Nash and Stackelberg Equilibria in Games with Pay-Off Functions Constructed by Minimum Convolutions of Antagonistic and Private Criteria

Abstract

AbstractThe paper proposes game models with pay-off functions being convolutions by the operation of taking minimum of two criteria one of which describes competition of players in some common (external) sphere of activity and the other describes private achievements of each player (in internal sphere). Strategies of players are distributions of resources between external and internal spheres. The first criterion of each player depends on strategies of all players; the second depends only on the strategy of given player. It is shown that under some natural assumptions of monotony of criteria such n-person games have good properties, namely, Nash equilibrium exists, is strong, stable and Pareto optimal. For two-person games, in Stackelberg equilibrium both the leader and the follower gain no less than in the best Nash equilibrium and the last belongs to \(\gamma \)-core. KeywordsPay-off functionsStackelberg equilibriumNash equilibriumMinimum convolutionsExternal sphereInternal sphere