Preface xi Acknowledgments xv Part I Examples in Discrete Time 1 1 Borel’s Law of Large Numbers 5 1.1 A Protocol for Testing Forecasts 6 1.2 A Game-Theoretic Generalization of Borel’s Theorem 8 1.3 Binary Outcomes 16 1.4 Slackenings and Supermartingales 18 1.5 Calibration 19 1.6 The Computation of Strategies 21 1.7 Exercises 21 1.8 Context 24 2 Bernoulli’s and De Moivre’s Theorems 31 2.1 Game-Theoretic Expected Value and Probability 33 2.2 Bernoulli’s Theorem for Bounded Forecasting 37 2.3 A Central Limit Theorem 39 2.4 Global Upper Expected Values for Bounded Forecasting 45 2.5 Exercises 46 2.6 Context 49 3 Some Basic Supermartingales 55 3.1 Kolmogorov’s Martingale 56 3.2 Doléans’s Supermartingale 56 3.3 Hoeffding’s Supermartingale 58 3.4 Bernstein’s Supermartingale 63 3.5 Exercises 66 3.6 Context 67 4 Kolmogorov’s Law of Large Numbers 69 4.1 Stating Kolmogorov’s Law 70 4.2 Supermartingale Convergence Theorem 73 4.3 How Skeptic Forces Convergence 80 4.4 How Reality Forces Divergence 81 4.5 Forcing Games 82 4.6 Exercises 86 4.7 Context 89 5 The Law of the Iterated Logarithm 93 5.1 Validity of the Iterated-Logarithm Bound 94 5.2 Sharpness of the Iterated-Logarithm Bound 99 5.3 Additional Recent Game-Theoretic Results 100 5.4 Connections with Large Deviation Inequalities 104 5.5 Exercises 104 5.6 Context 106 Part II Abstract Theory in Discrete Time 109 6 Betting on a Single Outcome 111 6.1 Upper and Lower Expectations 113 6.2 Upper and Lower Probabilities 115 6.3 Upper Expectations with Smaller Domains 118 6.4 Offers 121 6.5 Dropping the Continuity Axiom 125 6.6 Exercises 127 6.7 Context 131 7 Abstract Testing Protocols 135 7.1 Terminology and Notation 136 7.2 Supermartingales 136 7.3 Global Upper Expected Values 142 7.4 Lindeberg’s Central Limit Theorem for Martingales 145 7.5 General Abstract Testing Protocols 146 7.6 Making the Results of Part I Abstract 151 7.7 Exercises 153 7.8 Context 155 8 Zero-One Laws 157 8.1 Lévy’s Zero-One Law 158 8.2 Global Upper Expectation 160 8.3 Global Upper and Lower Probabilities 162 8.4 Global Expected Values and Probabilities 163 8.5 Other Zero-One Laws 165 8.6 Exercises 169 8.7 Context 170 9 Relation to Measure-Theoretic Probability 175 9.1 Ville’s Theorem 176 9.2 Measure-Theoretic Representation of Upper Expectations 180 9.3 Embedding Game-Theoretic Martingales in Probability Spaces 189 9.4 Exercises 191 9.5 Context 192 Part III Applications in Discrete Time 195 10 Using Testing Protocols in Science and Technology 197 10.1 Signals in Open Protocols 198 10.2 Cournot’s Principle 201 10.3 Daltonism 202 10.4 Least Squares 207 10.5 Parametric Statistics with Signals 212 10.6 Quantum Mechanics 215 10.7 Jeffreys’s Law 217 10.8 Exercises 225 10.9 Context 226 11 Calibrating Lookbacks and p-Values 229 11.1 Lookback Calibrators 230 11.2 Lookback Protocols 235 11.3 Lookback Compromises 241 11.4 Lookbacks in Financial Markets 242 11.5 Calibrating p-Values 245 11.6 Exercises 248 11.7 Context 250 12 Defensive Forecasting 253 12.1 Defeating Strategies for Skeptic 255 12.2 Calibrated Forecasts 259 12.3 Proving the Calibration Theorems 264 12.4 Using Calibrated Forecasts for Decision Making 270 12.5 Proving the Decision Theorems 274 12.6 From Theory to Algorithm 286 12.7 Discontinuous Strategies for Skeptic 291 12.8 Exercises 295 12.9 Context 299 Part IV Game-Theoretic Finance 305 13 Emergence of Randomness in Idealized Financial Markets 309 13.1 Capital Processes and Instant Enforcement 310 13.2 Emergence of Brownian Randomness 312 13.3 Emergence of Brownian Expectation 320 13.4 Applications of Dubins–Schwarz 325 13.5 Getting Rich Quick with the Axiom of Choice 331 13.6 Exercises 333 13.7 Context 334 14 A Game-Theoretic Itô Calculus 339 14.1 Martingale Spaces 340 14.2 Conservatism of Continuous Martingales 348 14.3 Itô Integration 350 14.4 Covariation and Quadratic Variation 355 14.5 Itô’s Formula 357 14.6 Doléans Exponential and Logarithm 358 14.7 Game-Theoretic Expectation and Probability 360 14.8 Game-Theoretic Dubins–Schwarz Theorem 361 14.9 Coherence 362 14.10 Exercises 363 14.11 Context 365 15 Numeraires in Market Spaces 371 15.1 Market Spaces 372 15.2 Martingale Theory in Market Spaces 375 15.3 Girsanov’s Theorem 376 15.4 Exercises 382 15.5 Context 382 16 Equity Premium and CAPM 385 16.1 Three Fundamental Continuous I-Martingales 387 16.2 Equity Premium 389 16.3 Capital Asset Pricing Model 391 16.4 Theoretical Performance Deficit 395 16.5 Sharpe Ratio 396 16.6 Exercises 397 16.7 Context 398 17 Game-Theoretic Portfolio Theory 403 17.1 Stroock–Varadhan Martingales 405 17.2 Boosting Stroock–Varadhan Martingales 407 17.3 Outperforming the Market with Dubins–Schwarz 413 17.4 Jeffreys’s Law in Finance 414 17.5 Exercises 415 17.6 Context 416 Terminology and Notation 419 List of Symbols 425 References 429 Index 455

Game-theoretic probability and finance come of age Glenn Shafer and Vladimir Vovk's Probability and Finance, published in 2001, showed that perfect-information games can be used to define mathematical probability. Based on fifteen years of further research, Game-Theoretic Foundations for Probability and Finance presents a mature view of the foundational role game theory can play. Its account of probability theory opens the way to new methods of prediction and testing and makes many statistical methods more transparent and widely usable. Its contributions to finance theory include purely game-theoretic accounts of Itô stochastic calculus, the capital asset pricing model, the equity premium, and portfolio theory. Game-Theoretic Foundations for Probability and Finance is a book of research. It is also a teaching resource. Each chapter is supplemented with carefully designed exercises and notes relating the new theory to its historical context. Praise from early readers "Ever since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure-theoretic. In this ground-breaking work, Shafer and Vovk give a game-theoretic foundation instead. While being just as rigorous, the game-theoretic approach allows for vast and useful generalizations of classical measure-theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades." –Peter Gr??nwald, CWI and University of Leiden "Shafer and Vovk have thoroughly re-written their 2001 book on the game-theoretic foundations for probability and for finance. They have included an account of the tremendous growth that has occurred since, in the game-theoretic and pathwise approaches to stochastic analysis and in their applications to continuous-time finance. This new book will undoubtedly spur a better understanding of the foundations of these very important fields, and we should all be grateful to its authors." –Ioannis Karatzas, Columbia University