Game-Theoretic Probability : Theory and Applications to Prediction, Science, and Finance

Cover -- Title Page -- Copyright -- Contents -- Preface -- Acknowledgments -- Part I Examples in Discrete Time -- Chapter 1 Borel's Law of Large Numbers -- 1.1 A Protocol for Testing Forecasts -- 1.2 A Game‐Theoretic Generalization of Borel's Theorem -- 1.3 Binary Outcomes -- 1.4 Slackenings and Supermartingales -- 1.5 Calibration -- 1.6 The Computation of Strategies -- 1.7 Exercises -- 1.8 Context -- Chapter 2 Bernoulli's and De Moivre's Theorems -- 2.1 Game‐Theoretic Expected value and Probability -- 2.2 Bernoulli's Theorem for Bounded Forecasting -- 2.3 A Central Limit Theorem -- 2.4 Global Upper Expected Values for Bounded Forecasting -- 2.5 Exercises -- 2.6 Context -- Chapter 3 Some Basic Supermartingales -- 3.1 Kolmogorov's Martingale -- 3.2 Doléans's Supermartingale -- 3.3 Hoeffding's Supermartingale -- 3.4 Bernstein's Supermartingale -- 3.5 Exercises -- 3.6 Context -- Chapter 4 Kolmogorov's Law of Large Numbers -- 4.1 Stating Kolmogorov's Law -- 4.2 Supermartingale Convergence Theorem -- 4.3 How Skeptic Forces Convergence -- 4.4 How Reality Forces Divergence -- 4.5 Forcing Games -- 4.6 Exercises -- 4.7 Context -- Chapter 5 The Law of the Iterated Logarithm -- 5.1 Validity of the Iterated‐Logarithm Bound -- 5.2 Sharpness of the Iterated‐Logarithm Bound -- 5.3 Additional Recent Game‐Theoretic Results -- 5.4 Connections with Large Deviation Inequalities -- 5.5 Exercises -- 5.6 Context -- Part II Abstract Theory in Discrete Time -- Chapter 6 Betting on a Single Outcome -- 6.1 Upper and Lower Expectations -- 6.2 Upper and Lower Probabilities -- 6.3 Upper Expectations with Smaller Domains -- 6.4 Offers -- 6.5 Dropping the Continuity Axiom -- 6.6 Exercises -- 6.7 Context -- Chapter 7 Abstract Testing Protocols -- 7.1 Terminology and Notation -- 7.2 Supermartingales -- 7.3 Global Upper Expected Values.