World cup group stage scenarios article post by Peterlloydlowe

FIFA World Cup Group Stage Scenarios By Peter Lowe, M.S. College Instruction: Mathematics, Eastern Washington University The FIFA (Federation International Futbol Association) World Cup is a tournament involving the top soccer teams in the world. It is held every four years, and currently uses a format which involves placing teams into groups of four. Each team must play a match (game) against all three other teams in its group. The team scoring the most goals wins the match with ties resulting in a draw. Teams are then awarded 3 points for a win, 1 point for a draw, and 0 points for a loss. The top two teams from each group (based on points earned) advance to the next stage. This article provides a framework for analyzing the possible occurrences within this type of group play. In a group of four teams the total number of games required for each team to play the others is six. The diagram below illustrates the six games (lines) between the four teams (dots). Each of the games could result either in a win for one team, a win for the other, or a draw. This makes three possibilities for each of six independent events. For one game the total number of possible outcomes would be three. Combining each of those three possibilities with the three possibilities for the second game we have nine different possible ways for two games to finish. As we continue to add games the number of possibilities continues to triple so the total number of permutations for six games would then be 36 = 729. This relatively large number of permutations makes analysis difficult; however, it takes into account a number of permutations that, for the purposes of analysis, are not significantly different. For example, if one team was to beat all three of the other teams, and the other teams were to draw against each other, that would be one possible scenario for the purpose of analysis. (This scenario is illustrated using an arrow to indicate that the team at the beginning of the arrow beat the team to whom the arrow points. Lines without arrows denote draws.) When counting permutations, however, this one scenario would represent 4 of the 729 possibilities because any of the four teams could finish as the dominant team with the others affecting draws indistinguishably. In the same way each of the 729 permutations is not truly unique when analyzing group scenarios because many of these permutations involve simply rearranging the names of the teams achieving certain wins and loses. This means that the 729 permutations can be grouped into scenarios which do not distinguish which of the four teams finishes 1st, 2nd, 3rd, and 4th. This new set of scenarios would be concerned instead with the pattern of wins and losses among the four teams. The task then becomes to illustrate all the group scenarios in such a way that, given the results of any group, one could put the names of the teams on one of these scenarios to illustrate the results of the six games. Determining the number of scenarios possible can be done with a group theoretic approach using the Burside-‐ Frobenius-‐Polya orbit counting formula and some attention to the number of arrangements fixed by the group action. Aside from being somewhat outside of the scope of most recreational mathematicians tools, this would only give us the number of scenarios and we are interested in illustrations of the scenarios themselves.

In finding all the possible scenarios it is important to note that just because a scenario is illustrated differently does not mean that it constitutes a new scenario. The diagram below, for example, looks different from the previously illustrated scenario, but still represents one team beating all three opponents, and those opponents affecting draws with each other. To avoid these duplications we will adopt the convention that the team scoring the most points (#1) will be placed in the center, and the #2, #3, and #4 teams will be placed clockwise starting in the upper left hand corner, so that the team finishing last is placed on the bottom. This does not eliminate the

problem when there are multiple ties, but it greatly reduces the confusion. We will also organize the scenarios by the number of wins, as scenarios with different numbers of wins cannot be duplicates. Another useful fact in avoiding repetition is that if two scenarios result in different scores, they must be distinct scenarios, though the converse is not true. Groupings which do result in identical scoring must be examined to determine if they constitute distinct scenarios. Finally, a list of scenarios can be confirmed as complete if each scenario is distinct and the number of permutations represented by each scenario sums to 729. The following list meets these criteria and so constitutes a complete list of scenarios. Line segments indicate draws. Arrows indicate that the team at the beginning of the arrow beat the team to whom the arrow points. Scenarios have been organized by the number of wins and, secondarily, by the top team’s point total. Methods for counting the number of permutations represented by each scenario are discussed in the appendix on counting. Information next to each illustration can be interpreted according to the key as follows: #3

#2 Finishing position:

Distinguishable Dominance Point totals (2/1/0 scoring)

Permutations Historical groups of this type Ties (2/1/0 scoring)

Distinguishable: Letters indicating which of the four positions (1-‐4 from left to right) are distinguishable for the purpose of counting permutations. ABBC, for example, means that either of the two teams placed in the number 2 and 3 positions could be placed first and the same permutation would be created. Point Totals: The number of points earned by each of the positions (1-‐4 from left to right) using the current FIFA scoring system which awards 3 points for a win, 1 for a draw, and zero for a loss. (2/1/0 scoring): The number of points earned by each of the positions (1-‐4 from left to right) using the old (pre-‐1994) FIFA scoring system which awarded 2 points for a win, 1 for a draw, and zero for a loss. Dominance: D-‐ The scenario exhibits the typical dominance relationship as discussed in the analysis portion of this article. N-‐ The scenario does not exhibit the typical dominance relationship. Permutations: The number of the 729 possible group stage permutations that are represented by this scenario. Tie: T-‐ This scenario represents a situation in which a tie breaker would be needed to determine the top two teams using the current scoring system (3 points for a win). If none is needed this field is left blank. (Tie): (T)-‐ this scenario represents a situation in which a tie breaker would be needed to determine the top two teams using the old scoring system (2 points for a win). If none is needed this field is left blank. Historical groups of this type: The year and group letter or number for each world cup four-‐team group stage that resulted in this scenario. Years charted are 1930 through 2014 with only the last two digits shown. During years which held two rounds of group play the convention of numbering the first round and lettering the second has been followed. *: Special considerations needed when counting these permutations, because the indistinguishable positions can be filled multiple ways. See appendix on counting.

#2 Finishing position:

O wins

Distinguishable Dominance Point totals (2/1/0 scoring)

Permutations Historical groups of this type Ties (2/1/0 scoring)

AAAA

3333

(3333)

(T)

1 wins

ABBC

5332

(4332)

(T)

‘90F, ‘82/1

2 wins

ABCC

‘98B

‘98E

‘10F

7322 (5322)

AABC 5531 (4431)

ABCD 5432 (4332)

AABB* 5522 (4422)

(T)

3 wins

ABBB

9222

(6222)

(T)

ABCD

7521

‘02A, ‘94C, ‘86A, ‘74/3, ‘66/1

(5421)

ABCD

‘58/3, ‘58/4

7431 (5331)

ABCD

(T)

7422 (5322)

ABCC 6522 (4422) AAAB 5550

(4440)

(T)

‘74/2

(3 wins)

ABCD

‘10C, ‘02F

5541 (4431)

AABC 5532 (4422)

ABCD 5442

(4332)

(T)

AAAB*

4 wins

4443

(3333)

(T)

ABCD

9421

7711 (5511)

‘14H, ‘06D, ‘06F ‘98C, ‘86E, ‘82/4

(6321) AABB

‘82/5, ‘70/2

(4 wins)

ABCD

‘74/1

‘10G, ‘06G, ‘02E

7621 (5421)

ABCD 7540

‘86B, ‘78/2

(5430)

ABCD

‘06B, ‘02H, ‘94B

‘78/4, ‘62/4

7531 (5421)

ABCD 7441

(5331)

(T)

ABBC

7441

(5331)

(T)

ABCD 7432 (5322)

‘14G, ‘10A

‘78A

(4 wins)

ABCD

6541

‘98A, ‘98D, ‘78/3 ‘58/2

(4431)

ABCD

6442

(4332)

(T)

ABCD

5443

(4332)

(T)

AAAA*

4444

(3333)

(T)

‘86F, ‘58/1

‘94E

5 wins

ABCC

9611

‘06H, ‘86D, ‘74A ‘70/4

(6411)

ABBC

‘02B, ‘02C, ‘82/6

9440

‘74/4

(6330)

(T)

(5 wins)

ABCD

9431

‘14C, ‘14F, ‘10B ‘66/4

(6321)

AABC

‘14A, ‘06C, ‘98F

7730

‘86C, ‘78B, ‘70/1

(5520)

‘66/2

ABCD

7640

‘14E, ‘90D, ‘90E ‘82/3, ‘62/2, ‘50/1

(5430)

ABCD

7631

‘14D, ‘06E, ‘98G ‘62/1, ‘50/final

(5421)

ABCD

‘02D, ‘02G, ‘62/3

‘10H

7433 (5322)

ABCD 6641 (4431)

(5 wins)

ABCD

6443

(4332)

(T)

ABBC

6443

(4332)

(T)

‘94A

‘10D, 90B

6 wins

ABCD

‘14B, ‘10E, ‘06A

9630

‘98H, ‘90A, ‘90C

(6420)

‘78/1, ‘74/B, ‘70/3

‘66/3, ‘30/1 ABBB*

9333

(6222)

(T)

AAAB*

6660

(4440)

(T)

ABCD

‘50/2

‘94D, ‘94F, ‘82/2

6633 (4422)

[42 scenarios]

[14-‐D, 38-‐N]

[729 perm]

[12/(14) ties]

[96 total groups]

Analysis As it turns out there are 42 different scenarios which model all 729 possible outcomes in a four-‐team FIFA group stage. The distribution of scenarios is somewhat symmetric, with 4 win groups occurring most frequently. Number of wins Number of scenarios Number of permutations

160

240

192

Ties The scenarios can be sorted into those which result in ties and those which do not, and then further sorted into those which result in ties involving the 2nd ranked team. These types of ties are of particular interest as they must be broken to determine who advances to the next stage of the tournament (ties between 2nd and 3rd) or the seeding of the teams in the next stage of the tournament (ties between 1st and 2nd). The 729 permutations are sorted by ties in the table below. No Ties

312

Some type of tie

417

Ties involving 1st and 2nd (but not 3rd) ranked teams

333

Ties involving at least 2nd and 3rd ranked teams

Ties involving 2nd rank team

126 207

Ties involving only 3 and 4 ranked teams

So 417 of the 729 permutations (57.2%) result in ties, and 333 of those (45.7%) must be broken to determine advancement (28.4%) or seeding (17.3%). The 207 permutations in which the 2nd and 3rd ranked teams are tied are the most critical since the loser of the tie breaker in those situations will be eliminated from the tournament. Under the old scoring system 2 additional scenarios resulted in 2nd and 3rd ranked ties for a total of 255 of the 729 possibilities (35.0%) so the change in scoring should theoretically have reduced the number of tie-‐breakers needed. This proved true in 2010 when the 2-‐win/5432 scenario occurred. The final standings of group F showed Paraguay winning the group with a win and two draws, followed by Slovakia and New Zealand: New Zealand with three draws and Slovakia with a win and a draw. Under the old scoring the latter two teams would have both earned three points, but under the current scoring system Slovakia, with one win and one draw, received the rank of 2nd without the need for a tiebreaker.

The scenarios involving the need for an advancement tie-‐breaker under the current scoring system involve two-‐way, three-‐way, and four-‐way ties for 2nd and 3rd rank as shown below. Two-‐way ties Three-‐way ties Four-‐way ties Number of scenarios 9 5 2

168

Number of permutations

FIFA’s criteria for ranking the teams are listed below.[1] Successive criteria are only considered if earlier criteria are inconclusive. Note: ‘points’ refers to those awarded for wins (3pts) and draws (1pt), while ‘goals’ refers to those scored during a match. a) b) c)

d) e) f)

Considering all six matches in the group stage: The team earning the most points is ranked the highest. The team with the highest ‘goal differential’ (total goals scored minus total goals allowed) ranks highest. The team with the most total goals scored ranks highest. Considering only matches between teams that are still tied after applying previous criteria: The team earning the most points is ranked the highest. The team with the highest ‘goal differential’ (total goals scored minus total goals allowed) ranks highest. The team with the most total goals scored ranks highest. If these criteria are inconclusive, team rankings will be determined by the drawing of lots.

One interesting conclusion that can be drawn from the analysis of these scenarios is that criteria (d) is redundant. By this we mean no group (regardless of the scores in the matches) would have rankings that differed if criteria (d) was removed. To prove this we consider the 16 scenarios that result in two-‐, three-‐, and four-‐way ties and also the relationship between criteria (d) and (e). We will show that in every case criteria (d) is either inconclusive or is equivalent to criteria (e). Two-‐way ties: In order for criteria (d) to be used the teams involved must be tied by criteria (a), (b) and (c). If that occurs, only matches involving the tied teams are considered. When there are only two teams involved in a tie the match between them will determine who advances. The winning team would be ranked highest by criteria (d) or (e) since you must score more goals than your opponent to win. In the event of a draw criteria (d) is inconclusive. The removal of criteria (d) would have no effect in these cases. Thus, in the 9 scenarios with two-‐ way ties, criteria (d) would be redundant. This is also true for any possible group where more than two teams are tied by criteria (a) but only two teams are left tied after criteria (b) and (c) are applied.

Three-‐way ties: By looking at the scenarios involving three-‐way ties, we see that if all three teams are still tied after criteria (b) and (c), they will all have the same number of points earned when considering only matches involving those teams (bolded below). The symmetric nature of the scenario that produced the three-‐way tie is preserved even when only the matches between those three teams are considered. Criteria (d) will then be inconclusive so it is redundant in all 5 of the three-‐way tie scenarios. 3-‐win/9222

3-‐win/5550

3-‐win/4443

6-‐win/9333

6-‐win/6660

Four-‐way ties: In the first four-‐way tie scenario (0-‐win/3333) criteria (d) would be inconclusive, since every match resulted in a draw. This leaves only the 4-‐win/4444 scenario. If all four teams are tied after criteria (b) and (c) are applied, then criteria (d) will be inconclusive as well since none of the matches would be eliminated. The situation we will analyze in depth is when criteria (b) or (c) eliminate one of the teams from the 4-‐win/4444 four-‐way tie, leaving only 3 tied teams. In these cases, criteria (d) always produces a ranking. We will show that this ranking must necessarily agree with the ranking assigned by criteria (e), making criteria (d) redundant in all cases.

0-‐win/3333

4-‐win/4444

Let us first demonstrate that the situation described can occur. In the diagram below scores have been added to the 4-‐win/4444 scenario. Numbers listed closest to each team indicate the number of goals they scored in the related match. Note that in this situation the top three teams have goal differentials of +1 and total goals scored of 4, while the bottom team has a goal differential of -‐3 with 2 goals scored. The 4-‐4-‐4-‐4 tie in points according to criteria (a) would then be broken by criteria (b), removing the bottom team, but the top three teams would remain tied by criteria (b) and (c). So it is possible to encounter a situation where the 4-‐ win/4444 scenario produces a three-‐way tie after applying criteria (b) and (c).

3 -‐ 0

2 -‐ 1

0 -‐

1 -‐

If any one team is removed from the 4-‐win/4444 scenario the remaining matches (those considered under criteria (d)-‐(f)) would have the relationship shown below. We will label the teams X, Y, and Z.

Criteria (d) which ranks teams by their points earned in these three matches (X-‐4, Y-‐3, Z-‐1) would produce a ranking of X-‐1st, Y-‐2nd, and Z-‐3rd, or XYZ. Our goal is to show that criteria (e), which ranks teams by goal differential in these three matches, would yield the same result. Consider the following facts: • • • •

The goal differentials of these three teams were equal when considering all six matches. The goal differential for team X will increase when considering only these three matches because the eliminated match involving them was a loss for team X. The goal differential for team Y will remain the same when considering only these three matches because the eliminated match involving them was a draw. The goal differential form team Z will decrease when considering only these three matches because the eliminated match involving them was a win for team Z.

If all three teams were tied in goal differential and team X’s increases, and team Y’s stays the same, while team Z’s decreases then the resulting ranking produced by criteria (e) must necessarily be XYZ.

We conclude that in every tie scenario the resulting ranking using the current FIFA criteria would not be changed if criteria (d) were removed entirely. While there is no harm in leaving it in place as it contributes to a certain symmetry and uniformity in the criteria, it is for practical purposes redundant. Dominance Another aspect of the group stage that becomes much more approachable when we view the 729 possibilities through the lens of the 42 scenarios is how well World Cup tournaments model the social concept of sports dominance. Sports dominance is an idea loosely held by fans that a team’s ability to beat another team should be transitive. That is, if team A beats team B, and team B beats team C then team A should beat team C. Indeed, the concept of a bracketed tournament is founded on this idea. We know that in practical terms this is not the case. Variability suggests that two teams matched against each other multiple times will not always yield the same result. Often in sports tournaments the competing teams will play a series of games against each other with the team winning the most games declared the victor. The question then posed is how many of the scenarios model this transitive dominance, and do the actual events in world cup history favor these scenarios? To answer these questions we will have to quantify the concept of dominance. The challenge in this case is to incorporate draws as well as wins and losses. While it would be simple to say “teams draw if they are of equal ability,” it would be more accurate to say “teams draw if they are of very similar abilities.” The concept of dominance can then be defined as follows. • • •

Every team has an unknown but quantifiable ability which can be used to put them on a spectrum of ability in comparison to other teams. If the gap between two teams’ abilities is smaller than a defined distance then the two teams will draw. If the gap is greater than (or equal to) the defined distance then the greater team will win.

To make this more manageable we will state the rule of dominance as “teams in a group can be given ability scores on a scale from 1 to 100 and a team will beat their opponent when the team’s ability score exceeds that of their opponent by 10 or more.” A simple example would then be a group in which the ability scores were 10, 30, 50 and 70. The top team would beat the other three. The next team would beat the two below them. The third team would beat the last team who would lose all three matches. The result would be the 6-‐win/9630 group which is the classic example of sports dominance. 50

30 70

A more complex example would involve draws. If four teams had abilities of 51, 44, 40, and 35, the top team would beat the bottom two, but draw with the second best team. All other matches would be draws. The result would be the 2-‐win/7322 scenario.

40 51

Some of the scenarios, however, do not exhibit this form of dominance. That is, there are no numbers which could be applied to the teams that would produce the scenario using the dominance rule. An obvious case would be one in which there was a nontransitive loop: A à B à C à A, but we also see non-‐dominance exhibited when two teams that draw have different non-‐draw results against a third team. This would look like a nontransitive loop with one win changed to a draw: A à B — C à A. If B and C are close enough in score to effect a draw, then it cannot be true that one is 10 or more points higher in ability than A, while the other is 10 or more points lower. This means that scenarios like the 4-‐win/7621 do not exhibit sports dominance.

Using this type of analysis we can categorize each of the 42 scenarios as either D if it exhibits our definition of dominance, or N if it does not. Data from this analysis is in the table below. Exhibits Dominance Does Not Exhibit Dominance Number of scenarios 14 28 Number of the 729 permutations 182 574 represented by these scenarios So 25% of the 729 permutations exhibit the dominance characteristic, but do the results from actual tournaments favor these scenarios more than their non-‐dominant counterparts? The answer is overwhelmingly ‘yes.’ If all matches were independent events with all outcomes equally likely we would expect 24 of the 96 group stages played out in the FIFA tournament since 1930 to exhibit dominance (25%). Instead we find that nearly half of them (47 of 96) exhibit dominance. Of the 7 scenarios that occur most frequently (five groups or more with that result) 5 of them exhibit dominance, again despite the relatively low

number Â of Â dominant Â scenarios. Â Â Finally, Â by Â far Â the Â most Â popular Â scenario, Â occurring Â 11 Â times Â (the Â next Â closest Â was Â 7), Â is Â the Â 6-Ââ&#x20AC;?win/9630 Â group Â which Â is Â the Â classic Â example Â of Â sports Â dominance. Â Â These Â results Â are Â somewhat Â to Â be Â expected Â considering Â that Â the Â process Â used Â by Â FIFA Â for Â creating Â the Â groups Â in Â many Â of Â the Â tournaments Â involved Â placing Â the Â best Â 8 Â teams Â into Â separate Â groups Â based Â on Â their Â qualifying Â performances. Â Â This Â greatly Â increases Â the Â chance Â of Â a Â group Â having Â a Â dominant Â team. Â Â Even Â so, Â we Â see Â that Â the Â average Â fanâ&#x20AC;&#x2122;s Â expectation Â that Â a Â group Â will Â follow Â the Â pattern Â of Â dominance Â is Â not Â entirely Â unwarranted. Â Appendix Â on Â Counting Â Â Calculating Â the Â number Â of Â permutations Â represented Â by Â each Â scenario Â was Â done Â using Â combinatorics. Â Â When Â positions Â were Â indistinguishable Â the Â number Â of Â possibilities Â was Â calculated Â using Â combinations. Â Â When Â positions Â were Â distinguishable Â permutations Â were Â used. Â Â Scenarios Â with Â an Â asterisk Â were Â treated Â individually. Â Scenario Â Type Â AAAA Â AAAB, Â ABBB Â AABB Â AABC, Â ABBC, Â ABCC Â ABCD Â

Calculations Â Used Â !đ??ś! = 1 Â !đ??ś! Ă&#x2014; !đ?&#x2018;&#x192;! = 4 Â !đ??ś! Ă&#x2014; !đ??ś! = 6 Â !đ??ś! Ă&#x2014; !đ?&#x2018;&#x192;! = 12 Â !đ?&#x2018;&#x192;! = 24 Â

Â AAAA*: Â This Â scenario Â involves Â ordering Â the Â four Â teams Â in Â a Â non-Ââ&#x20AC;?transitive Â loop. Â Â For Â any Â team Â W Â there Â are Â three Â teams Â which Â W Â might Â beat. Â Â For Â the Â next Â team Â in Â the Â sequence Â there Â are Â two Â remaining Â teams Â from Â which Â to Â choose. Â Â The Â last Â team Â must Â be Â fourth Â in Â the Â sequence, Â so Â there Â are Â 3 Â â&#x20AC;˘ Â 2 Â â&#x20AC;˘ Â 1 Â = Â 6 Â permutations Â for Â this Â scenario. Â Â Â WĂ ď&#x192; XĂ ď&#x192; YĂ ď&#x192; ZĂ ď&#x192; W Â Â WĂ ď&#x192; XĂ ď&#x192; ZĂ ď&#x192; YĂ ď&#x192; W Â Â

WĂ ď&#x192; YĂ ď&#x192; XĂ ď&#x192; ZĂ ď&#x192; W Â

WĂ ď&#x192; YĂ ď&#x192; ZĂ ď&#x192; XĂ ď&#x192; W Â

WĂ ď&#x192; ZĂ ď&#x192; XĂ ď&#x192; YĂ ď&#x192; W Â

WĂ ď&#x192; ZĂ ď&#x192; YĂ ď&#x192; XĂ ď&#x192; W Â

AAAB*, Â ABBB*: Â Â These Â scenarios Â involve Â choosing Â three Â teams Â and Â then Â arranging Â them Â in Â a Â non-Ââ&#x20AC;?transitive Â loop. Â Â There Â are Â !đ??ś! Â = Â 4 Â ways Â to Â choose Â three Â teams. Â Â For Â any Â given Â team Â there Â are Â two Â teams Â which Â it Â may Â beat Â when Â creating Â the Â non-Ââ&#x20AC;?transitive Â loop, Â so Â there Â are Â 4 Â â&#x20AC;˘ Â 2 Â = Â 8 Â permutations Â for Â these Â scenarios. Â Â

W, Â Â Â XĂ ď&#x192; YĂ ď&#x192; ZĂ ď&#x192; X Â

W, Â Â Â XĂ ď&#x192; ZĂ ď&#x192; YĂ ď&#x192; X Â

X, Â Â Â WĂ ď&#x192; YĂ ď&#x192; ZĂ ď&#x192; W Â

X, Â Â Â WĂ ď&#x192; ZĂ ď&#x192; YĂ ď&#x192; W Â

Y, Â Â Â WĂ ď&#x192; XĂ ď&#x192; ZĂ ď&#x192; W Â

Y, Â Â Â WĂ ď&#x192; ZĂ ď&#x192; XĂ ď&#x192; W Â

Z, Â Â Â WĂ ď&#x192; XĂ ď&#x192; YĂ ď&#x192; W Â

Z, Â Â Â WĂ ď&#x192; YĂ ď&#x192; XĂ ď&#x192; W Â

AABB*: Â This Â scenario Â involves Â choosing Â two Â teams Â for Â the Â top Â ranks Â ( !đ??ś! = 6) Â and Â then Â pairing Â them Â for Â a Â win Â with Â the Â two Â remaining Â teams Â ( !đ?&#x2018;&#x192;! = 2). Â Â Thus, Â there Â are Â 6 Â â&#x20AC;˘ Â 2 Â = Â 12 Â permutations Â for Â this Â scenario. Â

References 1. "Regulations of the 2014 FIFA World Cup" (PDF). FIFA.com. Fédération Internationale de Football Association. p. 50. Retrieved 20 March2015. Please send feedback and corrections to peterlloydlowe@gmail.com