Polynomial-time computation of exact correlated equilibrium in compact games

In a landmark paper, Papadimitriou and Roughgarden described a polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample correlated equilibria of concisely-represented games. Recently, Stein, Parrilo and Ozdaglar showed that this algorithm can fail to find an exact correlated equilibrium, but can be easily modified to efficiently compute approximate correlated equilibria. Currently, it remains an open problem to determine whether the algorithm can be modified to compute an exact correlated equilibrium. We show that it can, presenting a variant of the Ellipsoid Against Hope algorithm that guarantees the polynomial-time identification of exact correlated equilibrium. Also, our algorithm is the first to tractably compute correlated equilibria with polynomial-sized supports; such correlated equilibria are more natural solutions than the mixtures of product distributions produced previously, and have several advantages including requiring fewer bits to represent, being easier to sample from, and being easier to verify. Our algorithm differs from the original primarily in its use of a separation oracle that produces cuts corresponding to pure-strategy profiles. As a result, we no longer face the numerical precision issues encountered by the original approach, and both the resulting algorithm and its analysis are considerably simplified. Our new separation oracle can be understood as a derandomization of Papadimitriou and Roughgarden's original separation oracle via the method of conditional probabilities.

[1]  Wan Huang,et al.  Equilibrium Computation for Extensive Games , 2011 .

[2]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[3]  R. Rosenthal.  A class of games possessing pure-strategy Nash equilibria , 1973 .

[4]  Yishay Mansour,et al.  How long to equilibrium? The communication complexity of uncoupled equilibrium procedures , 2010, Games Econ. Behav..

[5]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[6]  Georg Gottlob,et al.  Pure Nash equilibria: hard and easy games , 2003, TARK '03.

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

[8]  Fabrizio Germano,et al.  Existence of Sparsely Supported Correlated Equilibria , 2005 .

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

[10]  Richard M. Karp,et al.  On Linear Characterizations of Combinatorial Optimization Problems , 1982, SIAM J. Comput..

[11]  Christos H. Papadimitriou,et al.  The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games , 2006, ICALP.

[12]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[13]  Christos H. Papadimitriou,et al.  Computing correlated equilibria in multi-player games , 2005, STOC '05.

[14]  Paul Erdös,et al.  On a Combinatorial Game , 1973, J. Comb. Theory, Ser. A.

[15]  Pablo A. Parrilo,et al.  Exchangeable equilibria contradict exactness of the Papadimitriou-Roughgarden algorithm , 2010, ArXiv.

[16]  Tim Roughgarden,et al.  Computing equilibria in multi-player games , 2005, SODA '05.

[17]  A. Talman,et al.  Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling , 1987 .

[18]  Kevin Leyton-Brown,et al.  Action-Graph Games , 2011, Games Econ. Behav..

[19]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

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

[21]  Paul W. Goldberg,et al.  Reducibility among equilibrium problems , 2006, STOC '06.

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

[23]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[24]  D. Griffel.  Linear programming 2: Theory and extensions , by G. B. Dantzig and M. N. Thapa. Pp. 408. £50.00. 2003 ISBN 0 387 00834 9 (Springer). , 2004, The Mathematical Gazette.

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

[26]  Bernhard von Stengel,et al.  Extensive-Form Correlated Equilibrium: Definition and Computational Complexity , 2008, Math. Oper. Res..

[27]  David Lindley,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 2010 .

[28]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[29]  Bernhard von Stengel,et al.  Computing an Extensive-Form Correlated Equilibrium in Polynomial Time , 2008, WINE.

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

[31]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[32]  Tim Roughgarden,et al.  Comment on “Computing Correlated Equilibria in Multi-Player Games” , 2010 .

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

[34]  Kevin Leyton-Brown,et al.  Computing Nash Equilibria of Action-Graph Games , 2004, UAI.

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

[36]  K. McCardle,et al.  Coherent behavior in noncooperative games , 1990 .

[37]  Christos H. Papadimitriou,et al.  Three-Player Games Are Hard , 2005, Electron. Colloquium Comput. Complex..

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