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On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

Yi-Ching Yao
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan, R. O. C. Email address: [email protected]
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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 by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ ½ ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.

Keywords

Gambling theoryprimitive casinored-and-blackroulettediscount factoroptimal strategy

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 91A60: Probabilistic games; gambling
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 212 - 225
Copyright
Copyright © Applied Probability Trust 2007 

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