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.
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