Ebook: Discrete Gambling and Stochastic Games

The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of chance. However, it was not until the middle of the twentieth century that mathematicians de­ veloped general techniques for maximizing the chances of beating a casino or winning against an intelligent opponent. These methods of finding op­ timal strategies for a player are at the heart of the modern theories of stochastic control and stochastic games. There are numerous applications to engineering and the social sciences, but the liveliest intuition still comes from gambling. The now classic work How to Gamble If You Must: Inequalities for Stochastic Processes by Dubins and Savage (1965) uses gambling termi­ nology and examples to develop an elegant, deep, and quite general theory of discrete-time stochastic control. A gambler "controls" the stochastic pro­ cess of his or her successive fortunes by choosing which games to play and what bets to make.

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The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of change. However, it was not until the middle of the twentieth century that mathematicians developed general techniques for maximizing the chances of beating a casino or winning against an intelligent opponent. These methods of finding optimal strategies are at the heart of the modern theory of stochastic control and stochastic games. This monograph provides an introduction to the ideas of gambling theory and stochastic games. The first chapters introduce the ideas and notation of gambling theory. Chapters 3 and 4 consider "leavable" and "nonleavable" problems which form the core theory of this subject. Chapters 5, 6, and 7 cover stationary strategies, approximate gambling problems, and two-person zero-sum stochastic games respectively. Throughout, the authors have included examples and there are problem sets at the end of each chapter.

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The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of change. However, it was not until the middle of the twentieth century that mathematicians developed general techniques for maximizing the chances of beating a casino or winning against an intelligent opponent. These methods of finding optimal strategies are at the heart of the modern theory of stochastic control and stochastic games. This monograph provides an introduction to the ideas of gambling theory and stochastic games. The first chapters introduce the ideas and notation of gambling theory. Chapters 3 and 4 consider "leavable" and "nonleavable" problems which form the core theory of this subject. Chapters 5, 6, and 7 cover stationary strategies, approximate gambling problems, and two-person zero-sum stochastic games respectively. Throughout, the authors have included examples and there are problem sets at the end of each chapter.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-3
Gambling Houses and the Conservation of Fairness....Pages 5-22
Leavable Gambling Problems....Pages 23-57
Nonleavable Gambling Problems....Pages 59-88
Stationary Families of Strategies....Pages 89-111
Approximation Theorems....Pages 113-170
Stochastic Games....Pages 171-225
Back Matter....Pages 227-244

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The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of change. However, it was not until the middle of the twentieth century that mathematicians developed general techniques for maximizing the chances of beating a casino or winning against an intelligent opponent. These methods of finding optimal strategies are at the heart of the modern theory of stochastic control and stochastic games. This monograph provides an introduction to the ideas of gambling theory and stochastic games. The first chapters introduce the ideas and notation of gambling theory. Chapters 3 and 4 consider "leavable" and "nonleavable" problems which form the core theory of this subject. Chapters 5, 6, and 7 cover stationary strategies, approximate gambling problems, and two-person zero-sum stochastic games respectively. Throughout, the authors have included examples and there are problem sets at the end of each chapter.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-3
Gambling Houses and the Conservation of Fairness....Pages 5-22
Leavable Gambling Problems....Pages 23-57
Nonleavable Gambling Problems....Pages 59-88
Stationary Families of Strategies....Pages 89-111
Approximation Theorems....Pages 113-170
Stochastic Games....Pages 171-225
Back Matter....Pages 227-244
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