THE CONSISTENT PREFERENCES APPROACH TO DEDUCTIVE REASONING IN GAMES

THEORY AND DECISION LIBRARY General Editors: W. Leinfellner (Vienna) and G Eberlein (Munich) Series A: Philosophy and Methodology of the Social Sciences Series B: Mathematical and Statistical Methods Series C: Game Theory, Mathematical Programming and Operations Research Series D: System Theory, Knowledge Engineering and Problem Solving

SERIES C: GAME THEORY, MATHEMATICAL PROGRAMMING AND OPERATIONS RESEARCH VOLUME 38

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THE CONSISTENT PREFERENCES APPROACH TO DEDUCTIVE REASONING IN GAMES by

GEIR B. ASHEIM University ofOslo, Norway

~ Springer

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To Kristin

Contents

Dedication List of Figures List of Tables Preface Copyright Permissions

1. INTRODUCTION 1.1 Conditions for Nash equilibrium 1.2 Modeling backward and forward induction 1.3 Integrating decision theory and game theory

v xi Xlll

xv xix

1 2 6 7

2. MOTIVATING EXAMPLES 2.1 Six examples 2.2 Overview over concepts

17

3. DECISION-THEORETIC FRAMEWORK 3.1 Motivation 3.2 Axioms 3.3 Representation results

21 22 26 29

4. BELIEF OPERATORS 4.1 From preferences to accessibility relations 4.2 Defining and characterizing belief operators 4.3 Properties of belief operators 4.4 Relation to other non-monotonic operators

37 40 44 46 48

5. BASIC CHARACTERIZATIONS 5.1 Epistemic modeling of strategic games 5.2 Consistency of preferences

53 53 58

11 11

CONSISTENT PREFERENCES

Vlll

5.3 Admissible consistency of preferences

62

6. RELAXING COMPLETENESS 6.1 Epistemic modeling of strategic games (cont.) 6.2 Consistency of preferences (cont.) 6.3 Admissible consistency of preferences (cont.)

69

7. BACKWARD INDUCTION 7.1 Epistemic modeling of extensive games 7.2 Initial belief of opponent rationality 7.3 Belief in each subgame of opponent rationality 7.4 Discussion

79

69

73 75

82 87 89

94

8. SEQUENTIALITY 8.1 Epistemic modeling of extensive games (cont .) 8.2 Sequential consistency 8.3 Weak sequential consistency 804 Relation to backward induction

104 107 113

9. QUASI-PERFECTNESS 9.1 Quasi-perfect consistency 9.2 Relating rationalizability concepts

115 116 118

10. PROPERNESS 10.1 An illustration 10.2 Proper consistency 10.3 Relating rationalizability concepts (cont .) lOA Induction in a betting game

121 123 124 127 129

11. CAPTURING FORWARD INDUCTION THROUGH FULL PERMISSIBILITY 11.1 Illustrating the key features 11.2 IECFA and fully permissible sets 11.3 Full admissible consistency 1104 Investigating examples 11.5 Related literature

99

101

133 135 138 142 149 152

Contents

IX

12. APPLYING FULL PERMISSIBILITY TO EXTENSIVE GAMES 12.1 Motivation 12.2 Justifying extensive form application 12.3 Backward indu ction 12.4 Forward indu ction 12.5 Concluding remarks

155 155 159 162 168 173

App endices A Proofs of result s in Chapter 4 B Proofs of results in Chapters 8- 10 C Proofs of results in Chapte r 11

175 175 181 193

References

195

Index

201

About the aut hor

203

List of F igures

2.1

G 1 ("battle-of-t he-sexes" ).

12

2.2

G2, illustrating ded uct ive reasoning. G3, illustrating weak domin ance.

13

2.3

13

G~ and a corresponding exte nsive form r~ (a "cent ipede" game) .

14

2.5

G~

16

2.6

G~

2.4

and a corresponding exte nsive form r~ .

and a corresponding exte nsive form r~ ("battleof-the-sexes with an outs ide option").

17

3.1

r 4 and its strategic form .

25

4.1

The basic st ructure of the analysis in Chapter 4.

39

7.1

r s (a four-legged "cent ipede" game).

93

8.1

I'6 and its st ra tegic form .

112

8.2

r~

and its pure st ra tegy reduced st rategic form .

112

10.1

G7 , illustrat ing common certain belief of proper consistency.

123

10.2

A bet ting game.

129

10.3

The st ra tegic form of the bet ting game.

130

11.1

Gs , illustrating that IEWDS may be problematic.

134

11.2

Gg, illustrating t he key features of full admissible consistency.

11.3 12.1

136

GlO , illustrating the relation between IECFA and IEWDS.

142

r 11 and its pure st rategy reduced strateg ic form.

156

xii

CONSISTENT PREFERENCES

12.2

12.3 12.4 12.5

Reduced form Ofr 12 (a 3-period "prisoners' dilemma" game) . G 13 (the pure strategy reduced strategic form of "burning money"). r~ and its pure strategy reduced strategic form. r 14 and its pure strategy reduced strategic form.

166 169 171 172

List of Tables

0.1 2.1 2.2 3.1 7.1 7.2 10.1 12.1

The main interactions between the chapters. XVI Relationships between different equilibrium concepts. 18 Relationships between different rationalizability concepts. 19 Relationships between different sets of axioms and their representations. 29 An epistemic model for G~ with corresponding ex89 tensive form r~ . An epistemic model for r5. 93 An epistemic model for the betting game. 131 Applying IECFA to "burn ing money" . 170

Preface

During the last decade I have explored the consequences of what I have chosen to call the 'consistent preferences' approach to deductive reasoning in games . To a great extent this work has been done in cooperation with my co-authors Martin Dufwenberg, Andres Perea, and Ylva Sovik, and it has lead to a series of journal articles. This book presents the results of this research program. Since the present format permits a more extensive motivation for and presentation of the analysis, it is my hope that the content will be of interest to a wider audience than the corresponding journal articles can reach. In addition to active researcher in the field, it is intended for graduate students and others that wish to study epistemic conditions for equilibrium and rationalizability concepts in game theory.

Structure of the book This book consists of twelve chapters. The main interactions between the chapters are illustrated in Table 0.1. As Table 0.1 indicates, the chapters can be organized into four different parts. Chapters 1 and 2 motivate the subsequent analysis by introducing the 'consistent preferences' approach, and by presenting examples and concepts that are revisited throughout the book. Chapters 3 and 4 present the decision-theoretic framework and the belief operators that are used in later chapters. Chapters 5, 6, 10, and 11 analyze games in the strategic form , while the remaining chapters-Chapters 7, 8, 9, and 12-are concerned with games in the extensive form. The material can, however, also be organized along the vertical axis in Table 0.1. Chapters 5, 8, 9, and 10 are concerned with players that are endowed with complete preferences over their own strategies. In con-

CONSISTENT PREFERENCES

XVI

Table 0.1. The ma in int eractions between the chapte rs.

Chapter 11 Chapter 1

Chapter 4

-ll

-,

Chapter 2

~

11' Chapter 3

~

11' Chapter 6

-,

i

~

Chapter 5

~

i ~

Motivation

Prelim inaries

Strategic games

Chapter 7

1 ~

Chapter 8

-ll

-ll Chapter 10

Chapter 12

f--

Chapter 9 Extensive games

strast, Chapters 4, 6, 7, 11, and 12 present analyses that allow players to have incomplete preferences, corresponding to an inability to assign subjective probabilities to the strategies of their opponents. The generalization to possibly incomp lete preferences is motivated in Section 3.1, and is an essential feature of the analysis in Chapter 11. Not e also that the concepts of Chapters 7, 8, 9, and 10 imply backward induction but not forward induction, while the concept of Chapters 11 and 12 promotes forward induction but not necessarily backward induction.

Notes on the history of t he research program While the arrows in Tab le 0.1 seek to guide the reader through the material presented here , they are not indicative of the chronological development of this work. I started my work on non-equilibrium concepts in games in 1993 by considering the games that are illustrated in Figures 12.1-12.4. After joining forces with Martin Dufwenberg-who had independently developed the same basic intuition about what deductive reasoning could lead to in these examples-we started in 1994 work on our joint papers "Admissibility and common belief" and "Deduct ive reasoning in extensive games" , pub lished in Games and Economi c Behavior and Economic Journal in 2003, and incorporated as Chapters 11 and 12 in this book. '

1 "Ded uct ive reasoning in extensive games" was awarded the Royal Econom ic Society Prize for the be st pap er publish ed in the Economic Journal in 2003 .

PREFACE

xvii

An important goal for our productive collaboration was to present an appropriate formalization of the intuition we had, both in terms of interactive epistemology and the underlying decision-theoretic framework . This endeavor lead to the more general methodological conclusion that interactive epistemology in games could fruitfully focus on the preferences of players, rather than their choice. This is precisely what is referred to as the 'consistent preferences' approach in this book. In 1998 I turned to the question: Could the 'consistent preferences' approach be used to characterize existing game-theoretic concepts? This lead to a characterization of proper rationalizability in "Proper rationalizability in lexicographic beliefs" , published in International Journal of Game Theory in 2001 and incorporated as Chapter 10, and conditions for the backward induction procedure in "On the epistemic foundation for backward induction', published in Mathematical Social Sciences in 2002 and incorporated as Chapter 7. Moreover , starting in 1999, Andres Perea and I asked whether the approach could be used to characterize equilibrium concepts for extensive games and define rationalizability concepts for such games. This lead to the paper "Sequential and quasi-perfect ratioanalizability in extensive games", published in Games and Economic Behavior in 2005 and incorporated as Chapters 8 and 9. This paper also contains a contribution to decision theory that is reported as Proposition 5 of Chapter 3. Finally, the research program spurred an interest in the belief operators involved in these and other contributions studying epistemic conditions for game-theoretic concepts. Hence, from 2000 on, Ylva Sevik and I investigated the semantics of such operators by means of accessibility relations, as used in modal logic. This is reported in "Preference-based belief operators", published in Moiliemaiicol Sociol Sciences in 2005 and incorporated in Chapter 4.

Acknowledgements My collaboration with Martin Dufwenberg, Andres Perea, and Ylva Sevik has been important for this work. Not only do I build on and reproduce joint research with them, but they have also contributed comments and suggestions on earlier drafts of this book . I am very grateful for the insights that I have obtained through the cooperation with my co-authors. I learnt game theory by visiting Stanford University (1985-86, sponsored by Peter Hammond, and again 2001-02, then invited by Kenneth Arrow and Lawrence Goulder) , Northwestern University (1995, made

xviii

CONSISTENT PREFERENCES

possible by Ehud Kalai and Jeroen Swinkels), and Harvard University (2000, sponsored by Martin Weitzman). It was a privilege to take part in lectures, seminars, and discussions during these visits, and the hospitality of these institutions are gratefully acknowledged. I also want to thank Joseph Greenberg for many discussions during visits to Haifa and McGill Universities, and Jorgen Weibull for inviting me to Stockholm in 1994, thus initiating my cooperation with Martin Dufwenberg . During the years that I have done research related to epistemic analysis of games , I have had the pleasure to have numerous exchanges of ideas with Pierpaolo Battigalli, Adam Brandenburger, Aanund Hylland, Stephen Morris, and Larry Samuelson. Their interest, advice, and support have been of great value. In addition to the help and input from the scholars mentioned above, I am also grateful to comments from and discussions with, among others, Horarcio Arlo-Costa, Kaushik Basu , Elchanan Ben-Porath, James Bergin, Giacomo Bonanno, Kristin Dale, Eddie Dekel, Yossi Feinberg , Amanda Friedenberg, Drew Fudenberg, Takako Fujiwara-Greve, Joseph Halpern, Mamoru Kaneko, Terje Lensberg, Barton Lipman, Andrew McLennan, George Mailath, Marco Mariotti, Xiao Luo, Frank Schuhmacher, Oliver Schulte , Yoam Shoham, Johan van Benthem, and Licun Xue. The challenges that have been offered by journal editors, associate editors, and referees are also, in retrospect , greatly appreciated. Finally, I am grateful to Hans Peters for having taken the initiative to this book project, to Springer for giving me the opportunity to publish the resulting manuscript, to Cathelijne van Herwaarden and Herma Drees at Springer for editorial assistance, to Elsevier , the Royal Economic Society and Springer for permission to incorporate material from published articles, to the Research Council of Norway and the Hewlett Foundation for financial support , and to my own institution, the Department of Economics at the University of Oslo, for providing me with working conditions that have made this project possible. Geir B. Asheim Oslo, March 2005

xix

Copyright Perm issions

1 G EIR B . ASHEIM

Proper rationalizability in lexicographic beliefs, International Journal of Gam e Th eory 30 (2001), 453-478. Incorp orat ed as Chapter 10, by permission of Springer. 2 G EIR B. A SHEIM

On the episte mic foundation for backward indu cti on, Math ematical Social Sciences 44 (2002), 121-144. Incorp orated as Chapt er 7, by permission of Elsevier.

3

G EIR

B.

A SHEIM AN D MART IN D UFWENBERG

Admissibility and common belief, Gam es and Economic B ehavior 42 (2003), 208-234. Incorporated as Chapter 11, by permission of Elsevier. 4

GEIR

B.

ASHEIM AND MARTI N D UFWENBERG

Deductive reasoning in exte nsive games, Econom ic Journal 113 (2003), 305-325. Incorporated as Chapter 12, by permission of the Royal Economic Society. 5

GEIR

B.

ASHEIM AND ANDRES PEREA

Sequential and quasi-perfect rationalizability in exte nsive games, Gam es and Economic B ehavior 53 (2005), 15-42. Incorporated as Chapters 8 and 9, by permission of Elsevier. 6 GEIR B . ASHEIM AND YLVA S 0 VIK

Preference-based belief operators , Math ematical Social Sciences 50 (2005) , 61- 82. Incorp orat ed as Chapter 4, by permission of Elsevier.

Chapter 1

INTRODUCTION

This book presents, applies, and synthesizes what my co-authors and I have called the 'consist ent preferences' approach to deductive reasoning in games . Briefly described, this means that the object of the analysis is the ranking by each player of his own strategies, rather than his choice. The ranking can be required to be consistent (in different senses) with his beliefs about the opponent's ranking of her strategies. This can be contrasted to the usual 'rational choice' approach where a player's strategy choice is (in different senses) rational given his beliefs about the opponent 's strategy choice. Our approach has turned out to be fruitful for providing epistemic conditions for backward and forward induction, and for defining or characterizing concepts like proper, quasi-perfect and sequential rationalizability. It also facilitates the integration of game theory and epistemic analysis with the underlying decision-theoretic foundation. The present text considers a setting where the players have preferences over their own strategies in a game, and investigates the following main question: What preferences may be viewed as "reasonable" , provided that each player takes into account the rationality of the opponent, he takes into account that the opponent takes into account the player 's own rationality, and so forth? And in the extension of this: Can we develop formal, intuitive criteria that eventually lead to a selection of preferences for the players that may be viewed as "reasonable"? The 'consistent preferences' approach as such is not new. It is firmly rooted in a thirty year old game-theoretic tradition where a strategy of a player is interpreted as an expression of the belief (or the "conject ure" )

2

CONSISTENT PREFERENCES

of his opponent; cf., e.g., Harsanyi (1973), Aumann (1987a), and Blume et al. (1991b). What is new in this book (and the papers on which it builds) is that such a 'consistent preferences' approach is used to characterize a wider set of equilibrium concepts and , in particular, to serve as a basis for various types of interactive epistemic analysis where equilibrium assumptions are not made . Throughout this book , games are analyzed from the subjective perspective of each player . Hence, we can only make subjective statements about what a player "will do" , by considering "reasonable" preferences (and the corresponding representation in terms of subjective probabilities) of his opponent. By having a subjective perspective this book follows recent contributions like Feinberg (2005a) and Kaneko and Kline (2004), which however differ from the present approach in many respects.! To illustrate the differences between the two approaches-the 'rational choice' approach on the one hand and the 'consistent preferences' approach on the other-in a setting that will be familiar to most readers, Section 1.1 will be used to consider how epistemic conditions for Nash equilibrium in a strategic game can be formulated within each of these approaches. The remaining Sections 1.2 and 1.3 will provide motivation for the 'consistent preferences' approach through the following two points: 1 It facilitates the analysis of backward and forward induction. 2 It facilitates the integration of game theory and epistemic analysis with the underlying decision-theoretic foundation.

1.1

Conditions for Nash equilibrium

To fix ideas, consider a simple coordination game, where two drivers must choose what side to drive on in order to avoid colliding. In an

1 In the present text, reasoning about hypothetical events will be captured by each player having an initial (interim - after having become aware of his own "type" ) system of conditional preferences; d. Chapters 3 and 4. This system encodes how the player will update his beli efs as actual play devel ops . In contrast, the subjective framework of Feinberg (2005a) does not represent the reasoning from such an interim viewpoint, and beliefs are not constrained to be evolving or revised. Instead, beli efs ar e represented whenever there is a decision to be

made based on the presumption that beliefs should only matter when a decision is made . In

Feinberg's framework, only the ex-post beliefs are present and all ex-post subjective views are equally modeled. Even though also Kaneko and Kline (2004) consider a player having a subjective view on the objective situation, their main point is the inductive derivation of this individual subjective view from individual experiences.

3

Introduction

equilibrium in the 'rational choice' approach, a driver chooses to drive on the right side of the road if he believes that his opponent chooses to drive on the right side of the road. This can be contrasted with an equilibrium in the 'consist ent preferences' approach, where a driver prefers to drive on the right side of the road if he believes that his opponent prefers to drive on the right side of the road. As mentioned, this follows a tradition in equilibrium analysis from Harsanyi (1973) to Blume et al. (1991b). This section presents, as a preliminary analysis, how these two interpretations of Nash equilibrium can be formalized. First, introduce the concept of a strategic game . A strategic twoplayer game G = (51 ,52 , UI, U2) consists of, for each player i, a set S, of pure strategies and a payoff function Ui : 51 x 52 - t JR.. Then, turn to the epistemic modeling. An epistemic model for a strategic game within the 'rational choice' approach will typically specify, for each player i, â€˘ a finite set T; of types, â€˘ a function and

Si :

T;

-t

S, that assigns a strategy choice to each type,

â€˘ for each type t; in Ti, a probability distribution J-l ti E t::..(Tj) on the set of opponent types, where t::..(Tj) denotes the set of probability distributions on Tj. When combined with i's payoff function, the function Sj and the probability distribution j.lt i determine player i's preferences at ti over his own strategies; these preferences will be denoted ?:ti:

Si ?:t i s~

iff

LJ-lti(tj)Ui(Si,Sj(tj)) 2:: LJ-lt i(tj)Ui(S~ ,Sj(tj)) . o 0

This in turn determines i 's set of best responses at ti, which will throughout be referred to as i 's choice set at ti :

Finally, in the context of the 'rational choice' approach, we can define the set of type profiles for which player i chooses rationally:

[rati] := {(tI, t2) E T I x T2 I si(td E Write [rat] := [ratI]

n [rat2].

5i

i

} .

4

CONSISTENT PREFERENCES

It is now straightforward to give sufficient epistemic conditions for a

pure strategy Nash equilibrium:

S: x S2 is a pure strategy Nash equilibrium if there exists an epistemic model with (t1 , t2) E [rat] such that (Sl , S2) = (81 (t1), 82(t2)) and , for each i, pti(tj) = 1 . (Sl, S2)

E

In words , (Sl , S2) is a pure strategy Nash equilibrium if there is mutual belief of a profile of types that rationally choose Sl and S2 . In fact, we need not require mutual belief of the type profile: in line with the insights of Aumann and Brandenburger (1995) (cf. their Preliminary Observation) it is sufficient that there is mutual belief of the strategy profile, as we need not be concerned with what one player believes that the other player believes (or any higher order beliefs). Consider next how to formulate epistemic conditions for a mixed strategy Nash equilibrium. Following, e.g., Harsanyi (1973), Armbruster and Boge (1979), Aumann (1987a) , Brandenburger and Dekel (1989), Blume et al. (1991b) , and Aumann and Brandenburger (1995), a mixed strategy Nash equilibrium is often interpreted as an equilibrium in beliefs. According to this rather prominent view, a player need not randomize in a mixed strategy Nash equilibrium, but may choose some pure strategy. However, the other player does not know which one, and the mixed strategy of the one player is an expression of the belief (or the "conject ure" ) of the other. The 'consistent preferences' approach is well-suited for formulating epistemic conditions for a mixed strategy Nash equilibrium according to this interpretation. An epistemic model for a strategic game within the 'consistent preferences' approach will typically specify, for each player i, â€˘ a finite set T; of types and â€˘ for each type ti in Ti, a probability distribution pti E b.(Sj x T j) on the set of opponent strategy-type pairs. Hence, instead of specifying a function that assigns strategy choices to types, each type's probability distribution is extended to the Cartesian product of the opponent's strategy set and type set . We can still determine type i's preferences at ti over his own strategies,

s, 'c. ti s~

LLpti(Sj ,tj)Ui(Si , Sj);::: LLpt i( Sj ,tj)U i(S~ ,Sj), ~ 0 ~ 0 and i 's choice set at ti: S;i := {s, E S, I Vs~ E Si , Si 'c. ti sa. iff

Introduction

5

However, we are now concerned with what i at ti believes that opponent types do, rather than with what i at ti does himself. Naturally, such beliefs will only be well-defined for opponent types that ti deems subjectively possible , i.e., for player j types in the set

i 路 t路) > TJ路ti '' = {t J. E TJ路II/t r: (S J' J

o}

,

where /-lt i (Sj, tj) := L SES /-lt i (Sj,tj) . Say that the mixed strategy p/iltj J J is induced for tj by ti if tj E T/ i, and for each Sj E Sj,

Finally, in the context of the 'consistent preferences' approach, we can define the set of type profiles for which ti induces a rational mixed strategy for any subjectively possible opponent type:

[iri] :=

{(t

l ,

t z)

E

TI x Tzi Vtj

E

T/ i, p/iit'i

E

b.(S/j) } .

If the true type profile is in [iriJ, then player i 's preferences over his strategies are consistent with the preferences of his opponent. Rather than player j actually being rational, it entails that player i believes that j is rational. Write [ir] := [irl] n [irz]. Through the event [ir] one can formulate sufficient epistemic conditions for a mixed strategy Nash equilibrium, interpreted as an equilibrium in beliefs:

(PI,PZ) E .6.(Sd x .6.(S2) is a mixed strategy Nash equilibrium if there exists an epistemic model with (tl' tz) E [ir] such that (PI ,PZ) = (Plt2 Itl ,pztllt2) and , for each i , /-lti(Sj , tj) = 1 . In words, (PI, pz) is a mixed strategy Nash equilibrium if there is mutual belief of a profile of typ es, where each type induces the opponent's mixed strategy for the other, and where any pure strategy in the induced mixed strategy is rational for the opponent type. Since any pure strategy Nash equilibrium can be viewed as a degenerate mixed strategy Nash equilibrium, these epistemic conditions are sufficient for pure strategy Nash equilibrium as well. Again , we need not require mutual belief of the type profile; it is sufficient that there is mutual belief of each player's belief about the strategy choice of his opponent. It is by no means infeasible to provide epistemic conditions for mixed strategy Nash equilibrium, interpreted as an equilibrium in beliefs, with-

6

CONSISTENT PREFERENCES

in the 'rational choice' approach. Indeed, this is what Aumann and Brandenburger (1995) do through their Theorem A in the case of twoplayer games. One can still argue for the epistemic conditions arising within the 'consistent preferences' approach. If a mixed strategy Nash equilibrium is interpreted as an expression of what each player believes his opponent will do, then one can argue-based on Occam's razor-that the epistemic conditions should specify these beliefs only, and not also what each player actually does. In particular, we need not require, as Aumann and Brandenburger (1995) do, that the players are rational.

1.2

Modeling backward and forward induction

This book is mainly concerned with the analysis of deductive reasoning in games-leading to rationalizability concepts-rather than the study of steady states where coordination problems have been solvedcorresponding to equilibrium concepts. Deductive reasoning within the 'consistent preferences' approach means that events like [ir] will be made subject to interactive epistemology, without assuming that there is mutual belief of the type profile. Backward induction is a prime example of deductive reasoning in games. To capture the backward induction procedure, one must believe that each player chooses rationally at every information set of an extensive game; also at information sets that the player 's own strategy precludes from being reached. As will be indicated through the analysis of Chapters 7-1O-based partly on joint work with Andres Perea-this might be easier to capture by analyzing events where each player believes that the opponent chooses rationally, rather than events where each player actually chooses rationally. The backward induction procedure can be captured by conditions on how each player revises his beliefs after "surprising" choices by the opponent. Therefore, it might be fruitful to characterize this procedure through restrictions on the belief revision policies of the players, rather than through restrictions on their behavior at all information sets (also at information sets that can only be reached if the behavioral restrictions at earlier information sets were not adhered to). As will be apparent in Chapters 7-10, the 'consistent preferences' approach captures the backward induction procedure through conditions imposed directly on the players' belief revision policies. In certain games-like the "bat tle-of-t he-sexes with outside option" game (d. Figure 2.6)-forward induction has considerable bite. To model forward induction, one must essentially assume that each player

Introduction

7

believes that any rational choice by the opponent is infinitely more likely than any choice that is not rational. Again , this might be easier to capture by analyzing events relating to the beliefs of the player, rather than events relating to the behavior of the opponent. Chapters 11 and 12 will report on joint work with Martin Dufwenberg that shows how the 'consist ent preferences ' approach can be used to promote the forward induction outcome. For ease of presentation only two-player games will be considered in this book. This is in part a matter of convenience , as much of the subsequent analysis can essentially be generalized to n-player games (with n > 2). In particular, this applies to the analysis of backward induction in Chapter 7, and to some extent, the analysis of forward induction in Chapters 11 and 12. On the other hand, in the equilibrium analysis of Chapters 5, 8, 9, and 10, a strategy of one player is interpreted as an expression of the belief of his opponent. This interpretation is straightforward in two-player games , but requires that the beliefs of different opponents coincide in games with more than two players-e.g., compare Theorems A and B of Aumann and Brandenburger (1995). Moreover, by only considering two-player games we can avoid the issue of whether (and if so, how) each player's beliefs about the strategy choices of his opponents are stochastically indep endent. Throughout, player 1 will be viewed as male (e.g., "he chooses among his strategies") , while player 2 will be viewed as female (e.g., "she believes that player 1 .. . " ). Also, in the examples, the strategies of player 1 will be denoted by upper case symbols (e.g., Land R), while the strategies of player 2 will be denoted by lower case symbols (e.g., â‚Ź and r) .

1.3

Integrating decision theory and game theory

When a player in a two-player strategic game considers what decision to make (i.e., what strategy to choose) , only his belief about the strategy choice of his opponent matters for his decision. However, in order to form a well-judged belief regarding the choice of his opponent, he should take her rationality into account . This makes it necessary for the player to consider his belief about her belief about his strategy choice. And so forth. Hence, the uncertainty faced by a player i concerns (a) the strategy choice of his opponent j , (b) j 's belief about i's strategy choice, and so on; d. Tan and Werlang (1988). A type of a player i corresponds to (a) a belief about j's strategy choice, (b) a belief about the pair of j's strategy choice and j's belief about i 's strategy choice, and so on.

8

CONSISTENT PREFERENCES

Models of such infinite hierarchies of beliefs-see, e.g., Boge and Eisele (1979), Mertens and Zamir (1985), Brandenburger and Dekel (1993), and Epstein and Wang (1996)-can, under given conditions, yield B1 x T 1 X B2 X T2 as the 'belief-complete' state space, where T; is the set of all feasible types of player i. Under such conditions, there is a homeomorphism for each i between T; and the set of beliefs on S, x Bj x Tj . In the decision problem of any player i, i's decision is to choose one of his own strategies. For the modeling of this problem, i's belief about his own strategy choice is not relevant and can be ignored . This does not mean that player i is not aware of his own choice. It signifies that such awareness plays no role in the analysis, and is thus redundant .f Hence, in the setting of a strategic game the belief of each type of player i can be restricted to the set of opponent strategy-type pairs, Bj x Tj. Combined with the payoff function specified by the strategic game, a belief on Bj x Tj yields preferences over player i's strategies. As discussed in Section 5.1, the above results on 'belief-complete' state spaces are not needed (since only finite games are treated without 'belief-completeness' being imposed) and not always applicable in the setting of the present text (since some of the analysis-e.g. in Chapters 6, 7, 11, and 12-allows for incomplete preferences). Indeed, infinite hierarchies of beliefs can be modeled by an implicit but 'belief-incomplete' model -with a finite type set T; for each player i-where the belief of a player corresponds to the player 's type, and where the belief of the player concerns the opponent's strategy-type pair. If we let each player be aware of his own type (as we will assume throughout), this leads to an epistemic model where the state space of player i is T; X Bj x Ti: For each player, this is a standard decisiontheoretic formulation in the tradition of Savage (1954), Anscombe and Aumann (1963), and Blume et al. (1991a): â€˘ Player i as a decision maker is uncertain about what strategy-type pair in Bj x Tj will be realized . â€˘ Player i's type ti determines his belief on Bj x Tj. â€˘ Player i's decision is to choose a (possibly mixed) strategy Pi E Ll(Bi ) ; each such strategy determines the (randomized) outcome of the game as a function of the opponent strategy Sj E Bj.3 2Tan and Werlang (1988) in their Sections 2 and 3 characterize rationalizable strategies without specifying beliefs about one's own choice. 3Hence, a strategy for a player corresponds to an Anscombe-A umann act, assigning a (possibly randomized) outcome to any uncertain state; cf. Chapter 3.

Introduction

9

The model leads, however , to a different state space for each player , which may perh aps be considered problemati c. In t he framework for epistemic modeling of games prop osed by Aumann (1987a)- appli ed by Aumann and Brandenburger (1995) and illustrat ed in Section 1.1-it is also explicitly modeled that each player is aware of his own decision (i.e., his strategy choice). This ent ails that , for each player i , t here is function Si from T; to S, t hat assigns Si(ti) to t.. Fur thermore, it means t hat t he relevant state space is T 1 x T2, which is identi cal for bot h players. In spite of its prevalence, Aumann's model leads to t he following pot enti al problem: If player i is of ty pe ti and in spite of t his were to choose some st rategy s, different from Si(ti), t hen the player would no longer be of ty pe ti (since only Si (ti) is assigned t o td . So what , st arting with a st ate where player i is of typ e ti , would player i believe about his opponent's st rategy choice if he were t o choose Si =J si(td ? In line with t he defense by Aumann and Brand enburger (1995) on pp . 1174-1175, one may argue t hat Aumann's framework is purely descriptive and contains enough informati on to det ermin e whether a player is rational and t hat we need not be concerned about what t he player would have believed if t he state were different. An alt ern ative is, however, to follow Board (2003) in arguing that ti'S belief about his opponent 's st rategy choice should remain unchanged in the counte rfact ual event t hat he were to choose s: =J s, (ti) . The above discussion can be interpreted as support for t he episte mic st ruct ure t hat will underlie t his book, and where t he st ate space of player i is Ii x 5j x T j . T his kind of epistemic model describ es t he fact ors t hat are relevant for each player as a decision maker (namely, what his opponent does and who his opponent is), while being silent about t he awareness of player i of his own decision. Also in this formulation, a different choice by player i cha nges the state, as an element of 51 x T 1 X 52 X T 2 , but it does not influence the type of player i , as a specific st rategy is not assigned to each type. Hence, a different choice by player i does not change his belief about what t he opponents do. In t his setting, t he episte mic analysis concerns t he ty pe profile, and not t he st rategy profile. As we have seen in Section 1.1, and which we will return to in Chapter 5, this is, however , sufficient to state and prove, e.g., a result that corresponds to Aumann and Brand enburger's (1995) Theorem A, provided that mutual belief of rationalit y is weakened to the condition that each player believes that his opponent is rational. As we will see in Chapters 5 and 6 it also facilit ates the int roduction of

10

CONSISTENT PREFERENCES

caution, which then corresponds to players having beliefs that take into account that opponents may make irrational choices, rather than players trembling when they make their choice. Chapters 3 and 4 are concerned with the decision-theoretic framework and epistemic operators derived from this framework . Chapter 3 spells out how the Anscombe-Aumann framework will be used as a decision-theoretic foundation. Following Blume et al. (1991a), continuity will be relaxed. Moreover, two different kinds of generalizations are presented. On the one hand, completeness will be relaxed, as this is not an integral part of the backward induction procedure, and cannot be imposed in the epistemic characterization of forward induction presented in Chapters 11 and 12. On the other hand, flexibility concerning how to specify a system of conditional beliefs will be introduced, leading to a structure that encompasses both the concept of a conditional probability system and conditionals derived from a lexicographic probability system. This flexibility turns out to be essential for the analysis of Chapters 8 and 9. Chapter 4 reports on joint work with Ylva Sevik which derives beliefoperators from the preferences of decision makers and develop their semantics. These belief operators will in later chapters be used in the epistemic characterizations. First, however, motivating examples will be presented and discussed in Chapter 2.

Chapter 2

MOTIVATING EXAMPLES

Through examples this chapter illuminates the features that distinguish the 'consistent preferences' approach from the 'rational choice' approach (cf. Chapter 1). The examples also illustrate issues of relevance when capturing backward and forward induction in models of interactive epistemology. The same examples will be revisited in later chapters. Section 2.1 presents six different games, and contains a discussion of how suggested outcomes in these games can be promoted by different solution concepts. This discussion leads in Section 2.2 to an overview of the solution concepts that will be covered in subsequent chapters. While Section 2.1 will illustrate how various concepts work in the different examples, Section 2.2 will relate the different concepts to each other, and provide references to relevant literature.

2.1

Six examples

Consider the "batt le-of-t he-sexes" game, G I , illustrated in Figure 2.1. This game has two Nash equilibria in pure strategies: (L, ÂŁ.) and (R, r) . In the 'rational choice' approach, the first of these Nash equilibrium is interpreted as player 1 choosing L and player 2 choosing ÂŁ., and these choices being mutual belief. It is a Nash equilibrium since there is mutual belief of the strategy choices and each player's choice is rational, given his belief about the choice of his opponent. In the 'consistent preferences' approach, in contrast, this Nash equilibrium is interpreted as player 1 believing that 2 chooses ÂŁ. and player 2 believing that 1 chooses L, and these conjectures being mutual belief. It is a Nash equilibrium since there is mutual belief of the conjectures about opponent choice and each

12

CONSISTENT PREFERENCES

Figure 2.1.

G 1 ("battle-of-the-sexes") .

player believes that the opponent chooses rationally given the opponent 's conjecture. The preferences of player I-that he ranks L about R-is consistent with the preferences of player 2-that she ranks £ above r, and vice versa. More precisely, that player 1 ranks Labove R is consistent with his beliefs about player 2, namely that he believes that she ranks £ above r and she chooses rationally (Le., chooses a top ranked strategy). The 'consistent preferences' interpretation of Nash equilibrium carries over to the mixed strategy equilibrium when interpreted as an equilibrium in beliefs-cf. the Harsanyi (1973) interpretation discussed in Section 1.1. If player 1 believes with probability 1/4 that 2 chooses £ and with probability 3/4 that 2 chooses r and player 2 believes with probability 3/4 that 1 chooses L and with probability 1/4 that 1 chooses R, and these conjectures are common belief, then the players' beliefs constitute a mixed-strategy Nash equilibrium. It is a Nash equilibrium since there is mutual belief of the conjectures about opponent choice and each player believes that the opponent chooses rationally given the opponent 's conjecture. Rationalizability concepts have no bite in the "bat t le-of-t he-sexes" game, G 1 : Interactive epistemology based on rationality alone cannot guide the players to one of the equilibria. Hence, to illustrate the force of deductive reasoning in games-leading to rationalizability conceptswe must consider other examples. In game G2 of Figure 2.2, there is a unique Nash equilibrium, (L , f) . Furthermore, deductive reasoning will readily lead player 1 to Land player 2 to £. In the 'rat ional choice' approach this works as follows: If player 1 chooses rationally, then he chooses L. This is independent of his conjecture about 2's behavior since L strongly dominates R (as 4 > 3 and 1 > 0). Therefore, if player 2 believes that 1 chooses rationally, and 2 chooses rationally herself, then she chooses £ (since 1 > 0) . This argument shows that L is the unique rationalizable strategy for player 1 and £ is the unique rationalizable strategy for player 2. In the 'consistent preferences' approach, we get: Player 1 ranks Labove R, independently of his conjecture about 2's behavior. If player 2 believes

.13

Mot ivating Examples

e

r

L 4, 1 1, 0

R 3, 0 0, 3

Figure 2.2. G2, illustrating dedu ctive reasoning .

Figure 2.3. G 3 , illust rating weak dom inanc e.

e

that 1 chooses rationally, t hen she believes that 1 chooses L and ranks above r . Therefore, if player 1 believes that 2 chooses rationally, and he believes t hat she believes that 1 chooses rationally, then he believes that 2 chooses e. As we will ret urn to in Chapters 5 and 6, t his is an alternative way to establish Land eas the players ' rationalizable strategies. In any case, t he deductive reasoning leading to rationalizability corresponds to iterated elimination of strongly dominated strategies (IESDS). In game G3 of Figure 2.3, there is also a uniq ue Nash equilibrium, (L, e). However, deductive reasoning is more prob lematic and inte resting in t he case of this game. For each player, both strategies are rationalizable, meaning that rationalizability has no bite in t his game. In pa rt icular , if player 1 deems it subjectively impossible t hat 2 may choose r , then R is a rational choice. Moreover, if player 2 believes that 1 chooses R , t hen r is a rational choice. St ill, we might argue t hat 1 should not rule out t he possibility t hat 2 might choose r , leading him to rank L above R (since L weakly dominat es R) and player 2 t o rank e above r . Such deductive reasoning leads to permissible strategies in t he terminology of Brandenburger (1992). Permissibility corresponds to one round of elimination of all weakly dominat ed strategies followed by iterated elimination of strongly dom inated strategies- the so-called Dekel-Fudenberg procedure, after Dekel and Fudenb erg (1990). It can be formalized in two different ways. On the one hand, within an analysis based on what players do, one can postulate that players make 'almost' rational choices by, in t he spirit of Selten (1975) and his "t rembling hand" , assuming that 'mist akes' are made with (infinitely) sma ll probability. Borgers (1994) shows how such

14

CONSIS TENT PREFEREN CES

r 2, 0 2, 0 Out InL 1,3 4, 2 InR 1, 3 3, 5

Figure 2.4.

1

2

1

Out l £1 21 03

LI 4 2

3

~5

C; and a corr esponding extensive form

r; (a

"cent ipede" ga me) .

an approach does indeed corres pond to t he Dekel-Fudenb erg proc edure and t hus characterizes permissibility. On t he other hand, within an analysis based on what players believe, one can require that players are 'caut ious' , in the sense of deeming no opponent strategy as subject ively impossible. This approach to permi ssibility- which is in t he spirit of Blume et al. (1991b) and Brandenburger (1992)-combines such caut ion with an assumption t ha t each player believes t hat t he opp onent is rational. It is shown in Cha pters 5 and 6 how t his yields an alternative cha racte rization of permi ssibility, where one need not consider whether players in fact are rational. Let us t hen t urn to an expanded version of G3 , namely t he game G~ illust rat ed in Figure 2.4 wit h a corresponding ext ensive form r~. Following Rosenthal (1981) r~ is often called a "cent ipede" game . Here, (Out, £) is norm ally suggested as a solutio n for t his game. In t he st rategic form G3 , t his suggestion can be obtained by it erat ed (maximal) eliminat ion of weakly domin ated st rategies (IEW DS), and in t he extensive form r~ , it is based on backward ind uction . Whil e epistemic conditions for t he procedure of IE WD S have been given by Brandenburger and Keisler (2002)-see also the related work by Batti galli and Siniscalchi (2002)IEWDS will fall outside the class of pro cedures t hat will be charac te rized in t his book. The pro cedure of backward induction , on t he ot her hand, will play a cent ra l role in Cha pte rs 7-10. Permissibility, which corresponds to the Dekel-Fudenb erg proc edure, does not promote only (Out, £) in t he games of Figure 2.4. While t he Dekel-Fudenb erg procedure eliminates the weakly domin ated st rategy InR, t his pro cedure does not allow for further round s of weak elimination. Hence, since r is not st rongly domin ated by £ even after t he eliminat ion of InR, r will not be elimina te d by t he Dekel-Fudenb erg procedure. Hence, InL as well as Out are permi ssible for player 1, and r as well as e are permissible for player 2. In t he extensive game , r~ , one can give t he following intui tion for how InL and r are compatible wit h t he deduct ive reasoning underlying

Motivating Examples

15

permissibility: If player 1 believes that player 2 will choose £, then he prefers Out to his two other strategies. Similarly, if player 2 assigns probability one to player 1 choosing Out, and revises her beliefs by assigning probability one to InL conditional on being asked to play, then she prefers £ to r. However, if player 2 assigns probability one to player 1 choosing Out, but revises her beliefs so that InL and InR are equally likely conditional on being asked to play, then she prefers r to eSo if player 1 assigns sufficient probability to player 2 being of the latter type and believes-conditional on her being of this type-that she will be rational by choosing her top-ranked strategy r, then he will prefer InL to his two other strategies. Following Ben-Porath (1997), Chapter 7 demonstrates within a formal epistemic model how such interactive beliefs are consistent with the assumptions underlying permissibility. As shown by Ben-Porath (1997), when permissibility is applied to an extensive game like r~, each player must believe that her opponent chooses rationally as long as the opponent's behavior is consistent with the player's initial beliefs. However, conditional on finding herself at an information set that contradicts her previous belief about his behavior, she is allowed to believe that he will no longer choose rationally. E.g., in r~ it is OK for player 2 to assign positive probability to the irrational strategy InR conditional on being asked to play, provided that she had originally assigned probability one to player 1 rationally choosing Out. An alternative is that the player should still believe that her opponent will choose rationally, even conditionally on being informed about "surprising" moves. Chapters 7-9 will consider the event that each player believes that her opponent chooses rationally at all his information sets within models of interactive epistemology. Building on joint work with Andres Perea, this provides • epistemic conditions for backward induction and • definitions for the concepts of sequential and quasi-perfect rationalizability. Note that requiring that a player believes that her opponent chooses rationally at all his information sets is a requirement imposed on her belief revision policy, not on her actual behavior. It therefore fits well within the 'consistent preferences' approach. If we move to an expanded version of G2 , namely the game G~ illustrated in Figure 2.5 with a corresponding extensive form r~ , not even the event that each player believes that the opponent chooses rationally at all his information sets, will be sufficient for reaching the solution that

16

CONSISTENT PREFERENCES

2 2

r Out 2, 2 2, 2 lnL 4, 1 1,0 lnR 3, 0 0, 3

Out

1

lnL

lnR

.......__... .._f . £ 4 1

. _

£

r

1

o

3 0

r

o 3

Figure 2.5. G~ and a corresponding extensive form r~ .

one would normally suggest , namely (lnL , f) . This outcome is supported by the following deductive reasoning: Since lnL strongly dominates lnR, implying that player 1 prefers the former strategy to the latter, player 2 should deem lnL much more likely than lnR conditional on being asked to play, and hence prefer £ to r. This in turn would lead player 1 to prefer lnL to his two other strategies if he believes that player 2 will be rational by choosing her top-ranked strategy £. However, the concepts of sequential and quasi-perfect rationalizability only preclude that player 2 unconditionally assigns positive probability to player 1 choosing lnR. If player 2 assigns probability one to player 1 choosing Out, then she may-when revising her beliefs conditional on being asked to play-assign sufficient probability to lnR so that r is preferred to £. If player 1 assigns sufficient probability to player 2 being of such a type, then he will prefer Out to his two other strategies. The outcome (lnL, £) can be promoted by considering the event that player 2 respects the preferences of her opponent by deeming one opponent strategy infinitely more likely than another if the opponent prefers the former to the latter. Respect of opponent preferences was first considered by Blume et al. (1991b) in their characterization of proper equilibrium. Being a requirement on the beliefs of players, it fits nicely into the 'consistent preferences' approach. Within a model of interactive epistemology, Chapter 10 characterizes the concept of proper rationalizability by considering the event that each player respects opponent preferences . Proper rationalizability implies backward induction. However, even though it yields conclusions that coincide with IEWDS in all of the examples above, this conclusion does not hold in general, as will be shown by the next example and further discussed in Chapter 10. Lastly, turn to an expanded version of Gl, namely the game Gi illustrated in Figure 2.6 with a corresponding extensive form ri. The exten-

17

Motivating Examples

2

Out

1

2

r Out 2, 2 2, 2 InL 3, 1 0, 0 InR 0, 0 1,3

InL

InR

..................~

£

r

3 1

o o

Figure 2.6. c; and a corresponding extensive form outside option") .

.

£ 0 0

ri

r 1 3

("battle-of-the-sexes with an

sive game f~ is referred to as the "battle-of-the-sexes with an outside option" game. This game was introduced by Kreps and Wilson (1982) (who credit Elon Kohlberg) and is often used to illustrate forward induction, namely that player 2 through deductive reasoning should figure out that player 1 has chosen InL and aims for the payoff 3 if 2 is being asked to play. Respect of preferences only requires player 2 to deem InR infinitely less likely than Out, since the latter strategy strongly dominates the former; it does not require 2 to deem InR infinitely less likely than InL and thereby prefer £ to r. In contrast, IEWDS eliminates all strategies except InL for player 1 and £ for player 2, thereby promoting the forward induction outcome. Chapter 11 contains a critical assessment of how iterated weak dominance promotes forward induction in this and other examples. Based on joint work with Martin Dufwenberg, it will be suggested how forward induction can be promoted by strengthening the concept of permissibility to our notion of full permissibility. Full permissibility is characterized by conditions levied on the beliefs of players, and therefore fits naturally into the 'consistent preferences' approach. In the final Chapter 12 this notion will be further illustrated through a series of extensive games, illustrating how it yields forward induction, while not always supporting backward induction (indeed, f 3 is an example of an extensive game where full permissibility does not promote the backward induction outcome) .

2.2

Overview over concepts

To provide a structure for the concepts that will be defined and characterized in the subsequent chapters, it might be useful as a roadmap to present an overview over these concepts and their relationships.

18

CONSISTENT PR EFEREN CES

Table 2.1.

Relationships between different equilibrium concept s.

Proper equilibrium Myerson (1978)

1 St rategic f orm perfect equil. Selten (1975) Na sh equi -

librium

<-

<-

Quasi-perfect equilibrium van Damme (1984)

1

1

Weak sequential equilibrium Reny (1992)

<-

Sequential equilibrium Kreps & Wilson (1982)

First, consider t he equilibrium concepts of Table 2.1. Here, weak sequential equilibrium refers to t he equilibrium concept- defined by Reny (1992)- t hat results when each player optimi zes only at information sets that the player 's own st rategy does not preclude from being reached. Moreover, quasi-perfect equilibrium is the concept defined by van Damme (1984) and which differs from Selt en's (1975) exte nsive form perfect equilibrium by having each player ignore the possibility of his own future mistakes. The arrows indicat e t hat any prop er equilibrium corresponds to a quasi-p erfect equilibrium and so forth. Nash equilibrium and (stra tegic form ) perfect equilibrium will be characterized in Chapter 5, while sequenti al equilibrium, quasi-perfect equilibrium, and proper equilibrium will be characterized in Chapte rs 8, 9, and 10, respectively. The non-equilibrium analogs to these equilibrium concepts are illustrated in Table 2.2. Again , the arrows indicate that proper rationalizability implies quasi-perfect rationalizability and so forth. Of course, the notion of rationalizability due to Bernheim (1984) and Pearce (1984) is a non-equilibrium analo g to Nash equilibrium. Likewise, t he noti on of permissibility due to Borgers (1994) and Brandenburger (1992) corresponds to Selten 's (1975) st rategic form perfect equilibrium, and the noti on of weak sequential ration alizability du e to Ben-Porath (1997)-coined 'weak extensive form rationalizablity' by Battigalli and Bonann o (1999)- is a non-equilibrium analog of weak sequent ial equilibrium. Fur thermore, sequenti al rationalizability due to Dekel et al. (1999, 2002), quasi-perfect rationalizability du e to Asheim and Perea (2005), and proper rati onal-

19

Motivating Examples

Table 2.2.

Relat ionships between different rationa lizability concepts.

Common cert. b eli ef that each p layer . . . . .. is cautious and respects preferences

. .. is cautious

. . . is not necessarily cautious

. .. believes the oppon. chooses rationa lly only init ially, in the whole game

.. . believes the oppon. chooses rationally at all reachable info. sets

[n .a.]

[n.a.]

[n.a.]

Permissibility Borgers (1994) Brandenb. (1992) Dek. & Fud . (1990) [Chapters 5-6]

Rationalizability Bernh. (1984) Pearce (1984) [Chapt ers 5-6]

Does not imply backward ind o

~

. . . believes the oppon. chooses rationally at all info. sets Proper rationalizability Schuhm. (1999) [Chapter 10]

1 ~

Quasi-perfect rationalizability Ash. & Per. (2004) [Chapter 9]

1

1

Weak sequential rationalizability Ben-Porath (1997) [Chapt er 8]

Sequential rationalizability Dekel et al. (1999,2002) [Chapter 8]

Does not imply backward ind o

~

Implies backward indo

izability due to Schuhmacher (1999) are non-equilibrium analogs to sequential equilibr ium , quasi-perfect equilibrium, and proper equilibrium, respectively. As ind icated by Tab le 2.2, t hese concepts will be treated in Chapters 5, 6, 8, 9, and 10, and t hey are characterized by â€˘ on t he one hand, whet her each player is cautious and respects opponent preferences, and â€˘ on the ot her hand, whether each player believes t hat his oppo nent chooses rationally only initi ally (in t he whole game), or at all reachable inform at ion sets, or at all information sets . T his taxonomy defines events which are made subject to common certain belief, where 'certain belief' is t he epistemic operator t hat will be used for t he interactive epistemology. T his operator is defined in Chapte r 4 and will have the following meaning: An event is said to be 'certainly believed' if the complement is deemed subjectively impossible.

20

CONSIS TEN T PREFERENCES

Throughout this book, we will analyze assumptions about players' prefere nces, leading to events t hat are subset s of type profiles. We can st ill make subjective statements about what a player "will do" , by considering t he preferences (and t he corresponding representation in te rms of sub jective probabilities) of t he other player. For t he concepts in t he left and cente r columns of Table 2.2, we can do more t han t his, if we so wish. E.g., when characterizing weak sequent ial rationalizability, we can consider t he event of rational pure choice at all reachable information sets, and assume t hat this event is commonly believed (where t he te rm 'belief' is used in t he sense of 'belief with probability one') . These assumptions yield subsets of stra tegy profiles, leading t o direct beh avioral implications within t he model. This does not carry over to the concepts in the right column. It is problematic to define the event of rational pur e choice at all inform ation sets, since reaching a non-reachable information set may cont radict rational choice at earlier information sets . Also, if we consider the event of (any kind of) rati onal pure choice, t hen we cannot use common certain belief, since t his-combined wit h rational choice- would pr event well-defined condit ional beliefs after irrational opponent choices. However , common belief (wit h probability one) of the event t hat each player believes his oppone nt chooses rationally at all information sets does not yield backward induction in generic perfect informat ion games, as shown in t he counte rexample illustrated in Figure 7.1. Common certain belief is essent ial for our analysis of the concepts in t he right column of Table 2; t his complicates obtaining direct behavioral implicati ons. Before defining t he various belief opera tors t hat will be used in t he lat er chapte rs, the decision-theoretic framework will be presented and analyzed in Chapter 3.

Chapter 3

DECISION-THEORETIC FRAMEWORK

In the 'consistent preferences' approach to deductive reasoning in games, the object of the analysis is each player's preferences over his own strategies, rather than his choice. The preferences can be required to be consistent (in different senses) with his beliefs about the opponent's preferences over her strategies. The player's preferences depend on his belief about the strategy choice of his opponent. Furthermore, in order for the player to consider the preferences of his opponent, her belief about his strategy profile matters, and so forth . What kind of decision-theoretic framework is suited for such analysis? This chapter spells out how the framework proposed by Anscombe and Aumann (1963) will be used as a decision-theoretic foundation . Following Blume et al. (1991a), the Archimedean property will be relaxed. Moreover, two different kinds of generalizations will be presented: (i) Completeness will be relaxed , as this is not an integral part of the backward induction procedure (ef. the analysis of Chapter 7), and cannot be imposed in the epistemic characterization of forward induction presented in Chapters 11 and 12.

(ii) Flexibility concerning how to specify conditional preferences, leading to a structure that encompasses both the concept of a conditional probability system and conditionals derived from a lexicographic probability system. This flexibility turns out to be essential for the analysis of Chapters 8 and 9. Section 3.1 motivates these generalizations, as well as providing reasons for the choice of the Ascombe-Aumann framework. Section 3.2 introduces the different sets of axioms that will be considered, while the final Section 3.3 presents the corresponding representation results.

22

3.1

CONSISTENT PREFERENCES

Motivation

Standard decision theory under uncertainty concerns two different kinds of decisions. 1 In the first kind, the object of choice is lotteries. There is a given set of outcomes, and a lottery is an objective probability distribution over outcomes. If the decision maker satisfies the von NeumannMorgenstern axioms-cf. von Neumann and Morgenstern (1947)then one can assign utilities to outcomes, so that the decision maker prefers one lottery to another if the former has higher expected utility. 2 In the second kind, the object of choice is acts. There is a given set of outcomes and a given set of uncertain states, and an act is a function from states to outcomes. If the decision maker satisfies the Savage (1954) axioms, then one can assign utilities to outcomes and subjective probabilities to states, so that the decision maker prefers one act to another if the former has higher (subjective) expected utility. An act in the sense of Anscombe and Aumann (1963) is a function from states to objective randomizations over outcomes.' By considering acts in this sense they are able to extend the von Neumann-Morgenstern theory so that the utilities assigned to outcomes are determined solely from preferences over lotteries, while the subjective probabilities assigned to states are determined when also acts are considered . A strategy in a game is a function that, for each opponent strategy choice, determines an outcome. A pure strategy determines for each opponent strategy a deterministic outcome, while a mixed strategy determines for each opponent strategy an objective randomization over the set of outcomes. Hence, a pure strategy is an example of an act in the sense of Savage (1954), while a mixed strategy is an example of an act in the generalized sense of Anscombe and Aumann (1963). Allowing for objective randomizations and using Anscombe-Aumann acts are convenient for two reasons in the present context: â€˘ The Anscombe-Aumann framework allows a player 's payoff function to be a von Neumann-Morgenstern (vNM) utility function determined from his preferences over randomized outcomes, independently 1 Ans combe and Aumann (1963) use the term 'roulet t e lottery ' for what we her e call 'lotteries' , 'horse lotteries' for functions from states to deterministic outcomes, i.e., acts in the Savage (1954) sense, and 'compound horse lotteries' for what we here refer to as Anscombe-Aumann acts.

Decision-theoretic Framework

23

of the likelihood that he assigns to the different strategies of his opponent . This is consistent with the way games are normally presented, where payoff functions for each player are provided independently of the analysis of the strategic interaction.e â€˘ When relaxing completeness, it turns out to be important to allow mixed strategies as objects of choice when determining maximal elements of a player's incomplete preferences, for similar reasons as domination by mixed strategies is needed for dominated strategies to correspond to strategies that can never be best replies. We will consider three kinds of generalizations of the AnscombeAumann framework.

First, as mentioned in the introduction to this chapter, throughout this book we will follow Blume et al. (1991a) by imposing the conditional Archimedean property (also called conditional continuity) instead of Archimedean property (also called continuity) . This is important for modeling caution, which requires a player to take into account the possibility that the opponent makes an irrational choice, while assigning probability one to the event that the opponent makes a rational choice. I.e., even though any irrational choice is infinitely less likely than some rational choice, it is not ruled out . Such discontinuous preferences will also be useful when modeling players' preferences in extensive games. Second, we will relax the axiom of completeness to conditional completeness. While complete preferences will normally be represented by means of subjective probabilities (ef. Propositions 1, 2, 3, and 5 of this chapter), incomplete preferences are insufficient to determine the relative likelihood of the uncertain states. One possibility is, following Aumann (1962) and Bewley (1986), to represent incomplete preferences by means of a set of subjective probability distributions. Subjective probabilities are not part of the most common deductive procedures in game theory-like IESDS, the Dekel-Fudenberg procedure, and the backward induction procedure. One can argue that, since they make no use of subjective probabilities, one should seek to provide epistemic conditions for such procedures without reference to subjec2This argument is in line with the analysis of Aumann and Dreze (2004), who however depart from the Anscombe-Aumann framework by considering preferences-not over all functions from states to randomized outcomes-but only on the subset of mixed strategies. The Ascombe-Aumann framework requires that the decision maker has access to objective probabilities; however , Machina (2004) points to how this requirement can be weakened.

24

CONSISTENT PREFERENCES

tive probabilities. Indeed, subjective probabilities play no role in the epistemic analysis of backward induction by Aumann (1995). In Chapters 6 and 7 we follow Aumann in this respect and provide epistemic conditions for IESDS, the Dekel-Fudenberg procedure, and the backward induction procedure through modeling players endowed with (possibly) incomplete preferences that are not represented by subjective probabilities. Moreover, for the modeling of forward induction in Chapters 11 and 12, it is a necessary part of the analysis that preferences are incomplete.

Third, we will allow for flexibility concerning how to specify conditional preferences. Such flexibility can be motivated in the context of the modeling of sequentiality and quasi-perfectness in Chapters 8 and 9. Sequential rationalizability will be defined and sequential equilibrium characterized by considering the event that each player believes that the opponent chooses rationally at all her information sets. Adding preference for cautious behavior to this event yields the concepts of quasiperfect rationalizability and equilibrium. For these definitions and characterizations, we must describe what a player believes both conditional on reaching his own information sets (to evaluate his rationality) and conditional on his opponent reaching her information sets (to determine his beliefs about her choices). In other words, we must specify a system of conditional beliefs for each player. There are various ways to do so. One possibility is a conditional probability system (CPS) where each conditional belief is a subjective probability distribution.' This is sufficient to model sequentiality. Another possibility, which is sufficient to model quasi-perfectness, is to apply a single sequence of subjective probability distributions-a so-called lexicographic probability system (LPS) as defined by Blume et al. (1991a)and derive the conditional beliefs as the conditionals of such an LPS . Since each conditional LPS is found by constructing a new sequence, which includes the well-defined conditional probability distributions of the original sequence, each conditional belief is itself an LPS. However, quasi-perfectness cannot always be modeled by a CPS since the modeling of preference for cautious behavior may require lexicographic probabilities. To see this, consider r 4 of Figure 3.1. In this game, if player 1 believes that player 2 chooses rationally, then player 1 3This is the terminology introduced by Myerson (1986) . In philosophical literature, related concepts are called Popper measures. For an overview over relevant literature and analysis, see Hammond (1994) and Halpern (2003) .

25

Decision-theoretic Fram ework

1

2

DI

dl

1 1

1 1

0

~O

Figure 3.1.

r4

d

f

F 1, 1 0, 0 D 1, 1 1, 1

and its st rategic form.

must assign probability one to player 2 choosing d. Hence, if each (condit ional) belief is associate d with a subject ive probability distribution- as is t he case with the concept of a CP S-and player 1 believes that his opponent chooses rationally, then player 1 is indifferent between his two st rategies. This is inconsistent with quasi-perfectness, which requir es players t o have preference for caut ious behavior , meaning that player 1 in r 4 prefers D t o F . Moreover , sequent iality cannot always be modeled by means of condit ionals of a single LPS since pr eference for caut ious behavior is induced. To see this, consider a modified version of r 4 where an additio nal subgame is substit uted for t he (0, O)- payoff, with all payoffs in t hat subgame being smaller t han 1. If player 1's conditional beliefs over st rategies for player 2 is derived from a single LPS , t hen a well-defined belief condit ional on reaching t he added subgame entails t hat player 1 deems possible t he event t hat player 2 chooses f , and hence, player 1 prefers D to F . This is inconsist ent with sequent iality, under which F is a rational choice. Therefore, this chapter will present a new way of describing a system of conditional beliefs, called a syste m of con di tional lexicographic probabilities (SCLP), and which is based on joint work with Andres Perea ; d . Asheim and Perea (2005). In cont rast to a CPS, an SCLP may induce conditional beliefs that are repr esent ed by LPSs rather than subject ive probability distributions. In cont rast to the system of condit ionals derived from a single LP S, an SCLP need not include all levels in the sequence of the original LPS when determining condit ional beliefs. Thus, an SCLP ensures well-defined conditional beliefs representing nontrivial conditional preferences, while allowing for flexibility w.r.t. whet her to assume preference for cautio us behavior.

26

3.2

CONSISTENT PREFERENCES

Axioms

Consider a decision maker und er uncertainty, and let F be a finite set of states. Th e decision maker is uncertain about what st at e in F will be realized. Let Z be a finite set of out com es. For each ¢ E 2F \ {0} , the decision maker is endowed with a binary relation (preferences) over all functions that to each element of ¢ assign an objective randomiz ation on Z . Any such function is called an act on ¢, and is the subj ect of analysis in the deciscion-theoretic framework introduced by Anscombe and Aumann (1963). Writ e P4> and q 4> for acts on ¢ E 2F \ { 0} . (For act s on F , writ e simply P and q .) A binary relation on the set of acts on ¢ is denoted by 'c. ¢, where P4> 'c.¢ q 4> means that P4> is preferred or in differe nt to q¢. As usual, let >-¢ (preferred to) and "'4> (indifferent to) denote t he asymmet ric and symmet ric parts of 'c. 4> . Consider the following five axioms, where the numb ering of axioms follows Blume et al. (1991a). A XIOM

1 (ORDER) 'c.¢ is compl ete and transitive.

AXIOM 2 (OBJECTIVE INDEPENDENCE) P~

>-¢ (resp. "'¢) P¢ iff ,),P~ +(1- ')')q¢ >-4> (resp. "'¢) ,),P¢ + (1- ')')q¢, whenever 0 < ')' < 1 and q¢ is arbit rary.

AXIOM

3

(NONTRIVIALITY)

Th ere exis t P4> and q 4> such that P4> >-4> q¢.

4 (ARCHIM ED EAN PROPERTY) If P¢ >-¢ q ¢ ':>-¢ P¢, then :30 < ')' < 0 < 1 such that op ¢ +(1 - o)p¢ >-¢ q ¢ >-¢ ,),P~ + (1 - ')')p¢ . AXIOM

Say that e E F is Savage-n ull if p{e} '" {e} q{e} for all acts p{e} and q{e} on {e}. Denot e by K, t he non-empty set of states that are n ot Savagenull; i.e., the set of states that the decision maker deems subj ectively possible. Writ e <P := {¢ E 2F \ { 0} I K, n ¢ =I- 0} . Refer to the collection {'c.4>1 ¢ E <p} as a system of con dition al preferen ces on the collect ion of sets of act s from from subsets of F t o outcomes. Whenever 0 =1= E ~ ¢, denot e by P€ the restriction of P¢ to E. iff P{j} and p{e,f} and q {e,f} satisfy P] e,f} (e) =

AXIOM 5 (NON-N ULL STATE INDEPENDENCE) P{e} >-{e} q{ e}

>- {j} q{j}, when ever e, f E P{ e,f } (I) and q{ e,f } (e)

K"

= q{ e,f} (I).

Define the condition al binary relation 'c. ,* by P¢ c:: ,* P¢ if, for some q ¢, (p~ , q ¢\ €) 'c.¢ (p~ , q ¢V)· By Axioms 1 and 2, this definition does not depend on q 4> . The following axiom st at es that preferences over acts on E, 'c. €, equal the condit ional of 'c.¢ on E, whenever 0 =1= E ~ ¢ .

27

Decision-theoretic Framework

AXIOM 6 (CONDITIONALITY) p, >-10 (resp. f"V4>le) q4>' whenever 0 =J E ~ <p.

f"V

e)

q, iff P4> >-4>110 (resp.

It is an immediate observation that Axioms 5 and 6 imply non-null state independence as stated in Axiom 5 of Blume et al. (1991a).

LEMMA 1 Assume that the system of conditional preferences {~4>1 <p E <p} satisfies Axioms 5 and 6. Then, V<p E <P, P4> >-4>I{e} q4> iffp4> >-4>I{f} q4> whenever e, f E K, n <p, and P4> and q4> satisfy P4>(e) = P4>(J) and q4>(e) = q4>(J) . TUrn now the relaxation of Axioms 1, 4, and 6, as motivated in the previous section. AXIOM I' (CONDITIONAL ORDER) ~4> is reflexive and transitive and, Ve E <p, ~4>I{e} is complete. AXIOM 4' (CONDITIONAL ARCHIMEDEAN PROPERTY) Ve E <p, if P~ >-4>1{e} q4> >-4>1{e} P;, then 30 < 'Y < 5 < 1 such that 5p~ +(1-5)p; >- 4>1{e} q4> >-4>I{e} 'YP~ + (1 - 'Y)p;, AXIOM 6' (DYNAMIC CONSISTENCY) Pe >-10 qe whenever P4> >-4>110 q4> and 0: =J E ~ <p. Since completeness implies reflexivity, Axiom I' constitutes a weakening of Axioms 1. This weakening is substantive since, in the terminology of Anscombe and Aumann (1963), it means that the decision maker has complete preferences over 'roulette lotteries' where objective probabilities are exogenously given, but not necessarily complete preferences over 'horse lotteries' where subjective probabilities, if determined, are endogenously derived from the preferences of the decision maker . Say that e E K, is deemed infinitely more likely than f E F (and write e » J) if p{e,J} >- {e,J} q{e,J } whenever p{e} >- {e} q{e}: Consider the following two auxiliary axioms. AXIOM 11 (PARTITIONAL PRIORITY) If e' or f » e" .

» e",

then

Vf

E F, e'

»

f

AXIOM 16 (COMPATIBILITY) There exists a binary relation ~p satisfying Axioms 1, 2, and 4' such that P >- PI4> q whenever P4> >-4> q4> and =J <p ~ F.

o

While it is straightforward that Axiom 1 implies Axiom 1', Axiom 4 implies Axiom 4' , and Axiom 6 implies Axiom 6', it is less obvious that

28

CONSIS TEN T PREFERENCES

• Axiom 1 tog ether with Axioms 2, 4' , 5, and 6 imply Axiom 11, and • Axiom 6 toget her with Axioms 1 2,4', and 5, imply Axiom 16. This is demon strated by the following lemma. 2 A ssume that (a) ~ ¢ sat isfies A xioms 1,2 and 4' if ¢ E 2F \ { 0} , and (b) the system of con diti onal pr efer ences {~ ¢I ¢ E <p} sati sfies A xioms 5 and 6. Th en {~ ¢ I¢ E <p} sat isfies A xiom s 11 and 16. LEMMA

Proof. Part 1: A xiom 11 is im plied. We must show , und er the given premise , that if e' » e", then, Vf E F , e' » f or f » e". Clearl y, e' » elf entails e' E K" implying that e' » f or f » elf if f ~ K, or elf ~ K,. The case where f = e' or f = elf is trivial. The case where f =1= e', f =1= e", f E K, and elf E K, remains . Assume that e' » f does not hold , which by complete ness (Axiom 1) entails the existe nce of p{ e' ,j} and q{e''/ } such that p{e''/ } :5{e''/ } q{e' ,f} and p{ e' } >- {e'} q{e' }. It suffices to show that f » elf is obtained ; i.e., P{j} >- {j} q{j } impli es P{e",/} >- {e"'/} q{e",/} ' Throughout we invoke Axiom 6 and Lemma 1, and choose ¢ E <P so that {e',e", J} E ¢. Let P¢ >-¢I{j} q ¢. Assume w.l.o.g. that p ¢(d) = q ¢(d) for d =1= f , elf, and p¢(d) = q ¢(d) for d =1= e', f. By transitivity (Axiom 1), P¢ :5¢I{e''/} ' >-¢I{e'} q ,¢ Imp . I ' -<¢l{f} q ,¢. However , since . >. fies q ,4> andP4> y P¢ - 4> sat is Axioms 2 and 4' , :3')' E (0,1) such that ,),P¢ + (1 - ')')p¢ >--¢I{j} ')'q¢ + (1 - ')')q¢ . Moreover, p ¢(e') = q ¢( e') and P¢ >-¢I{e'} q¢ entail that ,),P¢ + (1 - ')')p¢ >-¢I{e'} ,),q¢ + (1- ')')q¢ by Axiom 2, which implies that ,),P¢ + (1 - ')') p¢ >-¢I{e' ,e"} ')'q¢ + (1 - ')') q¢ since e' »e". Hence, by transitivity, ,),p¢ + (1-,),)p¢ >-¢I{e',e",f} ')'q¢ + (1-')')q¢ - or equivalent ly, ,),P¢ + (1 - ')')p¢ >-- ')'q¢ + (1 - ')')q¢ . Now, q¢ ~ ¢I{ e' ,f} P¢ mean s that ,),P¢ + (1 - ')')q¢ ~ ,),P¢ + (1 - ')')p¢ by Axiom 2, implying that I' P¢ + (1 - ')')q¢ >- ')'q¢ + (1 - ')')q¢ by t ran sitivity (Axiom 1), and P¢ >- q ¢ - or equivalently, P¢ >-¢I{e"'/} q¢ - by Axiom 2. Thus, P¢ >-¢I{j} q ¢ implies P¢ >-¢I{e",f} q ¢, meaning that f » c. Part 2: A xiom 16 is im plied. We must show, und er the given premise, that here exists a binary relation ~F satisfying Axioms 1, 2, and 4' such that P >- Fl¢ q whenever P¢ >-¢ q 4> and 0 =1= ¢ ~ F . Clearl y, since Axiom 6 is satisfi ed , ~F fulfil these requirements. _ We end this section by stating for later use an axiom which is impli ed by Axiom 4 and which impli es Axiom 4'. The numbering for this axiom also follows the one used by Blume et al. (1991a) .

29

Decision-theoretic Framework

Table 3.1.

Relationships between different sets of axioms and their representations.

Complete and continuous

123456 Prob. distr.

--.

1 23456' 16 CPS

1 Comp lete and partitionally continuous

1

1 234" 5 6 LCPS

1 Complete and discontinuous

1234' 56 LPS

--. 1 234' 56' 16 SCLP

1 Incomplete and discontinuous

I'll 234' 56 Conditionality

AXIOM

4"

Dynamic consistency

(PARTITIONAL ARCHIMEDEAN PROPERTY)

There is a parti-

tion {1f~,

,1f~I</>} of K, n e/> such that

• \f£ E {I , that 8p~

, Lle/>}, if p~ >-</>111'~ q</> >-</>111" P¢, then :30 < 'Y < 8 < 1 such + (1 - 8)p¢ >-</>111'~ q</> >-</>111'~ 'Y~~ + (1 - 'Y)p¢, and

• \f£ E {I, .. . ,LIe/> -I}, p </> >-"111" 'P e q</> implies p </> >-"111"u11" 'P e e+1 q</> .

Table 3.1 illustrates the relationships between the sets of axioms that we will consider. The arrows indicate that one set of axioms imp lies another. The figure indicates what kind of representations the different sets of axioms correspond to, as reported in the next section.

3.3

Representation res ults

In view of Lemma 1 and using the characterization result of Anscombe and Aumann (1963), we obtain the following resu lt under Axioms 1, 2, 3, 4, 5, and 6; cf. Theorem 2.1 of Blume et al. (I99Ia). For the statement of this and lat er resu lts , denote by v : Z --. lR a vNM utility function, and abuse notation slightly by writing v(p) = L::zEzp(z)v(z) whenever p E ~(Z) is an objective randomization. In this and later results, v is unique up to positive affine transformations. 1 (ANSCOMBE statements are equivalent .

PROPOSITION

AND AUMANN ,

1963) The following two

30

CONSISTENT PREFERENCES

1 (aJ 'c¢ satisfies Axioms 1, 2, and 4 if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I> , and (bJ the system of conditional preferences {'c¢1 ¢ E <I>} satisfies Axioms 5 and 6.

2 There exist a vNM utility function v :

~(Z) ~

JR and a unique subjective probability distribution /-l on F with support K such that,

V¢ E <I> ,

P¢ 'c¢ q¢ iff '~eE " ¢ /-ll ¢(e)v(p¢(e)) 2: '~eE¢ " /-ll¢( e)v(q¢( e)) , where /-ll¢ is the conditional of /-l on ¢.

In view of Lemma 1, and using Theorem 3.1 of Blume et al. (1991a) , we obtain the following result under Axioms 1, 2, 3, 4' , 5, and 6. For the statement of this and later results, we need to introduce formally the concept of a lexicographic probability system. A lexicographic probability system (LPS) consists of L levels of subjective probability distributions: If L 2: 1 and, W E {1, .. . , L} , /-le E ~(F) , then A = (/-lI ,' .. ,/-ld is an LPS on F. Denote by L~(F) the set of LPSs on F . Write supp A := U7=1 supp /-le for the support of A. If supp An ¢ =!= 0, denote by AI ¢ = (/-l~ , .. . /-l~I¢) the conditional of A on ¢.4 Furthermore, for two utility vectors v and w , denote by v 2:L W that, whenever we > ve , there exists k < £ such that Vk > Wk, and let> Land = L denote the asymmetric and symmetric parts, respectively. PROPOSITION 2 (BLUME ET . AL, 1991A) The following two statements

are equivalent.

1 (a) 'c¢ satisfies Axioms 1, 2, and 4' if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I> , and (b) the system of conditional preferences {'c¢1 ¢ E <I>} satisfies Axioms 5 and 6. 2 There exist a vNM utility function v : ~(Z) ~ JR and an LPS A on F with support K such that, V¢ E <I> ,

4I.e., "Ie E {I , . .. , LI¢>}, J1. ~ = J1.k£I <t> , wh ere the indices ke are given by ko = 0, ke = min{klJ1.k(¢) > 0 and k > ke-d for e > 0, and {klJ1.k( ¢) > 0 and k > k LI<t>} = 0, and where J1.k£l <t> is given by the usu al definition of condit ional probabilities; d . Definition 4.2 of Blume et al. (1991a) .

31

Decision- theoretic Fram ework

where Al q, = ( JL~ , .. . JL~ I q, ) is the condition al of A on ¢. In view of Lemm a 1 and using Theorem 5.3 of Blume et al. (I 99Ia) , we obtain t he following rusult und er Axioms 1, 2, 3, 4" , 5, and 6. For t he statement of this results, we need to introduce t he concept that is called a lexicographic conditional probability system in t he te rminology t hat Blume et al. (I99Ia) use in t heir Definiti on 5.2. A lexicographic conditional probability syst em (LCP S) consists of L levels of non-overlapping subjective probability distributions: If A = (JL1,' .. , JLd is an LPS on F and t he supports of t he JLe's are disjoint , t hen A is an LCPS on F. PROPOSITION 3 (BLUME ET. AL , I99IA) The following two statements are equivalent. 1 (a) 'e q, satisfies Axioms 1, 2, and 4" if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I> , and (b) the system of conditional preferences {'e q,! ¢ E <I>} satisfies Axioms 5 and 6.

2 There exist a vNM utility function v : ~ (Z ) A on F with support '" such that, V¢ E <I> ,

--t

lR and a unique LCPS

Pq, 'e q, qq, iff , ) Llq, ( , ) Llq, ( L eEq, JLe(e)v (p q, (e)) e=l ?'L L eEq, JLe(e)v (qq, (e)) e=l '

where Alq, = (JL~, ... JL~I q, ) is the conditional of A on ¢ (with the L CPS Alq, satisfying, \If E {I , . . . , LI ¢} , SUPPJLe = 7l"e)' 'Say t hat 'elf> is conditionally represented by a vNM utility function v if (a) 'e q, is non-trivial and (b) Pq, 'e q,I{e} q q, iff v( p q,(e)) ? v (qq, (e)) .whenever e is deemed subjectively possible. Under Axioms 1', 2, 3, 4', 5, and 6 condit ional repr esent ation follows directly from the vNM theorem of expecte d utility representation. PROPOSITION 4 Assum e that (a) 'e q, satisfies Axioms 1', 2, and 4' if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I> , and (b) the system of conditional preferences {'e q,1 ¢ E <I>} satisfies A xioms 5 and 6. Then there exists a vNM utility function v : ~ (Z ) --t lR such that, V¢ E <I> , Pq, 'e q,I {e} qq, iff v (pq, (e)) ? v (qq, (e)) whenever e E '" n ¢ . Und er Axioms 1, 2, 3, 4' , 5, 6', and 16 we obtain t he characterizat ion result of Asheim and Perea (2005). For t he st atement of t his result , we need to introduce t he concept of a syste m of conditio nal lexicographic probabilities. For t his definiti on,

32

CONSISTENT PREFERENCES

if A .- (f-t1 , . .. , f-td is an LPS and f E {l, ... ,L} , then write Ae := (f-t1' . .. , f-te) for the LPS that includes only the f top levels of the original sequence of probability distributions. DEFINITION 1 A system of conditional lexicographic probabilities (SCLP) (A, f) on F with support K, consists of • an LPS A = (f-tI, •• . , f-td on F with support

K"

and

• a function f : <I> ---+ {I, ... , L} satisfying (i) supp Ae(</J) n ¢ =1= 0, (ii) f(c:) 2 f(¢) whenever 0 =1= e S;;; ¢, and (iii) f( {e}) 2 f whenever e E supp f-te. The interpretation is that the conditional belief on ¢ is given by the conditional on ¢ of the LPS Ae(</J) , Ae(</J) I</J = (f-t~ , . .. f-t~(</J) I</J)' To determine preference between acts conditional on ¢ , first calculate expected utilities by means of the top level probability distribution, f-t~ , and then, if necessary, use the lower level probability distributions, f-t~ , . . . , f-t~(</J)I </J ' lexicographically to resolve ties . The function f thus determines, for every event ¢, the number of levels of the original LPS A that can be used , provided that their supports intersect with ¢ , to resolve ties between acts conditional on ¢ . Condition (i) ensures well-defined conditional beliefs that represent nontrivial conditional preferences. Condition (ii) means that the system of conditional preferences is dynamically consist ent, in the sense that strict preference between two acts would always be maintained if new information, ruling out st ates at which the two acts lead to the same outcomes, became available. To motivate condition (iii), note that if e E supp f-te and f( {e}) < f, then it follows from condition (ii) that f-te could as well ignore e without changing the conditional beliefs. PROPOSITION 5 (ASHEIM AND PEREA , 2005) Th e following two statements are equivalent. 1 (a) ~ </J satisfies Axioms 1, 2, and 4' if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I> , and (b) the system of conditional preferences {~</J I ¢ E <I>} satisfies Axioms 5, 6' , and 16 . 2 There exist a vNM utility fun ction v : ~(Z) on F with support K, such that, V¢ E <I> ,

---+

lR and an SeLF (A, f)

P </J ~ </J q </J iff

£(</J )I</J ( )e(</J)I<P ( L eE</Jf-te(e)v(p</J(e)) ) e=l 2£ L eE</Jf-te(e)v(q</J(e)) e=l '

33

Decision-theoretic Fram ework

where A£(</» I</> = (Il~ , . . ' Il~(</>)I</> ) is the conditional of A£(</» on c/>.

Proof. 1 i mplies 2. Since ?::-</> is trivial if c/> 1:- <1> , we may w.l.o.g. assume that Axiom 16 is satisfied with ?::-~I </> being t rivial for any c/> 1:- <1> . Consider any e E 11,. Since ?::-{e} satisfies Axioms 1, 2, 3, and 4' (implying Axiom 4 since {e} has only one state ), it follows from the vNM theorem of expect ed utility represent ation that there exists a vNM utility function v{e} : 6.(Z) --+ lR. such t hat V{e } represents ?::-{e}' By Axiom 5, we may choose a common vNM utility function v to represent ?::-{e} for all e E 11,. Since Axiom 16 implies, for any e E 11" ?::-~ I{ e} satisfies Axioms 1, 2, 3, and 4' , and furthermore, p ?- ~ {e} q whenever P{e} ?-{e} q{e} , we obtain that v represent s ?: - ~I{e} for alI e E 11,. It now follows that ?::-~ sat isfies Axiom 5 of Blume et al. (1991a) . By Theorem 3.1 of Blume et al. (1991a) ?::- ~ is represented by v and an LPS A = (Ill , ... , II L) on F with support 11,. Consid er any c/> E <1> . If p </> ?-</> q </> iff P ?-~ I</> q , then

where AI </> = (Il~ , · · · 1l~1</» is the condit ional of A on C/>, implying that we can set £(c/» = L. Otherwise, let £(c/» E {O, ... ,L - I} be the maximum £ for which it holds that

where t he r .h.s. is never satisfied if £ < min{ k [supp Akn c/> i- 0} , entailing t hat the implication holds for any such £. Define a set of pairs of acts on C/>, X, as follows: (p</> , q </» E I iff ,", ' ) £(</» I</> ( L..J eE</>Il£(e)v(p</> (e)) £= 1

= L

('"'

L..JeE</>Il~(e)v(q</>(e))

) £(</» I</> £= 1

'

with (p</> ,q</» E X for any acts p </> and q </> on c/> if £(c/» < min{£lsuppA£ nc/> i- 0}. Not e that X is a convex set . To show that v and A£( </» I</> represent ?::-</> ' we must establish that p </> ""</> q </> whenever (p </> , q</» E I . Hence, suppose there exists (p </> , q</» E X such that p </> ?-</> q</> . It follows

34

CONSISTENT PREFERENCES

from the definition of f(¢) and the completeness of there exists (p¢, q¢) E I such that

~¢

(Axiom 1) that

P¢ ~¢ q¢ and

L

eE

¢ /-le(¢)+l (e)v(p¢( e)) <

Objective independence of then

hence, by transitivity of

~¢

~¢

L

eE

¢ /-le(¢)+l (e )v( q¢( e)) .

(Axiom 2) now implies that, if 0 < "( < 1,

(Axiom 1), (3.1)

However, by choosing "( sufficiently small, we have that

Since I is convex so that ("(P¢ + (1 - "()p¢, "(q¢ + (1 - "()q¢) E I , this implies that "(P + (1 - "()p' ~~I¢ "(q + (1 - "()q' . (3.2) Since (3.1) and (3.2) contradict Axiom 16, this shows that P¢ "'¢ q¢ whenever (p¢,q¢) E I. This implies in turn that f(¢) ~ min{flsuppAe n¢ i- 0} since ~¢ is nontrivial. By Axiom 6', f(c) ~ f(¢) whenever i- e ~ ¢. Finally, since, v represents ~{e} for all e E K" it follows that Pie} >-{e} q{e} iff P >-~I{e} q. Hence, we can set f({e}) = L, implying f( {e}) ~ f whenever e E supp /-le. 2 implies 1. This follows from routine arguments. _

o

By strengthening Axiom 4' to Axiom 4, we get the following corollary. For the statement of this result, we need to introduce formally the concept of a conditional probability system. A conditional probability system (CPS) consists of a collection of subjective probability distributions: If, for each ¢ E ll> , /-l¢ is a subjective probability distribution on ¢, and {/-l¢ I ¢ E ll>} satisfies /-l£(<5) . /-l¢(E) = /-l¢(<5) whenever <5 ~ E ~ ¢ and E, ¢ E ll>, then {/-l¢ I ¢ Ell>} is a CPS on F with support K,.

35

Decision-theoretic Framework COROLLARY

1 The following three statements are equivalent.

1 (a) 'c.</> satisfies Axioms 1, 2, and 4 if ¢ E 2F \ {0} , and Axiom 3 if and only if ¢ E <I> , and (b) the system of conditional preferences {'c.</> I ¢ E <I>} satisfies Axioms 5, 6', and 16 . 2 There exist a vNM utility function v : ~(Z) ---+ 1R. and a unique LCPS A = (f-ll, .. . , f-lL) on F with support K, such that, V¢ E <I>,

where f-l</> is the conditional of f-le( </» on ¢ and £.(¢) # 0}.

= min {£. I su pPAe n

¢

3 There exist a vNM utility function v : ~(Z) ---+ 1R. and a unique CPS {f-l</> I ¢ E <I>} on F with support K, such that, V¢ E <I> ,

Proof. 1 implies 2. By Proposition 5, the system of conditional preferences is represented by an SCLP (A, £.) on F with support K,. By the strengthening Axiom 4' to Axiom 4, it follows from the representation result of Anscombe and Aumann (1963) that only the top level probability distribution is needed to represent each conditional preferences; i.e., for any ¢ E <I> , £.(¢) = min{£. I suppAe n ¢ # 0} . This implies that any overlapping supports in A can be removed without changing, for any ¢ E <I> , the conditional of Ae(</» on ¢, turning A into an LCPS. Furthermore, the LCPS thus determined is unique. 2 implies 1. This follows from routine arguments. 2 implies 3. {f-l</> I ¢ E <I>} is a CPS on F with support K, since f-lE(8) . f-l</>(E) = f-l</>(8) is satisfied whenever 8 ~ E ~ ¢ and E, ¢ E <I> . If an alternative CPS {tJ,</> I ¢ E <I>} were to satisfy, for any ¢ E <I> ,

then one could construct an alternative LCPS 5. = (tJ,1, "" tJ,L) such that, for any ¢ E <I> , tJ,</> is the conditional of tJ,Z(</» on 1>, where l(¢) := min {£. I supptJ,e n ¢ # 0}, contradicting the uniqueness of A. 3 implies 2. Construct the LCPS A = (f-ll, "" f-lL) by the following algorithm: (i) f-ll = f-lF, (ii) V£ E {2, ... , L}, f-le = f-l</>, where ¢ = F\Ui~\suPPf-lk # F\K" and (iii) Uf=lsUPPf-lk = K, . Then, for any ¢ E <I>a,

36

CONSIS TEN T PREFERENCES

J-L</> is the condit ional of J-Lec</» on ¢, where f (¢) := min{fl SUPPJ-Le n¢ and A is t he only LCPS having this property. _ A full support SCLP (i.e., an SCLP where

=1=

0},

=

F ) combines t he st ructur al implication of a full support LPS-namely t hat conditio nal preferences are nontrivial-with flexibility w.r.t. whether to assume t he behavioral implication of any condit ional of such an LPS-namely t hat t he conditional LPS 's full support induces preference for caut ious behavior. A full support SCLP is a genera lization of both K,

(1) conditional beliefs describ ed by a single full support LPS A = (J-Ll , .. . , J-Ld (cf. Propositi on 2): Let , for all ¢ E <1> , f (¢ ) = L. Then the condit ional belief on ¢ is describ ed by the conditional of A on ¢, AI </>. (2) condit ional beliefs describ ed by a CPS (d. Corollary 1): Let , for all ¢ E <1> , f (¢ ) = min{ f I suppx, n ¢ =1= 0}. Then, it follows from condit ions (ii) and (iii) of Definition 1 that the full support LPS A = (J-L l, " " J-Ld has non-overlapping supports - i.e., A is an LCPSand t he condit ional belief on ¢ is described by the top level pr obabili ty distribution of t he condit ional of A on ¢. This corresponds to t he isomorphism between CPS and LCPS noted by Blume et al. (1991a) on p. 72 and discussed by Hamm ond (1994) and Halpern (2003). However , a full support SCLP may describ e a syste m of condit ional beliefs t hat is not covered by t hese special cases. The following is a simple example: Let K, = F = {d ,e,j} and A = (J-L l , J-L2) , where J-Ll (d) = 1/2, J-Ll (e) = 1/2, and J-L2(J) = 1. If f( F) = 1 and f( ¢) = 2 for any ot her non-empty subs et ¢, t hen t he resulting SeLP falls outsi de cases (1) and

(2).

Chapter 4

BELIEF OPERATORS

Belief operators play an important role in epistemic analyses of games. For any event, a belief operator determines the set of states where this event is (in some precise sense) believed. Belief operators may satisfy different kinds of properties, like • if one event implies another, then belief of the former implies belief of the latter (monotonicity) , • if two events are believed, then the conjunction is also believed, • an event that is always true is always believed, • an event that is never true is never believed, • if an event is believed, then the event that the event is believed is also believed (positive introspection) , and • if an event is not believed, then the event that the event is not believed is believed (negative introspection) . Belief operators satisfying this list are called KD45 operators.' In epistemic analyses of games, it is common to derive belief operators from preferences, leading to what can be called subjective belief operators. Examples of subjective KD45 operators are 'belief with probability one', as used by, e.g., Tan and Werlang (1988), 'belief with primary probability one', as used by Brandenburger (1992), and 'condit ional belief with probability one', as used by Ben-Porath (1997). More recently, I A KD45 operator satisfies that beli ef of an event implies that the complement is not believed, but need not satisfy the t ruth axiom- i .e. that a believed event is always true.

38

CONSISTENT PREFERENCES

Brandenburger and Keisler (2002), Battigalli and Siniscalchi (2002) and Asheim and Dufwenberg (2003a) have proposed non-monotonic subjective belief operators called 'assumption', 'strong belief ' and 'full belief', respectively. With the exception of Asheim and Dufwenberg's (2003a) 'full belief ', these operators have in common that they are based on subjective probabilities-arising from a probability distribution, a lexicographic probability system, or a conditional probability system-that represent the preferences of the player as a decision maker . An alternative approach to belief operators, applied by e.g. Stalnaker (1996, 1998), is to define belief operators by means of accessibility relations, as used in modal logic. Of particular interest is Stalnaker's non-monotonic 'absolutely robust belief' operator. Reproducing joint work with Ylva Sevik-e-Asheim and Sovik (2005)this chapter integrates these two approaches by showing how accessibility relations can be derived from preferences and in turn be used to define and characterize belief operators; see Figure 4.1 for an illustration of the basic structure of the analysis in this chapter. These belief operators will in later chapters be used in the epistemic analysis. Morris (1997) observes that it is unnecessary to go via subjective probabilities to derive subjective belief operators from the preferences of a decision maker . This suggestion has been followed in Asheim (2002) and Asheim and Dufwenberg (2003a) , the content of which will be reproduced in Chapters 7 and 11. Epistemic conditions for backward induction are provided in Chapter 7 without the use of subjective probabilities (since one can argue that subjective probabilities play no role in the backward induction argument), while Chapter 11 provides epistemic conditions for forward induction within a structure based on incomplete preferences that cannot be represented by subjective probabilities. When deriving belief operators from preferences , it is essential that the preferences determine 'subjective possibility' (so that it can be determined whether an event is subjectively impossible) as well as 'epistemic priority' (so that one allows for non-trivial belief revision) . As we shall see, preferences need not satisfy completeness in order to determine 'subjective possibility' and 'epist emic priority'. This chapter shows how belief operators corresponding to those used in the literature can be derived from preferences that need not be complete. We assume that preferences satisfy Axioms I', 11, 2, 3, 4' , 5, and 6, entailing that preferences are (possibly) incomplete, but allow conditional representation (d. Proposition 4 of Chapter 3). Following the structure illustrated in Figure 4.1, Section 4.1 shows how a binary acces-

39

Belief operators

./

Preferences over acts (functions from states to randomized outcomes)

'Infinitely more likely'

,Admissibility'

1

1

Q Accessibility relation

(R 1 , . •• , Rd Vector of nested

of epistemic priority

accessibility relations

defines""

Figure 4.1.

Belief operators • certain belief • conditional belief • robust belief

. /cha rac t erizes

The basic structure of the analysis in Chapter 4 .

sibility relation of epistemic priority Q can be derived from preferences satisfying these axioms, by means of the 'infinitely-more-likely' relation. The properties of this priority relation are similar to but more general than those found, e.g., in Lamerre and Shoham (1994) and Stalnaker (1996, 1998) in that reflexivity of Q is not required.r Furthermore, it is shown how preferences through "admissibility" give rise to a vector of nested binary accessibility relations (R 1 , •• . , Rd, where, for each £, Re fulfills the usual properties of Kripke representations of beliefs; i.e., they are serial, transitive and Euclidean. Finally, we establish that the two kinds of accessibility relations yield two equivalent representations of 'subjective possibility' and 'epistemic priority'. In Section 4.2 we first use the accessibility relation of epistemic priority Q to define the following belief operators: • Certain belief coincides with what Morris (1997) calls 'Savage-belief' and means that the complement of the event is subjectively impossible. • Conditional belief generalizes 'conditional belief with probability one'.

2The term 'epistemic priority' will here be used to refer to what elsewhere is sometimes referred to as 'plausibility' or 'prejudice'; see , e.g., Friedman and Halpern (1995) and Lamerre and Shoham (1994) . This is similar to 'preference' among states (or worlds) in nonmonotonic logic- d. Shoham (1988) -leading agents towards some states and away from others. In contrast, we use the term 'preferences' in the decision-theoretic sense of a binary relation on the set of functions ('acts') from states to outcomes.

40

CONSISTENT PREFERENCES

• Robust belief coincides with what Stalnaker (1998) calls 'absolutely robust belief'. We then show how these operators can be characterized by means of the vector of nested binary accessibility relations (R 1 , . . . , Rd , thereby showing that the concept of 'full belief' as used by Asheim and Dufwenberg (2003a) coincides with robust belief. Section 4.3 establishes properties of these belief operators. In particular, the robust belief operator (while poorly behaved) is bounded by certain and conditional belief, which are KD45 operators. Section 4.4 shows how the characterization of robust belief corresponds to the concept of 'assumption' as used by Brandenburger and Keisler (2002), and observes how the definition of robust belief is related to the concept of 'strong belief ' as used by Battigalli and Siniscalchi (2002). We thereby reconcile and compare these non-standard notions of belief which have recently been used in epistemic analyses of games. The proofs of the results in this chapter are included in Appendix A.

4.1

From preferences to accessibility relations

The purpose of this section is to show how two different kinds of accessibility relations-see, e.g., Lamerre and Shoham (1994) and Stalnaker (1996, 1998)-can be derived from preferences . Consider the decision-theoretic framework of Chapter 3. However, as motivated below, assume that the decision maker's preferences may vary between states. Hence, denote by t~ the preferences over acts on ¢ at state d, and use superscript d throughout in a similar manner. Assume for the rest of this chapter that, for each dE F , (a) t~ satisfies Axioms I' , 2, and 4' if ¢ E 2F \ {0}, and Axiom 3 if and only if ¢ E <I>d (recalling from Chapter 3 that <I>d denotes {¢ E 2F \ {0} I ",d n ¢ =I- 0}), and (b) the system of conditional preferences {t~ I ¢ E <I>d} satisfies Axioms 5, 6, and 11. Axiom 6 implies that P t}lq, q is equivalent to Pq, t~ q q, . In view of this we simplify notation and write P t~ q

instead of p t}lq, q ,

and simplify further by substituting t d for t}. By Proposition 4, t d is conditionally represented: There exist a vNM utility function v d : ~(Z) -+ lR such that p tte} q iff vd(p(e)) ~ vd(q(e)) whenever e E ",d . If E ~ F, say that PE weakly dominates qE at d if, 'lie E E , vd(PE(e)) ~ vd(qE(e)), with strict inequality for some fEE . Say that t d is

41

B elief operators

admissible on E if E is non-empty and p >-d q whenever PE weakly dominates qE at d. The following connect ion between admissibility on subsets and the infinit ely-more-likely relati on is import ant for relating t he two kinds of accessibility relati ons derived from preferences below; the one kind is based on the infinit ely-mor e-likely relation, while t he other is based on admissibility on subsets . Writ e -, E for F \E. PROPOSITION 6 Let E =1=

e E E and

f

E -,E im ply

0

and -,E

=1=

0.

?::d

is adm issible on E iff

e Âť df .

An epistemic model. In a semantic formulation of belief operators one can , following Aumann (1999), start with an information partition of F , and then assum e that the decision maker , for each element of the partition, is endowed with a probability distribution that is concent rate d on this element of t he partition. Since all states within one element of t he partition are indistinguishable, t hey are assigned t he same probability distribution, which however differ from t he probability distributions assigned t o st ates outside t his element . In part icular , probability dist ribut ions assigned to two states in different elements of the partition have disjoint supports. Hence, in Aumann's (1999) formul ation, the decision maker 's probability distribution depend s on in which element of t he information partit ion t he t rue state is. This is consistent wit h the approach chosen here, where t he prob ability distribution-or more genera lly, t he preferences- of t he decision maker will be different for states in different elements of t he information parti tion, and be t he same for all states within t he same element . However, in line with our subject ive perspective, we will const ruct the information partition from t he preferences of the decision maker , so that each element of the partition is defined as a maxim al set of states where t he decision maker's preferences are the same , havin g the int erpretation that states within this set are indistinguishable. Moreover , Aumann's (1999) assumpt ion that the probability distribution is concent rated within the corresponding element of t he partition will in our framework be captured by t he property t hat all states outside (and possibly some st ates inside) t he element are deemed subjectively impossible. Thus, for each d E F , let T d := {e E F I p ?::e q iff P ?::d q} be the set of states t hat are su bjectiv ely distinguishable, and write d ~ e if e E T d . Note t hat ~ is a reflexive, transitive, and symmet ric binary relation; Le., ~ is an equivalence relati on t hat par titions F into equivalence classes (or "types" ).

42

CONSISTENT PREFERENCES

Moreover, ",d denotes the set of states that are subjectively possible (i.e., not Savage-null) at d. In line with the above discussion, assume that , for each d E F , ",d ~ T d â€˘ This assumption will ensure that the preference-based operators satisfy positive and negative introspection; it corresponds to "being aware of one's own type" . Refer to the collection {~dl d E F} as an epistemic model for the decision maker. Axiom 6 implies that p ~~ q {:} P ~d q whenever ",d ~ 1> c F; in particular, p ~~d q {:} P ~d q. The interpretation is the decision maker's preferences at d are not changed by ruling out states that he can distinguish from the true state at d. Hence, we can adopt an interim perspective where the decision maker has already become aware of his own preferences (his own "type") ; in particular, the decision maker's unconditional preferences are not obtained by conditioning "ex ante preferences" on his type. Accessibility relation of epistemic priority. Consider the following definition of the accessibility relation Q. DEFINITION 2 dQe ("d does not have higher epistemic priority than e") if (1) d :::::; e, (2) e is not Savage-null at d, and (3) d is not deemed infinitely more likely than e at d.

7 The relation Q is serial,3 transitive, and satisfies forward linearit y4 and quasi-backward linearity. 5

PROPOSITION

A vector of nested accessibility relations. Consider the collection of all sets E satisfying that ~d is admissible on E. Since ~d is admissible on ",d, it follows that the collection is non-empty as it is contains ",d. Also, since no e E E is Savage-null at d if ~d is admissible on E, it follows that any set in this collection is a subset of ",d . Finally, since e Âťd f implies that f Âťd e does not hold, it follows from Proposition 6 that E' ~ E" or E" ~ E' if ~d is admissible on both E' and E" , implying that the sets in the collection are nested. Hence, there exists a vector of nested sets, (p1, , P1d), on which ~d is admissible, satisfying:

o=1= p1 c

c

p~

c ...

C

P1d =

",d

~

(where C denotes ~ and =1=). 3Vd, 3e such that dQe . 4dQe and dQf imply eQf or fQe. 5If 3d' E F such that d'Qe, then dQf and eQf imply dQe or eQd.

T

d

43

Belief operators

If we assume t hat t d satisfies not only Axiom I' but also Axiom 1, so t hat, as report ed in Propositi on 2, t d is repr esented by v d and an LPS , d A d = (J.L 1, . . . , J.Lfd)- i.e., a sequence of L levels of subjective probability dist ributions- then (p1,.. . ,pfd ) can , in an obvious way, be derived from t he supports of t hese probabil ity distributi ons:

\If E {I , .. . , Ld },

Ped =

Ue

k= l

McLenn an (1989a) develops an ordering of

d sUPPJ.Lk' ",d

t hat is relat ed to

(p1,

... , Pfd) in a context where a system of conditional probabilities is taken as primitive. In a similar contex t , van Fraassen (1976) and Arlo-Cost a and Parith (2003) prop ose a concept of (belief/ probability) cores that correspond to the sets p1,..., Pfd' Grove (1988) spheres and Spohn's (1988 ) ordinal con ditional fun ctions are also related to these sets. For d E F with L d < L := maxeEF U , let Pfd = p1 = ", d for £ E {L d + 1, .. . , L} . The collection {p1 I d E F} of sets defines an accessibility relation ti; DEFINITION 3 dRee ("at d, e is deemed possible at t he epistemic level

£" ) if e E

p1.

PROPOSITION 8 Th e vect or (R 1 , .. . , R d of relations has the f ollowing properti es: For each £ E {I , .. . , L} , R e is seria l, tran sitive, and Euclidean .6 For each £ E {1, . . . , L - I}, (i) dRee implies dR e+ l e and (ii) (3f su ch tha t dRe+ d and eRe+d ) im plies (31' such that dRef' an d eR ef' ) . The correspondence between Q and (R 1 , . .. , R d. That d is not Savage-null at d can be interpreted as d being deemed subjectively possible (at some epistemic level) at any state in the same equivalence class. By part (i) of t he following result , d being not Savage-null at d has two equivalent repr esent ations in t erms of accessibility relations: dQd and dRLd. Likewise, e » d d can be interpret ed as e having higher episte mic priority than d. By part (ii) of the following result , e » d d have two equivalent represent ati ons: (dQe and not eQd ) and (:3£ E {I , . .. , L} such t hat dRee and not eR ed) . Thus, both Q and (R 1 , .. . , RL) capt ure 'subject ive possibility' and 'epistemic priority' as implied by t he preferences of the preference system. PROPOSITION 9 (i) dQd iff dRLd. (ii) (dQe and no t eQ d) iff (3£ E {I, . .. , L} suc h th at dR ee an d not eRed) . 6dR ee and dRd imply eR e! .

44

CONSISTENT PREFERENCES

If Axiom 4 is substituted for Axiom 4'-so that the conditional Archimedean property is strengthened to the Archimedean property-then e being deemed infinitely more likely than f at d implies that f is Savagenull. Hence, L = 1, and by Definitions 2 and 3, Q = RI. Hence, we are left with a unique serial, transitive, and Euclidean accessibility relation if preferences are continuous.

4.2

Defining and characterizing belief operators

In line with the basic structure illustrated in Figure 4.1, we now use the accessibility relations of Section 4.1 to define and characterize belief operators. Defining certain, conditional, and robust belief. Consider the accessibility relation Q of epistemic priority having the properties of Proposition 7. In Asheim and Sevik (2003) we show how equivalence classes can be derived from Q with the properties of Proposition 7, implying that Q with such properties suffices for defining the belief operators. In particular, we show that the set of states that are subjectively indistinguishable at d is given by T

d

= {e E

F 13f E F such thatdQf and eQ!},

and the set of states that are deemed subjectively possible at d equals ",d

=

{e E

T

d

I 3f E F

such that fQe} = {e E

T

d

I eQe} ,

where ",d f 0 since Q is serial, and where the last equality follows since, by quasi-backward linearity, eQe if fQe . Define 'certain belief' as follows . DEFINITION

4 At d the decision maker certainly believes E if d EKE,

where KE := {e E F

I ",e

~

E}.

Hence, at d an event E is certainly believed if the complement is deemed subjectively impossible at d. This coincides with what Morris (1997) calls 'Savage-belief' . 'Conditional belief' is defined conditionally on sets that are subjectively possible at any state; Le., sets in the following collection:

Hence, a non-empty set ¢ is not in <I> if and only if there exists d E F such that ",d n ¢ = 0. Note that FE <I> and, tI¢ E <I>, 0 f ¢ ~ F.

45

Belief operators

Since every ¢ E <I> is subjectively possible at any state, it follows that , V¢ E <I> ,

is non-empty, as demonstrated by t he following lemma. LEMMA 3 If ",d n ¢

i= 0, then :3e E T d n ¢

such that Vf E

Td

n ¢, fQ e.

Define 'condit ional belief' as follows. DEFINITION 5 At d the decision maker believes E conditional on ¢ if dE B(¢)E, where B(¢)E := {e E F I (3e(¢ ) ~ E}. Hence, at d an event E is believed conditional on ¢ if E contains any state in T d n ¢ with at least as high epist emic priority as any other state in T d n ¢ . This way of defining condit ional belief is in t he tradition of, e.g., Grove (1988), Boutilier (1994), and Lamerre and Shoham (1994). Let <I> E be the collection of subject ively possible events ¢ having t he prop erty t hat E is subjectively possible conditi onal on ¢ whenever E is subject ively possible: <I> E :=

n

dE F

<I>~ ,

where Vd E F ,

Hence, a non-empty set ¢ is not in <I> E if and only if (1) there exists d E F such th at ;;,d n ¢ = 0 or (2) t here exists d E F such that E n ", d i= 0 and En ",d n ¢ = 0. Note t hat <I> E is a subset of <I> t hat satisfies F E <I> E ; hence, 0 =1= <I> E <;:;; <I>. Define 'robust belief' as follows. DEFINITION 6 At d the decision maker robustly believes E if d E BO E , where BOE := n ¢E<l>EB(¢)E. Hence, at d an event E is robustly believed in t he following sense: E is believed condit ional on any event ¢ t hat does not make E subjectively impossible. Indeed , BO coincides with what St alnaker (1998) calls 'a bsolute ly robust belief' when we specialize to his setting where Q is also reflexive. The relati on between this belief operator and t he operators 'full belief', 'assumpt ion', and 'st rong belief', introdu ced by Asheim and Dufwenberg (2003a) , Brandenburger and Keisler (2002), and Bat tigalli and Siniscalchi (2002), respectively, will be discussed at t he end of this section as well as in Section 4.4.

46

CONSISTENT PREFERENCES

Characterizing certain, conditional, and robust belief. Consider the vector of nested accessibility relations (R 1 , .. . , RL) having the properties of Proposition 8 and being related to Q as in Proposition 9. In Asheim and Sevik (2003) we first derive (R 1 , ..• , RL) from Q and then show how (R 1 , , RL) characterizes the belief operators. In particular, for all £ E {1, , L}, T

d

= {e E F

131 E F such thatdRd and eRe!} ,

and Furthermore,

The latter observations yield a characterization of certain belief. PROPOSITION

10 KE = {d E F

I pi

~ E}.

Proposition 10 entails that certain belief as defined in Definition 4 corresponds to what Arlo-Costa and Parith (2003) call 'full belief '. Furthermore, by the next result , (unconditional) belief, B(F), corresponds to what van Fraassen (1995) calls 'full belief'. PROPOSITION 11 V1J E (/) =1=

p1 n 4> r; E}.

ll>, B(1J)E = {d E F

I 3£ E

{l, . . . , L} such that

Finally, by Proposition 9(ii) and the following result, E is robustly believed iff any subjectively possible state in E has higher epistemic priority than any state in the same equivalence class outside E. PROPOSITION

12 BOE

= {d

E

F

I

3£ E {l,oo.,L} such that

p1

=

En/\,d} . Asheim and Dufwenberg (2003a) say that an event A is 'fully believed ' at a if the preferences at a are admissible on the set of states in A that are deemed subjectively possible at a. It follows from Proposition 12 that this coincides with robust belief as defined in Definition 6.

4.3

Properties of belief operators

The present section presents some properties of certain, conditional, and robust belief operators. We do not seek to establish sound and complete axiomatic systems for these operators; this should , however, be

47

Belief operators

standard for the certain and conditional belief operators, while harder to establish for the robust belief operator. Rather, our main goal is to show how the non-monotonic (and thus poorly behaved) robust belief operator is bounded by the two KD45 operators certain and conditional belief. While the results certain belief and conditional belief are included as a background for the results on robust belief, the latter findings in combination with the results of Sections 4.2 and 4.4 shed light on the non-standard notions of belief recently used in epistemic analyses of games.

Properties of certain and conditional belief. Note that certain belief implies conditional belief since, by Definitions 4 and 5, (3d( ¢) ~ ",d n ¢. PROPOSITION 13 For all

¢

E

<P, KE

~

B(¢)E.

Furthermore, combined with Proposition 13 the following result implies that both operators K and B( ¢) correspond to KD45 systems. PROPOSITION 14 For all

KEnKE'

¢

E

<P, the following properties hold:

= K(EnE')

B(¢)E n B(¢)E'

= B(¢)(E n E') B(¢)0 = 0

KF=F KE~KKE

,KE

~

K(,KE)

B(¢)E ~ KB(¢)E ,B(¢)E

~

K(,B(¢)E).

Note that K0 = 0, B(¢)F = F, B(¢)E ~ B(¢)B(¢)E and ,B(¢)E ~ B(¢)(,B(¢)E) follow from Proposition 14 since KE ~ B(¢)E. Since an event can be certainly believed even though the true state is an element of the complement of the event, it follows that neither certain belief nor conditional belief satisfies the truth axiom (i.e. KE ~ E and B(¢)E ~ E need not hold).

Belief revision. Conditional belief satisfies the usual properties for belief revision as given by Stalnaker (1998); see also Alchourron et al. (1985). To show this we must define the set, (3d , that determines the decision maker's unconditional belief at the state d: (3d := {e E r d

I Vf E r d ,

fQe} ,

i.e, (3d = (3d(F) . Then the following result can be established. PROPOSITION 15

1 (3d(¢) ~ ¢.

48 2 If {3d

CONSISTENT PREFERENCES

n ¢> =1= 0, then (3d(¢»

3 If ¢> E <P, then (3d( ¢» =1=

= {3d

n ¢>.

0.

4 If (3d(¢» n ¢>' =1= 0, then (3d(¢> n ¢>')

= (3d(¢»

n ¢>' .

c

Properties of robust belief. It is easy to show that certain belief implies robust belief, which in turn implies (unconditional) belief. PROPOSITION

16 KE ~ BO E ~ B(F)E.

Even though robust belief is thus bounded by two KD45 operators, robust belief is not itself a KD45 operator. PROPOSITION

17 The following properties hold: BO E n BO E' ~ BO (E n E')

BOE ~ KBoE

-,BoE

~

K(-,BoE).

Note that B00 = 0, BOF = F, BOE ~ BOBoE and -,BoE ~ BO(-,BOE) follow from Propositions 14 and 17 since KE ~ BO E ~ B(F)E. However, even though the operator BO satisfies BO E ~-, B°-,E as well as positive and negative introspection, it does not satisfy monotonicity since E ~ E' does not imply B E ~ B E'. To see this let = {d} and p~ = ",d = {d, e, f} for some d E F. Now let E = {d} and E' = {d , e}. Clearly, E ~ E', and since p~ = En",d we have d E BO E. However, since neither = E' n ",d nor p~ = E' n ",d , d ¢:. B 0 E' .

°

°

pf

pf

4.4

Relation to other non-monotonic operators

The purpose of this section is to show how robust belief corresponds to the 'assumption' operator of Brandenburger and Keisler (2002) and is related to the 'strong belief' operator of Battigalli and Siniscalchi (2002).

The 'assumption' operator. Brandenburger and Keisler (2002) consider an epistemic model which • is more general than the one that we consider in Section 4.1, since the set of states need not be finite, and • is more special than ours, since, for all d E F, Axioms I', 11, and 4' are strengthened to Axioms 1 and 4", so that completeness and the partitional Archimedean property are substituted for conditional completeness, partitional priority, and the conditional Archimedean property.

Belief operators

49

Within our setting with a finite set of states, F, it now follows, as reported in Proposition 3, that ~d is represented by v d and an LCPS )..d = (flY, . . . ,flfd)-Le., a sequence of L d levels of non-overlapping subjective probability distributions. Hence, W E {I, . .. , L d } , sUPPfl1 = rrff, where (rrt , .. . , rrfd) is a partition of ",d. In their Appendix B, Brandenburger and Keisler (2002) employ an LCPS to represent preferences in their setting with an infinite set of states. Provided that completeness and the partitional Archimedean property are satisfied, Brandenburger and Keisler (2002) introduce the following belief operator in their Definition Bl; see also Brandenburger and Friedenberg (2003). DEFINITION 7 (BRANDENBURGER AND KEISLER, 2002) At d the decision maker assumes E if ~~ is nontrivial and p >-~ q implies p >-d q. PROPOSITION 18 Assume (in addition to the assumptions made in Section 4.1) that, for each d E F, ~d satisfies Axioms 1 and 4" if 1> E 2F \ {0}. Then BOE = {d E FIE is assumed atd} . Proposition 18 shows that the 'assumption' operator coincides with robust belief (and thus with Stalnaker's 'absolutely robust belief') under completeness and the partitional Archimedean property. However, if the partitional Archimedean property is weakened to the conditional Archimedean property, then this equivalence is not obtained. To see this, let ",d = {d, e, f}, and let the preferences ~d, in addition to the properties listed in Section 4.1, also satisfy completeness. It then follows from Proposition 2 that ~a is represented by va and a LPS-Le., a sequence of subjective probability distributions with possibly overlapping supports. Consider the example provided by Blume et al. (1991a) in their Section 5 of a two-level LPS , where the primary probability distribution, flY, is given by flY(d) = 1/2 and fly(e) = 1/2, and the secondary probability distribution, fl~, used to resolve ties , is given by fl~(d) = 1/2 and fl~(f) = 1/2. Consider the acts p and q, where vd(p(d)) = 2, vd(p(e)) = 0, and vd(p(f)) = 0, and where vd(q(d)) = 1, vd(q(e)) = 1, and vd(q(f)) = 2. Even though ~d is admissible on {d,e}, and thus {d, e} is robustly believed at d, it follows that {d, e} is not 'assumed' at d since p>-d{d,e}q w hil 1 e p-: d q. Brandenburger and Keisler (2002) do not indicate that their definition - as stated in Definition 7-should be used outside the realm of preferences that satisfy the partitional Archimedean property. Hence, our

50

CONSISTENT PREFERENCES

definition of robust belief-combined with the characterization result of Proposition 12 and its interpretation in term of admissibility-yields a preference-based generalization of Brandenburger and Keisler (2002) operator (in our setting with a finite set of states) to preferences that need only satisfy the prop erties of Section 4.1.

The 'strong belief' operator. In the setting of extensive form games , Battigalli and Siniscalchi (2002) have suggested a non-monotonic 'strong belief ' operator. We now show how their 'strong belief' operator is related to robust belief, and thereby, to 'absolutely robust belief' of Stalnaker (1998), 'full belief ' of Asheim and Dufwenberg (2003a), and 'assumpt ion' of Brandenburger and Keisler (2002). Battigalli and Siniscalchi (2002) base their 'st rong belief ' operator on a conditional belief operator derived from an epistemic model where , at each state d E F, the decision maker is endowed with a system of conditional preferences n:~ I ¢ E <I>d} (with , as before, <I>d denoting {¢ E 2F\ {0} I /'i,d n ¢ # 0}). However, Battigalli and Siniscalchi (2002) assume that, if the true state is d, then the decision maker's system of I ¢ E <I>d}. conditional preferences is represented by v d and a CPS Since a CPS does not satisfy conditionality as specified by Axiom 6, we must embed their conditional belief operator in the framework of the present chapter. We can do so using Corollary 1 of Chapter 3.

V4

One the one hand, Battigalli and Siniscalchi (2002) and Ben-Porath (1997) define 'condit ional belief with probability one' in the following way: At d the decision maker believes E conditional on ¢ E <I> if sUPPP~ ~ E, where {p~ I ¢ E <I>d} is a CPS on F with support /'i,d . On the other hand, according to Definition 5 of the present chapter, at d the decision maker believes E conditional on ¢ E <I> if {3d(¢ ) ~ E . If, however, Axioms 1', 4' , and 11 are strengthened to Axioms 1 and 4", so that by Proposition 3 ?:d is represented by v d and an LCPS , >..d = (pf, . .. ,P1d) , on F with support /'i,d, then Lemma 14 of Appendix A implies that {3d(¢) = suppp~n¢, where £-:= min{k I suppp%n¢ # 0}. Hence, by Corollary 1, 'condit ional belief with probability one' as defined by Battigalli and Siniscalchi (2002) and Ben-Porath (1997) is isomorphic to the conditional belief operator B (¢) derived from an epistemic model satisfying the assumptions of Section 4.1 of the present chapter. Given that the conditional belief operator of Battigalli and Siniscalchi (2002) thus coincides with the B (¢) operator of the present paper, we

Belief operators

51

can define their 'strong belief ' operat or as follows: Let <PH ( ~ <p ) be some non-empty subcollection of t he collection of subsets that are subjectively possible at any state; e.g., in an exte nsive game <PH may consist of t he subsets t hat correspond to subgames. Then <PH n <P E is t he collection of subsets ¢ satisfying ¢ E <P H and having t he property t hat E is subjectively possible condit ional on ¢ whenever E is subject ively possible. DEFINITION 8 (BATTIGALLI AND SINISCALCHI , 2002 ) At d t he decision maker strongly believes E if d E n ¢>E<l> Hn <l> E B (¢ )E. Hence, at d an event E is strongly believed if E is robustly believed in th e following sense: E is believed condit ional on any subset ¢ in <PH that does not make E subject ively impossible. Since <P E 2 <PH n <P E 2 {F} , it follows that the 'st rong belief ' operator is bounded by t he robust belief and (uncondit ional) belief operators. PROPOSITION 19 B OE ~ {d E FIE is strongly believed atd } ~ B(F )E. As suggested by Bat t igalli and Bonanno (1999), t he 'strong belief' operator may also be defined w.r.t. other subcollect ions of <P t han t he collection of subsets t hat correspond to subgames, and may be seen as a generalizat ion of robust belief by not necessarily requiring belief to be "absolutely robust" in t he sense of Stalnaker (1998). However, provided that F is included, Proposition 19 st ill holds. In any case, t he 'st rong belief' operator shares t he prop erties of robust belief: Also 'st rong belief' satisfies t he properties of P roposition 17, but is not monotoni c.

Chapter 5

BASIC CHARACTERIZATIONS

In this chapter we present characterizations of basic game-theoretic concepts. After presenting the concept of an epistemic model of a strategic game form in Section 5.1, we turn to the characterizations of Nash equilibrium and rationalizability in Section 5.2 and characterizations of (strategic form) perfect equilibrium and permissibility in the final Section 5.3. The characterizations of Nash equilibrium and rationalizability will be done by means of the event that each player has preferences that are consistent with the game and the preferences of the opponent. Likewise, the characterizations of (strategic form) perfect equilibrium and permissibility will be done by means of the event that each player has preferences that are admissibly consistent with the game and the preferences of the opponent. Hence, the chapter illustrates the 'consistent preferences' approach and set the stage for the analysis of subsequent chapters. Note that the results of this chapter are variants of results that can be found in the literature. In particular, the characterizations of Nash equilibrium and (strategic form) perfect equilibrium are variants of Propositions 3 and 4 of Blume et al. (1991b).

5.1

Epistemic modeling of strategic games

The purpose of this section is to present a framework for strategic games where each player is modeled as a decision maker under uncertainty. The analysis builds on the two previous chapters and introduces

54

CONSISTENT PREFERENCES

the concept of an epistemic model for a strategic game form. In this chapter preferences are assumed to be complete, an assumption that will be relaxed in Chapter 6.

A strategic game form. Denote by 5 i player i 's finite set of pure strategies, and let z : 5 ---t Z map strategy profiles into outcomes, where 5 = 51 X 52 is the set of strategy profiles and Z is the finite set of outcomes. Then (51,52, z) is a finite strategic two-player game form. An epistemic model. For each player i , any of i's strategies is an act from strategy choices of his opponent j to outcomes. The uncertainty faced by a player i in a strategic game form concerns (a) j's strategy choice, (b) j's preferences over acts from i's strategy choices to outcomes, and so on (d. the discussion in Section 1.3). A type of a player i corresponds to (a) preferences over acts from j's strategy choices, (b) preferences over acts from pairs of j's strategy choice and j's preferences over acts from i 's strategy choices, and so on. For any player i , i's decision is to choose one of his own strategies. As the player is not uncertain of his own choice, the player 's preferences over acts from his own strategy choices is not relevant and can be ignored. Hence, in line with the discussion in Section 1.3, consider an implicit model-with a finite type set T; for each player i-where the preferences of a player corresponds to the player 's type, and where the preferences of the player are over acts from the opponent's strategy-type pairs to outcomes. If we let each player be aware of his own type (as we will assume throughout), this leads to an epistemic model where the state space of player i is T; X 5 j x Tj , and where, for each ti E Ti ,

constitutes an equivalence class, being the set of states that are indistinguishable for player i at ti, and a non-empty subset K,ti of {td x 5 j x Tj is the set of states that player i deems subjectively possible at ti. DEFINITION 9 An epistemic model for the finite strategic two-player game form (51, 52, z) is a structure

(51, Tl ' 52, T2), where, for each type t, of any player i , t i corresponds to a system of conditional preferences on the collection of sets of acts from elements of

Basic characterizations

55

to ~(Z), where ""ti is a non-empty subset of {td x 5j x Tj. An implicit model with a finite set of types for each player, as considered throughout this book, does not allow for 'preference-completeness', where, for each player i, there exists some type of i for any feasible preferences that i may have. 1 Still, even a finite implicit model gives rise to infinite hierarchies of preferences, and - in effect - we assume that each player as a decision maker is able to represent his subjective hierarchy of preferences by means of a finite implicit model. Then, at the true profile of types, the two players' subjective hierarchies can be embedded in a single implicit model that includes the types of the two players that are needed to represent each player's hierarchy. Such a construction can fruitfully be used to analyze a wide range of game-theoretic concepts, as will be demonstrated throughout this book . However, when embedding the two player's subjective hierarchies into a single implicit model , it is illegitimate to require that player i deems the true type of his opponent j subjectively possible. Rather, we cannot rule out that, at the true type profile, player j's true type is not needed to represent player i's subjective hierarchy of preferences; this is particularly relevant for the analysis of non-equilibrium game-theoretic concepts. Hence, when applying finite implicit models for interactive analysis of games, it is important to allow-as we do in the framework of the present text-the decision maker to hold objectively possible opponent preferences as subjectively impossible . Throughout this book we will consider two different kinds of epistemic models that differ according to the kind of assumption imposed on the set of conditional preferences that ti determines. For the present chapter, as well Chapters 8, 9, and 10, we will make the following assumption. 1 For each ti of any player i, (a) t~ satisfies Axioms 1, ~ T; X 5j x Tj, and Axiom 3 if and only if Â˘ E <p t i , (b) the system of conditional preferences {t~1 Â˘ E <p ti } satisfies Axioms 5, 6', and 16, and (c) there exists a non-empty subset T/i of opponent types such that ""ti = {ti} x 5j x T/i.

ASSUMPTION

2, and 4' if 0 =1= Â˘

1 'P reference-complet eness' is needed for the interactive epistemic analyses of, e.g. , Brandenburger and Keisler (2002) and Battigalli and Siniscalchi (2002), but not for the analysis presented in this book. Brandenburger and Keisler (1999) show that there need not exist a 'preference-complet e' interactive epistemic model when preferences are not representable by subjective probabilities, implying that ' preference-com plet eness' may be inconsistent with the analysis of Chapters 6, 7, 11, and 12, where Axiom 1 is not imposed .

56

CONSISTENT PREFERENCES

In this assumption, T/i is the non-empty set of opponent types that player i deems subjectively possible at ti. The assumption explicitly allows for preferences over acts from subsets ¢ of T; x Sj x Tj , where projsj ¢ may be a strict subset of Sj. This accommodates the analysis of extensive game concepts in Chapters 8 and 9 and will permit the concepts in Tables 2.1 and 2.2 to be treated in a common framework. Write ?::t i for player i 's preferences conditional on being of type ti ; i.e., for ?::~ when ¢ = {td x Sj x T j . We will refer to ?::t i as player i 's unconditional preferences at ti . Under Assumption 1 it follows from Proposition 5 that, for each type ti of any player i , i's system of conditional preferences at ti can be represented by a vNM utility function vii : ~(Z) -----+ IR and an SCLP ( ).ti , £ti) on T, x S·J x TJ with support K,ti = {t l·} X S·J X TJ·ti. Throughout , we will adopt an interim perspective, where player i has already become aware of his own type. This entails that we can w.l.o.g. assume that , for any ¢ E (pt i, £(¢) = £(¢ n ({ til x Sj x T j)) . The interpretation is player i 's preferences at ti are not changed by ruling out states that i can distinguish from the true state at ti. Consequently, for expositional simplicity we choose to let the SCLP ().t, , £t i) be defined on Sj x Tj with support Sj x T/i ~

Preferences over strategies. It follows from the above assumptions that , for each type ti of any player i , player i 's unconditional preferences at ti, ?::t i , are a complete and transitive binary relation on the set of acts from Sj x Tj to ~(Z) that is represented by a vNM utility function vii and an LPS ).~i = (f.li\ .. . , f.l~i ) , where £ = £(Sj x Tj) . Since each pure strategy Si E S, is a function that assigns the deterministic outcome Z(Si, Sj) to any (Sj, tj) E Sj x Tj and is thus an act from Sj x T, to ~(Z) , we have that ?::t i determines complete and transitive preferences on i 's set of pure strategies, Si. Player i's choice set at ti, sii, is player i's set of rational pure strategies at ti:

Since ?::t i is complete and transitive and satisfies objective independence, and S, is finite, it follows that the choice set sii is non-empty, and that the set of rational mixed strategies equals ,6. (Sf;).

A strategic game. Let , for each i , Ui : S -----+ IR be a vNM utility function that assigns payoff to any strategy profile. Then G = (S1, S2, U1, U2)

57

Basic characterizations

is a finite strategic two-player game. Assume that, for each i, there exist s = (Sl,S2), S' = (s~,s~) E S such that Ui(S) > Ui(S'). The event that i plays the game G is given by

lUi] := {(t1' t2) E T1XT21 vfiozis a positive affine transformation of u.}, while [u] := [U1] n [U2] is the event that both players play G. Denote by Pi, qi E ~(Sd mixed strategies for player i, and let Sj (~ Sj) be a non-empty set of opponent strategies. Say that Pi strongly dominates qi on Sj if, VSj E Sj, Ui(Pi,Sj) > Ui(qi,Sj) . Say that qi is strongly dominated on Sj if there exists Pi E ~(Sd such that Pi strongly dominates q i on Sj. Say that Pi weakly dominates qi on Sj if, VSj E Sj, Ui(Pi,Sj) 2: Ui(qi ,Sj) with strict inequality for some sj E Sj. Say that qi is weakly dominated on Sj if there exists Pi E ~(Sd such that Pi weakly dominates q i on Sj. The following two results will be helpful for some of the proofs. 4 Let G = (Sl, S2, U1, U2) be a finite strategic two-player game. For each i, Pi E ~ (Si) is strongly dominated on Sj if and only if there

LEMMA

does not exist J1 E ~(Sj) with sUPPJ1 ~ Sj such that, Vs~ E Si,

L

J1(Sj)Ui(Pi, Sj) 2:

~E~

L

J1(Sj)Ui(S~, Sj) .

~E~

Proof. Lemma 3 of Pearce (1984). â€˘ 5 Let G = (Sl, S2, U1, U2) be a finite strategic two-player game. For each i, Pi E ~ (Si) is weakly dominated on Sj if and only if there does not exist J1 E ~(Sj) with sUPPJ1 = Sj such that, Vs~ E Si,

LEMMA

L ~E~

J1(Sj)Ui(Pi, Sj) 2:

L

J1(Sj)Ui(S~, Sj) .

~E~

Proof. Lemma 4 of Pearce (1984). â€˘ Certain belief. In the present chapter, as well in Chapters 8, 9, and 10, we will apply the certain belief operator (cf. Definition 4 of Chapter 4) for events that are subsets of the set of type profiles, T 1 x T2 . In Assumption 1 we allow for the possibility that each player deems some opponent types subjectively impossible, corresponding to an SCLP that does not have full support along the type dimension. Therefore, certain belief (meaning that the complement is subjectively impossible) can be

CONSISTENT PREFERENCES

58

derived from the epistemic model and defined for events that are subsets of T I x T2. For any E <;;; T I X T2, say that player i certainly believes the event E at ti if ti E projTiKiE, where

KiE := {(tl,t2)

E

T I X T2 I projTlxT2Kti

=

{td x T/; <;;; E}.

Say that there is mutual certain belief of E at (tl, t2) if (tl, t2) EKE, where KE := KIE n K2E . Say that there is common certain belief of E at (tI, t2) if (tl, t2) E CKE, where CKE:= KE n KKE n KKKE n ... . As established in Proposition 14, K, corresponds to a KD45 system. Moreover, the mutual certain belief operator, K, has the following properties, where we write KOE := E, and for each g ~ 1, KgE := KKg- I E. PROPOSITION 20 (i) For any E <;;; T I X T2 and all g > 1, KgE <;;; Kg- IE. If E = E I n E 2, where, for each i, E i = projT;E i X T j, then KE <;;; E . (ii) For any E <;;; T I X T2, there exists g' ~ 0 such that KgE = CKE for g ~ g', implying that CKE = K CKE. Proof. Part (i). If E = E I nE2, where, for each i, E i = proj--, E i x T j, then KE = KIE n K2E <;;; KIE I n K 2E 2 = E I n E 2 = E , establishing the second half of part (i). Since, for any E <;;; TI X T2, KE = KIE n K2E, where, for each i , KiE = proh;KiE x Tj , the first half of part (i) follows from the result of the second half. Part (ii) is a consequence of part (i) and T I x T2 being finite . â€˘

5.2

Consistency of preferences

In the present section we define the event of consistency of preferences and show how this event can be used to provide characterizations of mixed-strategy Nash equilibrium and mixed rationalizable strategies.

Inducing rational choice. In line with the discussion in Section 1.1, and following a tradition from Harsanyi (1973) to Blume et al. (1991b), a mixed strategy will interpreted, not as an object of choice, but as an expression of the beliefs for the other player . Say that the mixed strategy p/;It j is induced for tj by ti if tj E T/; and, for all 8j E Sj, .tiltj

PJ

t; (

)

(8.) = J-lf, 8j, tj J J-lf,t; (Sj ,tj ) ,

where J-l~;(Sj, tj) := LSjESj J-l~i(8j, tj), and where I! denotes the first level I! of Ati for which J-l~i (Sj, tj) > O. Furthermore, define the set of type profiles for which ti induces a rational mixed strategy for any subjectively

59

Basic characterizations

possible opponent type:

[iri] := {(t l ' t2) E TI x T21 Vtj E Tt, p/iltj E t::.(S/j) } . Write [ir] := [irl] n [ir2]' Say that at ti player i 's preferences over his strategies are consistent with the game G = (Sl, S2,UI, U2) and the preferences of his opponent, if ti E projTi([Ui] n [iri])' Refer to [u] n [ir] as the event of consistency.

Characterizing Nash equilibrium. In line with the discussion in Section 1.1, we now characterize the concept of a mixed-strategy Nash equilibrium as profiles of induced mixed strategies at a type profile in [u] n [ir] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Before doing so, we define a mixed-strategy Nash equilibrium. DEFINITION 10 Let G = (SI ,S2,UI,U2) be a finite strategic two-player game. A mixed strategy profile p = (PI,P2) is a mixed-strategy Nash equilibrium if for each i,

Ui(Pi ,Pj) = maxui(p~,Pj) .

p;

The characterization result-which is a variant of Proposition 3 of Blume et al. (1991b)-can now be stated. PROPOSITION 21 Consider a finite strategic two-player game G. A profile of mixed strategies p = (PI,P2) is a mixed-strategy Nash equilibrium if and only if there exists an epistemic model with (tl, t2) E [u] n fir] such that (1) there is mutual certain belief of {(tl, t2)} at (h , t2), and (2) for each i, Pi is induced for ti by tj. Proof. (Only if.) Let (PI,P2) be a mixed-strategy Nash equilibrium. Construct the following epistemic model. Let T I = {t d and T2 = {t2}' Assume that, for each i,

â€˘ v/t satisisfies t hat

Vit '

0

Z

= iu,

â€˘ the SCLP (>, ti, i ti ) has the properties that >.. ti = (J-"ii, . .. , J.LZ) with support Sj x {tj} satisfies that , VS j E Sj, J.Lii(sj, tj) = Pj(Sj), and i ti satisfies that i( Sj x {tj}) = 1. Then, it is clear that (tl, t2) E [u], that there is mutual certain belief of {(tl, t2)} at (tl , t2), and that, for each i , Pi is induced for ti by tj. It

60

CONSISTENT PREFERENCES

remains to show that (t1' t2) E [irJ, i.e. , for each i, Pi E ~(Sii) . Since , by Definition 10, it holds for each i that, Vs~ E Si, Ui (pi ,Pj) ~ u; (S~, Pi), it follows from the construction of (Ati, £ti) that Pi E ~ (Sii ). (If.) Suppose that there exists an epistemic model with (tI, t2) E [u] n [ir] such that there is mutual certain belief of {(t1, t2)} at (t1 ' t2), and, for each i, Pi is induced for ti by tj. Then, for each i , ?:t i is represented by Vii satisfying that vii o z is a positive affine transformation i i of Ui and an LPS A~i = , .. . ,f.l~i), where VS j E Sj , (Sj, tj) = Pi(Sj), and where £ = £(Sj x Tj) ~ 1. Suppose, for some i and p~ E ~(Sd , u;(pi,Pj) < Ui (p~, Pj). Then there is some Si E S, with Pi (sd > 0 and some s~ E Si such that u;(Si ' Pj) < u; (s~ , Pj), or equivalently

(f.li

L

sJ

f.li

f.lii(sj, tj)Ui(Si, Sj) <

L

sJ

f.lii(sj , tj)Ui(S~, Sj).

This means that Si rt Sii, which, since Pi(sd > 0, contradicts (t1,t2) E [irj]. Hence, by Definition 10, (P1,P2) is a Nash equilibrium. _ For the "if" part of Proposition 21, it is sufficient that there is mutual certain belief of the beliefs that each player has about the strategy choice. We do not need the stronger condition that (1) entails. Hence, higher order certain belief plays no role in the characterization, in line with the fundamental insights of Aumann and Brandenburger (1995) .

Characterizing rationalizability. We now turn to analysis of deductive reasoning in games and present a characterization of (ordinary) rationalizability. Since we are only concerned with two-player games, there is no difference between rationalizability, as defined by Bernheim (1984) and Pearce (1984), and correlated rationalizability, where conjectures are allowed to be correlated. As it follows that rationalizability in two-player games corresponds to IESDS, we use the latter procedure as the primitive definition. For any (0 f) X = Xl X X 2 ~ S, X S2, write c(X) := C1(X2) x C2(X 1), where

Ci(Xj ) :=

s, \ {Si E s, I :3Pi E ~(Si) s.t,pistrongly dominates s.on Xj} .

11 Let G = (Sl,S2 ,U1,U2) be a finite strategic two-player game. Consider the sequence defined by X(O) = Sl X S2 and, Vg ~ 1, X (g) = c(X (g - 1)). A pure strategy Si is said to be rationalizable if DEFINITION

Si E

R; :=

n°o

9=0

Xi (g) .

A mixed strategy Pi is said to be rationalizable if Pi is not strongly dominated on R j •

B asic characteriza tions

61

While any pure strat egy in t he support of a rati onalizable mixed strategy is it self rationali zable (d ue t o what P earce calls t he pure strategy prop erty), t he mixture on a set of rationalizable pure st rategies need not b e rationalizable. The following lemma is a straightforward implicati on of Definition 11. L EM MA 6 (i) For each i, R; =1= 0. (ii) R = c(R). (iii) For each i, s, E R; if and only if there exists X = Xl X X 2 with Si E R, such that X ~ c(X ).

We next characterize the conce pt of rationalizable mixed st rategies as induced mixed strategies under common certain b elief of [u] n [ir].

A mixed strategy Pi for i is rationalizable in a finit e strategic two-player game G if and only if there exists an epistemic model with (tl ' t2) E CK([u] n [ir]) such that Pi is induced for ti by tj.

PROPOSITION 22

Proof. Part 1: If pi is rationalizable, then there exists an epistemic model with (ti , t 2) E CK([u] n [ir]) such that pi is induced for ti by t; . Step 1: Construct an epistemic model with T I x T2 ~ CK( [u] n [ir]) such that for each s; E R; of any player i , there exists ti E T; with, s; E sii . Construct an epistemic mod el with, for eac h i, a bij ection s, : Ti -> R; from t he set of types t o t he t he set of rat ion ali zable pure strategies. Assume t hat, for each ti E T; of any player i , vi i satisfies t hat (a) vi i 0 z

= Ui

(so t hat T I x T2 ~ [u]),

and t he SeLP (>\ ti , £ti) on Sj x T j has the proper ti es t hat (b ) )..ti = (fLii , . . . , fLZ) with support Sj x Tl i satisfies t hat sUPPfLi l n (Sj x {t j}) = {(Sj (tj ),tj)} for all tj E Tl i (so t hat, Vtj E Tl i , pl iltj (sj(tj)) = 1), (c) £ti satisfies £ti(Sj x T j) = 1. Property (b) entails t hat t he support of the m argin al of fLii on Sj is included in Rj. By properties (a ) and (c) and Lemmas 4 and 6(ii) , we can still choose fLii (a nd Titi) so that s, (ti ) E sii. This combined wit h property (b) means t hat T I x T2 ~ [ir]. Furthermor e, T I x T2 ~ CK( [u] n [ir]) since Tp ~ T j for each i, E T i of any player i . Sinc e, for eac h pl ayer i , s i is ont o Ri' it follows t hat , for each s: E R; of any pl ayer i , t here exists t ; E T, wit h s, E sii. Step 2: Add type ti to t; Assume t hat vli satisfies (a) and ()..ti, £ti) satisfies (b) and (c). T he n fL lti ca n b e chose n so t hat pi E t::. (Sl i ).

62

CONSIS TEN T PREFERENCES

tt: U {tn ) x Tj {tn ) x Tj ~ CK([u] n fir ]).

Furthermore,

~ [u]

n fir],

and since Tp. ~ Tj ,

ir. U

Step 3: Add type tJ* to Tj. Assume that v]J satisfies (a) and t he SCLP r: t: t* t* t* ( ,\ i , f j ) on S, x (Ti U {ti} ) has t he property that ,\ j = (J.L 1i ; . . . ,J.L L j ) with support S, x {ti} satisfies t hat, \::fSi E Si, PIt; (Si, tT) = pi (Si), so that pi iSinducedJorti bytj. Fur thermore, (1iU{tn) x (TjU{tj }) ~ [u] n [ir], and since Tb ~ TiU{ti}, (TiU{tn ) x (Tj u { tj} ) ~ CK([u] n[i r]). Hence, (ti , t 2) E CK([u] n fir]) and pi is induced for ti by tj. Part 2: If there exists an epistemic model with (ti , t 2) E CK([u] n [ir]) such that pi is induced for ti by tj , then pi is rationalizable. Assum e that there exists an episte mic model with (ti , t 2) E CK([u]: n [ir']) such that pi is induced for ti by tj . In particular , CK([u] n [ir]) =J 0. Let , for each i , t: := projTiCK([u] n [ir]) and Xi := UtiET1S;i. By Proposition 20(ii) , for each ti E Tf of any player i , ti deems (S j ,tj) subject ively impossible if tj E Tj\Tj since CK([u] n [ir]) = KCK([u] n [ir]) ~ KiCK([u] n [ir]), implying T/ i ~ Tj. By the definitions of [u] and [ir], it follows that , for each ti E Tf of any player i, ?:t i is represent ed by sat isfying t hat 0 z is a positive affine t ransformation of Ui i and an LPS ,\~i = (J.L i , . .. , J.L~i ) , where f = f( Sj x Tj ) ~ 1, and where sUPPJ.Lii ~ X j x Tj . Hence, by Lemma 4, for each ti E T[ of any player i , if Pi E 6.(S;i), t hen no st rategy in t he support of Pi is st rongly dominated on X j , since it follows from Pi E 6.(S;i) and sUPPJ.Lii ~ X j x Tj that , \::fsi E SUPPPi and \::fs~ E Si,

Vii

Vii

L L S jE X j

tjETj

J.Lii(sj ,tj )Ui(Si, Sj ) ~

L L S j E Xj

J.Lii ( Sj , tj ) Ui (S~ , Sj ) .

tjETj

This implies X ~ c(X ), ent ailing by Lemma 6(iii) t hat , for each i, Xi ~ u; Furthermore, since (ti , t 2) E CK([u] n [ir]) and the mixed st rategy induced for ti by tj, pi , satisfies pi E 6.(S8) , it follows that pi is not strongly dominated on Xj ~ Rj . By Definition 11 this implies that pi is a rationalizable mixed strategy. _

5.3

Admissible consistency of preferences

We next refine t he event of consiste ncy of preferences and show how t his leads to char act erizations of (strategic form) perfect equilibrium and mixed permissible st rategies. Caution. Player i has preference for caut ious behavior at ti if he takes into account all opponent st rategies for any opponent ty pe t hat is deemed subjectively possible.

63

Basic characterizations

Throughout this chapter as well as chapters 8, 9, and 10, we assume that Assumption 1 is satisfied so that K,ti = {td x Sj x T/i. Under Assumption 1 player i is cautious at ti if {~~ I f/J i= ¢ ~ <I>t i} satisfies Axiom 6. Because then it follows from Proposition 2 that player i 's unconditional preferences at ti, ~ti, are represented by V;i and an LPS Ati with support Sj x T/ i. Since thus (Sj, tj) E suppx" for any (Sj, tj) satisfying tj E Tli, player i at ti takes into account all opponent strategies for any opponent type that is deemed subjectively possible. Hence, under Assumption 1, we can define the event

[caUi]:= {(tl,t2) E t, x T21 {~~ I f/J

i= ¢

~ <I>t i} satisfies Axiom 6}.

In terms of the representation of the system of conditional preferences I f/J i= ¢ ~ <I>t i} by means of a vNM utility function and an SeLP (d. Proposition 5), caution imposes the additional requirement that for each type ti of any player i the full LPS Ati is used to form the conditional {~~

beliefs over opponent strategy-type pairs. Formally, if L denotes the number of levels in the LPS At i , then

[caui]

= {(tt , t2) E Tl

x T21 £ti(Sj x T j)

= L}.

Since £ti is non-increasing w.r.t. set inclusion , ti E ProjTi [caui] implies that £ti (projsj «r, ¢) = L for all subsets ¢ of {td x Sj x T j with welldefined conditional beliefs. Since it follows from Assumption 1 that Ati has full support on Sj, ti E ProjTJcaui] means that i's choice set at ti never admits a weakly dominated strategy, thereby inducing preference for cautious behavior. Write [caul := [caud n [cau2]' Say that at ti player i 's preferences over his strategies are admissibly consistent with the game G = (SI, S2, Ul, U2) and the preferences of his opponent, if ti E ProjTi ([ud n [iri] n [caui])' Refer to [u] n [ir] n [caul as the event of admissible consistency. Characterizing perfect equilibrium. We now characterize the concept of a strategic form (or "t rembling-hand" ) perfect equilibrium as profiles of induced mixed strategies at a type profile in [u] n [ir] n [caul where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Before doing so, we define a (strategic form) perfect equilibrium. DEFINITION 12 Let G = (SI, S2, Ul, U2) be a finite strategic two-player game. A mixed strategy profile p = (Pl,P2) is a (strategic form) perfect equilibrium if there is a sequence (p(n) )nEN of completely mixed strategy

64

CONSISTENT PREFERENCES

profiles converging to P such that for each i and every n EN,

Ui(Pi ,pj(n)) = maxui(p~ ,pj(n)) .

P:

The following holds in two-player games. 7 Let G = (Sl, S2, UI , U2) be a finite strategic two-player game. A mixed strategy profile P = (PI ,P2) is a (strategic form) perfect equilib-

LEMMA

rium if and only if P is a mixed-strategy Nash equilibrium and, for each i, Pi is not weakly dominated .

Proof. Proposition 248.2 of Osborne and Rubinstein (1994). • The characterization result-which is a variant of Proposition 4 of Blume et al. (1991b)-can now be stated.

Consider a finite strategic two-player game G. A profile of mixed strategies P = (PI,P2) is a (strategic form) perfect equilibrium if and only if there exists an epistemic model with (tl ,t2) E [u] n [ir] n [caul such that (1) there is mutual certain belief of {(tl ' t2)} at (tl ' t2), and (2) for each i, Pi is indu ced for ti by tj . PROPOSITION 23

Proof. (Only if.) Let (PI,P2) be a (strategic form) perfect equilibrium. Then, by Lemma 7, (PI,P2) be a mixed-strategy Nash equilibrium and, for each i, Pi is not weakly dominated. Construct the following epistemic model. Let T I = {t I} and T 2 = {t2}' Assume that, for each i ,

• v/t satisisfies t hat • the SCLP port Sj x \:ISj E Sj, SUppf.t~i =

V it ' 0 Z = tu,

()\ti , £ti) has the properties that ).. ti = (f.tii, f.t~i) with sup{tj} has two levels, with the first level chosen so that, f.tii(sj , tj) = Pj(Sj) , and the second level chosen so that Sj X {tj} and, \:Is~ E Si,

L

sJ

f.t~i (Sj, tj )Ui(Pi, Sj) ~

L

sJ

f.t~i (Sj, tj )Ui(s~ , Sj)

(which is possible by Lemma 5 since Pi is not weakly dominated) , and £ti satisfies that £ti(Sj x Tj ) = 2. Then, it is clear that (tl , t2) E [u] n [caul, that there is mutual certain belief of {(tl, t2)} at (tI, t2) , and that, for each i , Pi is induced for t, by tj ' It remains to show that (tl, t2) E [irl, i.e., for each i, Pi E tl(S;i). Since, by Lemma 7, it holds for each i that , \:Is~ E Si, Ui(Pi,Pj) ~ Ui(S~,Pj), it follows from the construction of ().. ti, £ti) that Pi E tl (S;i).

65

Basic characterizations

(If.) Suppose that there exists an epistemic model with (iI, t2) E [u] n [ir] n [caul such that there is mutual certain belief of {(tl ' t2)} at (tl, t2), and, for each i, Pi is induced for ti by tj' Then, for each i, ?:t i is represented by Vfi satisfying that Vfi oz is a positive affine transformation of Ui and an LPS )..ti = (/1ii, ... , /12) , where \:fs j E Sj, /1ii (sj , tj) = Pj(Sj), and where suppx ti = Sj X {tj}. Suppose first that (PI, P2) is not a Nash equilibrium; i.e., for some i and p~ E L\(Si), Ui(Pi,Pj) < Ui(p~,Pj). Then there is some Si E S, with Pi (sd > 0 and some s~ E S, such that Ui (Si' Pj) < u ; (s~ , Pj), or equivalently

L

sJ

/1ii (sj , tj)Ui(Si, Sj) <

L

sJ

/1ii(Sj,tj)Ui(S~ , Sj).

This means that Si ¢:. S;i, which, since Pi(Si) > 0, contradicts (tl, t2) E [irj]. Suppose next that , for some i , Pi is weakly dominated. Since supp.V: = Sj x {tj} , this also implies that s, ¢:. S;i for some s, E S, with Pi(Si) > 0, again contradicting (tl, t2) E [irj] . Hence, by Lemma 7, (PI, P2) is a (strategic form) perfect equilibrium. _ As for Proposition 21, higher order certain belief plays no role in this characterization. Characterizing permissibility. We now turn to the non-equilibrium analog to (strategic form) perfect equilibrium, namely the concept of permissibility; d. Borgers (1994) as well as Brandenburger (1992) who coined the term 'permissibility'. To define the concept of permissible strategies, we use the equivalent Dekel-Fudenberg procedure as the primitive definition. For any (0 =1=) X = Xl X X 2 ~ Sl X S2, write a(X) := 0,1 (X2) x a2(XI), where

ai(Xj ) := Si \ {s,

E

s, I ::JPi E L\(Sd s.t ,pistrongly dominates s.on X, or Pi weakly dominates Sion Sj} .

DEFINITION 13 Let G = (Sl , S2, Ul , U2) be a finite strategic two-player game . Consider the sequence defined by X(O) = Sl X 8 2 and, \:fg ~ 1, X (g) = 0,( X (g - 1)) . A pure strategy s, is said to be permissible if

s, E Pi :=

n°o

9==0

Xi (g) .

A mixed strategy Pi is said to be permissible if Pi is not strongly dominated on Pj and not weakly dominated on Sj. While any pure strategy in the support of a permissible mixed strategy is itself permissible, the mixture over a set of permissible pure strategies need not be permissible.

66

CONSISTENT PREFERENCES

The following lemma is a straightforward implication of Definition 13.

8 (i) For eachi, Pi =1= 0. (ii) P = a(p). (iii) For each i, Si E Pi if and only if there exists X = Xl X X 2 with s, E Pi such that X ~ a(X).

LEMMA

We next characterize the concept of permissible mixed strategies as induced mixed strategies under common certain belief of [u] n [ir] n [cau].

A mixed strategy Pi for i is permissible in a finite strategic two-player game G if and only if there exists an epistemic model with (tl, t2) E CK([u] n [ir] n [cau]) such that Pi is induced for t; by tj'

PROPOSITION 24

Proof. Part 1: If pi is permissible, then there exists an epistemic model with (ti, t 2) E CK([u] n [ir] n [cau]) such that Pi is induced for ti by tj. Step 1: Construct an epistemic model with T I x T2 ~ CK([u] n [ir] n [cau]) such that for each s, E Pi of any player i, there exists ti E T; with, s: E Sfi. Construct an epistemic model with , for each i, a bijection s, : T; --t Pi from the set of types to the the set of permissible pure strategies. Assume that, for each ti E T; of any player i, V;i satisfies that (a) V;i 0

Z

= Ui (so that TI x T2

~ [u]),

and the SCLP (A ti, gt i) on Sj x T j has the properties that (b) Ati = (ttii, tt~i) with support Sj x Tli has two levels and satisfies that supptti1 n (Sj x {tj}) = {(Sj(tj),tj)} for all tj E Tli (so that, Vtj E Tli, pliltj(sj(tj)) = 1), (c) gti satisfies

r-is,

x T j)

= 2 (so that T I x T2

~

[cau]).

Property (b) entails that the support of the marginal of tti i on Sj is included in Pj. By properties (a) and (c) and Lemmas 4, 5 and 8(ii), we can still choose tti i and tt~i (and Tit i) so that Si(ti) E S;i . This combined with property (b) means that T I x T2 ~ [ir]. Furthermore, T I x T2 ~ CK([u] n [ir] n [cau]) since Tp ~ Tj for each ti E T; of any player i . Since, for each player i, Si is onto Pi, it follows that, for each Si E Pi of any player i, there exists ti E T; with Si E S;i. Step 2: Add type ti to Ti. Assume that v[i satisfies (a) and (Ati , gti) satisfies (b) and (c). Then ttlti and ttl i can be chosen so that pi E ~(Sli) . Furthermore, tt: u {tn) x T j ~ [u] n [ir] n [caul, and since T/i ~ Tj , (Ti u {tn) x Tj ~ CK([u] n [ir] n [cau]) . ~tep. 3: Add type tj to Tj . Assume that v}J satisfie~ (a) an~ the SCL.P (Atj, gt j) on s, x tr; u {tn) has the property that Atj = (ttlt j , .. . ,ttLtj)

67

Basi c characterizations

with support S, x {in satisfies that, '\lSi E Si, f.Ll tj (Si ' ri) = pi(Si), so that pi is induced for ti by tj , and £tj satisfies that £tj n; U {tn) = L. Fu;thermore, (Ii U {tn) x tr, U {tj}) ~ [u] n [ir] n [caul, and since Iitj ~ t: U {tn , U {tn) x (Tj U {tj}) ~ CK([u] n [ir] n [cau]). Hence, (ti, (2) E CK([u] n [ir] n [cau]) and pi is induced for ti by tj. Part 2: If there exists an epistemic model with (ti, t 2) E CK ([u] n [ir] n [cau]) such that pi is induced for ti by tj , then pi is permissible. Assume that there exists an epistemic model with (ti, t 2) E CK([u] n [ir] n [cau]) such that pi is induced for ti by tj . In particular, CK([u] n [ir] n [cau]) =1= 0. Let , for each i , T[ := projTiCK([u] n [ir] n [cau]) and X i := UtiETiS;i. By Proposition 20(ii), for each t ; E T[ of any player i, ti deems (Sj, tj) subjectively impossible if tj E Tj\ Tj since CK([u] n [ir] n [cau]) = KCK([u] n [ir] n [cau]) ~ KiCK([u] n [ir] n [cau]) , implying Tli ~ Tj. By the definitions of [u] , [ir] , and [caul, it follows that, for each ti E T[ of any player i , ~ti is represented by V;i satisfying that V;i oz is a positive affine transformation of Ui and an LPS )..ti = (f.Lii , . . . , f.LYJ, and where sUPPf.Lii ~ X j x Tj, and where suppx" = Sj X Tli. Hence, by Lemma 4, for each ti E T[ of any player i , if Pi E 6.(S;i), then no strategy in the support of Pi is strongly dominated on X j , since it follows from Pi E 6.(S;i) and SUPPf.Lii ~ Xj x Tj that, '\lSi E SUPPPi and '\Is~ E Si,

n:

L L SjEXj

tjETj

f.Lii (Sj, tj)Ui(Si, Sj) ~

L L s jEXj

f.Lhsj , tj)Ui(S~ , Sj) .

tjETj

Furthermore, since the projection of )..ti on Sj has full support , no strategy in the support in Pi is weakly dominated on Sj. This implies X ~ a(X) , entailing by Lemma 8(iii) that , for each i, X i ~ Pi. Finally, since (ti, t 2) E CK ([u] n [i r] n [cau]) and the mixed strategy induced for ti by tj , pi, satisfies pi E 6. (s8 ), it follows that pi is not strongly dominated on Xj ~ Pj and pi is not weakly dominated on Sj. By Definition 13 this implies that pi is a permissible mixed strategy. •

Chapter 6

RELAXING COMPLETENESS

In the pr evious chapter, we have pr esented epistemic charac te rizati ons of rat ionalizabili ty and permi ssibili ty. For t hese non-equilibrium dedu cti ve concepts, we have used , resp ecti vely, IESDS and t he DekelFudenb erg procedure (one round of weak elimination followed by iterated strong dominati on ) as t he primitive definiti ons. Neit her of t hese pr ocedures rely on players having sub jective pr obabili ti es over t he st rategy choice of t he opponent . In cont rast , t he epist emic characterizations- by relying on Assumpti on l -e-requi re t hat players have complete preferences t hat are representable by mea ns of subjective probabiliti es. In t his cha pter we show how rati onalizabili ty and permissibili ty can be epistemically char act erized wit hout requiring t hat players have complete pr eferences t hat are represent abl e by means of sub jective probabili ti es. The resulting st ructure will also be used for t he epistemic analysis of ba ckward inducti on in Chapter 7 and forward inducti on in Chapter 11. Hence, even though the results of the pr esent chapte r may hav e limited interest in their own right , t hey set the stage for later analysis.

6.1

Epistemic modeling of strategic games (cont.)

The purpose of this section is to pr esent a framework for st rategic games where each player is modeled as a decision maker under un certainty with preferences t hat are allowed to be incomplete.

An epistemic model. Consider an epistemic model for a finit e st rategic game form (81 ,82 , z) as formalized in Definit ion 9, with a finit e type set T; for each player i, and where t he pr eferences of a player corres ponds to t he player 's type. Hence, for each ty pe t i of any player i ,

70

CONSISTENT PREFERENCES

• {ti } X Sj x Tj is the set of states t hat are indistinguishable for player i at ti , • a non-empty subset K,ti of {ti } x Sj x Tj is t he set of states t hat player i deems subjectively possible at ti , and

• ti corresponds to a system of condit ional preferences on the collection of sets of acts from subsets of T; x Sj x Tj whose intersection with K,t i is non-empty to ~ (Z ) . However , inst ead Assumption 1, imp ose t he following assumption, where \f)t i st ill denot es { <p ~ t: x x T j I K,t i n <p i- 0}.

s,

2 For each ti of any player i, (a) ~ ~ satisfies Axioms 1', 2, and 4' if 0 i- <p ~ T; X Sj x Tj , and Axiom 3 if and only if <p E \f)t i , and (b) the system of conditional preferences {~ ~ I <p E \f) t i} satisfies Axioms 5, 6, and 11 .

A SSUMPTION

As before, write ~t i for player i's unconditional preferences at ti ; Le., for ~ ~ when <p = {td x Sj x T j. W .l.o.g. we may consider ~ ti to be pr eferences over acts from Sj x Tj to ~ (Z ) (instead of acts from {td x Sj x T j to ~ (Z ) ) . Under Assumpti on 2 it follows from Prop osition 4 t hat, for each t ; of any player i , i 's unconditional preferences at ti can be conditionally represent ed by a vNM ut ility functi on Vii : ~ (Z) --+ IR. Conditional represent at ion implies t hat st rong and weak dominance are is well-defined: Let E j ~ Sj x T j . Say that one act P e, strongly dominates another act q Ej at ti if,

\:f(Sj , tj ) E E j , Vii(PEj (Sj, tj ))) > Vii (qEj (Sj , tj))) . Say that P e, weakly dominates q Ej at i , if,

\:f(Sj, tj) E Ej, Vii(PEj (Sj , tj))) ~ Vii (qEj (Sj, tj ))) , with st rict inequality for some (sj , tj) E E j. Say that ~ti is admissible on {t i} x E j if E j is non-empty and P >- t i q whenever P e, weakly dominat es q Ej at ti. Assumption 2 entails t hat ~ti is admissible on K,ti . Ind eed, as i shown in Section 4.1, t here exists a vector of nest ed sets, Z), , ... , P on which ~ t i is admissible, satisfying:

(pi

oi- pi

i

C . . . C p~i C . ..

(where c denot es

~

and

c PZ =

K,ti

~ {ti}

X

s, x Tj

i-).

Preferences over strategies. It follows from the above assumptions t hat, for each ty pe ti of any player i , player i 's uncondit ional preferences

71

Relaxing completeness

at ti , ~t i , is a reflexive and transitive binary relation on acts from Sj x T j to ~(Z) that is conditionally represented by a vNM utility function Vfi. Since each mixed strategy Pi E ~(Si) is a function that assigns the randomized outcome Z(Pi,Sj) to any (Sj,tj) E Sj x T j and is thus an act from Sj x T j to ~(Z), we have that ~ti determines reflexive and transitive preferences on i's set of mixed strategies, ~(Si) ' Player i 's choice set at ti , Sfi, is player i's set of maximal pure strategies at ti : Hence, a pure strategy, Si, is in i's choice set at ti if there is no mixed strategy that is strictly preferred to Si given i's (possibly incomplete) preferences at ti . If i 's preferences at ti are complete, then ~Pi E ~(Si) , Pi r..ti Si is equivalent to Vs~ E Si , Si ~ti s~, and the definition of Sfi coincides with the one given in Section 5.1. Since ~ti is reflexive and transitive and satisfies objective independence , and S, is finite, it follows that the choice set Sfi is non-empty and supports any maximal mixed strategies: If qi E ~(Sd and ~Pi E ~(Si) such that Pi '(-ti qi, then qi E ~ (Sfi ). On the other hand, with incomplete preferences, it is not the case that all mixed strategies in ~ (Sfi) are maximal. We may have that :JPi E ~(Si) such that Pi '(-ti qi even though qi E ~(Sfi) . As an illustration, consider the case where ~ti is defined by P

'(-ti q if and only if

Pproj sxTKi J

J

weakly dominates qprojsXTKi at ti, J

J

and P rv ti q if and only if vfi(p(sj, tj))) = Vfi(q(Sj, tj))) for all (Sj, tj) E projsJÂˇ xTJ KiÂˇ Since a mixed strategy qi may be weakly dominated by a pure strategy s; that does not weakly dominate any pure strategy in the support of qi, this illustrates the possibility that a non-maximal mixed-strategy qi is supported by maximal pure strategies. The event that player i is rational is defined by

A strategic game. As before, G = (Sl,S2,U1,U2) denotes a finite strategic two-player game, where S = Sl X S2 is the set of strategy profiles and , for each i, Ui : S ---t IR is a vNM utility function that assigns payoff to any strategy profile. Assume that, for each i, there exist S = (Sl' S2), S' = (si, si) E S such that Ui(S) > Ui(S'). As in Chapter 5-but transferred to Sl X T 1 X S2 X T2 space-the event that

72

CONSISTENT PREFERENCES

i plays the game G is given by

[Ui]:= {(Sl,tl,s2,t2) E 51 x T l X 52 x T21 0 Z is a positive affine transformation of Ui} ,

Vii

while [u] := [Ul]

n [U2]

is the event that both players play G.

Belief operators. Since Assumptions 2 is compatible with the framework of Chapter 4, we can in line with Section 4.2 define belief operators as follows. For these definitions, say that E ~ 51 X T, X 52 X T2 does not concern player i 's strategy choice if E = S, x projTiXSjxTjE. If E does not concern player i's strategy choice, say that player i certainly believes the event E at ti if ti E ProjTiKiE, where

KiE := {( Sl, tl, S2, t2) E 51 x Tl

X

52

X

T2 I /'i,ti ~ proj-, XS2 XT2E} .

If E does not concern the strategy choice of either player, say that there is mutual certain belief of E at (tl' t2) if (tl , t2) E projTixT2KE, where KE := KIE n K 2E. If E does not concern the strategy choice of either player, say that there is common certain belief of E at (tl, t2) if (tl ' t2) E projTixT2CKE, where CKE:= KE n KKE n KKKE n ... . If E does not concern player i 's strategy choice, say that player i (unconditionally) believes the event E at ti if ti E projTiBiE, where

BiE := {(Sl,tl,s2,t2) E 51 x Tl X 52 x T21 (3t i ~ ProhlXS2XT2E} , and where (3t i := pii denotes the smallest set on which ~ti is admissible. If E does not concern the strategy choice of either player, say that there is mutual belief of E at (tl, t2) if (tl, t2) E ProjTiXT2BE, where BE := BIE n B 2E. If E does not concern the strategy choice of either player, say that there is common belief of E at (iI, t2) if (tl, t2) E ProjTiXT2CBE, where CBE := BE n BBE n BBBE n ... . As established in Proposition 14, K, and B, correspond to KD45 systems. Moreover, the mutual certain belief and mutual belief operators, K and B, have the following properties, where we write KO E := E and BO E := E, and for each 9 ~ 1, K9E := KK9- l E and B9E := BB9- l E. PROPOSITION 25 (i) For any E ~ Tl X T2 and all 9 > 1, K9E ~ K9- l E and B9E ~ B9- l E. If E = E l n E 2, where, for each i , E i = S, x ProjTiEi x s, x Tj, then KE ~ E and BE ~ E . (ii) For any E ~ T l X T 2, there exist g' and gil ~ 0 such that K9E = CKE for 9 ~ g' and B9E = CBE for 9 ~ gil, implying that CKE = KCKE and CBE = BCBE. Proof. See the proof of Proposition 20. â€˘

73

Relaxing completeness

6.2

Consistency of preferences (cont.)

In the present section we define the event of consistency of preferences in the case described by Assumption 2, where preferences need not be complete, and use this event to characterize the concept of rationalizable pure strategies. Belief of opponent rationality. In the context of the present chapter, define as follows the event that player i's preferences over his strategies are consistent with the game G = (Sl, S2,U1, U2) and the preferences of his opponent : C, := lUi] n Bilratj] . Write C := C 1 n C2 for the event of consistency. Characterizing rationalizability. We now characterize the concept of rationalizable pure strategies (d. Definition 11 of Chapter 5) as maximal pure strategies under common certain belief of consistency. 26 A pure strategy Si for i is rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model for some (h,t2) E projTlxT2CKC. with s, E

PROPOSITION

Sfi

To prove Proposition 26, it is helpful to establish a variant of Lemma 6. Write, for any (0 1') X = Xl XX2 ~ s. X S2, c(X) := C1(X2) x C2(X 1), where

Ci(Xj) := {s, E

s, I :3(0 1') Yj

~

x, such that , VPi E ~(Sd,

Pi does not weakly dominate s; on Yj} . LEMMA 9 (i) R = c(R) . (ii) For each i, Si E R ; if and only if there exists X = Xl X X 2 with s, E R such that X ~ c(X). Proof. In view of Lemma 6, it is sufficient to show that, for any (0 1') ~ Ci(Xj) = Ci(Xj) . Part 1: Ci(Xj) ~ Ci(Xj). If Si 1- Ci(Xj) , then :3Pi E ~(Sd s.t . Pi strongly dominates s, on X j . From this it follows that V (0 1') Yj ~ X j , :3Pi E D.(Si) s.t. Pi weakly dominates Si on Yj, implying that Si 1- Ci(Xj) . Part 2: Ci(Xj) ;2 Ci(Xj), If s; E Ci(Xj) , then there does not exist Pi E D.(Si) s.t . Pi strongly dominates Si on X]. Hence, by Lemma 4, there exists a subjective probability distribution f-t E D.(Sj) with suppu ~ X j such that Si is maximal in D.(Sd w.r.t . the preferences represented by the vNM utility function Ui and the subjective probability distribution

x, s;

74

CONSIS TENT PREFERENCES

/1. Then t here does not exist Pi E ~ (Si ) S.t. Pi weakly dominat es s, on sUPP/1 (~ X j ), implying t hat s ; E Ci( X j) . â€˘ Proof of Proposition 26. Part 1: If Si is rationalizable, then there exists an epistemic model with Si E Sii for some (tl , t2) E ProjTl XT2 CK C . It is sufficient t o const ruct a belief system with SI x T, X S2 X T2 ~ CKC such t hat, for each Si E ~ of any player i, t here exists ti E ~ with Si E Sii. Construct a belief system with, for each i , a bijection s, : T; ---t R; from the set of types to t he the set of rationalizable pure st ra tegies. By Lemm a 9(i) we have that , for each i , E T; of any player i, t here exists Yjti ~ R; such that there does not exist Pi E ~ ( Sd such t hat Pi weakly dominat es s, (ti) on Yjti . Determine the set of opponent typ es that ti deems subjectively possible as follows: T/i = {t j E Tj I Sj(tj) E Yjti} . Let , for each ti E ~ of any player i, ti satisfy

e:

1. Vii 0 Z = Ui (so t hat SI x T l

X

S2 X T2 ~ [u]) , and

2. p >-ti q iff PEj weakly dominates qEj for E j = E/i := { (Sj , tj ) I Sj Sj (tj) and tj E TJi} , which implies that {3t i = ",ti = {td X E/i.

=

By t he const ruct ion of E/ i, t his means t hat Si i :3 Si (ti) since, for any acts p and q on Sj x Tj satisfying t hat t here exist mixed st rategies Pi, qi E ~ (Sd such t hat , V(Sj , tj) E Sj x Tj , p (Sj , tj ) = Z(Pi, Sj) and q (Sj ,tj) = Z(qi, Sj) , P >-ti q iff PEj weakly domin at es q Ej for E j = Yjti X T j . This in t urn implies, for each ti E T; any player i ,

3. {3ti ~ projTi XSj XT) ratj] (so t hat SI x i; Bj[rati])'

X

S2

X

T2 ~ Bi[ratj] n

Furthermore, SI x i, X S2 X T2 ~ CKC since T/ i ~ T j for each ti E t; of any player i. Since, for each player i, Si is onto R i , it follows that , for each Si E R; of any player i, there exists ti E T; with s, E Si i. Part 2: If there exists an epistemic model with si E sF for some (ti, t 2) E ProjT1 XT2CKC , then si is rationalizable. Assume that there exists an episte mic model with si E S/: for some (ti ,t 2) E projT 1 XT2CKC. In particular, CKC i= 0. Let , for each i, T! := projTi CKC and X i := U tiETlSi i. It is sufficient to show t hat, for each i , X i ~ ~ . By Proposition 25(ii), for each ti E T! of any player i , {3ti ~ ",ti ~ {td x Sj x Tj since CKC = KCKC ~ KiCKC. By t he definition of C, it follows that, for each ti E Tf of any player i , 1.

e: ti

is conditionally represented by Vii satisfying t hat Vii positive affine t ra nsformation of Ui, and

0

z is a

75

Relaxing completeness

2. P >-t; q if PEj weakly domin ates qEj for Ej = E/ ; := projsj XTj,st;, where ,st; ~ ProjT;XSj XTj[ratj]. Writ e Yjt; := projsjE/ ; = projsj,st;, and not e t hat ,st; ~ ({ti } XSj xTj)n projT;XSj XTj[ratj ] implies Yjt; ~ X i. It follows t hat , for any acts P and q on Sj x T j satisfying that t here exist mixed strategies Pi, qi E ~ ( Si ) such t hat, V(Sj , tj ) E Sj x T j , p(Sj , tj ) = Z(Pi, Sj ) and q(sj ,tj) = Z(qi, Sj ), P >- t; q if P s, weakly domin at es q Ej for E j = Yjt; X T j. Hence, if Si E S;;, then t here does not exist Pi E ~ ( Sd such t hat Pi weakly dominates s; on Yjt;. Since this holds for each t, E T! of any player i , we have that X ~ c(X). Hence, Lemma 9(ii) entails that , for each i , Xi ~ Ri, • Proposition 26 is obtained also if CBC is used inst ead of CKe.

6.3

Admissible consistency of preferences (cont.)

In the pr esent sect ion we define t he event of admissible consistency of preferences in t he case considered by Assumption 2, where preferences need not be complete , and use t his event to characterize the concept of permissible pure st rategies.

Caution. As in Secti on 5.3, player i has preference for caut ious behavior at ti if he takes into account all opponent st rategies for any opponent ty pe t hat is deemed sub jectively possible. Throughout t his chapter , as well as Chapt er 7 and 11, we assume that Assumption 2 is satisfied, so t hat t he system of conditional preferences { ~ ~ I ¢ E <p t ; } satisfies Axiom 6, where <p t ; denot es {¢ ~ t: x x Tj I /),t ; n ¢ t= 0}, and where Kt;- the set of states t hat player i deems subjectively possible at ti- satisfies 0 t= K t; ~ {td x Sj x T j . Hence, TI ; := ProjTj K t; is the set of opponent types t hat player i deems subjectively possible. Und er Assumption 2, player i is caut ious at ti if K t ; = {td x Sj x Tt . Because then, player i at ti t akes into account all opponent st rategies for any opponent type t hat is deemed subject ively possible. This means that i's choice set at ti never admits a weakly dominat ed st rategy, thereby inducing preference for cautio us behavior. Hence, und er Assumption 2 we can define t he event

s,

[caUi]:= {(Sl ,tl , S2,t2) ESl x T l X S2 x T2 1 3T/ ; such t hat /),t ; = {t i} x Sj x T/ ; } . Write [caul := [caul ] n [cau2]' In t he context of the present cha pter, define as follows t he event t hat player i 's pr eferences over his st rategies are admissibly consist ent with

76

CONSISTENT PREFERENCES

the gam e G = (Sl, S2, Ul , U2) and the preferences of his opponent :

Writ e A := Al

n A 2 for the event

of admissible consistency.

Characterizing permissibility. We now characterize t he concept of permissible pure st rategies (cf. Definiti on 13 of Chapter 5) as maxim al pure st rategies und er common certain belief of admissible consiste ncy. 27 A pure strategy Si for i is permissi ble in a fini te strategic two-player game G if and only if there exists an epistemic model with Si E S fi for some (tl, t2) E projT1xT2CKA. PRO POSITION

To prove Proposition 27, it is helpful to establish a vari ant of Lemma 8. Define, for any (0 =F) Yj ~

s;

Di(Yj ) := {s, E S, 13pi E

~ ( Si )

such that

Pi weakly dominates s, on Yj or Sj } , and write, for any (0 =F) X = Xl a2(Xd , where

X

X 2 ~ Sl

X

S2, a(X ) := al( X 2) x

10 (i) P = a(P) . (ii) For each i , Si E Pi if and only if there exists X = Xl X X 2 with s, E Pi such that X ~ a(X) .

LEMM A

Proof. In view of Lemma 8, it is sufficient to show t hat , for any (0 =F) x, ~ s; ai( Xj) = ai(Xj) . Part 1: ai(Xj) ~ ai( X j) . If Si ~ ai(X j) , t hen 3Pi E ~ ( Si ) s.t . Pi st rongly dominat es Si on X j or Pi weakly dominates Si on Sj. From thi s it follows that V(0 =F ) Yj ~ Xj , 3Pi E ~(Sd s.t . Pi weakly dominates s, on Yj or Sj , implying that V(0 =F ) Yj ~ X j , s, E Di(Yj ). This means that s, ~ ai( X j) . Part 2: ai(X j) ;2 ai(X j) . If Si E ai( X j) , then t here does not exist Pi E ~ ( Sd s.t . Pi st rongly dominates Si on X j or Pi weakly dominates Si on Sj . Hence, by Lemmas 4 and 5, t here exists an LPS A = (f-l l , f-l2) E L~ ( Sj ) with SUPPf-ll ~ X j and SUPPf-l2 = Sj such t hat s, is maxim al in ~ ( Sd w.r.t. t he preferences represented by the vNM utility function Ui and t he LPS A. Th en there does not exist Pi E ~ ( Si) s.t . Pi weakly domin at es s, on SUPPf-ll (~ X j) or SUPPf-l2 (= Sj) , implyin g that Si ~ Di(Yj) for Yj = SUPPf-l l ~ X j. Thi s means t hat Si E ai(Xj ). â€˘

77

Relaxing completeness

Proof of Proposition 27. Part 1: If Si is permissible, then there exists an epistemic model with s, E Sii for some (tl, t2) E proj - , XT2CKA . It is sufficient to const ruct a belief system with S1 x T 1 X S2 X T2 ~ CKA such t hat , for each s; E Pi of any player i , there exists ti E T; with Si E Sii. Construct a belief syste m with, for each i, a bijection Si : T; ---. Pi from t he set of types to the t he set of permissible pure stra tegies. By Lemma lO(i) we have t hat, for each ti E Ti of any player i , th ere exists Yjti ~ Pi such that Si(ti) E Si\ Di(Yjti). Determine t he set of opponent types t hat ti deems subject ively possible as follows: Tl i = {t j E T, I Sj (tj) E Yl i}. Let , for each ti E T; of any player i, t ti satisfy 1. Vii

0 Z

= u; (so that S1 x T 1 X S2

X

T2 ~ [u]) , and

2. p >-- ti q iff PEj weakly dominat es qEj for Ej = El i := {(Sj ,tj) I Sj = Sj (tj) and tj E TJi} or E j = Sj X Tli, which implies that {3t i = {ti} x El i and ",ti = {td x Sj x Tl i (so that S1 x T 1 X S2 X T2 ~ [cau]). By th e const ruct ion of Eli, thi s means that Sii = Si\ Di(Yjti) 3 si(td since, for any acts P and q on Sj x T j satisfying t hat there exist mixed st rategies Pi, qi E .6.(Si) such t hat, V(Sj, tj ) E Sj x Tj , p (Sj , tj) = Z(Pi, Sj) and q (Sj , tj ) = Z(qi' Sj), P >--ti q iff PEj weakly dominates qEj for E j = Yjti X Tj or E j = Sj x Tj. This in t urn implies, for each ti E T; any player i ,

3. {3ti ~ projTi XSj XT) ratj] (so that S1 x T 1 Bj[rati]) '

X

S2

X

T2 ~ Bi[ratj] n

Furthermore, S1 X T 1 X S2 X T2 ~ CKA since Tl i ~ T j for each t i E t: of any player i . Since, for each player i , s, is onto Pi, it follows that , for each s; E Pi of any player i, th ere exists ti E T; with Si E Sii. Part 2: If there exists an epistemic model with si E sF for some (ti , t 2) E projT 1 XT2 CKA , then si is permissible. Assume t hat there exists an episte mic model with si E st: for some (ti, t 2) E projTl xT2CKA. In particular , CKA # 0. Let , for each i , T! := projTi CKA and X i := Ut iET1Si i. It is sufficient to show that , for each i , X i ~ Pi. By Proposition 25(ii), for each ti E T! of any player i , {3ti ~ ",ti ~ {td x x Tj since CKA = KCKA ~ KiCKA . By t he definition of A, it follows t hat, for each ti E T! of any player i,

s,

1. t ti is conditionally represented by Vii satisfying t hat Vii positive affine t ra nsformation of U i, and

0 Z

is a

78

CONSISTENT PREFERENCES

2. P ,;:--ti q if PEj weakly dominates qEj for E, = El i := Proj Sj XTj,eti or s, = X Tli, where ,eti S;;; projTiXSj XTj[ratj).

s,

Writ e Y/ i := projsjEl i = projsj,eti , and not e that ,eti S;;; ({ til x Sj xTj)n ProjTi XSjXTj[ratj) implies Yjti S;;; X i' It follows th at , for any acts P and q on Sj x Tj satisfying t hat there exist mixed st rategies Pi, qi E b.(Sd such that , V(Sj, tj) E Sj x T j, p(Sj , tj) = Z(Pi, Sj) and q( Sj, tj) = Z(qi, Sj), P ,;:--ti q if PEj weakly dominates qEj for Ej = Yjti X Tj or E j = Sj x Tj . Hence, S;i S;;; Si\ Di (Yjti) . Since this holds for each ti E T! of any player i , we have th at X S;;; a(X ). Hence, Lemma lO(ii) entails that , for each i , X i S;;; Pi. â€˘ Proposition 27 is obtained also if CBA is used inst ead of CKA ; this is essent ially the corresponding result by Brand enburger (1992). One may argue that the result above is more complicated as it involves two different epistemic operators. Still , it yields the insight that the essential feature in a characterization of t he Dekel-Fudenberg procedure is to let irrational opponent choice be deemed subjectively possible. It also turns out to be a useful benchmark for the analysis of backward indu ction in Section 7.3 where the certain belief operator K, - rather than the belief operator B, - must be used for the interactive epistemology (d. the analysis of f 5 illustrated in Figure 7.1).

Chapter 7

BACKWARD INDUCTION

In recent years, two influential contributions on backward induction in finite perfect information games have appeared, namely Aumann (1995) and Ben-Porath (1997). These contributions-both of which consider generic perfect information games (where all payoffs are different) reach opposite conclusions: While Aumann establishes that 'common knowledge of rationality' implies that the backward induction outcome is reached, Ben-Porath shows that the backward induction outcome is not the only outcome that is consistent with 'common certainty of rationality' . The models of Aumann and Ben-Porath are different. One such difference is that Aumann makes use of 'knowledge' in the sense of 'true knowledge', while Ben-Porath's analysis is based on 'certainty' in the sense of 'belief with probability one'. Another is that the term 'rationality' is used in different senses: Aumann imposes rationality in all subgames, while Ben-Porath assumes rationality initially, in the whole game, only (not after a "surprise" has occurred). The present chapter, which reproduces Asheim (2002), shows how the conclusions of Aumann and Ben-Porath can be captured by imposing requirements on the players within the same general framework. Furthermore, the interpretations of the present analysis correspond closely to the intuitions that Aumann and Ben-Porath convey in their discussions. Hence, the analysis of this chapter may increase our understanding of the differences between the analyses of Aumann and Ben-Porath, and thereby enhance our understanding of the epistemic conditions underlying backward induction. For ease of presentation, the analysis will be limited to two-player games, as in the rest of the book. In this chap-

80

CONSISTENT PREFERENCES

ter, this is purely a matter of convenience as everything can directly be generalized to n-player games (with n > 2). Among the large literature on backward induction during the last couple of decades,1 Reny's (1993) impossibility result is of special importance. Reny associates a player's 'rationality' in an extensive game with perfect (or almost perfect) information with what is called 'weak sequential rationality'; i.e., that a player chooses rationally in all subgames that are not precluded from being reached by the player's own strategy. He shows that there exist perfect information games where the event that both players satisfy weak sequential rationality cannot be commonly believed in all subgames. E.g., in the "centipede" game that is illustrated in Figure 2.4, common belief of weak sequential rationality cannot be held in the subgame defined by 2's decision node. The reason is that if 1 believes that 2 is rational in the subgame, and if 1 believes that 2 believes that 1 will be rational in the subgame defined by 1's second decision node , then 1 believes that 2 will choose e, implying that only Out is a best response for 1. Then the fact that the subgame defined by 2's decision node has been reached, contradicts 2's belief that 1 is rational in the whole game. As a response, Ben-Porath (1997) imposes that common beliefof weak sequential rationality is held initially, in the whole game, only. However, backward induction is not implied if weak sequential rationality is commonly believed initially, in the whole game, only. In the "centipede" game of Figure 2.4, the strategies Out and InL for player 1 and e and r for player 2 are consistent with such common belief, while backward induction implies that down is played at any decision node. In order to obtain an epistemic characterization of backward induction , Aumann (1995) considers 'sequential rationality' in the sense that a player chooses rationally in all subgames (see also footnote 3 of this chapter) . However, the event that players satisfy sequential rationality is somewhat problematic. If-in the "cent ipede" game of Figure 2.4-1 believes or knows that 2 chooses e, then only by choosing the strategy OutL will 1 satisfy sequential rationality. However, what does it mean that 1 chooses OutL in the counterfactual event that player 2's decision node were reached? It is perhaps more natural-as suggested by Stal-

1 Among contributions that ar e not otherwise referred to in this chapter ar e Basu (1990) , Bicchieri (1989) , Binmore (1987, 1995) , Bonanno (1991, 2001) , Clausing and Wilks (2000) , Dufwenberg and Lind en (1996) Feinberg (2005a) , Gul (1997), Kan eko (1999) , Rabinowicz (1997) , and Rosenthal (1981) .

Backward induction

81

naker (1998)-to consider 2's belief about l 's subsequent action if 2's decision node were reached. Since Aumann (1995) assumes knowledge of rational choice in an S5 partition structure, such a question of belief revision cannot be asked within Aumann's model. By imposing a full support restriction by considering players of types in projT\ XT2[cau] (cf. the definition of [caul in Section 6.3), the present chapter ensures that each player takes all opponent strategies into account, having the structural implication that conditional beliefs are welldefined and the behavioral implication that a rational choice in the whole game is a rational choice in all subgames that are not precluded from being reached by the player's own strategy. Hence, by this restriction, we may consider 'rationality' instead of 'weak sequential rationality' (as shown by Lemma 11 and the subsequent text). The main distinguishing feature of the present analysis is, however, to consider the event that a player believes in opponent rationality rather than the event that the player himself chooses rationally. This is of course in line with the 'consistent preferences' approach that is the basis for this book. As is shown by Proposition 27 of Chapter 6, permissible pure strategies-strategies surviving the Dekel-Fudenberg procedure , where one round of weak elimination is followed by iterated strong elimination-can be characterized as maximal strategies when there is common certain belief that each player believes initially, in the whole game, that the opponent chooses rationally ('belief of opponent rationality '). For generic perfect information games, Ben-Porath shows that the set of outcomes consistent with common belief of weak sequential rationality corresponds to the set of outcomes that survives the DekelFudenberg procedure. Hence, maximal strategies when there is common certain belief of 'belief of opponent rationality' correspond to outcomes that are promoted by Ben-Porath's analysis. An extensive game offers choice situations, not only initially, in the whole game, but also in proper subgames . In perfect information games (and, more generally, in multi-stage games) the subgames constitute an exhaustive set of such choice situations. Hence, in perfect information games one can replace 'belief of opponent rationality' by 'belief in each subgame of opponent rationality': Each player believes in each subgame that his opponent chooses rationally in the subgame. The main results of the present chapter (Propositions 28 and 29 of Section 7.3) show how, for generic perfect information games, common certain belief of 'belief in each subgame of opponent rationality' is possible and uniquely deter-

82

CONSIS TENT PR EFER ENCES

mines t he backward indu ction outco me. Hence, by substit ut ing 'belief in each subgame of opponent rationality' for 'belief of opponent rat ionality ', the present analysis provides an alternative route to Aumann's conclusion , namely that common knowledge (or certain belief) of an appropriate form of (belief of) rat ionality implies backward induct ion. T his epistemic foun dation for backward induct ion requires common certain belief of 'belief in each subga me of opponent rationality' , where t he term 'cert ain belief' is being used in t he sense t hat an event is certainly believed if t he complement is subjectively impossible. As shown by a counte rexample in Section 7.3, t he characterizat ion does not obtain if inst ead common belief is applied.f Fur thermore, the event of which t here is common certain belief - namely 'belief in each subgame of opponent rationality' - cannot be further restricted by taking the intersecti on with t he event of 'rat ionality' . The reason is t hat the full support restri ction (i.e., t hat players are of ty pes in projTl XT2 [cauJ) is inconsist ent with certain belief of opponent 'rationality' , as t he latt er prevents a player from taking into account irrati onal opponent choices and rules out a well-defined theo ry of belief revision.

7.1

Epistemic modeling of extensive games

T he purpose of this sect ion is to present a framework for exte nsive games of almost perfect information where each player is modeled as a decision maker und er uncert ainty, with preferences that are allowed to be incomplet e. An extensive game form. Inspired by Dubey and Kane ko (1984) and Chapter 6 of Osborne and Rubinstein (1994), a finite exte nsive twoperson game form of almost perfect information with M - 1 stages can be describ ed as follows. The set of histories is det ermined inductively: The set of histories at t he beginnin g of the first stage 1 is HI = {0}. Let H '" denote the set of histories at the beginning of stage m . At h E H '" let , for each player i, i's act ion set be denoted Ai( h), where i is inacti ve at h if Ai (h) is a singleton. Writ e A(h) := AI( h) x A2 (h). Define t he set of hist ories at the beginning of stage m + 1 by H m + 1 := {( h, a) I h E H'" and a E A(h)} . This concludes t he indu ction. Denot e by H := u:;;~; H '" t he set of subgames and denote by Z := H M t he set of outcomes.

2For definitions of t he certain belief ope ra tor K, and th e belief operator B, in th e cur rent conte xt, see Section 6.1.

83

Backward induction

A pure strategy for player i is a function s; that assigns an action in Ai(h) to any h E H . Denote by S, player i's finite set of pure strategies, and let z : 8 ---+ Z map strategy profiles into outcomes , where 8 := 8 1 X 82 is the set of strategy profiles.' Then (81 ,82, z) is the corresponding finite strategic two-person game form. For any h E H u Z , let 8(h) = 8 1(h) x 8 2(h) denote the set of strategy profiles that are consistent with h being reached. Note that 8 (0) = 8 . For any h, h' E H U Z, h (weakly) precedes h' if and only if 8(h) :2 8(h'). If Si E S, and h E H, let Si Ih denote the strategy in S, (h) having the following properties: (1) at subgames preceding h, silh determines the unique action leading to h, and (2) at all other subgames , silh coincides with s.. Epistemic modeling. Since the extensive game form determines a finite strategic game form, we may represent the strategic interaction by means of an epistemic model as defined by Definition 9 of Chapter 5. Since backward induction is a procedure-like IESDS and the DekelFudenberg procedure-that does not rely on subjective probabilities, the analysis will allow for incomplete preferences. Hence, the epistemic model is combined with Assumption 2 of Chapter 6. In this respect the present analysis follows Aumann (1995) who presents a characterization of backward induction where subjective probabilities play no role. Conditional preferences over strategies. Write?::~ for player i 's preferences at ti conditional on subgame h E H being reached ; i.e., for ?::~ when Â˘ = {til x 8 j(h) x Tj. W .l.o.g. we may consider ?::~ to be preferences over acts from 8j(h) x Tj to ~(Z) (instead of acts from {td x Sj(h) x Tj to ~(Z)) . Denote by

the set of subgames that i deems subjectively possible at ti. Under Assumption 2 it follows from Proposition 4 that, for each t; of any player i and all h E Ht i, i 's conditional preferences at ti in subgame h can be conditionally represented by a vNM utility function ~(Z) ---+ IR that does not depend on h.

V;i:

3A pure strategy Si E S i can be viewed as an act on S j that assigns Z(S i,Sj) E Z to any Sj E Sj . The set of pure strategies S, is partitioned into equivalent classes of acts sin ce a pure strategy Si also determines actions in subgames which s, prevents from being reached . Each such eq uivalent class corresponds to a plan of action, in the sense of Rubinstein (1991). As there is no need here to differentiate between identical acts in the pr esent analysis, the concept of a plan of action would have sufficed .

84

CONSISTENT PREFERENCES

Hence, for each type ti of any player i, player i 's conditional preferences at ti in subgame h, c~, is a reflexive and transitive binary relation on acts from Sj(h) x Tj to ~(Z) that is conditionally represented by a vNM utility function Vii if h E ttÂť. Since each mixed strategy Pi E ~(Si (h)) is a function that assigns the randomized outcome Z(Pi, Sj) to any (Sj, tj) E Sj(h) x Tj and is thus an act from Sj(h) x Tj to ~(Z), we have that c~ determines reflexive and transitive preferences on i's set of mixed strategies, ~(Sd. Player i's choice function at ti is a function shÂˇ) that assigns to every h E H player i's set of maximal pure strategies at ti in subgame h:

Hence, a pure strategy, Si , is in the set determined by i's choice function at ti in subgame h ifthere is no mixed strategy in ~(Si(h)) that is strictly preferred to s; given i 's (possibly incomplete) conditional preferences at ti in subgame h. Refer to Sii (h) as player i 's choice set at ti in subgame h, and write Sii = Sii (0), thereby following the notation of Chapter 6. Since c~ is reflexive and transitive and satisfies objective independence, and S, (h) is finite, it follows that the choice set Sii (h) is nonempty and supports any maximal mixed strategies: If qi E ~(Si(h)) and $Pi E ~(Si(h)) such that Pi >-~ qi, then qi E ~(S;i (h)) . By the following lemma , if Si is maximal at ti in subgame h, then Si is maximal at ti in any later subgame that Si is consistent with.

If s, E S;i(h), then s, E S;i(h') for any h' E H with Si E Si(h') ~ Si(h) .

LEMMA 11

Proof. The proof of this lemma is based on the concept of a 'strategically independent set ' due to Mailath et al. (1993). The set S' ~ S is strategically independent for player i in a strategic game G = (Sl, S2, U1, U2) if s' = Si x S~ and , \fsi, s~ E S~, :Js~' E S~ such that Ui(S~', Sj) = Ui(S~, Sj) for all Sj E Sj and Ui(S~', Sj) = Ui(Si, Sj) for all Sj E Sj \ Sj. It follows from Mailath et al. (Definitions 2 and 3 and the 'if' part of Theorem 1) that S(h) is strategically independent for i for any subgame h in a finite extensive game of almost perfect information, and this does not depend on the vNM utility function that assigns payoff to any outcome. The argument is based on the property that :Js~' E Si (h) such that z(s? , Sj) = z(s~, Sj) for all Sj E Sj(h) and z(s~', Sj) = Z(Si, Sj) for all Sj E Sj\Sj(h). The point is that i's decision conditional on j choosing a strategy consistent with hand i 's decision conditional on j choosing a strategy inconsistent with h can be made independently.

85

Backward induction

Suppose that Si is not a maximal strategy at ti in the subgame h'. Then there exists s~ E Si(h') such that s~ >-k Si . As noted above, S(h') is strategically independent for i. Hence, :Js~' E S, (h') such that z( s~', Sj) = z(s~ ,Sj) for all Sj E Sj(h') and z(s~',Sj) = Z(Si,Sj) for all Sj E Sj\Sj(h') . By Assumption 2 this implies that s~' >-k Si, which contradicts that s; is most preferred at t; in the subgame h. â€˘ The event that player i is rational in subgame h is defined by

[rati(h)]:= {(SI ,tl,s2 ,t2)

E

SI x T,

X

S2 x T21 Si

E

Sii(h)}.

Write [rati] = [rati(0)], thereby following the notation of Chapter 6. The imposition of a full support restriction by considering players of types in ProjTIXT2[cau] (cf. the definition of [caul in Section 6.3) has the structural implication that, for all h, the conditional preferences, are nontrivial. Moreover, by Lemma 11 it has the behavioral implication that any choice s; that is rational in h is also rational in any later subgame that Si is consistent with. This means that 'rationality' implies 'weak sequential rationality'. In fact, is admissible on {ti} x Sj(h) x T/i (cf. Section 6.1), implying that any strategy that is weakly dominated in h cannot be rational in h. Thus, preference for cautious behavior is induced . However, in the context of generic perfect information games (cf. Section 7.2 of the present chapter) such admissibility has no cutting power beyond ensuring that 'rationality' implies 'weak sequential rationality'; see, e.g., Lemmas 1.1 and 1.2 of Ben-Porath (1997). Hence, in the class of games considered in our main results it is of no consequence to use 'rationality' combined with full support rather than 'weak sequential rationality'.

?:k,

?:k

An extensive game. Consider an extensive game form, and let, for each i, Vi : Z -+ IR be a vNM utility function that assigns a payoff to any outcome. Then the pair of the extensive game form and the vNM utility functions (VI ,V2) is a finite extensive two-player game of almost perfect information, r. Let G = (SI ,S2 ,Ul,U2) be the corresponding finite strategic game, where for each i, the vNM utility function Ui : S -+ IR is defined by Ui = Vi 0 Z (i.e., Ui(S) = Vi(Z(S)) for any S = (SI, S2) E S). Assume that, for each i, there exist S = (SI ,S2), s' = (s~,s;) E S such that Ui(S) > Ui(S'). As before, the event that i plays the game G is given by

[Ui] := {(SI,tl,s2,t2)

vii 0

Z

SI x t, X S2 x T2 1 is a positive affine transformation of ud ,

E

86

CONSISTENT PREFERENCES

while [u] := [U1]

n [U2]

is the event that both players play G.

Conditional belief. As before, say that E ~ 8 1 X T1 X 8 2 X T2 does not concern player i's strategy choice if E = S, x ProjTiXSjxsjE. If E does not concern player i's strategy choice and h is deemed subjectively possible by i at ti (i.e., h E Ht i), say that player i at ti believes the event E conditional on subgame h if ti E probBi(h)E, where l

Bi( h)E := {( 81, t1, 82, t2) such that

E 81

x T 1 X 82 X T2

I 3ÂŁ E

{I, ... , L}

0 =I- p~i n ir; x 8j(h) x Tj) ~ ProjTiXSjXTjE} ,

and (pii , . . . , pZ) is the vector of nested sets on which ?::t i is admissible. By writing, for each h E tr-, (3t i(h) := p~i n (~ x 8 j(h) x Tj ) , where ÂŁ:= min{k E {l, . . . ,L} I pt n (Ti x 8 j(h) x Tj) =I- 0}, we have that

Bi(h)E = {(Sl' ii , 82, t2) E 81 x T1

X

82

X

T2 I (3t i (h) ~ ProjTiXSjXTjE} .

It follows from the analysis of Chapter 4 that, for each h E Hti, ?::~ is admissible on (3t i(h), and there is no smaller subset of {til x 8 j(h) x Tj on which ?::~ is admissible.' The collection of sets {(3t i (h) I h E Ht i} is a system of conditional filter generating sets as defined in Section 5 of Brandenburger (1998). Although completeness of preferences is not imposed under Assumption 2, ?::t i may encode more information about i's preferences at ti that what is recoverable from such a system of conditional filter generating sets . It follows from the full support restriction imposed by considering players of types in proj-. XT2 [caul (cf. the definition of [caul in Section 6.3) that ?::t i has full support on 8j, implying in turn that Hti = H and, at ti, i's belief conditional on the sub game h is "well-defined" (in the sense that the non-empty set (3;i (h) is uniquely determined) for all h E H. Hence, a "well-defined" belief conditional on h is implied by full support alone ; it does not require that h is actually being reached. This means that a requirement on i 's belief conditional on the subgame h is a requirement on the preferences (the type) of player i only; it does not impose that i makes a strategy choice consistent with h. Since the conditional belief operator is used only for objectively knowable events that are subjectively possible, we do not consider hypothetical events . Hence, hypothetical epistemic operators of the kind developed by Samet (1996) are not needed in the present framework. 4The existe nce of such a smaller subset would contradict Propositions 6 and 9(ii) .

Backward in duction

7.2

87

Initial belief of opponent rationality

A finit e exte nsive game is â€˘ ... of perfect info rmation if, at any h E H , there exists at most one player t ha t has a non-singleton act ion set. â€˘ ... generi c if, for each i , Vi (Z) i- Vi( Z') whenever Z and z' are different outcomes.

Generic exte nsive games of perfect inform ation have a unique subgameperfect equilibrium. Moreover , in such games t he pro cedure of backward induction yields in any subgame t he unique subgame-perfect equilibrium outco me. If s' denot es t he unique subgame-perfect equilibrium, then, for any subgam e h, Z(S* lh) is the backward induction outcome in the subgame h, and S(Z(S*lh)) is t he set of strat egy vect ors consiste nt with t he backward induction outcome in the sub game h . Both Aumann (1995) and Ben-Porath (1997) analyze generic extensive games of perfect inform ation. As alrea dy pointed out, while Aumann establishes t hat commo n (true) knowledge of (sequent ial) rat ionality 5 implies t hat t he backwar d inducti on outcome is reached, Ben-Porath shows t hat t he backward induction outcome is not t he only outc ome t hat is consiste nt wit h common belief (in t he whole game) of (weak sequential) rationality. The purpose of t he present sect ion is to int erpret the analysis of Ben-Porath by applying Propositi on 27 to t he class of generic perfect inform ation games.

Applying admissible consistency to extensive games. Recall t hat the event of admissible consiste ncy is defined as A .- Al n A 2 , where Ai := lUi] n Bilratj] n [caui] . Again note that a full support restriction is imp osed by considering players of types in projTl XT2 [caul , ensuring that each player takes all opponent st rateg ies int o account . In Proposition 27 of Chapt er 6 we have est ablished t ha t t he concept of permissible pure st rategies can be charac te rized as maxim al pure st rategies und er common certain belief of admissible consiste ncy. Recall also that permissible st rategies (d. Definition 13 of Chapt er 5) correspond t o (1995) uses the t erm substantive rat ionality, meaning that for all histories h, if a player wer e to rea ch h, then the player would choose rati onally at h. See Aumann (1995, pp . 14- 16) and Aum ann (1998) as well as Hal pern (2001) and St alnaker (1998, Section 5) .

5 Aumann

88

CONSISTENT PREFERENCES

strategies surviving the Dekel-Fudenberg procedure, where one round of weak elimination is followed by iterated strong elimination. In the context of generic perfect information games, Ben-Porath (1997) establishes through his Theorem 1 that the set of outcomes consistent with common belief (initially, in the whole game) of (weak sequential) rationality corresponds to the set of outcomes that survive the Dekel-Fudenberg procedure. Hence, by Proposition 27, maximal strategies when there is common certain belief of admissible consistency correspond to the outcomes promoted by Ben-Porath's analysis.

An example. To illustrate how common certain belief of admissible consistency is consistent with outcomes other than the unique backward induction outcome, consider the strategic game G3, with corresponding extensive form r 3; i.e., the "centipede" game illustrated in Figure 2.4. Here, backward induction implies that down is being played at any decision node. Let T 1 = {ti, and T2 = {t~, t~}. Assume that the preferences of each type ti of any player i are represented by a vNM utility satisfying 0 Z = Ui and a 2-level LPS on 8 j x Tj . In Tafunction ble 7.1, the first numbers in the parentheses express primary probability distributions, while the second numbers express secondary probability distributions. The strategies OutL and OutR are merged as their relative likelihood does not matter; see footnote 3. Note that all types are in proj--, xTz [caul, implying that players take all opponent strategies into account. With these 2-level LPSs each type's preferences over the player's own strategies are given by

tn

v;;

v;;

Out >-A InL >--A InR InL ,;>-t~ Out InR e ,;>-t~ r r ,;>-t~ e

.:ÂŤ

It is easy to check that both players satisfy 'belief of opponent rationality' at each of their types; e.g., both t~ and t~ assign positive (primary) probability to an opponent strategy-type pair only if it is a maximal strategy for the opponent type (i.e., Out in the case of ti and InL in the case of tn. Thus, 8 1 x T 1 X 82 X T2 ~ A. Since, for each ti E T; of any player i, ;;,t ; ~ {td x s, x 7j, it follows that 8 1 x T 1 X 8 2 X T2 ~ CKA. Hence, preferences consistent with common certain belief of admissible consistency need not reflect backward induction since TnL and rare maximal strategies. Note that, conditional on player 2's decision node being reached (i.e. 1 choosing InL or InR), player 2 at t~ updates her beliefs about the type

89

Backward induction

Table 1.1. An episte mic mod el for

t f1

t f2

.•

(!,

f r

t f2

(0,

170) 110)

t f1

•

'

Out InL InR

(~ ,t)

(0, ~) (0, s)

til2 (0, 110)

C; with til1 ..

(~, ~)

(2't) (O's)

t f2

f r

(! , lo) til1

corresponding extensive form

til2

G, \0)

(~, 1~) (5 ' 10)

t f1

til1 (0,0) (0,0) (0,0)

(0, 10)

til2 •.

(1,

Out InL InR

i)

(°'1) (°'4)

r;.

of player 1 and assigns (primary) probability one to player 1 being of assigns type t~. Consequently, the conditional belief of player 2 at (primary) probability one to player 1 choosing InL. Player 2 at t~, on the other hand, does not admit the possibility that 1 is of another type than t~ . Since the choice of In at 1's first decision node is not rational for player 1 at t~ , there is no restriction concerning the conditional belief of player 2 at t~ about the choice at 1's second decision node. In the terminology of Ben-Porath, a "surprise" has occurred. Subsequent to such a surprise, a player need not believe that the opponent chooses rationally among his remaining strategies.

t;

7.3

Belief in each subgame of opponent rationality

A simultaneous move game offers only one choice situation. Hence, for a game in this class, it seems reasonable that belief of opponent rationality is held in the whole game only, as formalized by the requ irement 'belief of opponent rati onality'. An extensive game with a nontrivial dynamic structure, however, offers such choice situations, not only init ially, in the whole game, but also in proper subgames. Moreover, for extensive games of almost perfect information, the subgames constitute an exhaustive set of such choice situations. This motivates imposing belief in each subgame of opponent rationality. Hence, consider the event that i believes conditional on subgame h E H t ; that j is rational in h:

Bi(h)[ratj(h)] = {(SI ' tl, S2, t2) E 8 1 x t,

X

(sj, tj) E projsj «r,

82

X

T2 I

e-(h) implies sj E 8/j (h)} ,

90

CONSISTENT PREFERENCES

Since H t ; = H whenever ti E projTJcaui], it follows that, if ti E proj- , [( nh EHt ; B, (h )[ratj(h)]) n [cauiJ] , then at ti player i believes conditional on any subgame h that j is rational in h. In other words,

is the event that player i believes in each subgame h that the opponent j is rational in h. 6 Consider a finite extensive two-player game r of almost perfect information with corresponding strategic game G. Say that at ti player i's preferences over his strategies are admissibly subgame consistent with the game r and the preferences of his opponent if ti E projT;Ai , where

Refer to A * := Ai n A 2 as the event of admissible subgame consistency. This definition of admissible subgame consistency can be applied to any finite extensive game of almost perfect information. However, in order to relate to Aumann's (1995) Theorems A and B, the following analysis is concerned with generic perfect information games.

The example revisited. In the belief system of Table 7.1, player 2 at type t~ does not satisfy 'belief in each subgame of opponent rationality '. By 'belief in each subgame of opponent rationality', player 2 must believe, conditional on the subgame defined by 2's decision node, that 1 chooses his maximal strategy, InL, in the subgame. This means that player 2 prefers ÂŁ to r , implying that player 1 must prefer Out to InL if he satisfi es 'belief in each subgame of opponent rationality'. Thus, common certain belief of admissible subgame consistency entails that any types of players 1 and 2 have the preferences Out r..tl

e

InL >--tl InR

>--t2 r

6Note that the requirement of su ch 'belief in each subgame of opponent rat ionality ' allow s a player to update his b elief a bout the type of his opponent. Hen ce, there is no assumption of 'epist em ic independence' between differ ent age nts in the sens e of Stalnak er (1998) ; cr. the remark after the proof of Proposition 28 as well as Section 7.4 . Still, the requirement can be considered a non-inductive analog to 'forward knowledge of rationality' as defined by Balkenborg and W inter (1997), and it is related to the requirement in Section 5 of Samet (1996) that each player hypothesizes that if h were re ach ed , then the opponent would behave rationally at h.

B ackward induction

91

respectively, meaning t hat if a player chooses a maxim al st rat egy in a subgame, then his choice is made in accorda nce with backward induction. Demonstrating that t his conclusion holds in genera l for generic perfect information games const it utes t he main results of t he present chapte r.

Main results. In analogy with Aumann's (1995) Theorems A and B, it is est ablished t hat • ... any vector of maximal st rategies in a subgame of a generic perfect information game, in a state where t here is common cert ain belief of admissible subgame consiste ncy, leads to the backward induction out come in the subgame (P roposit ion 28). Hence, by substituting nhEHti Bi(h)[ratj(h)] for Bdratj], the present analysis yields support to Aumann's conclusion , namely that if there is common knowledge (or certain belief) of an appropriate form of (belief of) rationality, then backward indu ction results. • ... for any generic perfect information game, common cert ain belief of admi ssible subgame consiste ncy is possible (P roposit ion 29). Hence, t he result of Propositi on 28 is not empty. 28 Consi der a finite generic extensive two-player game of perfect information r with corresponding strategic game G. If, for some epistemic model, (tl , t2) E proj T1 XT2 C K A *, then, for each h E H , S~I( h) x S~2 (h) ~ S (Z(S*lh))' where s* denotes the uni que subgameperfect equilibrium . PROPOSITION

Proof. In view of properties of the certain belief operator (cf. Proposit ion 25(ii)) , it suffices to show for any g = 0, . . . , M - 2 that S~I (h) x S~2(h) ~ S( Z(S*lh)) for any h E HM-1-g if there exists an episte mic model with (tl ,t2) E ProjTl xT2KgA*. This is established by induction. (g = 0) Let h E H M - 1 • First, consider j with a singleton action set at h. Then trivially S/ j(h) = Sj(h) = Sj( Z(S*lh))' Now, consider i with a non-singleton acti on set at h ; since r has perfect information, there is at most one such i. Let ti E proj --K" A* = probA*. Then it follows tha t Sii (h) = Si(z( s*lh)) since r is ~eneric and A/~ lUi] n [caui] ' (g = 1, ... , M - 2) Suppose t hat it has been est ablished for g' = 0, . . . , g - 1 t hat S~I (hi) x S~2 (hi) ~ S (z (s*Ih' )) for any hi E H M-1-g' if t here exists an epistemic model with (t l , t2) E ProhIXT2Kg'A*. Let h E H M-1- g. Part 1. Consider j with a singleto n action set at h. Let t j E projTjKg- 1A*. Then Sj( h) = Sj (h , a) and, by Lemma 11 and t he premise, S/ j (h) ~ S/j (h , a) ~ Sj( Z(S*ICh,a ))) if a is a feasible act ion

92

CONSISTENT PR EFEREN CES

vect or at h. This implies that S/ j (h) ~

n a

Sj( Z(S* I(h,a))) ~ Sj( Z(S* lh)) .

Hence, if Sj E S/ j (h) , then Sj is consiste nt with t he backward inducti on outcome in any subgame (h, a) immediat ely succeeding h. Part 2. Consider i with a non-singlet on act ion set at h; since r has perfect information, t here is at most one such i. Let ti E ProjTi KgA *. The preceding argument implies t hat S/j (h) ~ a Sj( Z(S*I(h ,a))) whenever tj E Tp since ti E ProjTi KgA * ~ ProjTi KiKg- 1 A *. Let s, E S, (h) be a strategy t hat differs from si Ih by assigning a different act ion at h (i.e., z(si, sj lh) =!= Z(S* lh) and si (h' ) = si lh(h') whenever Si(h) :J Si(h')) . Let P and q be acts on Sj X T, satisfying t hat, V(Sj, tj ) E Sj x Tj , p( Sj , tj) = z(si, Sj) and q(Sj , tj) = Z(Si, Sj). Then ,

n

Pnasj (Z(pl(h.a»))xTj

st rongly dominat es qnasj (Z(pl (h,a»))xTj

by backward induction since r is generic and t i E projTiKgA* ~ lUi ]' Since S/j( h) c n aSj (Z(S* I(h,a))) whenever tj E T/ i , it follows t hat, Vtj E T/ i , PS/ j (h )X{tj }

st rongly dominat es qS/ j (h)x{tj } '

and, thus, t i E prohiKgA* ~ Bi(h)[ratj(h)] P

n [caui]

implies that

>-,;t · q .

It has t hereby been established t hat Si E Si(h)\Sii( h) if Si differs from backward indu cti on only by t he act ion t aken at h. However, the premise t hat Sii (h, a) ~ Si( Z(S*j(h,a))) if a is a feasible act ion vector at h, it follows t hat any Si E Sii( h) is consiste nt with t he backward inducti on outcome in t he subgame (h, (si( h) , aj)) immediately succeeding h when i plays the act ion si( h) at h (since s, E Si (h, (sj (h), aj)) and, by Lemma 11, s, E S;i (h, (si(h) , aj )). Hence, Shh) ~ Si( z(plh))' • It follows from t he proof of Proposition 28 t hat , for a generic perfect inform ati on game with M -1 stages, it is sufficient with M - 2 order mut ual certain belief of admissible subgame consiste ncy in order to obtain backward induct ion. Hence, K M - 2 A * can be substituted for CK A *. Backward induction will not be obtained, however , if CBA* is substituted for CK A *. T his can be shown by considering a counte r-example that build s on the four-legged centi pede game of Figure 7.1 and the epistemi c model of Table 7.2. In the t able the preferences of each typ e ti of any player i are represent ed by a vNM utility functi on Vii satisfying

93

Backward inducti on

1

212

2

1 3

6

~4 Outl el LI ÂŁ'1

o Figure 7.1.

Table 7.2.

(a four-legged "cent ipede" game) .

An episte mic model for

r 5.

til1 ..

t~ : t~

7 7) e (45' 10' 12 re' rr'

r5

43 2 5

(0, to, i2) (0,0,i2)

t~

til' 1 ..

e

t~

(0, to, i2)

e3

e

(to) (to) (to) (to)

t~

t~

rr'

(fa)

(fa)

3) rf.'

5 '10 '12

(o,o, i2)

t il2 ..

t~ :

Ou t InL InR

t~

5 5 ) (0, to, i2) e 5' 10'12 e5'10'12 l l ) rf.' (0, to, i2) (0,0, i2) rr' (0,0, i2)

t~

t~

t~'

(o ,~ ,~)

(0, i , ~) ell) 2'3'4

(0,0,0) (0,0,0) (0,0,0)

e2' 3'l 4l) (O, O , ~)

(O, O , ~)

t~

Out InL InR

( 1,~ ,1) ( o ,~ , ~)

(0 ,0, i)

t~

t~'

(0, 0, 0) (0, 0, i2) (0, 0,0) (0,0,i2) (0,0,0) (o ,~ , i)

V~i 0 Z = U i and a 1 or 3-level LPS on 5 j x Tj , where T 1 = {t~ , t~ , t~'} and T2 = {t~ , t~}. While all types are in ProjTI X T2 [caul, implying t hat players t ake all opponent st rategies into accoun t , inspection shows t hat A* = 51 x {t~ , x 52 x { t~ , t~} , since player 1 at t~' does not satisfy 'belief in each subgame of opp onent rationalit y'. Furtherm ore, each player i believes at t~ or t~' that the opp onent is of a typ e in {t j , t'J}. This implies that CBA * = A *. Since InL is the maximal strategy for 1 at t~ and is the maximal strategy for 2 at t~ , it follows that preferences consistent with common belief of admissible sub game consiste ncy need not reflect backward induction. However , 2 does not certainly believe at t~ that the opp onent is not of typ e t~' . Therefore, KA * = A * = 5 x {t~ , x {t~} , while KKA * = 0. Hence, preferences that yield maxim al st rate gies in cont ra diction with backward induction are not consiste nt with common certain belief of admi ssible subgame consiste ncy. The example shows t hat t i E ProjTi Ai is consiste nt with player i at t i updating his beliefs about the preferences of his opponent condit ional on a subga me being reached. I.e., 1 at t~ assigns initi ally, in t he whole, (primary) probability ~ to 2 being of ty pe t~ with preferences >>- rr' ,

tn

re'

tn

e re'

94

CONSISTENT PREFERENCES

while in the subgame defined by 1's second decision node 1 at ti assigns (primary) probability one to 2 being of type t~ with preferences r£' >-- £- '" rr. This shows that Stalnaker's (1998) assumption of 'epistemic ind ependence' is not made; a player is in principle allowed to learn about the type of his opponent on the basis of previous play. However, in an epistemic model with CKA * i= 0, t, E ProjTi CKA * implies that 1 cert ainly believes at ti that 2 is of a type with preferences £- >-- r£' >-- rr', In other words, if there is common certain belief of admissible subgame consistency, there is essentially nothing to learn about the opponent. 29 For any finite generic two-player extensive game of perfect information r with corresponding strategic game G, there exists a belief system for G with CKA * i= 0. PROPOSITION

Proof. Construct an epistemic model with one type of each player:

T 1 = {t1} and T2 = {t2}' Write, for each player i , \:1m E {I, ... ,M -I} , S1 m := {silh I h E Hm} and , S1 M := s; Let , for each player i , Ati = (J11 i, . .. , J1~) E L~(Sj x {tj}) satisfy the following requirement: \:1m E {I, ... , M}, SUPPJ1~ = S1 m x {tj}. By letting tt i be represented by a vNM utility function V~i satisfying V~i 0 Z = Ui and the LPS Ati , then (1) lUi] n [caui] = Si x T 1 X S2 X T2. Let , \:Ih E H, Ati Ih = (J1~ti, .. . J1';;lh) denote the conditional of Ati on Sj(h) xTj. By the properties of a subgame-perfect equilibrium, \:Ih E H, J1~ti(s;lh ,tj) = 1 and silh E S;i(h). Hence, since likewise sjlh E S/ j(h) , we have that (2) nhEHti Bi(h)[ratj(h)] = Sl x T1 X S2 X T2. As (1) and (2) hold for both players, it follows that CKA* = A* = Sl X T 1 X S2 X T 2 i= 0. •

7.4

Discussion

In this section we first interpret our analysis in view of Aumann (1995) and then present a discussion of the relationship to Battigalli (1996a). Adding belief revision to Aumann's analysis. Consider a generic perfect information game. Say that a player 's preferences (at a given type) are in accordance with backward induction if, in any subgame, a strategy is a rational choice only if it is consistent with the backward induction outcome. Using this terminology, Proposition 28 can be restated as follows: Under common certain belief of admissible subgame consistency, players are of types with preferences that are in accordance with backward induction. Furthermore, common certain belief of admissible subgame consistency implies that each player deems it subjectively

Backward induction

95

impossible that the opponent is of a type with preferences not in accordance with backward induction. However, since admissible subgame consistency is imposed on preferences , reaching 2's decision node and l 's second decision node in the centipede game of Figure 2.4 does not contradict common certain belief of admissible subgame consistency. Of course, these decision nodes will not be reached if players choose rationally. But that players satisfy 'belief in each sub game of opponent rationality' is not a requirement concerning whether their own choice is rational; rather, it means that they believe (with probability one) in any subgame that their opponent will choose rationally. Combined with the assumption that all types are in proj --, XT2 [caul, which entails that each player deems any opponent strategy subjectively possible , this means that belief revision is well-defined. Hence, on the one hand, we capture the spirit of a conclusion that can be drawn from Aumann's (1995) analysis, namely that when being made subject to epistemic modeling backward induction corresponds to each player having knowledge (or being certain) of some essential feature of the opponent. In Aumann's case, each player deems it impossibleunder common (true) knowledge of (sequential) rationality-that the opponent makes an action inconsistent with backward induction. The analogous result in the present case is that each player deems it subjectivly impossible-under common certain belief of admissible subgame consistency-that the opponent has preferences not in accordance with backward induction. On the other hand, we are still able to present an explicit analysis of how players revise their beliefs about the opponent's subsequent choice if surprising actions were to be made. As noted in the introduction to this chapter , this fundamental issue of belief revision cannot formally be raised within Aumann's framework. Stalnaker (1998) argues-contrary to statements made by Aumann (1995, Section 5f)-that an assumption of belief revision is implicit in Aumann's motivation, namely that information about different agents of the opponent is treated as epistemically independent. In the reformulation by Halpern (2001),7 this means that in a state "closest" to the current state when a player learns that the opponent has not followed

7See Halpern (2001) for an instructive discussion of the differences between Aumann (1995) and St alnaker (1998) , as well as how these relate to Samet (1996).

96

CONSISTENT PREFERENCES

her strategy, he believes that the opponent will follow her strategy in the remaining subgame. There is no assumption of 'epist emic independence' in the current interpretation of Aumann's result. Instead, we have changed statements 'about opponents' from being concerned with strategy choice to being related to preferences. While it is desirable when modeling backward induction to have an explicit theory of revision of beliefs about opponent choice, a theory of revision of beliefs about opponent preferences is inconsistent with maintaining both (a) that preferences are necessarily revealed from choice, and (b) that there is common certain belief of the game being played (i.e., consider the case where A i (0) is non-singleton, and a; E A i (0) ends the game and leads to an outcome that is preferred by i to any other outcome) . Here we have kept the assumption that there is common certain belief of the game , meaning that the game is of 'complet e information', while requiring only conditional belief in each subgame of opponent rationality, meaning that irrational opponent choices-although being probability zero events-are not subjectively impossible. We have shown how common certain belief of admissible subgame consistency implies that each player deems it impossible that the opponent has preferences not in accordance with backward induction and thus interprets any deviation from the backward induction path as the opponent not having made a rational choice. In this way we present a model that combines a result that resembles Aumann (1995) by associating backward induction with certainty about opponent type, with an analysis that unlike Aumann's yields a theory of belief revision about opponent choice. Rationality orderings. The constructive proof of Proposition 29 shows how common certain belief of admissible subgame rationality may lead player i at ti to have preferences over i 's strategies that are represented by a vNM utility function V;i satisfying V;i 0 Z = u; and an LPS Ati = (,Ai, ..., ,}jJ E L~(Sj x T j) with more than two levels of subjective probability distributions (i.e., L > 2). E.g ., in the "cent ipede" game of Figure 2.4, common certain belief of admissible subgame consistency implies that player 2 at any type t2 has preferences that can be represented by V~2 satisfying V~2 0 Z = U2 and At2 = (lti2, 1t~2 , 1t~2) satisfying projslsuPPJLi2 = {Out} , projslsuPPJL~2 = {Out,InL}, and projsl SUPPJL~2 = SI Âˇ One may interpret â€˘ projsjsuPPJLii to be j's "most rational" strategies,

Backward induction

97

• projsjsuPPItZ\Uk<L projSjSUPPIt~ to be j's "completely irrational" strategies, and • projSjSUPPIt~i\Uk<e projSjSUPPIt~, for £ = 2, . . . , L - 1, to consist of strategies for j that are at "intermediate degrees of rationality" . This illustrates that (projsjsuPPltii, ... ,projsjsuPPItZ\ Uk<L projSjsuPPIt~) corresponds closely to what Battigalli (1996a) calls a rationality ordering for j. However, the present construction of such a rationality ordering differs from the one proposed by Battigalli. This difference is along two dimensions: 1 Battigalli considers best responses in reachable subgames only (see his Definition 2.1), while here belief of opponent rationality is held in all subgames (cf., 'belief in each subgame of opponent rationality'). 2 Battigalli considers best responses given beliefs where opponent strategies that are less than "most rational" are given positive probability, while here each player always believes that the opponent chooses rationally. This difference has the following consequences: • Although Battigalli's construction of rationality orderings also yields the backward induction outcome in any generic perfect information game, his proof (cf. Battigalli, 1997) is not tied to the backward induction procedure. • Battigalli's construction promotes the forward induction outcome (lnL, £) in the "battle-of-the-sexes with an outside option" game illustrated in Figure 2.6. This conclusion is not reached in the present analysis since there is no choice situation in which 1 under all circumstances will have a particular preference between his "battle-ofthe-sexes" strategies.f This also indicates how the epistemic foundation for the backward induction procedure offered here differs from the epistemic foundation for backward (and forward) induction outcomes provided by Battigalli and Siniscalchi (2002). 8Cha pt er 11 will , following Asheim and Dufwenberg (2003a) , demonstrate how the conc ept of admissible consistency can be strengthened so that the forward induction outcome is promoted in the "bat tl e-of-t he-sexes with an outside option" game.

Chapter 8

SEQUENTIALITY

One major problem in t he t heory of extensive games is t he following: How should a player react when he finds himself at an information set t hat contradicts his previous belief abo ut t he oppo nent 's st rategy choice? Different approaches have been proposed to t his problem. As mentioned in Chapter 2, Ben-P orath (1997) and Reny (1992) have formul at ed rationalizability and equilibrium not ions based on weak sequentiality , in which a player is allowed to believe, in t his sit uation, t hat his opponent will no longer choose rat ionally. Bat tigalli and Siniscalchi (2002) have shown t hat Pear ce's (1984) extensive f orm rationalizability can be characterized by assuming t hat a player, in such a sit uation, should look for t he highest degree of "strategic sophistication" t hat is compat ible with the event of reaching t his information set , and stick to this degree until it is cont radicted lat er on in t he game. Perea (2002, 2003) suggest s t hat t he player , in such a sit uat ion, may revise his conjecture about the opponent 's utility function in order to rationalize her "surprising" move, while maintaining common belief of rational choice at all information sets . The most prominent positi on, however , is that t he player should st ill believe that his oppone nt will choose rationally in t he remaind er of t he game; t his und erlies concepts t hat promot e backward indu ction. Such concepts will be present ed in t his and t he next chapter, which reproduce joint work with Andres Perea, cf. Asheim and Perea (2005). We define sequent ial rationalizability by imp osing common certain belief of t he event t hat each player believes t hat the opponent chooses rationally at all her information sets . Since t his is a non-equilibrium concepts, each player need not be certain of t he beliefs t hat t he oppo-

100

CONSISTENT PREFERENCES

nent has about the player's own action choice. However, by assuming that each player is certain of the beliefs that the opponent has about the player's own action choice, we obtain an epist emic characterization of the corresponding equilibri um concept: sequential equilibrium. When applied to generic games with perfect information, sequ ential rationalizability yields the backward induction procedure. As elsewhere, to avoid the issue of whether (and if so, how) each player's beliefs about the action choice of his opponents are stochastically independent , all analysis is limited to two-player games. The assumption is essential in the present context where a behavior strategy of a player will be interpreted as an expression of the belief of his opponent. For the above mentioned definition and characterization, we must describe what a player believes both conditional on reaching his own information sets (to evaluate his rationality) and conditional on his opponent reaching her information sets (to determine his beliefs about her choices). Hence, we must specify a system of conditional beliefs for each player. For reasons given in Section 3.1, this will be done by means of our concept of a system of conditional lexicographic probabilities (SCLP) as defined in Definition 1 and characterized in Proposition 5. We embed the notion of an SCLP in an epistemic model, as defined by Definition 9 of Chapter 5, by invoking Assumption 1. For each type ti of any player i , ti is described by an SCLP, inducing a behavior strategy for each opponent type tj that is deemed subjectively possible by k The event that 'player i believes that the opponent j chooses rationally at each information set ' can then be defined as the event where player i is of a type t; that, for each subjectively possible opponent type tj , induces a behavior strategy which is sequentially rational given tj 'S own SCLP. The characterization of sequential equilibrium reported in Proposition 30 is included in order to motivate the analogous non-equilibrium concept , namely sequential rationalizability. The result may, however , be of interest in its own right and in comparison with other such epistemic characterizations; see, e.g., Theorem 2 of Feinberg (2005b) . The concept of sequential rationalizability as stated in Definition 15 is related to various other concepts proposed in the literature. Already in Bernheim (1984) there are suggestions concerning how to define nonequilibrium concepts that involve rational choice at all information sets. By requiring rationalizability in every subgame, Bernheim defines the concept of subgame rationalizability -which coincides with our definition of sequential rationalizability for games of almost perfect information-

Sequentiality

101

but no epistemic characterization is offered. On p. 1022 Bernheim claims that it is possible to define a concept of sequential rationalizability, but does not indicate how this can be done . After related work by Greenberg (1996), sequential rationalizability was finally defined by Dekel et al. (1999, 2002), whose concept coincides with ours in our two-player setting. Our definition of quasi-perfect rationalizability is new. Dekel et al. (1999) and Greenberg et al. (2003) consider also extensive game concepts that lie between equilibrium and rationalizability; such concepts will not be considered here.

8.1

Epistemic modeling of extensive games (cont. )

The purpose of this section is to present a framework for a general class of extensive games where each player is modeled as a decision maker under uncertainty with complete preferences.

An extensive game form. Consider a finite extensive two-player game form without chance moves. Assume that the extensive game form satisfies perfect recall. Denote by Hi the finite collection of information sets controlled by player i. For every information set h E Hi, let A(h) be the set of actions available at h. A pure strategy for player i is a function s; which assigns to every information set h E Hi some action si(h) E A(h). Denote by Si the set of pure strategies for player i, where, in the subsequent analysis, there is no need to differentiate between pure strategies in S, that differ only at non-reachable information sets. Write S = SI X S2, denote by Z the set of outcomes (or terminal nodes) , and let z : S ----t Z map strategy profiles into terminal nodes. Then (S1 , S2, z) is the corresponding finite strategic two-player game form. For any h E HI U H2, let Si(h) be the set of strategies s, for which there is some strategy Sj such that (Si' Sj) reaches h. For any h and any node x E h, denote by S(x) = SI(X) X S2(X) the set of pure strategy profiles for which x is reached, and write S(h) := UXEh S(x). By perfect recall , it holds that S(h) = SI(h) x S2(h) for all information sets h. For any h, h' E Hi, h (weakly) precedes h' if and only if S(h) ~ S(h'). For any hE Hi and a E A(h), write Si(h,a) := {s, E Si(h) I si(h) = a} . A behavior strategy for player i is a function O"i that assigns to every hE Hi some randomization O"i(h) E Ll(A(h)) on the set of available actions. If t.ÂŤ Hi, denote by O"ilh the behavior strategy with the following properties: (1) at player i information sets preceding h, O"i Ih determines with probability one the unique action leading to h, and (2) at all other

102

CONS IS TENT PREFEREN CES

player i information sets , O"ilh coincides with O"i. Say t hat a, is outcomeequivalent t o a mixed st rategy Pi (E ~ (Sd ) if, for any Sj E Sj , a, and Pi induce the same probability distribution over te rminal nodes. For any h E Hi, O"i lh is outcome-equivalent to some Pi E ~ ( Si (h) ).

Epistemic modeling. Since t he extensive game form det ermines a finit e st rategic game form , we may repr esent t he st rategic interaction by means of an episte mic model as defined by Definition 9 of Chap ter 5. Since a behavior st rategy of a player will be interpr eted as an expression of t he belief of his opponent, it is essent ial that the analysis assumes complete preferences. Hence, the epist emic model is combined with Assumption 1 of Chap ter 5. Und er Assumption 1 it follows from Proposition 5 t hat, for each typ e ti of any player i , i' s syste m of condit ional preferences at ti can be ~ (Z ) ~ IR and an SCLP repr esent ed by a vNM utility function ti (>' ti, f ), which for expositional simplicity is defined on Sj x T j with support Sj x T/ i (instead of being defined on T; x Sj x T j with support K,ti = {t d x Sj x T/ i ). Hence, writing t~ for player i 's preferences at ti conditional on player i information set h E Hi being reached , we consider w.l.o.g. t~ to be preferences over acts from Sj( h) x Tj to ~ (Z ) (instead of acts from {td x Sj(h) x T j to ~ (Z ) ).

V;i :

Conditional preferences over strategies. It follows t hat , for each ti of any player i and all h E Hi, i's conditional preferences at ti in subgame h can be repr esented by t he vNM utility function ~ (Z) ~ IR t hat does not depend on h, and an LPS

V;i:

>'~(Sj (h) x Tj ) ISj (h) x Tj

=

(J1-~ , ... J1-~(Sj (h) xTj )ISj(h) XTj )

derived from the SCLP (>.ti,fti) on Sj x T; with support Sj x T/ i. Recall from Assumption 1 that player i deems an opponent strategytype pair (Sj, tj) subjectively possible at ti if and only if Sj E Sj and tj E T/i. This means that condit ional preferences are non-trivial for an event E j ( ~ Sj x Tj) if and only if e, n is, x TJi ) =1= 0. Note that {Sj (h) x t, I h E Hd is the set of events that are obj ectively observable by i. Hence, conditional preferences are always non-trivial for such events since, for x T/ i ) = (Sj( h) X T/ i) =1= 0. any h E Hi, (Sj( h) x T j ) n Since, for all h E Hi, each pur e st rategy s, E Si( h) is a function t hat assigns t he det erministic outco me Z(Si, Sj) to any (Sj , tj) E Sj(h) x Tj , it follows that Si E Si(h) is an act from Sj( h) x Tj to ~ (Z ) , and we have t hat t~ determines complete and tra nsitive preferences on i's set of pure st rategies, Si(h), conditional on h E Hi being reached.

is,

103

S equentiality

Player i's choice function at ti is a function Sfi(.) that assigns to every h E Hi player i 's set of rational pure strategies at ti conditional on h E Hi:

Sfi(h) := {Si E Si(h) I \:js~

E

Si(h), si ?::~: sa .

Refer to Sfi (h) as player i 's choice set at ti conditional on player i information set h, and write Sf i = Sfi (0), thereby following the notation of Chapter 5. Since ?::~ is complete and transitive and satisfies objective independence, and S; (h) is finite, it follows that the choice set Sfi (h) is nonempty, and that the set of rational mixed strategies equals 6.( Sfi(h)). Note that Lemma 11 does not hold under Assumption 1, unless also caution is imposed (cf. Section 9.1). The assumption that player i 's system of conditional preferences at t, is representable by means of an SCLP where the set of subjectively possible opponent types equals Sj x T/i has the structural implication that, for all h E Hi, the conditional preferences, ?::~, are nontrivial, even without imposing caution. However, representation by means of an SCLP does not have the behavioral implication that any choice s, that is rational conditional on h is also rational in any later player i information set that Si is consistent with. This means that 'rationality' does not imply 'weak sequential rationality' if caution is not imposed. An extensive game. As in Chapter 7, a finite extensive two-player game consists of the pair of the extensive game form and the vNM utility functions (VI, V2), with G = (Sl, S2, Ul, U2) denoting the corresponding finite strategic game, where for each i, the vNM utility function Ui : S ---+ IR is defined by u; = Vi 0 z . As before-i-but transferred to T l x T2 space-s-the event that i plays the game G is given by

lUi] := {(tl, t2) E T l xT2 Vfi OZ is a positive affine transformation of ud, 1

while [u] := [Ul]

n [U2]

is the event that both players play G.

Certain belief. As in Chapter 5, say for any E ~ T l X T2 that player i certainly believes the event E at ti if ti E probKiE, where t

KiE:= {(tl, t2)

E

T, x T2 I projT1XT2 h: t i = {td x T/i ~ E}.

Say that there is mutual certain belief of E at (tl, t2) if (t1, t2) EKE, where KE := KIE n K2E . Say that there is common certain belief of E at (iI, t2) if (tl, t2) E CKE, where CKE:= KE n KKE n KKKE n ....

104

CONSISTENT PREFERENCES

8.2

Sequential consistency

In this section , we use the episte mic model of the previous section based on the concept of an SeLP-to formalize the requirement that each player believes that the opponent chooses rationally at each of her information sets, given her preferences at these information sets. This enables us • to characterize sequential equilibrium (Kreps and Wilson, 1982), and • to define sequential rationalizability as a non-equilibrium analog to the concept of Kreps and Wilson (1982).

Inducing sequential rationality. In our setting a behavior strategy is not an object of choice, but an expression of the system of beliefs of the other player . Say that the behavior strategy o/il tj is induced for tj by ti if tj E T/i and, for all h E H, and a E A(h),

,tiltj (h)( ) '= J-l~i (Sj(h, a), tj) O'J a . ti ' J-l£ (Sj(h) , tj) where £ is the first level £ of Ati for which J-l~i(Sj(h), tj) > 0, implying that J-l~i restricted to Sj (h) x {tj} is proportional to the top level probability distribution of the LPS that describes ti'S conditional belief on Sj(h) x {tj}. Here, J-l~i(Sj(h) , tj) := L:sES(h) J-l~i(Sj , tj) and t, ti J J J-l/(Sj(h, a), tj) := L-sjESj(h,a) J-l£ (Sj, tj) . Say that the behavior strategy a, is sequentially rational for i at ti if, Vh E Hi, O'ilh is outcome-equivalent to some mixed strategy in .6. (Si i(h)). Define the event that player i is of a type that induces a sequentially rational behavior strategy for any opponent type that is deemed subjectively possible:

[i sri] := {( t i , t2) E T 1 x T2 IVtj

E

T/;,

O'/iltj is sequentially rational for tj} . Write [isr] := [isrl] n [isr2] for the event where both players are of such a type. Say that at ti player i 's preferences over his strategies are sequentially consistent with the game r and the preferences of his opponent, if ti E proj-, ([Ui] n [isri])' Refer to [u] n [isr] as the event of sequential consistency. Note that the behavior strategy induced for tj by ti specifies i 's belief revision policy at ti about the behavior of tj, as it defines probability

Sequentiality

105

distributions also at player j information sets that are unreachable given i's initial belief at ti about tj's behavior. Hence, if ti E projTi [isri] , then player i believes at ti that each subjectively possible opponent type tj chooses rationally also at player j information sets that contradict ti's initial belief about the behavior of tj. The above observation explains why we can characterize a sequential equilibrium as a profile of induced behavior strategies at a type profile in [isr] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible).

Characterizing sequential equilibrium. We first define sequential equilibrium. Player i's beliefs over past opponent actions at i 's information sets is a function f3i that to any h E Hi assigns a probability distribution over the nodes in h. An assessment (a,f3) = ((a1,a2) , (f31,f32)) , consisting of a pair of behavior strategies and a pair of beliefs, is consistent if there is a sequence (a(n), f3(n))nEN of assessments converging to (a, (3) such that for every n, a(n) is completely mixed and f3(n) is induced by a(n) using Bayes' rule. If a; and aj are any behavior strategies for i and i , and f3i are the beliefs of i, then let , for each hE Hi, ui(ai, aj; f3dlh denote i 's expected payoff conditional on h, given the belief f3i(h), and given that future behavior is determined by ai and aj. DEFINITION 14 An assessment (a,f3) = ((a1,a2) , (f31,f32)) is a sequential equilibrium if it is consistent and it satisfies that for each i and every hE Hi,

The characterization result can now be stated; it is proven in Appendix B. PROPOSITION 30 Consider a finite extensive two-player game r . A profile of behavior strategies a = (aI, (2) can be extended to a sequential equilibrium if and only if there exists an epistemic model with (t 1, t2) E [u] n [isr] such that (1) there is mutual certain belief of {(t1 , t2)} at (t1, t2), and (2) for each i, a, is induced for ti by tj ' For the "if" part, it is sufficient that there is mutual certain belief of the beliefs that each player has about the action choice of his opponent at each of her information sets . We do not need the stronger condition that (1) entails. Hence, higher order certain belief plays no role in the characterization, in line with the fundamental insights of Aumann and Brandenburger (1995).

106

CONSIS TENT PREFEREN CES

Defining sequential rationalizability. We next define t he concept of sequent ially ration alizable behavior st rategies as indu ced behavior st rategies und er common certain belief of [isr] . D EFI NITIO N 15 A behavior st rategy a, for i is sequenti ally rationalizable in a finit e extensive two-player game r if there exists an episte mic model with (t l, t2) E CK ( [u] n [isr]) such that a; is induced for ti by tj.

It follows from Propositi on 30 that a behavior strategy is sequent ially rationalizable if it is part of a profile of behavior st rategies that can be exte nded to a sequenti al equilibrium. Since a sequent ial equilibrium always exists, we obtain as an immediate consequence that sequentially rationalizable behavior st rategies always exist. For the concept of sequent ial rationalizability- as indeed , throughout the book-we restrict our at te nti on to games with two players. A natural question which arises is whether , and if so how, t he present analysis can be exte nded to the case of t hree or more players. In order to illustrat e t he pot ential difficulties of such an exte nsion, consider a t hree player game in which player 3 has an information set h with two nodes, x and y , where x is preceded by the player 1 act ion a and the player 2 act ion c, and y is preceded by t he player 1 act ion b and the player 2 action d. Suppose t hat player 3 views band c as subo pt imal choices, and hence player 3 deems a infinit ely more likely t han b, and deems d infinit ely more likely t han c. Then , player 3's LPS at h over player l 's strategy choice and player 3's LPS at h over player 2's st rategy choice do not provide sufficient inform ation to derive player 3's relati ve likelihoods attached to nod es x and y, and t hese relative likelihoods are crucial t o assess player 3's rational behavior at h . Hence, in addition t o t he two LPS s mentioned ab ove, we need another aggregated LPS for player 3 at h over his opponents' collect ive strat egy profiles. The key problem would then be what restrictions t o impose upon the connect ion between the LPSs over individual st rateg ies on the one hand and t he aggregate LPS over st rategy profiles on the other hand. Both classes of LPSs are needed, since the former are crucial in order t o evaluate t he beliefs about rationality of individual players, and the latter are needed in order to det ermin e the condit ional preferences of each player, as shown above. This issue is closely related to t he probl em of how to cha racterize consiste ncy of assessments in algebraic te rms, without t he use of sequences; ef. McLennan (1989a, 1989b), Bat tigalli (1996b) , Kohlb erg and Reny (1997), and Perea et al. (1997). In t hese pap ers, t he consistency requir ement for assessments has been characterized by

107

Sequentiality

means of conditional probability systems, relative probability systems and lexicographic probability syst ems, satisfying some appropriate additional conditions. Perea et al. (1997), for inst ance, use a refinement of LPS in which, at every inform ation set , not only an LPS over t he available act ions is defined , but moreover the relative likelihood level between act ions is "quant ified" by an addi tional paramet er , whenever one action is deemed infinit ely more likely than the ot her . This additional parameter makes it possible to derive a unique aggregate LPS over action profiles (and hence also over strategy profiles). A similar approach can be found in Govindan and Klumpp (2002). Such an approach could possibly be useful when exte nding our analysis of, e.g., sequent iality to the case of more t han two players. For the moment, we leave this issue for future research.

8.3

Weak sequential consistency

In the previous section we have shown how imp osing that each player believes that the opp onent chooses rationally at all her information sets can be used to characterize sequent ial equilibrium and define sequential rationalizability. Tabl e 2.2 suggests the following claim: Imposing t hat each player believes that the opponent chooses rationally only at her reachable information sets can be used to cha racterize the notion of weak sequential rationalizability, du e to Ben-Porath (1997) and coined 'weak exte nsive form rationalizablity' by Batti galli and Bonanno (1999). In this sect ion we verify t his claim and shed light on t he difference between sequent iality and weak sequentiality. I nduci ng weak sequential r a t io nalit y. Recall from Chapter 5 that the mixed st rategy pl iltj is induced for tj by ti if tj E Tl i and, for all Sj E Sj ,

p .tiltj(s ') = J

J

J.L~i(Sj ,tj)

J.L~i(Sj , tj) ,

where e is the first level e of Ati for which J.L~i(Sj, tj) > O. Say that a mixed st ra tegy Pi is weak sequentially rational for i at ti if, Vh E Hi s.t. SUPPPi n Si(h) =/: 0, sUPPPi n Si(h) ~ S;i(h) , and define the event that player i is of a type t hat induces a weakly sequent ially rational mixed st rategy for any opponent typ e t hat is deemed subject ively possible:

[iwriJ

:=

{(tl , t2) E T I x T 2 IVtj

p/

i

E

T/ ;,

Iti is weak sequent ially rational for tj} .

108

CONSIS TENT PR EFERENCES

Write [iwr] := [iwr1] n [iwr2]' Say t hat at ti player i 's preferences over his st rategies are weak sequentially consistent with the game r and the preferences of his opponent, if ti E prohi ([Ui ] n [iwri]) ' Refer to [u]n [iwr] as the event of weak sequential consistency. Note t hat t he mixed st ra tegy induced for tj by ti may be int erpret ed as i 's initial belief at ti about the behavior of tj . In cont rast to the behavior strategy induced for tj by t. , as defined in t he previous section, t he induced mixed st rategy gives no information about how i at ti revises his belief about t he behavior of tj at player j information sets that are unreachable given i 's initial belief at ti ab out tj's behavior. Hence, if ti E projTi [iwri] , then player i believes at ti that each subj ectively possible opponent typ e tj chooses rationally at player j information sets that do not contradict i 's initial belief at ti about the behavior of tj . However, and this is the crucial difference when compared t o t he case where ti E projTi [isri]: t ; E ProjTJiwri] entails no restri ction on how i at ti revises his beliefs about tj 's behavior condit ional on tj reaching "sur prising" inform ation sets. The above observation explains why weak sequent ially rati onalizable mixed st rategies can be shown to correspond t o induced mixed st rategies und er common certain belief of [u] n [iwr]. Characterizing weak sequential rationalizability. We first define weak sequent ial rationalizability. Since weak sequent ial rationalizability in two-player games corresponds t o it erated elimination of st rategies t hat are st rongly dominated at some reachable information set , we use t he latt er pro cedure as t he primi tive definition. For any (0 =J) X = Xl X X 2 ~ S , writ e b(X ) := b1(X2) x b2(Xd , where

bi(Xj ) := S, \ {s,

E

S, I:3Pi E ~ (Si) s.t. Pi strongly dominat es s; on X, or :3h E Hi with Si(h) :3 Si and qi E ~(Si(h)) s.t . qi strongly dominates s; on Sj(h)} .

If Pi is a mixed strategy and h E Hi satisfies t hat SUPPPi n Si(h ) then let Pi Ih be defined by

=J 0,

if s; E Si( h) ot herwise . D EFI NITIO N 16 Let r be a finite exte nsive two-player game. Consider t he sequence defined by X (O ) = Sl X S2 and , \:Ig 2: 1, X (g) = b(X (g - 1)). A pure strategy Si is said to be weak sequentially rational-

Sequentiality

109

izable if s, E

w, :=

nÂ°o

g=O

X i(g) .

A mixed strategy Pi is said to be weak sequentially rationalizable if Pi is not strongly dominated on W j and there does not exist h E Hi with SUPPPi n Si(h ) i= 0 such t hat Pilh is strongly domin at ed on Sj (h ). While any pure strategy in t he support of a weak sequent ially rationalizable mixed strategy is itself weak sequent ially rationalizable, the mixture over a set of weak sequenti ally rationalizable pure st rategies need not be weak sequent ially rationalizable. The following lemma is a st raight forward implication of Definition 11. 12 (i) For each i, Wi i= 0. (ii) W = b(W) . (iii) For each i, s, E Wi if and only if there exists X = XI X X 2 with s, E Wi such that X ~ b(X).

LEMMA

We next characte rize t he concept of weak sequent ially rationalizable mixed strategies as induced mixed st rategies und er common cert ain belief of [u] n [iwr]. 31 A mixed strategy Pi for i is weak sequentially rationalizable in a finite extensive two-player game r if and only if there exists an epistemi c model with (t1' t2) E CK([u] n [iwr]) such that Pi is induced for t, by tj . PROPOSITION

Proof. Part 1: If pi is weak sequentially rationalizable, then there exists an epistemic model with ÂŤt.t 2) E CK( [u] n [iwr]) such that p: is induced for ti by tj. Step 1: Constru ct an epistemic model with T 1 x T 2 ~ CK([u] n [iwr]) such that for each s, E Wi of any player i , there exists ti E T; with, s, E S;i. Construct an episte mic model with, for each i, a bijection s, : Ti ~ Wi from the set of types to the t he set of weak sequent ially rationalizable pure strat egies. Assume that , for each ti E T; of any player i , V;i satisfies that (a) V;i 0

Z

=

Ui (so that T 1 x T2 ~ [u]) ,

and t he SCLP (Ati , eti ) on Sj x Tj has t he properties that (b) Ati = (JLii , . . . , JLZ) with support Sj x Tl i satisfies t hat sUPPJLil n (Sj x {tj} ) = {(Sj (tj ), tj )} for all tj E Tl i (so t hat, Vtj E Tl i, plilti(sj(tj)) = 1),

110

CONSISTENT PREFERENCES

(c) VEj E Sj x Tj such that E j n (Sj x T/i) =1= 0, Rti(Ej) = min{R I SUPPA~i =1= 0} (so that, by Corollary 1, the SCLP corresponds to a CPS) . Property (b) entails that the support of the marginal of J.1ii on Sj is included in Wj. By properties (a) and (c) and Lemmas 4 and 12(ii) , we can still choose fAi (and Titi) so that s, (td E S;i. Since information sets correspond to strategically independent sets (d. the discussion in connection with Lemmas 11 and 13) we have that, Vh E Hi s.t. Si(h) :3 si(td and sUPPJ.1ii n s j (h) =1= 0, si(td E Sti(h) , while, Vh E Hi s.t . Si(h) :3 si(td and sUPPJ.1ii nSj(h) = 0, Si(ti) E Shh) by choosing the lower levels of Ati appropriately (again invoking properties (a) and (c) and Lemmas 4 and 12(ii)). This combined with property (b) means that TI XT2 ~ [iwr]. Furthermore, TI x T2 ~ CK([u] n [iwr]) since T/i ~ Tj for each ti E t: of any player i. Since, for each player i, s, is onto Wi , it follows that, for each Si E Wi of any player i, there exists ti E T; with Si E S;i. Step 2: Add type ti to Ti . Assume that v8 satisfies (a) and (Ati , Rti ) satisfies (b) and (c). Then J.1lti can be chosen so that pi E 6.(S8), and consequently, Vh E Hi s.t. sUPPPi n Si(h) =1= 0 and SUPPJ.1lti n Sj(h) =1= 0, sUPPPi n Si(h) ~ S/i (h), while, Vh E Hi s.t. supppi n Si(h) =1= 0 and SUPPJ.1lti n Sj(h) = 0, supppl n Si(h) ~ S/i (h) by choosing the lower levels of Ati appropriately. Furthermore, tt: U {tn) x Tj ~ [u] n [iwr], and since Tli ~ Tj , tt; u {tn) x Tj ~ CK([u] n [iwr]). Step 3: Add type tJ~ to Tj. Assume that vi; satisfies (a) and the SCLP (At*j , Rt:j ) on S, x (Ti u {ti}) has the property that Ar:j = (/-llt*j , .. . , /-l Lt*j ) with support S, x {ti} satisfies that, Vs, E Si, /-lIt; (Si' ti) = pi (s.), so that pi is induced for tt by tj. Furthermore, (Ti U{tn) x tr, U {tj}) ~ [u] n [iwr], and since Titj ~ TiU{tn, (TiU{tn) x (TjU{tj}) ~ CK([u]n[iwr]). Hence, (ti, t 2) E CK([u] n [iwr]) and Pi is induced for ti by tj. Part 2: If there exists an epistemic model with (ti, t 2) E CK ([u] n [iwr]) such that pi is induced for ti by tj, then pi is weak sequentially

rationalizable. Assume that there exists an epistemic model with (ti, t 2) E CK([u] n [iwr]) such that pi is induced for ti by tj . In particular, CK([u]n[iwr]) =1= 0. Let, for each i, TI := ProjTiCK([u] n [iwr]) and

Xi

:=

Uti

,{Si

ET i

E

Si I Vh E Hi s.t. Si(h):3 Si,Si

E

S;i(h)}.

By Proposition 20(ii), for each ti E TI of any player i, ti deems (Sj, tj) subjectively impossible if tj E Tj \Tj since CK([u] n [iwr]) = KCK([u] n [iwr]) ~ KiCK([U] n [iwr]) , implying T/i ~ Tj. By the definitions of

111

Sequentiality [u] and [iwr], it follows that, for each ti

TI of any player i ,

e: t i

is represented by satisfying that oz is a positive affine transformation of Ui and an LPS A~i = (/A\ . . . ,f-t~i) , where f = f(Sj x Tj) ~ 1, and where sUPPf-tii ~ Xj x Tj. Hence , by Lemma 4, for each ti E T! of any player i , if Pi E ~(Sd satisfies that , Vh E Hi s.t. SUPPPi n Si(h) =1= 0, SUPPPi n Si(h) ~ S;i(h) , then

V;i

V;i

E

• no strategy in the support of Pi is strongly dominated on X j , since then Pi E ~(S;i), and it follows from Pi E ~(S;i) and suppf-tii C X j x T j that , VSi E SUPPPi and Vs~ E Si ,

L L SjEX j tjETj

f-tii(sj ,tj)Ui(Si,Sj) ~

L L

f-tii(Sj ,tj)Ui(S~ ,Sj),

sj E Xj t jETj

• Vh E Hi s.t. SUPPPi n Si(h) =1= 0, no strategy in SUPPPi n Si(h) is strongly dominated on Sj(h) since SUPPPi n Si(h) ~ S;i(h). This implies X ~ b(X) , entailing by Lemma 12(iii) that , for each i, Xi ~ Wi . Furthermore, since (ti , t 2) E CK([u] n [iwr]) and the mixed strategy induced for ti by tj , pi , satisfies that , Vh E Hi s.t. supppi n Si(h) =1= 0, supppi n Si(h) ~ Sli(h), it follows that pi is not strongly dominated on X j ~ Wj and there does not exist h E Hi with supppi n Si(h) =1= 0 such that Pilh is strongly dominated on Sj(h) . By Definition 16 this implies that pi is a weak sequentially rationalizable mixed strategy. • The following observation (which is stated without proof) can now be used to establish the relationships between the rationalizability concepts on the lower row of Table 2.2. PROPOSITION

32 For any epistemic model and for each player i ,

Since [iwri] ~ [iri], Propositions 31 and 22 entail that weak sequential rationalizability refines (ordinary) rationalizability, and since [isri] ~ [iwri], Definition 15 and Proposition 31 entail that sequential rationalizability refines weak sequential rationalizability. That the two latter inclusions can be strict , is illustrated by r 6 and r~ of Figures 8.1 and 8.2, respectively. In T 6 rationalizability does not have any bite, while weak sequential rationalizability promotes that player 1 plays F and player 2 plays f. In r~-introduced by Reny (1992, Figure I) -weak sequential rationalizability only precludes the play of D at 1's second decision node . This can be established by applying the Dekel-Fudenberg procedure (i.e., one round of weak elimination followed by iterated strong elimination)

112

CONS ISTENT PREFERENCES

2

1

-1

D0I

1

0

3

~3

Figure 8.1.

1

2

1

DI

dl

DI

2 2

1 1

0 0

r6

and it s st rategic form.

3

~3

Figure 8.2.

r; and its

d f D 2, 2 2,2 FD 1, 1 0, 0 FF 1, 1 3,3

pure st rategy reduced strategic form .

which eliminates a st rategy if and only if it is not permissible. Since all terminal nodes yield different payoffs, weak sequential rationalizability leads to the same conclus ion.1 However , only t he play of F at both of 1's decision nodes and the play of f at 2's single decision node are sequentially rationalizable. This follows from Proposition 33 of the next section, showing that the latter concept implies the backward induction procedure. Extensi ve form rationalizability (EFR) , d . Pearce (1984) as well as Battigalli (1997) and Battigalli and Siniscalchi (2002), is an iterative deletion proce dure where , at any information set reached by a remaining strategy, any deleted st rat egy is deemed infinit ely less likely t ha n some remaining st rategy. Even though EFR only requires players to choose rat ionally at reachable information sets and preference for caut ious behav ior is not imp osed, EFR is different from weak sequent ial rationalizab ility. Unlike all concepts in Tables 2, EFR yields forward induction in common exam ples like t he "bat t le-of-t he-sexes with an outside option"

ITo see how the charac te rizatio n in Proposit ion 31 of weak seq uential rati onalizability is cons iste nt with (D, d) in r~ , let T I = {tJl with A tl = (( 1, 0) ,(0, 1)) (assigning probabi lit ies t o (d, t2) and (/, t2) resp ectively ), and T2 = {t 2} with At 2 = (( 1, 0, 0), (0, 1,0), (0, 0, 1)) (assigni ng pr obabiliti es to (D, til, (F D , til , and (F F, til resp ectively). Then , ind ep endently of how et l and et 2 are speci fied, (tl , t2) E CK([u] n [iwr]) , and , for each i , Pi is ind uce d for t i by t j , wh ere PI (D) = 1 and P2(d ) = 1.

Sequentiality

113

game, see Figure 2.6.2 EFR also leads to the backward induction outcome. However, unlike sequential rationalibility, EFR need not promote the backward induction procedure.

8.4

Relation to backward induction

The following result shows how sequential rationalizability implies the backward induction procedure in perfect information games. A finite extensive game I' , as introduced in Section 8.1, is of perfect information if, at any information set h E HI U H 2, h = {x} ; i.e., h contains only one node. It is generic if, for each i , Vi (z) =1= Vi (z') whenever z and z' are different outcomes. A generic extensive game of perfect information has a unique subgame-perfect equilibrium in pure strategies. Moreover, in such games the backward induction procedure yields in any subgame the unique subgame-perfect equilibrium outcome. 33 Consider a finite generic extensive two-player game of perfect information f. If there exists an epistemic model with (t1 ' t2) E CK([u]n[isr]) and, for eachi, a, is uuluced jor i; bytj , then o = ((J1,(J2) is the subgame-perfect equilibrium. PROPOSITION

Proof. In a perfect information game, the action a E A(h) taken at the information set h determines the immediate succeeding information set , which can thus be denoted (h, a). Also, any information set h E HI U H2 determines a subgame. Set H- 1 = Z (Le. the set of terminal nodes) and determine H9 for 9 ~ 0 by induction: h E H9 if and only if h satisfies max{g' 13h'

E

H9' and a

E

A(h) such that h' = (h,an = 9 -1.

In words, h E H9 if and only if 9 is the maximal number of decision nodes between h and a terminal node in the subgame determined by h. If (J is a profile of behavior strategies and h E HI U H2, denote by (Jlh the strategy profile with the following properties: (1) at information 2By strengthening permissibility, Asheim and Dufwenberg (2003a) define a rationalizability concept, fully permissible sets , which is d ifferent from those of Table 2.2 as well as EFR, as it yields forward induction, but does not alway s promote backward induction. This concept will be pres ented in Ch apters 11 a nd 12. Different kinds of strengthening of permissibility will be considered in Chapters 9 and 10, leading to the concepts of quasi-perfectly rationalizable and properly rationalizable strategies . These con cepts, wh ich are included in Table 2.2, imply backward induction, without promoting forward induction. It is of int erest to note, as we will do subsequent to Proposition 42, that the episte mic condit ions for proper rationalizability are incompatible with the epist emic condit ions for full permissibility.

114

CONSISTENT PREFERENCES

sets preceding h, (Jlh determines with probability one the unique action leading to h, and (2) at all other information sets, (Jlh coincides with (J . Say that (J' is outcome-equivalent to (J" if (J' and (J" induce the same probability distribution over terminal nodes. In view of properties of the certain belief operator (cf. Proposition 20 of Chapter 5), it is sufficient to show for any 9 = 0, . . . , max{g' I Hg' i= 0} that ifthere exists an epistemic model with (iI, t2) E K9([u] n [isr]) and , for each i , (Ji is induced for ti by tj , then, Vh E H9, (Jlh is outcomeequivalent to (J* Ih' where (J* = ((Ji, (J2) denotes the subgame-perfect equilibrium. This is established by induction. (g = 0) Let (tl , t2) E KO([u] n [i sr]) = [u] n [isr] and, for each i, (Ji be induced for ti by tj' Let h E HO and assume w.l.o.g. that h E Hi. Since (tl , t2) E lUi] n [isrj] and j takes no action at h, (Jlh is outcome equivalent to (J* Ih. (g = 1, , max{g' I H9' i= 0}) Suppose that it has been established , g -1 that if there exists an epistemic model with (tl , t2) E for g' = 0, Kg'([u] n [isr]) and , for each i , a, is induced for ti by tj , then, Vh' E H9', (Jlh' is outcome-equivalent to (J*lh" Let (tl' t2) E K9([u] n [isr]) and, for each i , a, be induced for ti by tj' Let h E H9 and assume w.l.o.g. that ti e Hi. Since (tl , t2) E K iK9- I [isr ], it follows from the premise of the inductive step that ti's SCLP ()..ti ,fti) satisfies , Vtj E TJi , Vh' E H j succeeding h, and Va' E A(h') , /lti (S Âˇ(h' a') t')

re

J

,

' j

J.L~i(Sj(h') , tj)

= (J~(h')(a') J

'

where f is the first level f of ).. ti for which J.L~i (Sj (h') , tj) > O. Since r is generic , (Ji is sequentially rational for ti only if a, (h) = (J; (h). Since (tl, t2) E lUi] n [isrj] and j takes no action at h, it follows from the premise that (Jlh is outcome-equivalent to (J* Ih . â€˘ Since sequentially rationalizable strategies always exist , there is an epistemic model with (tl, t2) E CK([u] n [isr]), implying that the result of Proposition 33 is not empty.

Chapter 9

QUASI-PERFECTNESS

In Chapter 5 we saw how the characterizations of Nash equilibrium and rationalizability lead to characterizations of (strategic form) perfect equilibrium and permissibility by adding preference for cautious behavior. In this chapter we show that the characterization of sequential equilibrium leads to a characterization of quasi-perfect equilibrium by adding caution. The concept of a quasi-perfect equilibrium, proposed by van Damme (1984), differs from Selten's (1975) extensive form perfect equilibrium by the property that, at each information set, the player taking an action ignores the possibility of his own future mistakes. So, paralleling Chapter 8, we define quasi-perfect rationalizability by imposing common certain belief of the event that each player has preference for cautious behavior (i.e., at every information set, one strategy is preferred to another if the former weakly dominates the latter) and believes that the opponent chooses rationally at all her information sets. Moreover, by assuming that each player is certain of the beliefs that the opponent has about the player's own action choice, we obtain an epistemic characterization of the corresponding equilibrium concept: quasi-perfect equilibrium. Since quasi-perfect rationalizability refines sequential rationalizability, it follows from Proposition 33 that also the former concept yields the backward induction procedure. By embedding the notion of an SCLP in an epistemic model with a set of epistemic types for each player, we are able to model quasi-perfectness as a special case of sequentiality. For each type t i of any player i, ti is described by an SCLP, which under the event that "player i believes that the opponent j chooses rationally at each information set" induces ,

116

CONSISTENT PREFERENCES

for each opponent type tj that is deemed subjectively possible by t.; a behavior strategy which is sequentially rational given tj 'S own SCLP. An SCLP ensures well-defined conditional beliefs representing nontrivial conditional preferences, while allowing for flexibility w.r.t. whether to assume preference for cautious behavior. Preference for cautious behavior, as needed for quasi-perfect rationalizability, is obtained by imposing the following additional requirement on ti's SCLP for each conditioning event: If an opponent strategy-type pair (Sj, tj) is compatible with the event and tj is deemed subjectively possible by ti, then (Sj, tj) is in the support the LPS that represents type ti's conditional preferences. This chapter's definition of quasi-perfect rationalizability was proposed by Asheim and Perea (2005).

9.1

Quasi-perfect consistency

In this section, we add preference for cautious behavior to the analysis of Chapter 8. This enables us to • characterize quasi-perfect equilibrium (van Damme, 1984), and • define quasi-perfect rationalizability as a non-equilibrium analog to the concept of van Damme (1984). The epistemic modeling is identical to the one given in Section 8.1; hence, this will not be recapitulated here. Caution. Under Assumption 1 it follows from Proposition 5 that, for each type ti of any player i, i's system of conditional preferences at ti can be represented by a vNM utility function ~(Z) ~ JR and an SCLP (>.ti, gt i) on Sj x Tj with support Sj x Tli. Recall from Section 5.3 that caution imposes the additional requirement that for each type ti of any player i the full LPS Ati is used to form the conditional beliefs over opponent strategy-type pairs. Formally, if L denotes the number of levels in the LPS At i , then

vii:

[caui] = {(tl,t2)

E

T l x T21 gti(Sj x Tj) = L} .

Since gti is non-increasing w.r.t . set inclusion, ti E proj-, [caui] implies that gti (projsj XT/P) = L for all subsets ¢J of {td x Sj x T, with welldefined conditional beliefs. Since it follows from Assumption 1 that At ; has full support on Sj, ti E projT; [cau,d means that i's choice function at t ; never admits a weakly dominated strategy, thereby inducing preference for cautious behavior. As before, write [caul := [caUl] n [cau2].

Quasi-perfectness

117

Say that at ti player i's preferences over his strategies are quasiperfectly consistent with the game r and the preferences of his opponent, if ti E proj-, ([Ui] n [isri] n [caui]) ' Refer to [u] n [isr] n [caul as the event of quasi-perfect consistency. Characterizing quasi-perfect equilibrium. We now characterize the concept of a quasi-perfect equilibrium as profiles of induced behavior strategies at a type profile in [u] n [isr] n [caul where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible) . To state the definition of quasi-perfect equilibrium, we need some preliminary definitions. Define the concepts of a behavior representation of a mixed strategy and the mixed representation of a behavior strategy in the standard way, cf., e.g., p. 159 of Myerson (1991). If a behavior strategy aj and a mixed strategy Pj are both completely mixed, and aj is a behavior representation of Pi or Pj is the mixed representation of aj , then, \fh E H j, \fa E A(h),

If a; is any behavior strategy for i and aj is a completely mixed behavior strategy for i , then abuse notation slightly by writing, for each h E Hi,

where Pi is outcome-equivalent to ai!h and Pj is the mixed representation of aj. DEFINITION 17 A behavior strategy profile a = (al,0"2) is a quasiperfect equilibrium if there is a sequence (o'(n) )nEN of completely mixed behavior strategy profiles converging to a such that for each i and every n E Nand h E Hi,

ui (a., aj ( n)) Ih = max Ui (a~ , a j ( n )) Ih

a;

.

The characterization result can now be stated; it is proven in Appendix B. PROPOSITION 34 Consider a finite extensive two-player game r. A profile of behavior strategies a = (aI, (2) is a quasi-perfect equilibrium if and only if there exists an epistemic model with (tl,t2) E [u] n [isr] n [caul such that (1) there is mutual certain belief of {(tl, t2)} at (tl, t2), and (2) for each i, ai is induced for ti by tj'

118

CONSISTENT PREFERENCES

As for Proposition 31, higher order certain belief plays no role in this characterization.

Defining quasi-perfect rationalizability. We next define the concept of quasi-perfectly rationalizable behavior strategies as induced behavior strategies under common certain belief of [u] n [isr] n [caul . 18 A behavior strategy a, for i is quasi-perfectly rationalizable in a finite extensive two-player game r if there exists an epistemic model with (tl' t2) E CK([u] n [isr] n [caul) such that (Ti is induced for DEFINITION

ti by tj'

It follows from Proposition 34 that a behavior strategy is quasi-perfectly rationalizable if it is part of a quasi-perfect equilibrium. Since a quasiperfect equilibrium always exists, we obtain as an immediate consequence that quasi-perfectly rationalizable behavior strategies always exist . Propositions 30 and 34 imply the well-known result that every quasiperfect equilibrium can be extended to a sequential equilibrium, while Definitions 15 and 18 imply that the set of quasi-perfectly rationalizable strategies is included in the set of sequentially rationalizable strategies. To illustrate that this inclusion can be strict, consider I'4 of Figure 3.1. Both concepts predict that player 2 plays d with probability one. However , only quasi-perfect rationalizability predicts that player 1 plays D with probability one. Preferring D to F amounts to preference for cautious behavior since by choosing D player 1 avoids the risk that player 2 may choose f . Since quasi-perfect rationalizability is thus a refinement of sequential rationalizability, it follows from Proposition 33 that quasi-perfect rationalizability implies the backward induction procedure in perfect information games.

9.2

Relating rationalizability concepts

The following result helps establishing some of the remaining relationships between the rationalizability concepts of Table 2.2. PROPOSITION

35 For any epistemic model and for each player i ,

To prove Proposition 35 we need the following lemma.

119

Quasi-perfectness

projTiKi[caUj], then, for each tj E T/ i and any h E H j , Sj E Sj( h)\S/j(h) implies that there exists sj E Sj(h) such that sj '(-t j Sj'

L EM M A

13 If ti

E

Proof. As for Lemma 11 the proof of t his lemma is based on t he concept of a st rategically independent set due to Mailath et al. (1993). It follows from Mailath et al. (Definitions 2 and 3 and t he 'if' part of Thm. 1) t hat S(h) is st rategically indepen dent for j at any player j informat ion set h in a finit e exte nsive game , and t his does not depend on the vNM ut ility functi on t hat assigns payoff to any outcome. If ti E projTiKi[cauj], then t he following holds for each tj E T/ i: Pl ayer j's system of condit ional preferences at tj satisfies Axiom 6 (Conditi onality). Suppose Sj E Sj( h)\S/j(h). Then there exists sj E Sj(h) such t hat sj '(- ~ Sj' As noted above, S(h) is a st rategically independent set for j . Hence, sj can be chosen such t hat z(sj, s.) = z(Sj , Si) for all S i E Si\Si( h). By Axiom 6 (Conditio nality), t his implies sj '(-t j Sj. â€˘ Proof of Proposition 35. Consider any epistemic model wit h

Suppose ti ~ projTJiwri]; i.e., the re exist tj E T/i and h E Hj such t hat p/iltj(sj) > 0 for some Sj E Sj(h)\S/j( h). Since ti E projTiKi[caUj], it follows from Lemma 13 that :Jsj E Sj(h) s.t . sj '(-t j Sj' Hence,

p/iltj ~ t::. (S] j ) , cont rad ict ing ti E projTJiri ]. This shows t hat ti E projTJiwri!- â€˘ Since [iri] n Kdcauj ] ~ [iwri], t he cell in Table 2.2 to t he left of 'permissibility' is not applicable, and permissibility refines weak sequent ial rationalizability. Figure 3.1 shows t hat t he inclusion can be st rict: Permissibility, but not weak sequent ial rati onalizability, precludes that player 1 plays F in r 4 . Since [isri] ~ [iri], Definiti on 18 and Propositi on 24 entail that quasiperfect rationalizability refines permissibility. That t he latt er inclusion can be strict is illust rated by r~ of Figure 8.2. Since t his is a generic extensive game, imposing preference for cautio us behavior has no bite, and the difference between permissibility and quasi-perfect rat ionalizability corresponds to the difference between weak sequent ial rati onalizabil ity and sequential rationalizability, as discussed in Section 8.3.

Chapter 10

PROPERNESS

Most contributions on the relation between common knowledge/belief of rationality and backward induction in perfect information games perform the analysis in the extensive form of the game. Indeed, the analyses in Chapters 7 and 8 of this book are examples of this. An exception to this rule is Schuhmacher (1999) who-based on Myerson 's (1978) concept of a proper equilibrium, but without making equilibrium assumptions-defines the concept of proper rationalizability in the strategic form and shows that proper rationalizable play leads to backward induction. Schuhmacher defines the concept of z-proper rationalizability by assuming that players make mistakes , but where more costly mistakes are made with a much smaller probability than less costly ones. A properly rationalizable strategy can then be defined as the limit of a sequence of e-properly rationalizable strategies as E goes to zero. For a given E, Schuhmacher offers an epistemic foundation for e-proper rationalizability. However, this does not provide an epistemic foundation for the limiting concept, i.e. proper rationalizability. It is one purpose of the present chapter, which reproduces Asheim (2001), to establish how proper rationalizability can be given an epistemic characterization in strategic two-player games, within an epistemic model where preferences are represented by a vNM utility function and an SCLP (i.e., an epistemic model satisfying Assumption 1 of Chapter 5). Blume et al. (1991b) characterize proper equilibrium as a property of preferences . When doing so they represent a player's preferences

122

CONSISTENT PREFERENCES

by a vNM utility function and an LPS, whereby the player may deem one opponent strategy to be infinitely more likely than another while still taking the latter strategy into account. In two-player games, their characterization of proper equilibrium can be described by the following two properties. 1 Each player is certain of the preferences of his opponent , 2 Each player 's preferences satisfies that the player takes all opponent strategies into account ('caution') and that the player deems one opponent strategy to be infinitely more likely than another if the opponent prefers the one to the other ('respect of opponent preferences'). The present characterization of proper rationalizability in two-player games drops property 1, which is an equilibrium assumption; instead it will be assumed that there is common certain belief of property 2, which will be referred to as proper consistency. Since, in the present framework , a player is not certain of the preferences of his opponent, player i's preferences must be defined on acts from Sj x Tj, where Sj denotes the set of opponent strategies and Tj denotes the set of opponent types. Under Assumption 1, each type of player i corresponds to a vNM utility function and an SCLP on Sj x Tj. As before, a player i has preference for cautious behavior at ti if he takes into account all strategies of any opponent type that is deemed subjectively possible . Moreover, a player i is said to respect opponent preferences at ti if, for any opponent type that is deemed subjectively possible, he deems one strategy of the opponent type to be infinitely more likely than another if the opponent type prefers the one to the other. At ti player i's preferences are said to be properly consistent with the game and the preferences of his opponent if at ti i both has preference for cautious behavior and respects opponent preferences. Hence, the present analysis follows the 'consistent preferences ' approach by imposing requirements on the preferences of players rather than their choice. In this chapter it is first shown (in Proposition 36) how the event of proper consistency combined with mutual certain belief of the type profile can be used to characterize the concept of proper equilibrium. It is then established (in Proposition 37) that common certain belief of proper consistency corresponds to Schuhmacher's (1999) concept of proper rationalizability. Furthermore, by relating 'respect of preferences' to 'inducement of sequential rationality' in Proposition 38, it follows by comparing Proposition 37 with Proposition 33 of Chapter 8 that only strategies leading to the backward induction outcome are properly ra-

123

Properness

c r 1, 1 1, 1 U 1, M 1, 1 2, 2 2, 2 D 0, 1 2, 2 3, 3

Â°

Figure 10.1. G7, illustrating common certain belief of proper consistency.

tionalizable in the strategic form of a generic perfect information game . Thus, Schuhmacher's Theorem 2 (which shows that the backward induction outcome obtains with "high" probability for any given "small" E) is strengthened, and an epistemic foundation for the backward induction procedure, as an alternative to Aumann's (1995) and others, is provided. Lastly, it is illustrated through an example how proper rationalizability can be used to test the robustness of inductive procedures.

10.1

An illustration

The symmetric game of Figure 10.1 is an example where common certain belief of proper consistency is sufficient to determine completely each player 's preferences over his or her own strategies. The game is due to Blume et al. (1991b, Figure 1). In this game, caution implies that player 1 prefers M to U since M weakly dominates U. Likewise, player 2 prefers c to e. Since 1 respects the preferences of 2 and, in addition, certainly believes that 2 has preference for cautious behavior, it follows that 1 deems c infinitely more likely than e. This in turn implies that 1 prefers D to U. Likewise, since 2 respects the preferences of 1 and, in addition, certainly believes that 1 has preference for cautious behavior, it follows that 2 prefers r to e. As a consequence, since 1 respects the preferences of 2, certainly believes that 2 respects the preferences of 1, and certainly believes that 2 certainly believes that 1 has preference for cautious behavior, it follows that 1 deems r infinitely more likely than e. Consequently, 1 prefers D to M . A symmetric reasoning entails that 2 prefers r to c. Hence, if there is common certain belief of proper consistency, it follows that the players' preferences over their own strategies are given by 1's preferences: D >- M 2's preferences: r >- c >-

>- U

e.

The facts that D is the unique most preferred strategy for 1 and r is the unique most preferred strategy for 2 mean that only D and r are properly

124

CONSISTENT PREFERENCES

rationalizable; cf. Proposition 37 of Section 10.2. By Proposition 36 of the same section , it then follows that the pure strategy profile (D, r) is the unique proper equilibrium, which can easily be checked. However, note that in the argument above, each player obtains certainty about the preferences of his opponent through deductive reasoning; i.e. such certainty is not assumed as in the concept of proper equilibrium. The concept of proper rationalizability yields a strict refinement of (ordinary) rationalizability (cf. Definition 11 of Chapter 5). All strategies for both players are rationalizable, which is implied by the fact that, in addition to (D,r), the pure strategy profiles (U,£) and (M,c) are also Nash equilibria. The concept of proper rationalizability yields even a strict refinement when compared permissibility (cf. Definition 13 of Chapter 5), corresponding to the Dekel-Fudenberg procedure, where one round of weak elimination followed by iterated strong elimination. When the Dekel-Fudenberg procedure is employed, only U is eliminated for 1, and only £ is eliminated for 2, reflecting that also the pure strategy profile (M, c) is a strategic form perfect equilibrium. It is a general result that proper rationalizability refines the Dekel-Fudenberg procedure; this follows from Section 10.3 as well as Theorem 4 of Herings and Vannetelbosch (1999).

10.2

Proper consistency

In this section, we add respect for opponent preferences to the analysis of Chapter 5. This enables us to characterize • proper equilibrium (Myerson, 1978), and • proper rationalizability (Schuhmacher, 1999). The epistemic modeling is identical to the one given in Section 5.1; hence, this will not be recapitulated here.

Respect of opponent preferences. Player i respects the preferences of his opponent at ti if the following holds for any opponent type that is deemed subjectively possible: Player i deems one opponent strategy of the opponent type to be infinitely more likely than another if the opponent type prefers the one to the other. To capture this, define the event

[resPi]

:=

{(tI' t2) E T I x T2 I(Sj , tj) »t i (sj, tj) whenever t j,EtT· i ' and

Sj ~ t'j Sj'} ,

where the notation » ti means "infinitely more likely at ti", as defined in Section 3.2.

125

Properness

Write [resp] := [resPl] n [resP2]' Say that at ti player i 's preferences over his strategies are properly consistent with the game G = (5 l,52 , U l , U2 ) and the preferences of his opponent, if ti E projTi ([Ui] n [resPi] n [caui])' Refer to [u] n [resp] n [caul as the event of proper consistency.

Characterizing proper equilibrium. We now characterize the concept of a proper equilibrium as profiles of induced mixed strategies at a type profile in [u] n [resp] n [caul where there is mutual certain belief of the type profile (Le., for each player, only the true opponent type is deemed subjectively possible) . Before doing so, we define a proper equilibrium. DEFINITION 19 Let G = (51,52 , Ul, U2) be a finite strategic two-player game . A completely mixed strategy profile p = (Pl,P2) is a e-proper equilibrium if, for each i,

cPi(Si) ~ Pi(S~) whenever Ui(Si,Pj) > Ui(S~,Pj). A mixed strategy profile p = (PI, P2) is a proper equilibrium if there is a sequence (p(n))nEN of c(n)-proper equilibria converging to p, where c(n) - t 0 as n - t 00. The characterization result-which is a variant of Proposition 5 of Blume et al. (1991b)-can now be stated. For this result , recall from Sections 5.2 and 8.3 that the mixed strategy p/ i1tj is induced for tj by ti if tj E T/i and, for all Sj E 5j,

.ti1tj (S .) _ fJeti (Sj, t) j PJ J ti(S . t .)' fJe J ' J where ÂŁ. is the first level ÂŁ. of Ati for which fJ~i(5j, tj) > O. PROPOSITION 36 Consider a finite strategic two-player game G. A profile of mixed strategies P = (PI, P2) is a proper equilibrium if and only if there exists an epistemic model with (tl' t2) E [u] n [resp] n [caul such that (1) there is mutual certain belief of {(tl , t2)} at (tl ' t2), and (2) for each i, Pi is induced for ti by tj' The proof is contained in Appendix B. As for similar earlier results, higher order certain belief plays no role in this characterization.

Characterizing proper rationalizability. We now turn to the non-equilibrium analog to proper equilibrium, namely the concept of proper rationalizability; cf. Schuhmacher (1999). To define the concept of properly rationalizable strategies, we must introduce the following

126

CONSISTENT PREFERENCES

variant of an epistemic model, with a mixed strategy pl; being associated to each type ti of any player i, where pl; is completely mixed. DEFINITION 20 An Âť-epistemic model for the finite strategic two-player game form (Sl, S2, z) is a structure

(S1, T1, S2, T2) , where , for each type ti of any player i, ti corresponds to (1) mixed strategy pl;, where supppl; = Si, and (2) a system of conditional preferences on the collection of sets of acts from elements of

<I>t; := {<p ~ t:

X

s, x Tj I ",t; n <p i= 0}

to ~(Z), where ",t; is a non-empty subset of {td x Sj x T j. Moreover, Schuhmacher (1999) in effect makes the following assumption. ASSUMPTION 3 For each ti of any player i, (aj ~~ satisfies Axioms 1, 2, and 4 if 0 i= <p ~ Ti X Sj x T j, and Axiom 3 if and only if <P E <I>t;, (bj the system of conditional preferences {~~ I <P E <I>t;} satisfies Axioms 5 and 6, and (c) there exists a non-empty subset of opponent types, T/;, such that ",t; = {td x Sj x T/;. Under Assumption 3 it follows from Proposition 1 that, for each type ti of any player i, i's system of conditional preferences at ti can be represented by a vNM utility function ~(Z) - t lR and a subjective probability distribution /-It; which for expositional simplicity is defined on Sj x T j with support Sj X T/; (instead of being defined on T; X Sj X Tj with support ",t ; = {td x Sj x T/;). Hence, as before we consider w.l.o.g. i's unconditional preferences at ti , >=t;, to be preferences over acts from Sj x Tj to ~(Z) (instead of acts from {ti} x Sj x T, to ~(Z)) . The combination of ",t; having full support on Si and Axiom 6 (Conditionality) being satisfied means that all opponent strategies are taken into account for any opponent type that is deemed subjectively possible, something that is reflected by /-It; having full support on Sj . Hence, preference for cautious behavior need not be explicitly imposed. Rather, following Schuhmacher (1999) we consider the following events . First, define the set of type profiles for which ti, for any subjectively possible opponent type, induces that type's mixed strategy:

vf;:

[indi] := {(t l ' t2)

E TI x

T21 'tit}

E

T/;, p/;Itj = p/j} .

Write lind] := [ind1 ] n [ind2]. Furthermore, define the set of type profiles for which ti, according to his mixed strategy pli, plays a pure strategy

Properness

127

with much greater probability than another if player i at ti prefers the former to the latter: [c-prop tremi]:= {(t l,t2 ) ETI x

T21

cp t路ti(s路) t

> 路_ ,;-ti Sf} . _ p.ti(s t t路) whenever s t t

If ti E ProjTJC-prop tremi], then player i is said to satisfy the e-proper trembling condition at k Schuhmacher's (1999) definition of s-proper rationalizability can now be formally stated. DEFINITION 21 (SCHUHMACHER, 1999) A mixed strategy Pi for i is cproperly rationalizable in a finite strategic two-player game G if there n exists an *-epistemic model with pli = Pi for some ti E projrCK([u] , lind] n [c-prop treml). A mixed strategy Pi for i is properly rationalizable if there exists a sequence (pi(n) )nEN of c(n )-properly rationalizable strategies converging to Pi, where E(n) --+ 0 as n --+ 00. We next characterize the concept of properly rationalizable strategies as induced mixed strategies under common certain belief of [u] n [resp] n [caul. The result is proven in Appendix B. PROPOSITION 37 A mixed strategy Pi for i is properly rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with (tl ' t2) E CK([u] n [resp] n [caul) such that Pi is induced for ti by tj . It follows from Propositions 36 and 37 that any mixed strategy is properly rationalizable if it is part of a proper equilibrium. Since a proper equilibrium always exists, we obtain as an immediate consequence that properly rationalizable strategies always exist.

10.3

Relating rationalizability concepts (cont.)

As shown by van Damme (1984), any proper equilibrium in the strategic form corresponds to a quasi-perfect equilibrium in the extensive form. The following result shows, by Propositions 34 and 36, this relationship between the equilibrium concepts and establishes, by Definition 18 and Proposition 37, the corresponding relationship between the rationalizability concepts. Furthermore, it means that the two cells in Table 2.2 to the left of 'proper rationalizability' are not applicable. PROPOSITION 38 For any epistemic model and for each player i,

[resPi]

n Ki[cauj] ~

[isri] .

128

CONSISTENT PREFEREN CES

Proof. Consider any epistemic model wit h

Suppose ti ~ projTi[isri] ; i.e. , t here exist tj E T/i and h E Hj such that (J/ iltj lh is out come equivalent to Pj , where Pj (Sj ) > 0 for some Sj E Sj( h)\S/ j (h). Since ti E projTiKdcauj], it follows from Lemma 13 t hat :lsj E Sj( h) s.t . sj ,;-tj Sj . Since ti E projTi [resPi], t his means t hat :lsj E Sj( h) s.t . (sj , tj) » ti (Sj ,tj) . Furtherm ore, Pj(Sj) > 0 impli es l1~i ( Sj , tj) > 0, where is t he first level of Ati for which l1~i ( Sj (h ) , tj) > O. Since t hen is also t he first level of Ati for which l1~i ({Sj , sj} , tj) > 0, this contra dicts (sj, tj ) » ti (Sj,tj ) and shows t hat ti E projTi [isri ]. •

e

e

e

e

Since pr oper rationalizability is thus a refinement of qu asi-p erfect rat ionalizability, which in t urn is a refinement of sequential rati onalizability, it follows from P rop ositi on 33 t hat pr op er rati onalizabili ty impli es t he backward inducti on pr ocedure in perfect informati on games . E.g., in t he "cent ipede" game illustrat ed in r~ of Figure 2.4, common certain belief of pr oper consiste ncy implies t hat t he players' pr eferences over t heir own st ra tegies are given by 1's pr eferences: Out ';- lnL ';- OutR 2's pr eferences: r.

e';-

This pr op erty of pr oper rationalizabili ty has been discussed by both Schuhmacher (1999) and Asheim (2001). From t he pr oof of Propositi on 1 in Mailath et al. (1997) one can conjecture t hat quas i-perfect rati onalizabili ty in every extensive form corresponding to a given st rategic game coincides wit h pr oper rati onalizabili ty in t ha t game. However , for any given extensive for m t he set of proper rationalizabl e strategies can be a strict subset of t he set of qu asi-p erfect rationalizable strat egies, as illustrated by r~ of Figure 2.5. Here, quasi-p erfect rationalizabili ty only pr ecludes t he play of lnR with positi ve probability. However , since lnL st rongly dominates lnR, it follows t hat 2 prefers to r if she respects 1's preferences. Hence, only with probability one is pr operly rationalizab le for 2, which implies t hat only lnL wit h probabil ity one is pr operly rat ionalizabl e for 1. As for t he concepts of sequent ial and quasi-p erfect rat ionalizabili ty, we do not pr ovide an algorit hm leading to the set of properly rati onalizabl e st rategies (alt hough part 1 of t he proof of Prop osition 37 is based on such an algorithm ). The analyses of Schulte (2003) and Perea (2004) indicate how the concept of proper rati onalizability can be cha racterized

e

e

129

Properness abc

Playerl~

Player2~ 1/ 3

Figure 10.2.

1/ 3

1/ 3

A betting game.

by means of algorithms. In their papers, they define algorithms leading to notions that are closely related to proper rationalizability.

10.4

Induction in a betting game

The games G7 (of Figure 10.1), r~ (of Figure 2.4), and r~ (of Figure 2.5) have in common that the properly rationalizable strategies coincide with those surviving iterated (maximal) elimination of weakly dominated strategies (IEWDS). In the present section it will be shown that this conclusion does not hold in general. Rather, the concept of proper rationalizability can be used to test the robustness of IEWDS and other inductive procedures. Figure 10.2 illustrates a simplified version of a betting game introduced by Sonsino et al. (2000) for the purpose of experimental study; Sevik (2001) has subsequently repeated their experiment in alternative designs. The two players consider to bet and have a common and uniform prior over the states that determine the outcome of the bet. If the state is a, then 1 looses 9 and 2 wins 9 if betting occurs. If the state is b, then 1 wins 6 and 2 looses 6 if betting occurs. Finally, if the state is c, then 1 looses 3 and 2 wins 3 if betting occurs. Player 1 is informed of whether the state of the bet is equal to a or in the set {b, c}. Player 2 is informed of whether the state of the bet is in the set {a, b} or equal to c. As a function of their information, each player can announce to accept the bet or not. For player 1 the strategy YN means to accept the bet if informed of a and not to accept the bet if informed of {b, c}, etc. For player 2 the strategy yn means to accept the bet if informed of {a, b} and not to accept the bet if informed of c, etc. Betting occurs if and only if both players have accepted the bet. This yields the strategic game of Figure 10.3.

An inductive procedure. If player 2 naively believes that player 1 is equally likely to accept the bet when informed of a as when informed of {b, c}, then 2 will wish to accept the bet when informed of {a, b}.

130

CONSISTENT PREFERENCES

yy yn ny -2,2 -1, 1 -1, 1 YY YN -3,3 -3,3 0, 0 NY 1, -1 2, -2 -1, 1

nn

0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 NN Figure 10.3. The strategic form of th e betting gam e.

However, the following, seemingly intuitive, inductive procedure appears to indicate that 2 should never accept the bet if informed of {a, b}: Player 1 should not accept the bet when informed of a since he cannot win by doing so. This eliminates his strategies YYand YN. Player 2, realizing this, should never accept the bet when informed of {a, b}, since-as long as 1 never accepts the bet when informed of a-she cannot win by doing so. This eliminates her strategies yyand yn. This in turn means that player 1, realizing this, should never accept the bet when informed of {b, c}, since-as long as 2 never accepts the bet when informed of {a, b}-he cannot win by doing so. This eliminates his strategy NY. This inductive argument corresponds to IEWDS , except that the latter procedure eliminates 2's strategies yn and nn in the first round. The argument seems to imply that player 2 should never accept the bet if informed of {a, b} and that player 1 should never accept the bet if informed of {b, c}. Is this a robust conclusion?

Proper rationalizability in the betting game. The strategic game of Figure 10.3 has a set of Nash equilibria that includes the pure strategy profiles (NN, ny) and (NN, nn) , and a set of (strategic form) perfect equilibria that includes the pure strategy profile (N N , ny). However , there is a unique proper equilibrium where player 1 plays NN with probability one, and where player 2 mixes between yy with probability 1/5 and ny with probability 4/5. It is instructive to see why the pure strategy profile (NN, ny) is not a proper equilibrium. If 1 assigns probability one to 2 playing ny, then he prefers YN to NY (since the more serious mistake to avoid is to accept the bet when being informed of {b, c}). However, if 2 respects l 's preferences and cert ainly believes that 1 prefers YN to NY, then she will herself prefer yy to ny, und ermining (NN, ny) as a proper equilibrium. The mixture between yy and ny in the proper equilibrium is const ruct ed so that 1 is indifferent between YN and NY.

131

Propern ess

Table 10.1. An epistemic mod el for the betting game.

t~

(0, 0, 0, 0) (0, 0,0, 0) (0,0,0,0) (0, 0, 0, 0)

t'1 YY (0,0,0,0) YN (0,0,0,0) NY (0,0,0,0) NN (0,0,0 ,0)

t"1 (0,0,1,0) (0,0,0,1) (1 ,0,0 ,0) (0,1 ,0,0)

yy yn ny nn

t'2

t~

t'2 (0,0, 1, 0) (0,0,0, 1) (1,0,0, 0) (0, 1, 0,0)

til1 yy yn ny nn

til2 YY

YN NY NN

t'2 (0,0,0, 0) (0, 0, 0,0) (0, 0, 0, 0) (0,0, 0, 0)

til 2 (1,0,0, 0) (0, 1,0, 0) (0, 0, 1, 0) (0,0, 0, 1)

t'1 (0,0,0, 1) (0,1,0,0) (0,0 ,1,0) (1,0,0, 0)

til1 (0,0,0,0) (0,0,0,0) (0,0,0,0) (0,0,0,0)

Since any mixed st rategy is properly rationalizable if it is part of a proper equilibrium, it follows t hat both yy and yn are properly rationalizable pure strategies for 2. Moreover , if 1 certainly believes t hat 2 is of a ty pe with only yy as a most preferr ed st rategy, t hen NY is a most preferr ed st rategy for 1, implying t hat N Y in addit ion to NN is a properly rationalizable st rategy for 1. That t hese st rategies are in fact properly rati onalizable is verified by t he epistemic model of Table 10.1. In t he table the preferences of any player i at each type ti are represent ed by a vNM utility function V;i satisfying V;i 0 Z = Ui and a 4-level LPS on 5j x {tj, tj}, with t he first numb ers in t he parantheses expressing prim ary probability distribut ions, t he second numb ers expressing seconda ry probability distributions, etc. It can be checked t hat {t~ , x {t~ , t~} ~ [u] n [resp] n [caul, which in turn implies {t~ , x {t~ , tD ~ CK([u] n [resp] n [caul) since, for each ti E t: of any player i , Tl i ~ {tj ,t'j}. Since each typ e's preferences over his/her own st rategies are given by

tn

tn

NN NY ny

~.t; Y N ~.t; NY ~t~ YY >--t~ NN ~t~ YY ~t~ Y N ~t~ nn ~t~ yy ~t~ yn

~ ~~ ~ ~~ ~ ~~ ~,

it follows t hat NY and NN are prop erly rationalizable for player 1 and yy and ny are properly rationalizable for player 2. Note t hat YY and YN for player 1 and yn and nn for player 2 cannot be prop erly rat ionalizable

132

CONSISTENT PREFERENCES

since these strategies are weakly dominated and , thus, cannot be most preferred strategies for cautious players . The lesson to be learned from this analysis is that is not obvious that deductive reasoning should lead players to refrain from accepting the bet in the betting game. The experiments by Sonsino et al. (2000) and Sevik (2001) show that some subjects do in fact accept the bet in a slightly more complicated version of this game. By comparison to Propositions 33 and 38, the analysis can be used to support the argument that backward induction in generic perfect information games is more convincing than the inductive procedure for the betting game discussed above .

Chapter 11

CAPTURING FORWARD INDUCTION THROUGH FULL PERMISSIBILITY

The procedure of iterated (maximal) elimination of weakly dominated strategies (IEWDS) has a long history and some intuitive appeal, yet it is not as easy to interpret as iterated elimination of strongly dominated strategies (IESDS). IESDS is known to be equivalent to common belief of rational choice; cf. Tan and Werlang (1988) as well as Propositions 22 and 26 of this book. IEWDS would appear simply to add a requirement of admissibility, i.e., that one strategy should be preferred to another if the former weakly dominates the latter on a set of strategies that the opponent "may choose" . However, numerous authors-in particular, Samuelson (1992)-have noted that it is not clear that we can interpret IEWDS this way. To see this, consider the following two examples . The left-hand side of Figure 2.6 shows G~ , the pure strategy reduced strategic form of the "bat t le-of-t he-sexes with an outside option" game. Here IEWDS works by eliminating InR, r, and Out, leading to the forward induction outcome (InL , e). This prediction appears consistent: if 2 believes that 1 will choose InL, then she will prefer ÂŁ to r as 2's preference over her strategies depends only on the relative likelihood of InL and InR. The situation is different in Gs of Figure 11.1, where IEWDS works by eliminating D , r , and M , leading to (U, ÂŁ). Since 2 is indifferent at the predicted outcome, we must here appeal to admissibility on a superset of {U} , namely {U, M} , to justify the statement that 2 must play L. However, it is not clear that this is reasonable. Admissibility on {U, M} means that 2's preferences respect weak dominance on this set and implies that M is deemed infinitely more likely than D (in the sense of Blume et al., 1991a, Definition 5.1; see also Chapter 3). However, why

134

CONSISTENT PREFERENCES

r 1, 1 1, 1 U M 0,1 2, 0 D 1,0 0, 1

Figure 11.1. G s , illustrating that IEWDS may be problematic.

should 2 deem M more likely than D? If 2 believes that 1 believes in the prediction that 2 plays £ (as IEWDS suggests), then it seems odd to assume that 2 believes that 1 considers D to be a less attractive choice than M. A sense in which D is "less rational" than M is simply that it was eliminated first. This hardly seems a justification for insisting on the belief that D is much less likely than M. Still , Stahl (1995) has shown that IEWDS effectively assumes this: a strategy survives IEWDS if and only if it is a best response to a belief where one strategy is infinitely less likely than another if the former is eliminated at an earlier round than the latter. Thus, IEWDS adds extraneous and hard-to-justify restrictions on beliefs, and may not appear to correspond to the most natural formalization of deductive reasoning under admissibility. So what does? Reproducing joint work with Martin Dufwenberg, cf. Asheim and Dufwenberg (2003a), this chapter presents the concept of 'fully permissible sets' as an answer . In G~ this concept agrees with the prediction of IEWDS, as seems natural. The procedure leading to this prediction is quite different, though, as is its interpretation. In G«, however, full permissibility predicts that 1's set of rational choices is either {U} or {U, M} , while 2's set ofrational choices is either {£} or {£, r}. This has interesting implications. If 2 is certain that 1's set is {U}, then-absent extraneous restrictions on beliefs-one cannot conclude that 2 prefers £ to r or vice versa . On the other hand, if 2 considers it possible that 1's set is {U, M} , then £ weakly dominates r on this set and justifies {£} as 2's set of rational choices. Similarly, one can justify that U is preferred to M if and only if 1 considers it impossible that 2's set is {£, r}. Thus, full permissibility tells a consistent story of deductive reasoning under admissibility, without adding extraneous restrictions on beliefs. This chapter is organized as follows . Section 11.1 illustrates the key features of the requirement-called 'full admissible consistency'-that is imposed on players to arrive at full permissibility. Section 11.2 formally defines the concept of fully permissible sets through an algorithm that

Capturing forward induction through full permissibility

135

eliminates strategy sets under full admissible consistency. General existence as well as other properties are shown. Section 11.3 establishes epistemic conditions for the concept of fully permissible sets, and checks that these conditions are indeed needed and thereby relates full permissibility to other concepts. Section 11.4 investigates examples, showing how forward induction is promoted and how multiple fully permissible sets may arise. Section 11.5 compares our epistemic conditions to those provided in related literature. As elsewhere in this book , the analysis will be limited to two-player games. In this chapter (and the next) , this is for ease of presentation, as everything can essentially be generalized to n-player games (with n > 2).

11.1

Illustrating the key features

Our modeling captures three key features : 1 Caution. A player should prefer one strategy to another if the former weakly dominates the latter. Such admissibility of a player's preferences on the set of all opponent strategies is defended, e.g., in Chapter 13 of Luce and Raiffa (1957) and is implicit in procedures that start out by eliminating all weakly dominated strategies. 2 Robust belief of opponent rationality. A player should deem any opponent strategy that is a rational choice infinitely more likely than any opponent strategy not having this property. This is equivalent to preferring one strategy to another if the former weakly dominates the latter on the set of rational choices for the opponent. Such admissibility of a player 's preferences on a particular subset of opponent strategies is an ingredient of the analyses of weak dominance by Samuelson (1992) and Borgers and Samuelson (1992), and is essentially satisfied by 'ext ensive form rationalizability' (EFR; cf. Pearce, 1984 and Battigalli, 1996a, 1997) and IEWDS. 3 No extran eous restrictions on beliefs . A player should prefer one strategy to another only if the former weakly dominates the latter on the set of all opponent strategies or on the set of rational choices for the opponent. Such equal treatment of opponent strategies that are all rational-or all irrational-have in principle been argued by Samuelson (1992, p. 311), Gul (1997), and Mariotti (1997).

These features are combined as follows. A player's preferences over his own strategies leads to a choice set (i.e., a set of maximal pure strategies; cf. Section 6.1). A player 's preferences is said to be fully

136

CONSISTENT PREFERENCES

r 1, 1 1, 1 U M 1, 1 1,0 D 1,0 0, 1

Figure 11.2. Gg, illustrating the key features of full admissible consist ency.

admissibly consistent with the game and the preferences of his opponent if one strategy is preferred to another if and only if the former weakly dominates the latter • on the set of all opponent strategies, or • on the union of the choice sets that are deemed possible for the opponent. A subset of strategies is a fully permissible set if and only if it can be a choice set when there is common certain belief of full admissible consistency. Hence, the analysis yields a solution concept that determines a collection of choice sets for each player. This collection can be found via a simple algorithm, introduced in the next section . We use G g of Fig. 11.2 to illustrate the consequences of imposing 'caut ion' and 'robust belief of opponent rationality'. Since 'caution' means that each player takes all opponent strategies into account, it follows that player l 's preferences over his strategies will be U rv M ?- D (where and ?- denote indifference and preference , respectively). Player 1 must prefer each of the strategies U and M to the strategy D , because the former strategies weakly dominate D. Hence, U and M are maximal, implying that l 's choice set is {U, M}. The requirement of 'robust belief in opponent rationality' comes into effect when considering the preferences of player 2. Suppose that 2 certainly believes that 1 is cautious and therefore (as indicated above) certainly believes that {U, M} is l's choice set . Our assumption that 2 has robust belief of l's rationality captures that 2 deems each element of {U, M} infinitely more likely than D. Thus, 2's preferences respect weak dominance on l's choice set {U, M}, regardless of what happens if 1 chooses D. Hence, 2's preferences over her strategies will be £ ?- r. Summing up, we get to the following solution for G g : I"V

l 's preferences : U M ?- D 2's preferences : t » r I"V

Capturing forward induction through full permissibility

137

Hence, {U, M} and {£'} are the players ' fully permissible sets . The third feature of full admissible consistency- 'no extraneous restrictions on beliefs'-means in G g that 2 does not assess the relative likelihood of 1's maximal strategies U and M. This does not have any bearing on the analysis of Gg , but is essential for capturing forward induction in G~ of Figure 2.6. In this case the issue is not whether a player assesses the relative likelihood of different maximal strategies, but rather whether a player assesses the relative likelihood of different non-maximal strategies. To see the significance in G~ , assume that 1 deems r infinitely more likely than £', while 2 deems Out infinitely more likely than InR and InR infinitely more likely than InL. Then the players rank their strategies as follows: 1's preferences: Out >- InR >- InL 2's preferences : r >- £' Both 'caution' and 'robust belief of opponent rationality' are satisfied and still the forward induction outcome (InL , £') is not promoted. However, the requirement of 'no extraneous restrictions on beliefs' is not satisfied since the preferences of 2 introduce extraneous restrictions on beliefs by deeming one of 1's non-maximal strategies, InR, infinitely more likely than another non-maximal strategy, InL. When we return to G~ in Sections 11.4 and 11.5, we show how the additional imposition of 'no extraneous restrictions on beliefs' leads to (InL , £') in this game. Several concepts with natural epistemic foundations fail to match these predictions in G~ and Gg . In the case of rationalizability-cf. Bernheim (1984) and Pearce (1984)-this is perhaps not so surprising since this concept in two-player games corresponds to IESDS. It can be understood as a consequence of common belief of rational choice without imposing caution, so there is no guarantee that a player prefers one strategy to another if the former weakly dominates the latter. In G g , for example , all strategies are rationalizable. It is more surprising that the concept of 'p ermissibility' does not match our solution of G«. Permissibility can be given rigorous epistemic foundations in models with cautious players-cf. Borgers (1994) and Brandenburger (1992), who coined the term 'permissible'; see also Ben-Porath (1997) and Gul (1997) as well as Propositions 24 and 27 of this book. In these models players take into account all opponent strategies, while assigning more weight to a subset of those deemed to

138

CONSISTENT PREFERENCES

be rational choices. As noted earlier , permissibility corresponds to the Dekel-Fudenberg procedure where one round of elimination of all weakly dominated strategies is followed by iterated elimination of strongly dominated strategies. In Gg, this means that 1 cannot choose his weakly dominated strategy D. However, while 2 prefers £ to r in our solution, permissibility allows that 2 chooses r , To exemplify using Brandenburger's (1992) approach, this will be the case if 2 deems U to be infinitely more likely than D which in turn is deemed infinitely more likely than M. The problem is that 'robust belief of opponent rationality' is not satisfied : Player 2 deems D more likely than M even though M is in 1's choice set, while D is not . In Section 11.3 we establish in Proposition 40 that the concept of fully permissible sets refines the Dekel-Fudenberg procedure.

11.2

IECFA and fully permissible sets

We present in this section an algorithm-'iterated elimination of choice sets under full admissible consistency' (IECFA) -leading to the concept of 'fully permissible sets'. This concept will in turn be given an epistemic characterization in Section 11.3 by imposing common certain belief of full admissible consistency. We present the algorithm before the epistemic characterization for different reasons: • IECFA is fairly accessible. By defining it early, we can apply it early, and offer early indications of the nature of the solution concept we wish to promote. • By defining IECFA, we point to a parallel to the concepts of rationalizable strategies and permissible strategies. These concepts are motivated by epistemic assumptions, but turn out to be identical in 2-player games to the set of strategies surviving simple algorithms: respectively, IESDS and the Dekel-Fudenberg procedure. • Just like IESDS and the Dekel-Fudenberg procedure, IECFA is easier to use than the corresponding epistemic characterizations. The algorithm should be handy for applied economists, independently of the foundational issues discussed in Section 11.3. IESDS and the Dekel-Fudenberg procedure iteratively eliminate dominated strategies. In the corresponding epistemic models, these strategies in turn cannot be rational choices, cannot be rational choices given that other players do not use strategies that cannot be rational choices, etc.

Capturing forward in duction through full permissibility

139

IECFA is also an elimination procedure. However , the interpret ation of the basic item t hrown out is not t hat of a st rategy that cannot be a rational choice, but rat her that of a set of strategies t hat cannot be a choice set for any preferences that are in a given sense consiste nt with t he preferences of the opponent. The specific kind of consiste ncy involved in IE CFA- which will be defined in Section 11.3 and referr ed to as 'full admissible consiste ncy'-requires t hat a player's preferences are characterized by the properties of 'caut ion' , 'robust belief of opponent rationality' and 'no extraneous restrictions on beliefs'. Thus, IEC FA does not start with each player's strategy set and t hen iteratively eliminat es st rategies. Rather , IE CFA starts with each player 's collection of non-empty sub sets of his st rategy set and t hen it eratively eliminates subsets from this collection.

Definition. Consider a finit e st rategic two-player game G = (81,82 , U1, U2), and recall t he following not ation from Chapter 6: For any (0 #) Yj ~

s;

Di (Yj ) := {s,

E

S, I :3Pi

E

.6.(8i ) such t hat Pi weakly dominates s, on Yj or 8 j }

.

Interp ret Yj as t he set of strategies t hat player i deems to be t he set of rational choices for his opponent. Let i 's choice set be equal to S, \ Di (Yj), entailing that i 's choice set consists of pure strategies that are not weakly dom inated by any mixed st rategy on Yj or 8j . In Section 11.3 we show how this corresponds to a set of maximal st rategies given t he player's preferences over his own st rategies. Let I: = I: 1 X I: 2 , where I: i := 2S i \ { 0} denotes t he collection of nonempty subsets of 8 i . Write a; (E I:d for a subset of pure st rategies. For any (0 #) 3 = 3 1 X 3 2 ~ I:, write 0(3) := 01( 3 2) x 02( 3 1), where

oi(3 j ) := {O"i

E

I: i I :3(0 #) iI!j ~

z, s.t. a; =

8i\ D i( Ua j EWj O"j )}.

Hence, 0i (3 j ) is t he collection of st rategy subsets t hat can be choice sets for player i if he associates Yj- the set of rational choices for his oppo nent-with t he union of t he strategy subsets in a non-empty subcollection of 3 j . We can now define t he concept of a fully permissible set. D EFI NITI ON 22 Let G = (8 1,82 , U 1 , U2) be a finite strategic two-player game . Consider the sequence defined by 3 (0) = I: and, Vg ~ 1, 3 (g) = 0(3(g - 1)). A non-empty st rategy set a; is said to be fully permissible if

140

CONSISTENT PREFERENCES a; E

n°o

9=0

3 i (g) .

Let II = III X II2 denote the collection of profiles of fully permissible sets. Since 0 i= Qi(3j) ~ Qi(3'j) ~ Qi(L;j) whenever 0 i= 3j ~ 3'1 ~ L;j and since the game is finite, 3(g) is a monotone sequence that converges to II in a finite number of iterations. IECFA is the procedure that in round 9 eliminates sets in 3(g - 1)\3(g) as possible choice sets. As defined in Definition 22, IECFA eliminates maximally in each round in the sense that, vs ~ 1, 3(g) = Q(3(g - 1)). However, it follows from the monotonicity of Qi that any non-maximal procedure, where 3g ~ 1 such that 3(g - 1) :::) 3(g) :::) Q(3(g - 1)), will also converge to II. A strategy subset survives elimination round 9 if it can be a choice set when the set of rational choices for his opponent is associated with the union of some (or all) of opponent sets that have survived the procedure up till round 9 - 1. A fully permissible set is a set that survives in this way for any g. The analysis of Section 11.3 justifies that strategy subsets that this algorithm has not eliminated by round 9 be interpreted as choice sets compatible with 9 - 1 order of mutual certain belief of full admissible consistency.

Applications. We illustrate IECFA by applying it . Consider Gg of Figure 11.2. We get: 3(0) = L;l X L;2 3(1) = {{U,M}} x L;2 II = 3(2) = {{U, M}} x {{ £} } .

Independently of Y2, 51 \D l (Y2) = {U, M}, so for 1 only {U, M} survives the first elimination round, while 52 \D2( {U, M}) = {z}, 52 \D2( {D}) = {r} and 52 \D 2 ( {U}) = {£, r}, so that no elimination is possible for player 2. However, in the second round only {e} survives since £ weakly dominates r on {U, M} , implying that 52 \D2( {U, M}) = {£}. Next , consider G~ of Figure 2.6. Applying IECFA we get: 3(0) = L;l X L;2 3(1) = {{Out}, {InL} , {Out,InL}} x L;2 3(2) = {{Out}, {InL} , {Out, InL}} x {{£}, {£, r}} 3(3) = {{InL}, {Out, InL}} x {{£},{e,r}} 3(4) = {{InL}, {Out, InL}} x {{£}} II = 3(5) = {{InL}} x {{e}}.

Captu ring forward induc ti on through full perm issibility

141

Again the algorithm yields a uniqu e fully permissible set for each player. Fin ally, apply IECFA to Gs of Figure 11.1: 3 (0) = ~1 X ~2 3 (1) = {{U}, {M} , {U, M}}

X ~2

3 (2) = {{U}, {M} , {U,M}} x {{f} ,{e,r}} II = 3 (3) = {{U}, {U, M}} x {{ f }, {e, r }} .

Here we are left with two fully permissible sets for each player. There is no further elimination, as {U} = 8 1\D 1 ({e}), {U,M} = 8 1\D 1 ({e,r }), {e} = 82\ D2 ( {U, M}) , and {e,r} = 82 \ D2 ( {U}). Th e elimination pro cess for G~ and Gs is explained and interpreted in Section 11.4. Results. The following proposition characterizes th e strategy subsets th at survive IECFA and thus are fully permi ssible, and is a straightforward implication of Definition 22 (keeping in mind that ~ is finite and, for each i, a i is monot one).

39 (i) For each i, IIi =1= 0. (ii) II = a(II). (iii) For each i, a, E IIi if and only if there exists 3 = 3 1 X 3 2 with a, E 3 i such that 3 ~ a(3) . PROPOSITION

Propositi on 39(i) shows existence, but not uniqu eness, of each player 's fully permissible set(s) . In addition t o G2, games with multiple strict Nash equilibria illustrat e t he possibility of such multiplicity; by Proposition 39(iii) any st rict Nash equilibrium corresponds to a profile of fully permissible sets . Propositi on 39(ii) means t hat II is a fixed point in t erms of a collect ion of profiles of st rategy sets as illustrated by G 2 above. By Proposition 39(iii) it is t he largest such fixed point . We close this section by recordin g some connections between IECFA on the one hand, and IESDS, the Dekel-Fudenb erg procedur e (i.e., permissibility) , and IEWDS on the other. First , we not e through the following Proposition 40 th at IECFA has more bit e t han the Dekel-Fudenberg procedur e. Both G 1 and G3 illustrat e that thi s refinement may be strict . PROPOSITION

permissible set

40 A pure strategy Si is permissible if there exists a fully O"i such that s, E a.,

Proof. Using Proposit ion 39(ii), t he definition s of a(Âˇ) (given above) and a(Âˇ) (given in Chapter 6) imply, for each i ,

142

CONSISTENT PREFERENCES

f UU 1, 1 UD 1, 1 DU 0, 1 DD 0, 0

e

r

1, 1 0, 1 0, 0 0, 1

0, 0 1,0 2, 0 0, 2

Figure 11.3. G IO , illust ratin g t he relat ion between IECFA and IEWD S.

Since p O ~ a(P O) implies each i , U Ui EIIiO"i ~ Pi. â€˘

p O

~

P , by Lemm a 10(iii), it follows t hat, for

It is a corollary that IECFA has also more cut t ing power than IESDS . However , neither of IECFA and IEWDS has more bite than the other , as demonstrated by t he game G lO of Fig. 11.3. It is st ra ight forward t o verify t hat UU and UD for player 1 and for player 2 surv ive IEWDS, while { UU} for 1 and {e, c} for 2 survive IE CFA and are t hus t he fully permissible sets , as shown below:

e

3 (0) = ~l X ~2 3 (1) = {{ UU} , {DU}, {UU, UD}, { UU, DU} , { UD, DU}, { UU, UD, DU}} x {{ e},{r} ,{e,c}, {e,r }, {c,r }, {f ,c,r }} 3 (2) = {{ UU}, {DU}, {UU, UD}, { UU, DU}, {UD, DU}, { UU, UD, DU}} x {{f}, {g, e}} 3 (3) = {{ UU} , { UU, UD}} x {{ f }, {f,c}} 3 (4) = {{ UU}, {UU, UD}} x {{g, e}} II = 3 (5) = {{ UU}} x {{e, c}} . St rategy UD survives IEWDS but does not appear in any fully permissible set . Strat egy c appears in a fully permissible set but does not survive IEWDS.

11.3

Full admissible consistency

When ju stifying rati onalizable and permissible st rategies t hrough epistemic conditions, players are usually modeled as decision makers under uncert ainty. Tan and Werlang (1988) chara cterize rationalizable st rategies by common belief (with prob ability one) of t he event that each player chooses a maximal st rategy given preferences t hat are represented by a subjective probability distribution. Hence, preferences ar e both complete and conti nuous (d. Propositi on 1). Brand enburger (1992) characterizes permissible st rategies by common belief (wit h pr i-

Capturing forward induction through full permissibility

143

mary probability one) of the event that each player chooses a maximal strategy given pr eferences t hat are represented by an LPS with full support on the set of opponent st rate gies (d. Proposition 2). Hence, pr eferences ar e st ill complete, but not cont inuous du e to the full support requirement . Since pr eferences are complete and repr esentable by a probabili ty distribution or an LPS , t hese epistemic justifications differ significantly from t he corresponding algori thms, IESDS and t he DekelFud enb erg pro cedure, neither of which makes reference to subjective pr obabili ti es.' When doing analogously for fully permissible sets, not only must continuity of preferences be relaxed to allow for 'caut ion' and 'robust belief of opponent rationality' , as discus sed in Section 11.1. One must also relax completeness of pr eferences to accommodate 'no extraneous rest rict ions on beliefs' , which is a requirement of minimal completeness and impli es that pr eferences are expressed solely in te rms of admissibility on nest ed sets . Hence, preferences are not in genera l repr esentable by subjective probabilities (except t hrough t reat ing incompl et e pr eferences as a se t of complete pr eferences; d . Aumann, 1962; Bewley, 1986). This means t hat episte mic op erat ors must be derived directl y from t he underlying pr eferences- as observed by Morris (1997) and explored further in Ch apt er 4 of t his book- since there is no probabili ty distribution or LPS t hat repr esent s the pr eferences. It also entails t hat t he resulting characterization, given in Proposition 41, must be closely related to t he algorithm used in t he definiti on of fully permissible sets . There is anot her fund ament al difference. Wh en cha racterizing rationalizable and permi ssible st ra tegies within the 'rational choice' approach, the event that is mad e subject to int eractive episte mology is defined by requiring that each player 's strategy choice is an element of his choice set (i.e. his set of maximal st rateg ies) given his belief about the opponent 's st rate gy choice.t In contrast , in the characte rizat ion of Proposition 40, the event that is made subject to int eractive episte mology is defined by imposing requirement s on how each player's choice se t is relat ed t o his belief about the opponent 's choice set . Since a player 's choice set equals t he set of maximal st rategies given t he ranking that t he player has over his st rateg ies, t he imp osed requirements relate a player 's ranking over

1 However , as shown by Proposit ions 26 and 27 of this bo ok, epistemic characterizat ion of rationalizability and per m issibility can be provided wit hout using subject ive probabilities. 2 As illustrated in Chapt ers 5 and 6 of t his book, it is also possible to charact erize rati onalizable and permissible stra tegies wit hin the 'co nsistent preferen ces' approach .

144

CONSISTENT PREFERENCES

his strategies to the opponent's ranking. Hence, fully permissible sets are characterized within the 'consistent preferences' approach. The epistemic modeling is identical to the one given in Section 6.1; hence, this will not be recapitulated here. Recall, however, that ""ti (~ {ti} X Sj x T j) denotes the set of states that player i deems subjectively possible at t., that (3t i (~ ""t i ) denotes the smallest set of states on which player i's preferences at ti, tt i , are admissible, and that Assumption 2 is imposed so that preferences are conditionally represented by a vNM utility function (cf. Proposition 4). Characterizing full permissibility. To characterize the concept of fully permissible sets, consider for each i , -0

Bdratj] := {(SI,tI ,s2,t2) E SI x TI X S2 x T21 (3t i = (ProjTiXSjXT)ratj]) n ""t i , and p ?- ti q only if p E j weakly dominates qEj for EJo= ProjsxT(3t i or EJo= ProjsxT""t i}, J

J

J

J

Define as follows the event that player i's preferences over his strategies are fully admissibly consistent with the game G = (SI, S2, UI , U2) and the preferences of his opponent: -0

-0

Ai := lUi] n B i [ratj] n [caud . Write AO

:= A~

n Ag for the event of full admissible consistency.

41 A strategy set a, for i is fully permissible in a finite strategic two-player game G if and only if there exists an epistemic model with a, = Sf i for some (tI ,t2) E ProjTIXT2CKAo .

PROPOSITION

Proof. Part 1: If a; is fully permissible, then there exists an epistemic model with a; = Sfi for some (tI , t2) E ProjTIXT2CKAo. It is sufficient to construct a belief system with SI x T I X S2 X T2 ~ CKAo such that , for each a, E IIi of any player i, there exists ti E T; with a; = Sfi. Construct a belief system with, for each i, a bijection a; : T i -----7 IIi from the set of types to the the collection of fully permissible sets . By Proposition 39(ii) we have that, for each ti E T; of any player i, there exists W/ i ~ II j such that Ui(ti) = Si\Di(Y/i), where Y/ i := {Sj E Sj I 3(jj E W/ i s.t. Sj E (jj}. Determine the set of opponent types that ti deems subjectively possible as follows: T/i = {tj E T j I Uj (tj) E WIi} . Let , for each ti E T; of any player i, tt i satisfy 1. Vfi

0

Z

=

Ui (so that SI x T I

X

S2

X

T2 ~ [u]), and

Capturing forward induction through full permissibility

145

2. pr.tiq iffpEj weakly dominates qEj for E j = E/i:= {(Sj,tj) I Sj E (7 j (tj) and tj E TJi} or E j = Sj X T/i, which implies that j3t i = {td X E/i and /'i,ti = {td x x T/i (so that Sl x T 1 X S2 X T2 ~ [cau]) .

s,

By the construction of E/i, this means that S;i = S, \Di(Y/i) = (7i(td since, for any acts P and q on Sj x T j satisfying that there exist mixed strategies Pi, qi E ~(Si) such that, V(Sj,tj) E SjxTj, p(Sj,tj) = Z(pi,Sj) and q(Sj, tj) = z(% Sj), P )-ti q iff PEj weakly dominates qE j for E j = Y/i X Tj or E j = Sj x Tj. This in turn implies, for each ti E T; any player i , 3. j3t i = (probXSXT[ratj])n/'i,ti (so that, in combination with 2., Sl x t J -oj -0 T 1 X S2 X T2 ~ Bdratj] n Bj[rati])' Furthermore, Sl x T 1 X S2 X T2 ~ CKAo since T/i ~ T j for each ti E t: of any player i. Since, for each player i, a, is onto IIi , it follows that , for each a; E IIi of any player i, there exists ti E T; with cri = S/i. Part 2: If there exists an epistemic model with cri = Sri for some (tt , t z) E projTj XT2 CKAo , then cri is fully permissible. Assume that there exists an epist emic model with cri = sF for some (tt,t z) E ProjyjXT2CKAo. In particular, CKAo i=- 0. Let , for each i , T[ := projTi CKAo and 3 i := {Si i I t, E Tn . It is sufficient to show that, for each i , 3 i ~ IIi. By Proposition 25(ii), for each ti E T! of any player i, j3t i ~ /'i,ti ~ {td x x Tj since CKAo = KCKAo ~ KiCKAo . By the definition of AO, it follows that, for each ti E T! of any player i , 1. ?:t i is conditionally represented by Vii satisfying that vii 0 Z is a

s,

positive affine transformation of Ui , and 2. p )-ti q iff PEJ weakly dominates qEJ· for E J· = EJl; := proj , J xTj3t; J or E j = Sj X T/i, where j3t i = (projTiXSjxT)ratj]) n /'i,ti.

{S/j I tj E T/ i} and Y/i := {Sj E Sj I ::Jcrj E wl i s.t . Sj E crj}, and note that /'i,ti ~ {td x s, x Tj implies wl i ~ 3j. It follows that , for any acts P and q on Sj x T j satisfying that there exist mixed strategies Pi , qi E ~(Si) such that, V(Sj, tj) E Sj x T j, p(Sj, tj) = Z(Pi, Sj) and q(Sj, tj) = Z(qi , Sj) , P )-ti q iff PEj weakly dominates qEj for Ej = Yjti X Tj or E j = Sj x T j. Hence, S;i = S, \Di (Yjti). Since this holds for each ti E T! of any player i, we have that 3 ~ a(3). Hence, Proposition 39(iii) entails that , for each i, 3 i ~ IIi. •

Write

wl i

:=

Interpretation. We now show how the event used to characterize fully permissible sets-full admissible consistency-can be interpreted in terms of the requirements of 'caut ion', 'robust belief of opponent ratio-

146

CONSISTENT PREFERENCES

nality' , and 'no extraneous restrictions on beliefs '. Following a common procedure of the axiomatic method , this will in turn be used to verify that these requirements are indeed needed for the characterization in Proposition 41 by investigating the consequences of relaxing one requirement at a time. These exercises contribute to the understanding of fully permissible sets by showing that the concept is related to properly rationalizable, permissible, and rationalizable pure strategies in the following manner: • When allowing extraneous restrictions on beliefs , we open for any properly rationalizable pure strategy, implying that forward induction is no longer promoted in G~ of Figure 2.6.3 • When weakening 'robust belief of opponent rationality' to 'belief of opponent rationality', we characterize the concept of permissible pure strategies independently of whether a requirement of 'no extraneous restrictions on beliefs ' is retained . • When removing 'caution', we characterize the concept of rationalizable pure strategies independently of whether extraneous restrictions on beliefs are allowed and robust belief of opponent rationality is weakened. Since it is clear that [caul = [caul] n [cau2] corresponds to caution (cf. Section 6.3), it remains to split B~[rat2]nB~[ratl] into 'robust belief of opponent rationality' and 'no extraneous restrictions on beliefs '. To state the condition of 'robust belief of opponent rationality' we need to recall the robust belief operator as defined and characterized in Chapter 4. Since Assumption 2 is compatible with the framework of Chapter 4, we can in line with Section 4.2 define robust belief as follows. If E does not concern player i 's strategy choice (i.e. , E = 8 i x prohxs .xs.E), say that player i robustly believes the event E at ti • J J if t i E projTiB?E , where

B?E:= {(Sl ,l},s2 ,t2) E 8 1 x T1 X 82 x T21 :3£ E {l , . . . , L} s.t . p~i

= proj-. XS2XT2E n ",t i }

,

3To relax 'no ext ra neous restrictions on beliefs ' we need an episte m ic model-as the one introduced in Section 6.1-that is versatile enough t o allow for prefer ences that are mor e complete than being determined by admissibility on two nested sets.

Capturing forw ard induction through fu ll permissibility

147

and where (pii, . . . , pZ) is t he profile of nest ed sets on which ?:t i is admissible, and which satisfies:

o=1= (3ti = pii C . . . C p~i C ... c pZ= ",ti ~ {td x s, x Tj (where c denotes ~ and =1=). If ti E projTiB?[ratj], t hen i robustly believes at ti t hat j is rational. By Proposition 6 t his means t hat any (Sj,tj) t hat is deemed subjectively possible and where Sj is a rational choice by j at tj is considered infinitely more likely t han ~ny (sj , tj) where sj is not a rational choice by j at tj. As ti E projTiB?[ratj] entails t hat (3ti = (projTiXSj XT)ratj ]) n ",ti, it follows t hat B?[ratj] ~ B?[ratj]. Hence, relative to B~[rat2] n Bg[ratl ], B~[rat2] nBg[ratl] is obtained by imposing minim al complete ness, which under robu st belief of opponent rationality yields the requirement of 'n o extraneous restrictions on beliefs'. As established in Section 4.3, robust belief B? is a non-monot one operator which is bounded by t he two KD45 operators, namely belief B, and certain belief K i . Furth ermore, as shown in Chapter 4, the robust belief operator coincides with 'absolute ly robust belief ', as int rodu ced by Stalnaker (1998), and 'assumpt ion' , as proposed by Brand enburger and Keisler (2002), and is closely related to 'strong belief', as used by Bat tigalli and Siniscalchi (2002). However, in contrast to the use of nonmonotonic operators in th ese cont ribut ions, our non-monotonic operator B? is used only to interpret 'full adm issible consistency', while t he KD45 operator K, is used for t he interactive epistemology. The import ance of t his will be discussed in Section 11.5. There we also comment on how t he present requirement of 'no ext ra neous restrictions on beliefs' is related to Brand enburger and Keisler 's and Bat ti galli and Siniscalchi's use of a 'preference-complete' episte mic model. Allowing extraneous restrictions on beliefs. In view of the previous discussion, we allow ext ra neous restrictions on beliefs by replacing, for each i, B?[ratj] by B?[ratj]. Hence, let for each i ,

A? := lUi]

n B?[ratj] n [caui].

The following result is proven in Appe ndix C and shows t hat any properly rationalizable pure strategy is consistent wit h common certain belief of AD := A~ nAg . P ROP OSITIO N 42 Consider a finite strategic two -player game G. If a pure strategy s: f or i is properly rationaliza ble, then there exists an episf or some (tl ' t2) E proj- , X T2 CKAD. temic model with S i E

S;i

148

CONSISTENT PREFERENCES

Note that both Out and r are properly rationalizable pure strategies (and, indeed, (Out,r) is a proper equilibrium) in Gi, the 'bat tl e-of-t hesexes-with-an-outside-option' game of Figure 2.6, while neither Out nor r is consistent with common certain belief of full admissible consistency. Hence, Proposition 42 demonstrates that 'no extraneous restrictions on beliefs' is needed for the characterization in Proposition 41 of the concept of fully permissible sets , which in Gi promotes only the forward induction outcome (InL,£') (ef. the analysis of Gi in Sections 11.2 and 11.4). This also shows that 'no extraneous restrictions on beliefs' is incompatible with 'respect of opponent preferences', used to characterize proper rationalizability in Proposition 37.

Weakening robust belief of opponent rationality. By applying the belief operator Bi , as defined in Section 6.1, we can weaken B~[rat2]n Bg[ratl] (i.e., robust belief of opponent rationality) to B l [rat2] nB2[ratl] (i.e., belief of opponent rationality). Moreover, we can weaken B~[rat2] n Bg[ratl] to Bdrat2] n B2[ratl], where for each i,

Bi[ratj]:= {(SI,tl , s2 ,t2) E 8 1 x Tl X 8 2 x T2 ! j3t i ~ (projTiXSjXT)rat j]) , and p >- t i q only if p E j weakly dominates

qEj

i

i } for EJ· = projsxrj3t or EJ· = projsxrf\;t J J J J

,

Relative to BJ[rat2]nB2[ratl], :8 1 [rat2]nIh [ratlJ is obtained by imposing minimal completeness, which under belief of opponent rationality yields the requirement of 'no extraneous restrictions on beliefs'. To impose 'caution ' and 'belief of opponent rationality', recall from Section 6.3 that A = Al n A 2 is the event of admissible consistency where, for each i ,

To add 'no extraneous restrictions on beliefs', consider for each i ,

Ai

:=

lUi]

n Bilratj] n [caui] ,

and write A := ih n A2. Since A ~ A, the following proposition implies that permissibility (i.e., the Dekel-Fudenberg procedure; see Definition 13) is characterized if 'robust belief of opponent rationality' is weakened to 'belief of opponent rationality', independently of whether a requirement of 'no extraneous restrictions on beliefs' is retained. This result, which strenghthens Proposition 27 and is proven in Appendix C, shows

149

Capturing forward induction through full permissibility

that 'robust belief of opponent rationality' is needed for the characterization in Proposition 41 of the concept of fully permissible sets. PROPOSITION 43 Consider a finite strategic two-player game G.

If a pure strategy s: for i is permissible, then there exists an epistemic model with Si E for some (tl , t2) E proj-, XT2 CKA. A pure strategy s, for i is permissible if there exists an epistemic model with s, E for some (tl,t2) E projTlxT2CKA.

S;i

S;i

Removing caution. Recall from Section 6.2 that C the event of consistency where, for each i ,

= C 1 n C2 is

To add 'no extraneous restrictions on beliefs' and 'robust belief of opponent rationality', consider for each i ,

and write eO := ep neg. Since eO ~ C, the following strengthening of Proposition 25 means that the removal of 'caut ion' leads to a characterization ofrationalizability (i.e., IESDS; see Definition 11), independently of whether extraneous restrictions on beliefs are allowed and robust belief of opponent rationality is weakened. Thus, 'caution' is necessary for the characterization in Proposition 41. PROPOSITION 44 Consider a finite strategic two-player game G. If a pure strategy s; for i is rationalizable, then there exists an epistem ic for some (tl,t2) E projT1 xT2 CKe o . A pure strategy model with Si E Si for i is rationalizable if there exists an epistem ic model with S i E for some (tl,t2) E projTlxT2CKC.

S;i

S;i

Also the proof of this result is contained in Appendix C.

11.4

Investigating examples

The present section illustrates the concept of fully permissible sets by returning to the previously discussed games G~ and Gg . Here, G~ will serve to show how our concept captures aspects of forward induction , while Gg will be used to interpret the occurrence of multiple fully permissible sets. The two examples will be used to shed light on the differences between, on the one hand , the approach suggested here and , on the other hand ,

150

CONSISTENT PREFERENCES

IEWDS as characterized by Stahl (1995): A strategy survives IEWDS if and only if it is a best response to a belief where one strategy is infinitely less likely than another if the former is eliminated at an earlier round than the latter." Forward induction. Reconsider G~ Figure 2.6, and apply our algorithm IECFA to this "battle-of-the-sexes with an outside option" game. Since InR is a dominated strategy, InR cannot be an element of 1's choice set . This does not imply, as in the procedure of IEWDS (given Stahl's, 1995, characterization) , that 2 deems InL infinitely more likely than InR. However, 2 certainly believes that only {Out} , {InL} and {Out,InL} are candidates for l 's choice set. This excludes {r} as 2's choice set, since {r} is 2's choice set only if 2 deems {InR} or {Out, InR} possible. This in turn means that 1 certainly believes that only {e} and {£', r} are candidates for 2's choice set , implying that {Out} cannot be 1's choice set. Certainly believing that only {InL} and {Out, InL} are candidates for 1's choice set does imply that 2 deems InL infinitely more likely than InR. Hence, 2's choice set is {e} and , therefore, l's choice set {InL}. Thus, the forward induction outcome (InL , £') is promoted. To show how common certain belief of the event AO is consistent with the fully permissible sets {InL} and {e}-and thus illustrate Proposition 41-consider an epistemic model with only one type of each player; i.e., T 1 x T2 = {td x {t2}' Let , for each i, t i satisfy that V; i 0 Z = Ui. Also, let (3t 1 = {tIl x {£'} x {t2} ",tl = {tIl X 8 2 x {t2} 2 (3t = {t2} x {lnL} x {tIl ",t2 = {t2} x 8 1 x {t1}'

e

Finally, let for each i , p ,;-ti q if and only if PEj weakly dominates qE j i or E ' = projsxr",t i . Then for E J' = projsxr(3t J J J J J 8~1 = {InL}

Inspection will verify that CKAo = AO = 8 1

X

T1

X

82

X

T2 ,

Multiple fully permissible sets. Let us also return to Gs of Figure 11.1, where IEWDS eliminates D in the first round , r in the second round, and M in the third round , so that U and £' survive. Stahl's (1995) characterization of IEWDS entails that 2 deems each of U and M infinitely more likely than D . Hence, the procedure forces 2 to deem 4Cf, Brandenburger and Keisler (2002 , Theorem 1) as well as Battigalli (1996a) and Rajan (199S) , See also Bicchieri and Schulte (1997) , who give con ceptually related interpretations ofIEWDS ,

151

Capturing f orward induction through fu ll permissibility

M infinitely more likely than D for t he sole reason t hat D is eliminated before M , even tho ugh both M and D are event ually eliminated by t he procedure. Applying our algorithm IECFA yields the following result . Since D is a weakly dominated strategy, D cannot be an element of 1's choice set . Hence, 2 certainly believes that only {U}, {M } and {U, M} are candidates for 1's choice set. This excludes {1'} as 2's choice set , since {1'} is 2's choice set only if 2 deems {D} or {U, D} possible. T his in t urn means t hat 1 certainly believes that only {f} and {f ,1'} are candidates for 2's choice set , implying that {M} cannot be 1's choice set . There is no further elimination. T his means that 1's collection of fully permissible sets is {{U} , {U, M} } and 2's collect ion of fully permissible sets is {f}, {f, r }}. Thus, common certain belief of full admissible consiste ncy implies t hat 2 deems U infinitely more likely t ha n D since U (respect ively, D) is an element of any (respectively, no) fully perm issible set for 1. However, whether 2 deems M infinitely more likely t han D depends on the type of player 2. To show how common cert ain belief of t he event jIo is consistent with the collections of fully permissible sets {{U} , {U, M} } and {{f }, {f ,1'}}and thus illustrate P ropositi on 41 also in the case of Gs-consider an epistemic model with two types of each player; i.e., T 1 x T2 = {t~ , tn x {t~ , tD . Let , for each type ti of any player i , >:: t; satisfy that vi; o z = iu , Moreover , let {3t; = {tD x {f} x {t~} {3t;' = {tn x {( f , t~), (f , t~) , (1', t~)} {3t~ = {t~ } x {(U, t~) , (U, {3t~ = { t~} x {U} x {tD

82 x { t~ } K, 7 = {tn x 82 x T2

fA = {tD

X

t

tn, (M , t~)}

F inally, let for each ty pe ti of any player i, p >--t; q if and only if PEJ weakly domin at es qEJÂˇ for E JÂˇ = proj s J xTJ {3t; or E JÂˇ = proj s J XTJ K,t;. Then

8~; = {U }

8~7 = {U,M}

8~; = {f}

8~~ = {f ,1' } .

Inspect ion will verify t hat C K jIo = jIo = 8 1 X T 1 X 8 2 X T2 Our analysis of Gs allows a player to deem an opponent choice set to be subject ively impossible even when it is the true choice set of the opponent . E.g., at (t~ , t~) , player 1 deems it subjectively impossible t hat player 2's choice set is { f , 1'} even though this is the true choice set of player 2. Likewise, at (t~ , t~) , player 2 deems it subject ively impossib le t hat player 1's choice set is {U, M} even though this is the true choice set

152

CONSISTENT PREFERENCES

of player 1. This is an unavoidable feature of this game as there exists no pair of non-empty strategy subsets (Y1, Y2 ) such that Y1 = 51 \D 1(Y2 ) and Y2 = 5 2\D2(Y1 ) . It implies that under full admissible consistency we cannot have in Gs that each player is certain of the true choice set of the opponent. Multiplicity of fully permissible sets arises also in the strategic form of certain extensive games in which the application of backward induction is controversial, e.g. the 'centipede' game r~ illustrated in Figure 2.4. For more on this , see Chapter 12 where the concept of fully permissible sets is used to analyze extensive games.

11.5

Related literature

It is instructive to explain how our analysis differs from the epistemic foundations of IEWDS and EFR provided by Brandenburger and Keisler (2002) (BK) and Battigalli and Siniscalchi (2002) (BS), respectively. It is of minor importance for the comparison that EFR makes use of the extensive form, while the present analysis is performed in the strategic form. The reason is that, by 'caution', a rational choice in the whole game implies a rational choice at all information sets that are not precluded from being reached by the player's own strategy (d. Lemma 11). To capture forward induction players must essentially deem any opponent strategy that is a rational choice infinitely more likely than any opponent strategy not having this property. An analysis incorporating this feature must involve a non-monotonic epistemic operator, which is called robust belief in the present analysis (ef. Section 11.3), while the corresponding operators are called 'assumption' and 'strong belief' by BK and BS, respectively (see Chapter 4 for an analysis of the relationship between these non-monotonic operators). We use robust belief only to define the event that the preferences of each player is 'fully admissibly consistent' with the preferences of his opponent, while the monotonic certain belief operator is used for the interactive epistemology: â€˘ each player certainly believes (in the sense of deeming the complement subjectively impossible) that the preferences of his opponent are fully admissibly consistent, â€˘ each player certainly believes that his opponent certainly believes that he himself has preferences that are fully admissibly consistent, and so on. As the examples of Section 11.4 illustrate, it is here a central question what opponent types (choice sets) a player deems subjectively

Capturing forward induction through full permissibility

153

possible. Consequently, the certain belief operator is appropriate for the interactive epistemology. In contrast, BK and BS use their non-monotonic operators for the interactive epistemology. In the process of defining higher order beliefs both BK and BS impose that lower order beliefs are maintained. This is precisely how BK obtain Stahl's (1995) characterization which-e.g., in Gs of Figure ILl-seems to correspond to extraneous and hard-tojustify restrictions on beliefs. Stahl's characterization provides an interpretation of IEWDS where strategies eliminated in the first round are completely irrational, while strategies eliminated in later rounds are at intermediate degrees of rationality. Likewise, Battigalli (1996a) has shown how EFR corresponds to the 'best rationalization principle', entailing that some opponent strategies are neither completely rational nor completely irrational. The present analysis, in contrast, differentiates only between whether a strategy is maximal (i.e., a rational choice) or not. As the examples of Section 11.4 illustrate, although a strategy that is weakly dominated on the set of all opponent strategies is a "stupid" choice, it need not be "more stupid" than any remaining admissible strategy, as this depends on the interactive analysis of the game . The fact that a non-monotonic epistemic operator is involved when capturing forward induction also means that the analysis must ensure that all rational choices for the opponent are included in the epistemic model. BK and BS ensure this by employing 'preference-complete' epistemic models, where all possible epistemic types of each player are represented. Instead, the present analysis achieves this by requiring 'no extraneous restrictions on beliefs', meaning that the preferences are minimally complete (d. Section 11.3). Since an ordinary monotonic operator is used for the interactive epistemology, there is no more need for a 'preference-complete' epistemic model here than in usual epistemic analyses of rationalizability and permissibility. The analysis of the present chapter has a predecessor in Samuelson (1992) , who also presents an epistemic analysis of admissibility that leads to a collection otsetsfor each player, called a 'generalized consistent pair'. Samuelsonrequires that a player's choice set equals the set of strategies that are not weakly dominated on the union of choice sets that are deemed possible for the opponent; this implies our requirements of 'robust belief of opponent rationality' and 'no extraneous restrictions on beliefs' (d. Samuelson , 1992, p. 311). However, he does not require that each player deems no

154

CONSISTENT PREFERENCES

opponent strategy impossible, as implied by our requirement of 'caution'. Hence, his analysis does not yield {{U, M}} x {{t'}} in G9 of Figure 11.2. Furthermore, he defines possibility relative to a knowledge operator that satisfies the truth axiom, while our analysis-as illustrated by the discussion of G8 in Section 11.4-allows a player to deem an opponent choice set to be subjectively impossible even when it is the true choice set of the opponent. This explains why we in contrast to Samuelson obtain general existence (ef. Proposition 39(i)). If each player is certain of the true choice set of the opponent, one obtains a 'consistent pair' as defined by Borgers and Samuelson (1992), a concept that need not exist even when a generalized consistent pair exists. Ewerhart (1998) modifies the concept of a consistent pair by adding 'caution'. However, since he allows extraneous restrictions on beliefs to ensure general existence, his concept of a 'modified consistent pair' does not promote forward induction in G~. A 'self-admissible set' in the terminology of Brandenburger and Friedenberg (2003) is a Cartesian product of strategy subsets, where each player's subset consists of strategies that weakly dominated neither on the subset of opponent strategies nor on the set of all opponent strategies. Also Brandenburger and Friedenberg allow extraneous restrictions on beliefs. Hence, 'modified consistent pairs ' and 'self-admissible sets' need not correspond to profiles of fully permissible sets . However, if there is a unique fully permissible set for each player, then the pair constitutes both a 'modified consistent pair' and a 'self-admissible set' . Basu and Weibull's (1991) 'tight curb" set ' is another variant of a consistent pair that ensures existence without yielding forward induction in G~, as they impose 'caution' but weaken 'robust belief of opponent rationality' to 'belief of opponent rationality'. In particular, the set of permissible strategy profiles is 'tight curb*' . 'Caution' and 'robust belief of opponent rationality' are admissibility requirements on the preferences of players, thus positioning the analysis of the present chapter in the 'consistent preferences' approach. Moreover, by imposing 'no extraneous restrictions on beliefs' as a requirement of minimal completeness, preferences are not in general representable by subjective probabilities, thus showing the usefulness of an analysis that relax completeness. 5

5By not employing subjective probabilities, the analysis is related to the filter model of beliefs presented by Brandenburger (1997 , 1998) .

Chapter 12

APPLYING FULL PERMISSIBILITY TO EXTENSIVE GAMES

In many economic contexts decision makers interact and take actions that extend through time. A bargaining party makes an offer, which is observed by the adversary, and accepted, rejected or followed by a counter-offer. Firms competing in markets choose prices , levels of advertisement, or investments with the intent of thereby influencing the future behavior of competitors. One could add many examples. The standard economic model for analyzing such situations is that of an extensive game. Reproducing joint work with Martin Dufwenberg, d. Asheim and Dufwenberg (2003b), this chapter revisits a question that was already posed in Chapters 7-10: What happens in an extensive game if players reason deductively by trying to figure out one another's moves? We have in Asheim and Dufwenberg (2003a), incorporated as Chapter 11 of this book, proposed a model for deductive reasoning leading to the concept of 'fully permissible sets ', which can be applied to many strategic situations. In the present chapter we argue that the model is appropriate for analyzing extensive games and we apply it to several such games.

12.1

Motivation

There is already a literature exploring the implications of deductive reasoning in extensive games , but the answers provided differ and the issue is controversial. Much of the excitement concerns whether or not deductive reasoning implies backward induction in games where that principle is applicable. We next discuss this issue, since it provides a useful backdrop against which to motivate our own approach.

156

CONSISTENT PREFERENCES

1

2

1

0

~3 103 020 Figure 12.1 .

d f 1,0 1,0 D FD 0, 2 3, 0 FF 0,2 0, 3

I'i i and its pure strategy reduced strategic form .

Consider the 3-stage "take-it-or-leave-it", introduced by Reny (1993) (a version of Rosenthal's, 1981, "centipede" game, see r~ of Figure 2.4), and shown in Figure 12.1 together with its pure strategy reduced strategic form.' What would 2 do in r11 if called upon to play? Backward induction implies that 2 would choose d, which is consistent with the following idea: 2 chooses d because she "figures out" that 1 would choose D at the last node. Many models of deductive reasoning support this story, starting with Bernheim's concept of 'subgame rationalizability' and Pearce's concept of 'extensive form rationalizability' (EFR). More recently, Battigalli and Siniscalchi (2002) provide a rigorous epistemic foundation for EFR, while Chapters 7-10 of this book epistemically model rationalizability concepts that resemble 'subgame rationalizability'. However, showing that backward induction can be given some kind of underpinning does not imply that the underpinning is convincing. Indeed, skepticism concerning backward induction can be expressed by means of r11. Suppose that each player believes the opponent will play in accordance with backward induction; i.e., 1 believes that 2 chooses d if asked to play, and 2 believes that 1 plays D at his initial note. Then player 1 prefers playing D to any of his two other strategies F D and FF. Moreover, if 2 is certain that 1 believes that 2 chooses d if she were asked to play, then 2 realizes that 1 has not chosen in accordance with his preferences if she after all is asked to play. Why then should 2 believe that 1 will make any particular choice between his two less preferred strategies, FD and FF, at his last node? So why then should 2 prefer d to f? This kind of perspective on the "t ake-it-or-leave-it " game is much inspired by the approach proposed by Ben-Porath (1997), where similar

1 We need not consider what players plan to do at decision nodes that their own strategy precludes them from reaching (d. Section 12.2).

Applying full permissibility to ext ensive games

157

objections against backward inductive reasoning are raised . We shall discuss his contribution in some detail, since the key features of our approach can be appreciated via a comparison to his model. Applied to f l l , Ben-Porath's model captures the following intuition: Each player has an initial belief about the opponent 's behavior. If this belief is contradicted by the play (a "surprise" occurs) he may subsequently entertain any belief consistent with the path of play. The only restriction imposed on updated beliefs is Bayes' rule. In f l l , Ben-Porath's model allows player 2 to make any choice. In particular, 2 may choose f if she initially believes with probability one that player 1 will choose D, and conditionally on D not being chosen assigns sufficient probability on FF. This entails that if 2 initially believes that 1 will comply with backward induction, then 2 need not follow backward induction herself. In f l l , our analysis captures much the same intuition as Ben-Porath's approach, and it has equal cutting power in this game. However, it yields a more structured solution as it is concerned with what strategy subsets that are deemed to be the set of rational choices for each player. While agreeing with Ben-Porath that deductive reasoning may lead to each of D and FD being rational for 1 and each of d and f being rational for 2, our concept of full permissibility predicts that l 's set of rational choices is either {D} or {D , F}, and 2's set of rational choices is either {d} or {d, f}. This has appealing features. If 2 is certain that l's set is {D}, then-unless 2 has an assessment of the relative likelihood of l's less preferred strategies FD and FF-one cannot conclude that 2 prefers d to f or vice versa; this justifies {d, f} as 2's set of rational choices. On the other hand, if 2 considers it possible that l 's set is {D, FD}, then d weakly dominates f on this set and justifies {d} as 2's set of rational choices. Similarly, one can justify that D is preferred to FD if and only if 1 considers it impossible that 2's set is {d, f}. This additional structure is important for the analysis of f~ , illustrated in Figure 8.2. This game is due to Reny (1992, Figure 1) and has appeared in many contributions. Suppose in this game that each player believes the opponent will play in accordance with backward induction by choosing FF and f respectively. Then both players will prefer FF and f to any alternative strategy. Moreover, as will be shown in Section 12.3, our analysis implies that {FF} and {f} are the unique sets of rational choices. Ben-Porath's approach, by contrast, does not have such cutting power in f~ , as it entails that deductive reasoning may lead to each of the strategies D and FF being rational for 1 and each of the strategies d and

158

CONSISTENT PREFERENCES

!

being rational for 2. The intuition for why the strategies D and dare admitted is as follows : D is 1's unique best strategy if he believes with probability one that 2 plays d. Player 1 is justified in this belief in the sense that d is 2's best strategy if she initially believes with probability one that 1 will choose D, and if called upon to play 2 revises this belief so as to believe with sufficiently high probability (e.g., probability one) that 1 is using FD. This belief revision is consistent with Bayes' rule , and so is acceptable. Ben-Porath's approach is a very important contribution to the literature, since it is a natural next step if one accepts the above critique of backward induction. Yet we shall argue below that it is too permissive, using r~ as an illustration. Assume that 1 deems d infinitely more likely than t, while 2 deems D infinitely more likely than FD and FD infinitely more likely than FF. Then the players rank their strategies as follows: l's preferences: D >- FF>- FD 2's preferences: d>-! This is in fact precisely the justification of the strategies D and d given above when applying Ben-Porath's approach to r~. Here, 'caution' is satisfied since all opponent strategies are taken into account; in particular, FF is preferred to FD as the former strategy weakly dominates the latter. Moreover, 'robust belief of opponent rationality' is satisfied since each player deems the opponent's maximal strategy infinitely more likely that any non-maximal strategy. However, the requirement of 'no extraneous restrictions on beliefs', as described in Chapter 11, is not satisfied since the preferences of 2 introduce extraneous restrictions on beliefs by deeming one of l 's non-maximal strategies, FD, infinitely more likely than another non-maximal strategy, FF. When we return to G~ in Section 12.3, we show how the additional imposition of 'no extraneous restrictions on beliefs' means that deductive reasoning leads to the conclusion that {FF} and {f} are the players' choice sets in this game. As established in Chapter 11, our concept of fully permissible sets is characterized by 'caut ion', 'robust belief of opponent rationality' , and 'no extraneous restrictions on beliefs'. In Section 12.2 we prove results that justify the claim that interesting implications of deductive reasoning in a given extensive game can be derived by applying this concept to the strategic form of that game. Sections 12.3 and 12.4 are concerned with such applications, with the aim of showing how our solution concept gives new and economically relevant insights into the implications of deductive reasoning in extensive

Applying full permissibility to extensive games

159

games. The material is organized around two central themes: backward and forward induction. Other support for forward induction through the concept of EFR and the procedure of IEWDS precludes outcomes in conflict with backward induction; see, e.g., Battigalli (1997). In contrast, we will show how the concept of fully permissible sets promotes forward induction in the "bat t le-of-t he-sexes with an outside option" and "burning money" games as well as an economic application from organization theory, while not insisting on the backward induction outcome in games (like r n and the 3-period prisoners' dilemma) where earlier contributions, like Basu (1990), Reny (1993) and others, have argued on theoretical grounds that this is problematic. Still, we will show that the backward induction outcome is obtained in r~ , and that our concept has considerable bite in the 3-period "prisoners' dilemma" game. Lastly, in Section 12.5 we compare our approach to related work.

12.2

Justifying extensive form application

The concept of fully permissible sets, presented and epistemically characterized in Chapter 11 of this book, is designed to analyze the implications of deductive reasoning in strategic form games. In this chapter, we propose that this concept can be fruitfully applied for analyzing any extensive game through its strategic form. In fact , we propose that it is legitimate to confine attention to the game 's pure strategy reduced strategic form (cf. Definition 23 below), which is computationally more convenient . In this section we prove two results which, taken together, justify such applications.

An extensive game. A finite ext ensive two-player game r (without chance moves) includes a set of terminal nodes Z and, for each player i , a vNM utility function V i : Z ~ IR that assigns a payoff to any outcome. For each player i , there is a finite collection of information sets H i, with a finite set of actions A(h) being associated with each h E Hi . A pure strategy for player i is a function s ; that to any h E Hi assigns an action in A(h). Let 5 i denote player i's finite set of pure strategies, and let 5 = 51 X 52. As before , write Si (E 5 i ) for pure strategies and Pi and qi (E ~ (5d) for mixed strategies. Define U i : 5 ~ IR by u ; = V i 0 Z , and refer to G = (51 ,52 , Ul , U2) as the strategic form of the extensive game r . For any h E HI U H2 , let 5(h) = 5 1(h) x 52(h) denote the set of strategy profiles that are consistent with h being reached . Weak sequential rationality. Consider any strategy that is maximal given preferences that satisfy that one strategy is preferred to an-

160

CONSISTENT PREFERENCES

other if and only if the one weakly dominates the other on lj - the set of strategies that player i deems to be the set of rational choices for his opponent - or Sj - the set of all opponent strategies. Hence, the strategy is maximal at the outset of a corresponding extensive game. Corollary 2 makes the observation that this strategy is still maximal when the preferences have been updated upon reaching any information set that the choice of this strategy does not preclude. Assume that player i's preferences over his own strategies satisfy that Pi is preferred to qi if and only if Pi weakly dominates qi on lj or Sj. Let, for any h E Hi, lj(h) := lj n Sj(h) denote the set of strategies in lj that are consistent with the information set h being reached. If Pi, qi E ~ (S (ih)), then i's preferences conditional on the information set h E Hi being reached satisfy that Pi is preferred to qi if and only if Pi weakly dominates qi on lj (h) or Sj (h) (where it follows from the definition that weak dominance on lj(h) is not possible if lj(h) = 0). Furthermore, i 's choice set conditional on h E Hi, S?j (h), is given by

Srj(h):= Si(h) \ {s, E Si(h) I :3xi E ~(Si(h)) s.t . Xi weakly dominates Si on lj (h) or Sj (h)} . Write stj := S?j (0) (= s, \Di(lj) in earlier notation). By the result below, if Si is maximal at the outset of an extensive game, then it is also maximal at later information sets for i that Si does not preclude. 2 Let (0 i) lj ~ Sj. If s, any h E Hi with Si(h) 3 s. ,

COROLLARY

E

S{J, then Si

E

S{j(h) for

Proof. This follow from Lemma 11 by letting i's preferences (at ti) on i 's set of mixed strategies satisfy that Pi is preferred to qi if and only if Pi weakly dominates qi on }j or Sj. â€˘

By the assumption of 'caut ion', each player i takes into account the possibility of reaching any information set for i that the player's own strategy does not preclude from being reached. Hence, 'rationality' implies 'weak sequential rationality' ; i.e., that a player chooses rationally at all information sets that are not precluded from being reached by the player 's own strategy. Reduced strategic form. It follows from Proposition 45 below that it is in fact sufficient to consider the pure strategy reduced strategic form when deriving the fully permissible sets of the game. The following definition is needed.

161

Applying full perm issibility to extensive gam es

DEFINITION 23 Let G = (Sl , S 2, U1 , U2) be a finite strat egic two-player game. The pure strategies Si and s~ (E Sd are equivalent if for each player k , Uk( S~ , Sj) = Uk( Si , Sj) for all Sj E Sj. The pure strategy reduced strategic form (PRSF) of G is obtained by letting, for each player i , each class of equivalent pure strategies be represented by exact ly one pure strategy. Since the maximality of one of two equivalent strategies implies that t he other is maximal as well, the following observation holds: If s, and s~ are equivalent and a; is a fully permissible set for i , then s: E (Ji if and only if s~ E a. , To see this formally, note that if Si E a, for some fully permissible set (Ji , then, by Proposition 39(ii) , there exists (0 #) W j ~ II j such that s; E (Ji = S r j for }j = Uo"jEIJI j (Jj . Since Si and s~ ar e equivalent, s~ E S r j = a.. This observation explains why the following result can be established. PROPOSITION 45 Let 0 = (81,82 , i'iI, U2) be a finit e strategic two-player gam e where s; and s~ are two equivalent strategies for i. Consider G = (Sl ,S2 ,U1 ,U2) where S, = 8i \{sa and Sj = 8j fo r j # i , and where, for each play er k , Uk is the restriction of Uk to S = Sl X S2. Let, fo r each play er k , Ilk (th) denote the collecti on of f ully permissible sets for k in G (0) . Th en IIi is obtain ed from TI i by remo ving s~ from any (Yi E TI i with s, E (Yi , while, for j # i , II j = TIj .

Proof. By Proposition 39(iii) it suffices to show that 1 If:3 ~ a (:3 ) for 0, then :3 ~ a(:3) for G, where s, is obtained from 2i by _removing s~ from any (Yi E 2i with Si E (Yi, while, for j # i, 2 If 3 ~ a( 3 ) for G, then 2 ~ a(2) for 0 , where 2i is obtained from 3 i by adding s~ to any (Ji E 3 i with Si E a., while, for j # i , 2j = 3 j . Part 1. Assume 2 ~ a(2) . By the observation preceding Proposition 45, if (Yi E TI i , _then s~ E (Yi if and only if s; E (Yi . Pick any player k and any (Yk E Ilk . Let f denot e the other player. By the definition of ak(') ' there exists (0 #) We ~ TIe such th at (y~ = for fe = Ua~Eq,e(J~. Construct W i by removing s~ from any (Yi E W i with Si E (Yi and replace 8i by S i , while, for j # i , W j = Wj and Sj = Let 1ÂŁ = Ua~ElJle(J~ . Then it follows from the definition of S kYe that SkYe = (Yk\ {sD if k = i and SkYe = (Yk if k # i . Since, for each player k, (0 #) W k ~ 3k , we have that 3 ~ a (3 ). Pa rt 2 is shown similarly. _

s/e s;

162

CONSISTENT PREFERENCES

Proposition 45 means that the PRSF is sufficient for analyzing common certain belief of full admissible consistency, which is the epistemic foundation for the concept of fully permissible sets . Consequently, in the strategic form of an extensive game, it is unnecessary to specify actions at information sets that a strategy precludes from being reached. Hence, instead of fully specified strategies, it is sufficient to consider what Rubinstein (1991) calls plans of action. For a generic extensive game , the set of plans of action is identical to the strategy set in the PRSF. In the following two sections we apply the concept of fully permissible sets to extensive games. We organize the discussion around two themes: backward and forward induction. Motivated by Corollary 2 and Proposition 45, we analyze each extensive game via its PRSF (cf. Definition 23), given in conjunction to the extensive form. In each example, each plan of action that appears in the underlying extensive game corresponds to a distinct strategy in the PRSF.

12.3

Backward induction

Does deductive reasoning in extensive games imply backward induction? In this section we show that the answer provided by the concept of fully permissible sets is "somet imes, but not always". 'Sometimes.' There are many games where Ben-Porath's approach does not capture backward induction while our approach does (and the converse is not true) . Ben-Porath (1997) assumes 'init ial common certainty of rationality' in extensive games of perfect information. As discussed in Chapter 7 he proves that in generic games (with no payoff ties at terminal nodes for any player) the outcomes consistent with that assumption coincide with those that survive the Dekel-Fudenberg procedure (where one round of elimination of all weakly dominated strategies is followed by iterated elimination of strongly dominated strategies). It is a general result that the concept of fully permissible sets refines the Dekel-Fudenberg procedure (cf. Proposition 40). Game r~ of Figure 8.2 shows that the refinement may be strict even for generic extensive games with perfect information, and indeed that fully permissible sets may respect backward induction where Ben-Porath's solution does not. The strategies surviving the Dekel-Fudenberg procedure, and thus consistent with 'initial common certainty of rationality', are D and FF for player 1 and d and f for player 2. In Section 12.2 we gave an intuition for why the strategies D and d are possible. This is, however, at odds with the implications of common certain belief of full admissible consistency.

Applying full permissibility to extensive gam es

163

Recall from Section 11.3 that t he collection of fully permissible sets is determined by t he algorit hm IECFA. Applying IECFA t o the PRSF of r~ of Figure 8.2 yields:

3 (0) = L;1

X

L;2

3 (1) = {{D} , {FF} , {D , FF}} x L;2 3(2) = {{D} , {FF} , {D ,FF}} x {{f} ,{d,f}} 3(3) = {{FF} , {D , FF}} x {{f} , {d, f}} 3(4) = {{FF} , {D , FF}} x {{f}} IT = 3(5)

= {{FF}} x {{fn

Int erpretation: 3(1): 'Caut ion' implies that FD cannot be a maximal strategy (i.e., an element of a choice set) for 1 since it is weakly dominated (in fact , even strongly dominated). 3(2): Player 2 certainly believes that only {D} , {FF} and {D , FF} are candidates for l 's choice set. By 'robust belief of opponent rationality' and 'no ext raneous rest rictions on beliefs' thi s excludes {d} as 2's choice set, since d weakly dominat es f only on {FD} or {D ,FD}. 3 (3): 1 certainly believes that only {f} and {f, d} are candidates for 2's choice set . By 'robust belief of opponent rationality' and 'no ext ra neous restrictions on beliefs' this excludes {D} as l 's choice set , since D weakly dominates FD and FFonly on {d}. 3 (4): Player 2 certainly believes t hat only {FF} and {D , FF} are candidates for l 's choice set . By 'robust belief of opponent rationality' t his implies that 2's choice set is {f} since f weakly dominates d on both {FF} and {D , FF}. 3 (5): 1 certainly believes that 2's choice set is {fl . By 'robust belief of opponent rati onality' t his implies that {FF} is l 's choice set since FF weakly domin at es D on {fl . No further eliminat ion of choice sets is possible, so {FF} and {f} are the respective players' unique fully permissible sets. 'Not always.' While fully permissibl e sets capt ure backward induction in r~ and other games, the concept does not capt ure backward induction in certain games where the procedure has been considered controversial. 2 The background for t he controversy is th e following paradoxical aspect : Why should a player believe th at an opponent 's future play will sat isfy backward indu ction if t he opp onent 's previous play is incompatibl e with backward indu ction? A prot otypical game for cast-

2See dis cussion a nd refer ences in Chapter 7 of the pr esen t book .

164

CONSISTENT PREFERENCES

ing doubt on backward induction is the "take-it-or-leave-it" game Figure 12.1, which we next analyze in detail. Applying IECFA to the PRSF of f ll of Figure 12.1 yields:

3(0) = ~1 X ~2 3(1) = {{D} , {FD}, {D, FD}} x

fll

of

~2

3(2) = {{D} ,{FD},{D,FD}} x {{d} ,{d,f}} II

= 3(3) = {{D} ,{D,FD}} x {{d} ,{d,f}}

Interpretation: 3(1): FF cannot be a maximal strategy for 1 since it is strongly dominated. 3(2): Player 2 certainly believes that only {D}, {FD} and {D, FD} are candidates for l 's choice set. This excludes {f} as 2's choice set since {f} is 2's choice set only if 2 deems {FF} or {FD, FF} subjectively possible. 3(3): 1 certainly believes that only {d} and {d, f} are candidates for 2's choice set , implying that {FD} cannot be l's choice set. No further elimination of choice sets is possible and the collection of profiles of fully permissible sets is as specified. Note that backward induction is not implied. To illustrate why, we focus on player 2 and explain why {d, f} may be a choice set for her. Player 2 certainly believes that l's choice set is {D} or {D , FD}. This leaves room for two basic cases. First, suppose 2 deems {D , FD} subjectively possible. Then {d} must be her choice set, since she must consider it infinitely more likely that 1 uses FD than that he uses FF. Second, and more interestingly, suppose 2 does not deem {D , FD} subjectively possible. Then conditional on 2's node being reached 2 certainly believes that 1 is not choosing a maximal strategy. As player 2 does not assess the relative likelihood of strategies that are not maximal (d. the requirement of 'no extraneous restrictions on beliefs') , {d, f} is her choice set in this case. Even in the case where 2 deems {D} to be the only subjectively possible choice set for 1, she still considers it subjectively possible that 1 may choose one of his non-maximal strategies FD and FF (d. the requirement of 'caution') , although each of these strategies is in this case deemed infinitely less likely than the unique maximal strategy D. Applied to (the PRSF of) f ll , our concept permits two fully permissible sets for each player . How can this multiplicity of fully permissible sets be interpreted? The following interpretation corresponds to the underlying formalism : The concept of fully permissible sets , when applied to f ll , allows for two different types of each player . Consider player 2. Either she may consider that {D, FD} is a subj ectively possible choice set for 1, in which case her choice set will be {d} so that she complies

Applying full permissibilit y to extensive games

165

wit h backward induction. Or she may consider {D} to be the only subjectively possible choice set for 1, in which case 2's choice set is {d, f}. Intuitively, if 2 is cert ain that 1 is a backward indu cter , t hen 2 need not be a backward inducter herself! In t his game, our model captures an intuition t hat is very similar to that of Ben-Porath's model. Reny (1993) defines a class of "belief consiste nt" games, and argu es on epistemic grounds t hat backward induction is problemat ic only for games that are not in this class. It is int eresting to not e t hat t he game where our concept of fully permissible sets differs from Ben-Porath 's analysis by promoting backward induction, f~ , is belief-consist ent. In contrast , t he game where the present concept coincides with his by not yielding backward induction, T 11 , is not belief-consistent. There are examples of games that are not belief consist ent, where full permissibility still implies backward induction, meaning that belief consist ency is not necessary for thi s conclusion. It is, however , an as-of-yet unproven conject ure that belief consiste ncy is sufficient for t he concept of fully permissible sets to promote backward inducti on. We now compare our results to t he very different findings of Aumann (1995), cf. also Section 5 of St alnaker (1998) as well as Chapter 7 of thi s book. In Aumann's model, where it is crucial t o specify full strategies (rather than plans of act ions) , common knowledge of rational choice implies in f n that all st rategies for 1 but DD (where he takes a payoff of 1 at his first node and a payoff of 3 at his last nod e) are impossible. Hence, it is impossible for 1 to play FD or FF and t hereby ask 2 t o play. However, in the counterfact ual event that 2 is asked to play, she optimizes as if player 1 at his last node follows his only possible st rategy DD, implying t hat it is impos sible for 2 to choose f (cf. Aumann 's Sections 4b, 5b, and 5c). Thus, in Aumann's anal ysis, if there is common knowledge of rational choice, then each player chooses the backward induction strategy. By cont rast, in our analysis player 2 being asked to play is seen to be incompatible with 1 playing DD or DF. For the determination of 2's preference over her st rategies it is the relative likelihood of FD versus FF t hat is important to her. As seen above, t his assessment depends on whether she deems {D , FD} as a possible candidate for l 's choice set . Prisoners' dilemma. We close this section by consider ing a finitely repeat ed "prisoners' dilemma" game. Such a game does not have perfect information, but it can st ill be solved by backward induction to find the unique subgame perfect equilibrium (no one cooperates in t he last period , given t his no one cooperates in t he penultimat e period , etc .).

166

CONSISTENT PREFERENCES

s!fT s f T 7,7 sfV 8, 4 sfE 8, 4 T 5, 5 v 6, 2 E 6, 2

sr sr sr

Figure 12.2.

s!fV 4, 8 5, 5 5, 5 8,4 9, 1 9, 1

Reduced form of

r 12

s!f E 4, 8 5,5 5, 5 5, 5 6, 2 6, 2

S~T S~V S~E

5, 5 4, 8 5, 5 3,3 2,6 3, 3

2, 6 1,9 2, 6 6, 2 5, 5 6, 2

2,6 1,9 2,6 3, 3 2, 6 3, 3

(a 3-period "prisoners' dilemma" game) .

This solution has been taken to be counterintuitive; cf, e.g. Pettit and Sugden (1989). We consider the case of a 3-period "prisoners' dilemma" game (f 12 ) and show that , again, the concept of fully permissible sets does not capture backward induction. However, the fully permissible sets nevertheless have considerable cutting power. Our solution refines the Dekel-Fudenberg procedure and generates some special "structure" on the choice sets that survive. The payoffs of the stage game are given as follows, using Aumann's (1987b, pp . 468-9) description: Each player decides whether he will receive 1 (defect) or the other will receive 3 (cooperate) . There is no discounting. Hence, the action defect is strongly dominant in the stage game , but still, each player is willing to cooperate in one stage if this induces the other player to cooperate instead of defect in the next stage. It follows from Proposition 45 that we need only consider what Rubinstein (1991) calls plans of action. There are six plans of actions for each player that survive the DekelFudenberg procedure. In any of these, a player always defects in the 3rd stage, and does not always cooperate in the 2nd stage. The six plans of actions for each player i are denoted sf'T ,sf'v, sf'E, sfT, sfV and sfE, where N denotes that i is nice in the sense of cooperating in the 1st stage, where R denotes that i is rude in the sense of defecting in the 1st stage, where T denotes that i plays tit-for-tat in the sense of cooperating in the 2nd stage if and only j =1= i has cooperated in the 1st stage, where V denotes that i plays inverse tit-for-tat in the sense of defecting in the 2nd stage if and only if j =1= i has cooperated in the 1st stage, and where E denotes that i is exploitive in the sense of defecting in the 2nd stage independently of what j =1= i has played in the 1st stage. The strategic form after elimination of all other plans of actions is given in Figure

167

Applying full permissibility to extensive games

12.2. Note that none of these plans of actions are weakly dominated in the full strategic form. Proposition 40 shows that any fully permissible set is a subset of the set of strategies surviving the Dekel-Fudenberg procedure. Hence, only subsets of can be i's choice set under common certain belief of full admissible consistency. Furthermore, under common certain belief of full admissible consistency, we have for each player i that

sf'

• any choice set that contains sf'T must also contain E, since sf'T is a maximal strategy only if E is a maximal strategy,

sf'

sf'

E, since sf'v • any choice set that contains sf'v must also contain is a maximal strategy only if E is a maximal strategy,

sf'

• any choice set that contains sfT must also contain sfE, since sfT is a maximal strategy only if sfE is a maximal strategy, • any choice set that contains sf V must also contain sfE, since sf V is a maximal strategy only if sfE is a maximal strategy, Given that the choice set of the opponent satisfies these conditions, this implies that • if sf'E is included in i 's choice set, only the following sets are candisNE sRT sRE}' {sNV sNE sRV sRE} dates for i's choice set·. {sNT Z'z'Z'Z z'z'Z'z' or {sf'E, sfE}. The reason is that E is a maximal strategy only if i considers it subjectively possible that j's choice set contains sfT (and hence, E) or sfT (and hence, sfE).

sf'

sf

sf'

• if sfE , but not E, is included in i's choice set , only the following sets are candidates for i 's choice set: {sfT , sfE}, {sf V, sfE}, or {sfE}. The reason is that sfE is a maximal strategy only if i considers it subjectively possible that j's choice set contains V, E, sf V, or sfE.

sf sf

This in turn implies that • i 's choice set does not contain sf'v or sf V since any candidate for j's choice set contains s~E, implying that E is preferred to sf'v and sfE is preferred to si v.

sf'

Hence, the only candidates for i's choice set under common certain belief of full admissible consistency are {sNT sNE sRT sRE} {sNE sRE} z 'z'z'Z' Z'z' {sfT, sfE}, and {sfE} . Moreover, it follows from Proposition 39(iii) that all these sets are indeed fully permissible since

168

CONSISTENT PREFERENCES

T sRE} but not • {SNT SNE SRT sRE} is i's choice set if he deems {sfl Z'z'Z'Z J'J' sNE sRT sRE} as possible candidates for )"s sRE} and {sNT { sNE J 'J J'J'J'J' choice set,

T sNE sRT sRE} as a • {sNE s!tE} is i's choice set if he deems {sf:l Z'z J'J'J'J possible candidate for j's choice set , • {sfT, sfE} is i's choice set if he deems {sfE} as the only possible candidate for j's choice set ,

• {sRE} is i's choice set if he deems {sNE sRE} Z J ' sRE} J ' but not {sRT J 'J and {SfT, sfE,sfT, sfE}, as possible candidates for j's choice set . While play in accordance with strategies surviving the Dekel-Fudenberg procedure does not provide any prediction other than both players defecting in the 3rd stage, the concept of fully permissible sets has more bite. In particular, a player cooperates in the 2nd stage only if the opponent has cooperated in the 1st stage. This implies that only the following paths can be realized if players choose strategies in fully permissible sets:

((cooperate, cooperate), (cooperate, cooperate) , (defect, defect)) (( cooperate, cooperate), (cooperate, defect), (defect, defect)) and vice versa ((cooperate, defect), (defect, cooperate), (defect, defect)) and vice versa (( cooperate, cooperate) , (defect, defect), (defect, defect)) ((cooperate, defect), (defect, defect), (defect, defect)) and vice versa (( defect, defect), (defect, defect), (defect, defect)). That the path (( cooperate, defect), (cooperate, defect), (defect, defect)) or vice versa cannot be realized if players choose strategies in fully permissible sets can be interpreted as an indication that the present analysis seems to produce some element of reciprocity in the 3-period "prisoners' dilemma" game.

12.4

Forward induction

In Chapter 11 we have already seen how the concept of fully permissible sets promotes the forward induction outcome, (InL , f) , in the PRSF of the "battl e-of-t he-sexes with an outside option" game ri, illustrated in Figure 2.6. In this section we first investigate whether this conclusion carries over to two other variants of the "bat t le-of-t he-sexes" game, before testing the concept of fully permissible sets in an economic application.

Applying full permissibilit y to extensive gam es

NU

ND BU BD

Figure 12.3.

169

ff fr r e r r 3, 1 3, 1 0, 0 0, 0 0, 0 0, 0 1,3 1,3 2, 1 -1, 0 2,1 -1, 0 -1,0 0, 3 -1,0 0, 3

G I 3 (t he pure st rategy reduced st rategic form of "burn ing money" ).

The "battle-of-the-sexes" game with variations. Consider first the "burn ing money" game due t o van Damme (1989) and Ben-Porath and Dekel (1992). Game G 13 of Figure 12.3 is the PRSF of a "battleof-th e-sexes" game with the addition that 1 can publicly destroy 1 unit of payoff before the "battle-of-t he-sexes" game starts. BU (NU) is the st rat egy where 1 burns (does not burn), and then plays U , et c., while f r is t he strategy where 2 responds with f conditional on 1 not burning and r conditional on 1 burning, etc . The forward indu ction out come (support ed e.g. by IEWDS) involves implementation of 1's preferr ed "battleof-t he-sexes" outcome, with no payoff being burnt. One might be skeptical to the use of IEWDS in t he "burning money" game, because it effectively requir es 2 to infer t hat B U is infinit ely more likely t han BD based on t he sole premise t hat BD is eliminated before BU, even though all st rategies involving burning (i.e. both BU and BD ) are event ually eliminated by t he pro cedure. On t he basis of thi s premise such an inference seems at best to be questi onable. As shown in Table 12.1, the application of our algorit hm IECFA yields a sequence of it erations where at no stage need 2 deem B U infinit ely more likely tha n BD , since {NU} is always included as a candidate for 1's choice set. The procedure uniquely determines {NU} as 1's fully permissible set and {ee, f r } as 2's fully permissible set . Even though the forward induction out com e is obtained, 2 does not have any assessment concerning the relative likelihood of opp onent st rate gies condit ional on burning; hence, she need not int erpret burning as a signal that 1 will play according with his preferr ed "battle-of-t he-sexes" outcome.i'

3 Also

Batt igalli (1991), Asheim (1994), and Dufwenb erg (1994) , as well as Hurkens (1996) in a different context, ar gue t hat (NU, f r ) in addition to (NU, eel is viable in "burn ing money" .

170

CONSISTENT PREFERENCES

Table 12.1.

Applying IECFA to "burning money" .

2(0) 2(1)

= ~l X ~2 = {{ NU}, {ND} , {BU}, {NU, ND}, {ND, BU}, {NU, BU}, {NU, ND, BU}}

2(2)

= {{NU}, {ND}, {BU}, {NU, ND}, {ND, BU}, {NU, BU}, {NU, ND, BU}}

x ~2 x {{ff}, {rf}, {ff,fr} ,{rf,rr}, {ff,rf}, {ff,fr,rf,rr}} = {{NU}, {BU}, {ND, BU}, {NU, BU}, {NU, ND, BU}} x {{ff} ,{rf}, {ff,fr}, {rf ,rr}, {ff,rf}, {ff,fr,rf,rr}} 2(4) = {{NU} , {BU} , {ND, BU}, {NU, BU}, {NU, ND, BU}} x {{ff} ,{rf} ,{ff,fr} ,{ff,rf}} 2(5) = {{NU},{BU}, {NU, BU}} x {{ff}, {rf} ,{€f,rf}, {€f,rf}} 2(6) = {{NU} ,{BU},{NU,BU}} x {{€f},{ff,fr}, {€f,rf}} 2(7) = {{NU}, {NU, BU}} x {{€f}, {ff,fr}, {€f,rf}} 2(8) = {{NU}, {NU, BU}} x {{€f} ,{€f,fr}} 2(9) = {{NU}} x {{€f},{€f,fr}} II = 2(10) = {{NU}} x {{€f,fr}}

2(3)

We next turn now to a game introduced by Dekel and Fudenberg (1990) (cf. their Figure 7.1) and discussed by Hammond (1993), and which is reproduced here as r~ of Figure 12.4. It is a modification of r~ which introduces an "extra outside option" for player 2. In this game there may seem to be a tension between forward and backward induction: For player 2 not to choose out may seem to suggest that 2 "signals" that she seeks a payoff of at least 3/2, in contrast to the payoff of 1 that she gets when the subgame structured like r~ is considered in isolation (as seen in the analysis of rD. However, this intuition is not quite supported by the concept of fully permissible sets . Applying our algorithm IECFA to the PRSF of r~ yields:

= ~1 X ~2 3(1) = {{Out}, {InL},{Out ,InL}} 3(0)

x {{out} , {inr}, {out, inz}, {out, inr}, {in£, inr}, {out, in£, inb}} 3(2) = {{Out}, [In.L}, {Out, rnL}} x {{out}, {out, in£}, {in£, inr}} 3(3)

= {{InL},{Out,InL}} x {{out} , {out,in£}, {in£,inr}}

II = 3(4) = {{rnL} , {Out, rnL}} x {{out}, {out, in£}} . The only way for Out to be a maximal strategy for player 1 is that he deems {out} as the only subjectively possible candidate for 2's choice

171

Applying full permissibility to extensive games 3

2

2 3 2

out

2 2

Out

III

1 I nR

InL

e 3 1

._-_

__._

r

o o

~. _-_

_-_ .

R 0 0

r

1 3

out inR 3 3 Out 2' 2 2, 2 3 3 InL 2' 2 3, 1 3 3 I nR 2' 2 0, 0

inr 2, 2 0, 0 1,3

Figure 12.4. r~ and its pure strategy reduced strategic form .

set, in which case 1's choice set is {Out, InL}. Else {InL} is 1's choice set. Furthermore, 2 can have a choice set different from {out} only if she deems {Out, InL} as a subjectively possible candidate for 1's choice set. Int uit ively this means that if 2's choice set differs from {out} (i.e., equals {out, inâ‚Ź} ), then she deems it subjectively possible that 1 considers it subjectively impossible that inR is a maximal strategy for 2. Since it is only under such circumstances that inÂŁ is a maximal element for 2, perhaps this strategy is better thought of in terms of "strategic manipulation" than in terms of "forward induction". Note that the concept of fully permissible sets has more bite than the Dekel-Fudenberg procedure; in addition to the strategies appearing in fully permissible sets also inr survives the Dekel-Fudenberg procedure. An economic a p plication. Finally, we apply the concept of fully permissible sets to an economic model from organization theory. Schotter (2000) discusses in his Chapter 8 incentives schemes for firms and the moral hazard problems that may plague them. "Revenue-sharing contracts", for example, often invite free-riding behavior by the workers, and so lead to inefficient outcomes. However, Schotter points to "forcing contracts" - incent ive schemes of a kind introduced by Holmst rom (1982)-as a possible remedy: Each worker is paid a bonus if and only if the collective of workers achieve a certain level of total production. If incentives are set right, then there is a symmetric and efficient Nash equilibrium in which each worker exerts a substantial effort. Each worker avoids shirking because he feels that his role is "pivotal", believing that any reduction in effort leads to a loss of the bonus.

172

CONSISTENT PREFERENCES

1

In

Out

------------------~------------------

out

III

out

W W

W W

W

W

S

o o Figure 12.5.

III

1

out

s

H

------------------~-----------------h S

o

-c

r 14

-c

0

h b-c b-c

inh Out W ,W W,W W,W InS W,W 0, 0 0, -c InH W ,W -c,O b-c,b-c IllS

and its pure strategy reduced strategic form.

However, forcing contracts are often problematic in that there typically exists a Nash equilibrium in which no worker exerts any effort at all. How serious is this problem? Schotter offers the following argument in support of the forcing-contract (p. 302): "While the no-work equilibrium for the forcing-contract game does indeed exist , it is unlikely that we will ever see this equilibrium occur. If workers actually accept such a contract and agree to work under its terms, we must conclude that they intend to exert the necessary effort and that they expect their coworkers to do the same . Otherwise, they would be better off obtaining a job elsewhere at their opportunity wage and not wasting their time pretending that they will work hard." Schotter appeals to intuition, but his argument has a forward induction flavor to it. We now show how the concept of fully permissible sets lends support. Consider the following situation involving a forcing contract: A firm needs two workers to operate. The workers simultaneously choose shirking at zero cost of effort , or high effort at cost c > O. They get a bonus b > c if and only if both workers choose high effort. As indicated above, this situation can be modeled as a game with two Nash equilibria (S, s) and (H, h), where (H, h) Pareto-dominates (S, s). However, let this game be a subgame of a larger game . In line with Schotter's intuitive discussion, add a preceding stage where each worker simultaneously decides whether to indicate willingness to join the firm with the forcing contract, or to work elsewhere at opportunity wage w, o < W < b - c. The firm with the forcing contract is established if and only if both workers indicate willingness to join it .

Applying full permissibility to extensive games

173

This situation is depicted by the extensive game f 14 . Again, we analyze the PRSF (d. Figure 12.5). Application of IECFA yields: 3(0) = ~1 X ~2 3(1) = {{Out}, {InH} , {Out, InH}} x {{out}, {inh}, {out,inh}} 3(2)

= {{InH} ,{Out,InH}} x {{inh} ,{out,inh}}

II = 3(3) = {{InH}} x {{inh}} . Interpretation: 3(1): Shirking cannot be a maximal strategy for either worker since it is weakly dominated. 3(2): This excludes the possibility that a worker's choice set contains only the outside option. 3(3): Since each worker certainly believes that hard work is, while shirking is not, an element of the opponent's choice set, it follows that each worker deems it infinitely more likely that the opponent chooses hard work rather than shirking. This means that, for each worker, only hard work is in his choice set, a conclusion that supports Schotter's argument .

12.5

Concluding remarks

In this final chapter of the book we have explored the implications of the concept of fully permissible sets in extensive games. In Chapter 11 we have already seen-based on Asheim and Dufwenberg (2003a)-that this concept can be characterized as choice sets under common certain belief of full admissible consistency. Full admissible consistency consists of the requirements â€˘ 'caution', â€˘ 'robust belief of opponent rationality', and â€˘ 'no extraneous restrictions on beliefs', and entails that one strategy is preferred to another if and only if the former weakly dominates the latter on the union of the choice sets that are deemed possible for the opponent, or on the set of all opponent strategies. The requirement of 'robust belief of opponent rationality' is concerned with strategy choices of the opponent only initially, in the whole game, not with choices among the remaining available strategies at each and every information set. To illustrate this point , look back at fn and consider a type of player 2 who deems {D} as the only subjectively possible choice set for 1. Conditional on 2's node being reached it is clear that 1 cannot be choosing a strategy that is maximal given his preferences. Conditional on 2's node being reached, the modeling of the

174

CONSISTENT PREFERENCES

current chapter imposes no constraint on 2's assessment of likelihood concerning which non-maximal strategy FF or FD that 1 has chosen. This crucially presumes that 2 assesses the likelihood of different strategies as chosen by player 1 initially, in the whole game. It is possible to model players being concerned with opponent choices at all information sets. In r 11 this would amount to the following when player 2 is of a type who deems {D} as the only possible choice set for 1: Conditional on 2's node being reached she realizes that 1 cannot be choosing a strategy which is maximal given his preferences. Still , 2 considers it infinitely more likely that 1 at his last node chooses a strategy that is maximal among his remaining available strategies given his conditional preferences at that node. In Section 12.2 we argued with Ben-Porath (1997) that this is not necessarily reasonable, a view which permeates the working hypotheses on which the current chapter in grounded. Yet, research on the basis of this alternative approach is illuminating and worthwhile. Indeed, Chapters 7-9 of this book have reproduced the epistemic models of Asheim (2002) and Asheim and Perea (2005) where each player believes that his opponent chooses rationally at all information sets. The former model yields an analysis that is related to Bernheim's (1984) subgame rationalizability, while the latter model demonstrates how it-in accordance with Bernheim's conjecture-is possible to define sequential rationalizability. Moreover, Chapter 10 has considered the closely related strategic form analyses of Schuhmacher (1999) and Asheim (2001) that define and characterize prop er rationalizability as a non-equilibrium analog to Myerson's (1978) proper equilibrium. Analysis that goes in this alternative direction promotes concepts that imply backward induction without yielding forward induction. Thus, they lead to implications that are significantly different from those of the current final chapter, where forward induction is promoted without insisting on backward induction in all games. The tension between these two approaches to extensive games cannot be resolved by formal epistemic analysis alone. It is worth noting, though, that the analysis-independently of this issue-makes use of the 'consistent preferences' approach to deductive reasoning in games .

Appendix A Proofs of results in Chapter 4

P roof o f P roposi tion 6. Only if. Assume that >::d is admissible on E. Let e E E and f E -,E. It now follows dir ectl y that e is not Savage-null at d and that p >-'{e} q impli es p >-'{e.n q . If. Assume that e E E and f E -,E imp ly e »" f . Let p and q satisfy that p E weakl y domin ates q E at d. Then there exists eo E E such t hat vd( p(eo)) > vd( q( eo)) . Write -,A = {fJ , . . . , f n} . Let , for m E {O, . .. , n} ,

n:~~m p(d' ) + nT~ 1 q (d' ) if d' m(d' ) = P

{

= eo

p (d' ) q (d' )

if d' E E \ eo if d'

= f m'

and m' E {I , .. . , m}

p (d' )

if d'

= f m'

and m' E {m+ l , . . . , n }.

Then p = p o, p m- l >-d p m for all m = {I , ... , n } (since e E E and f E -,E imp ly th at e »d J) , and p (n ) >-d q (since p (n ) weakl y dominates q at d with vd( pn(eo)) > vd( q(eo)) ). By t ra nsit ivity of >:: e, it follows t hat p >-d q . • P r o of of Proposi tion 7. (Q seri ol.} If d is Savage-null at d, t hen there exists e E T d such that e is not Savage-null at d since >:: d is nontrivial (Axiom 3) . Clearl y, d is not infinitely mor e likely than e at d, and dQe. If d is not Savage-nu ll at d, then dQd since d is not infinitely more likely t han itself at d. (Q transitive.) We must show that dQe and eQ f imp ly dQf . Clearl y, dQe and eQ f imply d ::::; e ::::; f , and that f is not Savage-nu ll at d. It remains to be shown t ha t d »d f do es not hold if dQ e and eQ f . Suppose to t he cont rary that d »d f. It suffices to show that dQe cont radicts eQ f. Since f is not Savage-null at d ::::; e, e » d f is needed to cont ra dict eQ f . This follows from partitional priority (Axiom 11) becau se dQe entails that d » d e does not hold . (Q satisfies forward lin earity.) We must show th at dQ e and dQf imp ly eQ f or fQ e. From dQ e and dQf it follows that d ::::; e ::::; f and that both e and f ar e not Savage-null at e::::; f . Since e » e f and f » f e cannot both hold , we have that eQ f or fQ e. (Q satisfies quasi-backward linearity .) We must show t hat dQf and eQ f imp ly dQ e or eQd if :Jd' E F such t hat d'Q e. From dQf and eQ f it follows t hat d ::::; e ::::; f ,

176

CONS IS TEN T PREFERENCES

while d' Q e implies that e is not Savage-null at d' ::::; d ::::; e. If d is Savage-null at d, then d »d e cannot hold , implying that dQ e. If d is not Savage-null at d ::::; e, then d » " e and e »e d cannot both hold , implying that dQ e or eQd. • Proof of Proposition 8. (R; serial .) For all d E F , pt i- 0. (R e tran sitive.) We must show that dRee and eli»] imply d.R»], Since d.Ree impli es that d se e, we have that pt = pi . Now, eRe! (i.e., f E pi) implies dRe! (i.e.,

f

E

pt)·

(Re Euclidean.) We must show that dRee and dRe! imply eR ef. Since dRee impli es that d::::; e, we have that pt = pi . Now, d.R»] (i.e., f E pt) implies eli»] (i.e., fEP f). (dRee implies dRe+ l e.) This follows from the property that pt ~ Pt+I' (3f such that dRe+d and eR e+ d) implies (3!' such that d.R» ] ' and eR e!' ) . Since dRe+d impli es that d::::; f and eR e+ d impl ies that e ::::; f , we have t ha t d::::; e and pt = pi· By the non-emptiness of this set , 3!, such t hat dRe!' and eR e!' . •

Proof of Proposition 9. (i) (dQd is equivalen t to d being no t Savage-null at d .) If dQd , t hen it follows dir ectly from Definition 2 that d is not Savage-null at d. If d is not Savage-null at d, t hen by Definition 2 it follows th at dQd since d ::::; d and not d »d d. (dR Ld is equiv alen t to d being n ot S avage-null at d .) By Definiti on 3, dR Ld iff d E pi = ",d, which dir ectl y est ablishes the result . (ii) Only if. Assum e that dQ e and not eQd. From dQ e it follows that d ::::; e an d e is not Savage-null at d, i.e. e E ",d ( ~ T d) . Consider E := {e' E F I eQe' }. Clearly, e E E ~ ",d ( ~ T d) and d E Td\ E i- 0. If e' E E and f E Td\E, t hen not «o], since ot herwise it would follow from eQ I' and the transitivity of Q t hat eQ f, t hereby cont radict ing f rf- E . If, on the one hand , f E ",d\E, t hen e' »d f since f is not Savage-null at d ::::; e' and e' Q f does not hold . If, on the ot her hand , f rf- ",d, then e' »d f since f is Savage-null at d and e' is not. Hence, e' E E and f E -, E imply e' »" f . By Proposition 6, ~ d is admissible on E, ent ailin g t hat 3£ E {I , .. . , L } such that pt = E. By Definition 3, dRee and not eRe d since e E E and dE Td\E. If. Assum e that 3£ E {I , . . . , L} such t hat dR ee and not eR ed. From dRee it follows that d ::::; e and e E pt(~ ",d); in particular , e is not Savage-null at d. Since eRed does not hold , however , d rf- pi = Pt. By constructio n, ~ d is admissible on pt , and it now follows from Propositi on 6 t hat e » d d. Fur therm ore, e »d d implies t ha t d »d e does not hold . Hence, dQ e since d ::::; e, e is not Savage-null at d and d »d e does not hold , while not eQd since e »" d. • Proof of Lemma 3. Since ",d = {e E T d I eQe }, it follows that 3e I E Td n if> such that el Qel if ",d n if> i- 0. Eith er , "If E Td n if>, f Qe l - in which case we are through - or not . In the latter case, 3e2 E T d n if> such t hat e2Qel does not hold . Since ei , e2 E T d, 3e~ E T d such that e l Qe~ and e2 Qe~ . Since ei Qel and not e2Qe l it now follows from quasi-backward linearity th at er Q e2. Moreover , not e2Qe l implies d e2 i- el · Either "If E T n if>, fQ e2 - in which case we are through - or not . In t he lat t er case we can, by repeating t he above argument and invokin g transitivity, show t he existe nce of some e3 E T d n if> such t hat e l Q e3, e2Qe3, and C3 f= e i , C2 . Since T d n if> is finite, this algorit hm converges to some e satis fying , "If E T d n if>, fQ e. • To prove Proposition 11 it suffices to show the following lemm a.

Appendix A : Proofs of results in Chapt er LEMMA

14 If </> E <P , then (3d( </»

4

177

= p1 n </>, where £ := min{k E {I , . . . , L } I p~ n </>

=1=

0}. Proof. ((3d(</» ~ p1 n </» Assume that (r d n </»\p1 =1= 0. Let e E (r d n </»\pt . Since pt n </> =1= 0, 3f E p1 n </>. Then , by Definition 3 eRe! and not f R Re, which by P rop ositi on 9(ii) impli es eQf and not fQ e. Hence, e E (r d n </»\(3d(</» , and pt n </> = (r d n </» np1 :;2 (r d n </» n (3d (¢ ) = (3d (</» . Assume t hen t ha t (r d n ¢ )\pt = 0. In this case , pt n </> = (r d n </» n pt = r d n </> :;2 (3d (</» . (p1 n </> ~ (3d (</» ) Let e E pt n </>. If f E pt n </>, t hen fR Lf since p1 ~ pi, and fQf by Prop osition 9(i) . Since e, f E r d and fQf , it follows by quas i-bac kwar d linearity of Q that fQf or eQ f. However , since by cons t ruc t ion, Yk E {I , . . . £ - I}, p~ n </> = 0, t here is no k E {I , . . . £ - I} such t hat f Rk e and not eR kf or vice versa , and P rop osition 9(ii) implies t hat both fQ e and eQ f must hold . In particular , fQ e. If, on the ot her hand , f E (r d n </» \pt , t hen by Definition 3 f RRe and not eRe!, implying by Propos itio n 9(ii) that fQ e. Thus, v! E r d n </>, fQ e, and e E (3d (¢ ) follows. • Proof of Proposition 12. Recall t hat BO E := n <PE<P EB(</»E, where <P E := ndEF <P~ is non-empty and defined by, Yd E F , <P~ := {</> E <p d l E n ",d n </> =1= 0ifE n ",d=l=0} . (If 3£ E {I , . .. , L} such that pt = E n ",d , then d E BOE .) Let p1 = E n ",d and consider any </> E <P E. We must show that d E B(¢)E. By the definition of <P E, E n ",d n </> =1= 0 since </> E <P E and E n ",d = p1 =1= 0. Since p1 n </> = E n ",d n </>, it follows t ha t 0 =1= p1 n </> ~ E , so by P ropos it ion 11, s « B(</» E. (If d E B OE, t hen 3£ E {I , . .. , L} such that pt = En ",d .) Let d E BOE ; i.e., Y¢ E <P E, dE B(</»E. We first show that p~ ~ E . Consider some </>' E <P E sat isfying r d n ¢' = (E n r d ) U pt . Since dE B(¢')E , 3k E {I , . . . , L} such t hat 0 =1= p~ n ¢ ' = p~ n (E U p~) ~ E . Since pt ~ p~, p~ ~ E. Let £ = max{k I p~ ~ E }. If e = L , t hen pt = «', and pt ~ E imp lies pt = E n ",d . If £ < L , t hen, sin ce p1 C pi = «', pt = pt n ",d ~ E n «' . To show t hat pt = E n ",d also in thi s case , suppose instead t hat (E n ",d )\ pt =1= 0, and consider some ¢I! E <P E sat isfyin g Tdn ¢1! = ((En",d)Upt+!)\pt. Since, Yk E {I, .. ,£}, p~ ~ pt , it follows from pt n ¢1! = 0 t ha t , Yk E {I , .., £}, p~ n ¢I! = 0. Since by const ruction, p1 ~ E, while p1+! ~ E does not hold , pt+! n ¢I! = p1+!\ p1 is not includ ed in E . Since pt c ... c pi, t here is no k E {O, . . . , L} such t hat 0 =1= p~ n ¢I! ~ E, cont ra dict ing by Proposition 11 t hat dE B(¢I!) E . Hence, p1 = En ",d . • Proof of Proposition 14. (KE n K E' = K( E n E' )) To prove KE n KE' ~ K( En E ') , let d E KE and d EKE'. Then, by Definition 4, ",d ~ E and ",d ~ E' and hence, ",d ~ E nE' , implying th at dE K(En E ') . To prove KE nKE':;2 K(En E'), let dE K(E n E' ). Then ",d ~ EnE' and hence, ",d ~ E and ",d ~ E' , implying t hat d E K E and d E KE'. (B( ¢) E n B(¢)E' = B(¢)( E n E' )) Using Definti on 5 the pro of of conjunct ion for B(¢) is ident ical to the one for K except that (3d (¢) is subst it uted for ",d . (K F = F ) KF ~ F is obvious . That KF :;2 F follows from Definition 4 since , Yd E F, ", d ~ r d ~ F. (B( </»0 = 0) This follows from Definition 5 since , Yd E F , (3d (¢ ) =1= 0, impl ying that t here exists no d E F such that (3d (¢ ) ~ 0.

178

CONSISTENT PREFERENCES

(KE ~ KKE) Let d EKE. By Definition 4, d E KE is equivalent to ",d ~ E. Sin ce \Ie E r", ", e = ",d, it follows that r d ~ KE . Henc e, ",d ~ r d ~ KE , implying by Definition 4 that d E KKE. (B( ¢»E ~ KB( ¢»E) Let d E B(¢»E . By Definition 5, d E B(¢»E is equivalent to (3d (¢» ~ E . Sin ce \Ie E r d, (3e(¢» = (3d (¢» , it follows that r d ~ B(¢»E . Hence, ",d ~ r d ~ B(¢»E , implying by Definition 4 that dE KB( ¢»E . (,KE ~ K( ,KE)) Let d E ,KE. By Definition 4, d E ,KE is equivalent t o ",d ~ E not holding. Since \Ie E r d, ", e = ",d, it follows that r d ~ ,KE. Hence, ",d ~ r d ~ ,KE, implying by Definition 4 that d E K(,KE) . (,B( ¢»E ~ K(,B( ¢»E)) Let d E ,B(¢»E. By Definition 5, d E ,B(¢»E is equivalent to (3d (¢» ~ E not holding. Sin ce \Ie E r", (3e(¢» = (3d (¢» , it follows that r d ,B(¢»E . Henc e, ",d ~ r d ~ ,B(¢»E , implying by Definition 4 that dE K(,B( ¢»E) . •

<

Proof of Proposition 15. (1.) (3d (¢» ~ ¢> follows by definition sinc e, \Ie E (3d(¢», e E ¢>. (2.) By Definitions 2 and 3 and Proposition 9, (3d = p~ . Hen ce, (3d n ¢> i- 0 implies p~ n ¢> i- 0 and min{ e I pt n ¢> i- 0} = 1. By Lemma 14, (3d (¢» = p~ n ¢> = (3d n ¢>. (3.) This follows directly from Lemma 3, sin ce ¢> E <P implies that, \ld E F , ",d

n ¢> i- 0.

(4.) Let (3d (¢» n ¢>' i- 0. By Lemma 14, (3d (¢» = pt n ¢> i- 0 where e := min{k I p% n ¢> =1= 0}. Likewis e, (3d (¢> n¢>' ) = pt, n ¢>n ¢>' , wher e l' := min{k I p%n ¢>n ¢>' i- 0}.

e

e :::;

It suffices to show that = l'. Obviously, (pt n ¢» n ¢>' = pt n ¢> n ¢>' implies that l' ?:. e.

l' . However , 0 i- (3d (¢» n ¢>' =

•

Proof of Proposition 16. That KE ~ B OE follows from Definition 4 and Propositions 9 and 12 since ",d ~ E implies that pi = ",d = ",d n E . That B OE ~ B(F)E follows from Definition 6 since FE <PE . • Proof of Proposition 17. (B O E n B O E' ~ B O(E n E')) Let d E B OE and O d E B E'. Then, by Proposition 12, there exist k such that pt = E n ",d and k' su ch that pt, = E' n «' , Sinc e p~ C .. . C pi, eit her pt ~ pt, or pt :2 pt" or equivalent ly, En ",d ~ E' n ",d or E n ",d :2 E' n ",d. Hen ce, eit her pt = E n ",d = EnE' n ",d or pt, = E' n ",d = EnE' n ",d , implying by Proposition 12 that a E B O(E n E'). (B OE ~ KB oE) Let d E B O E . By Proposition 12, d E B OE is equivalent to 3e E {1, . . . , L} such that pt = En «": Since \Ie E r " , Pe = pt and ", e = ", d , it follows that r d ~ B OE. Hence , ",d ~ r d ~ B OE, implying by Definition 4 that dE KB o E . (,B OE ~ K( ,B o E)) Let d E ,B o E . By Proposition 12, dE ,B oE is equivalent to ther e not existing k E {1, . . . , L} such that pt = E n ",d. Sin ce \Ie E r d , Pe = pt a nd « = ",d, it follows that r d ~ ,BoE. Hen ce, ",d < r d ~ ,B oE , implying by Definition 4 that dE K(,B oE) . • To prove Proposition 18 the following lemma is helpful.

15 Assume that t

d

satisfies A xioms 1 and 4" (in addition to the assumptions made in Section 4.1), and let e,l' E {1 , ... , L d } sat isfye < l' . Th en p>- ddq rr f implies p>- ddU d q .

LEMMA

tt P.

rr pI

Proof. This follows from Proposition 3. •

Append ix A : Proofs of results in Chapter

4

179

Pro of of Proposition 18 . (If E is assumed at d, then d E BOE .) Let E be assumed at d. Th en it follows that ?::'f,.. is nontrivial; hence, E n ",d # 0. Assume th at p En" d weakl y dominates q E n " d at d. Since En ", d # 0, we have that p ?-'f,.. q . Hence, it follows from the premise (viz., that E is assumed at d) that p ?-d q . This shows that ?::d is admissible on En ",d , and, by Proposition 12, dE B OE . (If d E B OE, then E is assumed at d .) Let dE B OE, so by Proposition 12 ?::d is admissible on E n ", d (# 0). Hence, by Proposition 6 , e E E n 'k and f E . (E n ",d) implies e » d f. By Axiom 4" thi s in t urn implies th at :J€ such that E n", d

= Uk= l 1l'kd , e

since the first property of Axiom 4" - t he Archimedean property of ?::d within each partitional element - rul es out that e and f are in t he sam e element of th e partition {1l'r , .. . , 1l'~d } if e » d f. Assum e that p ?-'f,.. q . Then p ?- ~ n "d q , and , by the above argument , d

p ?'-U ('=l 7r ~ q.

By complete ness and t he partition al Archimedean property, Lemm a 15 entails that :Je' E {I , ... , €} such t hat P ?-~ dq and, Vk E {I , . . . , €' - I } , p~~ d q . t'

k

d

By Lemma 15, p >" q since Ut=11l'~ = ", d. Hence, p ?-'f,.. q implies p >" q . Moreover , ?:: 'f,.. is nontrivial since E n ", d # 0, and it follows from Definition 7 th at E is assumed at d. •

Appendix B Proofs of results in Chapters 8-10

For t he pr oofs of Propositions 30, 34, 36, and 37 we need two results from Blume et al. (1991b) . To st ate these results, introduce the following notation. Let X = (P,l , ..., p,L) be an LPS on a finit e set F and let r = (ri , ..., r L-l) E (0,1 )L-l . Then , r D.\ denot es the probability distribution on F given by the nest ed convex combinat ion

The first is a rest atement of Prop osition 2 in Blume et al. (1991b) .

16 Let (x( n)) nEN be a sequence of probability distributions on a fin it e set F . Th en, th ere exists a subsequen ce x ( m) of (x(n) )nEN, an LPS .\ = (p,1, ..., P, L), an d a sequen ce r (m ) of vect ors in (0, I) L-l converging to zero such that x (m ) = r (m )D.\ fo r all m . LEMMA

The second is a vari ant of Proposition 1 in Blume et al. (1991b) .

17 Con sider a typ e ti of play er i whose preferences over acts on S j x Tj are represented by V;i - with V;i 0 z = Ui - and .\ti = (p,ii , .. . , p,i) E L~(Sj x T j) . Th en, for every sequence (r(n)) nEN in (0, I) L- l convergi ng to zero th ere is an n' su ch that, Vs i , s; E Si, Si -;- ti s; if and only if

LEMMA

L(r(m)D.\t i )(s j, t j )Ui( Si , Sj )

L Sj

>L Sj

tj

L(r(m)D.\t i )(Sj , t j )Ui( S;, Sj ) tj

fo r all n ~ n'.

Proof. Suppose that s, -;-ti s;. Then , there is some â‚Ź E {I, ..., L} such that

LLP,~i(Sj , tj)Ui(Si , Sj) 5j

for all k

tj

= LLP,~i(Sj ,tj)Ui(S; ,Sj) 5j

(8.1)

tj

< â‚Ź and LLP,~i(Sj ,tj)Ui(Si , Sj) Sj

tj

> LLP,~i(Sj,tj)Ui(S; , Sj) . Sj

tj

(8.2)

182

CONSIS TEN T PREFERENCES

Let (r(n)) nEN be a sequ ence in (O,l) L- l converging to zero. By (B.1) and (B .2),

~)r(n)DAt i)( Sj , t j )Ui(Si , Sj ) > L

L Sj

Sj

tj

L(r(n)DA ti )(sj, tj)Ui( S;, Sj ) tj

if n is large enough. Since S, is finit e, this is t rue if n is large enough for any Si, s; E S, satisfying s, >- i, The ot her dire ction follows from the proof of Proposition 1 in Blume et al. (1991b) . •

s;.

For the proofs of Propositions 30 and 34 we need the following definitions. Let the LPS Ai = (J1. L . . . , J1.D E L~(Sj) have full support on S j . Say that the behavior st ra tegy Uj is induced by Ai if for all n « Hj and a E A(h),

u(h)(a) '= J1.HS j(h, a)) J1.HSj(h)) , ) .

e

where = min{k I sUPP At n S j(h) =I- 0}. Moreover , say that player i 's beliefs over past oppo nent act ions /3i are induced by Ai if for all h E Hi and x E h, J1.~(Sj(X))

/3i (h )(x ) := J1. f.(Sj(h)) , where

e = min{k I sUPP Ak n Sj(h) =I- 0}.

Proof of Proposition 30. (Only if.) Let (u, (3) be a sequent ial equilibrium . Then (u, (3) is consistent and hence there is a sequ ence (u(n)) nEN of complete ly mixed behavior strategy profiles converging to a such that the sequence (/3(n)) nE N of indu ced belief systems converges to /3. For each i and all n , let Pi(n) E ~(S;) be the mixed representation of ui(n) . By Lemma 16, the sequence (pj(n)) nEN of probability distributions on S j contains a subsequence pj(m) such that we can find an LPS Ai = (J1.L . . . , J1.i) with full support on Sj and a sequ enc e of vectors rem) E (O,l) L-l converging to zero with pj(m) = r(m)DA i for all m . W. l.o.g., we assume that pj(n) = r(n)DA i for all n E N. We first show that Ai induces the behavior st rat egy Uj. Let Uj be the behavior strategy induced by Ai. By definition, Vh E H j , Va E A(h),

e

where = min {k I supp Ak n Sj (h) =I- 0}. For the fourth equat ion we used the fact that pj(n) is the mixed representation of uj(n) . Hence, for each i , Ai induces Uj. We then sh ow that Ai induces the beliefs /3i . Let /3i be player i's beliefs over past opponent actions induced by Ai. By definition, Vh E Hi, "Ix E h,

/3i (h )(x )

J1.HS j( x)) = lim r(n)DA'(Sj(x)) J1.HSj(h)) n-oo r(n)DN(Sj(h)) .

pj(n)(Sj( x))

lim Pi ()(S n j (h))

n -c-oc

. ()()( ) ()( ) = n-+ limoo /3i n h x ·= /3i h x ,

183

Appendix B: Proofs of results in Chapters 8-10

where f = min{k I sUPPA ~ n Sj(h) -I0} . For the fourth equality we used the facts that pj(n) is the mixed representation of O'j(n) and (3i(n) is induced by O'j(n). Hence, for each i , N induces (3i . We now define the following epistemic model. Let TI = {tJ} and T2 = {t2} . Let, for each i , V~ i satisfy V~ i 0 Z = Ui , and (Ati,ft i) be the SCLP with support Sj x {tj} , where (1) Ati coincides with the LPS Ai construct ed above, and (2) f ti(Ej) = min{f I supp A~i n e, -I 0} for all (0 -I) s, S;; s, x {t j} . Then, it is clear that (tl, t2) E [u], there is mutual cert ain belief of {(t I, t2)} at (tl , t2), and for each i , a , is induced for ti by tj . It remains to show that (tl ,t2) E [isr]. For this, it is sufficient to show , for each i , that a , is sequentially rational for ti . Suppose not . By the choice of f t i , it then follows that there is some inform ation set h E Hi and some mixed strategy Pi E f:l(Si(h)) that is outcome-equivalent to O'il h such that there exist s, E Si(h) with Pi(S;) > 0 and E Si(h) having the property that

s:

where f = min{k I sUPPA~i n (Sj(h) x {t j}) -I 0} and Jl~i ! Sj ( h ) E f:l(Sj(h)) is the conditional probability distribution on S, (h) induced by Jl~i . Recall that Jl~i is the f-th level of the LPS Ati . Since the beliefs (3i and the behavior strategy 0' j are induced by Ai, it follows that Ui(Si ,Jl~i ! Sj ( h )) = Ui(Si ,O'jj(3 i) !h and Ui(S: ,Jl~i! Sj (h )) = Ui(S:,O'j ;(3i)lh and hence

which is a contradiction to the fact that (0', (3) is sequentially rational. (If) Suppose that there is an epistemic model with (tl ' t2) E [u] n [isr] such that there is mutual certain belief of {(tl , t2)} at (tl ' t2), and for each i , a , is induced for i ; by tj . We show that 0' = (0'1 ,0'2) can be extended to a sequential equilibrium. For each i, let Ai = (JlJ., . . . , JlL) E Lf:l( Sj) be the LPS coinciding with Ati, and let (3i be player i 's beliefs over past opponent choices induced by Ai. Write (3 = ((31 , (32) . We first show that (0', (3) is consistent . Choos e sequences (r(n)) nEN in (0, 1)L-I converging to zero and let the sequences (pj(n)) nE N of mixed strategies be given by pj(n) = r(n)DA i for all n. Since Xi has full support on Sj for every n , Pj (n) is completely mixed. For every n , let O'j (n) be a behavior represent ation of pj(n) and let (3i (n ) be the beliefs induced by O'j(n). We show that (O'j(n)) nEN converges to O'j and that ((3i(n)) nEN converges to (3i , which imply consistency of (0',(3) . Note that the inducement of O'j by ti depends on Ati through, for each h E H j , Jl~" where e = min{ k I supp A~i n (Sj (h) x {t j}) -I 0}. This implies that O'j is induced by Ai . Since O'j(n) is a behavior representation of pj(n) and O'j is induced by At, we have, Vh E H ] , Va E A(h), lim O'j (n)( h)( a) n ~ CX)

wher e f

= min{k ! supp Ai n Sj(h) -I0}.

Hence, (O'j(n)) nEN converges to O'j .

CONSISTENT PREFERENCES

184

Since (3i(n) is induced by O"j(n) and O"j(n) is a behavior representation of pj(n), and furthermore, (3i is induced by Ai, we have, 'r/h E Hi, 'r/x E h, lim pj(n)(Sj(x)) = lim r(n)DA'(Sj(x)) n-oo pj(n)(Sj(h)) n-oo r(n)DN(Sj(h))

lim (3i (n)(h) (x)

n- oo

JlHSj(x)) Jl'e(Sj(h))

= (3.(h)( t

) x ,

where e = min{k I sUPPAk n Sj(h) =1= 0}. Hence, ((3i(n)) nEN converges to (3i. This establishes that (0" , (3) is consistent. It remains to show that for each i and 'r/h E Hi,

Suppose not . Then, Ui(O"i,O"j ;(3;)lh < ui(O";,O"j ;(3i) lh for some h E Hi and some 0"; . Let Pi E t:.(Si(h)) be outcome-equivalent to O"ilh. Then, there is some s, E Si(h) with pi(Si) > 0 and some s; E Si(h) such that

Ui(Si, O"j ;(3;)lh < Ui(S; , O"j ; (3i)k Since the beliefs (3i and the behavior strategy 0"j are induced by Ai, it follows (using the notation that has been introduced in the 'only if' part of this proof) that Ui(Si, O"j ; (3i)lh = ui(si ,Jl~;lsj (h )) and Ui(S;, O"j ; (3;)lh = ui(s: ,Jl~;lsj(h ))lh and hence

Ui(Si, Jl~i ISj( hÂť) < Ui(S;, Jl~i ISj(hÂť)' which contradicts the fact that proof of this proposition. _

a;

is sequentially rational for t;. This completes the

Proof of Proposition 34. (Only if.) Let (0"1,0"2) be a quasi-perfect equilibrium. By definition, there is a sequence (O"(n))nEN of completely mixed behavior strategy profiles converging to 0" such that for each i and every n E Nand h E Hi ,

For each j and every n, let pj(n) be the mixed representation of O"j(n) . By Lemma 16, the sequence (pj(n))nEN of probability distributions on Sj contains a subsequence pj(m) such that we can find an LPS A' = (Jli, ... , Jlt) with full support on Sj and a sequence of vectors r(m) E (0, l)L-I converging to zero with

pj(m)

= r(m)DA i

for all m . W.l.o.g., we assume that pj(n) = r(n)DA i for all n E N. By the same argument as in the proof of Proposition 30, it follows that Ai induces the behavior strategy 0"i - Now, we define an epistemic model as follows. Let T 1 = {t I } ti) be the SeLP and T 2 = {h}. Let , for each i, V;i satisfy 0 Z = ui, and (Ati,e ti with support S, x {tj}, where (1) A coincides with the LPS A' constructed above , and (2) (Sj x {tj}) = L . Then, it is clear that (tl , t2) E [u], there is mutual certain belief of {(t l ,t2)} at (tl,t2), and for each i, O"i is induced for t i by t j. It remains to show that (tl, t2) E [isr] n [cau] .

v;;

r-

185

Appendix B : Proofs of result s in Chapters 8- 10

Since , obviously, (h , t2) E [caul, it suffices to show, for each i , that a, is sequent ially rational for ti. Fix a player i and let h E Hi be given. Let Pi (E ~ ( Si (h ) ) ) be outcomeequivalent to uilh and let pj(n) be t he mixed represent ati on of u j (n ). Then , since (Ul' ( 2) is a qu asi-p erfect equilibrium, it follows t hat

for all n . Hence, Pi(Si)

2: sj ESj(h)

> 0 impli es t hat

2:

pj (n )lh(Sj )Ui(Si, Sj ) = ,max pj (n )lh(Sj )Ui(S;, Sj ) sÂˇESi(h) â€˘ sj ESj(h)

(B.3)

for all n . Let A ~i be i 's pr eferences at t, condit ional on h. Since t, E pro h i [cauiJ-SO t ha t i 's syst em of conditi onal pr eferences at t ; satisfies Axiom 6 (Conditionality)-and pj(n) = r(n)DprohAti for all n , there exist vectors r(n) lh converging to zero such that pj(n)lh = r (n)fh Dproh A~i for all n . Together with equa t ion (B .3) we obtain J that Pi(Si) > 0 imp lies

2:

(r(n)lhDproj s j A ~i )(Sj )Ui(Si' Sj )

sj ESj(h)

We show that Pi(Si ) > 0 impli es Si E S:i (h) . Suppose that s, E Si(h)\S:i (h). Then , t here is some s; E S, (h) with s; )- ~i s. . By applying Lemm a 17 in t he case of act s on Sj( h) x {tj} , it follows t hat r(n )lh has a subsequence r (m )lh for which

2:

2:

(r(m ) l h Dprohj A ~n( sj)u i(s ; , Sj ) > (r(m) lh DprohjA~i)(Sj)ui(s i ,Sj) sjESj(h) sj ESj(h) for all m, which is a cont ra diction to (8.4). Hence, s, E S:i (h ) whenever Pi(Si ) > 0, which impli es t hat Pi E ~ ( S: i (h ) ). Hence, uilh is out come equivalent to some Pi E ~ ( S: i (h) ) . This holds for every h E Hi, and hence a, is sequent ially rational for u, (If) Suppose , there is an episte mic mod el with (tI , h) E [u] n [isr]n [caul such that there is mutual certain belief of {( t l , t2)} at (tl , t2), and for both i , a , is induced for t i by t j . We show t ha t (Ul, ( 2) is a quasi-p erfect equilibrium. For each i, let Ai = (tL; ,... ,tLD E L~ ( Sj ) be the LPS coinciding with Ati . Cho ose sequences (r(n)) nEN in (0, I)L - l converging to zero and let the sequences (pj(n)) nEN of mixed strategies be given by pj(n) = r(n)DA i for all n . Since Ai has full support on Sj for every n , Pj (n) is complete ly mixed . For every n , let Uj (n ) be a behavior representation of pj(n ). Since Ai induces Uj , it follows t hat (uj(n)) nEN converges to Uj ; thi s is shown explicit ly und er t he 'if' part of Prop osition 30. Hence, to establish t hat (e r , ( 2) is a qu asi-p erfect equilibrium, we must show t ha t, for each i and "In E N and Vh E Hi, (B.5) Fix a player i and an informat ion set h E Hi. Let Pi (E equivalent to uilh. Then , equat ion (8 .5) is equivalent to

~ ( Si (h )))

be outco me-

186

CONSIS TENT PR EFERENCES

for all n . Hence, we must show that Pi(s;)

> 0 impli es that

for all n . In fact , it suffices t o show t his equat ion for infinitely many n , since in this case we can choose a subseque nce for which the above equation holds, and this would be sufficient to show that (0'1 ,0'2) is a qu asi-p erfect equilibrium . Since, by assumpt ion, a , is sequent ially rational for ti , O'i lh is outcome equivalent t o some mixed st rategy in .6. (S1; (h)) . Hence, Pi E .6. (S1; (h)) . Let Pi(S;) > O. By const ruc t ion, s, E (h). Suppose that s, would not sa t isfy (B.6) for infinit ely many n. Then , there exists some s; E Si (h) such t ha t

s1;

L

pj( n)l h(Sj)Ui(Si, Sj) <

sj ESj(h)

L

pj(n)l h(sj) ui( s;, Sj )

sj ESj (h)

for infinitely many n . Assume , w.l.o.g., that it is true for all n. Let A:,; be i's preferences at t, condit ional on h . Since ti E pro j - [caud - so that i's syste m of condit iona l preferences at t, sat isfies Axiom 6 (Conditio~ality)-and pj( n) = r (n )Dprojsj Ali for all n , there exist vectors r( n) Ih converging to zero such t hat Pi (n) lh = r (n ) IhDproj s J A~i for all n. This impli es that

L

(r(n)l hDproh j A:n( Sj )Ui(Si, Sj)

sj ESj(h)

<

L

(r(n) l hDproj sjA ~; )(Sj )Ui(S;, Sj ) sj ESj(h)

for all n . By applyin g Lemma 17 in the case of acts on S,(h) x {tj }, it follows t hat i at t, st rict ly pr efers s; t o s, condit ional on h, which contradicts the fact that s, E s1;(h) . Hence, Pi(Si) > 0 impli es (B.6) for infinitely many n, and as a consequence, (0'1 ,0'2) is a qu asi-p erfect equilibrium. _ Proof of Proposition 36. (O nly if.) Let (PI , P2) be a pr oper equilibr ium . Then , by Definition 7, t here is a sequ ence (p(n)) nEN of c: (n )-proper equilibria converging to P, where c:(n ) ---; 0 as n ---; 00 . By t he necessity part of Proposition 5 of Blume et al. (1991b) , there exist s an epistemic model with T 1 = {tIl and T 2 = {td where, for each i,

• vi; satisfies t hat vii •

0 Z

= Ui,

the SCLP (Ali , £t;) has t he prop er ties that Ati = (J.Lii, ... , J.L i) with support Sj x {t j} sa tisfies t hat, \fs j E Sj , J.Li;(sj ,tj) = Pj( Sj) , and satisfies that £t;(Sj x Tj ) = L ,

r-

such that (t l , t2) E [resp] . This argument involves Lemma 16 (which yields , for each i , the existe nce of AI i with full support on .6.(Sj x {t j}) by mean s of a subsequence pj (m ) of (pj( n)) nEN) and Lemm a 17 (which yields that , for m lar ge enough, i having the conjec t ure pj(m) lead s t o t he same pr eferences over i's strat egies as t ti) . The only-if part follows since it is clear t hat (t l , t2) E [u] n [cau], that t her e is mutual certain belief of {( t 1 , t2)} at (t 1 , h) , and t ha t, for each i, Pi is induced for i , by tj . (If .) Suppose t hat t here exists an episte mic mod el wit h (t 1 , h) E [u]n [resp] n [caul such th at there is mutual cert ain belief of {(t] ,t2)} at (tl ,t2), and , for each i , Pi is induced for t i by tj. Then , by the sufficiency part of Prop osit ion 5 in Blume et al. (1991b) , t here exists , for each i, a sequence of complete ly mixed st rateg ies

Appendix B: Proofs of result s in Chapters 8- 10

187

(pi(n))n EN converging to Pi, where, for each n , (Pl (n ),P2(n )) is an c( n) -proper equilibrium and E(n) ---; 0 as n ---; 00 . This argument involves Lemma 17 (which yields , for each j, t he existe nce of (x j(n)) nEN so that , for all n , i havin g t he conjecture pj (n ) lead s to t he same pr eferences over i 's st rategies as t ti ) . • P r o of of P roposit io n 37. Part 1: If pi is properly rationalizable, then there exists an epistemic model with (tj , t; ) E CK([u] n [resp] n [cau]) such that pi is induced for t; by tj. In th e definition of pr op er rat ionalizabili ty, 9 in K9[E-prop tr em ] goes to infinity for each s , and t hen e converges to O. The st ra tegy for the pr oof of t he 'only if' part of Proposition 37 is to reverse the ord er of 9 and e , by first noting th at e-proper rationalizabili ty impli es e-proper g-r ationalizabili ty for all g , then showing that e-prope r g-r ationali zability as e converges to 0 corre sponds t o th e gth round of a finit e algorithm , and finally pro ving that any mixed strategy sur viving all rounds of the algorithm is rational under common cert ain belief of [u] n [resp] n [caul in som e episte mic mode l. The algorithm elimina t es pr eference relations on th e players' strategy sets . It is related to , but differs from , Hammond's (2001) 'ra t iona lizable dominance relations', which ar e recursively const ructe d by gradually ext ending a single incompl et e bin ar y relation on each player 's strategy set . Say t hat a mixed stra te gy Pi for i is e -properbj g-rationalizable if there exist s an *-episte mic mod el with pl i = Pi for some ti E projyiK 9([u] n lind] n fe-prop tr em ]) . Since, for all g ,

CK [e-prop trem ]) <;;; K 9 [e-prop tr em ]) , it follows from Definiti on 21 t hat if pi is an e-properly rationalizabl e st rate gy, t hen, for all g, t here exists an »-epist emic mod el with pl i = pi for some i , E pr oj -. K9([u] n lind] n [e-prop tr em ]) . Consequent ly, if a mixed st rategy pi for i is *-pro~erly rat ionalizable, t hen, for all g, t here exists a sequence (pi(n ))nEN of e (n )-properly grati onalizabl e strateg ies converging to pi , where E(n ) ---; 0 as n ---; 00 . This means t hat it is sufficient to show t hat if pi satisfies that , for all g, t here exists a sequence (pi( n)) " EN of e(n )-prope rly g-r ationalizable st ra teg ies converging t o pi and E(n) ---; 0 as n ---; 00 , then pi is rational under common certain belief of [u] n [resp] n [caul in some episte mic mod el. This will in turn be shown in two st eps: 1 If a sequence of E(n) -properly g-rationali zable st rateg ies converges to pi , then pi sur vives the gth round of a finit e algorit hm. 2 Any mixed strateg y sur viving all rounds of the algorithm is rational under common cert ain belief of [u] n [resp] n [caul in some epistemic mod el. To const ru ct t he algorithm, not e that any complet e and tr ansitive binary relation on Si can be represent ed by a vect or of sets (Si (1), .. . , S, (L )) (with L ~ 1) that const it ute a partition of S i , The int erpretation is that s, is pr eferr ed or indifferent to s; if and only if Si E Si(€), s; E Si(e') and € :::; t . Let , for each i , E i := 2Si \ {0} be th e collect ion of non-empty subsets of S, and II i := {1l"i = (Si (1), .. . , S;(L" i )) E EF 'i

I {Si(I ), . . . , Si( L"i )} is a partition of S;}

denote the collect ion of vectors of sets t hat constit ute a par t ition of Si. Define the algorithm by, for each i, set t ing IIi l = IIi and det ermining, 'rig ~ 0, II; as follows: 1l"i = (S;(I) , ... , S; (L"i )) E if and only if 1l"i E II i and t here exists an LPS )."i E l L ~ ( Sj x II j ) with suppx "i = Sj X II j i for some IIt i <;;; IIJ- , satisfying t hat

rrr

(Sj ,1l"j ) >>

(s~ , 1l"j ) accordi ng to ~"i

188

CONSISTENT PREFERENCES

if and only if s, E Si(£) , s; E Si(e') and £ < r, where t 7Ti is represented by V ; i satisfying V; i 0 Z = u; and A7Ti. Write II := III x II2 and, '</g 2: 0, IIg = IIi x IIg. Since II o S;; II, it follows by induction that, '</g 2: 0, IIg S;; IIg-l . Moreover , since the finiteness of S = SI X S2 implies that II is finite , it follows that IIg converges to II oo in a finite number of rounds. Say that P i survives the gth round of the algorithm if there exists 7ri = (S i(l) , . . . , Si(L 7T i )) E II; with .6.(Si(l)) :1 Pi. Step 1. We first show that p; survives the gth round of the algorithm if there exists a sequence (pi(n) )nENof e( n )-properly g-rationalizable strategies converging to p; , where e( n) --> 0 as n --> 00 . Say that the probability distribution Jl E .6.(Sj x Tj) is an e-properlu g-rationalizable belief for i if there is an --epistemic model with Jl t i = Jl for some t, E proh;Kg([u] n lind] n fe-prop trem]). It is sufficient to establish the following result :

satisfies that there exists a sequence (Jl7T i(n))n EN of e(n)-properly g-rationalizable beliefs for i , where e(n) --> 0 as n --> 00, and where, for Ij tt;

= (S i(l) , ... ,Si(L 7T i))

all n ,

LL 5j

Jl

7T i

E IIi

(n )(sj , tj)Ui(Si, Sj) >

LL 5j

tj

Jl7T

i

(n )(sj , tj)Ui(S;,Sj)

(B .7)

tj

if and only if s, E S, (z), s; E S, (e') and £ < t' , then

7ri

E II;'

This result is established by induction. If (Jl7T i(n))nE Nis a sequence of e(n)-properly g-rationalizable beliefs for i , then, for each n, there exists an --epistemic model with T[ (n) x T2(n) as the set of type vectors, such that Jl 7T i (n ) E .6.(Sj x Tj(n)) . For the inductive proof we can w.l.o.g. partition Tj(n) into II j , where 7rj = (Sj(l), .. . , Sj (L 7T j )) E II j corresponds to the subset of j-types in T, (n) satisfying that

LL

Jl

tj

(n)(si, t;)Uj(Sj, Si) >

LL

Jl

tj

(n)(si, ti)Uj(Sj , s;)

if and only if Sj E Sj(£) , sj E Sj(e') and £ < e', since i 's certain belief of j's e(n)proper trembling only matters through j-types' preferences over j's pure strategies . Hence, we can w.l.o.g. assume that Jl7T i (n ) E .6.(Sj x II j ) . (g = 0) Let (p,7T i (n))nE N be a sequence of e(n)-properly O-rationalizable beliefs for i, where E(n) --> 0 as n --> 00, and where, for all n, (B.7) is satisfied. By Lemma 16, the sequence (Jl7T i (n))nEN contains a subsequence Jl7T i (m) such that one can find an LPS A7Ti E L.6.(Sj x II j ) and a sequence of vectors r 7Ti(m) E (0, l)L-I (for some L) converging to 0 with Il 7T i (m) = r 7T i (m)DA 7Ti for all m. By Definition 20, SUppA7Ti = s, X IIl i for some IIli S;; II j . Let t 7Ti be represented by V; i satisfying V;i 0 Z = u, and A7Ti . Since Definition 20 is the only requirement Oil (Jl7T'(n))nEN for 9 = 0, we may, for each 7rj E nr ', associate 7rj with (Sj(l), ,,,,Si(L 7T j )) E njl satisfying that (Sj ,7rj)>> (sj,7rj) according to t 7Ti if Sj E Sj(£) , sj E Sj(e') and < e'. By Lemma 2, t 7Ti yields the same preferences on S; as Jl7T i (n) (for any n). Hence , 7ri E II?

e

Appendix B: Proofs of resu lts in Chapters 8- 10

189

(g > 0) Suppose the resul t holds for g' = 0, .. . , g - 1. Let (Ji1r i (n))nEN be a sequence of c(n)-prope rly g-rationa lizable beliefs for i , where c (n ) - t 0 as n - t 00 , and where, for all n , (B.7) is satisfied. As for 9 = 0, use Lemm a 16 to const ruc t an LP S ).1r i E Lt.(Sj x II j ) , where supp.V" = S, X III i for some III' ~ II j , and where t 1ri is represented by V; i satisfying V; i 0 Z = Ui and )," i. Since g K g[f-prop trem] ~ K , ([f-prop tremj ] n K - 1 [f- prop tr em]) , 1

and (Sj ,Jr j) » (sj ,Jr j) according t he induct ion hypothesis implies that III i ~ IIr to t 1ri if Jrj = (Sj( l), .. . , Si (Uj )) E III i , Sj E Sj(f) , sj E Sj(1!') and f < e'. By Lemm a 17, t " i yields the same pr eferences on S, as Ji" i (n) (for any n). Hence, tt, E 11f. This concludes the inductio n and thereby St ep 1. Step 2. We then show that if a mixed st rategy pi sur vives all rounds of the algorit hm , then there exists an episte mic mod el with pi E t.( SF) for some t i E pr oj - , CK ( [u] n [resp] n [caul) . It is sufficient to show t hat one can const ruct an epistemic model with T 1 x T2 ~ CK ([u] n [resp] n [caul) such t ha t , for each i, VJri = (Si (1) , ... , S, tL" i )) E 11f , there exist s t i E T; satisfying that s, ~ ti if and only if s, E Si (f ), E Si (e') and f < e'. Construct an episte mic mod el with, for each i, a bijection tt i : T; - t IIf from t he set of typ es to the collect ion of vect ors in IIf . Since :lg' such tha t 11 g = II O<> for 9 ~ g' , it follows from t he definition of the algorit hm (IIg)g2: o that , for eac h i , 11f is characte rized as follows: Jri = (Si( l), ... , Si (£ " i)) E 11f if and only if there exist s ti E T, such t hat 7r i (t i ) = Jri , and an LP S ).ti = (Jiii , ... JiZ ) E Lt.(Sj x Tj ) with sup p x': = S , X T/ i for some T/ i ~ Tj , sat isfying for each tj E T/ i t hat

s:

s:

(Sj ,tj) >> (sj ,tj ) acco rding to t ti if 7r j (t j )

= (Sj( l) , ... , Sj( U

j (tj ») , Sj E Sj( f) , s j E Sj(e') and f

s:

< e. and

E Si(1!') and f < e', where Vi i satisfies Vii 0 Z = Ui and if and only if s, E Si(f) , t he SCLP ().ti , ft i) has the pr op erty t hat ft i sa t isfies ft i (Sj x Tj ) = L (so t hat t t i is repr esented by Vi i and ).t i ). Consider any Jri = (Si( l) , ... ,Si( £ " i)) E IIf . By th e construc t ion of the typ e sets, th ere exist s t , E T; such t hat 7r i(t ;) = Jri , and s, ~t i if and only if s , E Si(f), E Si (e') and f < e, in particular, Si (l ) = S;i . It remains t o be shown that , for each i , T 1 x T 2 ~ lUi ] n [resp;J n [caui], impl ying that T 1 x T2 ~ CK ([u] n [resp] n [caul) since T/ i ~ Tj for each t, E T; of any player i . It is clear that T 1 x T2 ~ lUi] n [caUi]' That t: x T2 ~ [r esp;J follows from the prop ert y that , for any ti E Ti, (s i- t j ) » (sj, t j) according to t ti whenever t j E T/ i if Sj E Sj(f) , s j E Sj (1!') and f < r, while Sj ~tj sj if and only if Sj E Sj(f) , s j E Sj(1!') and f < e' (where 7r j (t j ) = (Sj( l) , ... , Sj( L"" j (tj »)). This concludes Step 2. In t he const ruction in Step 2, let ti E T; satisfy t hat pi E t. (s F- ). To conclude Part 1 of t he pro of of Prop ositi on 37, add ty pe tj to T , havin g the prop erty that pi is induced for t i by t j . Assume that v/ j satisfies vp oz = Uj an d t he SC LP ().t i, f t i) on S, x T , with suppo rt S, x { ti } has the prop er ty that ).tj = (Jiltj, . . . ' JiLtj ) satisfies, "lSi E Si, Jil tj (Si , t i) = pi( si) and ft i satisfies f ti(Sj x T j ) = L (so t hat t tj is represent ed by v/ j and ).tj ). Fur ther more, assume t hat

s:

s:

(s ., ti)

»

(s: , t n according to t tj

190

CONSIS TEN T PREFERENCES

if 1Ti(ti) = (Si(l) , . . . , Si(L 7r i(til)) , s, E Si(£), s; E Si( f.') and £ < t . Then t; E pr ohj ([Uj] n [respj] n [cauj]), and since TfJ ~ Ti , t: x tr, U {tj}) ~ CK( [u] n [resp]n [cau]) . Hence, (t j , t 2) E CK([u] n [resp] n [cau]) and pi is induced for ti for t; . Part 2: If there exists an epistemic model with (tj , t 2) E CK([u] n [resp] n [cau]) such that p j is induced for tj by t 2, then pj is properly rationalizable. Schuhmac her (1999) considers a set of typ e profiles T = T 1 X T 2, where each ty pe ti of eit her player i plays a complete ly mixed strategy pl i and has a subjecti ve probabili ty distribution on Sj x Tj , for which t he condit ional distribution on S, x {t j} coincides with p/ j whenever t he condit ional distribution is defined . His form ulation impli es that all ty pes of a player agrees not only on the pr eferences but also on the relative likeliho od of the st ra tegie s for any given opp onent typ e. In cont ras t, the charac te rization given in Proposition 37 requires the typ es of a player only to agree on the pr eferences of any given opp onent typ e. This difference impli es that expanded ty pe sets must be const ructe d for the 'if' part of the pr oof of Proposition 37. Assume t hat there exists an episte mic mod el wit h (tj , t 2) E CK([u]n [resp] n [cau]) such that p j is induced for tj by t 2. In part icular , CK([u] n [resp] n [cau]) i- 0, and pj E b.(Sltj) since CK([u] n [resp] n [cau]) ~ [re sp2]' Let , for each i , T! := pr oj - , CK([u] n [resp] n [cau]). Not e that , for each ti E T! of any player i , i , deems (Sj, tj') subjectiv ely impossible if t j E Tj\ Tj since CK([u] n [ir] n [cau]) = KCK([u] n fir] n [cau]) ~ KiCK([u] n [ir] n [cau]), implying T/ i ~ Tj. We first const ruct a sequ ence, ind exed by n, of -- epistemic mod els. By Definition 20 and Assumption 3 this involves, for each n and for each player i , a finit e set of typ es-which we below denote by Tf' and which will not vary with n-and, for each n, for eac h i , and for eac h typ e r. E a mixed strategy and a probabil ity distribution (pTi(n) ,t-tTi(n)) E b.(Si) x b.(Sj x Tn that will var y with n . For eit her player i and each typ e ti E T! of the original epist emic model, make as many "clones" of t, as there are memb ers of Tj: For each i, Tf' := {Ti(ti, t j) I i; E T! and t j E Tj} , where Ti(t i, tj) is t he "clone" of t. , associated with tj . The term "clone" in the above st at ement reflect s t hat, Vtj E Tj , Ti(ti, t j ) is ass umed to "share" t he pr eferences of ti in the sense that

tt;

1 the set of opponent typ es that Ti(ti, tj) does not deem subject ively possible, T/ d ti.tj ), is equa l t o {Tj( tj ,ti) I t j E T/ i} (~Tj' since T/ i ~ Tj) , and 2 the likelihood of (Sj ,Tj( tj,t;)) according to ~ Td t i , tj ) is equal to t he likelihood of

(Sj ,tj) according to

~ti .

Since T/, (ti,tj ) = {Tj(tj ,ti) I t j E T/ i} is indep end ent of tj , but corres po nds to disjoint subsets of Tj' for different t i'S, we obtain t he following conclusion for any pair of ty pe vect ors (tl , t2), ( t~ , t~) E T{ x T~ : if if

ti = t, i-

t;, t;.

This ends the const ruct ion of typ e set s in the sequ ence of - -episte mic mod els. F ix a player i and consider any Ti E T f' . Since CK([u] n [resp] n [cau]) ~ lUi], ~ Ti

vr

vr

i sat is fy ing i 0 Z = Ui a n d a n LP g ,\Ti on Sj x 'I'[> , Since CK([u] n [resp] n [cau]) ~ [caui], thi s LPS yields , for each Tj E T'[> , a partition {Ej'(l) , .. . ,E/i (L Ti )} of Sj x T/ " wher e (Sj ,Tj) >> (Sj ,Tj) according to ~ Ti if and only (Sj,Tj) E E/ i( £), (Sj ,Tj) E E/ i(t) and £ < t . Since

can b e r epresented b y a v N M u tilit y function

191

Append ix B: Proofs of results in Chapt ers 8- 10

CK([u] n [resp] n [caul) ~ [respiJ, it follows that Sj is a most preferred st rategy for Tj in {sj E Sj I (Sj ,Tj) E Er(£) U .. · U El i (C i)} if (Sj,Tj) E Eli (£). Consider any i and Ti E T['. Construct the sequence (JiTi( n)) nENas follows. Choose 'VTi E {Ti(t i , t j) I tj E Tj} one common sequence (rTi(n)) nEN in (0, l) c i - 1 converging to 0 and let t he sequence of probability distributions (11Ti (n) )nENbe given by Ji r, (n) = rTi (n )D,\Ti . For all n , SUppJiTi (n ) = S, x T'[>. By Lemm a 17 (rTi(n)) nEN can be chosen such that , for all n ,

if and only if Si >-- Ti s;. Hence, for all n , t he belief JiTi (n) lead s to the sam e preferences over i 's st ra teg ies as ~ "' , T his ends the const ruc t ion of th e sequences (JiTi (n) )nEN in t he sequence of *-episte mic mod els. Consider now t he const ruct ion of th e sequence (pTi (n) )nEN for any i and r . E T['. There are two cases. Case 1: If th ere is Tj E Tj' such t hat t: E TTj , implying th at S, x {T;} ~ SUppJiTj (n ), t hen let pTi( n) be det ermined by

Ti ( )(S2.) -_ JiTJ(n)( si,Ti) 2 lIT](n)(Si, Ti)

p. n

•

Moreover , for each n, t here exists c(n ) such that , for each player i , t he c(n )-proper tr embling condition is satisfied at all such typ es in T!,: Since

pTi( n)( s;) JiTj( n)( s;,T;) T ' = Pi' (n)(S i) JiTj (n)( Si ,Ti)

0 ---+

as n

---+ 00

if (Si,T;) E E? (£), (S;,Ti) E E? (e') and £ < e', and since s , is a most preferred st ra te gy for Ti in {s; E s, I (S;,T;) E E;j (£) U·· · U E?(C j)} if (Si,T;) E E? (£), it follows t hat there exists a sequence (cTi(n)) nEN converging to 0 such that , for all n ,

cTi (n )pTi (n )(si) ~ pTi(n)(s;) whenever IL Ti (n)(sj , Tj )Ui(Si, Sj ) >

LL

L L JiTi (n)( sj ,Tj) Ui(S;, Sj) .

Let , for each n , c(n ) := max {cTl (n) I :JT2 E T~' s.t . T1 E T ;2} U {cT2(n) l:Jn E T{' s.t. T2 E T;l } . Since t he ty pe set s are finit e, c(n ) ---+ 0 as n ---+ 00 . Case 2: If there is no Tj E Tj' such th at Ti E TTj , t hen let pTi (n) be any mixed st rategy having the property that Ti sat isfies t he c(n )-proper t rembling cond ition given the belief /lTi(n) . This ends the const ruction of the sequences (pTi (n) )nEN in th e sequence of *-epist emic models. We then turn t o th e const ruc t ion of a sequence (p{i (n) )nENconverging to pi . Add typ e Tj to T{' havin g the property that /lT i(n) = /IT[(tj,t, )(n) for some t 2 E T~, but where p{i (n) = (1- ~ )pi + ~ p{tCtj ,t2)(n ). For all n , we have that the belief /lTi (n ) lead s to t he sam e preferences over l 's strategies as ~ t i. This in turn implies that i sat isfies th e c(n )-t rembling condition at Tj since pi E .0.(C 1t j ). Consider the sequence, indexed by n , of --episte mic models, • with T{' U {Tj} as th e typ e set for 1 and T~' as t he typ e set for 2,

192

CONSISTENT PREFERENCES

â€˘ with, for each type Ti of any player i, (pTi(n),j.lTi(n)) as the sequence of a mixed strategy and a probability distribution, as constructed above . Furthermore, it follows that, for all n , the E(n)-proper trembling condition is satisfied at all types in T{' U {Ti} and at all types in T~' , where E(n) ----> 0 as n ----> 00 . Hence, for all n, (T{' U {Tn) x T~' ~ CK[E(n)-prop irem] ; in particular, p{i (n) is E(n)-properly rationalizable. Moreover, (p{i (n))n EN converges to pi. By Definition 21, pi is a properly rationalizable strategy. â€˘

Appendix C Proofs of results in Chapter 11

Proof of Proposition 42. Assume that the pur e strategy s, for i is properly rat ionalizable in a finite strategic two-player gam e G. Then, there exists an episte mic model satisfying Assumption 1 with s, E S;i for some (t1,t2) E prohlx T2CK([u] n [resp] n [cau]) (this follows from Proposition 37 since CK([u] n [resp] n [cau]) = KCK([u] n [resp] n [cau]) S;; KjCK([u] n [resp] n [cauJ)). In particular, CK([u] n [resp] n [cau]) i- 0. By Proposition 20(ii) , for each i, CK([u] n [resp] n [cau]) = KCK([u] n [resp] n [cau]) S;; KiCK([u] n [resp] n [cau]). Hence, we can construct a new episte mic model (S1,T{ , S2, T~) where, for each i, T: := proj - , CK([u] n [resp] n [cau]), as for each ti E T: of any player i , I"ti = {t i} X S, X T} i S;; {til x Sj x Tj . Since T{ x T~ S;; [caul, according to th e definition of caut ion given in Section 5.3, it follows th at th e new episte mic model satisfies Axiom 6 for each i , E of any player i . Therefore, th e new epist emic model satisfies Assumption 2 with SI x T{ X S2 X T~ S;; [caul according to th e definition of caut ion given in Section 6.3. Also, S I x T{ X S2 X T~ S;; [u]. It remains to be shown that , for each i , S 1 x T{ X S2 X T~ S;; B?[rat jl, since by th e fact that I"ti S;; {til x Sj x Tj for each i , E T: of any player i, we then have an epist emic model with s, E S;i for some (t 1,t2) E projTlxT2CKAo.

T:

Since T{ x T~ S;; [resp], we have t hat, for each t , E T: of any player i , (sj , t j) »': (sj, t j ) whenever t j E tt: and Sj "e tj sj . In particular , for each ti E T: of any player i , (Sj, tj) » ti (sj, t j) whenever t j E ti«. Sj E Sf j and sj 1. By Proposition 6 this means that , for each t , E T: of any player i , "e ti is admissible on prohl XS2 XT2 [rat j] n I"ti , showing th at S I x T{ X S2 X T~ S;; (B?[rat2] n Bg[rat1]). •

s»,

Proof of Proposition 43 . Part 1: If s, is permissible, then there exists an epistemic model with s, E S;i for some (t1, t2) E prohl x T 2 CK.4. It is sufficient t o construct a belief syste m with S I x T 1 X S2 X T 2 S;; CKA such th at, for each s, E Pi of any player i, th ere exists i , E T; with s, E S;i. Construct a belief system with, for each i , a bijection s, : T: ~ Pi from th e set of types to th e the set of permissible pure st rategies. By Lemm a lO(i) we have th at , for each t, E T; of any player i , there exists Y} i S;; Pi such t ha t Si(t;) E Si\Di(Y/ i) . Det ermine th e set of opponent types that ti

194

CONSISTENT PR EFEREN CES

deems subject ively possible as follows: T/ i ti E T, of any player i, ~ t ; satisfy 1. v :i

0 Z

= Ui

(so t ha t 8 1 x T 1

X

= {tj

E

Tj

I Sj (tj ) E

Y/ i} . Let , for each

8 2 X T2 ~ [u]), and

2. P >- ti q iff P Ej weakl y dominat es q Ej for Ej = E/ i := {(Sj , tj ) I Sj = Sj (t j ) and tj E T]i} or E, = 8 j X T[« , which impli es t hat (3 ti = {t i} X E/ ; and " t; = {e.} x s, X T/ ; (so t hat 8 1 x T 1 X 8 2 X T2 ~ [cau]). By t he const ru ct ion of E/ i , t his means that 8: i = 8 i \ D i (Y/ i ) 3 Si(t ;) since, for any act s P and q on 8j x Tj sat isfying t hat there exist mixed st rategies Pi, qi E .6.(8 ;) such t hat, V(Sj,t j) E 8 j x Tj , p (Sj , tj ) = Z(pi , Sj ) and q( Sj ,tj ) = Z(qi, Sj ), p >- t; q iff P Ej weakl y dominates q Ej for Ej = Y/ ; x T, or Ej = 8 j x Tj. This in turn impli es, for each t, E T; any player i,

3. ~t i ~ projr i «s, XT) ratj ] (so t ha t , in combination with 2., 8 1 x T 1 X 8 2 Bdratj ] n Bj[rati]).

X

T2 ~

Furthermore, 8 1 x T 1 X 8 2 X T 2 ~ CK A since T/ ; ~ T , for each ti E Ti of any player i . Since , for each player i , s, is onto Pi , it follows that , for each Si E Pi of any player i, t here exists i , E T, with s, E Part 2: If there exists an epistemic model with Si E S l i for som e (t1, t2) E prohl XT2CKA , then s, is permissible. Par t 2 of the proof of P ropos it ion 27. •

sr.

Proof of Proposition 44. Part 1: If s, is rationalizable, then there exists an epistemic model with Si E 8 : i for some (t1, t2) E pr oh l XT2CK C O • It is sufficient t o construct a belief system wit h 8 1 x T 1 X 8 2 X T 2 ~ CK C such t hat, for each s, E R, of any player i , there exist s t i E T, wit h Si E 8 : i . Const ruct a belief system wit h, for each i, a biject ion s , : T, -> R; from t he set of types to the t he set of rat ionalizabl e pur e st rategies. By Lem ma 9(i) we have t hat, for each ti E Ti of any player i, t here exists Y/ i ~ R, such t hat t here does not exist Pi E .6.(8 i ) such that Pi weakly domin ates Si(ti) on y] i. Det ermine t he set of oppo nent typ es t hat i , deems subjectively possible as follows: T/ i = {t j E Tj I Sj(tj) E Y/ i }. Let , for each ti E T, of any player i, ~ t i sa t isfy 1. v:;

0

Z = Ui (so t hat 8 1 x T 1 X 82

X

T2 ~ [u]), and

2. P >-t ; q iff P Ej weakl y domin at es q Ej for E j = E ] ; := {(Sj , t j) I Sj = Sj (t j ) and t j E TJ;}, which impli es that (3t ; = " ti = {t;} X E] ;. By the const ruct ion of E/ i , thi s means that 8 : i 3 s, (t;) since , for any acts P and q on 8 j x Tj satisfying th at there exist mixed strategies Pi, qi E .6.(8 ;) such that, V(Sj, tj) E 8 j x T] , p (Sj ,tj ) = Z(pi,S j) and q (sj , tj ) = Z(qi,Sj ), p >- t; q iff p n, weakly dominat es q Ej for E j = Y/ i X Tj • This in turn impli es, for each ti E T; any player i ,

3. (3t; ~ ProjT;xS XT.[ratj ] (so t hat , in combina t ion wit h 2.,8 1 x T1 X 8 2 X T2 ~ - 0 - 0 J J B i [ratj] n Bj [rat d ). Fur therm ore, 8 1 x T 1 X 8 2 X T 2 ~ CKC o since T/ ; ~ Tj for each i , E T; of any player i . Since , for each player i , s , is onto Rs, it follows t hat, for each s, E R, of any player i, t here exist s i : E T; wit h s , E Part 2: If there exists an epistemic model with s, E S l i for some (iJ , t2) E proh l XT2 CKC, then s, is rationalizable. Part 2 of t he pr oof of Proposit ion 25. •

sr.

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Index

Accessibility relation, 39-44, 46 Act Ans cornb e-Aumann act , 8, 22, 26, 32-33, 39-40, 49, 54, 56, 70-71 , 74-75, 77-78,83-84,92, 102, 122, 126, 145, 181, 185-186, 194 Admissibility, 39 , 41-42, 46, 49- 50, 70, 72, 85-86, 133-135, 143-144, 147, 153-154,175-176,179,193 Ba ckward induction, 2, 6-7 , 10-11 , 14-17, 20-21 , 23-24, 38, 69, 78- 80, 82-83, 87-88, 91-97, 99-100, 112-113, 115, 118, 121, 123, 128, 132, 152, 155-1 59, 162-166,170,174 Belief op er ators a bsolute ly robust belief, 38, 40, 45, 49- 50, 147 assumption, 38, 40, 45, 48-50, 147, 152 cer tain beli ef, 19, 39, 44, 46-48, 57- 61, 63-66, 72-73, 76, 78, 81-82, 87-88, 90-96, 99, 103, 105-106, 108-109, 114-11 5, 117-118, 122-123, 125, 127-128, 136, 138, 140, 147-1 48, 150-153, 162, 167, 173, 183-188 conditional belief , 39- 40, 44-45, 47, 50, 86 full belief, 38, 40, 45-46, 50 robust belief, 40, 44- 46, 48- 51, 135-139, 143, 145-149, 152-154, 158, 163, 173 st ro ng belief , 38, 40, 45, 48, 50-51, 147, 152 Caution, 10, 14, 23, 62-63, 75, 103, 115-116, 123, 135-137, 139, 143, 145-146, 148- 149, 152, 154, 158, 160, 163-164, 173, 193 Consistency of preferences (ordinar y) consiste ncy, 5, 12, 53, 58- 59, 73,149

admissib le consisten cy, 53, 63, 75-76, 87-88, 97, 148 admissible subgame cons iste ncy, 90-96 full admissible cons iste ncy, 134-140, 144-1 45, 147-148, 151-152, 162, 167, 173 proper consiste ncy, 122-123, 125, 128 qu asi-perfect consi st ency, 117 sequential consistency, 104 weak sequential consis te ncy, 108 Consistent preferences approach, 1-7, 11-12, 15-17,21, 53,81 ,144,154,174 Epi stemi c ind ep end ence, 90, 94, 96 Epistemic model, 3-5, 8-9, 15, 41-42, 48, 50, 53-55, 58-62, 64-67, 69, 73-74, 76- 77, 83, 91-92, 94, 100, 102, 104-106,109- 111,11 3-115,117-119, 121, 125-128, 131, 138, 144-145, 14~ 149-1 51,1 53,174,183-190,193-194 Epistemic priority , 38-39, 42-46 Equilibrium Nash equilibrium, 2-6, 11-13, 18, 53, 58- 60, 64-65, 115, 124, 130, 141, 171-172 perfect equilibri um , 18, 53, 62-65, 115, 124, 130 proper equilibrium, 16, 18-19, 121-122, 124-125,127,130-131,148,174,186 quasi-perfect equilibrium , 18-19,24, 115-118, 127, 184-186 seque nt ia l equilibrium, 18-19, 24, 100, 104-107, 115, 118, 182-183 subgarne-perfect equilibrium, 87, 91, 94, 113-114 weak sequential equilibrium , 18 Forw ard induction , 2, 6-7,10-11 ,17,21,24, 38, 69, 97, 112-113, 133, 135, 137, 146, 148-1 50, 152-154, 159, 162,

202 168-172,174 Game ext ensive game, 6, 14-15, 17, 23, 50-51 , 56,80-85,87,89-91 ,94,99,101-103, 105-106,108-109,113,117-119,152, 155, 158- 160, 162, 173 of perfect information, 20, 79-82, 84-85, 87-92, 94, 97, 100-101 , 113, 118, 121, 123, 128, 132, 162 strategic game, 2-4, 7-8 , 53-54, 56-57, 59-61 , 63-66, 69, 71, 73, 76, 83-85 , 88,90-91 ,94, 101-103, 121, 125-130, 139, 144, 147, 149, 159, 161, 193 pure strategy reduced strategic form (PRSF), 133, 156, 159-164, 168-170, 173 Inducement (of rationality) of a rational mix ed strategy, 5, 58 of a sequentially rational behavior strategy, 104 of a weak sequ entially rational mixed strategy, 107 Iterated elimination Dekel-Fudenb erg procedure, 13-14,23-24, 65,69, 78, 81, 83, 88, 111, 124, 138, 141, 143, 148, 162, 166-168, 171 of choice sets under full admissible con sist ency (IECFA), 138-142, 150-151 ,163-164,169-170,173 of strongly dominated strategies (IESDS) , 13, 23-24 , 60, 69, 83, 133, 137-138, 141-143, 149 of weak ly dominated strategies (IEWDS), 14, 16-17, 129-130, 133-135 , 141-142, 150, 152-153, 159, 169 No extraneous restrictions on beli efs, 135, 137, 139, 143, 146-149, 153-154, 158, 163-164,173 Probability system conditional probability system (CPS), 24-25, 34-36, 50, 110

INDEX lexicographic conditional probability system (LCPS), 31, 35- 36, 49-50 lexicographic probability syst em (LPS) , 24-25, 30-33, 36, 43 , 49, 56, 60, 62-63, 65, 67, 76, 88, 93- 94, 96, 102, 104, 106-107, 111, 116, 122, 131, 143, 181-185, 187-190 system of condit ional lexicographic probabilities (SC LP) , 25, 32, 35- 36, 56-57,59, 61-64, 66, 100, 102-104, 109-110, 114-116, 121-122, 183-184, 186, 189 Rational choice a pproach, 1-3, 6, 11-12, 143 Rationalizability (ordinary) rationalizability, 8, 13, 18, 53, 60,69, 73, 124, 137-138, 142-143, 146, 149, 153 extensive form rationalizability, 99, 112-113, 135, 152-153, 156, 159 full permissibility, 17, 113, 134-146, 148-152, 154-155, 157-173 permissibility, 13-15, 17-18, 53, 62, 65-67, 69, 75-77, 81, 87, 112-113, 115, 119, 124, 137-138, 141- 143, 146, 148-149, 153-154, 193-194 proper rationalizability, 1, 16, 18-19, 113, 121-125,127-129,131 ,146-148,174, 187, 190, 192-193 quasi-p erfect rationalizability, 1, 15-16, 18,24,101, 113, 115-116, 118-119, 128 sequential rationalizability, 1, 15-16, 18, 99-101 ,104,106-107,111-115, 118-119,128,174 weak seq uential rationalizability, 18, 20, 107-112, 119 Strategic manipulation, 171 Strategica lly independent set, 84-85, 110, 119 Subjective possibility, 38- 39, 43

The consistent preferences approach to deductive reasoning in games

Published on Apr 15, 2018

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