Germeier’s Scalarization for Approximating Solution of Multicriteria Matrix Games

Abstract

In this paper, we study the properties of Germeier’s scalarization applied for solving multicriteria games. The equilibria and the equilibrium values of such games, as a rule, make sets, and the problems of parametrizing and approximating these sets arise. Shapley proved that Nash equilibrium of multicriteria matrix game can be found by solving a two-parametric family of scalar games obtained with the help of linear scalarization of the criteria vector. We show that Germeier’s scalarization parametrizes the equilibria of the multicriteria game by using one-parametric family of scalar games. Germeier’s scalarization has certain advantages over the linear one, and we suggest it for approximating the multicriteria game equilibria with a finite set. For two-criteria games with 2×2 matrices, we show by examples that there is no continuity of the values of scalar games in the scalarizing parameters. We prove one-sided (from the left or from the right) continuity for the game values. As a result, we come to convergence in Hausdorff metric for the set of equilibrium values obtained for ϵ-net on the simplex of scalarizing parameters to the value of the multicriteria game as ϵ→0. The constructed finite approximation may be helpful in practical applications, where players try to find a compromise in an iterative negotiating procedure under multiple criteria.