A Bayesian Theory of Games A Bayesian Theory of Games Iterative conjectures and determination of equilibrium JIMMY TENG Chartridge Books Oxford Hexagon House Avenue Station Lane Witney Oxford OX28 4BN, UK Tel: +44 (0) 1865 598888 Email: editorial@chartridgebooksoxford.com Website: www.chartridgebooksoxford.com Published in 2014 by Chartridge Books Oxford ISBN print: 978-1-909287-76-1 ISBN digital (pdf): 978-1-909287-77-8 ISBN digital book (epub): 978-1-909287-78-5 ISBN digital book (mobi): 978-1-909287-79-2 © J Teng 2014 The right of J Teng to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988 British Library Cataloguing-in-Publication Data: a catalogue record for this book is available from the British Library All rights reserved No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, 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in the UK and USA Contents Preface Acknowledgments About the author Introduction Sequential games with incomplete information and noisy inaccurate observation 2.1 Introduction 2.2 An inflationary game 2.3 Bayesian iterative conjecture algorithm as a Bayes decision rule 2.4 Conclusions Notes Sequential games with perfect and imperfect information 3.1 Introduction 3.2 The Bayesian iterative conjecture algorithm, sub-game perfect equilibrium and perfect Bayesian equilibrium 3.3 Solving sequential games of incomplete and perfect information 3.4 Multiple-sided incomplete information sequential games with perfect information 3.5 Conclusions Notes Simultaneous games 4.1 Introduction 4.2 Complete information simultaneous games 4.3 BEIC and refinements of Nash equilibrium 4.4 Simultaneous games with incomplete information 4.5 Conclusions Notes Conclusions References Index Preface This book introduces a new games theory equilibrium concept and solution algorithm that provide a unified treatment for broad categories of games that are presently solved using the different equilibrium concepts of Nash equilibrium, sub-game perfect equilibrium, Bayesian Nash equilibrium and perfect Bayesian equilibrium The new method achieves consistency in equilibrium results that current games theory at times fails to, such as those between Perfect Bayesian Equilibrium and backward induction (sub-game Perfect Equilibrium) The new equilibrium concept is Bayesian equilibrium by iterative conjectures (BEIC) and its associated algorithm is the Bayesian iterative conjecture algorithm BEIC requires players to make predictions on the strategies of other players using the Bayesian iterative conjecture algorithm The Bayesian iterative conjectures algorithm makes predictions starting from first order uninformative predictive distribution functions (or conjectures) and keeps updating with the Bayesian statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved Information known by the players such as the reaction functions are thereby incorporated into the higher order conjectures and help to determine the convergent conjectures and the associated equilibrium In a BEIC, conjectures are consistent with the equilibrium or equilibriums they support and so rationality is achieved for actions, strategies and conjectures and (statistical) decision rule The BEIC approach is capable of analyzing a larger set of games than current games theory, including games with noisy inaccurate observations and games with multiple-sided incomplete information games On the other hand, for the set of games analyzed by the current games theory, it generates smaller numbers of equilibriums and normally achieves uniqueness in equilibrium It treats games with complete and perfect information as special cases of games with incomplete information and noisy observations, whereby the variance of the prior distribution function on type and the variance of the observation noise term tend to zero Consequently, there is the issue of indeterminacy in statistical inference and decision-making in these games as the equilibrium solution depends on which variance tends to zero first It therefore identifies equilibriums in these games that have so far eluded current treatments Acknowledgments I thank D Banks, J Bono, P Carolyn, M Clyde, J Duan, I Horstmann, P.Y Lai, M Lavine, J Mintz R Nau, M Osborne, J Roberts, D Schoch, R Winkler, R Wolpert, F.Y Chiou, and G Xia for their comments I thank the students of my 2005, 2008 and 2009 games theory classes (at the Department and Graduate Institute of Political Science in the National Taiwan University in Taipei, Taiwan) for their enthusiasm in learning, and interesting questions raised in class I thank the participants of my three-day games theory workshop (at the Graduate Institute of Political Science in the National Sun Yat Sen University in Kaohsiung, Taiwan) for their questions I thank the students of my 2011, 2012 and 2013 microeconomic theory and advanced microeconomics classes (at the School of Economics in the University of Nottingham Malaysia Campus) for their questions About the author Jimmy Teng currently teaches at the School of Economics of the University of Nottingham (Malaysia Campus) He previously held research and teaching positions in Academia Sinica (Taiwan), National Taiwan University and Nanyang Technological University (Singapore) He was the recipient of Lee Foundation Scholarship (1991–1996) at the University of Toronto where he received his PhD in Economics He also earned a PhD in Political Science and a MS in statistics from Duke University, besides an LLM from the University of London He is the author of many articles and two books He is researching to give games theory a firm Bayesian statistical decision theoretic foundation Introduction ‘If a man will begin with certainties, he shall end in doubts; but if he will content to begin with doubts, he shall end in certainties.’ Francis Bacon (1561–1626) There is a lot of criticism against Nash equilibrium (and its myriad refinements) There are many alternative equilibrium concepts being proposed Yet, despite the criticism and the alternative equilibrium concepts, Nash equilibrium is still the dominant equilibrium concept In terms of popularity of usage, no alternative equilibrium concept comes close The most important reason is that Nash equilibrium is relatively easy to compute, while many of the alternative equilibrium concepts are algorithmically difficult to compute This book proposes a new equilibrium concept that overcomes many of the shortcomings of Nash equilibrium The proposed new equilibrium concept also has the nice algorithmic property of easy computation Consequently, not only could it solve games that the current approach based on Nash equilibrium could solve, it could also solve games that the current approach based on Nash equilibrium is unable to solve The new equilibrium concept is Bayesian equilibrium by iterative conjectures Current Nash equilibrium-based games theory solves a game by asking which combinations of strategies constitute equilibriums The implicit assumption is that agents know the strategies adopted by the other agents and which equilibrium they are in, for otherwise they will not be able to react specifically to the optimal strategies of other agents but must react to the strategies of the other agents they predicted or expected or conjectured.1 This implicit assumption reduces the uncertainty facing the agents and simplifies computation and gives Nash equilibrium its strong appeal Further refinements such as sub-game perfect equilibrium, Bayesian Nash equilibrium and Perfect Bayesian (Nash) equilibrium, though adding further requirements, not change this implicit assumption.2 Bayesian equilibrium by the iterative conjectures (BEIC) approach, in contrast, solves a game by assuming that the agents not know the strategies adopted or will be adopted by other agents, and have no idea which equilibrium they are in or will be in Therefore, to select a strategy they need to form predictions or expectations or conjectures about the strategies adopted or will be adopted by other agents and the equilibrium they are in or will be in, as well as conjectures about such conjectures, ad infinitum They so by starting with first order uninformative predictive probability distribution functions (or expectations or conjectures) on the strategies of the other agents The agents then keep updating their conjectures with game theoretic and Bayesian statistical decision theoretic reasoning until a convergence of conjectures is achieved.3 In BEIC, the convergent conjectures are consistent with the equilibrium they support BEIC therefore rules out equilibriums that are based on conjectures that are inconsistent with the equilibriums they support, as well as equilibriums supported by convergent conjectures that not start with first order uninformative conjectures This difference in solution approach is related to an ongoing argument in games theory about the relative validity of the two conflicting notions of rationality; Bayesian statistical decision theoretic rationality, and strategic rationality (as embodied by the game theoretic concepts of Nash equilibrium and its many refinements) Currently, games theorists think that the two concepts of rationality are incompatible.4 While most games theorists are steadfast to the concept of strategic rationality, this book undertakes the task of reconstructing the whole basic framework of non-cooperative games using the notion of Bayesian rationality The specification of the process of conjecture formation in BEIC strengthens the concept of rationality used in games The consequence is a new kind of game theoretic rationality that is based upon Bayesian rationality This new game theoretic rationality includes rationality in actions and strategies, rationality in prior and posterior beliefs and, rationality in statistical decision rule The resulting Bayesian strand of games theory has a statistical decision theoretic foundation It analyzes a larger set of games than the existing Nash equilibrium-based games theory given the same game theoretic structure It also acts as an equilibrium selection criterion for the subset of games that the existing Nash equilibrium-based games theory analyzes BEIC normally decreases the number of equilibriums to one, and selects the equilibrium that is most compelling The BEIC approach therefore increases the analytical power of games theory in current applications It also allows games theory to be applied to new areas, such as games of multiple-sided incomplete information.5 To comprehend the need for a new theory of non-cooperative games with Bayesian statistical decision theory as its foundation, one has to go back to the history of non-cooperative games theory and Bayesian statistical decision theory Non-cooperative games theory started with the works of John Nash in the 1950s Bayesian statistics resurgence started in the 1970s and 1980s Consequently, non-cooperative games theory developed largely independently of Bayesian statistical decision theory and had not started with a firm statistical decision theoretic foundation The works of Harsanyi (1967, 1968a, 1968b) came after the contributions of Nash (1950, 1951) Harsanyi’s (1967, 1968a, 1968b) works allow games of incomplete information to be analyzed When Harsanyi first proposed his transformation of games of incomplete information, Kadane and Larkey (1982a, 1982b) criticized that the Harsanyi Bayesian games were not really Bayesian as they involved just the use of Bayes rule, but without the use of subjective probability and Bayesian decision theory in the decision-making process of the players This book will use subjective probability and Bayesian statistical decision theory to reconstruct non-cooperative games theory Subjective probability is fundamental to BEIC In the BEIC approach, the process of iterative conjecturing starts with first order uninformative conjectures These first order uninformative conjectures are of course subjective, as are subsequent higher order conjectures What is the rationale to start with first order uninformative conjectures? Other than the assumption that the agents have no idea about the strategies adopted by other agents and the equilibrium they are in at the beginning of the conjecturing process, there are two further compelling reasons for starting with uninformative conjectures First is the motive to let the game solve itself and select its own equilibrium strategies and convergent conjectures The equilibrium so achieved is therefore not imposed or affected by informative conjectures arbitrarily chosen, but by the underlying structure and elements of the game, including the payoffs of the agents and the information they have Second is to ensure that all pathways and information sets have equal probabilities of being reached at the initial round of reasoning That is to say, the conjecturing process explores every pathway and information set (either on or off equilibrium) Order of Conjectures Probability X Probability Y U[0,1] U[0,1] 0 0 By the BEIC approach, when cooperation has more (less) than twice the returns of defect, the players choose cooperation (defect) A comparison with the payoff-dominance and risk-dominance refinements proposed by Harsanyi and Selten (1988) and Harsanyi (1995) reveals that for the stag game, the BEIC approach picks the payoff-dominance equilibrium if the difference between the return from the payoff-dominance equilibrium and the return from the risk-dominance equilibrium is large If the difference between the returns is small, then the BEIC approach chooses the risk-dominance equilibrium Iterated admissibility Iterated admissibility requires that players play only strategies that survive the iterated elimination of (weakly) dominated strategies The following are two examples that show that the BEIC approach does better than the iterated admissibility criterion In the game of Table 4.11, there are two pure strategy Nash equilibriums, (U, L) and (M, C) By the repeated iterative elimination of (weakly) dominated strategies, there is only an equilibrium (U, L) The BEIC solution: Table 4.11 Table 4.12 A 3-by-3 game BEIC for the game in Table 4.11 BEIC gives the same answer as the criterion of iterated admissibility In the next example, which is essentially a coordination game, the solutions of the two approaches differ: The three pure strategy NEs are (A, a) and (B, b) and (C, c) However, (A, a) survives iterated admissibility while (B, b) and (C, c) not Furthermore, (B, b) Pareto dominates (A, a) and (C, c) and, (C, c) Pareto dominates (A, a) Therefore, (B, b) is a natural focal point of the game The BEIC solution: Table 4.13 Table 4.14 A 3-by-3 game BEIC for the game in Table 4.13 The BEIC approach picks (B, b) as the equilibrium 4.4 Simultaneous games with incomplete information The solution of incomplete information simultaneous games proceeds in a likewise manner Consider the following investment entry game where firm 1, the incumbent, has two types, high investment costs (with probability a) or low investment costs (with probability 1-a) When the high investment cost firm encounters firm they have the following payoff matrix: (The complete information simultaneous game with the above payoff matrix has two pure strategy Nash equilibriums and a mixed strategy Nash equilibrium, (w=0, y=1), (w=1, y=0) and (w=1/2, y=1/5) The BEIC is (w=0, y=1).) When low investment cost firm encounters firm they have the following payoff matrix: (The complete information simultaneous game with the above payoff matrix has two pure strategy Nash equilibriums and a mixed strategy Nash equilibrium, (w=0, y=1), (w=1, y=0) and (w=1/2, y=1/2) The BEIC is (w∈[0,1], (y∈[0,1]) The conjectures fail to converge and remain as uninformative conjectures with uniform distribution.) Table 4.15 Investment entry game A Firm Firm (High Cost) Table 4.16 Enter (y) Refrain (1-y) Modern (w) 0, −2 7, Antique (1-w) 4, 6, Enter (y) Refrain (1-y) Modern (x) 3, −2 7, Antique (1-x) 4, 6, Investment entry game B Firm Firm (Low Cost) The Bayesian Nash equilibriums are: (w = = ), (w = 0, x = , x = 1, y = , a > ), (w = 0, x = 1, y ∈ ,a , y = , a < ), (w = 1, x = 1, y = 0, a ∈ [0,1]), (w = 0, x = 0, y = 1, a ∈ [0,1]) Besides multiple equilibriums, the set of Bayesian Nash equilibriums also has the following implausible features: When it is more likely to have a high cost firm 1, that is, a>1/2, firm enters with a lower probability, that is, y=1/5, and when it is more likely to have a low cost firm 1, that is, a y+ z, w(y, z) ∈[0,1] if = II x(y, z) = if > y+ z, x(y, z) ∈[0,1] if = III y(w, x) = if > w+ x, y(w, x) ∈[0,1] if = w+ x, y(w, x) = 0, if < w+ x IV z(w, x) = if > w+ x, z(w, x) ∈[0,1] if = w+ x, z(w, z) = 0, if < w+ x y+ y+ z, w(y, z) = 0, if < z, x(y, z) = 0, if < y+ y+ z z The solution by the Bayesian iterative conjecture algorithm is presented below: Table 4.22 BEIC solution of two-sided incomplete information investment entry game A Figure 4.4 Figure 4.5 E(w) E(x) Figure 4.6 Figure 4.7 E(y) E(z) So the Bayesian equilibrium by iterative conjectures is w=0, x=0, y=1, z=1 As the chance of meeting the low cost firm is very high, both types of firm choose antique As the chances of meeting the high cost firm is quite high, both types of firm choose enter Comparative statics The comparative statics exercise below illustrates that the BEIC approach generates results that are intuitive and compelling Now let the probability of firm being the low cost type be 3/4 and the probability of firm being the high cost type be 1/4 and, the probability of firm being the low cost type be 1/0 and the probability of firm being the high cost type be 9/10 The reaction functions are: I w(y, z) = if > y+ z, w(y, z) ∈[0,1] if = II x(y, z) = if > y+ z, x(y, z) ∈[0,1] if = III y(w, x) = if > w+ x, y(w, x) ∈[0,1] if = w+ x, y(w, x) = 0, if < w+ x IV z(w, x) = if > w+ x, z(w, x) ∈[0,1] if = w+ x, z(w, z) = 0, if < w+ x Table 4.23 y+ y+ z, w(y, z) = 0, if < z, x(y, z) = 0, if < y+ y+ z z BEIC solution of two-sided incomplete information investment entry game B So the Bayesian equilibrium by iterative conjectures is w=1, x=1, y=0, z=0 As the chances of meeting the low cost firm is very low now, both types of firm choose modern As the chances of meeting the low cost firm is quite high now, both types of firm choose refrain 4.5 Conclusions There are many refinements of Nash equilibrium and many alternative equilibrium concepts.3 One line of research focuses on modeling how players in a game conjecture about the strategic choices of other players Bayesian methodologies are typically involved in this line of research.4 The examples in this chapter show that the BEIC approach has the merit of narrowing down the equilibriums generally to one The BEIC selects the most compelling equilibrium In general, when there is a unique stable pure strategy Nash equilibrium, that Nash equilibrium is also the BEIC solution When there are multiple Nash equilibriums (which could be stable or unstable or mixed), or an unstable Nash equilibrium (as in the case of a 2-by-2 matrix game with only mixed strategy equilibrium), then the solutions of the two approaches differ The BEIC approach typically picks a unique equilibrium that is stable and rules out unstable equilibriums Notes Refer to Aumann (1985) for criticisms of mixed strategy equilibrium For examples, see Chatterjee and Samuelson (1987), Cramton (1984), Powell (1988), Schweizer (1989), Watson (1995) Refer to Govindan and Wilson (2008) For examples, see Mertens and Zamir (1985), Aumann (1987), Harsanyi and Selten (1988), Tan and Werlang (1988), Blume, Brandenburger and Dekel (1991), Brandenburger (1992), Harsanyi (1995), Aumann and Dreze (2008) Conclusions To conclude, the major differences between the BEIC approach and the current Nash equilibriumbased approach are: A unified solution algorithm The current Nash equilibrium-based games theory has different equilibrium concepts or refinements, and their associated solution algorithms for different types of games are: Nash equilibrium for complete information simultaneous games, sub-game perfect equilibrium for complete information sequential games, Bayesian Nash equilibrium for incomplete information simultaneous games, perfect Bayesian equilibrium for incomplete information sequential games, and many more.1 Consequently, there are inconsistencies in solutions from using different equilibrium concepts and solution algorithms when one uses the current Nash equilibrium-based games theory This is not the case for the Bayesian iterative conjecture approach It is a unified approach: the same equilibrium concept (Bayesian equilibrium by iterative conjectures) and solution algorithm (Bayesian iterative conjecture algorithm) applies to all the aforementioned types of games and more Use of reaction functions The current Nash equilibrium-based games theory solves for equilibriums by constructing reaction functions and looks for their intersections The BEIC approach constructs reaction functions as well However, it uses first order uninformative conjectures and reaction functions to derive higher and higher orders of conjectures until a convergence of conjectures is achieved Definition of rationality The Nash equilibrium-based approach does not have rationality in the processing of information and forming of conjectures or predictions, that is, rationality in (statistical) decision rule It deals with the issue of processing information and forming predictions in an ad hoc manner through perfect Bayesian equilibrium and its many refinements In contrast, rationality in the processing of information and forming of predictions is the very foundation of the BEIC Equilibrium in strategic space versus equilibrium in subjective probability space The Nash equilibrium approach defines equilibrium in the strategic/actions space The incorporation of beliefs in incomplete information games does not change that basic feature In contrast, the BEIC approach defines equilibrium in subjective probability space through the convergence of conjectures Of course, for conjectures to converge, they must also be consistent with the equilibrium they support, and so the BEIC’s equilibrium in subjective probability space includes equilibrium in strategic/action space as well Objective versus subjective probability distribution function The BEIC is based on the Bayesian view of subjective probability This allows the tracing of the updating of conjectures from first order uninformative conjectures to higher and higher orders of conjectures until convergence The Nash equilibrium-based approach largely sticks to the classical or frequentist view of probability However, it makes an exception in sequential games of incomplete information with pooling equilibriums by resorting to subjective probability in the specification of off equilibrium beliefs Notes Refer to Fudenberg and Tirole (1991), Osborne and Rubinstein (1994), Myerson (1997), Vega-Redondo (2003), Rasmusen (2006) and Gintis (2009) Refer to Harsanyi (1982a, 1982b), Kadane (1982a, 1982b) for an intellectual exchange between these two views of probability and games theory References Aumann, R.J (1985) ‘What is game theory trying to accomplish?’ in Arrow, K and Honkapohja, S (eds),Frontiers of Economics, (Oxford: Basil Blackwell) Aumann, R.J (1987) ‘Correlated Equilibrium as an Expression of Bayesian Rationality’ Econometrica 55 (1): 1–18 Aumann, R.J and Dreze, J.H (2008) ‘Rational Expectations in Games’ American Economic Review 98 (1): 72–86 Bagwell, K (1995) ‘Commitment and Observability in Games’ Games and Economic Behavior 8: 271–280 Berger, J.O (1980) Statistical Decision Theory and Bayesian Analysis 2nd edn, (New York: Springer-Verlag) Bhaskar, 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374– 400 Vega-Redondo, F (2003) Economics and the Theory of Games Cambridge University Press Watson, J (1995) ‘Alternating-Offer Bargaining with Two-Sided Incomplete Information’ Review of Economic Studies 65, 573–594 Index Backward induction, 20–1, 32, 41, 47 Bayesian equilibrium by iterative conjectures, 1, 5, 8, 17, 23, 26–9, 35–7, 42–3, 56, 60, 64, 81, 83, 85 BEIC, 2–6, 31, 33–42, 44–9, 53, 55–6, 60–3, 66–77, 79, 81–3, 85–6 Decision rule, 3, 6, 8–9, 23–8, 86 Focal point, 6, 60, 68, 73 Harsanyi, 4–6, 31, 72, 84, 87 Inaccurate observation, 5–10, 19–20, 22–3, 25, 31 Indeterminacy, 20, 22–3, 28 Infimum, 8–9, 25 Intuitive criterion, 40–1, 43–4, 46–7, 54 Iterated admissibility, 68, 72–3 Kadane and Larkey, 4, 31, 87 Mixed strategy, 60, 62–3, 70, 74, 83 Nash equilibrium, 1–3, 5, 7, 22, 29–30, 33–4, 39, 59–63, 65, 67–70, 72, 74–6, 83, 85–7 Non credible, 32–4, 56 Off equilibrium belief, 31, 41–2, 87 Pareto, 73 Payoff dominance, 68–70, 72, Perfect Bayesian equilibrium, 7, 22, 29–32, 35–7, 39–43, 50–1, 53, 85–6 Pooling equilibrium, 31–2, 44, 87 Risk dominance, 68–70 Schelling point, 6, 60 Separating equilibrium, 44, 46 Sequential equilibrium, 39–43 Sub-game perfect equilibrium, 2, 29, 32, 34, 41, 85 Subjective probability, 4, 86–7 Supremum, 8–9, 25 Uninformative, 2, 4–6, 9, 11, 16, 28, 31–2, 35, 37, 41–2, 60, 63, 67, 69, 75, 86–7 .. .A Bayesian Theory of Games A Bayesian Theory of Games Iterative conjectures and determination of equilibrium JIMMY TENG Chartridge Books Oxford Hexagon House Avenue Station Lane Witney... for a new theory of non-cooperative games with Bayesian statistical decision theory as its foundation, one has to go back to the history of non-cooperative games theory and Bayesian statistical... for actions, strategies and conjectures and (statistical) decision rule The BEIC approach is capable of analyzing a larger set of games than current games theory, including games with noisy inaccurate