Game Theory: A Modeling Approach (Textbooks in Mathematics)

343 Pages·2019·3.995 MB·English

by  Richard Alan Gillman, David Housman

Game Theory
A Modeling Approach
Textbooks in Mathematics
Series editors:
Al Boggess and Ken Rosen
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COMPLEX VARIABLES: A PHYSICAL APPROACH WITH APPLICATIONS, SECOND EDITION
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GAME THEORY: A MODELING APPROACH
Richard Alan Gillman and David Housman
https://www.crcpress.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
Game Theory
A Modeling Approach
Richard Alan Gillman
Valparaiso University, Valparaiso, Indiana
David Housman
Goshen College, Goshen, Indiana
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Library of Congress Cataloging-in-Publication Data
Names:Gillman,RichardAlan,author.|Housman,David,1957-author.
Title:Gametheory:amodelingapproach/RichardAlanGillmanandDavidHousman.
Description:Boca Raton : CRC Press,Taylor & Francis Group, 2019. |Series:Textbooks
inmathematics|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019000626|ISBN9781482248098(hardback:alk.paper)|
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Subjects:LCSH:Gametheory.|Decisionmaking--Mathematicalmodels.
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Contents
Preface vii
Acknowledgments xi
Chapter
1(cid:4) Models and Games 1
1.1 INTRODUCTIONTOMODELING 1
1.2 INTRODUCTIONTOGAMETHEORY 2
1.3 EXAMPLESOFGAMES 4
1.4 RATIONALITYASSUMPTION 7
Chapter
2(cid:4) Player Preferences 13
2.1 ORDINALUTILITIES 13
2.2 VONNEUMANN MORGENSTERNUTILITIES 17
2.3 CONSTRUCTINGUTILITIES 23
2.4 DETERMININGRISK 29
Chapter
3(cid:4) Simultaneous Play 39
3.1 STRATEGICGAMES 40
3.2 FINALJEOPARDY 48
3.3 MIXEDSTRATEGIES 52
3.4 NON MATRIXMODELS 61
3.5 LIMITATIONS 69
Chapter
4(cid:4) Bilateral Agreements 81
4.1 NEGOTIATIONS 81
4.2 BARGAININGINSTRATEGICGAMES 88
4.3 FAIRNESSPROPERTIES 92
Chapter
5(cid:4) Sequential Play 103
5.1 SEQUENTIALGAMES 104
5.2 SUBGAMEPERFECTEQUILIBRIA 112
5.3 COMBINATORIALGAMES 120
5.4 MORALHAZARD 133
v
vi (cid:4) Contents
Chapter
6(cid:4) Missing Information 145
6.1 IMPERFECTINFORMATION 146
6.2 ROMANSANDGERMANS 159
6.3 INCOMPLETEINFORMATION 163
6.4 BARGAININGGAMESWITHPRIVATEINFORMATION 170
6.5 INTERNATIONALCOLLABORATION 175
6.6 AUCTIONS 187
Chapter
7(cid:4) Repetitious Play 205
7.1 REPEATEDSOCIALDILEMMAS 206
7.2 MODELINGNOISE 215
7.3 EVOLUTIONARYGAMETHEORY 218
Chapter
8(cid:4) Multilateral Agreements 239
8.1 SHAPLEYARGUMENT 240
8.2 NUCLEOLUSARGUMENT 247
8.3 BARGAININGARGUMENT 252
8.4 VOTINGPOWER 257
Chapter
9(cid:4) Resource Allocation 275
9.1 RESOURCEALLOCATIONPROBLEM 275
9.2 BARGAININGANDCOALITIONMODELS 280
9.3 COMPLETEANDINCOMPLETEINFORMATIONMODELS 284
9.4 CONCLUDINGREMARKS 291
Appendix
A(cid:4) Selected Answers 295
Bibliography 317
Index 325
Preface
Game theory is a fascinating field. It is deeply mathematical, yet frequently studied by
people trained in other disciplines. These include economists, biologists, political scientists,
psychologists, and many others. It is accessible and popular among the general public,
wherezero-sumgamesandprisoner’sdilemmaareoftenusedascommon,everydayphrases.
Nonetheless it relies on deep mathematical arguments for its justification. The widespread
interestinthisfieldnodoubtstemsfromitsabilitytohelpusunderstandhumaninteractions
in a manner grounded in mathematical logic and truth.
This volume is intended to fill the gap between the popular treatments of game theory
and highly technical treatments of the subject. It is broad and introductory, but set at
a level that will engage and challenge advanced undergraduate mathematics majors and
graduate students in professional master’s programs. Such students are capable of under-
standing the mathematical underpinning of game theory but do not necessarily need a full
development of the underlying mathematical theory. Often, their primary interest will be
findingapplicationsofgametheorytoreal-worldproblemsthattheywillencounterintheir
work.
Hence this volume is inherently about modeling. It is, however, not quite a traditional
modeling text. Because students typically only take one course in game theory, we must
introduce the appropriate game theoretic concepts and tools as we engage in the modeling
process. This is very different from a canonical mathematical modeling text in which the
tools—calculus and differential equations—have been introduced and mastered in multiple
prior courses.
Unlike many game theory textbooks that treat the real world only in the abstract, we
take care to match conclusions to the real-world scenarios we set out to model. Through-
out the book, scenarios are described inside boxes. These scenarios—real or fictitious but
plausible— represent situations in which someone would want to make a strategic decision.
They form the basis of the modeling process developed in the text. Chapter 1 contains a
brief introduction to modeling that can be used as a review for students who have already
had a modeling course or as a beginning point for new modelers. Figure 1 summarizes the
prerequisite relationships among the chapters.
Chapter 1 also introduces the fundamental definition of a mathematical game and pro-
videsseveralbasicexamples.Itcloseswithabriefdiscussionofthelimitationsandassump-
tions of our approach, specifically the assumption of rationality that game theorists make
aboutgameparticipants.Thisassumptionismaintainedthroughoutthebook;however,we
interweave challenges to the rationality assumption which guide readers in understanding
both the limits and possibilities of game theoretic modeling.
Chapter 2 lays the foundation for mathematical game theory by introducing and devel-
oping the idea of a utility function. These functions transform vague personal preferences
intoquantifiableandmanipulableobjects.Understandingthisprocessisessentialtounder-
standing the applicability of game theory. The key theorems are developed in detail and
examples of many types of utility functions are provided. In particular, both discrete and
continuous utility functions are considered. This is a key step in moving readers beyond
vii
viii (cid:4) Preface
FIGURE1 Chapter Prerequisite Relationships.
popular treatments of game theory which restrict themselves to finite cases. It is a dual
perspective that is maintained throughout the volume.
Chapter 3 introduces strategic games, the simplest and most widely recognized game
theoretic model. Here you will find Prisoner’s Dilemma, Chicken, and Battle of the Sexes.
We also devote an entire section to analyzing a model for Final Jeopardy! which demon-
strates the difference between the colloquial and mathematical understanding of Prisoner’s
Dilemma.
Chapters4and5treatindependenttopics:bargainingandsequentialplay.Bothofthese
require the foundation laid in Chapters 1–3. The later chapters build on sophistication of
both topic and presentation. Chapter 6 relies on the study of sequential games in Chapter
5 to model missing information. Chapter 7, which covers repeated play, relies on the topics
of strategic games covered in Chapter 3. Chapter 8 models multilateral agreements, which
references material covered in Chapter 4.
Chapter9istrulyaculminatingchapter,drawingonthemultipleapproachespresented
in the book to study the problem of resource allocation. Readers should finish all Chapters
1–8 in order to fully engage with the material in Chapter 9.
Each of the chapters begins with sections which provide definitions, explain important
concepts, and provide illustrative examples. Sections 3.2, 5.3, 5.4, 6.2, 6.5, 6.6, and 8.4 are
each devoted to a single scenario, providing an in-depth modeling example to illustrate key
Preface (cid:2) ix
concepts. These could be included in core course material, used as independent studies or
projects,oromitted.Wealsoprovidemanycitationsthroughoutthetexttosuggestfurther
investigation. Each chapter ends with exercises that follow the order of the sections in the
text. Asterisks on exercises indicate problems that either move beyond the content of this
book or could serve as a student project. Select answers and solutions are included at the
back of the book, and an instructor’s solution manual is available from the authors.
In our experience, a one-semester course for advanced undergraduate mathematics ma-
jors can expect to cover about the first six chapters of this book, which has been designed
to be used in a classroom engaged in active learning, incorporating in-class discussions,
interactive simulations, and group work. Specifically, we anticipate that courses will end
withprojectsinwhichstudentsresearchanewscenarioofinteresttothem,andthenbuild,
analyze, and assess game theoretic models for their scenario. Examples from our previous
courseshaveincludedinvestigationsofpricingdecisionsinthevideogameconsolebusiness,
interactions between players and teams in crucial moments of basketball games, and nego-
tiations between governments and protestors. The material in Chapters 7, 8, and 9 can be
usedasstartingpointsforstudentprojectsthatextendthematerialcoveredintheprevious
chapters.
About the Cover Images
A multiple stage bicycle race can be represented by a variety of game theory models.
Identifying the individual cyclists as the players seems natural, but aggregating the cy-
clists on a team as a single player may make the resulting model easier to analyze without
a significant loss of fidelity. Although there may be several teams, a model with only two
teams could capture the essence needed given the goals of an analysis. The final place-
ments provide a reasonable first estimate of player preferences, but a mixture of personal,
team,andsportsculturalmotivationscanrefineandaddintensitiestotheutilitiesassigned.
Considerationofoverallapproachestointer-teamcompetitionsuggestsaone-shotstrategic
game or a Bayesian game if differences in information are relevant, while intra-team plan-
ning and collaboration suggest a bargaining or coalition game, and day-by-day decisions
with the mutual benefits of teams suggest a sequential or repeated game.