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Optimal Strategy for the Vardi Casino with Interest Payments

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ilie Grigorescu ,
Robert Chen  and
Larry Shepp
Ilie Grigorescu*
Affiliation:
University of Miami
Robert Chen*
Affiliation:
University of Miami
Larry Shepp*
Affiliation:
Rutgers University
*
∗ Postal address: Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA.
∗ Postal address: Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA.
∗∗∗∗ Postal address: Department of Statistics, Rutgers University, Piscataway, NJ 08855, USA. Email address: [email protected]
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Abstract

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A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

Keywords

Gambling problemVardi casinooptimal strategybold play

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. 199 - 211
Copyright
Copyright © Applied Probability Trust 2007 

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