Table of Contents, 1 Introduction 5, 1 1 What is Game Theory 5. 1 2 Examples 8, 1 3 Our Methodology 18, 2 Strategic Form Games with Complete Information 21. 2 1 Preliminaries 21, 2 2 Examples 23, 3 Strategic Form Solution Concepts 29. 3 1 Dominant Strategy Equilibrium 29, 3 2 Dominance Solvability 31. 3 3 Nash Equilibrium 38, 3 4 Nash Equilibrium and Dominant Dominated Actions 46. 3 5 Difficulties with Nash Equilibrium 48, 3 5 1 A Nash equilibrium may involve a weakly dominated action by some players 48. 4 Strategic Form Games Applications 55, 5 Mixed Strategy Equilibrium 71. 5 1 Introduction 71, 5 2 Mixed Strategies and Expected Payoffs 72. 5 3 Mixed Strategy Equilibrium 75, 5 4 Dominated Actions and Mixed Strategies 77. 6 Bayesian Games 79, 6 1 Preliminaries 79, 6 2 Bayesian Equilibrium 80. 6 3 Some Examples 81, TABLE OF CONTENTS 3, 6 3 1 Large Battle of the Sexes with incomplete information 81. 6 3 2 Cournot Duopoly with incomplete information 83. 7 Extensive Form Games with Perfect Information 85. 7 1 Extensive Form Games 85, 7 1 1 Game Trees 85, 7 1 2 Strategies 87. 7 2 Backward Induction Equilibrium 88, 7 2 1 Commitment and Mover Advantages 90. 7 3 Game Trees A More Formal Treatment 92, 7 4 Strategic Form of an Extensive Form Game 92. 7 5 Extensive Form Games with Imperfect Information 94. 7 5 1 Subgames and Subgame Perfect Equilibrium 96, 8 Extensive Form Games Applications 99. 8 1 Bargaining 99, 8 1 1 Ultimatum Bargaining 100, 8 1 2 Alternating Offers Bargaining 100. 8 1 3 Unique Subgame Perfect Equilibrium 102, 8 1 4 Properties of the Equilibrium 105. 9 Repeated Games 107, 9 1 Motivation 107, 9 2 Preliminaries 108. 9 3 Equilibria of Infinitely Repeated Games 110, 10 Auctions 117. 10 1 Independent Private Values 118, 10 1 1 Second Price Auctions 119. 10 1 2 First Price Auctions 119, 10 1 3 All Pay Auctions 121. 10 2 Revenue Equivalence 122, 10 3 Common Values and The Winner s Curse 124. 10 4 Auction Design 126, 10 4 1 Need for a Reserve Price 126. 10 4 2 Common Values 127, 10 4 3 Risk Averse Bidders 127. 4 TABLE OF CONTENTS, 10 4 4 Practical Concerns 127. 11 Extensive Form Games with Incomplete Information 131. 11 1 Introduction 131, 11 2 Perfect Bayesian Equilibrium 134. 11 3 Signalling Games 137, Introduction, 1 1 What is Game Theory. We humans cannot survive without interacting with other humans and ironically it some. times seems that we have survived despite those interactions Production and exchange require. cooperation between individuals at some level but the same interactions may also lead to disastrous. confrontations Human history is as much a history of fights and wars as it is a history of success. ful cooperation Many human interactions carry the potentials of cooperation and harmony as well. as conflict and disaster Examples are abound relationships among couples siblings countries. management and labor unions neighbors students and professors and so on. One can argue that the increasingly complex technologies institutions and cultural norms that. have existed in human societies have been there in order to facilitate and regulate these interactions. For example internet technology greatly facilitates buyer seller transactions but also complicates. them further by increasing opportunities for cheating and fraud Workers and managers have usu. ally opposing interests when it comes to wages and working conditions and labor unions as well as. labor law provide channels and rules through which any potential conflict between them can be ad. dressed Similarly several cultural and religious norms such as altruism or reciprocity bring some. order to potentially dangerous interactions between individuals All these norms and institutions. constantly evolve as the nature of the underlying interactions keep changing In this sense under. standing human behavior in its social and institutional context requires a proper understanding of. human interaction, Economics sociology psychology and political science are all devoted to studying human. behavior in different realms of social life However in many instances they treat individuals in. isolation for convenience if not for anything else In other words they assume that to understand. 6 Introduction, one individual s behavior it is safe to assume that her behavior does not have a significant effect on. other individuals In some cases and depending upon the question one is asking this assumption. may be warranted For example what a small farmer in a local market say in Montana charges for. wheat is not likely to have an effect on the world wheat prices Similarly the probability that my. vote will change the outcome of the U S presidential elections is negligibly small So if we are. interested in the world wheat price or the result of the presidential elections we may safely assume. that one individual acts as if her behavior will not affect the outcome. In many cases however this assumption may lead to wrong conclusions For example how. much our farmer in Montana charges compared to the other farmers in Montana certainly affects. how much she and other farmers make If our farmer sets a price that is lower than the prices. set by the other farmers in the local market she would sell more than the others and vice versa. Therefore if we assume that they determine their prices without taking this effect into account. we are not likely to get anywhere near understanding their behavior Similarly the vote of one. individual may radically change the outcome of voting in small committees and assuming that they. vote in ignorance of that fact is likely to be misleading. The subject matter of game theory is exactly those interactions within a group of individuals or. governments firms etc where the actions of each individual have an effect on the outcome that. Game theory studies strategic, is of interest to all Yet this is not enough for a situation to be a proper subject of game theory the. interactions, way that individuals act has to be strategic i e they should be aware of the fact that their actions. affect others The fact that my actions have an effect on the outcome does not necessitate strategic. behavior if I am not aware of that fact Therefore we say that game theory studies strategic. interaction within a group of individuals By strategic interaction we mean that individuals know. that their actions will have an effect on the outcome and act accordingly. Having determined the types of situations that game theory deals with we have to now discuss. how it analyzes these situations Like any other theory the objective of game theory is to organize. our knowledge and increase our understanding of the outside world A scientific theory tries to. abstract the most essential aspects of a given situation analyze them using certain assumptions and. procedures and at the end derive some general principles and predictions that can be applied to. individual instances, For it to have any predictive power game theory has to postulate some rules according to which. rules of the game, individuals act If we do not describe how individuals behave what their objectives are and how. they try to achieve those objectives we cannot derive any predictions at all in a given situation For. example one would get completely different predictions regarding the price of wheat in a local. market if one assumes that farmers simply flip a coin and choose between 1 and 2 a pound. compared to if one assumes they try to make as much money as possible Therefore to bring some. 1 1 What is Game Theory 7, discipline to the analysis one has to introduce some structure in terms of the rules of the game. The most important and maybe one of the most controversial assumption of game theory. which brings about this discipline is that individuals are rational We assume that individuals are. Definition An individual is rational if she has well defined objectives or preferences. over the set of possible outcomes and she implements the best available strategy to pursue. Rationality implies that individuals know the strategies available to each individual have com. plete and consistent preferences over possible outcomes and they are aware of those preferences. Furthermore they can determine the best strategy for themselves and flawlessly implement it. If taken literally the assumption of rationality is certainly an unrealistic one and if. applied to particular cases it may produce results that are at odds with reality We should. first note that game theorists are aware of the limitations imposed by this assumption. and there is an active research area studying the implications of less demanding forms. of rationality called bounded rationality This course however is not the appropriate. place to study this area of research Furthermore to really appreciate the problems with. rationality assumption one has to first see its results Therefore without delving into. too much discussion we will argue that one should treat rationality as a limiting case. You will have enough opportunity in this book to decide for yourself whether it produces. useful and interesting results As the saying goes the proof of the pudding is in the. The term strategic interaction is actually more loaded than it is alluded to above It is not. enough that I know that my actions as well as yours affect the outcome but I must also know that. you know this fact Take the example of two wheat farmers Suppose both farmer A and B know. that their respective choices of prices will affect their profits for the day But suppose A does not. know that B knows this Now from the perspective of farmer A farmer B is completely ignorant. of what is going on in the market and hence farmer B might set any price This makes farmer. A s decision quite uninteresting itself To model the situation more realistically we then have to. assume that they both know that they know that their prices will affect their profits One actually. has to continue in this fashion and assume that the rules of the game including how actions affect. the participants and individuals rationality are common knowledge. A fact X is common knowledge if everybody knows it if everybody knows that everybody. knows it if everybody knows that everybody knows that everybody knows it an so on This has. 8 Introduction, some philosophical implications and is subject to a lot of controversy but for the most part we will. We assume that the game and, avoid those discussions and take it as given. rationality are common, In sum we may define game theory as follows. Definition Game theory is a systematic study of strategic interactions among rational. individuals, Its limitations aside game theory has been fruitfully applied to many situations in the realm of. economics political science biology law etc In the rest of this chapter we will illustrate the main. ideas and concepts of game theory and some of its applications using simple examples In later. chapters we will analyze more realistic and complicated scenarios and discuss how game theory is. applied in the real world Among those applications are firm competition in oligopolistic markets. competition between political parties auctions bargaining and repeated interaction between firms. 1 2 Examples, For the sake of comparison we first start with an example in which there is no strategic inter. action and hence one does not need game theory to analyze. Example 1 1 A Single Person Decision Problem Suppose Ali is an investor who can invest his. 100 either in a safe asset say government bonds which brings 10 return in one year or he can. invest it in a risky asset say a stock issued by a corporation which either brings 20 return if the. company performance is good or zero return if the company performance is bad. Bonds 10 10, Stocks 20 0, Clearly which investment is best for Ali depends on his preferences and the relative likelihoods. of the two states of the world Let s denote the probability of the good state occurring p and that of. the bad state 1 p and assume that Ali wants to maximize the amount of money he has at the end. of the year If he invests his 100 on bonds he will have 110 at the end of the year irrespective. of the state of the world i e with certainty If he invests on stocks however with probability. 1 2 Examples 9, p he will have 120 and with probability 1 p he will have 100 We can therefore calculate his. average or expected money holdings at the end of the year as. p 120 1 p 100 100 20 p, If for example p 1 2 then he expects to have 110 at the end of the year In general if p 1 2. then he would prefer to invest in stocks and if p 1 2 he would prefer bonds. This is just one example of a single person decision making problem in which the decision. problem of an individual can be analyzed in isolation of the other individuals behavior Any A single person decision. problem has no strategic, uncertainty involved in such problems are exogenous in the sense that it is not determined or in interaction. fluenced in any way by the behavior of the individual in question In the above example the only. uncertainty comes from the performance of the stock which we may safely assume to be inde. pendent of Ali s choice of investment Contrast this with the situation illustrated in the following. Example 1 2 An Investment Game Now suppose Ali again has two options for investing his. 100 He may either invest it in bonds which have a certain return of 10 or he may invest it in. An Introduction to Game Theory Levent Koc kesen Koc University Efe A Ok New York University July 8 2007

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