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On Nonoptimality of Bold Play for Subfair Red-And-Black with a Rational-Valued House Limit

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

May-Ru Chen ,
Pei-Shou Chung ,
Shoou-Ren Hsiau  and
Yi-Ching Yao
May-Ru Chen*
Affiliation:
Academia Sinica
Pei-Shou Chung*
Affiliation:
National Hsinchu Girls' Senior High School
Shoou-Ren Hsiau*
Affiliation:
National Changhua University of Education
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
*
∗Current address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Road, Kaohsiung 804, Taiwan, R.O.C. Email address: chenmr@math.nsysu.edu.tw
∗∗Postal address: c/o Prof. S.-R. Hsiau, Department of Mathematics, National Changhua University of Education, Jin-De Campus, Chang-Hua 500, Taiwan, R.O.C. Email address: igo.tw@yahoo.com.tw
∗∗∗Postal address: Department of Mathematics, National Changhua University of Education, Jin-De Campus, Chang-Hua 500, Taiwan, R.O.C. Email address: srhsiau@cc.ncue.edu.tw
∗∗∗∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, R.O.C. Email address: yao@stat.sinica.edu.tw
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Abstract

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In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.

Keywords

Gambling theoryred-and-blackbold playoptimal strategy

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A60: Probabilistic games; gambling
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1024 - 1038
Copyright
Copyright © Applied Probability Trust 2008 

References

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