg.

14 36 36 101 122 136 187 176 235 290 158 54 11 1556

Recorded avg. preflop equity (%) 86.857 81.944 76.050 71.723 66.844 62.118 57.027 51.972 46.898 42.172 37.386 32.648 27.545 52.56

Should have won / split ... times 12.16 29.50 27.38 72.44 81.55 84.48 106.64 91.47 110.21 122.30 59.07 17.63 3.03 817.86

Did win split ... times 13 32 29 79 77 90 110 103 116 120 59 13 2 843

Recorded win _ split rate as % of expected

120

Recorded win + split rate as % of expected 106.91 108.48 105.92 109.06 94.42 106.53 103.15 112.60 105.25 98.12 99.88 73.74 66.01 103.07 y = 0.4983x + 70.825 R² = 0.4663

110

100

90

80

70

60

50

40

0

10

20

30

40

50

60

70

80

90

100

Recorded average pre-flop equity intervals

Categorically rejecting any speculation about randmoness, accidents, or coincidences, the linear correlation between the average recorded pre-flop equity series and the recorded as % of expected win plus split rate reaches, as graphically represented above, a positive 0.683, thus capturing a direct and intense mathematical relation as clearly as possible.

Such a figure indicates intentionality, premeditation. It does thus seem that the online allegations were once again correct.

165

Moreover, the recorded pattern is also completely opposite to the one discovered across cash games144.

Favorite's pre-flop equity interval (%) 90-94.99 85-89.99 80-84.99 75-79.99 70-74.99 65-69.99 60-64.99 55-59.99 50-54.99 45-49.99 40-44.99 35-39.99 30-34.99 25-29.99 Total/avg.

Cash games Tourneys Recorded Recorded Showwin + win + Showdowns split rate split rate downs (No.) (No.) as % of as % of expected expected 12 3 90.00 55 14 92.42 106.91 325 36 94.41 108.48 48 36 92.24 105.92 156 101 96.79 109.06 222 122 100.13 94.42 147 136 94.67 106.53 212 187 105.3 103.15 305 176 111.98 112.6 161 235 93.44 105.25 224 290 112.13 98.12 136 158 91.9 99.88 10 54 73.74 4 11 66.01 2017 1559

|Dif.| of percentages of expected

14.49 14.07 13.68 12.27 5.71 11.86 2.15 0.62 11.81 14.01 7.98

9.88

* p < .05

Put one next to the other, the figures are as contrastive as possible: ď&#x201A;ˇ of the eleven cases in which both series have admissible values145, in no less than eight, the color patterns are opposite: if the recorded % calculated as % of the expected one is less than 100 in one column, then in the other one it is above 100; ď&#x201A;ˇ the most clear opposition is registered among the three-four highest-placed equity intervals: whereas in cash games, the stronger favorite one is, the less he/she actually wins proportionally, in tournaments the situation is competely reversed; ď&#x201A;ˇ in no less than nine of the eleven comparable cases, the line-for-line differences between the percetages of percentages exceed 5%, with an average difference of no less than 9.88; in terms of the null hypothesis, there should ideally be no difference at all between the two columns. More realistically speaking in terms of statistical significance, the punctual and the overall difference should anyway not exceed 5% in relation to the normally expected 100 value for each column and line.

And, since it is said that a picture is worth a thousand words, here is the corresponding graphical representation of the data: 144

As a side note, it needs to be said that this is by no means the only significant difference registered when comparing cash games to tournaments. Allow me just two examples: 1.) whereas, among solely hands with a certain outcome, my average conversion rate of flopped flush draws into flushes by turn or river has reached a mere 33.1% in cash games, in tournaments it has been a spectacular 41.51%, albeit one measured on a relatively small sample of hands (53); 2.) whereas, as shown above, in cash games I actually won/split a mere 76.3% of my expected rate in all pre-flop engaged heads-up all ins, in tournaments, the equivalent % of % has reached a staggering 11.2% (specifically, I was expected to win plus split 130.4 of all 261 duels, whereas in reality I ended up winning plus splitting 145 times). Such huge differences may indicate, albeit preliminary and partially, a clear difference in the way Pokerstars' algorithm functions in cash games in comparison to tournaments.

145 Meaning over ten cases (i.e. showdowns) within the respective equity interval.

166

Graph. Recorded win plus split rate as % of expected in relation to pre-flop best positioned players' equities, in a cash games vs. tournaments comparison:

The correlation coefficients and corresponding R2 values are so high, the individual deviations directions so opposite, the slopes of the regression lines so stable, their directions so clearly divergent, the number of cases above or below the 100 threshold (depending on the series) so high, that Pokerstars can only be wished the best of luck in trying to once again invoke "randomness", "accidents" "variance", "coincidences", "bad beats", etc. In the context of this investigation, the findings are at least as important pieces of evidence as the ones collected during three other targeted analyses: of the "Riverstars" hypothesis in the previous chapter; of my limp-check-call experiment; of the series of pre-flop heads-up all ins. Together, and further supplemented by the dozens of other findings of the entire investigation, they expose an extremely hideous, though very carefully hidden, face of the game on the platform investigated. In light of them, any discussion about the "fairness" of the game provided

by Pokerstars to its clients remains rather a sinister joke.

V.6 Integrating the findings Building on the findings of the previous chapters, the analysis concluded here provided solid and multiple evidence (re-) confirming the leveling-the-field hypothesis.

Essentially, all the four items verified - pair vs. pair duels across all my 51,989 cash games, the limp-check-call experiment over 2.5k hands, the 106 pre-flop heads-up all-ins, and the 2k sample of showdowns - converge towards the same categorical conclusion: the playing "field" on Pokerstars is indeed leveled, and significantly so, the hypothesis being confirmed in all its three meanings or dimensions.

Thus, in the first meaning, i.e. in relation to one's opponent, across my entire series of 51,989 cash games played, the pre-flop favorite: 167

   



has been proven to win only 75.2% of his/her pair vs. pair duels, instead of the mathematically expected 81.5% win rate; flopped a set or better to a frequency of only 8.07% when encountering (at least) a lower pair at the table; oppositely however, when encountering a higher pair, the frequency increased by more than two times, up to 16.48%; Within my unusual experiment of limp-check-calling, over all duels involved, paired or not, the favorite won only 57.4% of the times, instead of 67.9%, as supposed to when judging by his average pre-flop equity.

Across the series of my 106 pre-flop heads-up all-ins, my actual, recorded, win rate has been a mere 37.7% instead of a mathematically expected 49.5%, meaning almost a quarter of my normal win rate somehow "evaporated" under the direct effect of the poker platform's "random" numbers generator.146 Finally, over a sample of over 2k showdowns analyzed, the pre-flop favorite, regardless who the person was, me or any of my thousands of opponents, the favorite has once again, and also significantly, been directly disadvantaged in relation to his/her pre-flop equity. Although, in terms of the boiling plus freezing water metaphor, the overall factual coverage of the expected win plus split rate is a statistically normal, and actually quite spectacular, 99.8%, this has been achieved by a mechanism of mutually compensating exceptions. Thus, at an in-depth look, the favorite has won less often than mathematically correct and as such expected, and significantly so, in two thirds of all situations differentiated on a ordinal scale on the basis of the pre-flop equity. Conclusively enough, when switching however from the favorite's perspective to my own, it turns out that my average underdog position corresponding to a 45% pre-flop equity has been directly accompanied by a coverage of my expected win plus split rate of 104.95%, meaning that, as an underdog, I won / would have won more often than normal.

Reinforcing in terms of the leveling-of-the-field mechanisms, most of these deviations have a direct correspondent in the hypothesis' second dimension, which states that the stronger one's hole cards, or draws, the smaller, proportionally, his recorded, actual, factual, win rate. Specifically, this research not only discovered multiple deviations placed miles above any threshold of statistical significance, but also stumbled upon a number of mathematically expressed linear correlations that, if assuming randomness, simply shouldn't be there, let alone reach the considerable values measured in this research:  for the first item analyzed for instance, there is a minus 0.69 correlation between a pocket pair's strength (expressed mathematically as it's expected win plus split rate in 6-players games), regardless if it was me or an opponent, or both, holding a pocket pair, and it's actual win rate recorded when facing a lower pair;  over the sample of almost 1k showdowns reached when I have been dealt pocket pairs, there is a correlation of minus 0.688 between the pair's strength and its actual win plus split rate expressed as % of its expected one;  from the favorites' perspective, across 2k showdowns sampled, there is a minus 0.525 correlation between one the one hand, how strong a favorite the player is exactly (meaning his/her specific pre-flop equity value across the averages of twelve

146

Allow me to repeat: this abnormality has nothing to do with sample size or "variance" or other excuses usually instrumented by Pokerstars representatives, since in seven out of eight types all-ins, the recorded significant deviations all work to my direct detriment, whereas, as said, under truly random circumstances, some of the anomalies are expected to work in my favor, others in my disfavor. It's like playing the roulette: if one keeps betting on red, in a series of attempts, and every time, but only when, he does it, the dice lands on black, things start to seem a little bit "fishy".

168

ordinally arranged equity intervals) and, on the other hand, the correspondingly recorded win plus split rate calculated as % of the mathematically expected rate.

Let's be clear about this: anomalies, deviations, "accidents" do indeed occur under random circumstances, in nature more generally. However, in the given case, as was the case with all the other depicted in the previous chapters, they a) cannot be attributed to randomness and b) nor can they be attributed, in an extrapolation attempt, to hands that have ended before showdowns. Whereas the latter has been demonstrated by the KK hands mini-case study and by my experiment as well, alongside a dozen previous examples which are simply mathematically impossible to admit any extrapolation, the former is supported not only by the very deviations' and correlations' values in themselves, but also, and crucially so, by the fact that the quasi-totality of anomalies described throughout this analysis manifest themselves in the same direction: to the flagrant and substantial detriment of the favorite, regardless who that was in person! It is, to simplify it, like a coin flip - one expects both heads and tails; analogously, if things would indeed be random, some of the recorded deviations should be to detriment of the favorite, others to his favor. And this is not what's happening on Pokerstars.

Finally, in regard to the third, spin-off dimension, addressing solely cash games, this research has identified at least nine distinct situations in which the recorded deviations come extremely close to the 10% figure implied by the "every tenth hand..." corollary147: 1) 11.65%, respectively 10.1%, of the pre-flop favorite's expected win rate, across the hands where I have been dealt KK or QQ, did not materialize; 2) along hands effectively played and with a certain outcome, 12.2% of the OESDs and DISDs mathematically expected to convert intro straights did not; 3) within the methodological matrix of played vs. not played and certain vs. uncertain outcome hands, in aggregated terms, 9.49% of the flopped sets that, under random conditions, should have turned into a full house or better, did not; 4) applying the same approach, 11.35% of the flopped flush draws expected to turn into flushes actually did not convert; 5) across flush draws within hands effectively played, 9.6% of the expected conversion rate into flushed did not materialize; 6) within the sample of pure heads-up pair vs. pair duels, 7.73% of the favorite's expected, normal, win rate simply went "missing in action"; 7) in the same experimental context, 10.19% of total hands played (equivalent to 15% of the expected win plus split rate) have been hijacked from their normal, mathematically expected course, towards the benefit of the underdog; 8) within the sample of 106 pre-flop heads-up all-ins, 11.73% of the duels have had an outcome contrary to the one expected if things were indeed random; 9) over an expanded sample of 2k showdowns, along six of the twelve ordinal categories of pre-flop equities, the deviations from the expected win rate recorded at showdowns, as well as the aggregated standard deviation within the series, lie within a Âą2.5% distance from the referential 10%.

Conclusively, Pokerstars' software acts in a (statistically) significant non-random manner, or, better said, a deliberately anti-random manner, in order to systematically, substantially and deliberately assist the underdog to the direct detriment of the top-dog. This has nothing to do with one person, meaning player, or another, but, more likely, with an insidiously business-motivated reasoning of the platform's owners. Its associated purpose seems threefold: 147

Which, needless to say, does by no means imply that in all other instances, things would have been "normal".

Quite the contrary, actually, in many cases the proportion of hands hijacked from their mathematically normal course has been higher: as shown, within my experiment, no less than 15% of my expected win rate "vanished"; within my all-ins, 23.7% of my expected win rate has not been covered in reality; etc.

169

i.

to prevent weaker players from getting demoralized and deserting the game on the platform; ii. directly related, to prevent better players from winning too much too fast, and withdrawing their profits, by deliberately adjusting their expected, normal, win rate downwards, thus forcing them to play more and more games, implicitly increasing the overall rake collected by the platform; iii. to keep the game more interesting, more spectacular, more "fun", thus potentially attracting more new players.

Obviously, these three elements act in a synergic manner towards the same causally chained goal: expand the player base increase the number of games played increase the rake collected at table increase the company's profit.

No matter how cynical it may seem, it is nevertheless as simple as that: "grinders" roaming freely means bad business for any poker platform, as they will rapidly cash in, clear the field before the collected rake is big enough, and discourage the vast majority of "recreational" players. Hence the need to "level the field", or "balance the odds" in favor of the latter category. And this is exactly what this chapter, and in extenso the entire investigation, have repeatedly proven across the two meanings / dimensions of and the corollary to the "leveling the field" hypothesis.

170

VI. CONCLUSIONS Roughly three decades ago, towards the end of the Cold War, the CIA started strongly suspecting that the Soviets had infiltrated a mole somewhere in the Agency's upper echelons. Years and years of internal investigation brought no result. The frustration grew steadily. True, there was this one particular agent, Aldrich Ames, who always remained among the top suspects, but, then again, he had successfully passed two polygraph tests. And then came the breakthrough. The investigators discovered the most interesting "correlation": during 1985-1986, Ames had had a serious of "random" contacts with Soviet personnel of the embassy in Washington, DC; after almost each of these meetings, right on the very next business day, he made large cash deposits into his bank accounts. There it was! Finally, the truth! What were the "odds", right?! None of the investigators had any doubt left whatsoever: Ames was the mole. And, indeed, it was him, and, soon after, he was eventually arrested and convicted for espionage to life imprisonment.

But guess what! Those financial reports, those "correlations", were not enough to convict him. In the eyes of justice, they were merely "circumstantial evidence", albeit the most compelling possible, and it was only after his written confession that Ames could finally be brought to justice.

On a different scale, the situation of this statistical report is the same. From within the dozens of findings brought forward by this research, let us briefly take a look at only ten linear (this time without quotation marks, meaning mathematically proper) correlation coefficients this investigation has bumped into over the analysis of the 55,320 hands played by me on Pokerstars: - 0.76 - 0.79

- 0.66

- 0.72 - 0.86

between the power rank of the suited connectors in 6 players-games and the recorded frequency of them being dealt to me as hole cards between the ordinally arranged probability of a flopped draw to convert into its corresponding targeted combination (e.g. flush draw into flush, 34.97%) and the actual conversion rate recorded between the default strength of 13 suited card lines as hole cards and the recorded rate of the flush draws turning into flushes among hands with a certain outcome between the power ranks of 78 suited hole cards, arranged in a descending order, and the cumulated relative frequency of making the flush by the river between the favorite player's weighted average equity recorded on the turn and his/her recorded win plus split rate calculated as a % of the expected rate (N = 774 showdowns of hands where I had been dealt pocket pairs)

- 0.82 - 0.69 - 0.69

between the same two types of series, but expanded (over N = 3,012 showdowns) between a pocket's pair power rank in 6-players games (i.e. its expected win plus split rate) and its actual recorded win rate against lower pocket pairs between a pocket's pair power rank and its actual recorded win rate against any opposition hole cards, regardless whether paired or not (expanded over 1k+ showdowns)

- 0.52

between the favorite's pre-flop recorded average equity and the factually recorded win plus split rate calculated as a % of the expected one (N = 2k+ showdowns of cash games)

+ 0.68

between the same variables, but covering tournament played hands (N = 1.5k+ showdowns)

How shall I put this in terms equivalent to Barry Greenstein's anecdotic story narrated in the Introduction of this document? Let's try it this way: 171

ď&#x201A;ˇ

a CIA investigator will tell you, "On the one side you have Ames' series of bank deposits, on the other, his meetings with the Soviets; ď&#x201A;ˇ a statistician will tell you, "I cannot state for certain that this represents the truth of Pokerstars being rigged", and then he will promise himself to never-ever play on this platform; ď&#x201A;ˇ a judge will tell you, "I see... But I still need something admissible as evidence in court!"

These alternative hypothetical perspectives bring us to the core of the issue: the here concluded research does not provide what a judicially admissible piece of evidence would legally suppose to when embarking on a journey of searching the "truth". Legally speaking, the ultimate, irrefutable, and probably universally accepted, "truth" of Pokerstars being "rigged" would require one thing only: access to the platform' algorithm of cards distribution. This, obviously, remains extremely unlikely, to say the least. The way I see it, it would mark the beginning of the end for of Pokerstars, and, further, it might also trigger a domino-effect on the entire online poker industry148.

Let's stick for another moment with the above listed correlations: as repeatedly explained throughout the text, if the distribution of cards on the platform would truly be random, then such correlations should simply not exist; their associated coefficients should necessarily gravitate around the 0.0 reference point. Well, in the factual reality recorded on the Pokerstars platform, not only are they not zero, not only are their values significantly high149, but, more importantly, all but one of them150 manifest themselves in the same single direction of confirming the - again, all but one - hypotheses and sub-hypotheses tested in this investigation. And these hypotheses converge themselves towards one single fundamental idea, namely that all the anomalies manifested on the poker platform (many of them longnoticed and indicated as such by thousands of poker players) have something in common: they implacably lead, in a synergic and systematically applied manner, towards an increase of the profit Pokerstars' owners make from the rake collected at the tables: as repeatedly explained in this paper, dealing better hole cards more often than normal to players will make them bet more, which increases the proportional rake the platform collects; a higher frequency of "coolers" ensures bet-raise-re-raise sequences that further increase the pot-size and subsequently the rake; a field-leveling to the favor of the underdog ensures more and more games are played, which once again translates into higher rakes, etc.

Moreover, the above selected correlations cover only a tiny fraction of what this research has discovered. Dozens of items investigated (78 specifically, when I last counted) have led to dozens of findings not only both reliable in terms of their statistical significance and highly indicative of deliberation, intent, premeditation in regard to Pokerstars, but also confirming five of the six (sub-)hypotheses formulated at the beginning of the research. Let us integrate here the main findings in their direct correspondence to the (sub-)hypotheses tested151.

148

Given the few references I made throughout of this paper in generalizing terms to the online poker in its entirety, I think I owe the reader an explanation: parallel to the 55k hands played on Pokerstars, I also played another cumulated 7-8 thousand or so on three other online platforms. True, such a sample of hands may look too small for any reliable conclusions, and it is, but what I can say for a fact after this little experience is: none of the verifications I made on the three platforms offered me a single reason to believe that things might be different from the ones happening on Pokerstars.

149 True, they are not, say, 0.90. Then again, I think both statisticians and poker players would agree with me when saying that, if such values would be reached, even the most inexperienced player would immediately notice something is definitely, and seriously, wrong with the way the cards are being dealt.

150 The last one, which however does not contradict any of this paper's conclusions. On the contrary, it yet again reconfirms an online allegation, namely that in tournaments, given Pokerstars' specific business interest, the favorite, oppositely to cash games, actually wins more often than expected to (for more details, see Chapter 5).

151 As explained, the here concluded analysis has been a pioneering effort. As such, it has on the one hand not addressed some issues that may otherwise fully deserve an investigation themselves, and on the other hand

172

Hypothesis H1. Non-randomness of hole cards' dealing: confirmed â&#x2C6;&#x161; In what has provided the first strong indicator of premeditation, the general statistical analysis (formulated in relation to a player's own cards and operated in terms of the overall normal, probabilistic, Gaussian, so "random", character of the distribution) actually captured the most normal distribution imaginable, meaning recorded deviations of the targeted parameters exiled somewhere in the spot of the second decimal152. However, a deepening of the analysis at the level of three specifically isolated items - six hierarchical groups of hole cards (arranged corresponding to their mathematically expected win plus split rate in 6-players games), suited hole cards, and pocket pairs - suddenly captured a completely different situation, one indicative of anything, but randomness.

Specifically, whereas the hierarchical groups of hole cards highlighted very slight, hardly visible, traces of non-randomness, the second item led to the first significant figure found: a negative .76 coefficient of correlation between how strong a hand of suited connectors is by default in a 6-players game and its recorded frequency of being dealt to me.

As far as pocket pairs are concerned, measurements taken at three separate moments in time, have illustrated that they have always remained the most frequently dealt group of hole cards, coincidentally also the strongest one, whereas suited connectors, the second best of the three groups, remained second in frequency. Thus, the better a type of hole cards, the more often it is being dealt in relation to its expected frequency.

It is this discrepancy between the figures at the two levels of investigation that is highly indicative of premeditation: in "profane" terms, it is like at the end of each day, someone at Pokerstars makes sure that overall, on the average, everything seems normal. But how exactly those final, average values end up looking absolutely normal is not achieved by opened new possible directions of research. Within the first category I would place the tournaments held on the poker platform. All partial findings collaterally gathered during the analysis seem to pre-confirm online allegations, in that they strongly suggest tournaments are also affected by multiple and significant anomalies from anything expectable under random circumstances, but in a pattern and sense completely opposite to the one affecting cash games (see for instance the intense negative relationship (correl. = -.68) between tournaments and cash games in terms of the recorded win/split rate of the favorite calculated as a percentage of the expected rate).

Thus, I would highly recommend that any further investigation of the poker platform operates an extended comparison of tournaments against cash games by the criteria commonly employed in this research.

A separate problem would be the one of the bots active on the platform - various scandals on this topic have affected Pokerstars in the recent years. In my guesstimate, at the lower stakes levels of 6-players rings, probably every 15th, if not every 10th, account, could be a bot. This issues crucially affects the very fairness of the game provided by the platform to its clients, thus deserving a solid, separate, investigation.

A recurrent, otherwise extremely serious, allegation I encountered has been the one that Pokerstars would obstruct, to not say directly "rob", certain players as persons via some particular function of the software. Such a statement seems impossible to verify unless, of course, given free access to the platform's software. All that this research has discovered has been a clear, irrefutable, consistent, assisting of the underdog to the detriment of the favorite. Thus, it may be that some players, the better, more experienced ones, found themselves more often during the games in the position of the favorite, which made them experience the leveling-of-the-field mechanisms more acutely or more often, which in turn may have led them to falsely believe they are personally obstructed. Still, certain findings of this research (e.g. single cards distribution, pocket pairs distribution, or the pre-flop heads-up all-ins) are peculiar enough to not entirely and apriorically dismiss the allegation.

Another issue not addressed has been the possibility of the alterations of randomness being actually conditioned by the stakes level: the higher the stakes, the less non-randomness. Interesting as it may sound, neither have I tried testing this, nor can I make any safe educated guess in regard to such an allegation.

Within the second category, I would suggest a third methodological perspective for the evaluation of the leveling-the-field mechanisms of Pokerstars' software: not of the player, and not of the favorite (on any of the first three streets), but of the actual winner of the hand - did the winner deservedly win, and, specifically, to what degree so in comparison to his/her equity on any of the first three streets of the game. This evaluation may lead to an even more accurate estimation of the proportion of hands "engineered" or "manipulated" from their mathematically correct, expectable outcome.

152 In fact, such an almost perfect distribution can hardly be expected even in nature, which is truly and entirely characterized by randomness.

173

truly implementing / mimicking randomness, but by what I have tried explaining by using the metaphor of mixing boiling and freezing water in order to eventually get a normal temperature: one loses significantly more than expected as a favorite, but wins proportionally more when being the underdog, if the correlation between suited cards and their dealing frequency is strongly negative, one's off-suite cards shall then be correlated positively, etc.

Sub-hypothesis H1*. The "coolers": confirmed √ In part overlapping the pair vs. pair duels category, the infamous "cooler" situations occur on the platform at frequencies spectacularly higher than the normal, mathematically expected ones. Across 55k hands I went through dozens of situations each mathematically expected to happen, for instance, once in every 650k, 1.5 million, 61 million, or even 85 million hands played at 6-ring table. As explained, the problem is not that such events happen, as they are indeed highly improbable, but nevertheless possible. Nor is it their considerable number (i.e. how many such "exceptionalities" is it actually normal to expect?). The problem is that:  on the one hand, all these probabilistic exceptionalities once again have one thing in common (an increase of the pot and subsequently of the rake) and,  on the other hand, conversely, no "non-cooler", or say we shall "dull", "boring", nonspectacular, situation, has ever been found in this analysis, nor ever reported by any player on Pokerstars for that matter, to have occurred at a frequency lower than its expected one. Say for instance playing 100 KK hands and not encountering Aces at a 6-players table; or playing 1,000 suited hole cards hands and not encounter a situation where three players simultaneously make a flush; or of having 2,000 flopped sets and not encountering an opposition flopped set; or playing 20,000 hands in a 6-ring and not going through the classic AA vs. KK vs. QQ situation; etc. How come? Well, such boring situations simply do not lead to bigger pots and rakes. It's that simple.

Here are some selective figures brought forward by the analysis:  the frequency of AA vs. KK duels: a minimum (proven, measured) 1.8 times higher than normal (respectively an estimated 2.2 times higher);  the frequency of equal pair duels: an extrapolated 2-4 times higher than normal;  the frequency of triple pair situations: 2.09 times higher than normal;  the overall frequency of pair vs. pair duels: a proven 1.75 times higher than normal (respectively 2.1 times higher when estimating it via three distinct methods).

The sub-hypothesis has also been collaterally, but very reliably, confirmed in relation to its coverage of the flops, which tend, statistically significant, to either aggravate the pre-flop-set coolers or bring on new ones: flopped sets vs. sets, two pairs vs. better two pairs vs. set, flush vs. straight, full house vs. full house, etc. (see the next hypothesis)

Hypothesis H2. The "second shuffle": confirmed √ Preliminary and partially pre-confirmed by a general analysis of 22k+ flops and then substantiated by the QQ hands case study (further expanded to incorporate the KK hands), the "second" shuffle hypothesis has been reliable and satisfyingly confirmed by the quadruple targeted analysis undertaken in the third chapter: the factually recorded frequency of flopping useful combinations (for exactly those players that have remained at the table after pre-flop) are higher, category-by-category, than their mathematically expected, normal, counterparts.

For methodological reasons, the first step could at most provide disparate indicators necessary to establish whether further testing is opportune153, which it fully did; an extremely selective analysis identified, over the 22k flops analyzed, materialized flop 153

In the meaning that an exhaustive analysis was impossible, considering that for each of the 1,326 possible hole cards, there are 19,600 flops possible, which then multiply exponentially towards billions of cases when adding into account the possible hole cards of the opponents at the table.

174

probabilities of, for instance, 0.104%, 0.237%, 0.296%, 0.57%, 0.079%, 0.019%, 0.059%, 0.13%, 0.046%, 0.039%, 0.0298%, 0.026%, 0.00015%, 0.0000403%, or 0.0000163%.154 These numbers simply confirm the partial findings of the QQ case study, where a mere 87 showdowns reached were enough to reveal some sensational flop-associated materialized probabilities of, for instance, 2.2%, 0.76%, 2.13%, 2.18%, 1.48%, 2.54%, 4.25%, 2.2%, 2.37%, 1.46%, 1.58%, 0.18%, 0.14%, 0.76%, 1.85%, 0.14%, 1.39%, 1.27%, 0.57%, 1.87%, 1.46%, or 2.08%. What did all such flops once again have in common, aside from their nanoscopic probabilities? A guaranteed induction of betting and (re-) raising, by almost always providing at least two players (sometimes three) with "bait-like", extremely attractive, useful combinations, thus subsequently ensuring an increase of the rake collected by Pokerstars.

The following is a selection of the main results of testing the hypothesis:  in relation to the opponent's hole cards:  the frequency of flopping trips: a minimum (recorded) 1.5, respectively a possible (projected155) 3.7 times higher than normal, i.e. mathematically expectable assuming randomness;  the frequency of flopping two pairs (using both hole cards): 1.6-2.8 times higher than normal;  the frequency of flopping the needed over-card against pocket Queens: 2.04 times higher than normal;  in relation to my own hole cards:  the frequency of flopping a royal flush draw: 2.5 times higher than normal (N = 998);  the frequency of flopping a set: 102.9% of the normal value (N = 2,132)156;  the frequency of flopping a full house: 1.1 times higher than normal (N = 2,132);  the frequency of flopping four of a kind (using both hole cards): 1.72 times higher than normal (N = 2,132);  the frequency of flopping a flush or a flush draw (using both hole cards): 105% of the normal value (N = 6,351);  the frequency of flopping another entirely mono-color flop: 106% of the normal value (N = 6,351);  the frequency of flopping a straight via middling connectors: 1.27 times higher than normal (N = 1,973);  the frequency of flopping two pairs (using both hole cards): 102.0% of the normal value (N= 19,921);  the frequency of flopping trips: 104.7% of the normal value (N= 19,921).157  in interactive player vs. player terms, emblematic of the platform's "coolers",  the recorded frequency of each a) flopped two pairs vs. two pairs and b) two pairs vs. sets situations is 1.41 times higher than normal158,  the recorded frequency of flopped two pairs vs. flush draws is 1.93 times higher;  the recorded frequency of set vs. set situations on the flop: 2.35 times higher; In addition to these results, the repeatedly employed methodological dichotomy between hands played and not played strongly reinforces the conclusions, along almost all 154

See the multiple selections within chapters II, III, and IV for these and more examples.

On the basis of the QQ case study.

156 Calculated in relation to opposition cards, of the 328 cases of pair vs. pair duels having reached showdown, the opponent's recorded frequency of flopping a set+ (n = 45) reaches even 107.95% of the expected value.

157 Essentially, over the entire list of measurements, there have been only two variables with recorded average values lower than their expected ones: flopping an OESD, respectively a GSSD when starting with middling connectors as hole cards (88.0, respectively 95.3% of expected).

158 The recorded values remain higher than normal even if absurdly taking into account all hands folded before showdown and assuming that in none of them did an opponent flop the targeted combination 155

175

parameters verified: the recorded frequency of flopping a useful/targeted combination depends directly on whether the player has effectively remained in the game (with the frequency significantly above normal) or he/she folded, with the hand being continued to showdown by two or more opponents (with the counterfactual projected frequency significantly below normal). Assuming there was indeed randomness, there should be no differences, or at least no statistically significant ones.

Conclusively: at first glance "helping" both the two (or more) players that remained at the table in the pre-flop phase, but the underdog significantly more so (see below), the flops on Pokerstars will bring along, spectacularly (more) often (than normal), combinations of cards of minuscule, infinitesimal probabilities, some of them happening once in tens of millions of hands played, and more generally useful, attractive cards, as long as they fulfill one business-motivated objective: players betting, raising and re-raising, hence bigger pots, hence more rake collected by the platform.

Hypothesis H3. "Leveling the field": confirmed √ Repeatedly pre-confirmed by the findings of the third and fourth chapters, the hypothesis, one of the most frequently and ardently invoked by online critics of Pokerstars, has been confirmed in its fullest, in both its dimensions, and beyond any reasonable doubt, by the separately undertaken, expanded and deepened, analysis undertaken within the fifth chapter: part of Pokerstars' business-motivated strategy159, the underdog is directly, significantly, systematically, and deliberately assisted (to the detriment of the favorite or "top dog"), making him/her end up winning significantly more often than mathematically expected to.  In its first dimension, i.e. in relation to an opponent, the leveling-the-field mechanisms are applied throughout the games, from flop to turn.  in pure heads up duels, the favorite wins only 75.2%, instead of the expected 81.5% of the times when facing a lower pocket pair (N = 197);  the recorded frequency of flopping a set when encountering a higher pocket pair at the table is 2.04 times higher than when facing a lower pair (16.48 vs. 8.07%; N = 343);  within the (open-) limp-check-call experiment over 2.5k hands, regardless of the exact type of the opposition's hole cards, the pre-flop favorite has been shown to win only 57.7%, instead of the expected 67.9% of times (N = 123 showdowns);  as shown by other 2k+ showdowns analyzed, the stronger a favorite a player is pre-flop, the less he/she ends up winning proportionally to the expected rate (correl. = -.50) (e.g. a 92.5% favorite factually wins + splits only 83.3% of times, whereas a 52.4% favorite ends up winning or splitting 58.7% of times, etc);

 as measured over another 3k+ showdowns, the stronger a favorite the player is on the turn, the lower his/her factually recorded win plus split rate calculated as a percentage of the expected rate (correl. = -.82, on a 16-points scale)160; The overall "normalization" of aggregated win rates is realized by the described mechanism of "mixing boiling and freezing water" (one wins significantly more than expected when being the underdog and significantly less when being the top dog).

159

As explained, the leveling-the-field mechanism has nothing to do with one person, meaning player, or another, but with an insidiously business-motivated reasoning of the platform's owners. Its associated purpose seems threefold: i) to prevent weaker players from getting demoralized and deserting the game on the platform; ii) directly related, to prevent better players from winning too much too fast, and withdrawing their profits, by deliberately adjusting their expected, normal, win rate downwards, thus forcing them to play more and more games, implicitly increasing the overall rake collected by the platform; iii) to keep the game more interesting, more spectacular, more "fun", thus potentially attracting more new players. Obviously, these three elements act in a synergic manner towards the same causally chained goal: expand the player base => increase the number of games played => increase the rake collected at table => increase the company's profit.

160 Calculated separately for 774 showdowns reached with pocket pairs, the correlation coefficient climbs to -.86 on a 8-point scale.

176



In the second dimension, the analysis showed that the stronger one's hole cards or draws, the lower the factually measured win/split or conversion/improvement rate161:  the stronger a flopped draw is, meaning the more probable its conversion is mathematically, the less it actually materializes (correl. = -.79).

For instance,  along 13 suited cards lines, the stronger the hole cards that which led to a flopped flush draw, the lower the conversion into flushes (correl. = -.66);  on a differently built scale, there is a correlation of -.72 between the power ranks of the 78 suited hole cards, arranged in a descending order, and the recorded cumulated relative frequency of making the flush by the river;  in pure heads up duels, the stronger a pocket pair, the lower its win plus split rate against lower pocket pairs (correl.= -.69 over 197 cases on a 13-points scale);  over the sample of almost 1k showdowns reached when I have been dealt pocket pairs, regardless of the opposition's hole cards type, there is a strong negative mathematical relation between the pair's strength and it's actual win/split rate expressed as % of its expected one (correl. = -.69)162.

Sub-hypothesis H3.A. "Riverstars": disconfirmed The analysis of 3,102 showdowns in relation to the equities recorded on the turn has produced results actually contrary to what the sub-hypothesis states stricto sensu: the turn favorite in fact ended up at a 102.5% effective win plus split rate calculated as a % of the expected rate.

For the particular 95 vs. 5% category, the coverage of the expected rate reached a statistically normal 99.7%. In context, on the transversal of this paper's findings, the resulting conclusion is as clear as possible: the most intense field-leveling to the favor of the underdog is operated not on the river, but directly on the flop. There, on that street, even if only because the associated mathematics remain so complex, hence the demonstration of any fraud so difficult, does most of the leveling "happen". Massively and systematically.

Corollary H3.B. "Every tenth hand is manipulated": substantiated √ Several specific figures identified in the research add substance to the corollary: 10) 11.65%, respectively 10.1%, of the pre-flop favorite's expected win rate, across the hands where I have been dealt KK or QQ, did not materialize; 11) along hands effectively played and with a certain outcome, 12.2% of the OESDs and 2xSDs mathematically expected to convert intro straights did not; 12) within the methodological matrix of played vs. not played and certain vs. uncertain outcome hands, in aggregated terms, 9.49% of the flopped sets that, under random conditions, should have turned into a full house or better, did not; 13) applying the same approach, 11.35% of the flopped flush draws expected to turn into flushes actually did not convert; 14) across flush draws within hands effectively played, 9.6% of the expected conversion rate into flushed did not materialize; 15) within the sample of pure heads-up pair vs. pair duels, 7.73% of the favorite's expected, normal, win rate simply went "missing in action"; 161

There has been a single exception discovered in the research: the frequency of flopping four types of draws, which in three cases seems genuinely independent from the strength of the hole cards, while in the fourth case (flopping at least one pair when starting with un-paired hole cards) there is actually an intense positive relation.

162 Once again hinting at premeditation, as implied by the "mixing boiling and freezing water" metaphor, the strongest eight pairs all win plus split below the expected frequency, while the lowest seven pairs all win above the expected frequency (10 out of 13 deviations recorded also being statistically significant).

177

16) in the same experimental context, 10.19% of total hands played (equivalent to 15% of the expected win plus split rate) have been hijacked from their normal, mathematically expected, course, towards the benefit of the underdog; 17) within the sample of 106 pre-flop heads-up all-ins, 11.73% of the duels have had an outcome contrary to the one expected if things were indeed random; 18) over an expanded sample of 2k showdowns, along six of the twelve ordinal categories of pre-flop equities, the deviations from the expected win rate recorded at showdowns, as well as the aggregated standard deviation within the series, lie within a Âą2.5% distance from the referential 10%.

As detailed in the text, this does not imply that in all the other cases things would have been normal. In dozens of other situations, the recorded deviations are smaller than |10%|, but nevertheless bigger than |5%|, whereas in other instances, they not rarely, and significantly, even surpass the |10%| threshold.

In summary: all three hypotheses, one of the two sub-hypotheses, respectively the corollary, may reliably be considered as having been confirmed, respectively substantiated, by the research, the single one disconfirmed being the "Riverstars" sub-hypothesis in its strict meaning.

In the form of an overarching conclusion: the analysis captured multiple, major, and statistically significant deviations from what a truly random distribution of cards should look like.

Aggravatingly, what strongly hints at intent, at deliberation, at premeditation on the part of Pokerstars, is not so much the frequency of such deviations (one anyhow too high by any methodological threshold selectable). Nor is it the recorded values in themselves of so many coefficients, parameters and indicators employed throughout the investigation (values also too high in any reasonable evaluation). It is rather the twofold fact that: ď&#x201A;ˇ at the end of the day, almost all general, overall, average parameters on the platform are statistically "the most normal" possible, with deviations exiled to the spot of second decimals - this is achieved by a carefully implemented mechanism of mutually compensating exceptions, which indicates a meticulous preoccupation for covering the tracks, and ď&#x201A;ˇ the quasi-totality of the anomalies discovered act in a systematical and mutually reinforcing manner, manifesting themselves in one single unifying direction - an increase of the size and/or number of the pots at the playing tables, which directly translates into an increase of the rake collected by Pokerstars, which in itself equates more profit for the platform's owners.

So... after everything said and done within this investigation, could it still not be possible, somehow, to some degree, by some contortionism of logic or hermeneutical ballet, that Pokerstars' software does though deal the cards randomly, and that everything this analysis has discovered is just an indeed-incredible-but-nevertheless-possible, very long chain of "exceptionalities"? Well, allow me to put it like this: yes, theoretically it is still possible, but it is way more likely for you to get out on the street after reading this and get hit by a meteorite. Not once, not twice, but three times. On the same day!

---------

178

APPENDIX 1 Hole cards' power ranking in 6-players game163 Hole cards AA AK-s AK-o AQ-s AQ-o AJ-s AJ-o AT-s AT-o A9-s A9-o A8-s A8-o A7-s A7-o A6-s A6-o A5-s A5-o A4-s A4-o A3-s A3-o A2-s A2-o KK KQ-s KQ-o KJ-s KJ-o KT-s KT-o K9-s K9-o K8-s K8-o K7-s K7-o K6-s K6-o K5-s K5-o K4-s K4-o 163

Power rank (%) 49.51 32.15 28.96 30.56 27.21 29.28 25.79 28.27 24.66 25.75 21.93 24.98 21.07 24.27 20.25 23.50 19.43 24.19 20.15 23.63 19.51 23.05 18.87 22.39 18.15 43.32 29.55 26.28 28.28 24.88 27.32 23.80 24.73 20.98 22.97 19.04 22.37 18.38 21.75 17.72 21.29 17.18 20.75 16.59

Hole cards K3-s K3-o K2-s K2-o QQ QJ-s QJ-o QT-s QT-o Q9-s Q9-o Q8-s Q8-o Q7-s Q7-o Q6-s Q6-o Q5-s Q5-o Q4-s Q4-o Q3-s Q3-o Q2-s Q2-o JJ JT-s JT-o J9-s J9-o J8-s J8-o J7-s J7-o J6-s J6-o J5-s J5-o J4-s J4-o J3-s J3-o J2-s J2-o

Power rank (%) 20.25 16.03 19.76 15.49 38.30 27.57 24.26 26.64 23.20 24.07 20.41 22.24 18.43 20.65 16.70 20.19 16.17 19.72 15.66 19.22 15.08 18.69 14.52 18.25 14.00 34.05 26.33 23.03 23.82 20.28 22.01 18.31 20.38 16.51 18.92 14.94 18.56 14.53 18.03 13.96 17.57 13.42 17.11 12.92

Hole cards TT T9-s T9-o T8-s T8-o T7-s T7-o T6-s T6-o T5-s T5-o T4-s T4-o T3-s T3-o T2-s T2-o 99 98-s 98-o 97-s 97-o 96-s 96-o 95-s 95-o 94-s 94-o 93-s 93-o 92-s 92-o 88 87-s 87-o 86-s 86-o 85-s 85-o 84-s 84-o 83-s 83-o 82-s

Power rank (%) 30.44 23.90 20.44 22.12 18.50 20.46 16.71 18.95 15.06 17.62 13.63 17.19 13.16 16.72 12.62 16.27 12.12 27.11 21.72 18.14 20.30 16.61 18.82 15.00 17.41 13.48 16.05 12.01 15.67 11.58 15.22 11.08 24.51 20.39 16.76 19.11 15.35 17.73 13.87 16.25 12.29 14.94 10.86 14.58

Hole cards 82-o 77 76-s 76-o 75-s 75-o 74-s 74-o 73-s 73-o 72-s 72-o 66 65-s 65-o 64-s 64-o 63-s 63-o 62-s 62-o 55 54-s 54-o 53-s 53-o 52-s 52-o 44 43-s 43-o 42-s 42-o 33 32-s 32-o 22

Power rank (%) 10.45 22.35 19.31 15.62 18.10 14.34 16.66 12.78 15.25 11.24 13.97 9.87 20.59 18.45 14.71 17.17 13.34 15.77 11.82 14.37 10.32 19.05 17.82 14.05 16.59 12.71 15.19 11.23 17.76 15.90 11.99 14.68 10.67 16.69 14.02 9.98 15.87

Figures taken from: https://wizardofodds.com/games/texas-hold-em/6-player-game/ (retrieved June 27, 2107).

179

APPENDIX 2 The best of Pokerstars coolers (all hands screen-captured on July 12-13, 2017, at two zoom tables ("Donati" and "Aludra") in a matter of a few hours)

20. Where two argue, the third wins:

19. The eleventh commandment: "Aces shall not win!"

180

18. "Did you just flop two pairs? A 2% probability? Fold! Fold now!!!"

17. " I told you, two pairs on the flop brings bad luck!"

181

16. "You flopped a set? You're not the only one!"

15. "Good evening! How many pairs would you like at this table? Three?

Coming right up!"

182

14. "Just a reminder: Aces shall not win, OK?!"

13. "What's better than a flush? Right, a straight flush!"

183

12. How to flop a straight and still lose:

11. Trip vs. full on the flop:

184

10. Just another reminder: "Do not flop yourself two pairs!"

9. Flopped trip vs. straight

185

8. "Waiter! I'll order the same as him!"

7 "My straight is better than yours!"

186

6. How to lose with a flopped set: how about quads on turn plus river?

5. "OK, as an exception, Aces are allowed to win today!"

187

4. Pokerstars level expert: how to lose with a flopped full house!

3. "How to lose with a flopped full - let's try something else, shall we?"

188

2. "OK, I'm running out of ideas. Let's allow the flopped trip to win for once?"

"And the winner is...: how to flop four of a kind and still lose!"

189

About the author:

Ionuţ APAHIDEANU, independent researcher.

PhD (University of Bucharest), MA (National School of Political Science and Public Administration, Bucharest) and BA in Political Science (Babeş-Bolyai University Cluj-Napoca), Selective publications: volume chapters in • Intercultural Conflict and Harmony in the Central European Borderlands, • Manual de relaţii internaţionale; • Crisis-management in Romania; • Armonie şi conflict intercultural în Banat şi Transilvania. Perspective cultural-istorice, 1650-1950.

Articles in • Journal for the Study of Religions and Ideologies, • South-East European Journal of Political Science, • EuroPolis, • Studia Politica. Romanian Political Science Review, • Romanian Journal of Society and Politics, • Strategic Impact, • Monitor Strategic, etc. contact: apahideanu [at] gmail.com

190