CONTENTS
1. INTRODUCTION
1. THE PROBLEM
2. PREVIEW
3. ANTECEDENTS
2. FORMULATION OF THE ABSTRACT GAMBLER’S PROBLEM
1. INTRODUCTION
2. FORTUNES AND HISTORIES
3. GAMBLES
4. GAMBLING HOUSE
5. STRATEGIES
6. THE INTEGRAL GENERATED BY A STRATEGY
7. FINITARY MAPPINGS
8. INDUCTIVELY INTEGRABLE FUNCTIONS
9. STOP RULES, POLICIES, AND TERMINAL FORTUNES
10. THE UTILITY OF FORTUNES, POLICIES, AND HOUSES
11. INCOMPLETE STOP RULES
12. FUNCTIONS THAT MAJORIZE U
13. DIGRESSION ABOUT COUNTABLE ADDITIVITY
14. U IS EXCESSIVE FOR T.
15. LIMITED PLAYING TIME
16. CONTINUOUS GAMBLING HOUSES
3. STRATEGIES
1. INTRODUCTION
2. THE UTILITY OF A STRATEGY
3. STRATEGIC UTILITY
4. STAGNANT STRATEGIES
5. OPTIMAL STRATEGIES
6. THRIFTY STRATEGIES
7. EQUALIZING STRATEGIES
8. OPTIMAL STRATEGIES AGAIN
9. STATIONARY FAMILIES
4. CASINOS WITH FIXED GOALS
1. INTRODUCTION
2. THE CASINO INEQUALITY
3. EXPLORATION OF THE CASINO INEQUALITY
4. THE THREE SPECIAL TYPES OF CASINOS
5. HOW NICE CASINOS ARE
6. THE CASINO-FUNCTION SEMIGROUP
7. RICH MAN’S, POOR MAN’S, AND INCLUSIVE CASINOS
8. THE SPECIAL CASINO INEQUALITIES DO NOT IMPLY THE GENERAL ONE.
5. RED-AND-BLACK
1. INTRODUCTION
2. THE UTILITY OF BOLD STRATEGIES
3. FOR w < 1/2, BOLD PLAY IS OPTIMAL.
4. OTHER OPTIMAL STRATEGIES FOR w < 1/2
5. LIMITED PLAYING TIME
6. THE TAXED COIN
6. PRIMITIVE CASINOS
1. INTRODUCTION
2. THE UTILITY OF BOLD STRATEGIES
3. BOLD PLAY IS OPTIMAL FOR SUBFAIR PRIMITIVE CASINOS.
4. DIGRESSION ABOUT N
5. OTHER OPTIMAL STRATEGIES FOR W < T
6. LIMITED PLAYING TIME
7. PRIMITIVE CASINOS IN RELATION TO GENERAL CASINOS
8. MORE ABOUT PRIMITIVE CASINOS FUNCTIONS
9. UNIFORM ROULETTE
7. SOME GENERAL PRINCIPLES
1. INTRODUCTION
2. COMPOSITION
3. PERMITTED SETS OF GAMBLES
4. FULL HOUSES
5. PERMITTED HOUSES
8. HOUSES ON THE REAL LINE
1. INTRODUCTION
2. DOMINATION
3. MONOTONE HOUSES
4. INCREASING FULL HOUSES
5. FULL-HOUSE CASINOES
6. STATIONARY FULL HOUSES
7. EXPONENTIAL HOUSES
9. THREE PARTICULAR KINDS OF CASINOS
1. INTRODUCTION
2. INCOME-TAX CASINOS
3. THE CASINO THAT TAKES A CUT OF THE STAKE
4. HOUSES OF INEQUITY
10. ONE-LOTTERY STRATEGIES
1. INTRODUCTION
2. LIMITED PLAYING TIME
3. CONVENTIONAL LOTTERIES
4. GENERAL LOTTERIES
5. PROPORTIONAL STRATEGIES
11. FAIR CASINOS
1. INTRODUCTION
2. FAIR LOTTERIES CAN GENERATE SUBFAIR CASINOS
3. WHEN IS A CASINO FAIR?
4. FAIR, SIMPLE-LOTTERY CASINOS
5. FAIR, ONE-GAME CASINOS
6. OPTIMAL STRATEGIES FOR FAIR CASINOS
12. THE SCOPE OF GAMBLER’S PROBLEMS
1. INTRODUCTION
2. FORTUNES
3. OPINION AS A COMPONENT OF FORTUNE
4. NONLEAVABLE HOUSES
5. DISCOUNTING THE FUTURE
6. A DIFFERENT APPROACH TO THE INDEFINITE FUTURE
7. DYNAMIC PROGRAMMING
8. BAYESIAN STATISTICS
9. STOCHASTIC PROCESSES
Bibliography
Bibliographic Supplement (1975)
Bibliographic Supplement (2014)
Index of Persons
Subject Index
Index of Symbols
Index of Open Problems
Remarks on Open Problems
Appendix on Finite Additivity
Each chapter leads most naturally to the next. Some chapters, however, lead fairly naturally to later ones, as this diagram indicates.
