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Improving on bold play when the gambler is restricted

Part of: Stochastic processes Game theory

Published online by Cambridge University Press:  14 July 2016

Jason Schweinsberg
Jason Schweinsberg*
Affiliation:
University of California, San Diego
*
∗Postal address: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. Email address: [email protected]
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Abstract

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Suppose that a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume that whenever the gambler stakes an amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 − w, where w < ½. Dubins and Savage showed that the optimal strategy, which they called ‘bold play’, is always to bet min{f, 1 − f}, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than ℓ at one time. We show that the bold strategy of always betting min{ℓ, f, 1 − f} is not optimal if ℓ is irrational, extending a result of Heath, Pruitt, and Sudderth.

Keywords

Bold playred-and-blackgamblingsupermartingale

MSC classification

Primary: 91A60: Probabilistic games; gambling
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 321 - 333
Copyright
© Applied Probability Trust 2005 

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