Epistemic Game Theory Adam Brandenburger J.P. Valles Professor, NYU Stern School of Business Distinguished Professor, NYU Tandon School of Engineering Faculty Director, NYU Shanghai Program on Creativity + Innovation Global Network Professor New York University
These particular uncertainties --- as to the other player s beliefs about oneself --- are almost universal, and it would constrict the application of a game theory fatally to rule them out -- Daniel Ellsberg (1959) Ann Bob Ann 7/12/17 3:17 PM 2
The Curious History of Game Theory Von Neumann (1928) maximin Nash (1951) equilibrium Arrow, Barankin, and Blackwell (1953), Gale (1953),..., Bernheim (1984), Pearce (1984) iterated dominance 7/12/17 3:17 PM http://www.pnas.org/site/classics/classics5.xhtml; 3 http://www.awesomestories.com/asset/view/john-nash-photo-as-a-young-man
What is Epistemic Game Theory (EGT)? Epistemic: of or relating to knowledge or cognition (en.wiktionary.org) EGT extends the expressive power of game theory as a language, in particular, by creating a language of hierarchies of beliefs about the game In EGT, notions such as rationality, belief, belief about rationality, belief about belief, etc., can be given precise meanings, and the implications of different epistemic assumptions on games can be worked out 7/12/17 3:17 PM 4
Baseline Assumptions (A1) Ann is rational (B1) Bob is rational (A2) Ann is rational and believes [Bob is rational] (B2) Bob is rational and believes [Ann is rational] (A3) Ann is rational and believes [Bob is rational] and believes [Bob is rational and believes [Ann is rational]] (B3) Under these assumptions, Ann and Bob will choose iteratively undominated (IU) strategies (This is a characterization of the IU strategies; see Brandenburger and Dekel, 1987, and Tan and Werlang, 1988) 7/12/17 3:17 PM 5
Epistemic Scenarios 1. Ann is not certain what Bob believes about her choice of strategy 2. Ann and Bob have different beliefs about the strategy chosen by a third player, Charlie 3. Charlie may have correlation in her belief about the strategies chosen by Ann and Bob, because she thinks their hierarchies of beliefs are correlated (even if they choose independently) All three scenarios can be described formally in epistemic game theory There is no intrinsic place for any of these scenarios in Nash equilibrium (for a precise statement, see the converse theorems in Aumann and Brandenburger, 1995) Each of these scenarios may be necessary for the play of IU strategies (under the preceding rationality assumptions) 7/12/17 3:17 PM 6
What Does EGT Predict (under the Baseline Assumptions)? IU may involve correlation in a player s beliefs about other players strategies The strategy Y is optimal for Charlie if she puts probability ½ : ½ on (U, L) : (D, R) There are no independent probabilities under which Y is optimal 7/12/17 3:17 PM 7
What Does EGT Predict contd.? IU may involve too much correlation --- i.e., correlation which cannot be explained via epistemic variables The strategy Y is optimal for Charlie if she puts probability ½ : ½ on (U, L) : (M, C) It is an IU strategy But it cannot be played under common-cause correlation using, as correlating variables, the hierarchies of beliefs held by Ann and Bob (Brandenburger and Friedenberg, 2008) 7/12/17 3:17 PM 8
Implications 1. EGT is not just a foundation for existing solution concepts 2. To the extent that EGT is empirically supported by cognitive science, it becomes important to develop new epistemically grounded solution concepts 3. This is true for game matrices, and we will now see it is true in an even stronger sense in game trees 7/12/17 3:17 PM 9
Epistemics in Game Trees The first guess is that the previous baseline rationality assumptions, suitably formulated for (perfect-information) game trees, are characterized by backward induction This is false, as this tree (Reny, 1992) shows Backward induction preserves I a - o a until the fourth round But I a - o a is dominated by O a on the first round (From Brandenburger, Danieli, and Friedenberg, 2017) 7/12/17 3:17 PM 10
Epistemics in Game Trees contd. The second guess is an iterated-dominance procedure tailored to the tree, called extensive-form rationalizability (Pearce, 1984; Battigalli and Sinischalchi, 2002) Round 1: Delete In - D Round 2: Delete R Round 3: Delete Out 7/12/17 3:17 PM 11
Epistemics in Game Trees contd. But suppose it is believed (technically, common full belief) that Bob is a bully --- i.e., that he will play R Then, it is consistent with the baseline (iterated) rationality assumptions for Ann to play Out For a characterization of the strategies that can be played under m-iterated rationality assumptions in the tree (for different m), see Brandenburger, Danieli, and Friedenberg (2017) 7/12/17 3:17 PM 12
The Future? 1. To do EGT, why don t we specify the sets of hierarchies of beliefs which the players in a game might hold (just as we specify their strategy sets and payoff functions) and make predictions relative to these variables? 2. The traditional view seems to have been that while strategy sets and payoff functions are observable, the players (hierarchies of) beliefs are unobservable 3. But game theory is changing 4. There are game experiments involving elicitation of beliefs (e.g., Costa- Gomes and Weizsa cker, 2008; Rey-Biel, 2009; and references) 5. With the rise of cognitive neuroscience, game theory can hope to acquire direct data about beliefs --- and even hierarchies of beliefs --- in games 7/12/17 3:17 PM 13
A Speculation on Belief Formation (from a Cognitive Science Perspective) Ann selects a candidate strategy choice (she anchors) She then examines her view as to whether or not Bob thinks she intends to make this choice (she adjusts) There is precedent for anchoring and adjusting processes in the Theory of Mind literature, e.g., in gauging another individual s preferences (Epley, Keysar, Van Boven, and Gilovich, 2004; Tamir and Mitchell, 2010) This can be thought of as an internal equilibrium vs. disequilibrium process It allows a simple case-counting argument for the complexity of higherorder beliefs (Brandenburger and Li, 2015) 7/12/17 3:17 PM 14
Theory of Games and Observation of Behavior Our knowledge of the relevant facts of economics is incomparably smaller than that commanded in physics at the time when the mathematization of that subject was achieved -- Von Neumann and Morgenstern (1944) 7/12/17 3:17 PM 15
References Arrow, K., E. Barankin, and D. Blackwell, Admissible Points of Convex Sets, in Kuhn, H., and A. Tucker (eds.), Contributions to the Theory of Games, Vol. II, Princeton University Press, 1953, 87-91. Aumann, R., and A. Brandenburger, Epistemic Conditions for Nash Equilibrium, Econometrica, 63, 1995, 1161-1180. Battigalli, P., and M. Siniscalchi, Strong Belief and Forward-Induction Reasoning, Journal of Economic Theory, 106, 2002, 356-391. Bernheim, D., Rationalizable Strategic Behavior, Econometrica, 52, 1984, 1007-1028. Brandenburger, A., A. Danieli, and A. Friedenberg, How Many Levels Do Players Reason? An Observational Challenge and Solution, 2017, at adambrandenburger.com. Brandenburger, A., and E. Dekel, Rationalizability and Correlated Equilibria, Econometrica, 55, 1987, 1391-1402. 7/12/17 3:17 PM 16
References contd. Brandenburger, A., and A. Friedenberg, Intrinsic Correlation in Games, Journal of Economic Theory, 141, 2008, 28-67. Brandenburger, A., and X. Li, Thinking About Thinking and Its Cognitive Limits, 2015, at adambrandenburger.com. Costa-Gomes, M., and G. Weizsa cker, Stated Beliefs and Play in Normal Form Games, Review of Economic Studies, 75, 2008, 729-762. Ellsberg, D., Rejoinder, Review of Economics and Statistics, 16, 1959, 42-43. Epley, N., B. Keysar, L. Van Boven, and T. Gilovich, Perspective Taking as Egocentric Anchoring and Adjustment, Journal of Personality and Social Psychology, 87, 2004, 327-339. Tamir, D., and J. Mitchell, Neural Correlates of Anchoring-and-Adjustment During Mentalizing, PNAS, 107, 2010, 10827-10832 Gale, D., A Theory of N-Person Games with Perfect Information, PNAS, 39, 1953, 496-501. 7/12/17 3:17 PM 17
References contd. Nash, J., Non-cooperative Games, Annals of Mathematics, 54, 1951, 286-295. Pearce, D., Rational Strategic Behavior and the Problem of Perfection, Econometrica, 52, 1984, 1029-1050. Reny, P., Backward Induction, Normal Form Perfection, and Explicable Equilibria, Econometrica, 60, 1992, 627-649. Rey-Biel, P., Equilibrium Play and Best Response to (Stated) Beliefs in Normal Form Games, Games and Economic Behavior, 65, 2009, 572-585. Tan, T., and S. Werlang, The Bayesian Foundations of Solution Concepts of Games, Journal of Economic Theory, 45, 1988, 370-391. Von Neumann, J., Zur Theorie der Gesellschaftsspiele, Mathematische Annalen, 100, 1928, 295-320. Von Neumann, J., and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944. 7/12/17 3:17 PM 18
Appendix 1 Ann is not certain what Bob believes about her choice of strategy L C R T 2 1 1 1 0 0 M 1 0 0 1 1 0 B 0 0 1 1 2 1 To play M, Ann must put probability ½ : ½ on L : R To play L, Bob must put probability 1 on T To play R, Bob must put probability 1 on B 7/12/17 3:17 PM 19
Appendix 2 Ann and Bob have different beliefs about the strategy chosen by a third player, Charlie L R L R T 3, 0, 3, 2, T 0, 3, 0, 2, B 2, 0, 2, 2, B 2, 3, 2, 2, W E To play T, Ann must put probability 2/3 on W To play L, Bob must put probability 2/3 on E 7/12/17 3:17 PM 20