# Discrete Gambling and Stochastic Games

## Maitra, Ashok P.; Sudderth, William D.

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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 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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Contenuti:

1 Introduction.- 1.1 Preview.- 1.2 Prerequisites.- 1.3 Numbering.- 2 Gambling Houses and the Conservation of Fairness.- 2.1 Introduction.- 2.2 Gambles, Gambling Houses, and Strategies.- 2.3 Stopping Times and Stop Rules.- 2.4 An Optional Sampling Theorem.- 2.5 Martingale Convergence Theorems.- 2.6 The Ordinals and Transfinite Induction.- 2.7 Uncountable State Spaces and Continuous-Time.- 2.8 Problems for Chapter 2.- 3 Leavable Gambling Problems.- 3.1 The Fundamental Theorem.- 3.2 The One-Day Operator and the Optimality Equation.- 3.3 The Utility of a Strategy.- 3.4 Some Examples.- 3.5 Optimal Strategies.- 3.6 Backward Induction: An Algorithm for U.- 3.7 Problems for Chapter 3.- 4 Nonleavable Gambling Problems.- 4.1 Introduction.- 4.2 Understanding u(?).- 4.3 A Characterization of V.- 4.4 The Optimality Equation for V.- 4.5 Proving Optimality.- 4.6 Some Examples.- 4.7 Optimal Strategies.- 4.8 Another Characterization of V.- 4.9 An Algorithm for V.- 4.10 Problems for Chapter 4.- 5 Stationary Families of Strategies.- 5.1 Introduction.- 5.2 Comparing Strategies.- 5.3 Finite Gambling Problems.- 5.4 Nonnegative Stop-or-Go Problems.- 5.5 Leavable Houses.- 5.6 An Example of Blackwell and Ramakrishnan.- 5.7 Markov Families of Strategies.- 5.8 Stationary Plans in Dynamic Programming.- 5.9 Problems for Chapter 5.- 6 Approximation Theorems.- 6.1 Introduction.- 6.2 Analytic Sets.- 6.3 Optimality Equations.- 6.4 Special Cases of Theorem 1.2.- 6.5 The Going-Up Property of $$\overline M$$.- 6.6 Dynamic Capacities and the Proof of Theorem 1.2.- 6.7 Approximating Functions.- 6.8 Composition Closure and Saturated House.- 6.9 Problems for Chapter 6.- 7 Stochastic Games.- 7.1 Introduction.- 7.2 Two-Person, Zero-Sum Games.- 7.3 The Dynamics of Stochastic Games.- 7.4 Stochastic Games with lim sup Payoff.- 7.5 Other Payoff Functions.- 7.6 The One-Day Operator.- 7.7 Leavable Games.- 7.8 Families of Optimal Strategies for Leavable Games.- 7.9 Examples of Leavable Games.- 7.10 A Modification of Leavable Games and the Operator T.- 7.11 An Algorithm for the Value of a Nonleavable Game.- 7.12 The Optimality Equation for V.- 7.13 Good Strategies in Nonleavable Games.- 7.14 Win, Lose, or Draw.- 7.15 Recursive Matrix Games.- 7.16 Games of Survival.- 7.17 The Big Match.- 7.18 Problems for Chapter 7.- References.- Symbol Index.

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## 1.DISCRETE GAMBLING AND STOCHASTIC GAMES

ISBN 10: 0387946284 ISBN 13: 9780387946283
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## 2.DISCRETE GAMBLING AND STOCHASTIC GAMES

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## 3.Discrete Gambling and Stochastic Games (Stochastic Modelling and Applied Probability)

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ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer, 1996. Hardcover. Condizione libro: New. 1996. This item is printed on demand. Codice libro della libreria DADAX0387946284

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## 4.Discrete Gambling and Stochastic Games (Stochastic Modelling and Applied Probability)

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Descrizione libro Springer Verlag, 1996. Hardcover. Condizione libro: Brand New. 1st edition. 244 pages. 9.75x6.50x0.75 inches. In Stock. Codice libro della libreria __0387946284

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## 5.Discrete Gambling and Stochastic Games (Stochastic Modelling and Applied Probability)

Editore: Springer (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer, 1996. Condizione libro: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: 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. Codice libro della libreria ABE_book_new_0387946284

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## 6.Discrete Gambling and Stochastic Games (Hardback)

Editore: Springer-Verlag New York Inc., United States (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer-Verlag New York Inc., United States, 1996. Hardback. Condizione libro: New. 1996 ed.. 236 x 156 mm. Language: English . Brand New Book. 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. Codice libro della libreria LIB9780387946283

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## 7.Discrete Gambling and Stochastic Games (Hardback)

Editore: Springer-Verlag New York Inc., United States (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer-Verlag New York Inc., United States, 1996. Hardback. Condizione libro: New. 1996 ed.. 236 x 156 mm. Language: English . Brand New Book. 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. Codice libro della libreria LIB9780387946283

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## 8.Discrete Gambling and Stochastic Games (Stochastic Modelling and Applied Probability)

Editore: Springer (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer, 1996. Hardcover. Condizione libro: New. book. Codice libro della libreria 0387946284

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## 9.Discrete Gambling and Stochastic Games

Editore: Springer Mrz 1996 (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer Mrz 1996, 1996. Buch. Condizione libro: Neu. 241x161x22 mm. Neuware - 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. 244 pp. Englisch. Codice libro della libreria 9780387946283

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## 10.Discrete Gambling and Stochastic Games

Editore: Springer-Verlag New York Inc. (1996)
ISBN 10: 0387946284 ISBN 13: 9780387946283
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Descrizione libro Springer-Verlag New York Inc., 1996. HRD. Condizione libro: New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria I1-9780387946283

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