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Strong Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Time-Dependent Parameters

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Yi-Ching Yao  and
May-Ru Chen
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
May-Ru Chen*
Affiliation:
Academia Sinica
*
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, R.O.C.
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, R.O.C.
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Abstract

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In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with suitably chosen stakes. This problem has recently been extended in a unifying framework to account for limited playing time as well as future discounting, under which bold play is known to be optimal provided that w ≤ ½ ≤ r. This paper is concerned with a further extension of the Dubins-Savage gambling problem involving time-dependent parameters, and shows that bold play not only maximizes the probability of reaching the goal, but also stochastically minimizes the number of plays needed to reach the goal. As a result, bold play also maximizes the expected utility, where the utility at the goal is only required to be monotone decreasing with respect to the number of plays needed to reach the goal. It is further noted that bold play remains optimal even when the time-dependent parameters are random.

Keywords

Gambling theoryprimitive casinodiscount factoroptimal 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 2 , June 2008 , pp. 403 - 416
Copyright
Copyright © Applied Probability Trust 2008 

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