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On Optimality of Bold Play for Primitive Casinos in the Presence of Inflation

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

Robert W. Chen ,
Larry A. Shepp ,
Yi-Ching Yao  and
Cun-Hui Zhang
Robert W. Chen*
Affiliation:
University of Miami
Larry A. Shepp*
Affiliation:
Rutgers University
Yi-Ching Yao*
Affiliation:
Academia Sinica
Cun-Hui Zhang*
Affiliation:
Rutgers University
*
∗Postal address: Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA.
∗∗Postal address: Department of Statistics, Rutgers University, Piscataway, NJ 08855, USA.
∗∗∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C. Email address: yao@stat.sinica.edu.tw
∗∗Postal address: Department of Statistics, Rutgers University, Piscataway, NJ 08855, USA.
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Abstract

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Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing the stake with probability 1 - w (r > 0, 0 < w < 1). Mr. G is permitted to gamble at the casino, but each time he places a bet, the amount of his debt is increased by a factor of 1 + α (α ≥ 0). How should Mr. G gamble to maximize his chance of reaching his (moving) target and thereby surviving? Dubins and Savage showed that an optimal strategy is to stake boldly if the primitive casino is subfair or fair (i.e. w(1 + r) ≤ 1) and the inflation rate α is 0. Intuitively, a positive inflation rate would motivate Mr. G to try to reach his goal as quickly as possible, so it seems plausible that the bold strategy is optimal. However, Chen, Shepp, and Zame found that, surprisingly, the bold strategy is no longer optimal for subfair primitive casinos with inflation if both r > 1 and α satisfies 1/r ≤ α < r. They also conjectured that the bold strategy is optimal for subfair primitive casinos with inflation if r < 1. It is shown in the present paper that this conjecture is true provided that w ≤ ½. Furthermore, by introducing an interesting notion of sharp strategy, additional results are obtained on optimality of the bold strategy.

Keywords

Gambling problemprimitive casinooptimal strategybold strategysharp strategy

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A60: Probabilistic games; gambling
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 121 - 137
Copyright
© Applied Probability Trust 2005 

References

Chen, R. (1978). Subfair ‘red-and-black’ in the presence of inflation. Z. Wahrscheinlichkeitsth. 42, 293–301.Google Scholar
Chen, R. W., Shepp, L. A. and Zame, A. (2004). A bold strategy is not always optimal in the presence of inflation. J. Appl. Prob. 41, 587–592.Google Scholar
Dubins, L. E. and Savage, L. J. (1965). How To Gamble If You Must: Inequalities for Stochastic Processes. McGraw-Hill, New York.Google Scholar
Dubins, L. E. and Teicher, H. (1967). Optimal stopping when the future is discounted. Ann. Math. Statist. 38, 601–605.CrossRefGoogle Scholar
Klugman, S. (1977). Discounted and rapid subfair red-and-black. Ann. Statist. 5, 734–745.CrossRefGoogle Scholar
Ramakrishnan, S. and Sudderth, W. (1998). Geometric convergence of algorithms in gambling theory. Math. Operat. Res. 23, 568–575.Google Scholar
Ruth, K. (1999). Favorable red and black on the integers with a minimum bet. J. Appl. Prob. 36, 837–851.Google Scholar