Refinement of pure Pareto Nash equilibria in finite multicriteria games using preference relations

Abstract

In this paper, we consider a noncooperative finite multicriteria two-person game G. We study the problem of refinement of Pareto Nash equilibria of G and we propose an approach based on modeling the preferences of the players by two binary relations. This approach follows three main steps: first, we associate to G another game $$\bar{G}$$ defined by the two sets of strategies and two binary relations over the set of the strategy profiles, where each binary relation expresses the preferences of the corresponding player. Second, we define a Noncooperative Equilibrium $$\textit{NCE}$$ for the game $$\bar{G}$$ and we prove that every $$\textit{NCE}$$ of $$\bar{G}$$ is a Pareto Nash equilibrium of G. Third, we propose a procedure for finding the set of $$\textit{NCE}$$ of $$\bar{G}$$ independently of how the binary relations are constructed. Moreover, we give three ways to model the preferences of the players by using scalarization and the outranking methods ELECTRE I and PROMETHEE II. All the steps of the proposed approach are completely illustrated through an accompanying example.