An Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games

This paper presents a technique for approximating, up to any precision, the set of subgame-perfect equilibria (SPE) in repeated games with discounting. The process starts with a single hypercube approximation of the set of SPE payoff profiles. Then the initial hypercube is gradually partitioned on to a set of smaller adjacent hypercubes, while those hypercubes that cannot contain any SPE point are gradually withdrawn. Whether a given hypercube can contain an equilibrium point is verified by an appropriate mixed integer program. A special attention is paid to the question of extracting players' strategies and their representability in form of finite automata.

[1]  Dilip Abreu.  On the Theory of Infinitely Repeated Games with Discounting , 1988 .

[2]  Shai Halevi,et al.  A Cryptographic Solution to a Game Theoretic Problem , 2000, CRYPTO.

[3]  H. Kuk.  On equilibrium points in bimatrix games , 1996 .

[4]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

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

[6]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[7]  E. Stacchetti,et al.  Towards a Theory of Discounted Repeated Games with Imperfect Monitoring , 1990 .

[8]  Jon Lee,et al.  Disjunctive Cuts for Non-convex Mixed Integer Quadratically Constrained Programs , 2008, IPCO.

[9]  G. Gordon,et al.  Finding Correlated Equilibria in General Sum Stochastic Games , 2007 .

[10]  K. Judd,et al.  Computing Supergame Equilibria , 2003 .

[11]  J. Nash.  THE BARGAINING PROBLEM , 1950, Classics in Game Theory.

[12]  Robert J. Aumann,et al.  Repeated Games with Incomplete Information , 1995 .

[13]  David G. Luenberger,et al.  Strongly Symmetric Subgame Perfect Equilibria in Infinitely Repeated Games with Perfect Monitoring and Discounting , 1994 .

[14]  M. Cronshaw,et al.  Algorithms for Finding Repeated Game Equilibria , 1997 .

[15]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[16]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[17]  R. McKelvey,et al.  Computation of equilibria in finite games , 1996 .

[18]  E. Kalai,et al.  Finite Rationality and Interpersonal Complexity in Repeated Games , 1988 .

[19]  Peter Stone,et al.  A polynomial-time nash equilibrium algorithm for repeated games , 2003, EC '03.

[20]  Yoav Shoham,et al.  Simple search methods for finding a Nash equilibrium , 2004, Games Econ. Behav..

[21]  B. Stengel,et al.  COMPUTING EQUILIBRIA FOR TWO-PERSON GAMES , 1996 .

[22]  G. Mailath,et al.  Repeated Games and Reputations: Long-Run Relationships , 2006 .

[23]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Xi Chen,et al.  Computing Nash Equilibria: Approximation and Smoothed Complexity , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[25]  A. Rubinstein,et al.  A Course in Game Theory , 1995 .

[26]  Charles L. Isbell,et al.  Solving Stochastic Games , 2009, NIPS.