Punchline algebra book b 18.4 answers

title: Strategies and Games : Theory and Practice author: Dutta, Prajit K.

publisher: MIT Press isbn10 | asin: 0262041693 print isbn13: 9780262041690 ebook isbn13: 9780585070223

language: English subject Game theory, Equilibrium (Economics)

publication date: 1999 lcc: HB144.D88 1999eb ddc: 330/.01/5193

subject: Game theory, Equilibrium (Economics) cover

Page III

Strategies and Games

Theory and Practice

Prajit K. Dutta

THE MIT PRESS CAMBRIDGE, MASSACHUSETTS LONDON, ENGLAND

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© 1999 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

This book was set in Melior and MetaPlus by Windfall Software using ZzTEX and was printed and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Dutta, Prajit K. Strategies and games: theory and practice / Prajit K. Dutta. p. cm. Includes bibliographical references and index. ISBN 0-262-04169-3 1. Game theory. 2. Equilibrium (Economics). I. Title. HB144.D88 1999 330'.01'15193dc21 98-42937

CIP

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MA AAR BABA KE

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BRIEF CONTENTS

Preface XXI

A Reader's Guide XXIX

Part One Introduction 1

Chapter 1 A First Look at the Applications 3

2 A First Look at the Theory 17

Two Strategic Form Games: Theory and Practice 33

3 Strategic Form Games and Dominant Strategies 35

4 Dominance Solvability 49

5 Nash Equilibrium 63

6 An Application" Cournot Duopoly 75

7 An Application: The Commons Problem 91

8 Mixed Strategies 103

9 Two Applications: Natural Monopoly and Bankruptcy Law 121

10 Zero-Sum Games 139

Three Extensive Form Games: Theory and Applications 155

11 Extensive Form Games and Backward Induction 157

12 An Application: Research and Development 179

13 Subgame Perfect Equilibrium 193

14 Finitely Repeated Games 209

15 Infinitely Repeated Games 227

16 An Application: Competition and Collusion in the NASDAQ Stock Market 243

17 An Application: OPEC 257

18 Dynamic Games with an Application to the Commmons Problem 275

Four Asymmetric Information Games: Theory and Applications 291

19 Moral Hazard and Incentives Theory 293

20 Games with Incomplete Information 309

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21 An Application: Incomplete Information in a Cournot Duopoly 331

22 Mechanism Design, the Revelation Principle, and Sales to an Unknown Buyer 349

23 An Application: Auctions 367

24 Signaling Games and the Lemons Problem 383

Five Foundations 401

25 Calculus and Optimization 403

26 Probability and Expectation 421

27 Utility and Expected Utility 433

28 Existence of Nash Equilibria 451

Index 465

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CONTENTS

Preface XXI

A Reader's Guide XXIX

Part One Indroduction 1

Chapter 1 A First Look at the Applications 3

1.1 Gabes That We Play 3

1.2 Background 7

1.3 Examples 8

Summary 12

Exercises 12

Chapter 2 A First Look at the Theory 17

2.1 Rules of the Game: Background 17

2.2 Who, What, When: The Extensive Form 18

2.2.1 Information Sets and Strategies 20

2.3 Who What, When: The Normal (or Strategic) Form 21

2.4 How Much: Von Neumann-Morgenstern Utility Function 23

2.5 Representation of the Examples 25

Summary 27

Exercises 28

Part Two Strategic Form Games: Theory and Practice 33

Chapter 3 Strategic Form Games and Dominant Strategies 35

3.1 Strategic Form Games 35

3.1.1 Examples 36

3.1.2 Equivalence with the Extensive Form 39

3.2 Case Study The Strategic Form of Art Auctions 40

3.2.1 Art Auctions: A Description 40

3.2.2 Art Auctions: The Strategic Form 40

3.3 Dominant Strategy Solution 41

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3.4 Cae Study Again A Dominant Strategy at the Auction 43

Summary 44

Exercises 45

Chapter 4 Dominance Solvability 49

4.1 The Idea 49

4.1.1 Dominated and Undominated Strategies 49

4.1.2 Iterated Elimination of Dominated Strategies 51

4.1.3 More Examples 51

4.2 Case Study Electing the United Nations Secretary General 54

4.3 A More Formal Definition 55

4.4 A Discussion 57

Summary 59

Exercises 59

Chapter 5 Nash Equilibrium 63

5.1 The Concept 63

5.1.1 Intuition and Definition 63

5.1.2 Nash Parables 64

5.2 Examples 66

5.3 Case Study Nash Equilibrium in the Animal Kingdom 68

5.4 Relation Between the Solution Concepts 69

Summary 71

Exercises 71

Chapter 6 An Application: Cournot Duopoly 75

6.1 Background 75

6.2 The Basic Model 76

6.3 Cournot Nash Equilibrium 77

6.4 Cartel Solution 79

6.5 Case Study Today's OPEC 81

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6.6 Variants on the Main Theme I: A Graphical Analysis 82

6.6.1 The IEDS Solution to the Cournot Model 84

6.7 Variants on the Main Theme II: Stackelberg Model 85

6.8 Variants on the Main Theme III: Generalization 86

Summary 87

Exercises 88

Chapter 7 An Application: The Commons Problem 91

7.1 Background: What is the Commons? 91

7.2 A Simple Model 93

7.3 Social Optimality 95

7.4 The Problem Worsens in a Large Population 96

7.5 Case Studies Buffalo, Global Warming, and the Internet 97

7.6 Averting a Tragedy 98

Summary 99

Exercises 100

Chapter 8 Mixed Strategies 103

8.1 Definition and Examples 103

8.1.1 What Is a Mixed Strategy? 103

8.1.2 Yet More Examples 106

8.2 An Implication 107

8.3 Mixed Strategies Can Dominate Some Pure Strategies 108

8.3.1 Implications for Dominant Strategy Solution and IEDS 109

8.4 Mixed Strategies are Good for Bluffing 110

8.5 Mixed Strategies and Nash Equilibrium 111

8.5.1 Mixed-Strategy Nash Equilibria in an Example 113

8.6 Case Study Random Drug Testing 114

Summary 115

Exercises 116

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Chapter 9 Tow Applications: Naturla Monopoly and Bankruptcy Law 121

9.1 Chicken, Symmetric Games, and Symmetric Equilibria 121

9.1.1 Chicken 121

9.1.2 Symmetric Games and Symmetric Equilibria 122

9.2 Natural Monopoly 123

9.2.1 The Economic Background 123

9.2.2 A Simple Example 124

9.2.3 War of Attrition and a General Analysis 125

9.3 Bankruptcy Law 128

9.3.1 The Legal Background 128

9.3.2 A Numerical Example 128

9.3.3 A General Analysis 130

Summary 132

Exercises 133

Chapter 10 Zero-Sum Games 139

10.1 Definition and Examples 139

10.2 Playing Safe: Maxmin 141

10.2.1 The Concept 141

10.2.2 Examples 142

10.3 Playing Sound: Minmax 144

10.3.1 The Concept and Examples 144

10.3.2 Two Results 146

10.4 Playing Nash: Playing Both Safe and Sound 147

Summary 149

Exercises 149

Part Three Extensive Form Games: Theory and Applications 155

Chapter 11 Extensive Form Games and Backward Induction 157

11.1 The Extensive Form 157

11.1.1 A More Formal Treatment 158

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11.1.2 Strategies, Mixed Strategies, and Chance Nodes 160

11.2 Perfect Information Games: Definition and Examples 162

11.3 Backward Induction: Examples 165

11.3.1 The Power of Commitment 167

11.4 Backward Induction: A General Result 168

11.5 Connection With IEDS in the Strategic Form 170

11.6 Case Study Poison Pills and Other Takeover Deterrents 172

Summary 174

Exercises 175

Chapter 12 An Application: Research and Development 179

12.1 Background: R&D, Patents, and Ologopolies 179

12.1.1 A Patent Race in Progress: High-Definition Television 180

12.2 A Model of R&D 181

12.3 Backward Induction: Analysis of the Model 183

12.4 Some Remarks 188

Summary 189

Exercises 190

Chapter 13 Subgame Perfect Equilibrium 193

13.1 A Motivating Example 193

13.2 Subgames and Strategies Within Subgames 196

13.3 Subgame Perfect Equilibrium 197

13.4 Two More Examples 199

13.5 Some Remarks 202

13.6 Case Study Peace in the World War I Trenches 203

Summary 205

Exercises 205

Chapter 14 Finitely Repeated Games 209

14.1 Examples and Economic Applications 209

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14.1.1 Three Repeated Games and a Definition 209

14.1.2 Four Economic Applications 212

14.2 Finitely Repeated Games 214

14.2.1 Some General Conclusions 218

14.3 Case Study Treasury Bill Auctions 219

Summary 222

Exercises 222

Chapter 15 Infinitely Repeated Games 227

15.1 Detour Through Discounting 227

15.2 Analysis of Example 3: Trigger Strategies and Good Behavior 229

15.3 The Folk Theorem 232

15.4 Repeated Games With Imperfect Detection 234

Summary 237

Exercises 238

Chapter 16 An Application: Competition and Collusion in the NASDAQ Stock Market243

16.1 The Background 243

16.2 The Analysis 245

16.2.1 A Model of the NASDAQ Market 245

16.2.2 Collusion 246

16.2.3 More on Collusion 248

16.3 The Broker-Dealer Relationship 249

16.3.1 Order Preferencing 249

16.3.2 Dealers Big and Small 250

16.4 The Epilogue 251

Summary 252

Exercises 252

Chapter 17 An Application: OPEC 257

17.1 Oil: A Historical Review 257

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17.1.1 Production and Price History 258

17.2 A Simple Model of the Oil Market 259

17.3 Oil Prices and the Role of OPEC 260

17.4 Repteated Games With Demand Uncertainty 262

17.5 Unobserved Quota Violations 266

17.6 Some Further Comments 269

Summary 270

Exercises 271

Chapter 18 Dynamic Games With An Application to the Commons Problem 275

18.1 Dynamic Games: A Prologue 275

18.2 The Commons Problem: A Model 276

18.3 Sustainable Development and Social Optimum 278

18.3.1 A Computation of the Social Optimum 278

18.3.2 An Explanation of the Social Optimum 281

18.4 Achievable Development and Game Equilibrium 282

18.4.1 A Computation of the Game Equilibrium 282

18.4.2 An Explanation of the Equilibrium 284

18.4.3 A Comparison of the Socially Optimal and the Equilibrium Outcomes 285

18.5 Dynamic Games: An Epilogue 286

Summary 287

Exercises 288

Part Four Asymmetric Information Games: Theory and Applications 291

Chapter 19 Moral Hazard and Incentives Theory 293

19.1 Moral Hazard: Examples and a Definition 293

29.2 A Principal-Agent Model 295

19.2.1 Some Examples of Incentive Schemes 297

19.3 The Optimal Incentive Scheme 299

19.3.1 No Moral Hazard 299

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19.3.2 Moral Hazard 299

19.4 Some General Conclusions 301

19.4.1 Extensions and Generalizations 303

19.5 Case Study Compensating Primary Care Physicians in an HMO 304

Summary 305

Exercises 306

Chapter 20 Games with Incomplete Information 309

20.1 Some Examples 309

20.1.1 Some Analysis of the Examples 312

20.2 A Complete Analysis of Example 4 313

20.2.1 Bayes-Nash Equilibrium 313

20.2.2 Pure-Strategy Bayes-Nash Equilibria 315

20.2.3 Mixed-Strategy Bayes-Nash Equilibria 316

20.3 More General Considerations 318

20.3.1 A Modified Example 318

20.3.2 A General Framework 320

20.4 Dominance-Based Solution Concepts 321

20.5 Case Study Final Jeopardy 323

Summary 326

Exercises 326

Chapter 21 An Application: Incomplete Information in a Cournot Duopoly 331

21.1 A Model and its Equilibrium 331

21.1.1 The Basic Model 331

21.1.2 Bayes-Nash Equilibrium 332

21.2 The Complete Information Solution 336

21.3 Revealing Costs to a Rival 338

21.4 Two-Sided Incompleteness of Information 340

21.5 Generalizations and Extensions 341

21.5.1 Oligopoly 341

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21.5.2 Demand Uncertainty 342

Summary 343

Exercises 343

Chapter 22 Mechanism Design, The Revelation Priciple, and Sales to an Unknown Buyer

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22.1 Mechanism Design: The Economic Context 349

22.2 A Simple Example: Selling to a Buyer With an Unknown Valuation 351

22.2.1 Known Passion 351

22.2.2 Unknown Passion 352

22.3 Mechanism Design and the Revelation Principle 356

22.3.1 Single Player 356

22.3.2 Many Players 357

22.4 A More General Example: Selling Variable Amounts 358

22.4.1 Known Type 359

22.4.2 Unknown Type 359

Summary 362

Exercises 362

Chapter 23 An Application: Auctions 367

23.1 Background and Examples 367

23.1.1 Basic Model 369

23.2 Second-Price Auctions 369

23.3 First-Price Auctions 371

23.4 Optimal Auctions 373

23.4.1 How Well Do the First- and Second-Price Auctions Do? 375

23.5 Final Remarks 376

Summary 377

Exercises 378

Chapter 24 Signaling Games and the Lemons Problem 383

24.1 Motivation and Two Examples 383

24.1.1 A First Analysis of the Examples 385

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24.2 A Definition, an Equilibrium Concept, and Examples 387

24.2.1 Definition 387

24.2.2 Perfect Bayesian Equilibrium 387

24.2.3 A Further Analysis of the Examples 389

24.3 Signaling Product Quality 391

24.3.1 The Bad Can Drive Out the Good 391

24.3.2 Good Can Signal Quality? 392

24.4 Case Study Used CarsA Market for Lemons? 394

24.5 Concluding Remarks 395

Summary 396

Exercises 396

Part Five Foundations 401

Chapter 25 Calculus and Optimization 403

25.1 A Calculus Primer 403

25.1.1 Functions 404

25.1.2 Slopes 405

25.1.3 Some Formulas 407

25.1.4 Concave Functions 408

25.2 An Optimization Theory Primer 409

25.2.1 Necessary Conditions 409

25.2.2 Sufficient Conditions 410

25.2.3 Feasibility Constraints 411

25.2.4 Quadratic and Log Functions 413

Summary 414

Exercises 415

Chapter 26 Probability and Expectation 421

26.1 Probability 421

26.1.1 Independence and Conditional Probability 425

26.2 Random Variables and Expectation 426

26.2.1 Conditional Expectation 427

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Summary 428

Exercises 428

Chapter 27 Utility and Expected Utility 433

27.1 Decision Making Under Certainty 433

27.2 Decision Making Under Uncertainty 436

27.2.1 The Expected Utility Theorem and the Expected Return Puzzle 437

27.2.2 Details on the Von Neumann-Morgenstern Theorem 439

27.2.3 Payoffs in a Game 441

27.3 Risk Aversion 441

Summary 444

Exercises 444

Chapter 28 Existence of Nash Equilibria 452

28.1 Definition and Examples 451

28.2 Mathematical Background: Fixed Points 453

28.3 Existence of Nash Equilibria: Results and Intuition 458

Summary 460

Exercises 461

Index 465

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PREFACE

This book evolved out of lecture notes for an undergraduate course in game theory that I have taught at Columbia University for the past six years. On the first two occasions I took the straight road, teaching out of available texts. But the road turned out to be somewhat bumpy; for a variety of reasons I was not satisfied with the many texts that I considered. So the third time around I built myself a small bypass; I wrote a set of sketchy lecture notes from which I taught while I assigned a more complete text to the students. Although this compromise involved minimal costs to me, it turned out to be even worse for my students, since we were now traveling on different roads. And then I (foolishly) decided to build my own highway; buoyed by a number of favorable referee reports, I decided to turn my notes into a book. I say foolishly because I had no idea how much hard work is involved in building a road. I only hope I built a smooth one.

The Book's Purpose And Its Intended Audience

The objective of this book is to provide a rigorous yet accessible introduction to game theory and its applications, primarily in economics and business, but also in political science, the law, and everyday life. The material is intended principally for two audiences: first, an undergraduate audience that would take this course as an elective for an economics major. (My experience has been, however, that my classes are also heavily attended by undergraduate majors in engineering and the sciences who take this course to fulfill their economics requirement.) The many applications and case studies in the book should make it attractive to its second audience, MBA students in business schools. In addition, I have tried to make the material useful to graduate students in economics and related disciplinesPh.D. students in political science, Ph.D. students in economics not specializing in economic theory, etc.who would like to have a source from which they can get a self-contained, albeit basic, treatment of game theory.

Pedagogically I have had one overriding objective: to write a textbook that would take the middle road between the anecdotal and the theorem-driven treatments of the subject. On the one hand is the approach that teaches purely by examples and anecdotes. In my experience that leaves the students, especially the brighter ones, hungering for more. On the other hand, there is the more advanced approach emphasizing a rigorous treatment, but again, in my experience, if there are too few examples and applications it is difficult to keep even the brighter students interested.

I have tried to combine the best elements of both approaches. Every result is precisely stated (albeit with minimal notation), all assumptions are detailed, and at least a sketch of a proof is provided. The text also contains nine chapter-length applications and twelve fairly detailed case studies.

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Distinctive Features Of The Book

I believe this book improves on available undergraduate texts in the following ways.

Content a full description of utility theory and a detailed analysis of dynamic game theory

The book provides a thorough discussion of the single-agent decision theory that forms the underpinning of game theory. (That exercise takes up three chapters in Part Five.) More importantly perhaps, this is the first text that provides a detailed analysis of dynamic strategic interaction (in Part Three). The theory of repeated games is studied over two and a half chapters, including discussions of finitely and infinitely repeated games as well as games with varying stage payoffs. I follow the theory with two chapter-length applications: market-making on the NASDAQ financial market and the price history of OPEC. A discussion of dynamic games (in which the game environment evolves according to players' previous choices) follows along with an application to the dynamic commons problem. I believe many of the interesting applications of game theory are dynamicstudent interest seems always to heighten when I get to this part of the courseand I have found that every other text pays only cursory attention to many dynamic issues.

Style emphasis on a parallel development of theory and examples

Almost every chapter that introduces a new concept opens with numerical examples, some of which are well known and many of which are not. Sometimes I have a leading example and at other times a set of (small) examples. After explaining the exam-pies, I go to the concept and discuss it with reasonable rigor. At this point I return to the examples and analyze the just introduced concept within the context of the examples. At the end of a sectiona set of chapters on related ideasI devote a whole chapter, and sometimes two, to economic applications of those ideas.

Length and Organization bite-sized chapters and a static to dynamic progression

I decided to organize the material within each chapter in such a fashion that the essential elements of a whole chapter can be taught in one class (or a class and a half, depending on level). In my experience it has been a lot easier to keep the students engaged with this structure than with texts that have individual chapters that are, for example, over fifty pages long. The topics evolve in a natural sequence: static complete information to dynamic complete information to static incomplete information. I decided to skip much of dynamic incomplete information (other than signaling) because the questions in this part of the subject are a lot easier than the answers (and my students seemed to have little stomach for equilibrium refinements, for example). There are a few advanced topics as well; different instructors will have the freedom to decide which subset of the advanced topics they would like to teach in their course. Sections that are more difficult are marked with the symbol . Depending on level, some instructors will want to skip

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these sections at first presentation, while others may wish to take extra time in discussing the material.

Exercises

At the end of each chapter there are about twenty-five to thirty problems (in the Exercises section). In addition, within the text itself, each chapter has a number of questions (or concept checks) in which the student is asked to complete a part of an argument, to compute a remaining case in an example, to check the computation for an assertion, and so on. The point of these questions is to make sure that the reader is really following the chapter's argument; I strongly encourage my students to answer these questions and often include some of them in the problem sets.

Case Studies and Applications

At the end of virtually every theoretical chapter there is a case study drawn from real life to illustrate the concept just discussed. For example, after the chapter on Nash equilibrium, there is a discussion of its usage in understanding animal conflicts. After a chapter on backward induction (and the power of commitment), there is a discussion of poison pills and other take-over deterrents. Similarly, at the end of each cluster of similar topics there is a whole chapter-length application. These range from the tragedy of the commons to bankruptcy law to incomplete information Cournot competition.

An Overview And Two Possible Syllabi

The book is divided into five parts. The two chapters of Part One constitute an Introduction. Part Two (Chapters 3 through 10) covers Strategic Form Games: Theory and Practice, while Part Three (Chapters 11 through 18) concentrates on Extensive Form Games: Theory and Practice. In Part Four (Chapters 19 through 24) I discuss Asymmetric Information Games: Theory and Practice. Finally, Part Five (Chapters 25 through 28) consists of chapters on Foundations.

I can suggest two possible syllabi for a one-semester course in game theory and applications. The first stresses the applications end while the second covers all the theoretical topics. In terms of mathematical requirements, the second is, naturally, more demanding and presumes that the students are at a higher level. I have consequently included twenty chapters in the second syllabus and only eighteen in the first. (Note that the numbers are chapter numbers.)

Syllabus 1 (Applications Emphasis)

1. A First Look at the Applications

3. Strategic Form Games and Dominant Strategies

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4. Dominance Solvability

5. Nash Equilibrium

6. An Application: Cournot Duopoly

8. Mixed Strategies

9.Two Applications: Natural Monopoly and Bankruptcy Law

11. Extensive Form Games and Backward Induction

12. An Application: Research and Development

13. Subgame Perfect Equilibrium

15. Infinitely Repeated Games

16. An Application: Competition and Collusion in the NASDAQ Stock Market

17. An Application: OPEC

19. Moral Hazard and Incentives Theory

20. Games with Incomplete Information

22. Mechanism Design, the Revelation Principle, and Sales to an Unknown Buyer

23. An Application: Auctions

24.Signaling Games and the Lemons Problem

Syllabus 2 (Theory Emphasis)

2. A First Look at the Theory

27. Utility and Expected Utility

3. Strategic Form Games and Dominant Strategies

4. Dominance Solvability

5. Nash Equilibrium

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6. An Application: Cournot Duopoly

7. An Application: The Commons Problem

8. Mixed Strategies

10. Zero-Sum Games

28. Existence of Nash Equilibria

11. Extensive Form Games and Backward Induction

13. Subgame Perfect Equilibrium

14. Finitely Repeated Games

15. Infinitely Repeated Games

17. An Application: OPEC

18. Dynamic Games with an Application to the Commons Problem

20. Games with Incomplete Information

21. An Application: Incomplete Information in a Cournot Duopoly

22. Mechanism Design, the Revelation Principle, and Sales to an Unknown Buyer

23. An Application: Auctions

Prerequisites

I have tried to write the book in a manner such that very little is presumed of a reader's mathematics or economics background. This is not to say that one semester each of calculus and statistics and a semester of intermediate microeconomics will not help. However, students who do not already have this background but are willing to put in extra work should be able to educate themselves sufficiently.

Toward that end, I have included a chapter on calculus and optimization, and one on probability and expectation. Readers can afford not to read the two chapters if they already have the following knowledge. In calculus, I presume knowledge of the slope of a function and a familiarity with slopes of the linear, quadratic, log, and the square-root functions. In optimization theory, I use the first-order characterization of an interior

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optimum, that the slope of a maximand is zero at a maximum. As for probability, it helps to know how to take an expectation. As for economic knowledge, I have attempted to explain all relevant terms and have not presumed, for example, any knowledge of Pareto optimality, perfect competition, and monopoly.

Acknowledgments

This book has benefited from the comments and criticisms of many colleagues and friends. Tom Gresik at Penn State, Giorgidi Giorgio at La Sapienza in Rome, Sanjeev Goyal at Erasmus, Matt Kahn at Columbia, Amanda Bayer at Swarthmore, Rob Porter at Northwestern, and Charles Wilson at NYU were foolhardy

enough to have taught from preliminary versions of the text, and I thank them for their courage and comments. In addition, the following reviewers provided very helpful comments:

Amanda Bayer, Swarthmore College

James Dearden, Lehigh University

Tom Gresik, Penn State

Ehud Kalai, Northwestern University

David Levine, UCLA

Michael Meurer, SUNY Buffalo

Yaw Nyarko, NYU

Robert Rosenthal, Boston University

Roberto Serrano, Brown University

Rangarajan Sundaram, NYU

A second group of ten referees provided extremely useful, but anonymous, comments.

My graduate students Satyajit Bose, Tack-Seung Jun, and Tsz-Cheong Lai very carefully read the entire manuscript. Without their hawk-eyed intervention, the book would have many more errors. They are also responsible for the Solutions Manual, which accompanies this text. My colleagues in the community, Venky Bala, Terri Devine, Ananth