A Game of Nontransitive Dice

Summary. We consider a two-player, simultaneous-move game where each player selects any permissible n-sided die for a fixed integer n. For any n > 3, there is a unique Nash equilibrium in pure strategies in which each player throws the standard n-sided die. Our proof of uniqueness is constructive, and we introduce an algorithm with which, for any nonstandard die, we can generate another die that beats it. For any nonstandard die there exists a one-step die—a die that is obtained by transferring one dot from one side to another on the standard die—that beats it.