Computing Best-Response Strategies in Infinite Games of Incomplete Information

We describe an algorithm for computing best-response strategies in a class of two-player infinite games of incomplete information, defined by payoffs piecewise linear in agents' types and actions, conditional on linear comparisons of agents' actions. We show that this class includes many well-known games including a variety of auctions and a novel allocation game. In some cases, the best-response algorithm can be iterated to compute Bayes-Nash equilibria. We demonstrate the efficacy of our approach on existing and new games.

[1]  Daniel M. Reeves,et al.  Notes on Equilibria in Symmetric Games , 2004 .

[2]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[3]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[4]  Roy Radner,et al.  Equilibria of the Sealed- Bid Mechanism for Bargaining with Incomplete Information*, ' , 1987 .

[5]  R. Myerson.  Refinements of the Nash equilibrium concept , 1978 .

[6]  Paul R. Milgrom,et al.  Adaptive and sophisticated learning in normal form games , 1991 .

[7]  J. Harsanyi.  Games with Incomplete Information Played by “Bayesian” Players Part II. Bayesian Equilibrium Points , 1968 .

[8]  Michael P. Wellman,et al.  Combinatorial auctions for supply chain formation , 2000, EC '00.

[9]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[10]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[11]  J. Robinson.  AN ITERATIVE METHOD OF SOLVING A GAME , 1951, Classics in Game Theory.

[12]  Michael B. Gordy.  Computationally Convenient Distributional Assumptions for Common-Value Auctions , 1997 .

[13]  R. McAfee,et al.  Auctions and Bidding , 1986 .

[14]  Michael L. Littman,et al.  Graphical Models for Game Theory , 2001, UAI.

[15]  Robert Wilson,et al.  A global Newton method to compute Nash equilibria , 2003, J. Econ. Theory.

[16]  Robert Wilson,et al.  Structure theorems for game trees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[17]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .

[18]  D. Fudenberg,et al.  The Theory of Learning in Games , 1998 .

[19]  S. Vajda.  Some topics in two-person games , 1971 .

[20]  Avi Pfeffer,et al.  Representations and Solutions for Game-Theoretic Problems , 1997, Artif. Intell..

[21]  D. Koller,et al.  Efficient Computation of Equilibria for Extensive Two-Person Games , 1996 .

[22]  J. Neumann,et al.  Theory of Games and Economic Behavior. , 1945 .

[23]  Michael B. Gordy.  Computationally Convenient Distributional Assumptions for Common Value Auctions , 1997 .

[24]  Gerhard Weiß,et al.  Antisocial Agents and Vickrey Auctions , 2001, ATAL.

[25]  Pierfrancesco La Mura.  Game Networks , 2000, UAI.

[26]  A. Copeland.  Review: John von Neumann and Oskar Morgenstern, Theory of games and economic behavior , 1945 .

[27]  Dov Monderer,et al.  A Learning Approach to Auctions , 1998 .

[28]  Daphne Koller,et al.  A Continuation Method for Nash Equilibria in Structured Games , 2003, IJCAI.

[29]  M. Bacharach.  Two-person Cooperative Games , 1976 .

[30]  R. Rubinstein,et al.  On relaxation algorithms in computation of noncooperative equilibria , 1994, IEEE Trans. Autom. Control..

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

[32]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[33]  D. H. Macgregor,et al.  The Mathematical Principles of the Theory of Wealth, 1838. , 1929 .

[34]  Michael P. Wellman,et al.  Computing approximate bayes-nash equilibria in tree-games of incomplete information , 2004, EC '04.

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

[36]  Kalyan Chatterjee,et al.  Bargaining under Incomplete Information , 1983, Oper. Res..

[37]  Steven R. Williams,et al.  Bilateral trade with the sealed bid k-double auction: Existence and efficiency , 1989 .

[38]  A. Cournot.  Researches into the Mathematical Principles of the Theory of Wealth , 1898, Forerunners of Realizable Values Accounting in Financial Reporting.

[39]  Susan Athey,et al.  Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information , 1997 .

[40]  O. H. Brownlee,et al.  ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION , 1952 .