Risk and Reward: The Science of Casino Blackjack

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