Affiliations: [a] Department of Optimal Control, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia | [b] District Committee of Administrative District Marburg-Biedenkopf, Marburg, Germany | [c] Orekhovo-Zuevo Branch of Moscow State University of Technologies and Management Named After K.G. Rasumovskiy, Orekhovo-Zuevo, Russia | Minneapolis, USA
Correspondence:
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Corresponding author: Vladislav Zhukovskiy, Department of Optimal Control, 2nd Educational Building, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia. E-mail: [email protected].
Note: [1] To the memory of Konstantin Vaisman, who encouraged and inspired our research.
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.
Keywords: Probability measure, mixed strategy, weak compactness in itself, guarantee, Berge-Vaisman equilibrium, Nash equilibrium, maximin
