Nash equilibria in normal games via optimization methods

This paper is devoted to Nash equilibria of normal-form games. We give a survey on computing Nash equilibria via optimization methods. Further on we prove, that the Nash equilibria of a game coincide with the zeros of a nonlinear, almost everywhere smooth system of equations with the same dimension as the strategy space. Thus, we can apply tools for solving nonlinear systems of equations. We present an algorithm for computing Nash equilibria, which has shown very satisfactory results.

[1]  R. McKelvey.  A Liapunov Function for Nash Equilibria , 1998 .

[2]  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 .

[3]  Dolf Talman,et al.  Finding a Nash equilibrium in noncooperativeN-person games by solving a sequence of linear stationary point problems , 1994, Math. Methods Oper. Res..

[4]  Bapi Chatterjee,et al.  An optimization formulation to compute Nash equilibrium in finite games , 2009, 2009 Proceeding of International Conference on Methods and Models in Computer Science (ICM2CS).

[5]  Jein-Shan Chen,et al.  A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs , 2012 .

[6]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[7]  J. Rosenmüller.  On a Generalization of the Lemke–Howson Algorithm to Noncooperative N-Person Games , 1971 .

[8]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

[9]  P. Herings,et al.  Homotopy methods to compute equilibria in game theory , 2006 .

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

[11]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[12]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[13]  R. Fletcher.  Practical Methods of Optimization , 1988 .

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

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

[16]  Natalia Akchurina.  Multi-agent reinforcement learning algorithms , 2010 .

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

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

[19]  J. Nash.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[20]  Robert Wilson,et al.  Computing Equilibria of N-Person Games , 1971 .

[21]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[22]  Lars Mathiesen,et al.  An algorithm based on a sequence of linear complementarity problems applied to a walrasian equilibrium model: An example , 1987, Math. Program..

[23]  Ruchira S. Datta.  Universality of Nash Equilibria , 2003, Math. Oper. Res..

[24]  Richard J. Lipton,et al.  Nash Equilibria via Polynomial Equations , 2004, LATIN.

[25]  Ruchira S. Datta.  Finding all Nash equilibria of a finite game using polynomial algebra , 2006 .