Challenge Instances for NASH

The problem NASH is that of finding one Nash equilibrium of a bimatrix game. The computational complexity of this problem is a long-standing open question. The Lemke–Howson algorithm is the classical algorithm for NASH. In an earlier paper, this algorithm was shown to be exponential, even in the best case, for square bimatrix games using dual cyclic polytopes. However the games constructed there are easily solved by another standard method, support enumeration. In this paper we present “challenge instances” for NASH. We extend the use of dual cyclic polytopes and construct a class of games which are shown to be hard to solve for both the Lemke–Howson algorithm and support enumeration. Other general methods for NASH are discussed. It is not obvious that they could solve these games efficiently.