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

Enumeration of Nash equilibria for two-player games

Abstract This paper describes algorithms for finding all Nash equilibria of a two-player game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method lrsnash is based on the known vertex enumeration program lrs, for “lexicographic reverse search”. It enumerates the vertices of only one best-response polytope, and the vertices of the complementary faces that correspond to these vertices (if they are not empty) in the other polytope. The second method is a modification of the known EEE algorithm, for “enumeration of extreme equilibria”. We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of lrsnash and EEE, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses