sign in sign up
<<
Home , Mathematical game, Nimber, Shapley value

STUDENT MATHEMATICAL LIBRARY Volume 80

Game Theory A Playful Introduction

Matt DeVos Deborah A. Kent Theory A Playful Introduction

https://doi.org/10.1090//stml/080

STUDENT MATHEMATICAL LIBRARY Volume 80

Game Theory A Playful Introduction

Matt DeVos Deborah A. Kent

Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov

2010 Mathematics Subject Classification. Primary 91-01, 91A46, 91A06, 91B06.

For additional information and updates on this book, visit www.ams.org/bookpages/stml-80

Library of Congress Cataloging-in-Publication Data Names: DeVos, Matthew Jared, 1974- | Kent, Deborah A., 1978- Title: Game theory : a playful introduction / Matthew DeVos, Deborah A. Kent. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Student mathematical library ; volume 80 | Includes bibliographical references and index. Identifiers: LCCN 2016035452 | ISBN 9781470422103 (alk. paper) Subjects: LCSH: Game theory–Textbooks. | Combinatorial analysis–Textbooks. | AMS: Game theory, economics, social and behavioral sciences – Instructional exposition (textbooks, tutorial papers, etc.). msc | Game theory, economics, social and behavioral sciences – Game theory – Combinatorial . msc | Game theory, economics, social and behavioral sciences – Game theory – n-person games, n > 2. msc | Game the- ory, economics, social and behavioral sciences – – Decision theory. msc Classification: LCC QA269 .D45 2016 | DDC 519.3–dc23 LC record available at https://lccn.loc.gov/2016035452

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions ® of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copy- right ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes.

© 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16 Dedicated to my family, MD. For Mom, in memory with love, DK.

Contents

Preface xi

Chapter 1. Combinatorial Games 1 §1.1. Game Trees 3 §1.2. Zermelo’s Theorem 9 §1.3. 14 Exercises 19

Chapter 2. Normal-Play Games 25 §2.1. Positions and Their Types 27 §2.2. Sums of Positions 30 §2.3. Equivalence 36 Exercises 41

Chapter 3. Impartial Games 45 §3.1. Nim 46 §3.2. The Sprague-Grundy Theorem 52 §3.3. Applying the Principle 54 Exercises 59

vii viii Contents

Chapter 4. and Partizan Games 63 §4.1. Hackenbush 64 §4.2. Dyadic and Positions 71 §4.3. The Simplicity Principle 77 Exercises 83

Chapter 5. Zero-Sum Matrix Games 89 §5.1. Dominance 91 §5.2. Mixed Strategies 95 §5.3. Von Neumann Solutions 100 Exercises 104

Chapter 6. Von Neumann’s Theorem 111 §6.1. Equating the Opponent’s Results 113 §6.2. Two-Dimensional Games 118 §6.3. Proof of the Minimax Theorem 123 Exercises 128

Chapter 7. General Games 133 §7.1. Utility 135 §7.2. Matrix Games 139 §7.3. Game Trees 145 §7.4. Trees vs. Matrices 150 Exercises 155

Chapter 8. and Applications 161 §8.1. Nash Equilibrium 162 §8.2. Evolutionary Biology 169 §8.3. Cournot Duopoly 176 Exercises 182

Chapter 9. Nash’s Equilibrium Theorem 187 §9.1. Sperner’s Lemma 189 §9.2. Brouwer’s Fixed Point Theorem 192 Contents ix

§9.3. Strategy Spaces 198 §9.4. Nash Flow and the Proof 202 Exercises 208

Chapter 10. Cooperation 213 §10.1. The Negotiation Set 214 §10.2. Nash Arbitration 221 §10.3. Repeated Games and the Folk Theorem 228 Exercises 238

Chapter 11. 푛-Player Games 245 §11.1. Matrix Games 247 §11.2. Coalitions 251 §11.3. Shapley Value 260 Exercises 270

Chapter 12. Preferences and Society 275 §12.1. 277 §12.2. Stable Marriages 285 §12.3. Arrow’s Impossibility Theorem 290 Exercises 298

Appendix A. On Games and Numbers 301

Appendix B. Linear Programming 309 Basic Theory 310 A Connection to Game Theory 313 LP Duality 317

Appendix C. Nash Equilibrium in High Dimensions 323

Game Boards 331

Bibliography 335

Index of Games 339

Index 341

Preface

The story of this book began in 2002 when Matt, then a postdoc at Prince- ton University, was given the opportunity to teach an undergraduate class in game theory. Thanks largely to the 2001 release of a Hollywood movie on the life of the famous Princeton mathematician and (classical) game theorist John Nash, this course attracted a large and highly diverse audience. Princeton’s mathematics department featured not only Nash, but also John Conway, the father of modern combinatorial game theory. So it seemed only natural to blend the two sides of game theory, combi- natorial and classical, into one (rather ambitious) class. The varied back- grounds of the students and the lack of a suitable textbook made for an extremely challenging teaching assignment (that sometimes went awry). However, the simple fun of playing games, the rich mathematical beauty of game theory, and its significant real-world connections still made for an amazing class. Deborah adopted a variant of this material a few years later and fur- ther developed it for a general undergraduate audience. Over the ensu- ing years, Deborah and Matt have both taught numerous incarnations of this course at various universities. Through exchange and collaboration, the material has undergone a thorough evolution, and this textbook rep- resents the culmination of our process. We hope it will provide an intro- ductory course in mathematical game theory that you will find inviting, entertaining, mathematically interesting, and meaningful.

xi xii Preface

Combinatorial game theory is the study of games like and Checkers in which two opponents alternate moves, each trying to win the game. This part of game theory focuses on deterministic games with full information and is thus highly amenable to recursive analysis. Com- binatorial game theory traces its roots to Charles Bouton’s theory of the game Nim and a classification theorem attributed independently to Roland Sprague and Patrick Grundy. The 1982 publication of the classic Winning Ways for Your Mathematical Plays by Elwyn Berlekamp, John Conway, and Richard Guy laid a modern foundation for the subject— now a thriving branch of combinatorics. In contrast, classical game the- ory is an aspect of applied mathematics frequently taught in departments of economics. Classical game theory is the study of strategic decision- making in situations with two or more players, each of whom may af- fect the . and Oskar Morgenstern are com- monly credited with the foundation of classical game theory in their groundbreaking work Theory of Games and Economic Behavior published in 1944. This treatise established a broad mathematical framework for reasoning about rational decision-making in a wide variety of contexts and it launched a new branch of academic study. Although there have been many significant developments in this theory, John Nash merits mention for his mathematical contributions, most notably the Nash Equilibrium Theorem. Traditionally, the classical and combinatorial sides of game theory are separated in the classroom. A strong theme of strategic thinking nonetheless connects them and we have found the combination to re- sult in a rich and engaging class. The great fun we have had teaching this broad mathematical tour through game theory undergirded our de- cision to write this book. From the very beginning of this project, our goal has been to give an honest introduction to the mathematics of game theory (both combinatorial and classical) that is accessible to an early un- dergraduate student. Over the years, we have developed an approach to teaching combinatorial game theory that avoids some of the set-theoretic complexities found in advanced treatments yet still holds true to the sub- ject. As a result, we achieve the two cornerstones of the Sprague-Grundy Theorem and the Simplicity Principle in an efficient and student-friendly Preface xiii manner. The classical game theory portion of the book contains numer- ous carefully sculpted and easy-to-follow proofs to establish the theoret- ical of the subject (including the Minimax Theorem, Nash arbitra- tion, Shapley Value, and Arrow’s Paradox). Most significantly, Chapter 9 is entirely devoted to an extremely gentle proof of Nash’s Equilibrium Theorem. For the sake of concreteness, the chapter focuses on 2 × 2 matrices, but each argument generalizes and Appendix C contains full details. Sperner’s Lemma appears in this chapter as the first step of our proof and we offer an intuitive exposition of this lemma by treating itas a game of solitaire. More broadly, Sperner’s Lemma provides a touch- stone through other chapters. In addition to using it to prove Nash’s Equilibrium Theorem, we also call on it to show that the combinatorial game Hex cannot end in a draw. Later still, Sperner’s Lemma allows us to construct an envy-free division of cake. Beyond including both combinatorial and classical theory, we have sought to provide a broad overview of (both sides of) the subject. Within the world of combinatorial game theory, we begin at a very high level of generality with game trees and Zermelo’s Theorem—concepts that ap- ply to Chess, Checkers, and many other 2-player games. We also intro- duce some widely applicable ideas such as symmetry and strategy steal- ing before specializing in normal-play games to develop the heart of the theory. On the classical side, in addition to the essential mathematical concepts, we tour a variety of exciting supplementary topics including the Folk Theorem, cake cutting, and stable marriages. Furthermore, we have devoted considerable effort to connecting the theory with applica- tions. Chapter 7 focuses on the modeling capability of a game-theoretic framework in the context of sports, biology, business, politics, and more! One of our primary goals in this book is to enhance the mathemat- ical development of our student readers. Indeed, we aim to take advan- tage of the naturally stimulating subject of game theory to teach mathe- matics. We have found that blending combinatorial and classical game theory has great pedagogical advantages. Beginning with combinatorial games means that student pairs are playing and recursively analyzing games right from the start. These games are not only fun to play, but they provide a perfect environment for working with game trees, prov- ing theorems by induction, and starting to think strategically. This part xiv Preface of the book features numerous rich examples of proofs by induction and also a of interesting proofs by contradiction. Turning to clas- sical game theory, we encounter basic probability, linear algebra, and convexity in our study of zero-sum matrix games. Our later chapters on general games continue to emphasize probability and geometric meth- ods but also introduce questions of modeling as well as plentiful appli- cations. The proof of Nash’s Equilibrium Theorem involves a nice blend of combinatorial and continuous mathematics in addition to a taste of topology. Whenever a significant new mathematical concept is required, we pause to introduce it; accordingly, this book contains elementary in- troductions to proofs by induction, proofs by contradiction, probability, and convexity. We have constructed this textbook for a one-semester undergradu- ate course aimed at students who have already taken courses in differen- tial calculus and linear algebra. However, we have found this material adaptable to a variety of situations and a range of audiences. In particu- lar, most of the book does not directly call upon either calculus or linear algebra and is thus suitable for students who lack these prerequisites but have a similar level of sophistication. Indeed, calculus is used very rarely, and for a capable student without linear algebra, only the proofs of the Minimax and Equilibrium Theorems would be out of reach after a quick introduction to matrix multiplication. The complete book is likely more material than can be comfortably covered in a standard undergraduate semester 3-credit course. To allow the instructor considerable flexibility in content choices, we have limited dependencies between the chapters (see the diagram on page xv). These limited dependencies also allow for portions of this book to be used in other contexts. For instance, the first four chapters on combinatorial games provide an appealing theme for an introductory proofs course, Chapters 5 and 6 on zero-sum matrix games together with Appendix B on linear programming make a nice addition to a linear algebra course, and all three sections in Chapter 12 can be taught independently. Further to assist the instructor, each chapter ends with a generous supply of exercises. We have sought to include problems at a variety of levels from basic skills all the way up to challenging proofs, with espe- Preface xv

12 Preferences and Society 11 n-Player Games n-Player 2-Player

9:1 10 Cooperation

Cooperative Individual 9 Nash's Equilibrium Theorem

General Zero-Sum 8 Nash Equilibrium and Applications 6:2

7 General Games

Classical Combinatorial

4 Hackenbush and 6 Von Neumann's Partizan Games Minimax Theorem

3 Impartial 5 Zero-Sum Matrix Games Games

2 Normal-Play Games

1 Combinatorial Games

Figure 0.1. Implication Diagram xvi Preface cially difficult exercises marked with the symbol *. References toexer- cises in the same chapter are by exercise number, while those to exercises in another chapter also include the chapter number. In addition, game boards and further supplementary material can be found online at www.ams.org/bookpages/stml-80. This book owes its existence to the many amazing teachers from whom we have been fortunate to learn. Matt’s genesis as a combinato- rialist is thanks to his incomparable PhD supervisor, Paul Seymour. He also benefited from an inspiring introduction to combinatorial games from John Conway and a detailed initiation to the mathematics of clas- sical game theory under the guidance of Hale Trotter. Deborah deeply appreciates her inimitable dissertation advisor, Karen Parshall, who in- troduced her to the joys and labors of academic mathematics. She also thanks Tom Archibald for his generous support of this and her other postdoctoral projects. We are so grateful to many of our friends and colleagues who have influenced the development of this book either di- rectly or indirectly: Derek Smith, Drago Bokal, Francis Su, Claude Tardif, and Dave Muraki top this list, but there are countless others. We owe a debt of gratitude to the universities that made it possible for us to teach versions of this class and to the many students who helped to shape this material with their questions, comments, and corrections. We would also like to thank Ina Mette, Arlene O’Sean, Courtney Rose, and the rest of the editorial staff at the AMS whose careful work on our manuscript dramatically improved the final product. Finally, we thank our friends and especially our families for their amazing support throughout the ex- tensive process of creating this book. Although it has taken far more effort and energy than we could ever have foreseen, writing thisbook has been a labor of love for us. We hope you will enjoy it, too! Game Boards

Chop

Chomp

These game boards are available online at www.ams.org/bookpages/stml-80.

331

Louise (bLack) to play first

Richard (gRay) to play first

3

1 2 Bibliography

1. Michael Albert, Richard Nowakowski, and David Wolfe, Lessons in play: An introduction to combinatorial game theory, A K Peters, Ltd., 2007. 2. Layman E. Allen, Games bargaining: A proposed application of the theory of games to collective bargaining, Yale Law J. 65 (1955), 660. 3. Kenneth J. Arrow, Social choice and individual values, vol. 12, Yale Univer- sity Press, 2012. 4. Robert M. Axelrod, The evolution of cooperation, Basic Books, 2006. 5. , Launching “the evolution of cooperation”, Journal of Theoretical Bi- ology 299 (2012), 21–24. 6. Emmanual N. Barron, Game theory: An introduction, 2nd ed., John Wiley & Sons, 2013. 7. Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning ways for your mathematical plays. vol. 1, 2nd ed., A K Peters, Ltd., 2001. 8. Ken Binmore, Game theory: A very short introduction, Oxford University Press, 2007. 9. , Playing for real, Oxford University Press, 2007. 10. André Bouchet, On the Sperner lemma and some colorings of graphs, J. Com- binatorial Theory Ser. B 14 (1973), 157–162. 11. Charles L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math. (2) 3 (1901/02), no. 1-4, 35–39. 12. John H. Conway, On numbers and games, 2nd ed., A K Peters, Ltd., 2001. 13. Antoine Augustin Cournot and Irving Fisher, Researches into the mathemat- ical principles of the theory of wealth, Macmillan, 1897.

335 336 Bibliography

14. George B. Dantzig, Constructive proof of the min-max theorem, Pacific J. Math. 6 (1956), no. 1, 25–33. 15. Morton D. Davis, Game theory: A nontechnical introduction, Courier Dover Publications, 2012. 16. Avinash K. Dixit, Thinking strategically: The competitive edge in business, pol- itics, and everyday life, WW Norton & Company, 1991. 17. Avinash K. Dixit, Susan Skeath, and David Reiley, Games of strategy, Norton, 1999. 18. Thomas S. Ferguson, Game theory, 2nd ed., 2014. 19. Len Fisher, Rock, paper, scissors: Game theory in everyday life, Basic Books, 2008. 20. and , The folk theorem in repeated games with discounting or with incomplete information, Econometrica 54 (1986), no. 3, 533–554. 21. Drew Fudenberg and , Game theory, MIT Press, 1991. 22. David Gale, A curious nim-type game, Amer. Math. Monthly 81 (1974), 876– 879. 23. , The game of Hex and the Brouwer fixed-point theorem, Amer. Math. Monthly 86 (1979), no. 10, 818–827. 24. David Gale and Lloyd S. Shapley, College admissions and the stability of mar- riage, Amer. Math. Monthly 120 (2013), no. 5, 386–391. 25. Rick Gillman and David Housman, Models of conflict and cooperation, Amer- ican Mathematical Society, 2009. 26. Patrick M. Grundy, Mathematics and games, Eureka 2 (1939), 6–8. 27. William D. Hamilton and , The evolution of cooperation, Sci- ence 211 (1981), no. 27, 1390–1396. 28. Michael Henle, A combinatorial introduction to topology, Dover Publications, Inc., 1994. 29. John F. Banzhaf III, Weighted voting doesn’t work: A mathematical analysis, Rutgers Law Rev. 19 (1964), 317. 30. Ehud Kalai and Meir Smorodinsky, Other solutions to Nash’s bargaining problem, Econometrica 43 (1975), 513–518. 31. Donald E. Knuth, Surreal numbers, Addison-Wesley Publishing Co., 1974. 32. Alexander Mehlmann, The game’s afoot! Game theory in myth and paradox, vol. 5, American Mathematical Society, 2000. 33. Elliott Mendelson, Introducing game theory and its applications, Chapman & Hall/CRC, 2004. 34. Peter Morris, Introduction to game theory, Springer-Verlag, 1994. Bibliography 337

35. Roger B. Myerson, Game theory, Harvard University Press, 2013. 36. John F. Nash, Jr., The bargaining problem, Econometrica 18 (1950), 155–162. 37. , Equilibrium points in 푛-person games, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 48–49. 38. , Non-cooperative games, Ann. of Math. (2) 54 (1951), 286–295. 39. John von Neumann, Zur theorie der gesellschaftsspiele, Mathematische An- nalen 100 (1928), no. 1, 295–320. 40. John von Neumann and Oskar Morgenstern, Theory of games and economic behavior, anniversary ed., Princeton University Press, 2007. 41. Martin J. Osborne and , A course in game theory, MIT Press, 1994. 42. Guillermo Owen, Game theory, 3rd ed., Academic Press, Inc., 1995. 43. Yuval Peres and Anna R. Karlin, Game theory, alive, American Mathematical Society (to appear). 44. Benjamin Polak, Econ 159: Game theory (online lectures). 45. William Poundstone, Prisoner’s dilemma, Random House LLC, 2011. 46. Anatol Rapoport, Two-person game theory, Dover Publications, Inc., 1999. 47. , 푁-person game theory, Dover Publications, Inc., 2001. 48. Jim Ratliff, A folk theorem sampler, 1996. 49. Jack Robertson and William Webb, Cake-cutting algorithms: Be fair if you can, A K Peters, Ltd., 1998. 50. Alexander Schrijver, Theory of linear and integer programming, John Wiley & Sons, Ltd., 1986. 51. Lloyd S. Shapley, A value for n-person games, Ann. Math. Stud. 28 (1953), 307–317. 52. , Game theory, Notes for Mathematics 147 at UCLA, 1991. 53. Lloyd S. Shapley and Martin Shubik, A method for evaluating the distribution of power in a committee system, American Political Science Review 48 (1954), no. 03, 787–792. 54. John M. Smith, Evolution and the theory of games, Cambridge University Press, 1982. 55. John M. Smith and George R. Price, The logic of animal conflict, Nature 246 (1973), 15. 56. William Spaniel, Game theory 101: The basics, 2011. 57. Roland P. Sprague, Über mathematische kampfspiele, Tohoku Math. J. 41 (1936), 351–354. 58. Saul Stahl, A gentle introduction to game theory, vol. 13, American Mathe- matical Society, 1999. 338 Bibliography

59. Philip D. Straffin, Game theory and strategy, New Mathematical Library, vol. 36, Mathematical Association of America, 1993. 60. Francis E. Su, Rental harmony: Sperner’s lemma in fair division, Amer. Math. Monthly 106 (1999), no. 10, 930–942. 61. Carsten Thomassen, The rendezvous number of a symmetric matrix and a compact connected metric space, Amer. Math. Monthly 107 (2000), no. 2, 163– 166. 62. Hale Trotter, Game theory, unpublished notes. 63. Philipp von Hilgers, War games: A history of war on paper, MIT Press, 2012. 64. Douglas B. West, Introduction to graph theory, vol. 2, Prentice Hall, 2001. 65. Willem A. Wythoff, A modification of the game of Nim, Nieuw Archief voor Wiskunde 7 (1905), 199–202. 66. Ernst Zermelo, Über eine anwendung der mengenlehre auf die theorie des schachspiels, Proceedings of the Fifth International Congress of Mathemati- cians 2 (1912), 501–504. Index of Games

The bold page numbers in index entries are the pages on which the term is defined.

2/3 of the Average Game, 250, 251 Heap, 61 3D Chop, 61 Hex, 2, 17, 18, 209, 210

AKQ, 157 Infinite Nim, 61 Investing Dilemma, 249 Chomp, 2, 17, 55 Chop, 1, 15, 54, 56, 57 , 23 Coin Jack, 157 Coin Poker, 149, 150, 151 Moving Knife, 278 Coin Toss, 147 Newcomb’s Paradox, 160 Colonel Blotto, 104 Nim, 45, 46–48, 50, 51 Common Side-Blotched Lizard, 172, 176 Odd-Nim, 60 Competing Pubs, 142 Odd-Person-Out, 271 , 140, 173, 174, 176 Cut-Cake, 26, 81 Pascal’s Wager, 160 Pick-Up-Bricks, 1, 15, 16, 56 Dating Dilemma, 140, 145, 205, 218, Prisoner’s Dilemma, 134, 145, 229, 246 220 Probabilistic Repeated Prisoner’s Divide the Dollar, 256, 257–260 Dilemma, 229, 230–232, 234, 236 , 34, 81, 83 Push, 84

Empty and Divide, 59, 61 Rock-Paper-Scissors, 89, 90, 173 Euclid’s Game, 44 -Dynamite, 129 Even-Nim, 60 -Lizard-Spock, 106 -Superman-Kryptonite, 129 General Volunteering Dilemma, 248 Weighted, 129

Hackenbush, 63, 64, 65–71, 74 S-Pick-Up-Bricks, 60, 62 Hawk vs. Dove, 170, 171–173, 176 SOS, 24

339 340 Index of Games

Split and Choose, 281 , 141, 145, 218, 220, 249

Tic, 4 , 246 Triangle Solitaire, 188 Turning Turtles, 61 Two-Finger Morra, 90, 106

Volunteering Dilemma, 141, 145, 218, 220, 249 Voting Scenario, 276

Wythoff’s King, 60 Wythoff’s Queen, 62 Index

The bold page numbers in index entries are the pages on which the term is defined. arbitration scheme, 223 depth, 12, 12, 53 egalitarian, 240 domination, 92, 93, 142, 143, 249–250 Kalai-Smorodinsky, 240 iterated removal, 92–95, 115, 143, Nash, 224, 224, 226, 228 168, 250 Arrow’s axioms, 292–293 S-domination, 259 Arrow’s Impossibility Theorem, 292, strict, 92, 142, 249 290–298 dot product, 123, 124 Axelrod’s Olympiad, 236–237 dyadic number, 71, 72–74 balanced Nim position, 48, 49–50 envy-free division, 280 best pure response, 143, 163–167 equating results, 113, 114–115, 167, 168 , 163–167 equitable division, 278 binary expansion, 46, 47, 72, 72 equivalence Brouwer’s Fixed Point Theorem, 193, class, 43 196, 197, 325–327 coalitional game, 272 matrix game, 159 chance node, 146, 147 position, 36, 37–41 closure, 73, 86 relation, 37, 43 coalition, 251–256, 258–260 topological, 199, 200 coalitional form, 253, 255 evolutionary stability, 173, 175, 176 coalitional game, 255, 256, 258–260 expected payoff, 98, 99, 135–137, 163 combinatorial game, 3, 4–6, 9, 26, 301 expected value, 97, 98 contradiction, proof by, 16 extensive form, 151, 154 convex hull, 124, 125, 217 convex set, 124, 125 Fibonacci Sequence, 23 core, 259, 260 Fisher’s Principle, 184 Cournot Duopoly, 176, 178–181 fixed point, 192, 193 fixed point property, 193, 194–197, 199, demand curve, 177 201, 202, 207, 326–328

341 342 Index

Folk Theorem, 229, 234, 234–236 payoff, 90, 134, 135–137, 139 payoff matrix, 162, 163, 215–216 Gale-Shapley algorithm, 287, 287–289 payoff polygon, 216–217, 218, 219, 221, game tree, 3–9, 12, 14, 145–154 223 golden ratio, 23, 44, 62 position, 3, 27, 147–150, 301–308 guarantee, 100, 100–103 balanced (Nim), 48, 49–50 guarantee function, 119, 120–122 dyadic, 75, 75, 77–79, 81 equivalence, 36, 37–41 hyperplane, 123, 124, 131, 318 fractional, 69–71 , 26, 45–58 integral, 66, 67–68 imputation, 257, 258, 260, 261 negation, 65 induction, proof by, 10–12 sums of, 31 information set, 148, 148–149, 152, 154 terminal, 3 instant runoff, 276 type of, 28, 29–30 iterated removal of dominated probability space, 96, 96, 97, 99 strategies, 92–95, 115, 143, 168, 250 random variable, 96 linear programming, 309–313 S-domination, 259 saddle point, 94, 95, 104 matrix game, 134, 139–144 security level, 214, 215, 216, 219 Matrix-to-Tree Procedure, 153–154 Shapley Value, 261, 261–267 MEX (minimal excluded value), 52 Shapley’s axioms, 263 MEX Principle, 53, 54, 56, 82 Shapley-Shubik Index, 269, 269 Minimax Theorem, 313 simplex, 191, 192, 197, 325 mixed outcome, 214, 217–219 Simplicity Principle, 77, 78, 79, 80, 82 move rule, 3 , 258 movement diagram, 144–145 solution point, 223 Sperner’s Lemma, 189–191, 192, Nash arbitration, 224, 224, 226, 228 324–325 Nash equilibrium, 166, 167–169, Sprague-Grundy Theorem, 52–53, 53 187–188 stable set, 260, 260 Nash Equilibrium Theorem, 166, 167, status quo point, 221, 223 187–207, 246 , 108, 108–109 Nash flow, 202, 204, 205, 328–330 strategic form, 151–153 Nash’s axioms, 225 strategy negative of a Hackenbush position, 65 Alternating Trigger, 241 negotiation set, 219, 220 dominated, 92, 93, 142, 143, 249–250 Nim-sum, 50 drawing, 7, 12 nimber, 50, 51–53 evolutionarily stable, 175, 176 node, 4, 5, 7, 145, 146, 149 , 231, 232–233 normal form, 151 mixed, 98, 99, 101, 102, 162, 163–166 normal-play game, 3, 25–27 pure, 91, 98, 142, 231 outcome, 3, 4–5, 89, 96, 97, 99, 133–139, strictly dominated, 92, 142, 249 213–215, 217–220, 245 Tit-for-Tat, 237 winning, 7, 12 partition, 43, 44 strategy space, 198, 199–202, 204–207, partizan game, 26, 63–83 327–329 Index 343 strategy stealing, 16, 18 sum of positions, 31 surreal numbers, 307–308 symmetric Nash equilibrium, 172, 173, 175 symmetry, 15, 16, 65, 172, 225, 264

Tree-to-Matrix Procedure, 151, 152 type of a position, 28, 29–30 utility, 135–139 utility function, 278 valuation scheme, 261 value, 102 von Neumann and Morgenstern’s Lottery, 138–139 von Neumann Minimax Theorem, 102, 102, 111, 123–128 von Neumann solution, 102, 103, 113–115, 118–122 voting game, 268

W-L-D game tree, 5, 6–9, 12–14 win rule, 3 winning move, 56, 58, 84

Zermelo’s Theorem, 9, 12, 14, 28 zero-sum matrix game, 89, 90, 99–102, 111–112, 134 This book offers a gentle introduction to the mathematics of both sides of game theory: combinatorial and classical. The combination allows for a dynamic and rich tour of the subject united by a common theme of strategic reasoning.

The first four chapters develop combi- Photo courtesy of Radina Droumeva natorial game theory, beginning with an introduction to game trees and mathematical induction, then investigating the games of Nim and Hackenbush. The analysis of these games concludes with the cornerstones of the Sprague- Grundy Theorem and the Simplicity Principle. The last eight chapters of the book offer a scenic journey through the math- ematical highlights of classical game theory. This contains a thorough treatment of zero-sum games and the von Neumann Minimax Theorem, as well as a student-friendly development and proof of the Nash Equilibrium Theorem. The Folk Theorem, Arrow’s voting paradox, evolutionary biology, cake cutting, and other engaging auxiliary topics also appear. The book is designed as a textbook for an undergraduate mathematics class. With ample material and limited dependencies between the chapters, the book is adaptable to a variety of situations and a range of audiences. Instructors, students, and independent readers alike will appreciate the flexibility in content choices as well as the generous sets of exercises at various levels.

For additional information and updates on this book, visit www.ams.org/bookpages/stml-80

AMS on the Web STML/80 www.ams.org