Vector-valued games: characterization of equilibria in matrix games

In this paper, we provide a systematization of the equilibrium concepts for vector-valued games. In general abstract setting, we give the definitions of Nash equilibrium, weak Nash equilibrium, and proper Nash equilibrium, we investigate the relationships among them, and we provide existence results for proper Nash equilibria and weak Nash equilibria. The link between the introduced equilibrium concepts and Nash equilibria of a game with scalar payoffs obtained by means of linear scalarization is investigated. For a vector matrix game, we show that the notions of proper Nash equilibrium and Nash equilibrium are equivalent, we prove existence of both Nash equilibria and weak Nash equilibria, we give characterizations of proper and weak best correspondence sets respectively, by defining characterizing functions from the matrices and the preference cones, and finally we list all best correspondence sets. The given definitions are illustrated by means of examples.