Axiomatic Approach to Solutions of Games

We consider solutions of normal form games that are invariant under strategic equivalence. We consider additional properties that can be expected (or be desired) from a solution of a game, and we observe the following: • Even the weakest notion of individual rationality restricts the set of solutions to be equilibria. This observation holds for all types of solutions: in pure-strategies, in mixed strategies, and in correlated strategies where the corresponding notions of equilibria are pure-Nash, Nash and coarse-correlated. An action profile is (strict) simultaneous maximizer if it simultaneously globally (strictly) maximizes the payoffs of all players. • If we require that a simultaneous maximizer (if it exists) will be a solution, then the solution contains the set of pure Nash equilibria. • There is no solution for which a strict simultaneous maximizer (if it exists) is the unique solution. ∗Center for the Mathematics of Information, Department of Computing and Mathematical Sciences, California Institute of Technology. e-mail:[email protected].