On the Moments of the Asymmetric n-Player Gambler’s Ruin Problem

Abstract

In this article, we study the asymmetric n-player gambler’s ruin problem with ties allowed, where each player has equal capital of d units of money. In each round, the probabilities of winning from each player are fixed and the outcomes are independent from previous rounds. The game ends when at least one player run out of the money. The duration of the game is called the ruin time. For 1≤d<n, the formula of the moments of ruin time is found by Hashemiparast and Sabzevari (2011). Moreover, for d=n or n+1, the expectation and variance of ruin time were found by Hashemiparast and Sabzevari Hashemiparast and Sabzevari (2011) and Sabzevari(2017) respectively. In this work, we obtain the explicit formulas of the moments of ruin time for the cases d=n,n+1. Recurrence relation techniques are used to prove the results. Our results agree with the results from Monte carlo simulations. These novel formulas allow us to compute all the moments of the ruin time of the game, including expectation and variance.