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A mathematical analysis of the card game of betweenies through Kelly’s criterion. (English) Zbl 1200.91058

Summary: In the game of Betweenies, the player is dealt two cards out of a deck and bets on the probability that the third card to be dealt will have a numerical value in between the values of the first two cards. In this work, we present the exact rules of the two main versions of the game, and we study the optimal betting strategies. After discussing the shortcomings of the direct approach, we introduce an information-theoretic technique, Kelly’s criterion, which basically maximizes the expected log-return of the bet: we offer an overview, discuss feasibility issues, and analyze the strategies it suggests. We also provide some gameplay simulations.

MSC:

91A60 Probabilistic games; gambling
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References:

[1] Article on In Between, http://www.onlinepoker.net/Card-Games/In-Between.php.
[2] Article on Between the Sheets, http://www.bettercardgames.com/how-to-play-between-the-sheets-card-games.php.
[3] Article on Red Dog in Wikipedia, http://en.wikipedia.org/wiki/Red_Dog.
[4] Article on Acey Deucey in Wikipedia, http://en.wikipedia.org/wiki/Acey_Deucey_(card_game).
[5] J. L. Kelly Jr., “A new interpretation of information rate,” The Bell System Technical Journal, vol. 35, pp. 917-926, 1956.
[6] Article on Kelly’s criterion in Wikipedia, http://en.wikipedia.org/wiki/Kelly_criterion.
[7] E. O. Thorp, “The Kelly criterion in blackjack, sports betting, and the stock market,” in Proceedings of the 10th International Conference on Gambling and Risk Taking, Montreal, Canada, June 1997.
[8] T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Hoboken, NJ, USA, 2nd edition, 2006. · Zbl 1140.94001 · doi:10.1002/047174882X
[9] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1058.90049
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