The Two-Dimensional Asymmetric Gambler's Ruin Problem with n Players and Simulations

Abstract

In this article, the asymmetric n-player gambler's ruin problem is considered, when the players use equal initial fortunes of d dollars and d euros, 1 ≤ d ≤ n + 1. In each round an unfair coin is tossed to decide the currency. The expected ruin time and the individual ruin probabilities are computed. It is proved that the ruin time and which player is ruined are independent events. Finally, some special games are simulated. The simulation results verify the validity of the proposed formulas. As an innovation, the present study makes a combination of the n-player and multi dimensional games which can be viewed as a starting point for future studies.

Keywords:

• Asymmetric play
• 2-Dimensional
• Gambler's ruin
• Independence
• Ruin time

2000 Mathematics Subject Classification:

• 60G20, 60G50

Acknowledgments

The authors are very grateful to the Editor and anonymous referees for their comments and suggestions which led to the present improved version of the manuscript. The authors would also like to thank Mr. Hamid Reza Arjmandi for his help during the preparation of this article.

Notes

It can be observed that the results of the simulations, specially for large N accord completely to the results obtained from (9) and (19) which verify the validity of these equations.