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Optimal Play: Mathematical Studies of Games and Gambling

Stewart N. Ethier and William R. Eadington, editors
Institute for the Study of Gambling and Commercial Gaming
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Bollman
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 “Mathematicians look at issues differently than the rest of us.”
Optimal Play, p. xiii.

This sentence, with which many of us would quickly agree, opens Optimal Play. This book is the third is a series collecting scholarly papers on mathematical aspects of gambling, and this collection lives up to its claim to reflect “the best problem solving and analysis” of mathematicians looking at gambling.

As to how mathematicians’ perspective on gambling differs from “the rest of us”, William Eadington, an economics professor at the University of Nevada, observes that the mathematician looks at a game as a “conceptual construct, rather than a personal contest between the customer and the house” (p. xiii). This is, of course, both a less expensive and less lucrative perspective. With that in mind, we have in Optimal Play a fine set of papers on a wide range of mathematical topics related to casino gambling. A section on blackjack gives due attention to card counting, including a sequence of papers comprising a dialogue of sorts among authors and editor, but also looks at the insurance bet in considerable detail. Other card games are represented in a pair of subsequent sections, and ultimately the full range of casino gambling is subjected to intense mathematical scrutiny, complete with permutation groups, nondecreasing convex functions, and multiple summations.

This applies even, for example, to roulette — which I tend to dismiss when I teach gambling mathematics as a non-interesting game. From an elementary probability standpoint, every bet except one in American roulette has a house advantage of 5.26% (and that one bet is even worse for the player), and so elementary analysis stops there. However, examining the game in the twin contexts of covariance (for mathematics) and rebated losses (for casino management), as is done in a pair of papers here, shows that there’s a deeper level available for exploration. Not that the basic house advantage changes, of course — the game is still unbeatable in the long run.

If you have a favorite casino game — either as an actual pastime or a conceptual construct — someone has done considerable mathematical legwork to break it down and analyze it, and you’ll find some enlightenment in Optimal Play. If your game is backgammon, or if you prefer purely theoretical gambling, there’s also some fine mathematics on those subjects included here. If not, what you’ll find is a collection of fascinating excursions into applied mathematics, which is not in any sense a bad thing.

Mark Bollman ( is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.