Finding Gale Strings

The problem 2-Nash of finding a Nash equilibrium of a bimatrix game belongs to the complexity class PPAD. This class comprises computational problems that are known to have a solution by means of a path-following argument. For bimatrix games, this argument is provided by the Lemke–Howson algorithm. It has been shown that this algorithm is worst-case exponential with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterize the vertices of these polytopes. We define the combinatorial problem Another completely labeled Gale string whose solutions define the Nash equilibria of games defined by cyclic polytopes, including games where the Lemke–Howson algorithm takes exponential time. If this problem was PPAD-complete, this would imply that 2-Nash is PPAD-complete, in a much simpler way than the currently known proofs, including the original proof by Chen and Deng [Chen, X., and X. Deng (2006). Settling the complexity of two-player Nash equilibrium. Proc. 47th FOCS, 261–272]. However, we show that Another completely labeled Gale string is solvable in polynomial time by a simple reduction to Perfect matching in graphs, making it unlikely to be PPAD-complete. Although this result is negative, we hope that it stimulates research into combinatorially defined problems that are PPAD-complete and imply this property for 2-Nash.