# Game-Theoretic Foundations for Probability and Finance

Wiley Series in Probability and Statistics

1. Edition June 2019

480 Pages, Hardcover*Handbook/Reference Book*

**978-0-470-90305-6**

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 Ito's 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 Grünwald, 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

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

Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London.

Shafer and Vovk are the authors of Probability and Finance: It's Only a Game, published by Wiley and co-authors of Algorithmic Learning in a Random World. Shafer's other previous books include A Mathematical Theory of Evidence and The Art of Causal Conjecture.