Mixed-Integer Programming Methods for Finding Nash Equilibria

We present, to our knowledge, the first mixed integer program (MIP) formulations for finding Nash equilibria in games (specifically, two-player normal form games). We study different design dimensions of search algorithms that are based on those formulations. Our MIP Nash algorithm outperforms Lemke-Howson but not Porter-Nudelman-Shoham (PNS) on GAMUT data. We argue why experiments should also be conducted on games with equilibria with medium-sized supports only, and present a methodology for generating such games. On such games MIP Nash drastically outperforms PNS but not Lemke-Howson. Certain MIP Nash formulations also yield anytime algorithms for E-equilibrium. with provable bounds. Another advantage of MIP Nash is that it can be used to find an optimal equilibrium (according to various objectives). The prior algorithms can be extended to that setting, but they are orders of magnitude slower.

[1]  P. Pardalos,et al.  Optimization and optimal control , 2003 .

[2]  M. Padberg,et al.  Addendum: Optimization of a 532-city symmetric traveling salesman problem by branch and cut , 1990 .

[3]  J. M. Bilbao,et al.  Contributions to the Theory of Games , 2005 .

[4]  Anatoly Antipin.  Extragradient approach to the solution of two person non-zero sum games , 2003 .

[5]  N. Vorob’ev.  Equilibrium Points in Bimatrix Games , 1958 .

[6]  Todd R. Kaplan,et al.  A Program for Finding Nash Equilibria , 1993 .

[7]  M. Whinston,et al.  Coalition-Proof Nash Equilibria I. Concepts , 1987 .

[8]  O. Mangasarian.  Equilibrium Points of Bimatrix Games , 1964 .

[9]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet , 2001, ICALP.

[10]  Andrew McLennan,et al.  Asymptotic expected number of Nash equilibria of two-player normal form games , 2005, Games Econ. Behav..

[11]  Andrew McLennan,et al.  Gambit: Software Tools for Game Theory , 2006 .

[12]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[13]  Oleg Khamisov.  A Global Optimization Approach to Solving Equilibrium Programming Problems , 2003 .

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

[15]  Aranyak Mehta,et al.  Playing large games using simple strategies , 2003, EC '03.

[16]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[17]  Bernhard von Stengel,et al.  Exponentially many steps for finding a Nash equilibrium in a bimatrix game , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[18]  Vincent Conitzer,et al.  Complexity Results about Nash Equilibria , 2002, IJCAI.

[19]  Robert J. Aumann,et al.  16. Acceptable Points in General Cooperative n-Person Games , 1959 .

[20]  Yoav Shoham,et al.  Run the GAMUT: a comprehensive approach to evaluating game-theoretic algorithms , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

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