Equilibrium oddness theorem

A finite nondegenerate bimatrix game has a finite odd number of Nash equilibria. After shifting to positive payoffs, fully labelled vertex pairs in the bounded best-response polytopes correspond to equilibria plus one artificial zero pair. Dropping a fixed label gives a finite path graph whose endpoints are exactly these pairs. The number of endpoints is even, leaving an odd number of genuine equilibria. This is the endpoint-parity argument underlying the Lemke-Howson algorithm.