Three-Player Gambler’s Ruin Problem: Some Extensions
Abstract
Abstract Views: 276
For calculating the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. However, the martingale approach is not appropriate to determine the ruin time of asymmetric game. In general, ruin probabilities in both cases, that is, symmetric and asymmetric games as well as the expected ruin time for the asymmetric games are still need to be calculated. The current work is also about three-player gambler’s ruin problem with some extensions. We provide expressions for the ruin time with or without ties when all the players have equal or unequal initial fortunes. Finally, the validity of the asymmetric game is also tested through a Monte Carlo simulation study.
Copyright (c) The Authors
Downloads
References
Ogilvy CS. Tomorrow's Math; Unsolved Problems for the Amateur. Oxford University Press, USA; 1972.
Sandell D. A game with three players. Stat Probab Lett. 1988;7(1):61-3. https://doi.org/10.1016/0167-7152(88)90089-2
Rocha AL, Stern F. The gambler's ruin problem with n players and asymmetric play. Stat Probab Lett. 1999 Aug 1;44(1):87-95. https://doi.org/10.1016/S0167-7152(98)00295-8
Rocha AL, Stern F. The asymmetric n-player gambler's ruin problem with equal initial fortunes. Adv Appl Math. 2004;33(3):512-30. https://doi.org/10.1016/j.aam.2003.07.005
Hussain A, Cheema SA, Haroon S, Kifayat T. The ruin time for 3-player gambler’s problem: an approximate formula. Commun Stat Simul Comput. 2021:1-9. https://doi.org/10.1080/03610918.2021.1888996
Hussain A, Cheema SA, Hanif M. A game based on successive events. Stat. 2020;9(1):e274. https://doi.org/10.1002/sta4.274
Jones PW, Smith, P. Stochastic Processes: An Introduction. 3rd ed. CRC Press; 2018.
Copyright (c) 2021 Abid Hussain , Muhammad Hanif, Moazzam Naseer
This work is licensed under a Creative Commons Attribution 4.0 International License.
