Book review: Optimal Play

Reviewing this one hasn’t been easy, mostly because it’s got some very difficult math and I haven’t taken a math class since my junior year in high school. Still, it’s a valuable addition to the literature, and I figured I give it a shot. Read on for more…

Stewart N. Ethier and William R. Eadington, eds. Optimal Play: Mathematical Studies of Games and Gambling. Reno: Institute for the Study of Gambling and Commercial Gaming, 2008. 550 pages, hardcover.

Saying the Optimal Play is a tough read for non-mathematicians is like saying that the Mona Lisa’s a well-known painting. Understatement doesn’t even begin to describe it. So this book is clearly not geared towards your average player, unless the average casino visitor is now someone with a Ph.D. in math. If you’re not familiar with minimax, Bayesian analyses, and Markov chains, this is going to be a very frustrating read.

That said, this is a valuable book. Without these “esoteric” looks into the mathematics of gambling, there wouldn’t be card counting or many other “advantage” strategies. So while you personally might not be able to appreciate the significance of some of the equations in here, there’s a chance that, as these ideas are pursued, they will filter down to the casino floor.

The volume starts with a short preface by Professor Eadington. Eadington, the leading figure in gaming studies, is kind enough to warn the reader that “mathematicians look at issue differently than the rest of us,” before explaining just you’ve gotten yourself into. The preface then offers a roadmap for the rest of the book.

Blackjack is the keystone of gambling math–everyone in the field worth his salt has tackled twenty-one. So it’s fitting that Optimal Play starts with that game. Following an essay on advanced insurance play by R. Michael Canjar and “New Blackjack Basic Strategy for Players Who Modify Their Bets Based on the Count” by Hal Marcus, Richard Werthamer reconsiders basic strategy by using an analytic approach. According to Werthamer, “Counter Basic Strategy” offers better return that original playing strategy, at least under non-perfect conditions. Both Canjar and Ethier respond to Werthamer’s claim in comment sections, followed by Werthamer’s rebuttal.

The next section, poker, offers slightly more accessible articles: first a game-centric analysis of “The Endgame in Poker” by Chris and Tom Ferguson (Chris, AKA “Jesus,” is the 2000 WSOP champion), then Thomas Humphrey’s study of the use of a Passive Reverse Turing Test to detect poker-playing bots. That involves analyzing mouse movements, and seems to make some sense without hitting the math too hard. Following that, there are more poker articles, then a series of pieces on other games, including ye olde French favorite Le Her and two articles on baccara–not casino bacc, which is pretty brainless, but the old, social game, where the bank rotated and the player could choose whether to draw a third card or not.

After this comes one of my favorite articles in the collection: James Grosjean asks, “Are Casinos Paranoid,” and, instead of answering, “duh!” decides to use math to demonstrate that, counter to casino perceptions, players at games like blackjack, pai gow tiles, and Caribbean Stud Poker cannot get anything close to an advantage by sharing information. You might be amused simply by an author deriding pit bosses as dunderheads, then proving it with geometric logic. After that, Stewart Ethier weighs in with a lucid consideration of the math behind faro, a perennial favorite that has since fallen into disfavor because, as Ethier demonstrates, it has too low a house edge.

There are several articles about dice games, including Edward Throp’s 1975 study on the optimal strategy for backgammon’s pure running game. Then you’ve got pieces on a hodge podge of other betting opportunities: sports betting, roulette, horseracing, and even lotteries. The book closes with a section on “gambling theory” that includes two articles on Parrando’s Paradox, the little bit of mathematical sophistry that seems to prove that you can combine two negative expectation games into one positive expectation betting opportunity. It’s true, but with a HUGE “yeah, but…” and no casino offers a game that would fulfill Parrando’s requirements. Still, as Dr. Eadington warned us, this is what mathematicians do when they hit the casino.

If you’re conversant in the language of math and interested in gambling, this book will be a rare treasure. If you’re not confident in your ability to understand complicated mathematical equations and formulae, you might be better off reading something else.

I’m more than a little envious of mathematicians, because books in my discipline (history) written for advanced specialists usually only bore the lay reader. Complex math books, by contrast, can confuse and demoralize the intelligent non-specialist without even trying that hard. Advantage: math.