Hard‐to‐Solve Bimatrix Games

The Lemke-Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space. The construction is extended to nonsquare games where, in addition to exponentially long Lemke-Howson computations, finding an equilibrium by support enumeration takes on average exponential time. Copyright The Econometric Society 2006.

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