Discrete Gambling and Stochastic Games (Stochastic Modelling and Applied Probability)

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.

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 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 that form the core theory of this subject. Chapters 5, 6, and 7 cover stationary strategies, approximation results, 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.