The asymmetric n-player gambler's ruin problem with equal initial fortunes

Abstract

Formulas for the expected ruin time and the individual ruin probabilities are found for the asymmetric n-player gambler's ruin problem with equal initial fortunes of n+r dollars, for r a fixed non-negative integer, and n≥max(r,2); these are evaluated in several cases. Asymptotic results are obtained for the expected time to ruin for the asymmetric game, and for the individual ruin probabilities for the symmetric game. We also prove that the ruin time and which player is ruined are dependent events in the case where the players each start with n+2 dollars and n≥3. This differs from the asymmetric 2-player game with equal initial fortunes, as well as from the n-player game with n≥3 and equal initial fortunes less than or equal to n+1 dollars, in which the analogous events have been shown to be independent. © 2004 Elsevier Inc. All rights reserved.