A dice game, a multinomial walk, and the inverted Dirichlet distribution

TL;DR

This paper explores a simple dice game using probabilistic distributions, revealing surprising properties and analyzing asymptotic behavior and winning probabilities.

Contribution

It introduces a novel connection between the game and conjugacy relations of Gamma, Poisson, and inverted Dirichlet distributions, with new monotonicity results.

Findings

Monotonicity property of the regularized beta function established.

Asymptotic behavior of the game analyzed for large parameters.

Probability of being last in the game characterized.

Abstract

We consider a simple dice game, which leads to an intriguing study of multinomial walks, with surprising and seemingly paradoxical properties. The winning and losing probabilities of a general version of the game are investigated via conjugacy relations between Gamma and Poisson distributions, as well as between negative multinomial and inverted Dirichlet distributions. We show a monotonicity property of the regularized beta function, which implies a monotonicity property of the winning probability. Furthermore, the asymptotic behavior of the game for one or several parameters of the game tending to infinity is analyzed, as well as the probability of being last in the game.