Risk and Reward: The Science of Casino Blackjack

By N. Richard Werthamer

For decades, casino gaming has been steadily increasing in popularity worldwide. Blackjack is among the most popular of the casino table games, one where astute choices of playing strategy can create an advantage for the player.

RISK AND REWARD analyzes the game in depth, pinpointing not just its optimal strategies but also its financial performance, in terms of both expected cash flow and associated risk.

The book begins by describing the strategies and their performance in a clear, straightforward style. The presentation is self-contained, non-mathematical, and accessible to readers at all levels of playing skill, from the novice to the blackjack expert. Careful attention is also given to simplified, but still nearly optimal strategies that are easier to use in a casino. Unlike other books in the literature the author then derives each aspect of the strategy mathematically, to justify its claim to optimality. The derivations mostly use algebra and calculus, although some require more advanced analysis detailed in supporting appendices. For easy comprehension, formulae are translated into tables and graphs through extensive computation.

This book will appeal to everyone interested in blackjack: those with mathematical training intrigued by its application to this popular game as well as all players seeking to improve their performance.

• Language: English
• Publisher: Springer
• Release date: Jul 20, 2018
• ISBN: 9783319913858

Book preview

Part IStrategy

© Springer International Publishing AG, part of Springer Nature 2018

N. Richard WerthamerRisk and Rewardhttps://doi.org/10.1007/978-3-319-91385-8_1

1. The Game

N. Richard Werthamer¹ 

(1)

Chelsea Technologies, Sag Harbor, NY, USA

1.1 History of Casino Blackjack and Its Analysis

Casino gambling has been booming steadily for decades, throughout the world. In the United States, from a time before the 1930s when casinos were illegal everywhere, they have grown in number and in popularity. Legalized initially in Nevada, they took off after World War II, particularly in Las Vegas and Reno, then around Lake Tahoe. Regulated casinos later mushroomed in Puerto Rico and, beginning in the 1970s, in Atlantic City, NJ. Casinos then sprang up on river boats in parts of the Midwest and South. More recently, American Indian tribes have been using their distinctive legal status to develop casinos on their reservations, especially in Connecticut (the largest anywhere in the world) but also in Illinois, California, Washington and elsewhere. Older casinos in Las Vegas have even been bulldozed and replaced by much larger and grander ones. In almost every instance, each new casino has stimulated another increment in the public’s interest. Total casino gaming revenue has risen year after year, into the tens of billions of dollars in the U.S. alone, although evidence is growing that the market is saturating: some older venues are losing share to newer ones. The expansion has been worldwide as well; Macau and Singapore are particularly striking examples of the casino-building boom, and are drawing customers from their previous U.S. destinations.

By far the most popular game in practically every casino is the slot machine. Slots require no skill—in fact almost no thought—and treat the occasional winner to a celebration of sound and light that cannot escape the attention (and envy) of nearby patrons. Engineered with odds that heavily favor the casino (usually 8% or more on each spin of the wheels, several spins per minute), they are rapid and inexorable money transfer generators from their patrons to their operators.

The table games have also expanded in popularity, proportional to the growth of casinos. These games, chiefly craps and blackjack but also roulette and baccarat, vary both in the range of options they afford Player—and hence in the skills required to best take advantage of those options—and in their odds. Roulette is clearly the most straightforward: no strategy decisions are called for, and the odds on each spin of the wheel are uniformly 5.26% against Player (on tables with both zero and double zero), 2.70% on European tables with just the zero. Craps offers a number of different bets from which Player can choose, each with its own odds, but all favor the casino and none offer strategy decisions. Even the best choice, on the Pass Line, favors the casino by about 1.4%. The casino edge in baccarat, a particular favorite of Asian Players, is only slightly less: about 1.06% when betting on Bank.

Blackjack, on the other hand, requires Player decisions which have a great influence on the expected return. Apart from the added entertainment value of a game of strategy versus a game of mere chance, blackjack with optimal play has the potential not only to reduce the casino advantage virtually to zero but in fact to become favorable to an astute Player. In blackjack (and also poker, alone among casino games), skill creates the possibility, at least in the long run, to systematically win money. Nevertheless, achieving that edge is difficult: not only does Player have to apply concentrated intellectual effort to approach optimal performance, but the casino has several legitimate tools available to counter him.

The published literature about blackjack is considerable. Prior to the mid-’50s the game had not been carefully analyzed. The then-popular books about casino gambling (Scarne (1961) as one example) made assertions that were not much more than guesswork, some in fact quite wrong. A major breakthrough was the work of Baldwin et al. (1956) published in a scholarly statistics journal that for the first time presented a mathematical approach for analyzing blackjack and an optimized playing technique which (with later extensions and correction of their minor errors) has become known as Basic Strategy. Baldwin et al. were also the first to apply computational methods in their analysis, although their computers were of the mechanical adding-machine kind. However, Baldwin et al. were explicit, and correct, that Basic Strategy still has a negative expected return: the odds favor the casino for a game with multiple decks.

A few years later, Edward Thorp (1966) took this approach further, by realizing that the expected return for a given hand varies with the composition of the as-yet-undealt cards, and that a careful scan of the cards as they are dealt contains useful indications of that composition. Thorp proposed a specific scanning scheme, or count, which he claimed could identify instances when the odds were actually in favor of Player. Thorp concluded that Player, by increasing his bet when these instances were detected, could achieve a positive overall edge.

Thorp’s work, first published in 1962 as a book titled Beat the Dealer, created a sensation in both the gambling public and the casino industry. Gamblers rushed to apply his methods, while casinos quickly made rule and procedure changes to blunt what they felt was a serious threat to their viability. Some players, based on casino suspicions that they were counting, were politely but firmly asked to leave. Soon, however, the casinos’ panic subsided and the changes were reversed.

Thorp had also written a program for electronic computer that improved on the work of Baldwin, et al. Later, Julian Braun, an IBM staffer, wrote a more extensive and precise routine. The use of electronic computers, even though primitive then relative to today’s, gave results within hours rather than months. Over the next decade, Braun refined his methods, supplied results to other blackjack authors, and eventually published his own book (Braun 1980). Braun and others developed a different form of counting scheme than Thorp’s, one that was both simpler and more informative. Several variants of this form of counting, particularly one ascribed to Harvey Dubner, have been in wide use ever since. An extensive history of blackjack and developments in its analysis is given by Snyder (2006).

In the several decades since the work of Thorp, Braun, and others, dozens of books about blackjack strategy have appeared, mostly repackaging, extending and popularizing the conclusions of the pioneers. The more quantitative of the subsequent authors have included Peter Griffin and Stanford Wong. Griffin (1999) went beyond Thorp and Braun to analyze many other aspects of blackjack strategy. Wong (1994), on the other hand, undertook an exhaustive study of one favored card-counting method, using a large-scale computer simulation. Wong programmed a computer to play millions of hands of blackjack for each of many possible play strategies within that counting method. By comparing outcomes, he identified the most favorable of those strategies and the overall average return to Player. Subsequently, Grosjean (2000) has reported some quite sophisticated analyses of detailed aspects of blackjack and other casino