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Card Counting in Continuous Time

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  04 February 2016

Patrik Andersson
Patrik Andersson*
Affiliation:
Stockholm University
*
∗ Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: patrika@math.su.se
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Abstract

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We consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. The sampling without replacement makes it possible to take advantage of the changes in the expected value as the deck is depleted, making large bets when the game is advantageous. Using such a strategy, which is easy to implement, is known as card counting. We consider the case of a large number of decks, making an approximation to continuous time possible. A limit law of the return process is found and the optimal card counting strategy is derived. This continuous-time strategy is shown to be a natural analog of the discrete-time strategy where the so-called effects of removal are replaced by the infinitesimal generator of the card process.

Keywords

Sampling without replacementinvariance principlegambling theoryoptimal control

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 184 - 198
Copyright
© Applied Probability Trust 

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