Really though, the way to do this is to represent each game as a payout matrix A, which for this category of game will be skew-symmetric with -1/0/+1 entries. Then since this is a symmetric zero-sum game, you find the null space of that matrix, impose the constraint that probabilities must sum to 1 and individually be >= 0, and that gives you the endpoints of the Nash Equilibria. The optimal strategies (could be one unique one for odd n!) lie on the convex hull of those points.
Really though, the way to do this is to represent each game as a payout matrix A, which for this category of game will be skew-symmetric with -1/0/+1 entries. Then since this is a symmetric zero-sum game, you find the null space of that matrix, impose the constraint that probabilities must sum to 1 and individually be >= 0, and that gives you the endpoints of the Nash Equilibria. The optimal strategies (could be one unique one for odd n!) lie on the convex hull of those points.