Symmetry and p-Stability

A symmetry of a game is a permutation of the player set and their strategy sets that leaves the payoff functions invariant. In this paper we introduce and discuss two relatively mild symmetry properties for set-valued solution concepts (that are equivalent when the solution concepts are single-valued) and show using examples that stable sets satisfy neither version. These examples also show that for every integer q, there exists a game with an equilibrium component of index q.