Polynomial games and sum of squares optimization

We study two-person zero-sum games, where the payoff function is a polynomial expression in the actions of the players. This class of games was introduced by Dresher, Karlin, and Shapley in 1950. We show that the value of the game, and the corresponding optimal strategies, can be obtained by solving a single semidefinite programming problem. In addition, we show how the results extend, with suitable modifications, to a general class of semialgebraic games

[1]  Ioana Popescu,et al.  Optimal Inequalities in Probability Theory: A Convex Optimization Approach , 2005, SIAM J. Optim..

[2]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[3]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[4]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[5]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[6]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[7]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[8]  N. Z. Shor.  Class of global minimum bounds of polynomial functions , 1987 .

[9]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[10]  David Gale,et al.  A note on polynomial and separable games , 1958 .

[11]  S Karlin,et al.  Geometry of Reduced Moment Spaces. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Marc Nerlove,et al.  Mathematical Methods and Theory in Games, Programming, and Economics, Vol. I: Matrix Games, Programming, and Mathematical Economics. Vol. II: The Theory of Infinite Games. Samuel Karlin , 1960 .

[13]  Samuel Karlin,et al.  Mathematical Methods and Theory in Games, Programming, and Economics , 1961 .

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

[15]  Asuman E. Ozdaglar,et al.  Separable and low-rank continuous games , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[16]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[17]  Samuel Karlin,et al.  Mathematical Methods and Theory in Games, Programming, and Economics , 1961 .

[18]  I. Glicksberg,et al.  9. Notes on Games over the Square , 1953 .

[19]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[20]  L. Shapley,et al.  Geometry of Moment Spaces , 1953 .