We show that the problem of finding optimal strategies for both players in a simple stochastic game reduces to the generalized linear complementarity problem (GLCP) with a P-matrix, a well-studied problem whose hardness would imply NP = co–NP. This makes the rich GLCP theory and numerous existing algorithms available for simple stochastic games. As a special case, we get a reduction from binary simple stochastic games to the P-matrix linear complementarity problem (LCP).