Search for nash equilibria in bimatrix games with probability and quantile payoff functions

A bimatrix game with deterministic payoffs and mixed strategies is considered. The probability and quantile functions of the players’ losses (payoffs taken with the opposite sign) are defined. The problem of search for a Nash equilibrium is considered for these functions. It is shown that the game with probability criteria is reduced to a bimatrix game with expectation payoff functions. Necessary and sufficient conditions for the existence of an equilibrium in a game with a quantile criterion are obtained. A theorem on the relation between equilibria in games with quantile and probability criteria is proved. An algorithm for searching equilibria in the game with a quantile criterion is proposed. The algorithm is based on successively solving problems of searching for points belonging to sets described by quadratic nonconvex constraints. Approaches to finding these points are proposed. Results of calculating equilibrium pairs of strategies are given.