A Game of Nontransitive Dice
This resolves the equilibrium selection problem in nontransitive dice games for game theorists and mathematicians, though the result is incremental as it extends known properties of dice games.
The paper proves that for a two-player game where each player selects an n-sided die, the unique pure-strategy Nash equilibrium for n>3 is for both players to use the standard die with distinct numbers. The proof is constructive via an algorithm that generates a counter-die for any nonstandard die.
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.