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The Action Gambler and Equal-Sized Wagering

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

David Hartvigsen
David Hartvigsen*
Affiliation:
University of Notre Dame
*
∗Postal address: 354 Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556-5646, USA. Email address: dhartvig@nd.edu
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Abstract

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A gambler with an initial bankroll is faced with a finite sequence of identical and independent bets. For each bet, he may wager up to his current bankroll, and will win this amount with probability p or lose it with probability 1-p. His problem is to devise a wagering strategy that will maximize his final expected utility with the side condition that the total amount wagered (i.e. the total ‘action’) be at least his initial bankroll. Our main result is an expression that characterizes when the strategy of placing equal-sized wagers on all bets is optimal. In particular, for a given bankroll B, utility function f (concave, increasing, differentiable), and n bets, we show that it is optimal to wager b/n on each bet if and only if the probability of winning each bet is less than or equal to some value p⋆∈[½,1] (where p⋆ is an explicit function of B, f, and n). We prove the result by using a basic nonlinear programming technique.

Keywords

GamblingKelly criterionnonlinear programming

MSC classification

Secondary: 91A60: Probabilistic games; gambling 90C30: Nonlinear programming 60G40: Stopping times; optimal stopping problems; gambling theory 90C90: Applications of mathematical programming
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 35 - 54
Copyright
Copyright © Applied Probability Trust 2009 

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