A three-player gambler's ruin problem: some extensions

Abstract

For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.