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 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 winni [...]
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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.