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