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Preface15Illustrations) and 6 (Extensive Games with Perfect Information: Illustrations). You may add to this plan any combination of Chapters 4 (Mixed StrategyEquilibrium), 9 (Bayesian Games, except Section 9.8), 7 (Extensive Gameswith Perfect Information: Extensions and Discussion), 8 (Coalitional Gamesand the Core), and 16 (Bargaining). If you read Chapter 4 (Mixed Strategy Equilibrium) then you may in additionstudy any combination of the remaining chapters covering strategic games,and if you study Chapter 7 (Extensive Games with Perfect Information: Extensions and Discussion) then you are ready to tackle Chapters 14 and 15(Repeated Games).All the material should be accessible to undergraduate students. A one-semestercourse for third or fourth year North American economics majors (who have beenexposed to a few of the main ideas in first and second year courses) could coverup to about half the material in the book in moderate detail.Personal pronounsThe lack of a sex-neutral third person singular pronoun in English has led manywriters of formal English to use “he” for this purpose. Such usage conflicts withthat of everyday speech. People may say “when an airplane pilot is working, heneeds to concentrate”, but they do not usually say “when a flight attendant isworking, he needs to concentrate” or “when a secretary is working, he needs toconcetrate”. The use of “he” only for roles in which men traditionally predominate in Western societies suggests that women may not play such roles; I find thisinsinuation unacceptable.To quote the New Oxford Dictionary of English, “[the use of he to refer to refer toa person of unspecified sex] has become . . . a hallmark of old-fashioned languageor sexism in language.” Writers have become sensitive to this issue in the last halfcentury, but the lack of a sex-neutral pronoun “has been felt since at least as farback as Middle English” (Webster’s Dictionary of English Usage, Merriam-WebsterInc., 1989, p. 499). A common solution has been to use “they”, a usage that theNew Oxford Dictionary of English endorses (and employs). This solution can createambiguity when the pronoun follows references to more than one person; it alsodoes not always sound natural. I choose a different solution: I use “she” exclusively. Obviously this usage, like that of “he”, is not sex-neutral; but its use maydo something to counterbalance the widespread use of “he”, and does not seemlikely to do any harm.AcknowledgementsI owe a huge debt to Ariel Rubinstein. I have learned, and continue to learn, vastlyfrom him about game theory. His influence on this book will be clear to anyone

16Prefacefamiliar with our jointly-authored book A course in game theory. Had we not writtenthat book and our previous book Bargaining and markets, I doubt that I would haveembarked on this project.Discussions over the years with Jean-Pierre Benoı̂t, Vijay Krishna, Michael Peters, and Carolyn Pitchik have improved my understanding of many game theoretic topics.Many people have generously commented on all or parts of drafts of the book.I am particularly grateful to Jeffrey Banks, Nikolaos Benos, Ted Bergstrom, TilmanBörgers, Randy Calvert, Vu Cao, Rachel Croson, Eddie Dekel, Marina De Vos, Laurie Duke, Patrick Elias, Mukesh Eswaran, Xinhua Gu, Costas Halatsis, Joe Harrington, Hiroyuki Kawakatsu, Lewis Kornhauser, Jack Leach, Simon Link, BartLipman, Kin Chung Lo, Massimo Marinacci, Peter McCabe, Barry O’Neill, RobinG. Osborne, Marco Ottaviani, Marie Rekkas, Bob Rosenthal, Al Roth, MatthewShum, Giora Slutzki, Michael Smart, Nick Vriend, and Chuck Wilson.I thank also the anonymous reviewers consulted by Oxford University Pressand several other presses; the suggestions in their reviews greatly improved thebook.The book has its origins in a course I taught at Columbia University in the early1980s. My experience in that course, and in courses at McMaster University, whereI taught from early drafts, and at the University of Toronto, brought the book toits current form. The Kyoto Institute of Economic Research at Kyoto Universityprovided me with a splendid environment in which to work on the book duringtwo months in 1999.ReferencesThe “Notes” section at the end of each chapter attempts to assign credit for theideas in the chapter. Several cases present difficulties. In some cases, ideas evolvedover a long period of time, with contributions by many people, making their origins hard to summarize in a sentence or two. In a few cases, my research has ledto a conclusion about the origins of an idea different from the standard one. In allcases, I cite the relevant papers without regard to their difficulty.Over the years, I have taken exercises from many sources. I have attemptedto remember where I got them from, and have given credit, but I have probablymissed some.Examples addressing economic, political, and biological issuesThe following tables list examples that address economic, political, and biologicalissues. [SO FAR CHECKED ONLY THROUGH CHAPTER 7.]Games related to economic issues (THROUGH CHAPTER 7)

Preface17Exercise 31.1,Section 2.8.4,Exercise 42.1Provision of a public goodSection 2.9.4Collective decision-makingSection 3.1,Exercise 133.1Cournot’s model of oligopolySection 3.1.5Common propertySection 3.2,Exercise 133.2,Exercise 143.2,Exercise 189.1,Exercise 210.1Bertrand’s model of oligopolyExercise 75.1Competition in product characteristicsSection 3.5Auctions with perfect informationSection 3.6Accident lawSection 4.6Expert diagnosisExercise 125.2,Exercise 208.1Price competition between sellersSection 4.8Reporting a crime (private provision of a public good)Example 141.1All-pay auction with perfect informationExercise 172.2Entry into an industry by a financially-constrained challengerExercise 175.1The “rotten kid theorem”Section 6.2.2The holdup gameSection 6.3Stackelberg’s model of duopolyExercise 207.2A market gameSection 7.2Entry into a monopolized industrySection 7.5Exit from a declining industryExample 227.1Chain-store gameGames related to political issues (THROUGH CHAPTER 7)Exercise 32.2Voter participationSection 2.9.3VotingExercise 47.3Approval votingSection 2.9.4Collective decision-makingSection 3.3,Exercise 193.3,Exercise 193.4,Section 7.3Hotelling’s model of electoral competition

18PrefaceExercise 73.1Electoral competition between policy-motivated candidatesExercise 73.2Electoral competition between citizen-candidatesExercise 88.3Lobbying as an auctionExercise 115.3Voter participationExercise 139.1Allocating resources in election campaignsSection 6.4Buying votes in a legislatureSection 7.4Committee decision-makingExercise 224.1Cohesion of governing coalitionsGames related to biological issues (THROUGH CHAPTER 7)Exercise 16.1Hermaphroditic fishSection 3.4War of attritionTypographic conventions, numbering, and nomenclatureIn formal definitions, the terms being defined are set in boldface. Terms are set initalics when they are defined informally.Definitions, propositions, examples, and exercises are numbered according tothe page on which they appear. If the first such object on page z is an exercise, forexample, it is called Exercise z.1; if the next object on that page is a definition, it iscalled Definition z.2. For example, the definition of a strategic game with ordinalpreferences on page 11 is Definition 11.1. This scheme allows numbered items tofound rapidly, and also facilitates precise index entries.Symbol/termMeaning?Exercise?Hard exercise DefinitionPropositionExample: a game that illustrates a game-theoretic pointIllustrationA game, or family of games, that shows how the theory can illuminate observed phenomenaI maintain a website for the book. The current URL ishttp://www.economics.utoronto.ca/osborne/igt/.

Draft chapter from An introduction to game theory by Martin J. OsborneOsborne@chass.utoronto.ca; www.economics.utoronto.ca/osborneVersion: 00/11/6.c 1995–2000 by Martin J. Osborne. All rights reserved. No part of this book may be reCopyright produced by any electronic or mechanical means (including photocopying, recording, or informationstorage and retrieval) without permission in writing from Oxford University Press, except that onecopy of up to six chapters may be made by any individual for private study.1IntroductionWhat is game theory? 1The theory of rational choice1.1G4What is game theory?AME THEORY aims to help us understand situations in which decision-makersinteract. A game in the everyday sense—“a competitive activity . . . in whichplayers contend with each other according to a set of rules”, in the words of mydictionary—is an example of such a situation, but the scope of game theory isvastly larger. Indeed, I devote very little space to games in the everyday sense;my main focus is the use of game theory to illuminate economic, political, andbiological phenomena.A list of some of the applications I discuss will give you an idea of the rangeof situations to which game theory can be applied: firms competing for business,political candidates competing for votes, jury members deciding on a verdict, animals fighting over prey, bidders competing in an auction, the evolution of siblings’behavior towards each other, competing experts’ incentives to provide correct diagnoses, legislators’ voting behavior under pressure from interest groups, and therole of threats and punishment in long-term relationships.Like other sciences, game theory consists of a collection of models. A modelis an abstraction we use to understand our observations and experiences. What“understanding” entails is not clear-cut. Partly, at least, it entails our perceivingrelationships between situations, isolating principles that apply to a range of problems, so that we can fit into our thinking new situations that we encounter. Forexample, we may fit our observation of the path taken by a lobbed tennis ball intoa model that assumes the ball moves forward at a constant velocity and is pulledtowards the ground by the constant force of “gravity”. This model enhances ourunderstanding because it fits well no matter how hard or in which direction theball is hit, and applies also to the paths taken by baseballs, cricket balls, and awide variety of other missiles, launched in any direction.A model is unlikely to help us understand a phenomenon if its assumptions arewildly at odds with our observations. At the same time, a model derives powerfrom its simplicity; the assumptions upon which it rests should capture the essence1

2Chapter 1. Introductionof the situation, not irrelevant details. For example, when considering the pathtaken by a lobbed tennis ball we should ignore the dependence of the force ofgravity on the distance of the ball from the surface of the earth.Models cannot be judged by an absolute criterion: they are neither “right” nor“wrong”. Whether a model is useful or not depends, in part, on the purpose forwhich we use it. For example, when I determine the shortest route from Florenceto Venice, I do not worry about the projection of the map I am using; I work underthe assumption that the earth is flat. When I determine the shortest route fromBeijing to Havana, however, I pay close attention to the projection—I assume thatthe earth is spherical. And were I to climb the Matterhorn I would assume that theearth is neither flat nor spherical!One reason for improving our understanding of the world is to enhance ourability to mold it to our desires. The understanding that game theoretic modelsgive is particularly relevant in the social, political, and economic arenas. Studyinggame theoretic models (or other models that apply to human interaction) may alsosuggest ways in which our behavior may be modified to improve our own welfare.By analyzing the incentives faced by negotiators locked in battle, for example, wemay see the advantages and disadvantages of various strategies.The models of game theory are precise expressions of ideas that can be presented verbally. However, verbal descriptions tend to be long and imprecise; inthe interest of conciseness and precision, I frequently use mathematical symbolswhen describing models. Although I use the language of mathematics, I use fewof its concepts; the ones I use are described in Chapter 17. My aim is to take advantage of the precision and conciseness of a mathematical formulation withoutlosing sight of the underlying ideas.Game-theoretic modeling starts with an idea related to some aspect of the interaction of decision-makers. We express this idea precisely in a model, incorporatingfeatures of the situation that appear to be relevant. This step is an art. We wish toput enough ingredients into the model to obtain nontrivial insights, but not somany that we are lead into irrelevant complications; we wish to lay bare the underlying structure of the situation as opposed to describe its every detail. The nextstep is to analyze the model—to discover its implications. At this stage we need toadhere to the rigors of logic; we must not introduce extraneous considerations absent from the model. Our analysis may yield results that confirm our idea, or thatsuggest it is wrong. If it is wrong, the analysis should help us to understand whyit is wrong. We may see that an assumption is inappropriate, or that an importantelement is missing from the model; we may conclude that our idea is invalid, orthat we need to investigate it further by studying a different model. Thus, the interaction between our ideas and models designed to shed light on them runs intwo directions: the implications of models help us determine whether our ideasmake sense, and these ideas, in the light of the implications of the models, mayshow us how the assumptions of our models are inappropriate. In either case, theprocess of formulating and analyzing a model should improve our understandingof the situation we are considering.

1.1 What is game theory?3A N OUTLINE OF THE HISTORY OF GAME THEORYSome game-theoretic ideas can be traced to the 18th century, but the major development of the theory began in the 1920s with the work of the mathematicianEmile Borel (1871–1956) and the polymath John von Neumann (1903–57). A decisive event in the development of the theory was the publication in 1944 of thebook Theory of games and economic behavior by von Neumann and Oskar Morgenstern. In the 1950s game-theoretic models began to be used in economic theoryand political science, and psychologists began studying how human subjects behave in experimental games. In the 1970s game theory was first used as a tool inevolutionary biology. Subsequently, game theoretic methods have come to dominate microeconomic theory and are used also in many other fields of economicsand a wide range of other social and behavioral sciences. The 1994 Nobel prize ineconomics was awarded to the game theorists John C. Harsanyi (1920–2000), JohnF. Nash (1928–), and Reinhard Selten (1930–).J OHN VON N EUMANNJohn von Neumann, the most important figure in the early development of gametheory, was born in Budapest, Hungary, in 1903. He displayed exceptional mathematical ability as a child (he had mastered calculus by the age of 8), but his father, concerned about his son’s financial prospects, did not want him to become amathematician. As a compromise he enrolled in mathematics at the University ofBudapest in 1921, but immediately left to study chemistry, first at the Universityof Berlin and subsequently at the Swiss Federal Institute of Technology in Zurich,from which he earned a degree in chemical engineering in 1925. During his time inGermany and Switzerland he returned to Budapest to write examinations, and in1926 obtained a PhD in mathematics from the University of Budapest. He taughtin Berlin and Hamburg, and, from 1930 to 1933, at Princeton University. In 1933 hebecame the youngest of the first six professors of the School of Mathematics at theInstitute for Advanced Study in Princeton (Einstein was another).Von Neumann’s first published scientific paper appeared in 1922, when he was19 years old. In 1928 he published a paper that establishes a key result on strictlycompetitive games (a result that had eluded Borel). He made many major contributions in pure and applied mathematics and in physics—enough, according to Halmos (1973), “for about three ordinary careers, in pure mathematics alone”. Whileat the Institute for Advanced Study he collaborated with the Princeton economistOskar Morgenstern in writing Theory of games and economic behavior, the book thatestablished game theory as a field. In the 1940s he became increasingly involvedin applied work. In 1943 he became a consultant to the Manhattan project, whichwas developing an atomic bomb. In 1944 he became involved with the development of the first electronic computer, to which he made major contributions. He

4Chapter 1. Introductionstayed at Princeton until 1954, when he became a member of the US Atomic EnergyCommission. He died in 1957.1.2The theory of rational choiceThe theory of rational choice is a component of many models in game theory.Briefly, this theory is that a decision-maker chooses the best action according toher preferences, among all the actions available to her. No qualitative restrictionis placed on the decision-maker’s preferences; her “rationality” lies in the consistency of her decisions when faced with different sets of available actions, not in thenature of her likes and dislikes.1.2.1 ActionsThe theory is based on a model with two components: a set A consisting of allthe actions that, under some circumstances, are available to the decision-maker,and a specification of the decision-maker’s preferences. In any given situationthe decision-maker is faced with a subset1 of A, from which she must choose asingle element. The decision-maker knows this subset of available choices, andtakes it as given; in particular, the subset is not influenced by the decision-maker’spreferences. The set A could, for example, be the set of bundles of goods thatthe decision-maker can possibly consume; given her income at any time, she isrestricted to choose from the subset of A containing the bundles she can afford.1.2.2 Preferences and payoff functionsAs to preferences, we assume that the decision-maker, when presented with anypair of actions, knows which of the pair she prefers, or knows that she regardsboth actions as equally desirable (is “indifferent between the actions”). We assumefurther that these preferences are consistent in the sense that if the decision-makerprefers the action a to the action b, and the action b to the action c, then she prefersthe action a to the action c. No other restriction is imposed on preferences. In particular, we do not rule out the possibility that a person’s preferences are altruisticin the sense that how much she likes an outcome depends on some other person’swelfare. Theories that use the model of rational choice aim to derive implicationsthat do not depend on any qualitative characteristic of preferences.How can we describe a decision-maker’s preferences? One way is to specify,for each possible pair of actions, the action the decision-maker prefers, or to notethat the decision-maker is indifferent between the actions. Alternatively we can“represent” the preferences by a payoff function, which associates a number witheach action in such a way that actions with higher numbers are preferred. More1 SeeChapter 17 for a description of mathematical terminology.

1.2 The theory of rational choice5precisely, the payoff function u represents a decision-maker’s preferences if, forany actions a in A and b in A,u(a) u(b) if and only if the decision-maker prefers a to b.(5.1)(A better name than payoff function might be “preference indicator function”;in economic theory a payoff function that represents a consumer’s preferences isoften referred to as a “utility function”.)E XAMPLE 5.2 (Payoff function representing preferences) A person is faced withthe choice of three vacation packages, to Havana, Paris, and Venice. She prefersthe package to Havana to the other two, which she regards as equiva

Nov 06, 2000 · Game theoretic reasoning pervades economic theory and is used widely in other social and behavioral sciences. This book presents the main ideas of game theory and shows how they can be used to understand economic, social, political, and bi-ological phenomena. It