A Simple Adaptive Procedure Leading to Correlated Equilibrium

We propose a simple adaptive procedure for playing a game. In this procedure, players depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies (these measures are updated every period). It is shown that our adaptive procedure guaranties that with probability one, the sample distributions of play converge to the set of correlated equilibria of the game. To compute these regret measures, a player needs to know his payoff function and the history of play. We also offer a variation where every player knows only his own realized payoff history (but not his payoff function).

[1]  N. Megiddo On repeated games with incomplete information played by non-Bayesian players , 1980 .

[2]  Sergiu Hart,et al.  Existence of Correlated Equilibria , 1989, Math. Oper. Res..

[3]  Conditional Universal Consistency , 1999 .

[4]  R. Marimon Learning from learning in economics , 1996 .

[5]  L. Samuelson Evolutionary Games and Equilibrium Selection , 1997 .

[6]  A. Roth,et al.  Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria , 1998 .

[7]  J. Robinson An Iterative Method of Solving a Game , 1951 .

[8]  C. Sanchirico A Probabilistic Model of Learning in Games , 1996 .

[9]  Coherent behavior in noncooperative games , 1990 .

[10]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[11]  D. Blackwell An analog of the minimax theorem for vector payoffs. , 1956 .

[12]  Bayesian Rationality,et al.  CORRELATED EQUILIBRIUM AS AN EXPRESSION OF , 1987 .

[13]  J. Jordan Three Problems in Learning Mixed-Strategy Nash Equilibria , 1993 .

[14]  Fernando Vega-Redondo,et al.  Evolution, Games, and Economic Behaviour , 1996 .

[15]  On Pseudo-Games , 1968 .

[16]  A. Roth,et al.  Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term* , 1995 .

[17]  J. Hofbauer,et al.  Fictitious Play, Shapley Polygons and the Replicator Equation , 1995 .

[18]  A. Roth,et al.  Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria , 1998 .

[19]  Howard Raiffa,et al.  Games And Decisions , 1958 .

[20]  Nicolò Cesa-Bianchi,et al.  Gambling in a rigged casino: The adversarial multi-armed bandit problem , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[21]  Dean P. Foster,et al.  A Randomization Rule for Selecting Forecasts , 1993, Oper. Res..

[22]  R. Vohra,et al.  Calibrated Learning and Correlated Equilibrium , 1996 .

[23]  R. Aumann Subjectivity and Correlation in Randomized Strategies , 1974 .

[24]  R. Myerson Dual Reduction and Elementary Games , 1997 .

[25]  R. Aumann Correlated Equilibrium as an Expression of Bayesian Rationality Author ( s ) , 1987 .

[26]  V. Crawford Learning behavior and mixed-strategy Nash equilibria , 1985 .

[27]  D. Fudenberg,et al.  Consistency and Cautious Fictitious Play , 1995 .

[28]  William Feller,et al.  An Introduction to Probability Theory and its Applications. , 1958 .

[29]  James Hannan,et al.  4. APPROXIMATION TO RAYES RISK IN REPEATED PLAY , 1958 .

[30]  S. Zamir,et al.  Repeated Games. Part B: The Central Results , 1994 .

[31]  Dean P. Foster,et al.  Regret in the On-Line Decision Problem , 1999 .

[32]  R. Aumann,et al.  Epistemic Conditions for Nash Equilibrium , 1995 .

[33]  D. Saari,et al.  Effective Price Mechanisms , 1978 .

[34]  Camerer,et al.  Experience-Weighted Attraction Learning in Coordination Games: Probability Rules, Heterogeneity, and Time-Variation. , 1998, Journal of mathematical psychology.