Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball explores how paradoxi- cal challenges involving mathematical expectation often necessitate a reexamination of basic prem- ises. The author takes you through mathematical paradoxes associated with seemingly straightfor- ward applications of mathematical expectation and shows how these unexpected contradictions may push you to reconsider the legitimacy of the applications. Praise for Leonard Wapner’s first book, The Pea and the Sun The book requires only an understanding of basic algebraic operations and includes supplemental “What is presented in this book is maths for its own sake: beautiful, elegant, artistic, mathematical background in chapter appendices. astonishing...it would surely make a great present for a budding pure mathema- After a history of probability theory, it introduces tician—and what a present it would be, to give someone their first inkling of the the basic laws of probability as well as the defini- wonders that lie at the heart of pure mathematics.” tion and applications of mathematical expecta- —Helen Joyce, Plus Magazine, September 2005 tion/expected value (E). The remainder of the text Leonard Wapner has taught mathematics at El covers unexpected results related to mathemati- Camino College in Torrance, California, since 1973. cal expectation, including: He received his BA and MAT degrees in mathemat- “This book is sure to intrigue, fascinate, and challenge the mathematically inclined ics from the University of California, Los Angeles. reader.” —The Mathematics Teacher, May 2006 • The roles of aversion and risk in rational decision He is the author of The Pea and the Sun: A Math- making ematical Paradox and his writings on mathemat- “Very readable and aimed at non-technical readers … an ideal book for undergradu- • A class of expected value paradoxes referred to ics education have appeared in The Mathemat- ates or sixth form students.” as envelope problems ics Teacher (The National Council of Teachers of —Anthony C. Robin, The Mathematical Gazette, November 2006 • Parrondo’s paradox—how negative (losing) ex- Mathematics) and The AMATYC Review (The pectations can be combined to give a winning result American Mathematical Association of Two-Year “One does not need to have a degree in mathematics in order to follow the lively and Colleges). Len lives in Seal Beach, California, with readable, highly intriguing story of the paradox. Yet the exposition is serious, correct • Problems associated with imperfect recall his wife Mona and his Labrador Retriever Bailey. and comprehensive, and it presents a detailed proof of the result. The presentation • Non-zero-sum games, such as the game of is light-hearted, highly entertaining and illustrated with many examples, puzzles, etc. chicken and the prisoner’s dilemma This book is (already) a classic in an area that needed one.” • Newcomb’s paradox—a great philosophical par- —Newsletter of the European Mathematical Society, June 2006 adox of free will • Benford’s law and its use in computer design and fraud detection While useful in areas as diverse as game theory, quantum mechanics, and forensic science, math- ematical expectation generates paradoxes that frequently leave questions unanswered yet reveal K13061 interesting surprises. Encouraging you to embrace the mysteries of mathematics, this book helps you appreciate the applications of mathematical -ex pectation, “a statistical crystal ball.” Mathematics Unexpected Expectations This page intentionally left blank Unexpected Expectations The Curiosities of a Mathematical Crystal Ball Leonard M. Wapner CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20120202 International Standard Book Number-13: 978-1-4398-6767-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com For my mom, Lea Wapner For my wife, Mona Wapner This page intentionally left blank Table of Contents Acknowledgments xi The Crystal Ball xiii 1. Looking Back 1 Beating the Odds: Girolamo Cardano 3 Vive la France: Blaise Pascal and Pierre de Fermat 6 Going to Press: Christiaan Huygens 8 Law, but No Order: Jacob Bernoulli 10 Three Axioms: Andrei Kolmogorov 13 2. The ABCs of E 19 The Definition of Probability 20 The Laws of Probability 22 Binomial Probabilities 31 The Definition of Expected Value 32 Utility 35 Infinite Series: Some Sum! 37 Appendix 39 3. Doing the Right Thing 41 What Happens in Vegas 41 Is Insurance a Good Bet? 45 Airline Overbooking 47 vii viii Unexpected Expectations Composite Sampling 51 Pascal’s Wager 54 Game Theory 56 The St. Petersburg Paradox 63 Stein’s Paradox 64 Appendix 68 4. Aversion Perversion 71 Loss Aversion 72 Ambiguity Aversion 75 Inequity Aversion 78 The Dictator Game 78 The Ultimatum Game 80 The Trust Game 80 Off-Target Subjective Probabilities 82 5. And the Envelope Please! 91 The Classic Envelope Problem: Double or Half 92 The St. Petersburg Envelope Problem 94 The “Powers of Three” Envelope Problem 95 Blackwell’s Bet 97 The Monty Hall Problem 99 Win-Win 104 Appendix 105 6. Parrondo's Paradox: You Can Win for Losing 109 Ratchets 101 109 The Man Engines of the Cornwall Mines 110 Parrondo’s Paradox 112 Reliabilism 114 From Soup to Nuts 115 Parrondo Profits 116 Truels: Survival of the Weakest 117 Going North? Head South! 120 Appendix 123 7. Imperfect Recall 127 The Absentminded Driver 128 Unexpected Lottery Payoffs 131 Sleeping Beauty 134 Applications 137 8. Non-zero-sum Games: The Inadequacy of Individual Rationality 141 Pizza or Pâté 142 The Threat 145 Table of Contents ix Chicken: The Mamihlapinatapei Experience 146 The Prisoner’s Dilemma 154 The Nash Arbitration Scheme 163 Appendix 167 9. Newcomb's Paradox 169 Dominance vs. Expectation 170 Newcomb + Newcomb = Prisoner’s Dilemma 174 10. Benford's Law 177 Simon Newcomb’s Discovery 178 Benford’s Law 179 What Good Is a Newborn Baby? 184 Appendix 189 Let the Mystery Be! 191 Bibliography 193 Index 199 This page intentionally left blank Acknowledgments his book is inspired by family, friends, students, colleagues, and authors too numerous to list. I begin by thanking Martin Gardner for all the years of Trecreational mathematics he has given us. His passing is a great loss to all who study, teach, and enjoy mathematics. I thank Phil Everson of Swarthmore College, Robert Vanderbei of Princeton University, and my good friend Paul Wozniak of El Camino College for clarifying various mathematical issues. Several illustrations appearing in this book were created using Peanut Soft- ware, a wonderful collection of nine programs written by Rick Parris of Phillips Exeter Academy, Exeter, New Hampshire. The nine programs are freely distrib- uted, and I thank Rick for his generosity. Two years ago my friend Michael Carter phoned with the question, “What do you know about Benford’s law?” I thank him for the question and the answer now appears as Chapter 10 of this book. I wish to especially thank Jim Stein, Professor of Mathematics at California State University, Long Beach, California. Jim is the author of four popular math- ematics books, and we’ve had many good discussions of topics in this book while choking down BLTs at a local coffee shop. I’m grateful for his assistance, from the original proposal to the final manuscript. One of our discussions involved the James-Stein theorem, a discussion of which appears in Chapter 3. I had some questions and, after all, who better to ask about the James-Stein theorem than James Stein himself? As it turns out, and as Jim will readily admit, he is neither xi xii Unexpected Expectations the James nor the Stein of the James-Stein theorem. Coincidentally, Jim and Wil- lard James (the real James of James-Stein) were colleagues at CSULB many years ago. And, to my surprise, I’ve come to find out that Willard James was one of my professors. As a student, I never made the connection between Willard James and the James-Stein theorem. It was while choking down one of those BLTs that I put it all together.