Table Of ContentModels of Conflict
and Cooperation
http://dx.doi.org/10.1090/mbk/065
Models of Conflict
and Cooperation
Rick Gillman
David Housman
American Mathematical Society
Providence, RI
2000 Mathematics Subject Classification. Primary 00A06,
91–01, 91A10, 91A12, 91A40, 91B08.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mbk-65
Library of Congress Cataloging-in-Publication Data
Gillman,Rick.
Modelsofconflictandcooperation/RickGillmanandDavidHousman.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4872-2(alk.paper)
1. Game theory. 2. Decision making—Mathematical models. I. Housman, David, 1957–
II.Title.
QA269.G487 2009
519.3—dc22
2009014906
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
(cid:1)c 2009bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 141312111009
Contents
Preface vii
Chapter 1. Deterministic Games 1
1. A Very Simple Game 1
2. Rules of the Game 6
3. Heuristics and Strategies 17
4. Game Trees 26
5. A Solution for Nim 40
6. Theorems of Zermelo and Nash 49
Chapter 2. Player Preferences 55
1. Measurement 55
2. Ordinal Preferences 67
3. Cardinal Preferences 76
4. Ratio Scale Preferences 91
Chapter 3. Strategic Games 101
1. Tosca 101
2. Fingers and Matches 107
3. Four Solution Concepts 116
4. Once Again, Game Trees 130
5. Trees and Matrices 140
Chapter 4. Probabilistic Strategies 153
1. It’s Child’s Play 153
2. Mixed Strategy Solutions 160
3. Finding Solutions in 2×2 Games 176
4. Nash Equilibria in m×2 Games 189
5. Zero-Sum Games 196
Chapter 5. Strategic Game Cooperation 203
1. Experiments 203
2. The Prisoners’ Dilemma 217
3. Resolving the Prisoners’ Dilemma 224
Chapter 6. Negotiation and Arbitration 235
1. A Simple Negotiation 235
2. Bargaining Games 241
3. The Egalitarian Method 255
4. The Raiffa Method 269
5. The Nash Method 278
v
vi CONTENTS
Chapter 7. Coalition Games 287
1. A Savings Allocation Problem 287
2. Two Properties and Five Methods 293
3. The Shapley Method 305
4. The Nucleolus Method 316
5. You Can’t Always Get What You Want 329
Chapter 8. Fair Division 335
1. An Inheritance Problem 335
2. Fair Division Games and Methods 340
3. Fairness Properties 354
4. Choosing a Fair Method 361
Epilogue 371
Answers to Selected Exercises 377
Deterministic Games 377
Player Preferences 380
Strategic Games 381
Probabilistic Strategies 388
Strategic Game Cooperation 392
Negotiation and Arbitration 394
Coalition Games 397
Fair Division 401
Bibliography 409
Index 413
Preface
Decisions. We make them every day. They are made at the personal level when
resolving family tensions, deciding where to take family vacations, choosing a col-
lege, planning for retirement, or partitioning an inheritance. We make decisions at
aprofessional levelas wemake business decisions aboutproductstomarket, where
tolocateabusiness, whentotakeonapartner,orwheretoinvestscarceresources.
We make decisions on a sociopolitical level as well, as we work to resolve environ-
mentalandsocialproblemsininternationalpolitics, andmakedecisionsabouthow
we should be represented in government.
Almost all of these many decisions are made in situations where the resulting out-
come also depends on the decisions made by other people. For example, your
decisionabout where togotocollege isinfluenced by the decisions of your parents,
of financial aid counselors, and of your friends.
Game theory, defined in its broadest sense, is the mathematical field that studies
these types of decision making problems. Specifically, it asks the question: How
does a player maximize his or her return in a situation where the outcome is at
least partially determined by other, independently acting players? Game theory
was developed during the 1930s and 1940s in the context of the study of economic
behavior, but has since expanded in scope to include many social and political
contexts. In recent years, it has even been used by biologists to better understand
how behaviors evolve.
The process of representing a decision, behavior, or any other real-world phenome-
noninmathematicaltermsiscalledmathematicalmodeling. Theessentialfeatures
of this process are making explicit assumptions to create the model, using math-
ematical techniques to arrive at results, interpreting those results, and verifying
their reasonableness in the original context. A primary goal of this book is to help
students to learn, practice, and use this process.
To the Student
Thestudyofmathematicsisaparticipatoryprocess,requiringfullengagementwith
the material in order to learn it. READ WITH ANTICIPATION. As you read a
sentence, ask yourself: What does this tell me? Where do I think that this idea
leads me? How does it connect with previous material? Notice that we used
theword“sentence”ratherthan“paragraph”,“page”,or“chapter”intheprevious
sentence. This is because reading mathematics, unlike reading literature, is a
patient process. Individual sentences are meant to convey significant information
vii
viii PREFACE
and frequently need to be read two or three times for complete comprehension.
Furthermore, DO THE EXERCISES. Mathematical learning requires you to pick
up pencil and paper to solve problems.
As a game theory text, this book offers additional, less traditional, opportunities
to engage with the material. First among these is the opportunity to PLAY THE
GAMES described in almost all of the sections. In several sections, there are
activitiesthat encourage you to dothis. Find some friends with whom toplay the
games; you will find that the games are fun and thought provoking to play. As
you play a game, reflect, both personally and with the other players, on the “best”
ways to play them. Identify your goals and how you would attempt to achieve
these goals.
Each chapter opens with a “dialogue”, an idealized conversation between two or
more individuals about the topic of the chapter. Together with your friends, ACT
OUT THE DIALOGUES. On the surface, these dialogues are merely a somewhat
corny way to introduce the topic at hand, but at a deeper level they model the
quantitatively literate conversations we hope that you will have in your life.
A Brief Description of the Chapters
Each sectionincludes a succinct statement of the learning objectives and exercises.
Mostexercisesaredirectlylinkedtothelearningobjectivesbutsome provideextra
challenge. Following the dialogue section of each chapter, there are two or three
sectionsthatcontainthecoreideasofthechapter. Thesesectionscarefullylayout
thecriticalconcepts,methodsandalgebraictoolsdevelopingtheseideas. Thefinal
section of each chapter is usually an extension of the topics that could be omitted
in a first reading.
Chapters 1 and 2 are key to the entire book. Chapter 1 introduces the reader to
deterministic games and uses them as a means to ease the reader into quantita-
tive thinking. Readers are introduced to the distinct concepts of heuristics and
strategies in this chapter and are encouraged to use them to find ways of winning
several games, including the well-known game of Nim. They are also introduced
to the idea of a game tree and its application to identifying winning strategies for
a game. Chapter 2 develops the critical idea of player preferences. Before an
individual can make informed decisions about their actions in situations involving
conflict or cooperation, he or she must be able todetermine preferences among the
possible outcomes for all decision making. These preferences can be quantified
in a variety of ways and provide the numerical basis for much of the work in the
following chapters. This quantification of preferences is our first example of the
mathematical modeling process, and the diagram depicting this process is referred
to throughout the rest of the book.
Chapters3,4,and5areabasicstudyofproblemsofconflict. Chapter3introduces
the reader to strategic games and provides many opportunities for the reader to
developmodelsofsimplereal-worldscenarios. Thischapteralsointroducesvarious
solution concepts for strategic games, including the Nash equilibrium. Chapter 4
extends this work by developing the algebraic background necessary to find Nash
