Strategic Characterization of the Index of an Equilibrium

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− 1 for an m× n game, which is much lower than the dimension m+n of the polytopes that are classically used.