978-0-521-85012-4 - THE MATHEMATICS OF BEHAVIOR - by Earl Hunt

Frontmatter/Prelims

## The Mathematics of Behavior

Mathematical thinking provides a clear, crisp way of defining problems. Our whole technology is based on it. What is less appreciated is that mathematical thinking can also be applied to problems in the social and behavioral sciences. This book illustrates how mathematics can be employed for understanding human and animal behavior, using examples in psychology, sociology, economics, ecology, and even marriage counseling.

Earl Hunt is Professor Emeritus of Psychology at the University of Washington in Seattle. He has written many articles and chapters in contributed volumes and is the past editor of *Cognitive Psychology* and *Journal of Experimental Psychology*. His books include *Concept Learning: An Information Processing Problem*, *Experiments in Induction*, *Artificial Intelligence*, and *Will We Be Smart Enough?*, which won the William James Book Award from the American Psychological Association in 1996. His most recent book is *Thoughts on Thought*.

# The Mathematics of Behavior

EARL HUNT

*University of Washington, Seattle*

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University Press

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Information on this title: www.cambridge.org/9780521850124

© Cambridge University Press 2007

This publication is in copyright. Subject to statutory exception

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no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 2007

Printed in the United States of America

*A catalog record for this publication is available from the British Library.*

*Library of Congress Cataloging in Publication Data*

Hunt, Earl B.

The mathematics of behavior/Earl Hunt

p. cm.

Includes bibliographical references and index.

ISBN 0 521 85012 6 (hardcover) – ISBN 0 521 61522 4 (pbk.)

1. Psychology – Mathematical models. 2. Social sciences – Mathematical models. I. Title.

BF39.H86 2006

150.1′51–dc22 2005030591

ISBN-13 978-0-521-85012-4 hardback

ISBN-10 0-521-85012-6 hardback

ISBN-13 978-0-521-61522-8 paperback

ISBN-10 0-521-61522-4 paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

## Contents

Preface | page ix | ||

1 | INTRODUCTION | 1 | |

1.1. What’s in the Book? | 1 | ||

1.2. Some Examples of Formal and Informal Thinking | 2 | ||

1.3. A Bit of History | 4 | ||

1.4. How Big Is the Earth? Eratosthenes’ Solution | 5 | ||

1.5. A Critique of Eratosthenes | 12 | ||

1.6. Applications of Mathematics to Social and Behavioral Issues | 14 | ||

1.7. Statistics | 16 | ||

2 | APPLYING PROBABILITY THEORY TO PROBLEMS IN SOCIOLOGY AND PSYCHOLOGY | 18 | |

2.1. Introduction | 18 | ||

2.2. Defining Probability and Probability Measures | 19 | ||

2.3. How Closely Connected Are We? | 23 | ||

2.4. Conscious and Unconscious Memories | 27 | ||

2.5. Some Final Comments | 32 | ||

Appendix 2A. The Basis for Kolmogorov’s Axioms | 32 | ||

Appendix 2B. Some Important Properties of Probability Measures | 33 | ||

3 | FROM PHYSICS TO PERCEPTION | 42 | |

3.1. The Psychophysical Problem | 42 | ||

3.2. Weber’s Law | 44 | ||

3.3. Fechner’s Law | 47 | ||

3.4. Stevens’s Scaling Technique: Deriving the Psychophysical Function from Magnitude Estimation | 53 | ||

3.5. Judging Complex Objects | 61 | ||

3.6. A Comment on Measurement | 65 | ||

4 | WHEN SYSTEMS EVOLVE OVER TIME | 67 | |

4.1. Systems of Variables | 67 | ||

4.2. Differences and Differentiation | 68 | ||

4.3. Exponential Growth and Decay | 70 | ||

4.4. Numerical Analysis: The Transmission of Jokes and Colds | 76 | ||

4.5. Questions about Modeling | 81 | ||

4.6. Graphical Analysis: The Evolution of War and Peace | 86 | ||

4.7. Making Love, Not War: The Gottman-Murray Model of Marital Interactions | 96 | ||

4.8. Concluding Comments on Modeling Simple Systems | 101 | ||

Appendix 4A. A Proof of the Exponential Growth Equation | 103 | ||

5 | NON-LINEAR AND CHAOTIC SYSTEMS | 104 | |

5.1. Continuous Change and Sudden Jumps | 104 | ||

5.2. The Lotka-Volterra Model of Predator and Prey Interactions | 106 | ||

5.3. The Logistic Equation: Introduction and Behavior When k < 1 | 111 | ||

5.4. Non-zero Asymptotes and Cycles as k Increases | 116 | ||

5.5. Chaos | 121 | ||

5.6. Chaos and Network Models | 123 | ||

5.7. Closing Comments on Chaos | 130 | ||

6 | DEFINING RATIONALITY | 132 | |

6.1. Axiomatic Reasoning | 132 | ||

6.2. Decision Making under Risk | 133 | ||

6.3. The Concept of Utility | 135 | ||

6.4. Von Neumann and Morgenstern’s Axiomatic Approach to Decision Making | 139 | ||

6.5. The Utility of Money | 143 | ||

6.6. A Summary of the Argument | 148 | ||

6.7. Psychological Research on Decision Making | 151 | ||

6.8. The Problem of Voting | 158 | ||

6.9. Definition and Notation | 161 | ||

6.10. Arrow’s Axioms: The Restrictions on Social Welfare Functions | 162 | ||

6.11. Illustration of the Definitions and Concepts for the Three-Person Society | 164 | ||

6.12. A Proof of Arrow’s Theorem | 166 | ||

6.13. Commentary on the Implications of Arrow’s Theorem | 173 | ||

6.14. Summary Comments and Questions About Axiomatic Reasoning | 174 | ||

7 | HOW TO EVALUATE EVIDENCE | 176 | |

7.1. The Legacy of Reverend Bayes | 176 | ||

7.2. Bayes’ Theorem | 178 | ||

7.3. Some Numerical Examples | 180 | ||

7.4. Calculating the Odds | 184 | ||

7.5. Some Examples of Signal Detection | 185 | ||

7.6. A Mathematical Formulation of the Signal Detection Problem | 187 | ||

7.7. The Decision Analyst’s Problem | 191 | ||

7.8. A Numerical Example of ROC Analysis | 199 | ||

7.9. Establishing a Criterion | 203 | ||

7.10. Examples | 207 | ||

7.11. Four Challenge Problems | 213 | ||

8 | MULTIDIMENSIONAL SCALING | 216 | |

8.1. The Basic Idea | 216 | ||

8.2. Steps and Technique | 219 | ||

8.3. Extensions to Non-geometric Data | 222 | ||

8.4. Extending the Idea to Conceptual Classes | 223 | ||

8.5. Generalizations of Semantic Space Models | 227 | ||

8.6. Qualifications on the Semantic Space Model | 229 | ||

9 | THE MATHEMATICAL MODELS BEHIND PSYCHOLOGICAL TESTING | 231 | |

9.1. Introduction | 231 | ||

9.2. A Brief Review of Correlation and Covariance | 234 | ||

9.3. Predicting One Variable from Another: Linear Regression | 240 | ||

9.4. The Single Factor Model: The Case of General Intelligence | 244 | ||

9.5. Multifactor Theories of Intelligence and Personality | 249 | ||

9.6. Geometric and Graphic Interpretations | 254 | ||

9.7. What Sort of Results Are Obtained? | 255 | ||

Appendix 9A. A Matrix Algebra Presentation of Factor Analysis | 256 | ||

10 | HOW TO KNOW YOU ASKED A GOOD QUESTION | 259 | |

10.1. The Problem | 259 | ||

10.2. An Illustrative Case: Vocabulary Testing | 260 | ||

10.3. The Basics of Item Response Theory | 262 | ||

10.4. Standardization: Estimating Item and Person Parameters Simultaneously | 265 | ||

10.5. The Application Phase: Adaptive Testing | 267 | ||

10.6. More Complicated IRT Models | 269 | ||

10.7. Mathematics Meets the Social World: Mathematical Issues and Social Relevance | 272 | ||

Appendix 10A. The Adaptive Testing Algorithm | 274 | ||

Appendix 10B. An Exercise in Adaptive Testing | 275 | ||

11 | THE CONSTRUCTION OF COMPLEXITY | 277 | |

11.1. Some Grand Themes | 277 | ||

11.2. The Problem of Complexity | 278 | ||

11.3. Cellular Automata Can Create Complicated Constructions | 281 | ||

11.4. Is Capitalism Inherently Unfair? Reconstructing a Simple Market Economy | 283 | ||

11.5. Residential Segregation, Genocide, and the Usefulness of the Police | 289 | ||

11.6. Is This a New Kind of Science? | 294 | ||

12 | CONNECTIONISM | 297 | |

12.1. The Brain and the Mind | 297 | ||

12.2. Computation at the Neural Level | 299 | ||

12.3. Computations at the Network Level | 303 | ||

12.4. A Philosophical Aside | 307 | ||

12.5. Connectionist Architectures | 309 | ||

12.6. Simulating a Phenomenon in Visual Recognition: The Interactive Activation Model | 311 | ||

12.7. An Artificial Intelligence Approach to Learning | 313 | ||

12.8. A Biological Approach to Learning: The Hebbian Algorithm | 319 | ||

12.9. The Auto-associator | 321 | ||

12.10. A Final Word | 324 | ||

13 | L’ENVOI | 325 | |

References | 328 | ||

Index of Names | 333 | ||

Index of Subjects | 337 |

## Preface

Many, many years ago, when I was a graduate student at Yale University, I attended Professor Robert Abelson’s seminar on mathematical psychology. This was in the late 1950s, just as mathematical techniques were beginning to hit psychology. Subsequently I met Professor Jacob Marschak, an economist whose work on the economics of information was seminal in the field. After I received my doctorate in 1960 I had the great opportunity to work with Marschak’s group at the University of California, Los Angeles. Marschak set a gold standard for the use of mathematics to support clear, precise thinking. It is now almost 50 years later, near the end of my own career, and I have yet to meet someone whose logic was so clear. I have had the opportunity to see some people come close to Marschak’s standard, both in my own discipline of psychology and in other fields. This book is an attempt to let future students see how our understanding of behaviors, by both humans and non-humans, can be enhanced by mathematical analysis.

Is such a goal realistic today? Many people have deplored the alleged decline in mathematical training among today’s college students. I do not think that that is fair. On an absolute level, students at the major universities arrive far better trained than they were 50 years ago. High school courses in the calculus are common today; they were rare even 25 years ago. It is true that on a comparative basis American students have slipped compared to their peers abroad, but on an absolute basis the better students in all countries are simply better prepared than they used to be. I have set my sights accordingly. This book should be easily accessible to anyone who has a basic understanding of the calculus, and most of the book will not even require that. It will require the ability (the willingness?) to follow a mathematical argument. I hope that the effort will be rewarded. Curious about the mathematics of love? Or how unprejudiced people can produce a segregated society? Read on!

And to those of you on college and university faculties, consider teaching a course that covers topics like this; mathematics used to analyze important issues in our day, or important issues in the history of science. Don’t restrict it to your own discipline; think broadly. I hope you find this book helpful, but if you don’t, get some readings and teach the course anyway. I have been fortunate to teach such a course in the University of Washington Honors Program for the past several years, and the discussions among students pursuing majors from philosophy to biology and engineering have been informative and enjoyable.

No one prepares a book without a great deal of support. I have had it. I thank the Honors Program and, most especially, the students in my classes, for letting me lead the course. I also thank the Psychology Department for letting me lead a predecessor of this course, focusing somewhat more on psychology. Cambridge University Press provided assistance in book preparation that was far superior to that of any other press with which I have ever worked. I thank Regina Paleski for production editing assistance, and I particularly thank Phyllis Berk for a superb job of copyediting a difficult manuscript. I also thank the editor, Philip Laughlin, for his assistance, and in particular for his obtaining very high-quality editorial reviews. Naturally, the people who wrote them are thanked, too! The final review, by Professor Jerome Busemeyer of the University of Indiana, was a model of constructive criticism.

Every author closes with thanks to family … or at least, he should. My wife, Mary Lou Hunt, has supported me in this and all my scholarly work. I could not accomplish any efforts without her loving aid and assistance.

Earl Hunt

Bellevue, Washington, and

Hood Canal, Washington

February 2006

© Cambridge University Press