A Game of Nontransitive Dice

We consider a two player simultaneous-move game where the two players each select any permissible n-sided die for a fixed integer n. A player wins if the outcome of his roll is greater than that of his opponent. Remarkably, for n>3, there is a unique Nash Equilibrium in pure strategies. The unique Nash Equilibrium is for each player to throw the Standard n-sided die, where each side has a different number. Our proof of uniqueness is constructive. We introduce an algorithm with which, for any nonstandard die, we may generate another die that beats it.