Exponentially many steps for finding a Nash equilibrium in a bimatrix game

The Lemke–Howson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical “directed parity argument”, which puts NASH into the complexity class PPAD. This paper presents a class of bimatrix games for which the Lemke–Howson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring ­((µ3=4)d) many steps, where µ is the Golden Ratio. The “parity argument” for NASH is thus explicitly shown to be inefficient. The games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space.