Risk and Reward

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, nonmathematical, 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.

N. Richard Werthamer is retired from a successful career as a scientist and executive, most recently as the Executive Officer of the American Physical Society. He graduated summa cum laude from Harvard College before receiving his PhD in Theoretical Physics from the University of California at Berkeley. His original research has been published extensively in the world’s leading journals. In this book, he applies his scientific background to the analysis of blackjack.

Blackjack is hugely popular in casinos worldwide. This book analyzes the game in depth, covering its optimal strategies and its financial performance in terms of expected cash flow and associated risk. It is accessible to readers at all levels of skill.

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.

Contents Preface Introduction

Working with the Interactive Edition

Part I

1                    The Game

1.1              History of Casino Blackjack and Its Analysis

1.2              Rules, Procedures and Terminology

2                    Playing the Hand

2.1              Basic Strategy

2.1.1        Expected Return with Variant Rules and Procedures

2.1.2        Expected Return vs. Return on Investment

2.2              Composition-Dependent Play

3                    Tracking the Cards

3.1              Linear Counts

3.3              Unbalanced Counting Vectors

3.4              Relating the True Count to the Expected Return

4                    Betting

4.1              Yield, Risk, and Optimal Bet Strategies

4.11                Bet size in relation to true count

4.12                Other criteria

4.3                Multiple Hands

4.4                Back-Counting and Table-Hopping

5                    Playing the Hand When the Count and Bet Vary

5.1              Play Strategies that Vary with the Count

5.1.1        Reconsidering the Counting Vector

5.1.2        Count Dependence of the Play Parameters

5.1.3        The Insurance Bet

5.2             Counter Basic Strategy for the Variable Bettor

6                    Synthesis and Observations

6.1              A Practical, Nearly Optimal Strategy

6.2              Blackjack as a Recreation vs. a Profession

7                    Play Strategies

7.1              Basic Strategy, Large Number of Decks

7.1.1        Analytical Framework

7.1.2        Expected Player Return

7.1.4        Multiple Simultaneous Hands: Return, Variance, and Covariance

7.1.5        Expected Number of Cards Used per Round

7.2              Basic Strategy, Small Number of Decks

7.2.1        Analytical Framework

7.2.2        Expected Return, and Optimal Basic Strategy, vs. Number of Decks

7.2.3        Surrender; Insurance

7.3              Play Parameters Dependent on Identities of Initial Cards

7.3.1        Comparison with Previous Authorities

8                    Card Counting

8.1              Analytical Framework

8.1.1        Asymptotic Distribution

8.1.2        Expected Return; Invariance Theorem

8.2              Expected Return at Nonzero Depth

8.3              Optimizing the Counting Vectors

8.4              Optimizing the Counting Vectors: Many-Cards Limit

8.5              Computation of the Derivatives of the Expected Return

8.6              Unbalanced Counts

Appendix 8.A Asymptotic Distribution of Card Likelihoods Appendix 8.B Eigenmodes

9                    Bet Strategies

9.1              Risk and Capitalization

9.1.1        Risk in a Game with Fixed Return

9.1.2        Optimal Betting When Return Fluctuates: Expected Capital and Risk

9.1.3        Connections with Finance

9.1.4        Distribution of Capital

9.1.5        Properties of the Risk and Expected Capital Expressions

9.1.6        Optimal Betting When Return Fluctuates: Bet Strategy

9.1.7        Yield When the Bet Size Is Discrete; Wong Benchmark Betting

9.2              Betting Proportional to Current Capital

9.2.1        Mixed Additive and Multiplicative Betting

9.3              Betting When Playing Multiple Simultaneous Hands

9.4                Back-Counting and Table-Hopping

9.4.1       Entry

9.4.2       Entry and Exit

9.4.3       Entry and Departure

9.4.4       Entry, Exit, and Departure

Appendix 9.A   Distribution of Player’s Capital, Asymptotically for Large N

10                Play Strategies with Card Counting

10.1          Count-Dependent Playing Strategy

10.1.1    Counting Vector Optimal for Play Variation Alone

10.1.2    Single Counting Vector Optimal for Bet and Play Together

10.1.3    Two Distinct Counting Vectors

10.2          Counter Basic Strategy References