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Discrete gambling and stochastic games. (English) Zbl 0864.90148

Applications of Mathematics. 32. New York, NY: Springer. 244 p. (1996).
In 1965, L. E. Dubins and L. J. Savage published their classic book: ‘How gamble if you must’ (Zbl 0133.41402). The present book presents the state of the theory which has sprung from this highly original source. In contrast to the book of Dubins-Savage, probabilities in this monograph are countably additive. In almost all sections the state space is assumed countable in order to avoid technical issues of measurability, but there are some sections and remarks and many references guiding the reader who is interested in more generality.
Chapter 2 introduces the ideas and notation of gambling theory. It gives an optimal sampling theorem, discusses stop rule induction and treats the relation to martingale theory. Chapter 3 discusses “leavable” gambling houses, where the player can use a stop rule to quit playing. Topics are: the optimal return function, positive dynamic programming, upcrossings, timid play and other examples. Chapter 4 is devoted to nonleavable gambling problems. Chapter 5 discusses the problem when stationary strategies suffice. For example, if the state space is finite and at each state only finitely many gambles are available, then stationary strategies suffice.
In chapter 6 the payoff is bounded and real valued. Here, the proof of approximation theorems requires analytic sets, but an introduction is given. Finally, in chapter 7 there is a second player and the actions of both players determine the transitions. Two persons zero sum stochastic games are discussed. The highlight is the theorem of the authors stating that such games with a limsup payoff have a value.
Each chapter offers problems and references. The book is very well readable and well suited for a course for graduate students. The only prerequisite is a course on measure theoretic probability. The choice of examples is attractive. The authors have succeeded in producing an update of gambling theory, which will be very useful for anybody interested in this field.

MSC:

91A60 Probabilistic games; gambling
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91A15 Stochastic games, stochastic differential games

Citations:

Zbl 0133.41402
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