Abstract. We prove that an equilibrium of a nondegenerate bimatrix game has index +1 if and only if it can be made the unique equilibrium of an extended game with additional strategies of one player. The main tool is the “dual construction”. A simplicial polytope, dual to the common best-response polytope of one player, has its facets subdivided into best-response regions, so that equilibria are completely labeled points on the surface of that polytope. That surface has dimension m − 1foranm × n game, which is much lower than the dimension m+n of the polytopes that are classically used. 1

Static stability of equilibrium in strategic games differs from dynamic stability in not being linked to any particular dynamical system. In other words, it does not make any assumptions about off-equilibrium behavior. Examples of static notions of stability include evolutionarily stable strategy (ESS) and continuously stable strategy (CSS), both of which are meaningful or justifiable only for particular classes of games, namely, symmetric multilinear games or symmetric games with a unidimensional strategy space, respectively. This paper presents a general notion of local static stability, of which the above two are essentially special cases. It is applicable to virtually all

Static stability in strategic games differs from dynamic stability in only considering the players ’ incentives to change their strategies. It does not rely on any assumptions about the players ’ reactions to these incentives and it is thus independent of the law of motion (e.g., whether players move simultaneously or sequentially). Examples of static notions of stability include evolutionarily stable strategy (ESS) and continuously stable strategy (CSS), both of which are meaningful or justifiable only for particular classes of symmetric and population games, such as games with multilinear payoff functions or with unidimensional strategy spaces. This paper presents a general notion of static stability in symmetric (-player) games and population games and with non-discrete strategy spaces, of which ESS and CSS are essentially special cases. JEL Classification: C72.

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. 1