Discrete Gambling and Stochastic Games

TLDR: In this article, the authors present an algorithm for the value of a non-leavable game and the optimality equation for a two-player, zero-sum game.

Abstract: 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.