Game-Theoretic Foundations for Probability and Finance WILEY SERIES IN PROBABILITY AND STATISTICS Established by Walter A Shewhart and Samuel S Wilks Editors: David J Balding, Noel A C Cressie, Garrett M Fitzmaurice, Geof H Givens, Harvey Goldstein, Geert Molenberghs, David W Scott, Adrian F M Smith, Ruey S Tsay Editors Emeriti: J Stuart Hunter, Iain M Johnstone, Joseph B Kadane, Jozef L Teugels The Wiley Series in Probability and Statistics is well established and authoritative It covers many topics of current research interest in both pure and applied statistics and probability theory Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research A complete list of titles in this series can be found at http://www.wiley.com/go/wsps Game-Theoretic Foundations for Probability and Finance GLENN SHAFER Rutgers Business School VLADIMIR VOVK Royal Holloway, University of London This edition first published 2019 © 2019 John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions The right of Glenn Shafer and Vladimir Vovk to be identified as the authors of this work has been asserted in accordance with law Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com Wiley also publishes its books in a variety of electronic formats and by print-on-demand Some content that appears in standard print versions of this book may not be available in other formats Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make This work is sold with the understanding that the publisher is not engaged in rendering professional services The advice and strategies contained herein may not be suitable for your situation You should consult with a specialist where appropriate Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages Library of Congress Cataloging-in-Publication Data Names: Shafer, Glenn, 1946- author | Vovk, Vladimir, 1960- author Title: Game-theoretic foundations for probability and finance / Glenn Ray Shafer, Rutgers University, New Jersey, USA, Vladimir Vovk, University of London, Surrey, UK Other titles: Probability and finance Description: First edition | Hoboken, NJ : John Wiley & Sons, Inc., 2019 | Series: Wiley series in probability and statistics | Earlier edition published in 2001 as: Probability and finance : it’s only a game! | Includes bibliographical references and index | Identifiers: LCCN 2019003689 (print) | LCCN 2019005392 (ebook) | ISBN 9781118547939 (Adobe PDF) | ISBN 9781118548028 (ePub) | ISBN 9780470903056 (hardcover) Subjects: LCSH: Finance–Statistical methods | Finance–Mathematical models | Game theory Classification: LCC HG176.5 (ebook) | LCC HG176.5 S53 2019 (print) | DDC 332.01/5193–dc23 LC record available at https://lccn.loc.gov/2019003689 Cover design by Wiley Cover image: © Web Gallery of Art/Wikimedia Commons Set in 10/12pt, TimesLTStd by SPi Global, Chennai, India Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY 10 Contents Preface Acknowledgments Part I Examples in Discrete Time xi xv 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 16 18 19 21 21 24 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 31 33 37 39 45 v vi CONTENTS 2.5 Exercises 2.6 Context 46 49 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 55 56 56 58 63 66 67 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 69 70 73 80 81 82 86 89 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 93 94 99 100 104 104 106 Part II Abstract Theory in Discrete Time 109 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 111 113 115 118 121 125 CONTENTS 6.6 Exercises 6.7 Context vii 127 131 Abstract Testing Protocols 7.1 Terminology and Notation 7.2 Supermartingales 7.3 Global Upper Expected Values 7.4 Lindeberg’s Central Limit Theorem for Martingales 7.5 General Abstract Testing Protocols 7.6 Making the Results of Part I Abstract 7.7 Exercises 7.8 Context 135 136 136 142 145 146 151 153 155 Zero-One Laws 8.1 Lévy’s Zero-One Law 8.2 Global Upper Expectation 8.3 Global Upper and Lower Probabilities 8.4 Global Expected Values and Probabilities 8.5 Other Zero-One Laws 8.6 Exercises 8.7 Context 157 158 160 162 163 165 169 170 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 10.1 Signals in Open Protocols 10.2 Cournot’s Principle 197 198 201 viii CONTENTS 10.3 10.4 10.5 10.6 10.7 10.8 10.9 Daltonism Least Squares Parametric Statistics with Signals Quantum Mechanics Jeffreys’s Law Exercises Context 202 207 212 215 217 225 226 11 Calibrating Lookbacks and p-Values 11.1 Lookback Calibrators 11.2 Lookback Protocols 11.3 Lookback Compromises 11.4 Lookbacks in Financial Markets 11.5 Calibrating p-Values 11.6 Exercises 11.7 Context 229 230 235 241 242 245 248 250 12 Defensive Forecasting 12.1 Defeating Strategies for Skeptic 12.2 Calibrated Forecasts 12.3 Proving the Calibration Theorems 12.4 Using Calibrated Forecasts for Decision Making 12.5 Proving the Decision Theorems 12.6 From Theory to Algorithm 12.7 Discontinuous Strategies for Skeptic 12.8 Exercises 12.9 Context 253 255 259 264 270 274 286 291 295 299 Part IV Game-Theoretic Finance 305 13 Emergence of Randomness in Idealized Financial Markets 13.1 Capital Processes and Instant Enforcement 13.2 Emergence of Brownian Randomness 13.3 Emergence of Brownian Expectation 13.4 Applications of Dubins–Schwarz 13.5 Getting Rich Quick with the Axiom of Choice 309 310 312 320 325 331 ... developed since 2001: calibration of lookbacks and p-values, and defensive forecasting • Part IV, Game- Theoretic Finance, studies continuous-time game- theoretic probability and its application to finance. .. the publication of Probability and Finance, is developed in Chapter 12 Continuous-time game- theoretic finance Measure -theoretic finance assumes that prices of securities in a financial market follow... twentieth-century authors, including Peter Williams and Peter Walley As we explained in Probability and Finance, we can combine Williams and Walley’s picture of limited betting opportunities in individual situations