# A Game of Nontransitive Dice

## TL;DR

This paper analyzes a two-player game involving selecting n-sided dice, revealing a unique pure strategy Nash Equilibrium for n>3 where players choose standard dice, and provides an algorithm to improve nonstandard dice.

## Contribution

It proves the uniqueness of the Nash Equilibrium in pure strategies for n>3 and introduces an algorithm to generate dice that beat any given nonstandard die.

## Key findings

- Unique pure strategy Nash Equilibrium for n>3 is the standard n-sided die.
- An algorithm can generate a die that beats any nonstandard die.
- The equilibrium strategy involves both players choosing the standard n-sided die.

## Abstract

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.