Application of probability measures to the existence problem of Berge-Vaisman guaranteed equilibrium 1

Abstract

We formalize a guaranteed solution notion for a non-cooperative game of n persons under uncertainty. This notion is based on the appropriate modification of maximin and the Berge equilibrium. We obtain existence conditions for the guaranteed solution in the class of mixed strategies (probability measures). We prove existence of such solution in mixed strategies under standard (for the mathematical game theory) restrictions, such as continuity of payoff functions and compactness of the sets of players' strategies. We use a new method for construction of guaranteed solutions in pure and mixed strategies. We reduce construction of a guaranteed solution to construction of a saddle point of a special convolution of payoff functions.