On symmetric linear games

In this paper, we generalize classical von Neumann symmetrization of two-person zero-sum games to general linear games. We use this symmetrization to show that for a given general linear game there exists a symmetric linear game whose solution yields a solution to the underlying linear game. We define symmetric linear games of type gRPS (generalized Rock-Paper-Scissors) and prove that a symmetric linear game has a pure strategy equilibrium if and only if it is not a gRPS game. From this we deduce that a completely mixed symmetric linear game is gRPS.