Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T05:08:06.764Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Prob. 8, 745–764.Google Scholar
Ekström, E. and Wanntorp, H. (2009). Optimal stopping of a Brownian bridge. J. Appl. Prob. 46, 170–180.Google Scholar
Ethier, S. N. (2010). The Doctrine of Chances. Springer, Berlin.Google Scholar
Ethier, S. N. and Levin, D. A. (2005). On the fundamental theorem of card counting, with application to the game of trente et quarante. Adv. Appl. Prob. 37, 90–107.Google Scholar
Gottlieb, G. (1985). An analytic derivation of blackjack win rates. Operat. Res. 33, 971–988.Google Scholar
Griffin, P. (1976). The rate of gain in player expectation for card games characterized by sampling without replacement and an evaluation of card counting systems. In Gambling and Society: Interdisciplinary Studies on the Subject of Gambling, ed. Eadington, W. R., Charles C. Thomas Publishing, Springfield, IL, pp. 429–442.Google Scholar
Liese, F. and Miescke, K.-J. (2008). Statistical Decision Theory. Springer, New York.Google Scholar
McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Prob. 2, 620–628.Google Scholar
Milbrodt, H. (1987). An invariance principle for U-statistics in simple random sampling without replacement. Metrika 34, 195–200.Google Scholar
Thorp, E. O. (1962). Beat the Dealer: A Winning Strategy for the Game of Twenty-One. Random House, New York.Google Scholar
Thorp, E. O. and Walden, W. E. (1973). The fundamental theorem of card counting with applications to trente-et-quarante and baccarat. Internat. J. Game Theory 2, 109–119.Google Scholar