Equilibria and Compromises in Two-Person Zero-Sum Multicriteria Games

The problem is to formalize the solution of a two-person zero-sum multicriteria (MC) game, providing a payoff to both players with respect to their MC best guaranteed results. As the basic concept of the solution of the MC game, the Shapley equilibrium is selected. It is parametrized by the inverse logical convolution based on scalarization in the Germeyer sense, i.e., on the weighted maximin approach. We investigate the relation between the equilibrium value of the game and its one-sided values defined for each player as the best guaranteed result that does not depend on the order of the moves. We describe the possibilities of a compromise in zero-sum MC games. For such a finite game in mixed strategies, we introduce the notions of the compromise and negotiation values and establish their relation to the equilibrium and one-sided values of the game. We consider a special interpretation of the MC averaging of the result for players oriented on using the inverse logical convolution. For such a case, the nonemptiness of the negotiation set is analyzed. The obtained conclusions are demonstrated on a prototype example.