Optimal minimax strategy in a dice game

TL;DR

This paper models a two-player dice game as a stochastic game, proves the existence of a game value, and provides an algorithm to compute optimal strategies, demonstrating its effectiveness across different game variants.

Contribution

It formulates the dice game as a competitive Markov decision process, proves the existence of a value, and introduces an algorithm for optimal minimax strategy computation.

Findings

The game has a well-defined value.

The algorithm computes optimal strategies effectively.

Results are demonstrated on three game variants.

Abstract

Each of two players, by turns, rolls a dice several times accumulating the successive scores until he decides to stop, or he rolls an ace. When stopping, the accumulated turn score is added to the player account and the dice is given to his opponent. If he rolls an ace, the dice is given to the opponent without adding any point. In this paper we formulate this game in the framework of competitive Markov decision processes (also known as stochastic games), show that the game has a value, provide an algorithm to compute the optimal minimax strategy, and present results of this algorithm in three different variants of the game.