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Two New Models for the Two-Person Red-And-Black Game

Part of: Game theory

Published online by Cambridge University Press:  14 July 2016

May-Ru Chen  and
Shoou-Ren Hsiau
May-Ru Chen*
Affiliation:
National Sun Yat-sen University
Shoou-Ren Hsiau*
Affiliation:
National Changhua University of Education
*
∗Postal address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Road, Kaohsiung 804, Taiwan, R.O.C. Email address: mayru@faculty.nsysu.edu.tw
∗∗Postal address: Department of Mathematics, National Changhua University of Education, No. 1, Jin-De Road, Changhua 500, Taiwan, R.O.C. Email address: srhsiau@cc.ncue.edu.tw
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Abstract

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In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

Keywords

Two-person red-and-black gamewin probability functionbold strategytimid strategyNash equilibriumconvex function

MSC classification

Secondary: 91A15: Stochastic games 91A05: 2-person games 91A06: $n$-person games, $n>2$
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 97 - 108
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Chen, M.-R. (2009). Proportional three-person red-and-black games. Prob. Eng. Inf. Sci. 23, 37–50.CrossRefGoogle Scholar
[2] Chen, M.-R. and Hsiau, S.-R. (2006). Two-person red-and-black games with bet-dependent win probability functions. J. Appl. Prob. 43, 905–915.CrossRefGoogle Scholar
[3] Dubins, L. E. and Savage, L. J. (1976). Inequalities for Stochastic Processes (How to Gamble if You Must). Corrected republication of the 1965 edition. Dover, New York.Google Scholar
[4] Maitra, A. P. and Sudderth, W. D. (1996). Discrete Gambling and Stochastic Games. Springer, New York.CrossRefGoogle Scholar
[5] Pontiggia, L. (2005). Two-person red-and-black with bet-dependent win probability. Adv. App. Prob. 37, 75–89.CrossRefGoogle Scholar
[6] Pontiggia, L. (2007). Nonconstant sum red-and-black games with bet-dependent win probability function. J. Appl. Prob. 44, 547–553.CrossRefGoogle Scholar
[7] Ross, S. M. (1974). Dynamic programming and gambling models. Adv. App. Prob. 6, 598–606.CrossRefGoogle Scholar
[8] Secchi, P. (1997). Two-person red-and-black stochastic games. J. Appl. Prob. 34, 107–126.CrossRefGoogle Scholar