strategy: an introduction to game theory (second ed) instructor's manual

Strategy: An Introduction to Game TheorySecond Edition

Instructors’ Manual

Joel Watson with Jesse Bull

April 2008

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Strategy: An Introduction to Game Theory, Second Edition, Instructors’ Manual,version 4/2008.

c©Copyright 2008, 2002 by Joel Watson. This document is available, with the per-mission of W. W. Norton & Company, for use by textbook adopters in conjunctionwith the textbook. The authors thank Pierpaolo Battigalli, Takako Fujiwara-Greve,Michael Herron, David Miller, David Reinstein, Christopher Snyder, and CharlesWilson for pointing out some typographical errors in earlier versions of this manual.

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This Instructors’ Manual has four parts. Part I contains some notes on outlining andpreparing a game theory course that is based on the textbook. Part II contains moredetailed (but not overblown) materials that are organized by textbook chapter. PartIII comprises solutions to all of the exercises in the textbook. Part IV contains somesample examination questions.

Please report any typographical errors to Joel Watson ([email protected]). Also feelfree to suggest new material to include in the instructors’ manual or web site.

Instructors' Manual for Strategy:An Introduction to Game Theory

Copyright 2002, 2008 by Joel WatsonFor instructors only; do not distribute.

Contents

I General Materials 7

II Chapter-Specific Materials 12

1 Introduction 13

2 The Extensive Form 15

3 Strategies and the Normal Form 19

4 Beliefs, Mixed Strategies,and Expected Payoffs 22

5 General Assumptions and Methodology 24

6 Dominance and Best Response 25

7 Rationalizability and Iterated Dominance 28

8 Location and Partnership 30

9 Nash Equilibrium 32

10 Oligopoly, Tariffs, Crime, and Voting 34

11 Mixed-Strategy Nash Equilibrium 35

12 Strictly Competitive Gamesand Security Strategies 37

13 Contract, Law, and Enforcementin Static Settings 38

14 Details of the Extensive Form 41

15 Backward Inductionand Subgame Perfection 43

16 Topics in Industrial Organization 45

17 Parlor Games 46

3Instructors' Manual for Strategy:An Introduction to Game Theory


CONTENTS 4

18 Bargaining Problems 48

19 Analysis of Simple Bargaining Games 50

20 Games with Joint Decisions;Negotiation Equilibrium 52

21 Unverifiable Investment, Hold Up,Options, and Ownership 54

22 Repeated Games and Reputation 56

23 Collusion, Trade Agreements,and Goodwill 58

24 Random Events andIncomplete Information 60

25 Risk and Incentives in Contracting 63

26 Bayesian Nash Equilibriumand Rationalizability 65

27 Lemons, Auctions,and Information Aggregation 66

28 Perfect Bayesian Equilibrium 68

29 Job-Market Signaling and Reputation 70

30 Appendices 71

III Solutions to the Exercises 72

2 The Extensive Form 73

3 Strategies and the Normal Form 77

4 Beliefs, Mixed Strategies,and Expected Payoffs 82

6 Dominance and Best Response 84

7 Rationalizability and Iterated Dominance 86



CONTENTS 5

8 Location and Partnership 88

9 Nash Equilibrium 92

10 Oligopoly, Tariffs, Crime,and Voting 96

11 Mixed-Strategy Nash Equilibrium 100

12 Strictly Competitive Gamesand Security Strategies 105

13 Contract, Law, and Enforcementin Static Settings 106

14 Details of the Extensive Form 111

15 Backward Inductionand Subgame Perfection 112

16 Topics in Industrial Organization 116

17 Parlor Games 120

18 Bargaining Problems 124

19 Analysis of Simple Bargaining Games 127

20 Games with Joint Decisions;Negotiation Equilibrium 130

21 Unverifiable Investment, Hold Up,Options, and Ownership 133

22 Repeated Games and Reputation 136

23 Collusion, Trade Agreements,and Goodwill 139

24 Random Events andIncomplete Information 143

25 Risk and Incentives in Contracting 145

26 Bayesian Nash Equilibriumand Rationalizability 147



CONTENTS 6

27 Lemons, Auctions,and Information Aggregation 152

28 Perfect Bayesian Equilibrium 155

29 Job-Market Signaling and Reputation 157

30 Appendix B 162

IV Sample Examination Questions 163



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Part I

General MaterialsThis part contains some notes on outlining and preparing a game theory course forthose adopting Strategy: An Introduction to Game Theory.



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Sample Syllabi

Most of the book can be covered in a semester-length (13-15 week) course. Hereis a sample thirteen-week course outline:

Weeks Topics Chapters

A. Representing Games1 Introduction, extensive form, strategies, 1-3

and normal form1-2 Beliefs and mixed strategies 4-5

B. Analysis of Static Settings

2-3 Best response, rationalizability, applications 6-83-4 Equilibrium, applications 9-105 Other equilibrium topics 11-125 Contract, law, and enforcement 13

C. Analysis of Dynamic Settings

6 Extensive form, backward induction, 14-15and subgame perfection

7 Examples and applications 16-178 Bargaining 18-199 Negotiation equilibrium and problems of 20-21

contracting and investment10 Repeated games, applications 22-23

D. Information

11 Random events and incomplete information 2411 Risk and contracting 2512 Bayesian equilibrium, applications 26-2713 Perfect Bayesian equilibrium and applications 28-29

In a ten-week (quarter system) course, most, but not all, of the book can becovered. For this length of course, you can easily leave out (or simply not cover inclass) some of the chapters. For example, any of the chapters devoted to applications(Chapters 8, 10, 16, 21, 23, 25, 27, and 29) can be covered selectively or skipped



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without disrupting the flow of ideas and concepts. Chapters 12 and 17 contain ma-terial that may be regarded as more esoteric than essential; one can easily have thestudents learn the material in these chapters on their own. Instructors who prefernot to cover contract can skip Chapters 13, 20, 21, and 25.

Below is a sample ten-week course outline that is formed by trimming some ofthe applications from the thirteen-week outline. This is the outline that I use for myquarter-length game theory course. I usually cover only one application from eachof Chapters 8, 10, 16, 23, 27, and 29. I avoid some end-of-chapter advanced topics,such as the infinite-horizon alternating-offer bargaining game, I skip Chapter 25, and,depending on the pace of the course, I selectively cover Chapters 18, 20, 27, 28, and 29.

Weeks Topics Chapters

A. Representing Games1 Introduction, extensive form, strategies, 1-3

and normal form1-2 Beliefs and mixed strategies 4-5

B. Analysis of Static Settings

2-3 Best response, rationalizability, applications 6-83-4 Equilibrium, applications 9-105 Other equilibrium topics 11-125 Contract, law, and enforcement 13

C. Analysis of Dynamic Settings

6 Backward induction, subgame perfection, 14-17and an application

7 Bargaining 18-197-8 Negotiation equilibrium and problems of 20-21

contracting and investment8-9 Repeated games, applications 22-23

D. Information

9 Random events and incomplete information 2410 Bayesian equilibrium, application 26-2710 Perfect Bayesian equilibrium and an application 28-29



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Experiments and a Course Competition

In addition to assigning regular problem sets, it can be fun and instructive torun a course-long competition between the students. The competition is mainly forsharpening the students’ skills and intuition, and thus the students’ performance inthe course competition should not count toward the course grades. The competi-tion consists of a series of challenges, classroom experiments, and bonus questions.Students receive points for participating and performing near the top of the class.Bonus questions can be sent by e-mail; some experiments can be done by e-mail aswell. Prizes can be awarded to the winning students at the end of the term. Somesuggestions for classroom games and bonus questions appear in various places in thismanual.

Level of Mathematics and Use of Calculus

Game theory is a technical subject, so the students should come into the coursewith the proper mathematics background. For example, students should be verycomfortable with set notation, algebraic manipulation, and basic probability theory.Appendix A in the textbook provides a review of mathematics at the level used inthe book.

Some sections of the textbook benefit from the use of calculus. In particular, afew examples and applications can be analyzed most easily by calculating derivatives.In each case, the expressions requiring differentiation are simple polynomials (usuallyquadratics). Thus, only the most basic knowledge of differentiation suffices to followthe textbook derivations. You have two choices regarding the use of calculus.

First, you can make sure all of the students can differentiate simple polynomials;this can be accomplished by either (a) specifying calculus as a prerequisite or (b)asking the students to read Appendix A at the beginning of the course and thenperhaps reinforcing this by holding an extra session in the early part of the term toreview how to differentiate a simple polynomial.

Second, you can avoid calculus altogether by either providing the students withnon-calculus methods to calculate maxima or by skipping the textbook examples thatuse calculus. Here is a list of the examples that are analyzed with calculus in thetextbook:

• the partnership example in Chapters 8 and 9,

• the Cournot application in Chapter 10 (the tariff and crime applications in thischapter are also most easily analyzed using calculus, but the analysis is notdone in the book),

• the advertising and limit capacity applications in Chapter 16 (they are basedon the Cournot model),

• the dynamic oligopoly model in Chapter 23 (Cournot-based),



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• the discussion of risk-aversion in Chapter 25 (in terms of the shape of a utilityfunction),

• the Cournot example in Chapter 26, and

• the analysis of auctions in Chapter 27.

Each of these examples can be easily avoided, if you so choose. There are also somerelated exercises that you might avoid if you prefer that your students not deal withexamples having continuous strategy spaces.

My feeling is that using a little bit of calculus is a good idea, even if calculus isnot a prerequisite for the game theory course. It takes only an hour or so to explainslope and the derivative and to give students the simple rule of thumb for calculatingpartial derivatives of simple polynomials. Then one can easily cover some of the mostinteresting and historically important game theory applications, such as the Cournotmodel and auctions.



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Part II

Chapter-Specific MaterialsThis part contains instructional materials that are organized according to the chaptersin the textbook. For each textbook chapter, the following is provided:

• A brief overview of the material covered in the chapter;

• Lecture notes (including an outline); and

• Suggestions for classroom examples and/or experiments.

The lecture notes are merely suggestions for how to organize lectures of the textbookmaterial. The notes do not represent any claim about the “right” way to lecture.Some instructors may find the guidelines herein to be in tune with their own teach-ing methods; these instructors may decide to use the lecture outlines without muchmodification. Others may have a very different style or intent for their courses; theseinstructors will probably find the lecture outlines of limited use, if at all. I hope thismaterial will be of some use to you.



1 Introduction

This chapter introduces the concept of a game and encourages the reader to beginthinking about the formal analysis of strategic situations. The chapter contains ashort history of game theory, followed by a description of “non-cooperative theory”(which the book emphasizes), a discussion of the notion of contract and the related useof “cooperative theory,” and comments on the science and art of applied theoreticalwork. The chapter explains that the word “game” should be associated with anywell-defined strategic situation, not just adversarial contests. Finally, the format andstyle of the book are described.

Lecture Notes

The non-administrative segment of a first lecture in game theory may run asfollows.

• Definition of a strategic situation.

• Examples (have students suggest some): chess, poker, and other parlor games;tennis, football, and other sports; firm competition, international trade, inter-national relations, firm/employee relations, and other standard economic exam-ples; biological competition; elections; and so on.

• Competition and cooperation are both strategic topics. Game theory is a generalmethodology for studying strategic settings (which may have elements of bothcompetition and cooperation).

• The elements of a formal game representation.

• A few simple examples of the extensive form representation (point out the basiccomponents).

Examples and Experiments

1. Clap game. Ask the students to stand and then, if they comply, ask them toclap. (This is a silly game.) Show them how to diagram the strategic situationas an extensive form tree. The game starts with your decision about whether toask them to stand. If you ask them to stand, then they (modeled as one player)have to choose between standing and staying in their seats. If they stand, thenyou decide between saying nothing and asking them to clap. If you ask them toclap, then they have to decided whether to clap. Write the outcomes at terminalnodes in descriptive terms such as “professor happy, students confused.” Thenshow how these outcomes can be converted into payoff numbers.



1 INTRODUCTION 14

2. Auction the textbook. Many students will probably not have purchased thetextbook by the first class meeting. These students may be interested in pur-chasing the book from you, especially if they can get a good deal. However,quite a few students will not know the price of the book. Without announcingthe bookstore’s price, hold a sealed-bid, first-price auction (using real money).This is a common-value auction with incomplete information. The winning bidmay exceed the bookstore’s price, giving you an opportunity to talk about the“winner’s curse” and to establish a fund to pay students in future classroomexperiments.



2 The Extensive Form

This chapter introduces the basic components of the extensive form in a non-technicalway. Students who learn about the extensive form at the beginning of a course aremuch better able to grasp the concept of a strategy than are students who are taughtthe normal form first. Since strategy is perhaps the most important concept in gametheory, a good understanding of this concept makes a dramatic difference in eachstudent’s ability to progress. The chapter avoids the technical details of the extensiveform representation in favor of emphasizing the basic components of games. Thetechnical details are covered in Chapter 14.

Lecture Notes

The following may serve as an outline for a lecture.

• Basic components of the extensive form: nodes, branches. Nodes are wherethings happen. Branches are individual actions taken by the players.

• Example of a game tree.

• Types of nodes: initial, terminal, decision.

• Build trees by expanding, never converging back on themselves. At any placein a tree, you should always know exactly how you got there. Thus, the treesummarizes the strategic possibilities.

• Player and action labels. Try not to use the same label for different places wheredecisions are made.

• Information sets. Start by describing the tree as a diagram that an externalobserver creates to map out the possible sequences of decisions. Assume theexternal observer sees all of the players’ actions. Then describe what it meansfor a player to not know what another player did. This is captured by dashedlines indicating that a player cannot distinguish between two or more nodes.

• We assume that the players know the game tree, but that a given player maynot know where he is in the game when he must make any particular decision.

• An information set is a place where a decision is made.

• How to describe simultaneous moves.

• Outcomes and how payoff numbers represent preferences.



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Several examples should be used to explain the components of an extensive form.In addition to some standard economic examples (such as firm entry into an industryand entrant/incumbent competition), here are a few I routinely use:

1. Three-card poker. In this game, there is a dealer (player 1) and two potentialbetters (players 2 and 3). There are three cards in the deck: a high card, amiddle card, and a low card. At the beginning of the game, the dealer looks atthe cards and gives one to each of the other players. Note that the dealer candecide which of the cards goes to player 2 and which of the cards goes to player 3.(There is no move by Nature in this game. The book does not deal with movesof Nature until Part IV. You can discuss moves of Nature at this point, but itis not necessary.) Player 2 does not observe the card dealt to player 3, nor doesplayer 3 observe the card dealt to player 2. After the dealer’s move, player 2observes his card and then decides whether to bet or to fold. After player 2’sdecision, player 3 observes his own card and also whether player 2 folded orbet. Then player 3 must decide whether to fold or bet. After player 3’s move,the game ends. Payoffs indicate that each player prefers winning to folding andfolding to losing. Assume the dealer is indifferent between all of the outcomes(or specify some other preference ordering).

2. Let’s Make a Deal game. This is the three-door guessing game that was madefamous by Monty Hall and the television game show Let’s Make a Deal. Thegame is played by Monty (player 1) and a contestant (player 2), and it runs asfollows.

First, Monty secretly places a prize (say, $1000) behind one of threedoors. Call the doors a, b, and c. (You might write Monty’s actionsas a′, b′, and c′, to differentiate them from those of the contestant.)

Then, without observing Monty’s choice, the contestant selects oneof the doors (by saying “a,” “b,” or “c”).

After this, Monty must open one of the doors, but he is not allowedto open the door that is in front of the prize, nor is he allowed to openthe door that the contestant selected. Note that Monty does not havea choice if the contestant chooses a different door than Monty chosefor the prize. The contestant observes which door Monty opens. Notethat she will see no prize behind this door.

The contestant then has the option of switching to the other unopeneddoor (S for “switch”) or staying with the door she originally selected(D for “don’t switch”).

Finally, the remaining doors are opened and the contestant wins theprize if it is behind the door she chose. The contestant obtains a




payoff 1 if she wins, zero otherwise. Monty is indifferent between allof the outcomes.

For a bonus question, you can challenge the students to draw the extensive formrepresentation of the Let’s Make a Deal game or the Three-Card Poker game.Students who submit a correct extensive form can be given points for the classcompetition. The Let’s Make a Deal extensive form is pictured on the nextpage.



3 Strategies and the Normal Form

As noted already, introducing the extensive form representation at the beginning ofa course helps the students appreciate the notion of a strategy. A student that doesnot understand the concept of a “complete contingent plan” will fail to grasp thesophisticated logic of dynamic rationality that is so critical to much of game theory.Chapter 3 starts with the formal definition of strategy, illustrated with some examples.The critical point is that strategies are more than just “plans.” A strategy prescribesan action at every information set, even those that would not be reached because ofactions taken at other information sets.

Chapter 3 proceeds to the construction of the normal-form representation, startingwith the observation that each strategy profile leads to a single terminal node (anoutcome) via a path through the tree. This leads to the definition of a payoff function.The chapter then defines the normal form representation as comprising a set of players,strategy spaces for the players, and payoff functions. The matrix form, for two-player,finite games, is illustrated. The chapter then briefly describes seven classic normalform games. The chapter concludes with a few comments on the comparison betweenthe normal and extensive forms.

Lecture Notes


• Formal definition of strategy.

• Examples of strategies.

• Notation: strategy space Si, individual strategy si ∈ Si. Example: Si = {H, L}and si = H.

• Refer to Appendix A for more on sets.

• Strategy profile: s ∈ S, where S = S1 × S2 × · · · × Sn (product set).

• Notation: i and −i, s = (si, s−i).

• Discuss how finite and infinite strategy spaces can be described.

• Why we need to keep track of a complete contingent plan: (1) It allows theanalysis of games from any information set, (2) it facilitates exploring how aplayer responds to his belief about what the other players will do, and (3) itprescribes a contingency plan if a player makes a mistake.

• Describe how a strategy implies a path through the tree, leading to a terminalnode and payoff vector.

• Examples of strategies and implied payoffs.



3 STRATEGIES AND THE NORMAL FORM 20

• Definition of payoff function, ui : S → R, ui(s). Refer to Appendix A for moreon functions.

• Example: a matrix representation of players, strategies, and payoffs. (Use anyabstract game, such as the centipede game.)

• Formal definition of the normal form.

• Note: The matrix representation is possible only for two-player, finite games.Otherwise, the game must be described by sets and equations.

• The classic normal form games and some stories. Note the different strategicissues represented: conflict, competition, coordination, cooperation.

• Comparing the normal and extensive forms (translating one to the other).


1. Ultimatum-offer bargaining game. Have students give instructions to others asto how to play the game. Those who play the role of “responder” will have tospecify under what conditions to accept and under what conditions to reject theother player’s offer. This helps solidify that a strategy is a complete contingentplan.

2. The centipede game (like the one in Figure 3.1(b) if the textbook). As with thebargaining game, have some students write their strategies on paper and givethe strategies to other students, who will then play the game as their agents.Discuss mistakes as a reason for specifying a complete contingent plan. Thendiscuss how strategy specifications helps us develop a theory about why playersmake particular decisions (looking ahead to what they would do at variousinformation sets).

3. Any of the classic normal forms.

4. The Princess Bride poison scene. Show the “poison” scene (and the few minutesleading to it) from the Rob Reiner movie The Princess Bride. In this scene,protagonist Wesley matches wits with the evil Vizzini. There are two gobletsfilled with wine. Away from Vizzini’s view, Wesley puts poison into one ofthe goblets. Then Wesley sets the goblets on a table, one goblet near himselfand the other near Vizzini. Vizzini must choose from which goblet to drink.Wesley must drink from the other goblet. Several variations of this game can bediagrammed for the students, first in the extensive form and then in the normalform.




5. A 3 × 3 dominance-solvable game, such as the following.

The payoffs are in dollars. It is very useful to have the students play a gamesuch as this before you lecture on dominance and best response. This will helpthem to begin thinking about rationality, and their behavior will serve as areference point for formal analysis. Have the students write their strategiesand their names on slips of paper. Collect the slips and randomly select aplayer 1 and a player 2. Pay these two students according to their strategyprofile. Calculate the class distribution over the strategies, which you can lateruse when introducing dominance and iterated dominance.

6. Repeated Prisoners’ Dilemma. Describe the k-period, repeated prisoners’ dilemma.For a bonus question, ask the students to compute the number of strategies forplayer 1 when k = 3. Challenge the students to find a mathematical expressionfor the number of strategies as a function of k.



4 Beliefs, Mixed Strategies,

and Expected Payoffs

This chapter describes how a belief that a player has about another player’s behavioris represented as a probability distribution. It then covers the idea of a mixed strat-egy, which is a similar probability distribution. The appropriate notation is defined.The chapter defines expected payoff and gives some examples of how to compute it.At the end of the chapter, there are a few comments about cardinal versus ordinalutility (although it is not put in this language) and about how payoff numbers reflectpreferences over uncertain outcomes. Risk preferences are discussed in Chapter 25.

Lecture Notes


• Example of belief in words: “Player 1 might say ‘I think player 2 is very likelyto play strategy L.’”

• Translate into probability numbers.

• Other examples of probabilities.

• Notation: μj ∈ ΔSj, μj(sj) ∈ [0, 1],∑

sj∈Sjμj(sj) = 1.

• Examples and alternative ways of denoting a probability distribution: for Sj ={L, R} and μj ∈ Δ{L, R} defined by μj(L) = 1/3 and μj(R) = 2/3, we canwrite μj = (1/3, 2/3).

• Mixed strategy. Notation: σi ∈ ΔSi.

• Refer to Appendix A for more on probability distributions.

• Definition of expected value. Definition of expected payoff.

• Examples: computing expected payoffs.

• Briefly discuss how payoff numbers represent preferences over random outcomes,risk. Defer elaboration until later.



BELIEFS AND EXPECTED PAYOFFS 23


1. Let’s Make a Deal game again. For the class competition, you can ask thefollowing two bonus questions: (a) Suppose that, at each of his informationsets, Monty randomizes by choosing his actions with equal probability. Is itoptimal for the contestant to select “switch” or “don’t switch” when she hasthis choice? Why? (b) Are there conditions (a strategy for Monty) under whichit is optimal for the contestant to make the other choice?

2. Randomization in sports. Many sports provide good examples of randomizedstrategies. Baseball pitchers may desire to randomize over their pitches, andbatters may have probabilistic beliefs about which pitch will be thrown to them.Tennis serve and return play is another good example.1

1See Walker, M., and Wooders J. “Minimax Play at Wimbledon,” American Economic Review91 (2001): 1521-1538.



5 General Assumptions and Methodology

This chapter contains notes on (a) the trade-off between simplicity and realism informulating a game-theoretic model, (b) the basic idea and assumption of rationality,(c) the notion of common knowledge and the assumption that the game is commonlyknown by the players, and (d) a short overview of solution concepts that are discussedin the book. It is helpful to briefly discuss these items with the students during partof a lecture.



6 Dominance and Best Response

This chapter develops and compares the concepts of dominance and best response.The chapter begins with examples in which a strategy is dominated by another purestrategy, followed by an example of mixed strategy dominance. After the formaldefinition of dominance, the chapter describes how to check for dominated strategiesin any given game. The first strategic tension (the clash between individual and jointinterests) is illustrated with reference to the prisoners’ dilemma, and then the notionof efficiency is defined. Next comes the definition of best response and examples.The last section of the chapter contains analysis of the relation between the set ofundominated strategies and the set of strategies that are best responses to somebeliefs. An algorithm for calculating these sets is presented.

Lecture Notes


• Optional introduction: analysis of a game played in class. If a 3×3 dominance-solvable game (such as the one suggested in the notes for Chapter 4) was playedin class earlier, the game can be quickly analyzed to show the students what isto come.

• A simple example of a strategy dominated by another pure strategy. (Use a2 × 2 game.)

• An example of a pure strategy dominated by a mixed strategy. (Use a 3 × 2game.)

• Formal definition of strategy si being dominated. Set of undominated strategiesfor player i, UDi.

• Discuss how to search for dominated strategies.

• The first strategic tension and the prisoners’ dilemma.

• Definition of efficiency and an example.

• Best response examples. (Use simple games such as the prisoners’ dilemma, thebattle of the sexes, Cournot duopoly.)

• Formal definition of si being a best response to belief μ−i. Set of best responsesfor player i, BRi(μ−i). Set of player i’s strategies that can be justified as bestresponses to some beliefs, Bi.

• Note that forming beliefs is the most important exercise in rational decisionmaking.



6 DOMINANCE AND BEST RESPONSE 26

• Example to show that Bi = UDi. State formal results.

• Algorithm for calculating Bi = UDi in two-player games: (1) Strategies thatare best responses to simple (point mass) beliefs are in Bi. (2) Strategies thatare dominated by other pure strategies are not in Bi. (3) Other strategies canbe tested for mixed strategy dominance to see whether they are in Bi. Step 3amounts to checking whether a system of inequalities can hold.

• Note: Remember that payoff numbers represent preferences over random out-comes.

• Note that Appendix B contains more technical material on the relation betweendominance and best response.

The book does not discuss weak dominance until the analysis of the second-priceauction in Chapter 27. This helps avoid confusion (students sometimes interchangethe weak and strong versions) and, besides, there is little need for the weak dominanceconcept.


1. Example of dominance and best response. To demonstrate the relation betweendominance and best response, the following game can be used.

First show that M is the best response to L, whereas B is the best response to R.Next show that T is dominated by player 1’s strategy (0, 1/2, 1/2), which putsequal probability on M and B but zero probability on T. Then prove that thereis no belief for which T is a best response. A simple graph will demonstratethis. On the graph, the x-axis is the probability p that player 1 believes player 2will select L. The y-axis is player 1’s expected payoff of the various strategies.The line corresponding to the expected payoff playing T is below at least oneof the lines giving the payoffs of M and B, for every p.

2. The 70 percent game. This game can be played by everyone in the class, eitherby e-mail or during a class session. Each of the n students selects an integerbetween 1 and 100 and writes this number, along with his or her name, on a slip




of paper. The students make their selections simultaneously and independently.The average of the students’ numbers is then computed and the student whosenumber is closest to 70 percent of this average wins 20 dollars. If two or morestudents tie, then they share the prize in equal proportions. Ideally, this gameshould be played between the lecture on Best Response and the lecture onRationalizability/Iterated Dominance. The few students whose numbers fallwithin a preset interval of 70 percent of the average can be given bonus points.



7 Rationalizability and Iterated Dominance

This chapter follows naturally from Chapter 6. It discusses the implications of com-bining the assumption that players best respond to beliefs with the assumption thatthis rationality is common knowledge between the players. At the beginning of thechapter, the logic of rationalizability and iterated dominance is demonstrated with anexample. Then iterated dominance and rationalizability are defined more formally.The second strategic tension—strategic uncertainty—is explained.

Lecture Notes


• Example of iterated dominance, highlighting hierarchies of beliefs (“player 1knows that player 2 knows that player 1 will not select. . . ”).

• Common knowledge: information that each player knows, each player knows theothers know, each player knows the others know that they all know. . . . It is asthough the information is publicly announced while the players are together.

• Combining rationality (best response behavior, never playing dominated strate-gies) with common knowledge implies, and only implies, that players will playstrategies that survive iterated dominance. We call these the rationalizablestrategies.

• Formally, let Rk be the set of strategy profiles that survives k rounds of iterateddominance. Then the rationalizable set R is the limit of Rk as k gets large. Forfinite games, after some value of k, no more strategies will be deleted.

• Notes on how to compute R: algorithm, order of deletion does not matter.

• The second strategic tension: strategic uncertainty (lack of coordination be-tween beliefs and behavior).



7 RATIONALIZABILITY AND ITERATED DOMINANCE 29


1. The 70 percent game again. Analyze the game and show that the only rational-izable strategy is to select 1. In my experience, this always stimulates a livelydiscussion of rationality and common knowledge. The students will readilyagree that selecting 100 is a bad idea. However, showing that 100 is dominatedcan be quite difficult. It is perhaps easier to demonstrate that 100 is never abest response.

Note that one player’s beliefs about the strategies chosen by the other players is,in general, a very complicated probability distribution, but it can be summarizedby the “highest number that the player believes the others will play with positiveprobability.” Call this number x. If x > 1, then you can show that the player’sbest response must be strictly less than x (considering that the player believesat least one other player will select x with positive probability). It is a goodexample of a game that has a rationalizable solution, yet the rationalizable setis quite difficult to compute. Discuss why it may be rational to select a differentnumber if common knowledge of rationality does not hold.

2. Generalized stag hunt. This game can be played in class by groups of differentsizes, or it can be played over the Internet for bonus points. In the game, nplayers simultaneously and independently write “A” or “B” on slips of paper.If any of the players selected B, then those who chose A get nothing and thosewho chose B get a small prize (say, $2.00 or 10 points). If all of the playersselected A, then they each obtain a larger prize ($5.00 or 25 points). The gamecan be used to demonstrate strategic uncertainty, because there is a sense inwhich strategic uncertainty is likely to increase (and players are more likely tochoose B) with n.

A good way to demonstrate strategic uncertainty is to play two versions ofthe game in class. In the first version, n = 2. In this case, tell the studentsthat, after the students select their strategies, you will randomly choose twoof them, whose payoffs are determined by only each other’s strategies. In thesecond version of the game, n equals the number of students. In this case, tellthe students that you will pay only a few of them (randomly chosen) but thattheir payoffs are determined by the strategies of everyone in the class. That is,a randomly drawn student who selected A gets the larger prize if and only ifeveryone else in the class also picked A. You will most likely see a much higherpercentage of students selecting A in the first version than in the second.



8 Location and Partnership

This chapter presents two important applied models. The applications illustrate thepower of proper game theoretic reasoning, they demonstrate the art of construct-ing game theory models, and they guide the reader on how to calculate the set ofrationalizable strategies. The location game is a finite (nine location) version ofHotelling’s well-known model. This game has a unique rationalizable strategy profile.The partnership game has infinite strategy spaces, but it too has a unique rational-izable strategy profile. Analysis of the partnership game coaches the reader on howto compute best responses for games with differentiable payoff functions and contin-uous strategy spaces. The rationalizable set is determined as the limit of an infinitesequence. The notion of strategic complementarity is briefly discussed in the contextof the partnership game.

Lecture Notes

Students should see the complete analysis of a few games that can be solvedby iterated dominance. The location and partnership examples in this chapter areexcellent choices for presentation. Both of these require nontrivial analysis and leadto definitive conclusions. It may be useful to substitute for the partnership game ina lecture (one can, for example, present the analysis of the Cournot duopoly game inclass and let the students read the parallel analysis of the partnership game from thebook). This gives the students exposure to two games that have continuous actionspaces.


• Describe the location game and draw a picture of the nine regions. Si ={1, 2, 3, 4, 5, 6, 7, 8, 9}.

• Show that the end regions are dominated by the adjacent ones. Write thedominance condition ui(1, sj) < ui(2, sj). Thus, R1

i = {2, 3, 4, 5, 6, 7, 8}.• Repeat. R2

i = {3, 4, 5, 6, 7}, R3i = {4, 5, 6}, R4

i = {5} = R.

• Applications of the location model.

• Describe the partnership game (or Cournot game, or other). It is useful to drawthe extensive form.

• Player i’s belief about player j’s strategy can be complicated, but, for expectedpayoff calculations, only the average (mean) matters. Thus, write BR1(y) or,for the Cournot duopoly game, BRi(qj), etc.

• Differentiate (or argue by way of differences) to get the best response functions.

• Sets of possible best responses (Bi).



8 LOCATION AND PARTNERSHIP 31

• Restrictions: implications of common knowledge of rationality. Construct R1i ,

R2i , R3

i . Indicate the limit R.

• Concept of strategic complementarity.


1. Location games. You can play different versions of the location game in class(see, for instance, the variations in the Exercises section of the chapter).

2. Repeated play and convergence. It may be useful, although it takes time andprizes, to engage your class in repeated play of a simple matrix game. Thepoint is not to discuss reputation, but rather to see whether experimental playstabilizes on one particular strategy profile or subset of the strategy space.This gets the students thinking about an institution (historical precedent, inthis case) that helps align beliefs and behavior, which is a nice transition to thematerial in Chapter 9.

Probably the easiest way of running the experiment is to have just two studentsplay a game like the following:

A game like the one from Exercise 6 of Chapter 9 may also be worth trying.To avoid repeated-game issues, you can have different pairs of students play indifferent rounds. The history of play can be recorded on the chalkboard. Youcan motivate the game with a story.

3. Contract or mediation. An interesting variant on the convergence experimentcan be used to demonstrate that pre-play communication and/or mediationcan align beliefs and behavior. Rather than have the students play repeatedly,simply invite two students to play a one-shot game. In one version, they can beallowed to communicate (agree to a self-enforced contract) before playing. In asecond version, you or a student can recommend a strategy profile to the players(but in this version, keep the players from communicating between themselvesand separate them when they are to choose strategies).



9 Nash Equilibrium

This chapter provides a solid conceptual foundation for Nash equilibrium, based on(1) rationalizability and (2) strategic certainty, where players’ beliefs and behaviorare coordinated so there is some resolution of the second strategic tension. Strate-gic certainty is discussed as the product of various social institutions. The chapterbegins with the concept of congruity, the mathematical representation of some co-ordination between players’ beliefs and behavior. Nash equilibrium is defined as aweakly congruous strategy profile, which captures the absence of strategic uncertainty(as a single strategy profile). Various examples are furnished. Then the chapter ad-dresses the issue of coordination and welfare, leading to a description of the thirdstrategic tension—the specter of inefficient coordination. Finally, there is an aside onbehavioral game theory (experimental work).

Lecture Notes


• Discuss strategic uncertainty (the second strategic tension). Illustrate with agame (such as the battle of the sexes) where the players’ beliefs and behaviorare not coordinated, so they get the worst payoff profile.

• Institutions that alleviate strategic uncertainty: norms, rules, communication,etc.

• Stories: (a) repeated social play with a norm, (b) pre-play communication (con-tracting), and (c) an outside mediator suggests strategies.

• Represent as congruity. Define weakly congruous, best response complete, andcongruous strategy sets.

• Example of an abstract game with various sets that satisfy these definitions.

• Nash equilibrium (where there is no strategic uncertainty). A weakly congruousstrategy profile. Strict Nash equilibrium (a congruous strategy profile).

• Examples of Nash equilibrium: classic normal forms, partnership, location, etc.

• An algorithm for finding Nash equilibria in matrix games.

• Pareto coordination game shows the possibility of inefficient coordination. Dis-cuss real examples of inefficient coordination in the world. This is the thirdstrategic tension.

• Note that a institution may thus alleviate the second tension, but we shouldbetter understand how. Also, the first and third strategic tensions still remain.



9 NASH EQUILIBRIUM 33


1. Coordination experiment. To illustrate the third strategic tension, you canhave students play a coordination game in the manner suggested in the previ-ous chapter (see the repeated play, contract, and mediation experiments). Forexample, have two students play a Pareto coordination game with the recom-mendation that they select the inferior equilibrium. Or invite two students toplay a complicated coordination game (with, say, ten strategies) in which thestrategy names make an inferior equilibrium a focal point.

2. The first strategic tension and externality. Students may benefit from a discus-sion of how the first strategic tension (the clash between individual and jointinterests) relates the classic economic notion of externality. This can be illus-trated in equilibrium, by using any game whose equilibria are inefficient. Ann-player prisoners’ dilemma or commons game can be played in class. You candiscuss (and perhaps sketch a model of) common economic settings where a neg-ative externality causes people to be more active than would be jointly optimal(pollution, fishing in common waters, housing development, arms races).

3. War of attrition. A simple war of attrition game (for example, one in discretetime) can be played in class for bonus points or money. A two-player game wouldbe the easiest to run as an experiment. For example, you could try a game likethat in Exercise 9 of Chapter 22 with x = 0. Students will hopefully thinkabout mixed strategies (or at least, nondegenerate beliefs). You can presentthe “static” analysis of this game. To compute the mixed strategy equilibrium,explain that there is a stationary “continuation value,” which, in the gamewith x = 0, equals zero. If you predict that the analysis will confused thestudents, this example might be better placed later in the course (once studentsare thinking about sequential rationality).



10 Oligopoly, Tariffs, Crime, and Voting

This chapter presents six standard applied models: Cournot duopoly, Bertrand duopoly,tariff competition, a model of crime and police, candidate location (the median votertheorem), and strategic voting. Each model is motivated by an interesting, real strate-gic setting. Very simple versions of the models are described and the equilibria offour of the examples are calculated. Calculations for the other two models are left asexercises.

Lecture Notes

Any or all of the models can be discussed in class, depending on time constraintsand the students’ background and interest. Other equilibrium models can also bepresented, either in addition to or substituting for the ones in the textbook. In eachcase, it may be helpful to organize the lecture as follows.

• Motivating story and real-world setting.

• Explanation of how some key strategic elements can be distilled in a gametheory model.

• Description of the game.

• Overview of rational behavior (computation of best response functions, if ap-plicable).

• Equilibrium calculation.

• Discussion of intuition.


Students would benefit from a discussion of real strategic situations, especiallywith an eye toward understanding the extent of the first strategic tension (equilibriuminefficiency). Also, any of the applications can be used for classroom experiments.

Here is a game that can be played by e-mail, which may be useful in introducingmixed strategy Nash equilibrium in the next lecture. (The game is easy to describe,but difficult to analyze.) Students are asked to each submit a number from theset {1, 2, 3, 4, 5, 6, 7, 8, 9}. The students make their selections simultaneously andindependently. At a prespecified date, you determine how many students picked eachof the numbers and you calculate the mode (the number that was most selected). Forexample, if ten students picked 3, eight students picked 6, eleven students picked 7,and six students picked 8, then the mode is 7. If there are two or more modes, thehighest is chosen. Let x denote the mode. If x = 9, then everyone who selected thenumber 1 gets one bonus point (and the others get zero). If x is not equal to 9, theneveryone who selected x + 1 gets x + 1 bonus points (and the others get zero).



11 Mixed-Strategy Nash Equilibrium

This chapter begins with the observation that, intuitively, a randomized strategyseems appropriate for the matching pennies game. The definition of a mixed-strategyNash equilibrium is given, followed by instructions on how to compute mixed-strategyequilibria in finite games. The Nash equilibrium existence result is presented.

Lecture Notes

Few applications and concepts rely on the analysis of mixed strategies, so thebook does not dwell on the concept. However, it is still an important topic and onecan present several interesting examples. Here is a lecture outline.

• Matching pennies—note that there is no Nash equilibrium (in pure strategies).Ask for suggestions on how to play. Ask “Is there any meaningful notion ofequilibrium in mixed strategies?”

• Note the (1/2, 1/2), (1/2, 1/2) mixed strategy profile. Confirm understandingof “mixing.”

• The definition of a mixed-strategy Nash equilibrium—the straightforward exten-sion of the basic definition.

• Two important aspects of the definition: (a) what it means for strategies thatare in the support (they must all yield the same expected payoff) and (b) whatit means for pure strategies that are not in the support of the mixed strategy.

• An algorithm for calculating mixed-strategy Nash equilibria: Find rationalizablestrategies, look for a mixed strategy of one player that will make the other playerindifferent, and then repeat for the other player.

• Note the mixed-strategy equilibria of the classic normal form games.

• Mixed-strategy Nash equilibrium existence result.


1. Attack and defend. Discuss how some tactical choices in war can be analyzedusing matching pennies-type games. Use a recent example or a historical exam-ple, such as the choice between Normandy and the Pas de Calais for the D-Dayinvasion of June 6, 1944. In the D-Day example, the Allies had to decide atwhich location to invade, while the Germans had to choose where to bolstertheir defenses. Discuss how the model can be modified to incorporate morerealistic features.



11 MIXED-STRATEGY NASH EQUILIBRIUM 36

2. A socially repeated strictly competitive game. This classroom experiment demon-strates how mixed strategies may be interpreted as frequencies in a populationof players. The experiment can be done over the Internet or in class. Theclassroom version may be unwieldy if there are many students. The game canbe played for money or for points in the class competition.

For the classroom version, draw on the board a symmetric 2×2 strictly compet-itive game, with the strategies Y and N for each of the two players. Use a gamethat has a unique, mixed-strategy Nash equilibrium. Tell the students thatsome of them will be randomly selected to play this game against one another.Ask all of the students to select strategies (by writing them on slips of paperor using cards as described below). Randomly select several pairs of studentsand pay them according to their strategy profile. Compute the distribution ofstrategies for the entire class and report this to all of the students. If the classfrequencies match the Nash equilibrium, then discuss this. Otherwise, repeatthe gaming procedure several times and discuss whether play converges to theNash equilibrium.

Here is an idea for how to play the game quickly. With everyone’s eyes closed,each student selects a strategy by either putting his hands on his head (the Ystrategy) or folding his arms (the N strategy). At your signal, the students opentheir eyes. You can quickly calculate the strategy distribution and randomlyselect students (from a class list) to pay.

3. Another version of the socially repeated game. Instead of having the entire classplay the game in each round, have only two randomly selected students play.Everyone will see the sequence of strategy profiles and you can discuss how theplay in any round is influenced by the outcome in preceding rounds.

4. Randomization in sports. Discuss randomization in sport (soccer penalty shots,tennis service location, baseball pitch selection, American football run/passmix).

In addition to demonstrating how random play can be interpreted and form aNash equilibrium, the social repetition experiments also make the students familiarwith strictly competitive games, which provides a good transition to the material inChapter 12.



12 Strictly Competitive Games

and Security Strategies

This chapter offers a brief treatment of two concepts that played a major role inthe early development of game theory: two-player, strictly competitive games andsecurity strategies. The chapter presents a result that is used in Chapter 17 for theanalysis of parlor games.

Lecture Notes

One can present this material very quickly in class, or leave it for students to read.An outline for a lecture may run as follows.

• Definition of a two-player, strictly competitive game.

• The special case called zero-sum.

• Examples of strictly competitive and zero-sum games.

• Definition of security strategy and security payoff level.

• Determination of security strategies in some examples.

• Discuss the difference between security strategy and best response, and whybest response is our behavioral foundation.

• The Nash equilibrium and security strategy result.


Any abstract examples will do for a lecture. It is instructive to demonstratesecurity strategies in the context of some games that are not strictly competitive, sothe students understand that the definition applies generally.



13 Contract, Law, and Enforcement

in Static Settings

This chapter presents the notion of contract. Much emphasis is placed on how con-tracts help to align beliefs and behavior in static settings. It carefully explains howplayers can use a contract to induce a game whose outcome differs from that of thegame given by the technology of the relationship. Further, the relationship betweenthose things considered verifiable and the outcomes that can be implemented is care-fully explained. The exposition begins with a setting of full verifiability and completecontracting. The discussion then shifts to settings of limited liability and defaultdamage remedies.

Lecture Notes

You may find the following outline useful in planning a lecture.

• Definition of contract. Self-enforced and externally enforced components.

• Discuss why players might want to contract (and why society might want laws).Explain why contracts are fundamental to economic relationships.

• Practical discussion of the technology of the relationship, implementation, andhow the court enforces a contract.

• Definition of the induced game.

• Verifiability. Note the implications of limited verifiability.

• Complete contracting. Default damage rules: expectation, reliance, restitution.

• Liquidated damage clauses and contracts specifying transfers.

• Efficient breach.

• Comments on the design of legal institutions.


1. Contract game. A contract game of the type analyzed in this chapter canbe played as a classroom experiment. Two students can be selected to firstnegotiate a contract and then play the underlying game. You play the role ofthe external enforcer. It may be useful to do this once with full verifiabilityand once with limited verifiability. This may also be used immediately beforepresenting the material in Chapter 13 and/or as a lead-in to Chapter 18.



CONTRACT, LAW, AND ENFORCEMENT 39

2. Case study: Chicago Coliseum Club v. Dempsey (Source: 265 Ill. App. 542;1932 Ill. App.). This or a different case can be used to illustrate the variouskinds of default damage remedies and to show how the material of the chapterapplies to practical matters.2 First, give the background of the case and thenpresent a stylized example that is based on the case.

Facts of the Case:

Chicago Coliseum Club, a corporation, as “plaintiff,” brought its action against“defendant” William Harrison Dempsey, known as Jack Dempsey, to recoverdamages for breach of a written contract executed March 13, 1926, but bearingdate of March 6 of that year.

Plaintiff was incorporated as an Illinois corporation for the promotion of generalpleasure and athletic purposes and to conduct boxing, sparring and wrestlingmatches and exhibitions for prizes or purses. Dempsey was well known in thepugilistism world and, at the time of the making and execution of the contractin question, held the title of world’s Champion Heavy Weight Boxer.

Dempsey was to engage in a boxing match with Harry Wills, another well-known boxer. At the signing of the contract, he was paid $10. Dempsey was tobe paid $800,000 plus 50 percent of “the net profits over and above the sum of$2,000,000 in the event the gate receipts should exceed that amount.” Further,he was to receive 50 percent of “the net revenue derived from moving pictureconcessions or royalties received by the plaintiff.” Dempsey was not to engagein any boxing match after the date of the agreement and before the date of thecontest. He was also “to have his life and health insured in favor of the plaintiffin a manner and at a place to be designated by the plaintiff.” The ChicagoColiseum Club was to promote the event. The contract between the ChicagoColiseum Club and Wills was entered into on March 6, 1926. It stated thatWills was to be payed $50,000. However, he was never paid.

The Chicago Coliseum Club hired a promoter. When it contacted Dempsey con-cerning the life insurance, Dempsey repudiated the contract with the followingtelegram message.

BM Colorado Springs Colo July 10th 1926

B. E. Clements

President Chicago Coliseum Club Chgo Entirely too busy training formy coming Tunney match to waste time on insurance representativesstop as you have no contract suggest you stop kidding yourself andme also Jack Dempsey.

2For a more detailed discussion of this case, see Barnett, R., Contracts: Cases and Doctrine, 2dEd. (Aspen 1999), p.125.




The court identified the following issues as being relevant in establishing dam-ages:

First: Loss of profits which would have been derived by the plaintiffin the event of the holding of the contest in question;

Second: Expenses incurred by the plaintiff prior to the signing of theagreement between the plaintiff and Dempsey;

Third: Expenses incurred in attempting to restrain the defendantfrom engaging in other contests and to force him into a compliancewith the terms of his agreement with the plaintiff; and

Fourth: Expenses incurred after the signing of the agreement andbefore the breach of July 10, 1926.

The Chicago Coliseum Club claimed that it would have had gross receipts of$3,000,000 and expenses of $1,400,000, which would have left a net profit of$1,600,000. However, the court was not convinced of this as there were too manyundetermined factors. (Unless shown otherwise, the court will generally assumethat the venture would have at least broken even. This could be comparedto the case where substantial evidence did exist as to the expected profits ofChicago Coliseum.) The expenses incurred before the contract was signed withDempsey could not be recovered as damages. Further, expenses incurred inrelation to 3 above could only be recovered as damages if they occured beforethe repudiation. The expense of 4 above could be recovered.

Stylized Example

The following technology of the relationship shows a possible interpretationwhen proof of the expected revenues is available.

This assumes that promotion by Chicago Coliseum Club benefits Dempsey’sreputation and allows him to gain by taking the other boxing match. The strate-gies for Chicago Coliseum are “promote” and “don’t promote.” The strategiesfor Dempsey are “take this match” and “take other match.” This example canbe used to illustrate a contract that would induce Dempsey to keep his agree-ment with Chicago Coliseum. Further, when it is assumed that the expectedprofit is zero, expectations and reliance damages result in the same transfer.



14 Details of the Extensive Form

This chapter elaborates on Chapter 2’s presentation of the extensive form represen-tation. The chapter defines some technical terms and states five rules that mustbe obeyed when designing game trees. The concepts of perfect recall and perfectinformation are registered.

Lecture Notes

This material can be covered very quickly in class, as a transition from normalform analysis to extensive form analysis. The key, simply, is to bring the extensiveform back to the front of the students’ minds, and in a more technically completemanner than was needed for Part I of the book. Here is an outline for a lecture.

• Review of the components of the extensive form: nodes, branches, labels, infor-mation sets, and payoffs; initial, decision, and terminal nodes.

• Terms describing the relation between nodes: successor, predecessor, immediatesuccessor, and immediate predecessor.

• Tree rules, with examples of violations.

• Perfect versus imperfect recall.

• Perfect versus imperfect information.

• How to describe an infinite action space.


1. Abstract examples can be developed on the fly to illustrate the terms and con-cepts.

2. Forgetful driver. This one-player game demonstrates imperfect recall. Theplayer is driving on country roads to a friend’s house at night. The playerreaches an intersection, where he must turn left or right. If he turns right, hewill find a police checkpoint, where he will be delayed for the entire evening.If he turns left, he will eventually reach another intersection requiring anotherright/left decision. At this one, a right turn will bring him to his friend’s house,while a left turn will take him to the police checkpoint. When he has to makea decision, the player does not recall how many intersections he passed throughor what decisions he made previously. The extensive form representation ispictured on the next page.



14 DETAILS OF THE EXTENSIVE FORM 42



15 Backward Induction

and Subgame Perfection

This chapter begins with an example to show that not all Nash equilibria of a gamemay be consistent with rationality in real time. The notion of sequential rationalityis presented, followed by backward induction (a version of conditional dominance)and then a demonstration of backward induction in an example. Next comes theresult that finite games of perfect information have pure strategy Nash equilibria(this result is used in Chapter 17 for the analysis of parlor games). The chapterthen defines subgame perfect Nash equilibrium as a concept for applying sequentialrationality in general games. An algorithm for computing subgame perfect equilibriain finite games is demonstrated with an example.

Lecture Notes

An outline for a lecture follows.

• Example of a game featuring a Nash equilibrium with an incredible threat.

• The definition of Sequential rationality.

• Backward induction: informal definition and abstract example. Note that thestrategy profile identified is a Nash equilibrium.

• Result: every finite game with perfect information has a (pure strategy) Nashequilibrium.

• Note that backward induction is difficult to extend to games with imperfectinformation.

• Subgame definition and illustrative example. Note that the entire game is itselfa subgame. Definition of proper subgame.

• Definition of subgame perfect Nash equilibrium.

• Example and algorithm for computing subgame perfect equilibria: (a) draw thenormal form of the entire game, (b) draw the normal forms of all other (proper)subgames, (c) find the Nash equilibria of the entire game and the Nash equilibriaof the proper subgames, and (d) locate the Nash equilibria of the entire gamethat specify Nash outcomes in all subgames.


1. Incredible threats example. It might be useful to discuss, for example, the cred-ibility of the Chicago Bulls of the 1990s threatening to fire Michael Jordan.



BACKWARD INDUCTION AND SUBGAME PERFECTION 44

2. Grab game. This is a good game to run as a classroom experiment immediatelyafter lecturing on the topic of subgame perfection. There is a very good chancethat the two students who play the game will not behave according to backwardinduction theory. You can discuss why they behave differently. In this game,two students take turns on the move. When on the move, a student can eithergrab all of the money in your hand or pass. At the beginning of the game, youplace one dollar in your hand and offer it to player 1. If player 1 grabs thedollar, then the game ends (player 1 gets the dollar and player 2 gets nothing).If player 1 passes, then you add another dollar to your hand and offer the twodollars to player 2. If she grabs the money, then the game ends (she gets $2and player 1 gets nothing). If player 2 passes, then you add another dollar andreturn to player 1. This process continues until either one of the players grabsthe money or player 2 passes when the pot is $21 (in which case the game endswith both players obtaining nothing).



16 Topics in Industrial Organization

This chapter presents several models to explore various strategic elements of marketinteraction. The chapter begins with a model of advertising and firm competition,followed by a model of limit capacity. In both of these models, firms make a techno-logical choice before competing with each other in a Cournot-style (quantity selection)arena. The chapter then develops a simple two-period model of dynamic monopoly,where a firm discriminates between customers by its choice of price over time. Thechapter ends with a variation of the dynamic monopoly model in which the firm caneffectively commit to a pricing scheme by offering a price guarantee. The models inthis chapter demonstrate a useful method of calculating subgame perfect equilibriain games with infinite strategy spaces. When it is known that each of a class of sub-games has a unique Nash equilibrium, one can identify the equilibrium and, treatingit as the outcome induced by the subgame, work backward to analyze the game tree.

Lecture Notes

Any or all of the models in this chapter can be discussed in class, dependingon time constraints and the students’ background and interest. Other equilibriummodels, such as the von Stackelberg model, can also be presented or substituted forany in the chapter. With regard to the advertising and limit capacity models (as wellas with others, such as the von Stackelberg game), the lecture can proceed as follows.

• Description of the real setting.


• Description of the game.

• Observe that there are an infinite number of proper subgames.

• Note that the proper subgames at the end of the game tree have unique Nashequilibria. Calculate the equilibrium of a subgame and write its payoff as a func-tion of the variables selected by the players earlier in the game (the advertisinglevel, the entry and production facility decisions).

• Analyze information sets toward the beginning of the tree, conditional on thepayoff specifications just calculated.

The dynamic monopoly game can be analyzed similarly, except it pays to stressintuition, rather than mathematical expressions, with this game.


Students would benefit from a discussion of real strategic situations, especiallywith an eye toward understanding how the strategic tensions are manifested. Also,any of the applications can be used for classroom experiments.



17 Parlor Games

In this chapter, two results stated earlier in the book (from Chapters 12 and 15) areapplied to analyze finite, strictly competitive games of perfect information. Manyparlor games, including chess, checkers, and tic-tac-toe, fit in this category. A resultis stated for games that end with a winner and a loser or a tie. A few examples arebriefly discussed.

Lecture Notes


• Describe the class of two-player, finite games of perfect information that arestrictly competitive.

• Examples: tic-tac-toe, checkers, chess, etc.

• Note that the result in Chapters 12 and 15 apply. Thus, these games have (purestrategy) Nash equilibria and the equilibrium strategies are security strategies.

• Result: If the game must end with a “winner,” then one of the players has astrategy that guarantees victory, regardless of what the other player does. Ifthe game ends with either a winner or a tie, then either one of the players hasa strategy that guarantees victory or both players can guarantee at least a tie.

• Discuss examples, such as chess, that have no known solution.

• Discuss simpler examples.


1. Chomp tournament. Chomp is a fun game to play in a tournament format, withthe students separated into teams. For the rules of Chomp, see Exercise 5 inChapter 17. Have the students meet with their team members outside of classto discuss how to play the game. The teams can then play against each other atthe end of a few class sessions. Give them several matrix configurations to play(symmetric and asymmetric) so that, for fairness, the teams can each play therole of player 1 and player 2 in the various configurations. After some thought(after perhaps several days), the students will ascertain a winning strategy forthe symmetric version of Chomp. An optimal strategy for the asymmetricversion will elude them, as it has eluded the experts. You can award bonuspoints based on the teams’ performance in the tournament. At some point, youcan also explain why we know that the first player has a winning strategy, whilewe do not know the actual winning strategy.



17 PARLOR GAMES 47

2. Another tournament or challenge. The students might enjoy, and learn from,playing other parlor games between themselves or with you. An after-classchallenge provides a good context for meeting with students in a relaxed envi-ronment.



18 Bargaining Problems

This chapter introduces the important topic of negotiation, a component of many eco-nomic relationships and theoretical models. The chapter commences by noting howbargaining can be put in terms of value creation and division. Several elements ofnegotiation—terms of trade, divisible goods—are noted. Then the chapter describesan abstract representation of bargaining problems in terms of the payoffs of feasi-ble agreements and the disagreement point. This representation is common in thecooperative game literature, where solution concepts are often expressed as axiomsgoverning joint behavior. Transferable utility is assumed. Joint value and surplusrelative to the disagreement point are defined and illustrated in an example. Thestandard bargaining solution is defined as the outcome in which the players maximizetheir joint value and divide the surplus according to fixed bargaining weights.

Lecture Notes


• Examples of bargaining situations.

• Translating a given bargaining problem into feasible payoff vectors (V ) and thedefault payoff vector d, also called the disagreement point.

• Transferable utility. Value creation means a higher joint value than with thedisagreement point. Recall efficiency definition.

• Divisible goods, such as money, that can be used to divide value.

• An example in terms of agreement items x and a transfer t, so payoffs arev1(x) + t and v2(x) − t.

• The standard bargaining solution: summarizing the outcome of negotiation interms of bargaining weights π1, π2. Assume the players reach an efficient agree-ment and divide the surplus according to their bargaining weights. Descriptiveand predictive interpretations.

• Player i’s negotiated payoff is u∗i = di+πi(v

∗−d1−d2). This implies that x = x∗

(achieving the maximized joint value v∗) and t∗ = d1 +π1(v∗−d1−d2)−u1(x

∗).



18 BARGAINING PROBLEMS 49


1. Negotiation experiments. It can be instructive—especially before lecturing onnegotiation problems—to present a real negotiation problem to two or morestudents. Give them a set of alternatives (such as transferring money, gettingmoney or other objects from you, and so on). It is most useful if the alternativesare multidimensional, with each dimension affecting the two players differently(so that the students face an interesting “enlarge and divide the pie” problem).For example, one alternative might be that you will take student 1 to lunchat the faculty club, whereas another might be that you will give one of them(their choice) a new jazz compact disc. The outcome only takes effect (enforcedby you) if the students sign a contract. You can have the students negotiateoutside of class in a completely unstructured way (although it may be usefulto ask the students to keep track of how they reached a decision). Have thestudents report in class on their negotiation and final agreement.

2. Anonymous ultimatum bargaining experiment. Let half of the students be theofferers and the other half responders. Each should write a strategy on a slipof paper. For the offerers, this is an amount to offer the other player. For aresponder, this may be an amount below which she wishes to reject the offer(or it could be a range of offers to be accepted). Once all of the slips havebeen collected, you can randomly match an offerer and responder. It may beinteresting to do this twice, with the roles reversed for the second run, and totry the non-anonymous version with two students selected in advance (in whichcase, their payoffs will probably differ from those of the standard ultimatumformulation). Discuss why (or why not) the students’ behavior departs fromthe subgame perfect equilibrium. This provides a good introduction to thetheory covered in Chapter 19.



19 Analysis of Simple Bargaining Games

This chapter presents the analysis of alternating-offer bargaining games and showshow the bargaining weights discussed in Chapter 18 are related to the order of movesand discounting. The ultimatum game is reviewed first, followed by a two-period gameand then the infinite-period alternating-offer game. The analysis features subgameperfect equilibrium and includes the motivation for, and definition of, discount factors.At the end of the chapter is an example of multilateral bargaining in the legislativecontext.

Lecture Notes

A lecture can proceed as follows.

• Description of the ultimatum-offer bargaining game, between players i and j(to facilitate analysis of larger games later). Player i offers a share between 0and 1; player j accepts or rejects.

• Determination of the two sequentially rational strategies for player j (the re-sponder), which give equilibrium play in the proper subgames: (*) accept alloffers, and (**) accept if and only if the offer is strictly greater than 0.

• Strategy (**) cannot be part of an equilibrium in the ultimatum game. Notethat this observation will be used in larger games later.

• The unique subgame perfect equilibrium specifies strategy (*) for player j andthe offer of 0 by player i. Bargaining weight interpretation of the outcome.

• Discounting over periods of time. Examples. Representing time preferences bya discount factor δi.

• Description of the two-period, alternating-offer game with discounting. De-termining the subgame perfect equilibrium using backward induction and theequilibrium of the ultimatum game. Bargaining weight interpretation of theoutcome.

• Description of the infinite-period, alternating-offer game with discounting. Sketchof the analysis: the subgame perfect equilibrium is stationary; mi is player i’sequilibrium payoff in subgames where he makes the first offer.

• Bargaining weight interpretation of the equilibrium outcome of the infinite-period game. Convergence as discount factors approach one.

• Brief description of issues in multilateral negotiation.



19 ANALYSIS OF SIMPLE BARGAINING GAMES 51


For a transition from the analysis of simple bargaining games to modeling jointdecisions, you might run a classroom experiment in which the players negotiate acontract that governs how they will play an underlying game in class. This combinesthe negotiation experiment described in the material for Chapter 18 with the contractexperiment in the material for Chapter 13.



20 Games with Joint Decisions;

Negotiation Equilibrium

This chapter introduces, and shows how to analyze, games with joint decision nodes.A joint decision node is a distilled model of negotiation between the players; it takesthe place of a noncooperative model of bargaining. Games with joint decision nodescan be used to study complicated strategic settings that have a negotiation compo-nent, where a full noncooperative model would be unwieldy. Contractual relationshipsoften have this flavor; there are times when the parties negotiate to form a contractand there are times in which the parties work on their own (either complying withtheir contract or failing to do so). Behavior at joint decision nodes is characterized bythe standard bargaining solution. Thus, a game with joint decision nodes is a hybridrepresentation, with cooperative and noncooperative components.

The chapter explains the benefits of composing games with joint decisions and, intechnical terms, demarcates the proper use of this representation. The term “regime”generalizes the concept of strategy to games with joint decisions. The concept of anegotiation equilibrium combines sequential rationality at individual decision nodeswith the standard bargaining solution at joint decision nodes. The chapter illustratesthe ideas with an example of an incentive contract.

Lecture Notes

Here is an outline for a lecture.

• Noncooperative models of negotiation, such as those just analyzed, can be com-plicated. In many strategic settings, negotiation is just one of the key compo-nents.

• It would be nice to build models in which the negotiation component werecharacterized by the standard bargaining solution. Then we could examinehow bargaining power and disagreement points influence the outcome, whileconcentrating on other strategic elements.

• Definition of a game with joint decisions—distill a negotiation component intoa joint decision (a little model of bargaining).

• Example: the extensive form version of a bargaining problem, utilizing a jointdecision node. Recall the pictures and notation from Chapter 18.

• Always include a default decision to describe what happens if the players donot reach an agreement.

• Labeling the tree. Tree Rule 6.

• Definition of regime: a specification of behavior at both individual and jointdecision nodes.



JOINT DECISIONS AND NEGOTIATION EQUILIBRIUM 53

• Negotiation equilibrium: sequential rationality at individual decision nodes;standard bargaining solution at joint decision nodes.

• Example of a contracting problem, modeling by a game with a joint decision.

• Calculating the negotiation equilibrium by backward induction. First determinethe effort incentive, given a contract. Then, using the standard bargainingsolution, determine the optimal contract and how the surplus will be divided.


1. Agency incentive contracting. You can run a classroom experiment where threestudents interact as follows. Students 1 and 2 have to play a matrix game.Specify a game that has a single rationalizable (dominance-solvable) strategy sbut has another outcome t that is strictly preferred by player 1 [u1(t) > u1(s)]and has the property that t2 is the unique best response to t1. Student 1 isallowed to contract with student 3 so that student 3 can play the matrix gamein student 1’s place (as student 1’s agent). The contract between students 1and 3 (which you enforce) can specify transfers between them as a function ofthe matrix game outcome.

You can arrange the experiment so that the identities of students 1 and 3are not known to student 2 (by, say, allowing many pairs of students to writecontracts and then selecting a pair randomly and anonymously, and by payingthem privately). After the experiment, discuss why you might expect t, ratherthan s, to be played in the matrix game. To make the tensions pronounced,make s an efficient outcome.

2. Ocean liner shipping-contract example. A producer who wishes to ship a moder-ate shipment of output (say three or four full containers) overseas has a choice ofthree ways of shipping the product. He can contract directly with the shipper,he can contract with an independent shipping contractor (who has a contractwith a shipper), or he can use a trade association that has a contract with ashipper. The producer himself must negotiate if he chooses either of the firsttwo alternatives, but in the third the trade association has a non-negotiable feeof 45. Shipping the product is worth 100 to the producer. Suppose that theproducer only has time to negotiate with one of the parties because his productis perishable, but in the event of no agreement he can use the trade association.The shipper’s cost of the shipment is 20. The shipping contractor’s cost is 30.



21 Unverifiable Investment, Hold Up,

Options, and Ownership

This chapter applies the concept of joint decisions and negotiation equilibrium toillustrate the hold up problem. An example is developed in which one of the playersmust choose whether to invest prior to production taking place. Three variations arestudied, starting with the case in which a party must choose his/her investment levelbefore contracting with the other party (so here hold up creates a serious problem).In the second version, parties can contract up front; here, option contracts are shownto provide optimal incentives. The chapter also comments on how asset ownershipcan help alleviate the hold up problem.

Lecture Notes


• Description of tensions between individual and joint interests because of thetiming of unverifiable investments and contracting. Related to externality.

• Hold up example: unverifiable investment followed by negotiation over the re-turns, where agreement is required to realize the returns.

• Calculate the negotiation equilibrium by backward induction. Find the outcomeand payoffs from the joint decision node, using the standard bargaining solution.Then determine the rational investment choice.

• Note the incentive to underinvest, relative to the efficient amount.

• Consider up-front contracting and option contracts. Describe how option con-tracts work and are enforced.

• Show that a particular option contract leads to the efficient outcome. Calculateand describe the negotiation equilibrium.

• Extension of the model in which the value of the investment is tied to an asset,which has a value in the relationship and another value outside of the relation-ship.

• Ownership of the asset affects the disagreement point (through the outsidevalue) and thus affects the outcome of negotiation.

• Find the negotiation equilibrium for the various ownership specifications.

• Investor ownership is preferred if the value of the asset in its outside use riseswith the investment. This may not be true in general. If the outside asset valuerises too quickly with the investment, then the investor may have the incentiveto overinvest.



INVESTMENT AND HOLD UP 55


You can discuss real examples of hold up, such as those having to do with specifichuman capital investment, physical plant location, and unverifiable investments inlong-term procurement contracting. You can also present the analysis of, or run anexperiment based on, a game like that of the Guided Exercise in this chapter.



22 Repeated Games and Reputation

This chapter opens with comments about the importance of reputation in ongoingrelationships. The concept of a repeated game is defined and a two-period repeatedgame is analyzed in detail. The two-period game demonstrates that any sequence ofstage Nash profiles can be supported as a subgame perfect equilibrium outcome (aresult that is stated for general repeated games). The example also shows how a non-stage Nash profile can be played in equilibrium if subsequent play is conditioned sothat players would be punished for deviating. The chapter then turns to the analysisof infinitely repeated games, beginning with a review of discounting. The presenta-tion includes derivation of the standard conditions under which cooperation can besustained in the infinitely repeated prisoners’ dilemma. In the following section, amore complicated, asymmetric equilibrium is constructed to demonstrate that differ-ent forms of cooperation, favoring one or the other player, can also be supported. ANash-punishment folk theorem is stated at the end of the chapter.

Lecture Notes

A lecture may be organized according to the following outline.

• Intuition: reputation and ongoing relationships. Examples: partnerships, col-lusion, etc.

• Key idea: behavior is conditioned on the history of the relationship, so thatmisdeeds are punished.

• Definition of a repeated game. Stage game {A, u} (call Ai actions), played Ttimes with observed actions.

• Example of a two-period (non-discounted) repeated game.

• Diagram of the feasible repeated game payoffs and feasible stage game payoffs.

• Note how many subgames there are. Note what each player’s strategy specifies.

• The proper subgames have the same strategic features, since the payoff matricesfor these are equal, up to a constant. Thus, the equilibria of the subgames arethe same as those of the stage game.

• Characterization of subgame perfect equilibria featuring only stage Nash profiles(action profiles that are equilibria of the stage game).

• A reputation equilibrium where a non-stage Nash action profile is played in thefirst period. Note the payoff vector.

• Review of discounting.



22 REPEATED GAMES AND REPUTATION 57

• The infinitely repeated prisoners’ dilemma game.

• Trigger strategies. Grim trigger.

• Conditions under which the grim trigger is a subgame perfect equilibrium.

• Example of another “cooperative” equilibrium. The folk theorem.


1. Two-period example. It is probably best to start a lecture with the simplestpossible example, such as the one with a 3 × 2 stage game that is presented atthe beginning of this chapter. You can also run a classroom experiment basedon such a game. Have the students communicate in advance (either in pairsor as a group) to agree on how they will play the game. That is, have thestudents make a self-enforced contract. This will hopefully get them thinkingabout history-dependent strategies. Plus, it will reinforce the interpretation ofequilibrium as a self-enforced contract, which you may want to discuss near theend of a lecture on reputation and repeated games.

2. The Princess Bride reputation example. At the beginning of your lecture onreputation, you can play the scene from The Princess Bride in which Wesley isreunited with the princess. Just before he reveals his identity to her, he makesinteresting comments about how a pirate maintains his reputation.



23 Collusion, Trade Agreements,

and Goodwill

This chapter presents three applications of repeated game theory: collusion betweenfirms over time, the enforcement of international trade agreements, and goodwill.The first application involves a straightforward calculation of whether collusion canbe sustained using grim trigger strategies in a repeated Cournot model. This examplereinforces the basic analytical exercise from Chapter 22. The section on internationaltrade is a short verbal discussion of how reputation functions as the mechanism forself-enforcement of a long-term contract. On goodwill, a two-period game with asequence of players 2 (one in the first period and another in the second period) isanalyzed. The first player 2 can, by cooperating in the first period, establish a valuablereputation that he can then sell to the second player 2.

Lecture Notes

Any or all of the applications can be discussed in class, depending on time con-straints and the students’ background and interest. Other applications can also bepresented, in addition to these or substituting for these. For each application, it maybe helpful to organize the lecture as follows.

• Description of the real-world setting.


• (If applicable) Description of the game to be analyzed.

• Determination of conditions under which an interesting (cooperative) equilib-rium exists.


• Notes on how the model could be extended.


1. The Princess Bride second reputation example. Before lecturing on goodwill,you can play the scene from The Princess Bride where Wesley and Buttercupare in the fire swamp. While in the swamp, Wesley explains how a reputationcan be associated with a name, even if the name changes hands over time.



COLLUSION, TRADE AGREEMENTS, AND GOODWILL 59

2. Goodwill in an infinitely repeated game. If you want to be ambitious, you canpresent a model of an infinitely repeated game with a sequence of players 2who buy and sell the “player 2 reputation” between periods. This can followthe Princess Bride scene and be based on Exercise 4 of this chapter (which,depending on your students’ backgrounds, may be too difficult for them to doon their own).

3. Repeated Cournot oligopoly experiment. Let three students interact in a re-peated Cournot oligopoly. This may be set as an oil (or some other commodity)production game. It may be useful to have the game end probabilistically. Thismay easy to do if it is done by e-mail, but may require a set time frame if donein class. The interaction can be done in two scenarios. In the first, players maynot communicate, and only the total output is announced at the end of eachround. In the second scenario, players are allowed to communicate and eachplayer’s output is announced at the end of each round.



24 Random Events and

Incomplete Information

This chapter explains how to incorporate exogenous random events in the specificationof a game. Moves of Nature (also called the nonstrategic “player 0”) are made atchance nodes according to a fixed probability distribution. As an illustration, the giftgame is depicted in the extensive form and then converted into the Bayesian normalform (where payoffs are the expected values over Nature’s moves). Another abstractexample follows.

Lecture Notes


• Discussion of settings in which players have private information about strategicaspects beyond their physical actions. Private information about preferences:auctions, negotiation, etc.

• Modeling such a setting using moves of Nature that players privately observe.(For example, the buyer knows his own valuation of the good, which the sellerdoes not observe.)

• Extensive form representation of the example. Nature moves at chance nodes,which are represented as open circles. Nature’s probability distribution is notedin the tree.

• The notion of a type, referring to the information that a player privately ob-serves. If a player privately observes some aspect of Nature’s choices, then thegame is said to be of incomplete information.

• Many real settings might be described in terms of players already knowing theirown types. However, because of incomplete information, one type of player willhave to consider how he would have behaved were he a different type (becausethe other players consider this).

• Bayesian normal form representation of the example. Note that payoff vectorsare averaged with respect to Nature’s fixed probability distribution.

• Other examples.



RANDOM EVENTS AND INCOMPLETE INFORMATION 61


1. The Let’s Make a Deal game revisited. You can illustrate incomplete informationby describing a variation of the Let’s Make a Deal game that is described inthe material for Chapter 2. In the incomplete-information version, Nature pickswith equal probabilities the door behind which the prize is concealed and Montyrandomizes equally between alternatives when he has to open one of the doors.

2. Three-card poker. This game also makes a good example (see Exercise 4 inChapter 24 of the textbook).

3. Ultimatum-offer bargaining with incomplete information. You might present,or run as a classroom experiment, an ultimatum bargaining game in which theresponder’s value of the good being traded is private information (say, $5 withprobability 1/2 and $8 with probability 1/2). For an experiment, describe thegood as a soon-expiring check made out to player 2. You show player 2 theamount of the check, but you seal the check in an envelop before giving it toplayer 1 (who bargains over the terms of trading it to player 2).

4. Signaling games. It may be worthwhile to describe a signaling game that youplan to analyze later in class.

5. The Price is Right. The bidding game from this popular television game showforms the basis for a good bonus question. (See also Exercise 5 in Chapter 25for a simpler, but still challenging, version.) In the game, four contestants mustguess the price of an item. Suppose none of them knows the price of the iteminitially, but they all know that the price is an integer between 1 and 1, 000. Infact, when they have to make their guesses, the contestants all believe that theprice is equally likely to be any number between 1 and 1, 000. That is, the pricewill be 1 with probability 1/1000, the price will be 2 with probability 1/1000,and so on.

The players make their guesses sequentially. First, player 1 declares his/herguess of the price, by picking a number between 1 and 1, 000. The other playersobserve player 1’s choice and then player 2 makes her guess. Player 3 nextchooses a number, followed by player 4. When a player selects a number,he/she is not allowed to pick a number that one of the other players alreadyhad selected.

After the players make their guesses, the actual price is revealed. Then theplayer whose guess is closest to the actual price without going over wins $100.The other players get 0. For example, if player 1 chose 150, player 2 chose 300,player 3 selected 410, and player 4 chose 490, and if the actual price were 480,then player 3 wins $100 and the others get nothing.




This game is not exactly the one played on The Price is Right, but it is close.The bonus question is: Assuming that a subgame perfect equilibrium is played,what is player 1’s guess? How would the answer change if, instead of the winnergetting $100, the winner gets the value of the item (that is, the actual price)?



25 Risk and Incentives in Contracting

This chapter presents the analysis of the classic principal-agent problem under moralhazard, where the agent is risk-averse. There is a move of Nature (a random produc-tive outcome). Because Nature moves last, the game has complete information. Thus,it can be analyzed using subgame perfect equilibrium. This is why the principal-agentmodel is the first, and most straightforward, application covered in Part IV of thebook.

At the beginning of the chapter, the reader will find a thorough presentationof how payoff numbers represent preferences over risk. An example helps explainthe notions of risk aversion and risk premia. The Arrow-Pratt measure of relativerisk aversion is defined. Then a streamlined principal-agent model is developed andfully analyzed. The relation between the agent’s risk attitude and the optimal bonuscontract is determined.

Lecture Notes

Analysis of the principal-agent problem is fairly complicated. Instructors will notlikely want to develop in class a more general and complicated model than the onein the textbook. A lecture based on the textbook’s model can proceed as follows.

• Example of a lottery experiment/questionnaire that is designed to determinethe risk preferences of an individual.

• Representing the example as a simple game with Nature.

• Note that people usually are risk averse in the sense that they prefer the ex-pected value of a lottery over the lottery itself.

• Observe the difference between an expected monetary award and expected util-ity (payoff).

• Risk preferences and the shape of the utility function on money. Concavity,linearity, etc.

• Arrow-Pratt measure of relative risk aversion.

• Intuition: contracting for effort incentives under risk.

• The principal-agent model. Risk neutral principal.

• Incentive compatibility and participation constraints. They both will bind atthe principal’s optimal contract offer.

• Calculation of the equilibrium. Note how the contract and the agent’s behaviordepend on the agent’s risk preferences.



25 RISK AND INCENTIVES IN CONTRACTING 64

• Discussion of real implications.


You can illustrate risk-aversion by offering choices over real lotteries to the stu-dents in class. Discuss risk aversion and risk premia.



26 Bayesian Nash Equilibrium

and Rationalizability

This chapter shows how to analyze Bayesian normal form games using rationalizabilityand equilibrium theory. Two methods are presented. The first method is simply toapply the standard definitions of rationalizability and Nash equilibrium to Bayesiannormal forms. The second method is to apply the concepts by treating different typesof a player as separate players. The two methods are equivalent whenever all typesare realized with positive probability (an innocuous assumption for static settings).Computations for some finite games exemplify the first method. The second methodis shown to be useful when there are continuous strategy spaces, as illustrated usingthe Cournot duopoly with incomplete information.

Lecture Notes


• Examples of performing standard rationalizability and equilibrium analysis toBayesian normal form games.

• Another method that is useful for more complicated games (such as those withcontinuous strategy spaces): treat different types as different players. One canuse this method without having to calculate expected payoffs over Nature’smoves for all players.

• Example of the second method: Cournot duopoly with incomplete informationor a different game.


You can run a common- or private-value auction experiment or a lemons experi-ment in class as a transition to the material in Chapter 27. You might also considersimple examples to illustrate the method of calculating best responses for individualplayer-types.



27 Lemons, Auctions,

and Information Aggregation

This chapter focuses on three important settings of incomplete information: price-taking market interaction, auctions, and information aggregation through voting.These settings are studied using static models, in the Bayesian normal form, and thegames are analyzed using the techniques discussed in the preceding chapter. The“markets and lemons” game demonstrates Akerlof’s major contribution to informa-tion economics. Regarding auctions, the chapter presents the analysis of both first-price and second-price formats. In the process, weak dominance is defined and therevenue equivalence result is mentioned. The example of voting and informationaggregation gives a hint of standard mechanism-design/social-choice analysis and il-lustrates Bayes’ rule.

Lecture Notes

Any or all of these applications can be discussed in class, depending on timeconstraints and the students’ background and interest. The lemons model is quitesimple; a lemons model that is more general than the one in the textbook can easilybe covered in class. The auction analysis, on the other hand, is more complicated.However, the simplified auction models are not beyond the reach of most advancedundergraduates. The major sticking points are (a) explaining the method of assuminga parameterized form of the equilibrium strategies and then calculating best responsesto verify the form and determine the parameter, (b) the calculus required to calculatebest responses, and (c) double integration to establish revenue equivalence. One canskip (c) with no problem. The information aggregation example requires students towork through Bayes’ rule calculations.

For each application, it may be helpful to organize the lecture as follows.



• Description of the game to be analyzed.

• Calculations of best responses and equilibrium. Note whether the equilibriumis unique.





LEMONS, AUCTIONS, AND INFORMATION AGGREGATION 67


1. Lemons experiment. Let one student be the seller of a car and another be thepotential buyer. Prepare some cards with values written on them. Show thecards to both of the students and then, after shuffling the cards, draw one atrandom and give it to student 1 (so that student 1 sees the value but student 2does not). Let the students engage in unstructured negotiation over the terms oftrading the card from student 1 to student 2, or allow them to declare whetherthey will trade at a prespecified price. Tell them that whomever has the cardin the end will get paid. If student 1 has the card, then she gets the amountwritten on it. If student 2 has the card, then he gets the amount plus a constant($2 perhaps).

2. Stock trade and auction experiments. You can run an experiment in whichrandomly-selected students play a trading game like that of Exercise 8 in thischapter. Have the students specify on paper the set of prices at which they arewilling to trade. You can also organize the interaction as a common-value auc-tion, or run any other type of auction in class. You can discuss the importanceof expected payoffs contingent on winning or trading.



28 Perfect Bayesian Equilibrium

This chapter develops the concept of perfect Bayesian equilibrium for analyzing be-havior in dynamic games of incomplete information. The gift game is utilized through-out the chapter to illustrate the key ideas. First, the example is used to demonstratethat subgame perfection does not adequately represent sequential rationality. Thencomes the notion of conditional belief, which is presented as the belief of a player atan information set where he has observed the action, but not the type, of anotherplayer. Sequential rationality is defined as action choices that are optimal in responseto the conditional beliefs (for each information set). The chapter then covers thenotion of consistent beliefs and Bayes’ rule. Finally, perfect Bayesian equilibrium isdefined and put to work on the gift game.

Lecture Notes


• Example to show that subgame perfection does not adequately capture sequen-tial rationality. (A simple signaling game will do.)

• Sequential rationality requires evaluating behavior at every information set.

• Conditional belief at an information set (regardless of whether players origi-nally thought the information set would be reached). Initial belief about types;updated (posterior) belief.

• Sequential rationality: optimal actions given beliefs (like best response, but withactions at a particular information set rather than full strategies).

• Consistency: updating should be consistent with strategies and the basic defi-nition of conditional probability. Bayes’ rule. Note that conditional beliefs areunconstrained at zero-probability information sets.

• Perfect Bayesian equilibrium: strategies, beliefs at all information sets, suchthat (1) each player’s strategy prescribes optimal actions at all of his informationsets, given his beliefs and the strategies of the other players, and (2) the beliefsare consistent with Bayes’ rule wherever possible.

• Definition of pooling and separating equilibria.

• Algorithm for finding perfect Bayesian equilibria in a signaling game: (a) posita strategy for player 1 (either pooling or separating), (b) calculate restrictionson conditional beliefs, (c) calculate optimal actions for player 2 given his beliefs,and (d) check whether player 1’s strategy is a best response to player 2’s strategy.

• Calculations for the example.



28 PERFECT BAYESIAN EQUILIBRIUM 69


1. Conditional probability demonstration. Students can be given cards with differ-ent colors written on them, say “red” and “blue.” The colors should be given indifferent proportions to males and females (for example, males could be givenproportionately more cards saying red and females could be given proportion-ately more cards saying blue). A student could be asked to guess the color ofanother student’s card. This could be done several times, and the color revealedfollowing the guess. Then a male and female student could be selected, and astudent could be asked to guess who has, for example, the red card.

2. Signaling game experiment. It may be instructive to play in class a signalinggame in which one of the player-types has a dominated strategy. The variantof the gift game discussed at the beginning of Chapter 28 is such a game.

3. The Princess Bride signaling example. A scene near the end of The PrincessBride movie is a good example of a signaling game. The scene begins withWesley lying in a bed. The prince enters the room. The prince does not knowwhether Wesley is strong or weak. Wesley can choose whether or not to stand.Finally, the prince decides whether to fight or surrender. This game can bediagrammed and discussed in class. After specifying payoffs, you can calculatethe perfect Baysian equilibria and discuss whether it accurately describes eventsin the movie. Exercise 6 in this chapter sketches one model of this strategicsetting.



29 Job-Market Signaling and Reputation

This chapter presents two applications of perfect Bayesian equilibrium: job-marketsignaling and reputation with incomplete information. The signaling model demon-strates Michael Spence’s major contribution to information economics. The repu-tation model illustrates how incomplete information causes a player of one type topretend to be another type, which has interesting implications. This offers a glimpseof the reputation literature initiated by David Kreps, Paul Milgrom, John Roberts,and Robert Wilson.

Lecture Notes

Either or both of these applications can be discussed in class, depending on timeconstraints and the students’ background and interest. The extensive form tree of thejob-market signaling model is in the standard signaling-game format, so this modelcan be easily presented in class. The reputation model may be slightly more difficultto present, however, because its extensive form representation is a bit different andthe analysis does not follow the algorithm outlined in Chapter 28.

For each application, it may be helpful to organize the lecture as follows.



• Description of the game to be analyzed.

• Calculating the perfect Bayesian equilibria (using the circular algorithm fromChapter 28, if appropriate).




In addition to, or in place of, the applications presented in this chapter, you mightlecture on the problem of contracting with adverse selection. Exercise 9 of Chapter 29would be suitable as the basis for such a lecture. This is a principal-agent game, wherethe principal offers a menu of contracts to screen between two types of the agent. Youcan briefly discuss the program of mechanism design theory as well.



30 Appendices

Appendix A offers an informal review of the following relevant mathematical topics:sets, functions, basic differentiation, and probability theory. Your students can con-sult this appendix to brush up on the mathematics skills that are required for gametheoretic analysis. As noted at the beginning of this manual, calculus is used spar-ingly in the textbook and it can be avoided. In addition, where calculus is utilized,it usually amounts to a simple exercise in differentiating a second-degree polynomial.If you wish to cover the applications/examples to which the textbook applies differ-entiation, and if calculus is not a prerequisite for your course, you can simply teach(or have them read on their own) the short section entitled “Functions and Calculus”in Appendix A.

Appendix B gives some of the details of the rationalizability construction. Ifyou want the students to see some of the technical details behind the differencebetween correlated and uncorrelated conjectures, the relation between dominanceand best response, or the rationalizability construction, you can advise them to readAppendix B just after reading Chapters 6 and 7. Three challenging mathematicalexercises appear at the end of Appendix B.



72

Part III

Solutions to the ExercisesThis part contains solutions to all of the exercises in the textbook. Although weworked diligently on these solutions, there are bound to be a few typos here andthere. Please report any instances where you think you have found a substantialerror.



2 The Extensive Form

1.

2.

(a)

(b) Incomplete information. The worker does not know who has hiredhim/her.




3.

Note that we have not specified payoffs as these are left to the students.

4.

The order does not matter as it is a simultaneous move game.




5.

The payoffs below are in the order A, B, C.

6.




7.

8.



3 Strategies and the Normal Form

1.

Exercise 1: SL = {A,B}. SM = {Rr, Rg, Gr, Gg}. SJ = {Aa, Ab, Ba,Bb }. Exercise 4: Si = {R,P,S}, i = 1, 2.

2.

No, “not hire” does not describe a strategy for the manager. A strat-egy for the manager must specify an action to be taken in every contin-gency. However, “not hire” does not specify any action contingent uponthe worker being hired and exerting a specific level of effort.

3.

(a)

(b)




(c)

(d)

(e)




(f)

4. Player 2 has 4 strategies: {(c, f), (c, g), (d, f), (d, g)}.5.

The normal form specifies player, strategy spaces, and payoff functions.Here N = {1, 2}. Si = [0,∞). The payoff to player i is give by ui(qi, qj) =(2 − qi − qj)qi.

6.

N = {1, 2}. S1 = [0,∞). Player 2’s strategy must specify a choice ofquantity for each possible quantity player 1 can choose. Thus, player 2’sstrategy space S2 is the set of functions from [0,∞) to [0,∞).




7.

Some possible extensive forms are shown below and on the next page.

(a)




(b)



4 Beliefs, Mixed Strategies,

and Expected Payoffs

1.

(a) u1(U,C) = 0.

(b) u2(M,R) = 4.

(c) u2(D,C) = 6.

(d) For σ1 = (1/3, 2/3, 0) u1(σ1,C) = 1/3(0) + 2/3(10) + 0 = 6 2/3.

(e) u1(σ1,R) = 5 1/4.

(f) u1(σ1, L) = 2.

(g) u2(σ1, R) = 3 2/3.

(h) u2(σ1, σ2) = 4 1/2.

2.

(a)

(b) Player 1’s expected payoff of playing H is z. His expected payoff ofplaying L is 5. For z = 5, player 1 is indifferent between playing H or L.

(c) Player 1’s expected payoff of playing L is 20/3.

3.

(a) u1(σ1,I) = 1/4(2) + 1/4(2) + 1/4(4) + 1/4(3) = 11/4.

(b) u2(σ1,O) = 21/8.

(c) u1(σ1, σ2) = 2(1/4) + 2(1/4) + 4(1/4)(1/3) + 1/4(2/3) + 3/4(1/3) +14(2/3) = 23/12.

(d) u1(σ,σ2) = 7/3.



BELIEFS AND EXPECTED PAYOFFS 83

4.

Note that all of these, except “Pigs,” are symmetric games.

Matching Pennies: u1(σ1, σ2) = u2(σ1, σ2) = 0.

Prisoners’ Dilemma: u1(σ1, σ2) = u2(σ1, σ2) = 1 1/2.

Battle of the Sexes: u1(σ1, σ2) = u2(σ1, σ2) = 3/4.

Hawk-Dove/Chicken: u1(σ1, σ2) = u2(σ1, σ2) = 1 1/2.

Coordination: u1(σ1, σ2) = u2(σ1, σ2) = 1/2.

Pareto Coordination: u1(σ1, σ2) = u2(σ1, σ2) = 3/4.

Pigs: u1(σ1, σ2) = 3, u2(σ1, σ2) = 1.

5.

The expected profit of player 1 is (100 − 28 − 20)14 − 20(14) = 448.



6 Dominance and Best Response

1.

(a) B dominates A and L dominates R.

(b) L dominates R.

(c) 2/3 U 1/3 D dominates M. X dominates Z.

(d) none.

2.

(a) To determine the BR set we must determine which strategy of player 1yields the highest payoff given her belief about player 2’s strategy selec-tion. Thus, we compare the payoff to each of her possible strategies.

u1(U,θ2) = 1/3(10) + 0 + 1/3(3) = 13/3.

u1(M,θ2) = 1/3(2) + 1/2(10) + 1/3(6) = 6.

u1(D,θ2) = 1/3(3) + 1/3(4) + 1/3(6) = 13/3.

BR1(θ2) = {M}.(b) BR2(θ1) = {L,R}.(c) BR1(θ2) = {U,M}.(d) BR2(θ1) = {C}.

3.

Player 1 solves maxq1(100 − 2q1 − 2q2)q1 − 20q1. The first order conditionis 100 − 4q1 − 2q2 − 20 = 0. Solving for q1 yields BR1(q2) = 20 − q2/2.It is easy to see that BR1(0) = 20. Since q2 ≥ 0, it cannot be that 25 isever a best response. Given the beliefs, player 1’s best response is 15.

4.

(a) First we find the expected payoff to each strategy: u1(U, θ2) = 2/6+0+4(1/2) = 7/3; u1(M, θ2) = 3(1/6)+1/2 = 1; and u1(D, θ2) = 1/6+1+1 =13/6. As the strategy U yields a higher expected payoff to player 1, givenθ2, BR1(θ2) = {U}.(b) BR2(θ1) = {R}.(c) BR1(θ2) = {U}.(d) BR1(θ2) = {U, D}.(e) BR2(θ1) = {L, R}.




5.

(a) BR1(θ2) = {P}.(b) BR1(θ2) = {R, S}.(c) BR1(θ2) = {P}.(d) BR1(θ2) = S1.

6.

No. This is because 1/2 A 1/2 B dominates C.

7.

M is dominated by (1/3, 2/3, 0).

8.

From exercise 3, BR1(q2) = 20 − q2/2. So UD1 = [0, 20].



7 Rationalizability and Iterated Dominance

1.

(a) R = {U, M, D} × {L, R}.(b) Here there is a dominant strategy. So we can iteratively delete dom-inated strategies. U dominates D. When D is ruled out, R dominates C.Thus, R = {U, M} × {L, R}.(c) R = {(U, L)}.(d) R = {A, B} × {X, Y}.(e) R = {A, B} × {X, Y}.(f) R = {A, B} × {X, Y}.(g) R = {(D, Y)}.

2.

For “give in” to be rationalizable, it must be that x ≤ 0. The man-ager must believe that the probability that the employee plays “settle” is(weakly) greater than 1/2.

3.

R = {(x, c)}. The order does not matter because if a strategy is domi-nated (not a best response) relative to some set of strategies of the otherplayer, then this strategy will also be dominated relative to a smaller setof strategies for the other player.

4.

R = {(7:00, 6:00, 6:00)}.



7 RATIONALIZABILITY AND ITERATED DOMINANCE 87

5.

Yes. If s1 is rationalizable, then s2 is a best response to a strategy ofplayer 1 that may rationally be played. Thus, player 2 can rationalizestrategy s2.

6.

No. It may be that s1 is rationalizable because it is a best response to someother rationalizable strategy of player 2, say s2, and just also happens tobe a best response to s2.

7.

R = {(0, 0, 0, 0, 0, 0, 0, 0, 0, 0)}. Note that player u10 = (a − 10 − 1)s10

and that a − 11 < 0 since a is at most 10. So player 10 has a singleundominated strategy, 0. Given this, we know a will be at most 9 (ifeveryone except player 10 selects 9). Thus, a − 10 < 0 and so player 9must select 0. by induction, every player selects 0.



8 Location and Partnership

1.

We label the regions as shown below.

We first find the best response sets. Noticing the symmetry makes thiseasier. BRi(1) = {2, 4, 5}; BRi(2) = {5}; BRi(3) = {2, 5, 6}; BRi(4) ={5}; BRi(5) = {5}; BRi(6) = {5}; BRi(7) = {4, 5, 8}; BRi(8) = {5};and BRi(9) = {5, 6, 8}. It is easy to see that {1, 3, 7, 9} are never bestresponses. Thus, R1

i = {2, 4, 5, 6, 8}. Since player i knows that player jis rational, he/she knows that j will never play {1, 3, 7, 9}. This impliesR2

i = Ri = {5}.

2.

For x < 80 locating in region 2 dominates locating in region 1.

3.

(a) Yes, preferences are as modeled in the basic location game. When theeach player’s objective is to maximize his/her probability of winning, thebest response set is not unique. Suppose, for example, that player 2 plays1 then BR1 = {2, 3, 4, . . . , 8}.(b) Here, we should focus on R2

i = {3, 4, 5, 6, 7}. It is easy to see thatif the regions are divided in half between 5 and 6 that 250 is distributedto each half. So unlike in the basic location model there is not a singleregion that is “in the middle”. Thus, R = {5,6}× {5,6}. In any of theseoutcomes, each candidate receives the same number of votes.

(c) When x > 75, player i’s best response to 5 is 6, and his/her bestresponse to 6 is 6. Thus, R = {(6, 6)}.When x < 75, player i’s best response to 6 is 5, and his/her best responseto 5 is 5. Thus, R = {(5, 5)}.




4.

Recall from the text that BR1(y) = 1+cy, and BR2(x) = 1+cx. Assume−1 < c < 0. This yields the following graph of best response functions.

As neither player will ever optimally exert effort that is greater than 1,R1

i = [0, 1]. Realizing that player j’s rational behavior implies this, R2i =

[1 + c, 1]. Continuing yields R3i = [1 + c, 1 + c + c2]. Repeating yields

Ri = { 1+c1−c2

} = 11−c

.

Repeat of analysis for c > 1/4: Recall from the text that BR1(y) = 1+cy,and BR2(x) = 1 + cx. Assume 1/4 < c ≤ 3/4. This yields the followinggraph of best response functions.

Because player i will never optimally exert effort that is either less than1 or greater than 1 + 4c, we have R1

i = [1, 1 + 4c]. Because the players




know this about each other, we have R2i = [1+ c, 1+ c(1+4c)]. Repeating

yields Ri = { 1+c1−c2

} = 11−c

.

Next suppose that c > 3/4. In this case, the functions x = 1+ cy and y =1+cx suggest that players would want to select strategies that exceed 4 inresponse to some beliefs. However, remember that the players’ strategiesare constrained to be less than or equal to 4. Thus, the best responsefunctions are actually

BR1(y) =

{1 + cy if 1 + cy ≤ 44 if 1 + cy > 4

and

BR2(x) =

{1 + cx if 1 + cx ≤ 44 if 1 + cx > 4

.

In this case, the best response functions cross at (4, 4), and this is theonly rationalizable strategy profile.

5.

(a) u1(p1, p2) = [10 − p1 + p2]p1. u2(p1, p2) = [10 − p2 + p1]p2.

(b) ui(p1, p2) = 10pi − p2i + pjpi. As above, we want to solve for pi that

maximizes i’s payoff given pj. Solving for the first order condition yieldspi(pj) = 5 + 1/2pj.

(c) Here there is no bound to the price a player can select. Thus, wedo not obtain a unique rationalizable strategy profile. The best responsefunctions are represented below.

Similar to the above, we have R1i = [5,∞) and R2

i = [15/2,∞). Repeatingthe analysis yields Ri = [10,∞) for i = 1, 2.




6.

(a) No.

(b) σi = (0, p, 0, 0, 1 − p, 0) dominates locating in region 1, for all p ∈(1/2, 1).

7.

Player 1 chooses x to maximize u1(x, y) = 2xy − x2. The first ordercondition implies x = y. Thus, BR1(y) = y. Similarly, player 2 chooses yto maximize u2(x, y) = 4xy−y2. The first order condition implies y = 2x,but y ∈ [2, 8]. So

BR2(x) =

{2x if x ≤ 48 if x > 4

.

So R = {(8, 8)}.



9 Nash Equilibrium

1.

(a) The Nash equilibria (w,b) and (y,c).

(b) (y,c) is efficient.

(c) X is not congruent.

2.

(a) The set of Nash equilibria is {(B, L)} = R.

(b) The set of Nash equilibria is {(U, L),(M, C)}. R = {U, M, D} ×{L, C}.(c) The set of Nash equilibria is {(U, X)} = R.

(d) The set of Nash equilibria is {(U, L), (D, R)}. R = {U, D}× {L, R}.

3.

Figure 7.1: The Nash equilibrium is (B,Z).

Figure 7.3: The Nash equilibrium is (M,R).

Figure 7.4: The Nash equilibria are (stag,stag) and (hare,hare).

Exercise 1: (a) No Nash equilibrium. (b) The Nash equilibria are (U,R)and (M,L). (c) The Nash equilibrium is (U,L). (d) The Nash equilibria are(A,X) and (B,Y). (e) The Nash equilibria are (A,X) and (B,Y). (f) TheNash equilibria are (A,X) and (B,Y). (g) The Nash equilibrium is (D,Y).

Chapter 4, Exercise 2: The Nash equilibria are (Ea,aa′) and (Ea,an′).

Chapter 5, Exercise 1: The Nash equilibrium is (D,R).

Exercise 3: No Nash equilibrium.

4.

Only at (1/2, 1/2) would no player wish to unilaterally deviate. Thus,the Nash equilibrium is (1/2, 1/2).




5.

Player 1 solves maxs1 3s1−2s1s2−2s21. Taking s2 as given and differentiat-

ing with respect to s1 yields the first order condition 3−2s2−4s1 = 0. Re-arranging, we obtain player 1’s best response function: s1(s2) = 3/4−s2/2.player 2 solves maxs2 s2+2s1s2−2s2

2. This yields the best response functions2(s1) = 1/4+s1/2. The Nash equilibrium is found by finding the strategyprofile that satisfies both of these equations. Substituting player 2’s bestresponse function into player 1’s, we have s1 = 3/4−1/2[1/4+s1/2]. Thisimplies that the Nash equilibrium is (1/2, 1/2).

6.

(a) The congruous sets are S, {(z, m)}, and {w, y} × {k, l}.(b) They will agree to {w, y} × {k, l}.(c) No, there are four possible strategy profiles.

7.

(B,X) is a Nash equilibrium has no implications for x. (A,Z) is efficientrequires that x ≥ 4. For Y to be a best response to θ1 = (1

2, 1

2), we need

u2(θ1, Y ) = 3 ≥ u2(θ1, Z) = x/2 + 1. So we need x ≤ 4. Thus, for allthree statements to be true requires x = 4.

8.

(a) In the first round strategies 1, 2, 8, and 9 are dominated by 3 and7. Note that 3, 4, 5, 6, and 7 are all best responses to beliefs that putprobability .5 on 3 and probability .5 on 7, giving an expected payoff of3.5.

(b) The Nash equilibria are (3,7) and (7,3).

9.

(a) The Nash equilibria are (2, 1), (5/2, 2), and (3, 3).

(b) R = [2, 3] × [1, 3].




10.

Consider the following game, in which (H, X) is an efficient strategy profilethat is also a non-strict Nash equilibrium.

11.

(a) Play will converge to (D, D), because D is dominant for each player.

(b) Suppose that the first play is (opera, movie). Recall that BRi(movie)= {movie}, and BRi(opera) = {opera}. Thus, in round two, play will be(movie, opera). Then in round three, play will be (opera, movie). Thiscycle will continue with no equilibrium being reached.

(c) In the case of strict Nash equilibrium, it will be played all of the time.The non-strict Nash equilibrium will not be played all of the time. It mustbe that one or both players will play a strategy other than his part of sucha Nash equilibrium with positive probability.

(d) Strategies that are never best responses will eventually be eliminatedby this rule of thumb. Thus, in the long run si will not be played.

12.

It must be the case that {s∗1, t∗1} × {s∗2, t∗2} is weakly congruous. For{s∗1, t∗1} × {s∗2, t∗2} to be weakly congruous, we need s∗1 ∈ BR1(θ2), t∗1 ∈BR1(θ

′2), s∗2 ∈ BR2(θ1), and t∗2 ∈ BR2(θ

′1), where θ2, θ

′2 ∈ Δ{s∗2, t∗2} and

θ1, θ′1 ∈ Δ{s∗1, t∗1}. This is true for θ2 putting probability 1 on s∗2, θ′2

putting probability 1 on t∗2, etc., because s∗ and t∗ are Nash equilibria.




13.

(a)

(b) This game has no pure-strategy Nash equilibrium.

(c) Yes, it has a Nash equilibrium. To find equilibrium, look for a casein which the players are getting the same payoffs and none wished tounilaterally deviate. This requires γ = 2α = 3β. Thus, we need γ = 6,α = 3, and β = 2. It is an equilibrium if 6 players select Z, 3 select X,and 2 select Y. The number of equilibria is 4620.



10 Oligopoly, Tariffs, Crime,and Voting

1.

(a) Si = [0,∞). ui(qi, Q−i) = [a−bQ−i−bqi]qi−cqi, where Q−i ≡ ∑j �=i qj.

(b) Firm i solves maxqi[a − bQ−i − bqi]qi − cqi. This yields the first ordercondition a−bQ−i−c = 2bqi. Player i’s best response function is qi(Q−i) =(a − c)/2b − Q−i/2. This is represented in the graph below.

(c) By symmetry, total equilibrium output is Q∗ = nq∗, where q∗ is theequilibrium output of an individual firm. Thus, Q∗

−i = (n − 1)q∗. Soq∗ = [a − c − b(n − 1)q∗]/2b. Thus, q∗ = [a − c]/b(n + 1) and Q∗ =n[a − c]/b(n + 1). We also have

p∗ = a − bn[a − c]/b(n + 1) = n[a − c]/(n + 1)= [an + a − an + nc]/(n + 1) = [a + cn]/(n + 1].

and

u∗ = p∗q∗ − cq∗

= ([a + cn]/(n + 1])[n[a− c]/b(n + 1)] − cn[a − c]/b(n + 1)= (a − c)2/b(n + 1)2

.

(d) In the duopoly case qi(qj) = (a − c)/2b − qj/2. The Nash equilibriumis found by solving the system of two equations given by the best responsefunctions of the two players (alternatively, one can just set n = 2 in theabove result). Thus, q∗ = (a − c)/3b. By examining the best responsefunction, we can identify the sequence Rk

i and inspection reveals thatRi = {(a − c)/3b} for i = 1, 2.

2.

(a) Si = [0,∞], ui(pi, p−i) =

{1m

(a − pi)[pi − c] if pi = p0 if pi > p,

where m

denotes the number of players k ∈ {1, 2, . . . , n} such that pk = p.



OLIGOPOLY, TARIFFS, CRIME, AND VOTING 97

(b) The Nash equilibrium is: pi = c for all i. For n > 2, there are otherNash equilibria in which one or more players selects a price greater thanc (but at least two players select c).

(c) The notion of best response is not well defined. Let p−idenote the

minimum pj selected by any player j = i. If c < p−i, player i’s best

response is to select pi < p−i, but as close to p−i

as possible. Howeverthere is no such number.

3.

(a) BRi(xj) = 30 + xj/2.

(b) The Nash equilibrium is (60, 60).

(c) ui(60, 60) = 200. ui(0, 0) = 2000.

(d) The best response functions are represented below.

It is easy to see that player i will never set xi < 30 or xi > 80. Thus,R1

i = [30, 80], R2i = [45, 70], and so on. Thus, Ri = {60} for i = 1, 2.

4.

(a) G solves maxx −y2x−1 − xc4. This yields the first order conditiony2

x2 − c4 = 0. Rearranging, we find G’s best response function to be x(y) =y/c2. C solves maxy y1/2(1 + xy)−1. This yields the first order condition

12y1/2(1+xy)

− y1/2x(1+xy)2

= 0. Rearranging, we find C’s best response function

to be y(x) = 1/x. These are represented at the top of the next page.




(b) We find x and y such that x = y/c2 and y = 1/x. The Nash equilib-rium is x = 1/c and y = c.

(c) As the cost of enforcement c increases, enforcement x decreases andcriminal activity y increases.

5.

In equilibrium b1 = b2 = 15, 000. Clearly, neither player wishes to bidhigher than 15,000 as she will receive a negative payoff. Further, neitherdoes better by unilaterally deviating to a bid that is less than 15,000because this leads to a payoff of zero.

6.

(a) The normal form is given by N = {P, D}, ei ∈ [0,∞), uP (eP , eD) =8eP/(eP + eD) − eP , and uD(eP , eD) = −8 + 8eD/(eP + eD) − eD.

(b) The prosecutor solves maxeP8eP /(eP + eD) − eP . The first order

condition is 8/(eP + eD)− 8eP/(eP + eD)2 = 1. This implies 8(eP + eD)−8eP = (eP + eD)2, or 8eD = (eP + eD)2. Taking the square root of bothsides yields 2

√2eD = eP +eD. Rearranging, we find e∗P (eD) = 2

√2eD−eD.

Similarly, e∗D(eP ) = 2√

2eP − eP .

(c) By symmetry, it must be that e∗P = 2√

2e∗p − e∗P . Thus, e∗P = e∗D = 2.

The probability that the defendant wins in equilibrium is 1/2.

(d) This is not efficient.




7.

BR1(q2) = 5 − 12q2, and BR2(q1) = 4 − 1

2q1. The Nash equilibrium is

q∗1 = 4 and q∗2 = 2.

8.

(a) For α ≥ 13.

(b) For α ≤ 14.

9.

For L, voting for McClintock is dominated by voting for Bustamante.Knowing that L will not vote for McClintock, M does strictly better votingfor Schwarzenegger than by voting for McClintock, for any strategy pro-files of the others (aside from L voting for McClintock). So neither L nor Mwill vote for McClintock. We can then show that L does strictly better byvoting for Bustamante than voting for Schwarzenegger for any strategiesof the others (assuming M does not vote for McClintock). Knowing this,M will vote for Schwarzenegger. Thus, C will vote for Schwarzenegger.

10.

(a) All of the strategies are rationalizable. If player x selects G she gets1. If she selects F, she gets 2m. If player x believes that no one else willplay F, then her best response is G. If she believes that everyone else willplay F, then her best response is F.

There is a symmetric Nash equilibrium in which everyone plays F. Thereis another symmetric Nash equilibrium in which everyone plays G.

(b) Playing G yields 1; playing F yields 2m − 2x. Note that m ≤ 1. Ifafter some round of iterated dominance it is rational for at most m ofthe players to choose F, then any x with 2m − 2x < 1 will find that Gdominates F. Rearranging yields x > m− 1

2. This means that in the next

round, m has decreased by 12. After two rounds, we get that G is the only

rationalizable strategy for everyone.

(c) Every player x > 12

selects G, and every player x < 12

selects F, andx = 1

2selects either F or G.



11 Mixed-Strategy Nash Equilibrium

1.

(a) R = {X, Y } × {Q, Z}.(b) The Nash equilibrium is ((1

2, 1

2), (3

4, 0, 1

4)).

2.

There is enough information. It must be that u1(A, σ2) = 4, so we need6σ2(X) + 0σ2(Y ) + 0σs(Z) = 4. So σ2(X) = 2

3.

3.

(a) (N, L) and (L, N).

(b) Firm Y chooses q so that Firm X is indifferent between L and N. Thisyields −5q + (x − 15)(1 − q) = 10 − 10q. Rearranging yields q = 25−x

20−x.

Firm X chooses p so that firm Y is indifferent between L and N. Thisyields −5p + 15 − 15p = 10 − 10p. Rearranging yields p = 1/2.

(c) The probability of (L, N) = p(1− q) = (1/2)[20−x−25+x20−x

] = (1/2)[ 5x−20

].

(d) As x increases, the probability of (L, N) decreases. However, as xbecomes larger, (L, N) is a “better” outcome.

4.

(a) σ1 = (1/5, 4/5) σ2 = (3/4, 1/4).

(b) It is easy to see that M dominates L, and that (2/3, 1/3, 0) dominatesD. Thus, player 1 will never play D, and player 2 will never play L. Weneed to find probabilities over U and C such that player 2 is indifferentbetween M and R. This requires 5p + 5 − 5p = 3p + 8 − 8p or p =3/5. Thus, σ1 = (3/5, 2/5, 0). We must also find probabilities over Mand R such that player 1 is indifferent between U and C. This requires3q + 6 − 6q = 5q + 4 − 4q or q = 1/2. Thus, σ2 = (0, 1/2, 1/2).

5.

When x < 1, the Nash equilibria are (U, L) and ((0, 1/2, 1/2), (0, 1/2, 1/2)).When x > 1, Nash equilibrium is (U, L). Further, for 0 < x < 1, there isan equilibrium of ((1 − x, x/2, x/2), (1 − x, x/2, x/2)).




6.

(a) σi = (1/2, 1/2).

(b) (D, D)

(c) There are no pure strategy Nash equilibria. σ1 = (1/2, 1/2) and σ2 =(1/2, 1/2).

(d) (A, A), (B, B), and σ1 = (1/5, 4/5), σ2 = (1/2, 1/2).

(e) (A, A), (B, B), and σ1 = (2/3, 1/3), σ2 = (3/5, 2/5).

(f) Note that M dominates L. So player 2 chooses probabilities over Mand R such that player 1 is indifferent between at least two strategies.Let q denote the probability with which M is played. Notice that the qwhich makes player 1 indifferent between any two strategies makes himindifferent between all three strategies. To see this note that q = 1/2solves 4 − 4q = 4q = 3q + 1 − q. Thus, σ2 = (0, 1/2, 1/2). It remains tofind probabilities such that player 2 is indifferent between playing M andR. Here p denotes the probability with which U is played and r denotesthe probability with which C is played. Indifference between M and Rrequires 2p + 4r + 3(1 − p− r) = 3p + 4(1 − p − r). This implies r = 1/5.Thus, σ1 = (x, 1/5, y), where x, y ≥ 0 and x + y = 4/5.

7.

First game: The normal form is represented below.

Player 2 mixes over X and Y so that player 1 is indifferent between thosestrategies on which player 1 puts positive probability. Let q be the prob-ability that player 2 selects X. The comparison of 8q to 2q + 6 − 6q to5 shows that we cannot find a mixed strategy in which player 1 placespositive probability on all of his strategies. So we can consider each of thecases where player 1 is indifferent between two of his strategies. Clearly,at q = 5/8 player 1 is indifferent between A and C. Indifference between Aand B requires 8q = 6− 4q or q = 1/2. However, note that BR1(1/2, 1/2)= {C} and, thus, there is no equilibrium in which player 1 mixes between




A and B. Finally, indifference between B and C requires 6 − 4q = 5 orq = 1/4. Further, note that BR1(1/4, 3/4) = {B, C}.Turning to player 2’s incentives, there is clearly no equilibrium in whichplayer 1 mixes between A and C; this is because player 2 would strictlyprefer X, and then player 1 would not be indifferent between A and C.Likewise, there is no equilibrium in which player 1 mixes between B andC; in this case, player 2 would strictly prefer Y, and then player 1 wouldnot be indifferent between B and C. There are, however, mixed strategyequilibria in which player 1 selects C with probability 1 (that is, plays apure strategy) and player 2 mixes between X and Y. This is an equilibriumfor every q ∈ [1/4, 5/8].

Second game: The normal form of this game is represented below.

Clearly, there is no equilibrium in which player 1 selects ID with positiveprobability. There is also no equilibrium in which player 1 selects IU withpositive probability, for, if this were the case, then player 2 strictly prefersO and, in response, player 1 should not pick IU. Note that player 1 prefersOU or OD if player 2 selects O with a probability of at least 3/5. Further,when player 1 mixes between OU and OD, player 2 is indifferent betweenhis two strategies. Thus, the set of mixed strategy equilibria is describedby σ1 = (0, 0, p, 1 − p) and σ2 = (q, 1 − q), where p ∈ [0, 1] and q ≤ 2/5.

8.

(a) The symmetric mixed strategy Nash equilibrium requires that eachplayer call with the same probability, and that each player be indifferentbetween calling and not calling. This implies that (1 − pn−1)v = v − c or

p = (c/v)1

n−1 .

(b) The probability that at least one player calls in equilibrium is 1−pn =1− (c/v)

nn−1 . Note that this decreases as the number of bystanders n goes

up.




9.

(a) If the game has a pure-strategy Nash equilibrium, we are done.

(b) Assume the game has no pure-strategy Nash equilibrium, and proceedas follows. That (U,L) is not a Nash equilibrium implies e > a and/ord > b. That (U,R) is not a Nash equilibrium implies g > c and/or b > d.That (D,R) is not a Nash equilibrium implies c > g and/or f > h. That(D,L) is not a Nash equilibrium implies a > e and/or h > f . It is easyto see that if there is no pure strategy Nash equilibrium, then only oneof each of these pairs of conditions can hold. This implies that each purestrategy of each player is a best response to some other pure strategy ofthe other. Further, it must be that there is a mixture for each player isuch that the other player j is indifferent between his two strategies.

Consider player 1. It must be that either e > a and g > c or a > eand c > g. It is easy to show that there exists a q ∈ [0, 1] such thataq +c(1−q) = eq +g(1−q). Rearranging yields (a−e) = (g−c)(1−q)/q.It is the case that (a − e) and (g − c) have the same sign. The analogousargument can be made with respect to player 2.

10.

No, it does not have any pure strategy equilibria. The mixed equilibriumis ((1/3, 1/3, 1/3), (1/3, 1/3, 1/3)).

11.

(a) When θ2 > 2/3, 001 should choose route a. When θ2 < 2/3, 001should choose route d. When θ2 = 2/3, 001 should choose either route a,route c, or route d.

(b) It is advised that 001 never take route b. Route b is dominated by amixture of routes a and c. One such mixture is 2/3 probability on a and1/3 probability on c. It is easy to see that 12(2/3)+10(1/3) = 34/11 > 11,and 4(1/3) > 1.

(c) As 002’s payoff is the same, regardless of his strategy, when 001 choosesc, we should expect that the equilibrium with one player mixing and theother playing a pure strategy will involve 001 choosing c. Clearly 002 isindifferent between x and y when 001 is playing c. Further, 002 can mixso that c is a best response for 001. A mixture of 2/3 and 1/3 implies that001 receives a payoff of 8 from all of his undominated strategies. Thisequilibrium is s1 =c and σ2 = (2/3, 1/3).

Since b is dominated, we now consider a mixture by 001 over a and d. Infinding the equilibrium above, we noticed that 002’s mixing with proba-bility (2/3, 1/3) makes 001 indifferent between a, c, and d. Thus, we need




only to find a mixture over a and d that makes 002 indifferent betweenx and y. Let p denote the probability with which 001 plays a, and 1 − pdenote the probability with which he plays d. Indifference on the part of002 is reflected by 3 − 3p = 6p. This implies p = 1/3, which means that002 receives a payoff of 2 whether he chooses x or y. This equilibrium isσ = ((1/3, 0, 0, 2/3), (2/3, 1/3).

In considering whether there are any more equilibria, it is useful to noticethat in both of the above equilibria that 002’s payoff from choosing x isthe same as that from y. Thus we should expect that, so long as the ratioof a to d is kept the same, 001 could also play c with positive probability.Let p denote the probability with which 001 plays a, and let q denote theprobability with which he plays c. Since he never plays b, the probabilitywith which d is played is 1−p−q. Making 002 indifferent between playingx and y requires that 2q + 3(1 − p − q) = 6p + 2q. This implies that anyp and q such that 1 = 3p + q will work. One such case is (1/9, 6/9, 2/9),implying an equilibrium of ((1/9, 6/9, 2/9), (2/3, 1/3))

12.

(a)

(b) The pure strategy Nash equilibria are (X,Y,Y), (Y,X,Y), and (Y,Y,X).

(c) In equilibrium p =√

10−22

.



12 Strictly Competitive Games

and Security Strategies

1.

(a) No. Note that u1(A, Z) = u1(C, Z), but u2(A, Z) > u2(C, Z).

(b) Yes.

(c) Yes.

(d) No. Note that u1(D, X) > u1(D, Y), but u2(D, X) > u2(D, Y).

2.

(a) 1: C, 2: Z

(b) 1: C, 2: Z

(c) 1: A, 2: X

(d) 1: D, 2: Y

3.

Examples include chess, checkers, tic-tac-toe, and Othello.

4.

Let i be one of the players and let j be the other player. Because sis a Nash equilibrium, we have ui(s) ≥ ui(ti, sj). Because t is a Nashequilibrium, we have uj(t) ≥ uj(ti, sj); strict competition further impliesthat ui(t) ≤ ui(ti, sj). Putting these two facts together, we obtain ui(s) ≥ui(ti, sj) ≥ ui(t). Switching the roles of s and t, the same argumentyields ui(t) ≥ ui(si, tj) ≥ ui(s). Thus, we know that ui(s) = ui(si, tj) =ui(ti, sj) = ui(t) for i = 1, 2, so the equilibria are equivalent. To seethat the equilibria are also interchangeable, note that, because si is a bestresponse to sj and ui(s) = ui(ti, sj), we know that ti is also a best responseto sj. For the same reason, si is a best response to tj.



13 Contract, Law, and Enforcement

in Static Settings

1.

(a)

(I,I) can be enforced by setting α between −4 and −2.

(b)

No.

2.

(a) A contract specifying (I, I) can be enforced under expectations dam-ages because neither player has the incentive to deviate from (I, I).




(b) Yes.

(c) No, player 2 still has the incentive to deviate.

(d)

(e) c > 1.

(f) Consider (I,N). Player 1 sues if −c > −4 or c < 4. Consider (N,I).Player 2 sues if −c > −4 or c < 4. Thus, suit occurs if c < 4.

(g) c > 1/2.

3.

(a) 10

(b) 0

4.

(a) Now the payoff to i when no one calls is negative. Let d denote the finefor not calling. Consider the case where the fine is incurred regardless ofwhether anyone else calls. This yields the new indifference relationship of

(1−pn−1)v−d = v−c. This implies that, if c > d, then p = [(c−d)/v]1

n−1 .If c < d then p = 0 in equilibrium.




Now consider the case where the fine is incurred only when no one calls.The indifference relationship here implies (1 − pn−1)v − dpn−1 = v − c.

This implies p = [c/(d + v)]1

n−1 .

(b) (1) Given that if i doesn’t call then he pays the fine with certainty,the fine can be relatively low. (2) Here, if i doesn’t call then he pays thefine with a low probability. Thus, the fine should be relatively large.

(c) Either type of fine can be used to induce any particular p value, exceptfor p = 0 which results only if the type (1) fine is imposed. The requiredtype (2) fine may be much higher than the required type (1) would be.The type (2) fine may be easier to enforce, because in this case one onlyneeds to verify whether the pedestrian was treated promptly and who thebystanders were. The efficient outcome is for exactly one person to call.There are pure strategy equilibria that achieve this outcome, but it neverhappens in the symmetric mixed strategy equilibrium.

5.

Verifiability is more important. It must be possible to convey informationto the court in order to have a transfer imposed.

6.

Expectations damages gives the non-breaching player the payoff that heexpected to receive under the contract. Restitution damages takes fromthe breacher the amount of his gain from breaching. Expectations dam-ages is more likely to achieve efficiency. This is because it gives a playerthe incentive to breach when it is efficient to do so.

7.

(a) For technology A, the self-enforced component is to play (I, I). Theexternally-enforced component is a transfer of at least 1 from player 2to player 1 when (I, N) occurs, a transfer of at least 2 from player 1 toplayer 2 when (N, I) occurs, and none otherwise. For technology B, theself-enforced component is to play (I, I). The externally-enforced compo-nent is a transfer of at least 4 from player 1 to player 2 when (N, I) occurs,and none otherwise.

(b) Now for technology A, the self-enforced component is to play (N,N). There is no externally-enforced component. For B the self-enforcedcomponent is to transfer 4 from player 1 to player 2 when someone playsN, and no transfer when both play I.




(c) Expectations damages gives the non-breaching player the amount thathe expected to receive under the contract. The payoffs under this remedyare depicted for each case as shown here:

Reliance damages seek to put the non-breaching party back to where hewould have been had he not relied on the contract. The payoffs underreliance damages are depicted below.

Restitution damages take the gain that the breaching party receives dueto breaching. The payoffs under restitution damages are depicted below.

8.

(a)




(b) (H,H) and (L,L) are self-enforcing outcomes.

(c) The court cannot distinguish between (H,L) and (L,H).

(d) The best outcome the parties can achieve is (L,H). Their contract issuch that when (H,H) is played player 1 pays δ to player 2, and wheneither (H,L) or (L,H) is played, player 1 pays α to player 2. We need αand δ to be such that α > 2 and δ > α + 1.

(c) The best outcome the parties can achieve is (H,H). Their contract issuch that when (H,H) is played player 1 pays δ to player 2, when either(H,L) is played player 1 pays α to player 2, and when (L,H) is played player1 pays β to player 2. We need α, β, and δ to be such that α+2 < δ < β+1.

9.

(a) S1 = [0,∞), S2 = [0,∞). If y > x, then the payoffs are (0, 0). Ifx ≥ y, the payoffs are (y − Y, X − y).

(b) There are multiple equilibria in which the players report x = y = α,where α ∈ [Y, X]. There is another set of multiple equilibria in which theplayers report x (player 2) and y (player 1) such that x ≤ Y < X ≤ y.

(c) There are multiple equilibria; all satisfy x < y, y ≥ X, and x ≤ Y .

(d) It is efficient if an equilibrium in the first set of multiple equilibria ofpart (b) is selected. This is because the plant is shut down if and only ifit is efficient to do so.

10.

Examples include the employment contracts of salespeople, attorneys, andprofessors.



14 Details of the Extensive Form

1.

No general rule. Consider, for example, the prisoners’ dilemma. Clearly,the extensive form of this game will contain dashed lines. Consider Exer-cise 3 (a) of Chapter 4. The normal form of this does not exhibit imperfectinformation.

2.

Suppose not. Then it must be that some pure strategy profile inducesat least two paths through the tree. Since a strategy profile specifies anaction to be taken in every contingency (at every node), having two pathsinduced by the same pure strategy profile would require that Tree Rule 3not hold.

3.

4.

5.



15 Backward Induction

and Subgame Perfection

1.

(a) (AF, C)

(b) (BHJKN, CE)

(c) (I, C, X)

2.

(a) The subgame perfect equilibria are (WY, AC) and (ZX, BC). TheNash equilibria are (WY, AC), (ZX, BC), (WY, AD), (ZY, BC), and(WX, BD).

(b) The subgame perfect equilibria are (UE, BD) and (DE, BC). The Nashequilibria are (UE, BD), (DE, BC), (UF, BD), and (DE, AC).

3.

(a) (AHILN,CE)

(b) 6

4.

For any given x, y∗1(x) = y∗

2(x) = x; and x∗ = 2.

5.

(a)

(b) Working backward, it is easy to see that in round 5 player 1 will chooseS. Thus, in round 4 player 2 will choose S. Continuing in this fashion, wefind that, in equilibrium, each player will choose S any time he is on themove.

(c) For any finite k, the backward induction outcome is that player 1chooses S in the first round and each player receives one dollar.




6.

Payoffs in the extensive form representation are in the order RBC, CBC,and MBC.

In the subgame perfect equilibrium, MBC chooses 7, RBC chooses 76′,and CBC chooses 76′6′′7′′′. The outcome differs from the simultaneousmove case because of the sequential play.

7.

(a)

(b) If x > 3, the equilibria are (OA,A), (OB,A), (OA,B), (OB,B). IfX = 3, add (IA, A) to this list. If 1 < x < 3, the equilibria are (IA,A),(OA,B), (OB,B). If x = 1, add (IB, B) to this list. If x < 1, the equilibriaare (IA,A), (IB,B).




(c) If x > 3 any mixture with positive probabilities over OA and OB forplayer 1, and over A and B for player 2.

If 1 < x < 3, then IB is dominated. Any mixture (with positive proba-bilities) over OA and OB will make player 2 indifferent. Player 2 plays Awith a probability that does not exceed x/3.

Next consider the case in which 3/4 ≤ x ≤ 1. Let p denote the probabilitythat player 1 plays IA, let q denotes the probability with which she playsIB, and let 1 − p − q denote the probability that player 1 plays OA orOB. There is a mixed strategy equilibrium in which p = q = 0. Here,player 2 mixes so that player 1 does not want to play IA or IB, implyingthat player 2 can put no more than probability x/3 on A and no morethan x on B. There is not an equilibrium with p and/or q positive. Tosee this, note that for player 2 to be indifferent, we need p = 3q. We alsoneed player 2 to mix so that player 1 is indifferent between IA and IB,but (for x > 3/4) this mixture makes player 1 strictly prefer to select OAor OB.

For x < 3/4, OA and OB are dominated. In equilibrium, player 1 choosesIA with probability 3/4 and IB with probability 1/4. In equilibrium,player 2 chooses A with probability 1/4, and B with probability 3/4.

(d)

The pure strategy equilibria are (A, A) and (B, B). There is also a mixedequilibrium (3/4, 1/4; 1/4, 3/4).

(e) The Nash equilibria that are not subgame perfect include (OB, A),(OA, B), and the above mixed equilibria in which, once the proper sub-game is reached, player 1 does not play A with probability 3/4 and/orplayer 2 does not play A with probability 1/4.

(f) The subgame perfect mixed equilibria are those in which, once theproper subgame is reached, player 1 does plays A with probability 3/4and player 2 does plays A with probability 1/4.




8.

(a) Si = {A, B} × (0,∞) × (0,∞). Each player selects A or B, picks apositive number when (A, B) is chosen, and picks a positive number when(B, A) is chosen.

(b) It is easy to see that 0 < (x1 + x2)/(1 + x1 + x2) < 1, and that(x1 + x2)/(1 + x1 + x2) approaches 1 as (x1 + x2) → ∞. Thus, each hasa higher payoff when both choose A. Further, B will never be selected inequilibrium. The Nash equilibria of this game are given by (Ax1, Ax2),where x1 and x2 are any positive numbers.

(c) There is no subgame perfect equilibrium because the subgames follow-ing (A, B) and (B, A) have no Nash equilibria.



16 Topics in Industrial Organization

1.

From the text, z1(a) = a2/9−2a3/81. If the firms were to write a contractthat specified a, they would choose a to maximize their joint profit (withm set to divide the profit between them). This advertising level solvesmaxa 2a2/9 − 2a3/81, which is a∗ = 6.

2.

The subgame perfect equilibrium is a = 0 and p1 = p2 = 0.

3.

Because this is a simultaneous move game, we are just looking for theNash equilibrium of the following normal form.

The equilibrium is (L, L). Thus, in the subgame perfect equilibrium bothplayers invest 50,000 in the low production plant.

4.

(a) u2(q1, q∗2(q1)) = (1000 − 3q1 − 3q2)q2 − 100q2 − F . Maximizing by

choosing q2 yields the first order condition 1000 − 3q1 − 6q2 − 100 = 0.Thus, q∗2(q1) = 150 − (1/2)q1.

(b) u1(q1, q∗2(q1)) = (1000−3q1−3[150−1/2q1])q1−100q1−F . Maximizing

by choosing q1 yields the first order condition 550 − 3q1 − 100 = 0. Thus,q∗1 = 150. q∗2 = 150− (1/2)(150) = 75. Solving for equilibrium price yieldsp∗ = 100 − 3(150 + 75) = 325. u∗

1 = 325(150) − 100(150) = 33, 750 − F .u∗

2 = 325(75) − 100(75) − F = 16875 − F .

(c) Find q1 such that u2(q1, q∗2(q1)) = 0. We have

(1000 − 3q1 − 3[150 − (1/2)q1])[150 − (1/2)q1] − 100[150 − (1/2)q1] − F

= (900 − 3q1)[150 − (1/2)q1] − 3[150 − (1/2)q]12 − F

= 6[150 − (1/2)q1]2 − 3[150 − (1/2)q1]

2 − F= 3[150 − (1/2)q1]

2 − F.



16 TOPICS IN INDUSTRIAL ORGANIZATION 117

Setting profit equal to zero implies F = 3[150 − (1/2)q1]2 or (F/3)1/2 =

150 − (1/2)q1. Thus, q1 = 300 − 2(F/3)1/2. Note that

u1 = (1000 − 3[300 − 2(F/3)1/2])[300 − (F/3)1/2]−100[300 − 2(F/3)1/2] − F

= 900[300 − 2(F/3)1/2] − 3[300 − 2(F/3)1/2]2 − F.

d) (i) F = 18, 723 implies q1 = 142 < q∗1. So firm 1 will produce q∗1 andu1 = 48, 777. (ii) F = 8112: In this case, q1 = 300 − 2(8112/3)1/2 = 196and pi1 = 900(196) − 3(196)2 − 8, 112 = 53, 040. u∗

1 = 33, 750 − 8, 112 =25, 630. Thus firm 1 will produce q1 = 196, resulting in u1 = 53, 040. (iii)F = 1728: Here, q1 = 300 − 2(1, 728/3)1/2 = 252 and u1 = 900(252) −3(252)2 − 1, 728 = 34, 560. u1∗ = 33, 750 − 1, 728 = 32, 022. Thus, firm 1will produce q1 = 252, resulting in u1 = 34, 560. (iv) F = 108: In this case,q1 = 300−2(108/3)1/2 = 288 and u1 = 900(288)−3(288)2−108 = 10, 260.u∗

1 = 33, 750−108 = 33, 642. Thus, firm 1 will produce q1 = 150, resultingin u1 = 33, 642.

5.

(a) If Hal does not purchase the monitor in period 1, then p2 = 200 is notoptimal because p2 = 500 yields a profit of 500, while p2 = 200 yields aprofit of 400. The optimal pricing scheme is as follows. Set p1 = 1, 700(or just below to make Hal strictly want to buy). If one unit is sold in thefirst period (that is, Hal purchased) then set p2 = 200 to sell to Laurie.On the other hand, if there are no first-period sales (Hal deviated) thenset p2 = 500 to sell to Hal in the second period. With Hal buying in thefirst period and Laurie in the second, total revenue is 1, 900. Tony wouldnot benefit from being able to commit not to sell monitors in period 2.

(b) The optimal prices are p1 = 1, 400 and p2 = 200. Hal buys in period 1and Laurie buys is period 2. Here, Tony would not benefit from being ableto commit not to sell monitors in period 2, because the gain in extractingsurplus from Hal is more than offset by the loss of not selling to Laurie.

6.

(a) If enters against Firm 2, q∗1 = q∗2 = 3. If enters against Firm 3,q′1 = q′3 = 4. Firm 1 enters Firm 3’s industry.

(b) Yes. If Firm 3’s strategy is to choose q′3 = 14 if Firm 1 enters, thenFirm 1’s best response is to enter against Firm 2.




7.

The subgame perfect equilibrium is for player 1 to locate in region 5, andfor player 2 to use the strategy 234555678 (where, for example, 2 denotesthat player 2 locates in region 2 when player 1 has located in region 1).

8.

(a) Without payoffs, the extensive form is as follows.

In the subgame perfect equilibrium, player 1 selects E, player 2 choosesDE′, and the quantities are given by q1 = q2 = q3 = 3, q′1 = q′3 = 4,q′′2 = q′′3 = 4, and q′′′3 = 6.

(b) Player 1 enters.

9.

(a) The government solves maxp 30 + p− W − p/2− 30 or maxp p/2− W .This implies that they want to set p as high as possible, regardless of thelevel of W . So p∗ = 10.

Knowing how the government will behave, the ASE solves maxW −(W −10)2. The first order condition implies W ∗ = p∗ = 10. So in equilibriumy = 30.

(b) If the government could commit ahead of time, it would solve maxW −W/2.This implies that it would commit to p = 0 and the ASE would set W = 0.In (a) u = 0 and v = −5. Now, when commitment is possible, u = 0 andv = 0.

(c) One way is to have a separate central bank that does not have apolitically elected head that states its goals.




10.

For scheme A to be optimal, it must be that twice Laurie’s (the low type)value in period 1 is at least as great as Hal’s (high type) period 1 valueplus his period 2 value. An example of this is below.

For scheme B to be optimal, it must be that Laurie’s (low type) value inperiod 2 is at least as large as both Hal’s (high type) period 1 value andLaurie’s period 1 value. An example of this is below.



17 Parlor Games

1.

(a) Use backward induction to solve this. To win the game, a player mustnot be forced to enter the top-left cell Z; thus, a player would lose if hemust move with the rock in either cell 1 or cell 2 as shown in the followingdiagram.

A player who is able to move the rock into cell 1 or cell 2 thus wins thegame. This implies that a player can guarantee victory if he is on themove when the rock is in one of cells 3, 4, 5, 6, or 7, as shown in thediagram below.

We next see that a player who must move from cell 8, cell 9 or cell 10(shown below) will lose.



17 PARLOR GAMES 121

Continuing the procedure reveals that, starting from a cell marked withan X in the following picture, the next player to move will win.

Since the dimensions of the matrix are 5× 7, player 2 has a strategy thatguarantees victory.

(b) In general, player 2 has a winning strategy when m, n > 1 and both areodd, or when m or n equals 1 and the other is even. Otherwise, player 1has a winning strategy.

2.

If a player puts in the fifteenth penny—and no more, that player is assuredof winning because her opponent must add at least one penny. Similarly,if a player puts in exactly the tenth penny, that player is assured of beingable to put in exactly the fifteenth penny. Continuing with this, the playerwho puts in exactly the fifth penny is assured of winning. So player 2 hasa winning strategy, and that strategy involves always putting in enoughpennies to exactly put in the fifth one, the tenth one, and the fifteenthone.

3.

This can be solved by backward induction. Let (x, y) denote the statewhere the red basket contains x balls and the blue basket contains y balls.To win this game, a player must leave her opponent with either (0,1) or(1,0). Thus, in order to win, a player must not leave her opponent witheither any of the following (0, z), (1, z), (z, 1), or (z, 0), z > 1. So, to win,a player should leave her opponent with (2, 2). Thus, a player must notleave her opponent with either (w, 2) or (2, w), where w > 2. Continuingwith this logic and assuming m, n > 0, we see that player 2 has a winningstrategy when m = n and player 1 has a winning strategy when m = n.



17 PARLOR GAMES 122

4.

(a) In order to win, in the matrix below, a player must avoid entering acell marked with an X. As player 1 begins in cell Y, he must enter a cellmarked with an X. Thus, player 2 has a strategy that ensures a win.

(b) There are many subgame perfect equilibria in this game, because play-ers are indifferent between moves at numerous cells. There is a subgameperfect equilibrium in which player 1 wins, another in which player 2 wins,and still another in which player 3 wins.

5.

Player 1 has a strategy that guarantees victory. This is easily proved usinga contradiction argument. Suppose player 1 does not have a strategyguaranteeing victory. Then player 2 must have such a strategy. Thismeans that, for every opening move by player 1, player 2 can guaranteevictory from this point. Let X be the set of matrix configurations thatplayer 1 can create in his first move, which player 2 would then face. Aconfiguration refers to the set of cells that are filled in.

We have that, starting from each of the configurations in X, the nextplayer to move can guarantee victory for himself. Note, however, thatif player 1 selects cell (m, n) in his first move, then, whatever player 2’sfollowing choice is, the configuration of the matrix induced by player 2’sselection will be in X (it is a configuration that player 1 could have createdin his first move). Thus, whatever player 2 selects in response to hischoice of cell (m, n), player 1 can guarantee a victory following player 2’smove. This means that player 1 has a strategy that guarantees him awin, which contradicts what we assumed at the beginning. Thus, player 1actually does have a strategy that guarantees him victory, regardless ofwhat player 2 does.

This game is interesting because player 1’s winning strategy in arbitrarym × n Chomp games is not known. A winning strategy is known for the



17 PARLOR GAMES 123

special case in which m = n. This strategy selects cell (2, 2) in the firstround.

6.

(a) Yes.

(b) No.

(c) Player 1 can guarantee a payoff of 1 by choosing cell (2,1). Player 2will then rationally choose cell (1,2) and force player 3 to move into cell(1,1).



18 Bargaining Problems

1.

(a) v∗ = 50, 000; u∗J = u∗

R = 25, 000; t = 15, 000.

(b) Solving maxx 60, 000 − x2 + 800x yields x∗ = 400. This implies v∗ =220, 000, u∗

J = u∗R = 110, 000, vJ = −100, 000, and vR = 320, 000. Thus,

t = 210, 000.

(c) From above, x∗ = 400 and v∗ = 220, 000. u∗J = 40, 000 + (220, 000 −

40, 000−20, 000)/4 = 80, 000 and u∗R = 20, 000+(3/4)(220, 000−60, 000) =

140, 000. This implies t = 180, 000.

2.

John should undertake the activity that has the most impact on t, andhence his overall payoff, per time/cost. A one-unit increase in x will raiset by πJ . A one unit increase in w raises t by 1−πJ. Assuming that x andw can be increased at the same cost, John should increase x if πj > 1/2;otherwise, he should increase w.

3.

(a) x = 15, t = 0, and u1 = u2 = 15.




(b) x = 15, t = −1, u1 = 14, and u2 = 16.

(c) x = 15, t = −7, u1 = 8, and u2 = 22.

(d) x = 10, t = −175, u1 = 25, and u2 = 75.




(e) x = 12, t = 144π1 − 336, u1 = 144π1, and u2 = 144π2.

4.

The other party’s disagreement point influences how much of v∗ you getbecause it influences the size of the surplus.

5.

You should raise the maximum joint value if your bargaining weight ex-ceeds 1/2; otherwise, you should raise your disagreement payoff. In thelatter case, your decision is not efficient.

6.

Possible examples would include salary negotiations, merger negotiations,and negotiating the purchase of an automobile.



19 Analysis of Simple Bargaining Games

1.

(a) The superintendent offers x = 0, and the president accepts any x.

(b) The president accepts x if x ≥ min{z, |y|}.(c) The superintendent offers x = min{z, |y|}, and the president accepts.

(d) The president should promise z = |y|.

2.

(a) Here you should make the first offer, because the current owner isvery impatient and will be quite willing to accept a low offer in the firstperiod. More precisely, since δ < 1/2, the responder in the first periodprefers accepting less than one-half of the surplus to rejecting and gettingall of the surplus in the second period. Thus, the offerer in the first periodwill get more than half of the surplus.

(b) In this case, you should make the second offer, because you are patientand would be willing to wait until the last period rather than accepting asmall amount at the beginning of the game. More precisely, in the least,you can wait until the last period, at which point you can get the entiresurplus (the owner will accept anything then). Discounting to the firstperiod, this will give you more than one-half of the surplus available inthe first period.

3.

In the case of T = 1, player 1 offers m = 1 and player 2 accepts. If T = 2,player 1 offers 1 − δ in the first period and player 2 accepts, yielding thepayoff vector (1−δ, δ). For T = 3, the payoff vector is (1−δ(1−δ), δ(1−δ)).The payoff is (1−δ−δ2(1−δ), δ−δ2(1−δ)) in the case of T = 4. For T = 5,the payoff is (1 − δ − δ2(1− δ + δ2), δ − δ2(1 − δ + δ2)). As T approachesinfinity, the payoff vector converges to ([1 − δ]/[1 − δ2], [δ − δ2]/[1 − δ2],which is the subgame perfect equilibrium payoff vector of the infinite-period game.

4.

Note that BRi(mj) = 1 − mj. The set of Nash equilibria is given by{m1, m2 ∈ [0, 1] | m1 + m2 = 1}. One can interpret the equilibriumdemands (the mi’s) as the bargaining weights.




5.

6.

For simplicity, assume that the offer is always given in terms of the amountplayer 1 is to receive. Suppose that the offer in period 1 is x, the offerin period 2 it is y, and the offer in period 3 is z. If period 3 is reached,player 2 will offer z = 0 and player 1 will accept. Thus, in period 2,player 2 will accept any offer that gives her at least δ. Knowing this,in period 2 (if it is reached) player 1 will offer y such that player 2 isindifferent between accepting and rejecting to receive 1 in the next period.This implies y = 1 − δ. Thus, in period 1, player 2 will accept any offerthat gives her at least δ(1 − δ). In the first period, player 1 will offer xso that player 2 is indifferent between accepting and rejecting to receive1 − δ in the second period. Thus, player 1 offers x = 1 − δ + δ2 and it isaccepted.

7.

Player 3 accepts any offer such that his share is at least zero. Player 2substitutes an offer of x = 0, y = 1 for any offer made by player 1. Player 1




makes any offer of X and Y . Also, it may be that player 2 acceptsX = 0, Y = 1.

8.

(a)

(b) Player 2 accepts any m such that m + a(2m − 1) ≥ 0. This impliesaccepting any m ≥ a/(1 + 2a). Thus, player 1 offers a/(1 + 2a).

(c) As a becomes large the equilibrium split is 50:50. This is because,when a is large, player 2 cares very much about how close his share is toplayer 1’s share and will reject any offer in which a is not close to 1 − a.



20 Games with Joint Decisions;

Negotiation Equilibrium

1.

(a)

(b) When b ≥ x2.

(c) v8 = 16, surplus = 16, uM = 8, uW = 8, x∗ = 4, b∗ = 16, and t∗ = 8.

(d) This assumes verifiability of worker effort.

2.

(a)

Carina expends no effort (e∗ = 0) and Wendy sets t = 0.

(b) Carina solves maxe 800xe − e2. This yields the first order conditionof 800x = 2e. This implies e∗ = 400x. Wendy solves maxx 800[400x] −800x[400x]. This yields x∗ = 1/2.

(c) Given x and t, Carina solves maxe 800xe + t − e2. This impliese∗ = 400x. To find the maximum joint surplus, hold t fixed and solvemaxx 800[400x] − [400x]2. This yields x∗ = 1. The joint surplus is320, 000 − 160, 000 = 160, 000. Because of the players’ equal bargainingweights, the transfer is t∗ = −80, 000.




3.

(a) x∗ = 10p and y∗ = 5(1 − p).

(b) p∗ = .8, t∗ = −19.5, x∗ = 8, and y∗ = 1.

4.

(a) The players need enforcement when (H, L) is played. In this case,player 2 would not select “enforce.” For player 1 to have the incentive tochoose “enforce,” it must be that t ≥ c. Player 2 prefers not to deviatefrom (H, H) only if t ≥ 4. We also need t − c ≤ 2, or otherwise player 1would prefer to deviate from (H, H) and then select “enforce.” Combiningthese inequalities, we have c ∈ [t − 2, t] and t ≥ 4. A value of t thatsatisfies these inequalities exists if and only if c ≥ 2. Combining this withthe legal constraint that t ≤ 10, we find that (H, H) can be enforced (usingan appropriately chosen t) if and only if c ∈ [2, 10].

(b) We need t large to deter player 2, and t− c small to deter player 1. Itis not possible to do both if c is close to 0. In other words, the legal feedeters frivolous suits from player 1, while not getting in the way of justicein the event that player 2 deviates.

(c) In this case, the players would always avoid court fees by negotiatinga settlement. This prevents the support of (H, H).

5.

The game is represented as below. Note that m ∈ [0, (100 − q1 − q2)(q1 +q2)].

6.

(a) Since the cost is sunk, the surplus is [100 − q1 − q2](q1 + q2). Thus,ui = −10qi + πi[100 − q1 − q2](q1 + q2).

(b) u1 = (1/2)[100 − q1 − q2](q1 + q2) − 10q1 and u2 = (1/2)[100 − q1 −q2](q1 + q2) − 10q2.




(c) Firm 1 solves maxq1(1/2)[100 − q1 − q2](q1 + q2) − 10q1. The firstorder condition implies q∗1(q2) = 40 − q2. By symmetry q∗2(q1) = 40 − q1.In equilibrium, q1 + q2 = 40. Since there are many combinations of q1

and q2 that satisfy this equation, there are multiple equilibria. Each firmwants to maximize its share of the surplus less cost. The gain from havingthe maximum surplus outweighs the additional cost. Note that the totalquantity (40) is less than both the standard Cournot output and themonopoly output. Since it is less than the monopoly output, it is notefficient from the firms’ point of view.

(d) Now each firm solves maxqi πi[100 − qi − qj](qi + qj) − 10qi. Thisimplies best response functions given by q∗i (qj) = 50−5/πi−qj that cannotbe simultaneously satisfied with positive quantities. This is because theplayer with the smaller πi would wish to produce a negative amount. Inthe equilibrium, the player with the larger bargaining weight π produces50 − 5/π units and the other firm produces zero.

(e) The player with the smaller bargaining weight does not receive enoughgain in his share of the surplus to justify production.



21 Unverifiable Investment, Hold Up,

Options, and Ownership

1.

(a) The efficient outcome is high investment and acceptance.

(b) If p0 ≥ p1−5 then the buyer always accepts. The seller will not chooseH.

(c) In the case that L occurs, the buyer will not accept if p1 ≥ 5 + p0.In the case that H occurs, the buyer will accept if p1 ≤ 20 + p0. Thus,it must be that 20 + p0 ≥ p1 ≥ 10 + p0. Because the seller invests highif p1 ≥ 10 + p0, there are values of p0 and p1 that induce the efficientoutcome.

(d) The surplus is 10. Each gets 5. Thus, p1 = 15 and p0 ∈ [−5, 5]. Theseller chooses H. The buyer chooses A if H, and R if L.

2.

(a) Let x = 1 denote restoration and x = 0 denote no restoration. LettE denote the transfer from Joel to Estelle, and let tJ denote the transferfrom Joel to Jerry. The order of the payoffs is Estelle, Jerry, Joel. Hereis the extensive form with joint decisions:

The surplus is 900−500 = 400. The standard bargaining solution requiresthat each player i receive di + πi[v

∗ − di − dl − dk], where l and k denotethe other players. Thus, Joel buys the desk, Joel pays Estelle 400/3, Joelpays Jerry 1900/3, and Jerry restores the desk. This is efficient.




(b) Let t denote the transfer from Estelle to Jerry. Let m denote thetransfer from Joel to Estelle when the desk has been restored. Let b denotethe transfer from Joel to Estelle when the desk has not been restored.

In equilibrium, the desk is not restored and Joel buys the desk for 50.This is not efficient.

(c) Let tE denote the transfer from Joel to Estelle, and let tJ denote thetransfer from Joel to Jerry.

In equilibrium, Joel buys the desk for 125, and pays Jerry 650 to restoreit. This is efficient. However, Jerry’s payoff is greater here than in part(a) because Jerry can hold up Joel during their negotiation, which occursafter Joel has acquired the desk from Estelle.

(d) Estelle (and Jerry) do not value the restored desk. Thus, Estelle canbe held up if she has the desk restored and then tries to sell it to Joel.




3.

If the worker’s bargaining weight is less than 1, then he gets more of anincrease in his payoff from increasing his outside option by a unit thanfrom increasing his productivity with the single employer. Thus, he doesbetter to increase his general human capital.

4.

(a) The efficient investment level is the solution to maxx x − x2m whichis x∗ = 1/2.

(b) Player 1 selects x = 1/2. Following this investment, the players de-mand m2(1/2) = 0 and m1(1/2) = x. In the event that player 1 deviatesby choosing some x = 1/2, then the players are prescribed to make thedemands m2(x) = x and m1(x) = 0.

(c) One way to interpret this equilibrium is that player 1’s bargainingweight is 1 if he invests 1/2, but it drops to zero if he makes any otherinvestment. Thus, player 1 obtains the full value of his investment whenhe selects 1/2, but he obtains none of the benefit of another investmentlevel.

5.

(a) The union makes a take-it-or-leave-it offer of w = (R − M)/n, whichis accepted. This implies that the railroad will not be built, since theentrepreneur can foresee that it will lose F .

(b) The surplus is R − M . The entrepreneur gets πE[R − M ] and theunion gets nw + πU [R −M ]. The railroad is built if πE[R − M ] > F .

(c) The entrepreneur’s investment is sunk when negotiation occurs, so hedoes not generally get all of the returns from his investment. When he hasall of the bargaining power, he does extract the full return. To avoid thehold-up problem, the entrepreneur may try to negotiate a contract withthe union before making his investment.

6.

Stock options in a start-up company, stock options for employees, andoptions to buy in procurement settings are examples.

7.

If it is not possible to verify whether you have abused the computer ornot, then it is better for you to own it. This gives you the incentive totreat it with care, because you will be responsible for necessary repairs.



22 Repeated Games and Reputation

1.

(U, L) can be supported as follows. If player 2 defects ((U,M) is played)in the first period, then the players coordinate on (C, R) in the secondperiod. If player 1 defects ((C, L) is played) in the first period, then theplayers play (D, M) in the second period. Otherwise, the players play (D,R) in the second period.

2.

(a) To support cooperation, δ must be such that 2/(1−δ) ≥ 4+δ/(1−δ).Solving for δ, we see that cooperation requires δ ≥ 2/3.

(b) To support cooperation by player 1, it must be that δ ≥ 1/2. Tosupport cooperation by player 2, it must be that δ ≥ 3/5. Thus, we needδ ≥ 3/5.

(c) Cooperation by player 1 requires δ ≥ 4/5. Player 2 has no incentiveto deviate in the short run. Thus, it must be that δ ≥ 4/5.

3.

(a) The Nash equilibria are (B,X) and (B,Y).

(b) Yes. Player 1 plays A in period 1 and B in period 2. Player 2 plays Xin period 1. In period 2, player 2 plays X if player 1 played A in period1, and plays Y if player 1 played B in period 1.

4.

In period 2, subgame perfection requires play of the only Nash equilibriumof the stage game. As there is only one Nash equilibrium of the stage game,selection of the Nash equilibrium to be played in period 2 cannot influenceincentives in period 1. Thus, the only subgame perfect equilibrium is playof the Nash equilibrium of the stage game in both periods. For any finiteT , the logic from the two period case applies, and the answer does notchange.




5.

Alternating between (C, C) and (C, D) requires that neither player hasthe incentive to deviate. Clearly, however, player 1 can guarantee himselfat least 2 per period, yet he would get less than this starting in period 2if the players alternated as described. Thus, alternating between (C,C)and (C,D) cannot be supported.

On the other hand, alternating between (C,C) and (C,D) can be sup-ported. Note first that, using the stage Nash punishment, player 2 hasno incentive to deviate in odd or even periods. Player 1 has no incen-tive to deviate in even periods, when (D, D) is supposed to be played.Furthermore, player 1 prefers not to deviate in an even period if

7 +2δ

1 − δ≤ 3 + 2δ + 3δ2 + 2δ3 + 3δ4 + . . . ,

which simplifies to

7 +2δ

1 − δ≤ 3 + 2δ)

1 − δ2.

Solving for δ yields δ ≥√

45.

6.

A long horizon ahead.

7.

(a) The (pure strategy) Nash equilibria are (U, L, B) and (D, R, B).

(b) Any combination of the Nash equilibria of the stage game are subgameperfect equilibria. These yield the payoffs (8, 8, 2), (8, 4, 10), and (8, 6, 6).There are two other subgame perfect equilibria. In the first, the playersselect (U, R, A) in the first round, and then if no one deviated, they play(D, R, B) in the second period; otherwise, they play (U, L, B) in thesecond period. This yields payoff (9, 7, 10). In the other equilibrium, theplayers select (U, R, B) in the first round and, if player 2 does not cheat,(U, L, B) in the second period; if player 2 cheats, they play (D, R, B) inthe second period. This yields the payoff (8, 6, 9).




8.

(a) Player 2t plays a best response to player 1’s action in the stage game.

(b) Consider the following example. There is a subgame perfect equilib-rium, using stage Nash punishment, in which, in equilibrium, player 1plays T and player 2t plays D.

(c) Consider, for example, the prisoners’ dilemma. If only one playeris a long-run player, then the only subgame perfect equilibrium repeatedgame will involves each player defecting in each period. However, from thetext we know that cooperation can be supported when both are long-runplayers.

9.

(a) As x < 10, there is no gain from continuing. Thus, neither playerwishes to deviate.

(b) If a player selects S, then the game stops and this player obtains 0.Since the players randomize in each period, their continuation values fromthe start of a given period are both 0. If the player chooses C in a period,he thus gets an expected payoff of 10α − (1 − α). Setting this equal to 0(which must be the case in order for the players to be indifferent betweenS and C) yields α = 1/11.

(c) In this case, the continuation value from the beginning of each periodis αx. When a player selects S, he expects to get αz; when he chooses C,he expects 10α + (1 − α)(−1 + δαx). The equality that defines α is thusαz = 10α + (1 − α)(−1 + δαx).



23 Collusion, Trade Agreements,

and Goodwill

1.

(a) Consider all players selecting pi = p = 60, until and unless someonedefects. If someone defects, then everyone chooses pi = p = 10 thereafter.

(b) The quantity of each firm when they collude is qc = (110 − 60)/n =50/n. The profit of each firm under collusion is (50/n)60 − 10(50/n) =2500/n. The profit under the Nash equilibrium of the stage game is 0. Ifplayer i defects, she does so by setting pi = 60 − ε, where ε is arbitrarilysmall. Thus, the stage game payoff of defecting can be made arbitrarilyclose to 2, 500.

To support collusion, it must be that [2500/n][1/(1−δ)] ≥ 2500+0, whichsimplifies to δ ≥ 1 − 1/n.

(c) Collusion is “easier” with fewer firms.

2.

(a) The best response function of player i is given by BRi(xj) = 30+xj/2.Solving for equilibrium, we find that xi = 30+ 1

2[30+ xi

2] which implies that

x∗1 = x∗

2 = 60. The payoff to each player is equal to 2, 000− 30(60) = 200.

(b) Under zero tariffs, the payoff to each country is 2,000. A deviationby player i yields a payoff of 2, 000 + 60(30) − 30(30) = 2, 900. Thus,player i’s gain from deviating is 900. Sustaining zero tariffs requires that

2000

1 − δ≥ 2900 +

200δ

1 − δ.

Solving for δ, we get δ ≥ 1/3.

(c) The payoff to each player of cooperating by setting tariffs equal to kis 2000 + 60k + k2 − k2 − 90k = 2000 − 30k. The payoff to a player fromunilaterally deviating is equal to

2, 000 + 60[30 + k

2

]+

[30 + k

2

]k −

[30 + k

2

]2 − 90k

= 2, 000 +[30 + k

2

]2 − 90k.

Thus, the gain to player i of unilaterally deviating is

[30 +

k

2

]2

− 60k.




In order to support tariff setting of k, it must be that[30 +

k

2

]2

− 60k +200δ

1 − δ≤ [2000 − 30k]δ

1 − δ.

Solving yields the condition

[30 + k2]2 − 60k

1800 − 90k + [30 + k2]2

≤ δ.

3.

The Nash equilibria are (A, Z) and (B, Y). Obviously, there is an equilib-rium in which (A, Z) is played in both periods and player 21 sells the rightto player 22 for 8α. There is also a “goodwill” equilibrium that is like theone constructed in the text, although here it may seem undesirable fromplayer 21’s point of view. Players coordinate on (A, X) in the first periodand (A, Z) in the second period, unless player 21 deviated from X in thefirst period, in which case (B,Y) is played in the second period. Player 21

sells the right to player 22 for 8α if he did not deviate in the first period,whereas he sells the right for 4α if he deviated. This is an equilibrium(player 21 prefers not to deviate) if α > 3/4.

4.

(a) Each player 2t cares only about his own payoff in period t, so he willplay D. This implies that player 1 will play D in each period.

(b) Suppose players select (C, C) unless someone defects, in which case(D, D) is played thereafter. For this to be rational for player 1, we need2/(1 − δ) ≥ 3 + δ/(1 − δ) or δ ≥ 1/2. For player 2t, this requires that2 + δpG ≥ 3 + δpB , where pG is the price he gets with a good reputationand pB is the price he gets with a bad reputation. (Trade occurs at thebeginning of the next period, so the price is discounted). Cooperation canbe supported if δ(pG − pB) ≥ 1.

Let α be the bargaining weight of each player2t in his negotiation to sellthe right to player 2t+1. We can see that the surplus in the negotiationbetween players 2t and 2t+1 is 2 + δpG, because this is what player 2t+1

expects to obtain from the start of period t+1 if he follows the prescribedstrategy of cooperating when the reputation is good. This surplus is di-vided according to the fixed bargaining weights, implying that player 2t

obtains pG = α[2 + δpG]. Solving for pG yields pG = 2α/(1 − δα). Sim-ilar calculations show that pB = α/(1 − δα). Substituting this into thecondition δ(pG − pB) ≥ 1 and simplifying yields δα ≥ 1/2. In words,the discount factor and the owner’s bargaining weight must be sufficientlylarge in order for cooperation to be sustained over time.




5.

(a) The Nash equilibria are (x, x), (x, z), (z, x), and (y, y).

(b) They would agree to play (y,y).

(c) In the first round, they play (z, z). If no one defected in the first period,then they are supposed to play (y, y) in the second period. If player 1defected in the first period, then they coordinate on (z, x) in the secondperiod. If player 2 defected in the first period, then they coordinate on (x,z) in the second period. It is easy to verify that this strategy is a subgameperfect equilibrium.

(d) The answer depends on whether one believes that the players’ bar-gaining powers would be affected by the history of play. If deviation by aplayer causes his bargaining weight to suddenly drop to, say, 0, then theequilibrium described in part (c) seems consistent with the opportunityto renegotiate before the second period stage game. Another way of in-terpreting the equilibrium is that the prescribed play for period 2 is thedisagreement point for renegotiation, in which case there is no surplus ofrenegotiation. However, perhaps a more reasonable theory of renegotia-tion would posit that each player’s bargaining weight is independent ofthe history (it is related to institutional features) and that each playercould insist on some neutral stage Nash equilibrium, such as (x, x) or (y,y). In this case, as long as bargaining weights are positive, it would not bepossible to sustain (x, z) or (z, x) in period 2. As a result, the equilibriumof part (c) would not withstand renegotiation.

6.

(a) If a young player does not expect to get anything when he is old, thenhe optimizes myopically when young and therefore gives nothing to theolder generation.

(b) If player t − 1 has given xt−1 = 1 to player t − 2, then player t givesxt = 1 to player t − 1. Otherwise, player t gives nothing to player t − 1(xt = 0). Clearly, each young player thus has the incentive to give 1 tothe old generation.

(c) Each player obtains 1 in the equilibrium from part (a), 2 in the equilib-rium from part (b). Thus, a reputation-based intergenerational-transferequilibrium is best.

7.

(a) Any δ.

(b) δ ≥ 37.

(c) m = 43(1−δ)

.




8.

(a) Cooperation can be sustained for δ ≥ 23.

(b) Cooperation can be sustained for δ ≥ kk+1

.

(c) Cooperation can be sustained for δ ≥ 4(k−2)!4(k−2)!+k!

.



24 Random Events and

Incomplete Information

1.

2.

(a)




(b)

3.

4.



25 Risk and Incentives in Contracting

1.

Examples include stock brokers, commodities traders, and salespeople.

2.

The probability of a successful project is p. This implies an incentivecompatibility constraint of

p(w + b − 1)α + (1 − p)(w − 1)α ≥ wα

and a participation constraint of

p(w + b − 1)α + (1 − p)(w − 1)α ≥ 1.

Thus, we need

p(w + b − 1)α + (1 − p)(w − 1)α = 1 = wα.

This implies that b = p−1/α.

3.

(a) The wage offer must be at least 100 − y, so the firm’s payoff is 180 −(100 − y) = 80 + y.

(b) In this case, the worker accepts the job if and only if w + 100q ≥ 100,which means the wage must be at least 100(1 − q). The firm obtains200 − 100(1 − q) = 100(1 + q).

(c) When q = 1/2, it is optimal to offer the risky job at a wage of 50 ify ≤ 70, whereas the safe job at a wage of 100 − y is optimal otherwise.

4.

(a) Below is a representation of the extensive form for T = 1.



25 RISK AND INCENTIVES IN CONTRACTING 146

(b) Regardless of T , whenever player 1 gets to make the offer, he offersq2δ to player 2 (and demands 1 − q2δ for himself). When player 1 offersq2δ or more, then player 2 accepts. When player 2 gets to offer, she offersq1δ to player 1. When player 2 offers q1δ or more, player 1 accepts.

(c) The expected equilibrium payoff for player i is qi. Thus, the probabilitywith which player i gets to make an offer can be viewed as his bargainingweight.

(d) The more risk averse a player is, the lower is the offer that he is willingto accept. Thus, an increase in a player’s risk aversion should lower theplayer’s equilibrium payoff.

5.

(a) n2 = 6.

(b) n2 = 1 or 6.

(c) n1 = 2, 4, 6, or 7.



26 Bayesian Nash Equilibrium

and Rationalizability

1.

(a) The Bayesian normal form is:

(Z, V) is the only rationalizable strategy profile.

(b) The Bayesian normal form is:

XAYB is a dominant strategy for player 1. Thus, the rationalizable set is(XAYB,W).

(c) False.



BAYESIAN EQUILIBRIUM AND RATIONALIZABILITY 148

2.

Player 1’s payoff is given by

u1 = (x1 + x2L + x1x2L) + (x1 + x2H + x1x2H) − x21.

The low type of player 2 gets the payoff

u2L = 2(x1 + x2L + x1x2L) − 2x22L,

whereas the high type of player 2 obtains

u2H = 2(x1 + x2H + x1x2H) − 3x22H.

Player 1 solves

maxx1

(x1 + x2L + x1x2L) + (x1 + x2H + x1x2H) − x21.

The first-order condition is 1 + x2L − x1 + 1 + x2H − x1 = 0. This impliesthat x∗

1(x2L, x2H) = 1+ (x2L + x2H)/2. Similarly, the first-order conditionof the low type of player 2 yields x∗

2L(x1) = (1 + x1)/2. The first ordercondition of the high type of player 2 implies x∗

2H(x1) = (1+x1)/3. Solvingthis system of equations, we find that the equilibrium is given by x∗

1 = 177,

x∗2L = 12

7, and x∗

2H = 87.

3.

(a) The extensive form and normal form representations are:

The set of Bayesian Nash equilibria is equal to the set of rationalizablestrategies, which is {(Du, R)}.




(b) The extensive form and normal form representations in this case are:

The equilibrium is (D, u, R). The set of rationalizable strategies is S.

(c) Regarding rationalizability, the difference between the settings of parts(a) and (b) is that in part (b) the beliefs of players 1A and 1B do nothave to coincide. In equilibrium, the beliefs of player 1A and 1B must bethe same.

4.

Recall that player 1’s best response function is given by BR1(qL2 , qH

2 ) =1/2−qL

2 /4−qH2 /4. The low type of player 2 has a best response function of

BRL2 (q1) = 1/2−q1/2. The high type of player 2 has a best response func-

tion of BRH2 (q1) = 3/8−q1/2. If q1 = 0, then player 2’s optimal quantities




are qL2 = 1/2 and qH

2 = 3/8. Note that player 2 would never produce morethan these amounts. To the quantities qL

2 = 1/2 and qH2 = 3/8, player 1’s

best response is q1 = 5/16. Thus, player 1 will never produce more thanq1 = 5/16. We conclude that each type of player 2 will never producemore than her best response to 5/16. Thus, qL

2 will never exceed 11/32,and qH

2 will never exceed 7/32. Repeating this logic, we find that the ra-tionalizable set is the single strategy profile that simultaneously satisfiesthe best response functions, which is the Bayesian Nash equilibrium.

5.

(a) u1(p1, p2) = 42p1 + p1p2 − 2p21 − 220 − 10p2 and u2(p1, p2) = 922 +

2c)p2 + p1p2 − 2p22 − 22c − cp1.

(b) BR1(p2) = 42+p2

4and BR2(p1) = 22+2cp1

4.

(c) p∗1 = 14 and p∗2 = 14.

(d) p∗1 = 14, p∗2,c=14 = 16, and p∗2,c=6 = 12.

6.

(LL′, U).

7.

(a)

(b) (BA′, Y)

8.

It is easy to see that, whatever is the strategy of player j, player i’s bestresponse has a “cutoff” form in which player i bids if and only if his draw isabove some number αi. This is because the probability of winning when




i bids is increasing in i’s type. Let αj be player j’s cutoff. Then, bybidding, player i of type xi obtains an expected payoff of

b(xi, αj) =

{1 · αj + (1 − αj)(−2) if xi ≤ αj

1 · αj + (xi − αj)(2) + (1 − xi)(−2) if xi > αj

Note that, as a function of xi, b(·, αj) is the constant 3αj − 2 up to αj

and then rises with a slope of 4. Player i’s best response is to fold ifb(xi, αj) < −1 and bid if b(xi, αj) > −1. Note that if αj > 1/3 thenplayer i optimally bids regardless of his type (meaning that αi = 0),if αj < 1/3 then player i’s optimal cutoff is αi = (1 + αj)/4, and ifαj = 1/3 then player i’s optimal cutoff is any number in the interval[0, 1/3]. Examining this description of i’s best-response, we see that thereis a single Nash equilibrium and it has α1 = α2 = 1/3.

9.

The unique Nash equilibrium is (Bf,B). That is player 1 bids when he hasthe Ace and folds when he has the King, and player 2 always bids.



27 Lemons, Auctions,

and Information Aggregation

1.

There is always an equilibrium in this game. Note that, regardless ofp, there is an equilibrium in which neither the lemon nor the peach istraded (Jerry does not trade and Freddie trades neither car). When either1000 < p ≤ 2000 or p > 1000 + 2000q, the only equilibrium involves notrade whatsoever.

2.

Your optimal bidding strategy is b = v/3, you should bid b(3/5) = 1/5.

3.

To show that bidding vi is weakly preferred to bidding any x < vi, considerthree cases, with respect to x, vi, and the other player’s bid bj. In the firstcase, x < bj < vi. Here, bidding x causes player i to lose, but biddingvi allows player i to win and receive a payoff of vi − bj. Next considerthe case in which x < vi < bj. In this case, it does not matter whetherplayer i bids x or vi; he loses either way, and receives a payoff of 0. Finally,consider the case where bj < x < vi. Here, bidding either x or vi ensuresthat player i wins and receives the payoff vi − bj.

4.

(a) Colin wins and pays 82.

(b) Colin wins and pays 82 (or 82 plus a very small number).

(c) The seller should set the reserve price at 92. Colin wins and pays 92.

5.

As discussed in the text, without a reserve price, the expected revenue ofthe auction is 1000/3. With a reserve price r, player i will bid at least r ifvi > r. The probability that vi < r is r/1000. Thus, the probability thatboth players have a valuation that is less than r is (r/1000)2 . Consider,for example, setting a reserve price of 500. The probability that at leastone of the players’ valuations is above 500 is 1 − (1/2)2 = 3/4. Thus,the expected revenue of setting r = 500 is at least 500(3/4) = 385, whichexceeds 1000/3.




6.

The equilibrium bidding strategy for player i is bi(vi) = v2i /2.

7.

Let vi = 20. Suppose player i believes that the other players’ bids are10 and 25. If player i bids 20 then she loses and obtains a payoff of0. However, if player i bids 25 then she wins and obtains a payoff of20 − 10 = 10. Thus, bidding 25 is a best response, but bidding 20 is not.

8.

(a) Clearly, if p < 200 then John would never trade, so neither player willtrade in equilibrium. Consider two cases for p between 200 and 1000.

First, suppose 600 ≤ p ≤ 1, 000. In this case, Jessica will not trade if hersignal is x2 = 200, because she then knows that 600 is the most the stockcould be worth. John therefore knows that Jessica would only be willingto trade if her signal is 1, 000. However, if John’s signal is 1, 000 and heoffers to trade, then the trade could occur only when v = 1000, in whichcase he would have been better off not trading. Realizing this, Jessicadeduces that John would only be willing to trade if x1 = 200, but thenshe never has an interest in trading. Thus, the only equilibrium has bothplayers choosing “not,” regardless of their types.

Similar reasoning establishes that trade never occurs in the case of p <600 either. Thus, trade never occurs in equilibrium. Interestingly, wereached this conclusion by tracing the implications of common knowledgeof rationality (rationalizability), so the result does not rely on equilibrium.

(b) It is not possible for trade to occur in equilibrium with positive prob-ability. This may seem strange compared to what we observe about realstock markets, where trade is usually vigorous. In the real world, playersmay lack common knowledge of the fundamentals or each other’s rational-ity, trade may occur due to liquidity needs, and there may be differencesin owners’ abilities to run firms.

(c) Intuitively, the equilibrium strategies can be represented by numbersx1 and x2, where John trades if and only if x1 ≤ x1 and Jessica trades ifand only if x2 ≥ x2. For John, trade yields an expected payoff of

∫ x2

100(1/2)(x1 + x2)F2(x2)dx2 +

∫ 1000

x2

pF2(x2)dx2 − 1.

Not trade yields ∫ 1000

100(1/2)(x1 + x2)F2(x2)dx2.




Simplifying, we see that John’s trade payoff is greater than is his no-tradepayoff when

∫ 1000

x2

[p − (1/2)(x1 + x2)]F2(x2)dx2 ≥ 1. (∗)

For Jessica, trade implies an expected payoff of

∫ x1

100[(1/2)(x1 + x2) − p]F1(x1)dx1 − 1.

No trade gives her a payoff of zero. Simplifying, she prefers trade when

∫ x1

100[(1/2)(x1 + x2) − p]F1(x1)dx1 ≥ 1. (∗∗)

By the definitions of x1 and x2, (*) holds for all x1 ≤ x1 and (**) holdsfor all x2 ≥ x2. Integrating (*) over x1 < x1 yields

∫ x1

100

∫ 1000

x2

[p − (1/2)(x1 + x2)]F2(x2)F1(x1)dx2dx1 ≥∫ x1

100F1(x1)dx1.

Integrating (**) over x2 > x2 yields

∫ x1

100

∫ 1000

x2

[(1/2)(x1 + x2) − p]F2(x2)F1(x1)dx2dx1 ≥∫ 1000

x2

F2(x2)dx2.

These inequalities cannot be satisfied simultaneously, unless trade neveroccurs in equilibrium—so that x1 is less than 100 and x2 exceeds 1, 000,implying that all of the integrals in these expressions equal zero.

9.

(a) Player 1’s best-response bidding strategy is

bi(vi) =

{y1 for y1 ≥ 30 for y1 < 0

.

(b) Player i will bid up to yi.



28 Perfect Bayesian Equilibrium

1.

Let w = Prob(H | p) and let r = Prob(H | p).

(a) The separating equilibrium is (pp′, NE′) with beliefs w = 1 and r = 0.

(b) For q ≤ 1/2, there is a pooling equilibrium with strategy profile (pp′,NN′) and beliefs w = q and any r ≤ 1/2. There are also similar poolingequilibria in which the entrant chooses E and has any belief r ≥ 1/2. Forq > 1/2, there is a pooling equilibrium in which the strategy profile is(pp′, EE′) and the beliefs are w = q and any r ≤ 1/2. There are alsopooling equilibria in which the incumbent plays pp′.

2.

(a) No.

(b) Yes. (AA′, Y) with belief q ≤ 35

(c)

3.

(a) Yes, it is (RL′, U) with q = 1.

(b) Yes, it is (LL′,D) with q ≤ 1/3.

4.

Yes. Player 1’s actions may signal something of interest to the otherplayers. This sort of signaling can arise in equilibrium as long as, giventhe rational response of the other players, player 1 is indifferent or prefersto signal.



28 PERFECT BAYESIAN EQUILIBRIUM 156

5.

(a) The perfect Bayesian equilibrium is given by E0N1, y = 1, y = 0,q = 1, and y = 1.

(b) The innocent type provides evidence, whereas the guilty type doesnot.

(c) In the perfect Bayesian equilibrium, each type x ∈ {0, 1, . . . , K − 1}provides evidence and the judge believes that he faces type K when noevidence is provided.

6.

(a) c ≥ 2. The separating perfect Bayesian equilibrium is given by OB′,FS′, r = 0, and q = 1.

(b) c ≤ 2. The following is such a pooling equilibrium: OO′, SF′, r = 0,and q = 1/2.

7.

(a) If the worker is type L, then the firm offers z = 0 and w = 35. If theworker is type H, then the firm offers z = 1 and w = 40.

(b) Note that the H type would obtain 75+35 = 110 by accepting the safejob. Thus, if the firm wants to give the H type the incentive to acceptthe risky job, then the firm must set w1 so that 100(3/5) + w1 ≥ 110,which means w1 ≥ 50. The firm’s optimal choice is w1 = 50, which yieldsa higher payoff than would be the case if the firm gave to the H type theincentive to select the safe job.

(c) The answer depends on the probabilities of the H and L types. If thefirm follows the strategy of part (b), then it expects 150p + 145(1 − p) =145 + 5p. If the firm only offers a contract with the safe job and wants toemploy both types, then it is best to set the wage at 35, which yields apayoff of 145. Clearly, this is worse than the strategy of part (b). Finally,the firm might consider offering only a contract for the risky job, with theintention of only attracting the H type. In this case, the optimal wage is40 and the firm gets an expected payoff of 160p. This “H-only” strategyis best if p ≥ 145/155; otherwise, the part (b) strategy is better.

8.

In the perfect Bayesian equilibrium, player 1 bids with both the Ace andthe King, player 2 bids with the Ace and folds with the Queen. Whenplayer 1 is dealt the Queen, he bids with probability 1/3. When player 2is dealt the King and player 1 bids, player 2 folds with probability 1/3.



29 Job-Market Signaling and Reputation

1.

Education would not be a useful signal in this setting. If high types andlow types have the same cost of education, then they would have the sameincentive to become educated.

2.

Consider separating equilibria. It is easy to see that NE′ cannot be anequilibrium, by the same logic conveyed in the text. Consider the worker’sstrategy of EN′. Consistent beliefs are p = 0 and q = 1, so the firm playsMC′. Neither the high nor low type has the incentive to deviate.

Next consider pooling equilibria. It is easy to see that EE′ cannot be apooling equilibrium, because the low type is not behaving rationally inthis case. There is a pooling equilibrium in which NN′ is played, p = 1/2,the firm selects M′, q is unrestricted, and the firm’s choice between M andC is whatever is optimal with respect to q.

3.

(a) There is no separating equilibrium. The low type always wants tomimic the high type.

(b) Yes, there is such an equilibrium provided that p is such that theworker accepts. This requires 2p−(1−p) ≥ 0, which simplifies to p ≥ 1/3.The equilibrium is given by (OHOL, A) with belief q = p.

(c) Yes, there is such an equilibrium regardless of p. The equilibrium isgiven by (NHNL, R) with belief q ≤ 1/3.

4.

Clearly, the PBE strategy profile is a Bayesian Nash equilibrium. In fact,there is no other Bayesian Nash equilibrium, because the presence of theC type in this game (and rationality of this type) implies that player 2’sinformation set is reached with positive probability. This relation doesnot hold in general, of course, because of the prospect of unreached infor-mation sets.



29 JOB-MARKET SIGNALING AND REPUTATION 158

5.

As before, player 1 always plays S, I′, and B′. Also, player 2 randomizes sothat player 1 is indifferent between I and N , which implies that s = 1/4.Player 1 randomizes so that player 2 is indifferent between I and N . Thisrequires 2q − 2(1 − q) = 0, which simplifies to q = 1/2. However, q =p/(p+ r−pr). Substituting and solving for r, we get r = p/(1−p). Thus,in equilibrium, player 1 selects action I with probability r = p/(1 − p),and player 2 has belief q = 1/2 and plays I with probability 1/4.

If p > 1/2, then player 2 always plays I when her information set isreached. This is because 2p − 2(1 − p) = 4p − 2 > 0. Thus, equilibriumrequires that player 1’s strategy is II′SB′, that player 2 has belief q = p,and that player 2 selects I.

6.

In period 2 player 2 will accept p2 if and only if v ≥ p2. So player 1’s offerwill solve maxp2 p2prob(p2 < v). This yields p2 = c(p1)

2.

Player 2 accepts p1 if and only if v = c(p1). Using that player 2 withv = c(p1) is indifferent between accepting and rejecting p1, we find thatc(p1) = p1

1− δ2

. In period 1 player 1 maximizes

p1prob(p1 ≥ c(p1)) + δp2prob(p2 < v)prob(p1 < c(p1).

Substituting for player 2’s offer in period 2, we find that p1 = 2[1−δ/2]2

4−δ.

7.

(a) The extensive form is:

In the Bayesian Nash equilibrium, player 1 forms a firm (F) if 10p−4(1−p) ≥ 0, which simplifies to p ≥ 2/7. Player 1 does not form a firm (O) ifp < 2/7.




(b) The extensive form is:

(c) Clearly, player 1 wants to choose F with the H type and O with theL type. Thus, there is a separating equilibrium if and only if the types ofplayer 2 have the incentive to separate. This is the case if 10− g ≥ 0 and0 ≥ 5 − g, which simplifies to g ∈ [5, 10].

(d) If p ≥ 2/7 then there is a pooling equilibrium in which NN′ and F′

are played, player 1’s belief conditional on no gift is p, player 1’s beliefconditional on a gift is arbitrary, and player 1’s choice between F and Ois optimal given this belief. If, in addition to p ≥ 2/7, it is the case thatg ∈ [5, 10], then there is also a pooling equilibrium featuring GG′ andFO′. If p ≤ 2/7 then there is a pooling equilibrium in which NN′ andOO′ are played (and player 1 puts a probability on H that is less than 2/7conditional on receiving a gift).

8.

(a) A player is indifferent between O and F when he believes that theother player will choose O for sure. Thus, (O, O; O, O) is a BayesianNash equilibrium.

(b) If both types of the other player select Y, the H type prefers Y if10p−4(1−p) ≥ 0, which simplifies to p ≥ 2/7. The L type weakly prefersY, regardless of p. Thus, such an equilibrium exists if p ≥ 2/7.

(c) If the other player behaves as specified, then the H type expects −g +p(w + 10) + (1 − p)0 from giving a gift. He expects pw from not givinga gift. Thus, he has the incentive to give a gift if 10p ≥ g. The L typeexpects −g + p9w + 5) + (1− p)0 if he gives a gift, whereas he expects pw




if he does not give a gift. The L type prefers not to give if g ≥ 5p. Theequilibrium, therefore, exists if g ∈ [5p, 10p].

9.

(a) The manager’s optimal contract solves maxe,x e−x subject to x−αe2 ≥0 (which is necessary for the worker to accept). Clearly, the managerwill pick x and e so that the constraint binds. Using the constraint tosubstitute for x yields the unconstrained problem maxe e − αe2. Solvingthe first-order condition, we get e = 1/(2α) and x = 1/(4α).

(b) Using the solution of part (a), we obtain e = 4, x = 2, e = 4/3, andx = 2/3.

(c) The worker will choose the contract that maximizes x − αe2. Thehigh type of worker would get a payoff of −4 if he chooses contract (e, x),whereas he would obtain 0 by choosing contract (e, x). Thus, he wouldchoose the contract that is meant for him. On the other hand, the lowtype prefers to select contract (e, x), which gives him a payoff of 4/9,rather than getting 0 under the contract designed for him.

(d) The incentive compatibility conditions for the low and high types,respectively, are

xL − 1

8e2

L ≥ xH − 1

8e2

H

and

xH − 3

8e2

H ≥ xL − 3

8e2

L.

The participation constraints are

xL − 1

8e2

L ≥ 0

and

xH − 3

8e2

H ≥ 0.

(e) Following the hint, we can substitute for xL and xH using the equations

xL = xH − 1

8e2

H +1

8e2

L

and

xH =3

8e2

H.

Note that combining these gives xL = 14e2

H + 18e2

L. Substituting for xL andxH yields the following unconstrained maximization problem:

maxeL,eH

1

2

[eH − 3

8e2

H

]+

1

2

[eL − 1

4e2

H − 1

8eL2

].




Calculating the first-order conditions, we obtain e∗L = 4, x∗L = 54/25,

e∗H = 4/5, and x∗H = 6/25.

(f) The high type exerts less effort than is efficient, because this helps themanager extract more surplus from the low type.



30 Appendix B

1.

(a) Suppose not. Then it must be that B(Rk−1) = ∅, which implies thatBi(R

k−1) = ∅ for some i. However, we know that the best response setis nonempty (assuming the game is finite), which contradicts what weassumed at the start.

(b) The operators B and UD are monotone, meaning that X ⊂ Y impliesB(X) ⊂ B(Y ) and UD(X) ⊂ UD(Y ). This follows from the definitionsof Bi and UDi. Note, for instance, that any belief for player i that putspositive probability only on strategies in X−i can also be considered inthe context of the larger Y−i. Furthermore, if a strategy of player i isdominated with respect to strategies Y−i, then it also must be dominatedwith respect to the smaller set X−i. Using the monotone property, we seethat UD(S) = R1 ⊂ S = R0 implies R2 = UD(R1) ⊂ UD(R0) = R1. Byinduction, Rk ⊂ Rk−1 implies Rk+1 = UD(Rk) ⊂ Rk = UD(Rk−1).

(c) Suppose not. Then there are an infinite number of rounds in which atleast one strategy is removed for at least one player. However, from (b),we know strategies that are removed are never “put back,” which meansan infinite number of strategies are eventually deleted. This contradictsthat S is finite.

2.

This is discussed in the lecture material for Chapter 7 (see Part II of thismanual).

3.

(a) For any p such that 0 ≤ p ≤ 1, it cannot be that 6p > 5 and 6(1−p) >5.

(b) Let p denote the probability that player 1 plays U and let q denote theprobability that player 2 plays M. Suppose that C ∈ BR. Then it mustbe that the following inequalities hold: 5pq ≥ 6pq, −100(1 − p)q ≥ 0,−100p(1 − q) ≥ 0, and 5(1 − p)(1 − q) ≥ 6(1 − p)(1 − q). This requiresthat (1−p)q = p(1−q), which contradicts the assumption of uncorrelatedbeliefs.

(c) Consider the belief θ−1 that (U, M) is played with probability 1/2 andthat (D, N) is played with probability 1/2. We have that u1(C, θ−1) = 5and u1(B, θ−1) = u1(A, θ−1) = 3.



163

Part IV

Sample Examination QuestionsOn the following pages are some sample examination questions, in the form of twoexaminations used by Joel Watson for undergraduates at UCSD in 2007 (wild fires inSan Diego shortened the term in Fall 2007, so the examinations covered a relativelynarrow set of topics) and a group of questions from Jesse Bull. Instructors are welcometo send their own sample questions to Watson ([email protected]), who will add themto those shown here. Please also report errors to Watson.



Economics 109 Midterm Exam IProf. Watson, Fall 2007

You have 50 minutes to complete this examination. You maynot use your notes, calculators, or any books during the exami-nation. Write your answers, including all necessary derivations,in the spaces provided on the answer sheet that has been dis-tributed separately. You may use the scratch paper that hasbeen distributed but submit only your answer sheet. You donot need to show any work in your answers to questions 1-5;these questions will be graded only on the basis of whether yourfinal answers are correct.

1. Write your name in the designated space on the answer sheet.In the space marked “version,” write the following number: 4.

2. In the extensive-form game pictured on the right, how many(pure) strategies does player 1 have? Do not name the strategies;simply report how many there are.

3. In the normal-form game pictured on your answer sheet,suppose that player 1 believes that player 2 is equally likely toplay any of her strategies. What is player 1’s best response?

4. In the normal-form game pictured on your answer sheet,is player 1’s strategy M dominated? If so, describe a strategythat dominates it. If not, describe a belief to which M is a bestresponse.

5. Consider the normal-form game pictured on your answersheet.

(a) List all of the efficient strategy profiles in thisgame.

(b) Calculate the rationalizable set of strategy pro-files in this game.

1



6. Consider a game in which, simultaneously, player 1 selectsa number x ∈ [0, 20] and player 2 selects a number y ∈ [0, 20].The payoffs are given by:

u1(x, y) = 2xy − x2

u2(x, y) = 10y + xy − y2.

(a) Calculate and graph each player’s best-responsefunction, as a function of the opposing player’s purestrategy (equivalently, expected strategy).

(b) Determine the rationalizable strategy profiles forthis game. Show your logic.

7. Consider a strategic setting in which ten firms simultaneouslyand independently decide whether to locate in the city (X) orin the suburbs (Y). That is, there are ten players (n = 10) andtwo strategies for each player. Each firm’s payoff of locating inthe suburbs is 20, and this is independent of how many otherfirms locate there. However, if firm i locates in the city, then itspayoff is vi(m), where m is the total number of firms (includingfirm i) that locate in the city. Suppose that

v1(m) = v2(m) = v3(m) = v4(m) = 31 − m,

v5(m) = v6(m) = v7(m) = v8(m) = 31 − 3m,

andv9(m) = v10(m) = 31 − 2m.

You are to determine the set of rationalizable strategies in thisgame. To show that you have done this accurately, answer thefollowing questions:

(a) How many strategy profiles are contained in theset of rationalizable strategy profiles?

(b) Describe one of the rationalizable strategy pro-files.

(c) Describe the various values of m that can arisein a rationalizable outcome.

2



Economics 109 Midterm Examination IAnswer Sheet, Fall 2007, Prof. Watson

C1. Your name:_____________________________ Your student ID:_________________________________

2. Number of strategies that player 1 has (circle one): 1 2 3 4 5 6

7 8 10 16 64 256

3.

Player 1's best response:

4.

Is M dominated? Circle one: YES NO

If so, name a strategy that dominates it:

If not, name a belief to which M is a best response:

5.

(a) The efficient strategy profiles are:

(b) The rationalizable set is: R =

Version:

4, 5 2, 2

1, 3 8, 5

t

m

l c2

1 r

5, 2

1, 2

b 2, 1 3, 2 5, 2

8, 2 0, 0

3, 0 5, 1

K

L

X Y2

1

M 4, 2 3, 2

8, 2 2, 4

6, 2 4, 3

w

x

a b2

1 c

6, 8

4, 4

y 1, 1 3, 3 9, 2

2 3 4 5 6 7 Total

4 6 6 8 8 8 40



6. (a) Best-response functions

Graph:

BR1(y) =

BR2(x) =

(b) Rationalizable set: R =

7.

(a) Number of rationalizable strategy profiles (circle one): 0 1 2 4 8 16 256 1024

(b) One of the rationalizable strategy profiles is:

(c) Values of m that can arise in a rationalizable outcome:

20

00 20

x

y



Economics 109 Midterm Exam IIProf. Watson, Fall 2007

You have 50 minutes to complete this examination. You maynot use your notes, calculators, or any books during the exami-nation. Write your answers, including all necessary derivations,in the spaces provided on the answer sheet that has been dis-tributed separately. You may use the scratch paper that hasbeen distributed but submit only your answer sheet. You donot need to show any work in your answers to question 2; thisquestions will be graded only on the basis of whether your finalanswers are correct.

1. Write your name in the designated space on the answer sheet.In the space marked “version,” write the following number: 2.

2. For the normal-form game pictured on your answer sheet,find the pure strategy Nash equilibria and describe which, ifany, are efficient.

3. For the normal-form game pictured on your answer sheet,calculate the mixed strategy Nash equilibrium.

4. Consider the following two-player game. First, player 1selects a number x, which must be greater than or equal tozero. Player 2 observes x. Then, simultaneously and inde-pendently, player 1 selects a number y1 and player 2 selectsa number y2, at which point the game ends. Player 1’s pay-off is u1 = 2y1y2 + xy2 − y2

1 and player 2’s payoff is u2 =4y2 − 4xy2 − 2y1y2 − y2

2. Calculate the subgame perfect equilib-rium of this game and report the equilibrium strategies.

5. Consider a two-player game in which the strategy spaces areS1 = [0,∞) and S1 = [0,∞). That is, each player selects anumber that is greater than or equal to zero. Let s1 denote thestrategy of player 1 and let s2 denote the strategy of player 2.Suppose that the payoff functions are given by

u1(s1, s2) = 2s1 + 2as1s2 − s21

andu2(s1, s2) = 2s2 + 2as1s2 − s2

2,

where a is a constant parameter.

(a) Is there any value of a such that this game hasno Nash equilibrium? If so, provide such a value.In either case, explain your answer.

(b) Is there any value of a such that this game hasan efficient Nash equilibrium? If so, provide sucha value. In either case, explain your answer.

1



Economics 109 Midterm Examination IIAnswer Sheet, Fall 2007, Prof. Watson

1. Your name:_____________________________ Your student ID:_________________________________

2. Circle the cells that are pure-strategy Nash equilibria and put an asterisk (*) in the cell of each efficient Nashequilibrium.

3.

Mixed strategy equilibrium:

C

Version:

2 3 4 5 Total

8 10 12 10 40

3, 9 1, 8

0, 1 6, 3w

x

a b2

1 c

0, 3

5, 4

y

6, 8 5, 7 2, 4

d e

v

z

5, 6 2, 3

4, 3 7, 2

1, 1 0, 4

1, 5 6, 1 3, 2 7, 6 2, 6

7, 4 8, 3 4, 1 4, 0 3, 3

5, 6 2, 3

8, 2 1, 5

w

x

a b2

1 c

1, 5

4, 1

y 0, 3 6, 1 7, 2



4.

Equilibrium strategy profile:

5.

(a) Is there an a such that the game has no Nash equilibrium? Circle one: YES NO

If yes, a =

(b) Is there an a such that the game has an efficient Nash equilibrium? Circle one: YES NO

If yes, a =

_________________________________________________________________________________________Your comments on the course so far:



Economics 109 Final ExaminationProf. Watson, Fall 2007

You have two hours and fifty minutes to complete this examination. You may notuse your notes, a calculator, or any books during the examination. Keep your eyes onyour own examination sheets. Questions marked with an asterisk (*) will be gradedonly on the basis of your final answers (not your derivations); for these questions,write your final answers in the space provided on the separate answer sheet that youhave been given. For the other questions, write your complete answers (includingderivations) on the separate answer sheet. It is important that you include theessential derivations on the answer sheet, so your knowledge of the appropriatetechniques can be verified. Use scratch paper as you wish, but you may not submityour scratch paper. Submit only your answer sheet at the end of the examinationperiod.

ON THE FIRST PAGE OF YOUR ANSWER SHEET, PLEASE SIGN THEWAIVER IF YOU AGREE TO IT.

1.* Consider the normal form game pictured here:

(a) What is the set of rationalizable strategy profiles in this game?

(b) Determine the game’s pure strategy Nash equilibrium strategy profile(s).

(c) Does this game have a mixed strategy equilibrium in which both X and Y areplayed with positive probability?

2.* Consider the normal form game pictured here:

All of the payoff numbers are specified, with the exception of those denoted by xand y. Find numbers for x and y such that the following three statements are alltrue.

• (U, M) is a Nash equilibrium,

• (U, M) is an inefficient strategy profile, and

• For the belief μ1 = (13, 2

3), which puts probability 1/3 on U and 2/3 on D, the

set of best responses for player 2 is BR2(μ1) = {L, N}.



3.* Consider the following game.

(a) Solve the game by backward induction and report the resulting strategy profile.(b) How many proper subgames does this game have?

4. Consider the following stage game.

Suppose this is the stage game in an infinitely repeated game. Assuming thatthe players discount future payoffs according to the discount factor δ, under whatconditions is there a subgame perfect equilibrium in which (C, X) is played in eachperiod? (Use grim trigger strategies.)

5. Consider a dynamic pricing problem for a monopolist who faces two types ofcustomers (H type and L type), over two periods of time — as in the examplediscussed in class and in the textbook. Suppose the types of customers have valuesof consuming the durable good in each period as shown in the following table:

Suppose there is one H type customer and one L type customer. The following twoquestions refer to the subgame perfect Nash equilibrium (in which the three playersare behaving sequentially rationally).

(a) What price would the monopolist set in the second period if neither customerpurchased in the first period?

(b) What is the monopolist’s optimal first-period price?



6. Consider a four-period bargaining game in which player 1 would make the offerin periods 1 and 4, and player 2 would make the offer in periods 2 and 3. That is, inperiod 1 player 1 makes an offer m1 to player 2. If player 2 rejects player 1’s offer,then the game proceeds to period 2, where player 2 makes an offer m2 to player 1. Ifplayer 1 rejects this offer, then the game proceeds to period 3, where player 2 makesanother offer m3. If player 1 rejects this offer, then the game proceeds to period 4,where player 1 makes an offer m4.

If an agreement is reached in period t, then in this period the player who acceptedthe offer gets mt dollars and the other player gets 1−mt dollars. The dollar amountsare discounted relative to earlier periods, where δ is the discount factor for bothplayers per period. If an agreement is not reached by the end of the fourth period,then both players get 0.

(a) In the subgame-perfect equilibrium of this game, what is the offer that player 1makes in the fourth period (contingent on agreement not occuring earlier)?

(b) In the subgame-perfect equilibrium of this game, what is the offer that player 2makes in the third period?

(c) In the subgame-perfect equilibrium of this game, what is the offer that player 2makes in the second period?

(d) In the subgame-perfect equilibrium of this game, what is the offer that player 1makes in the first period?

7. Consider the following game with nature:

(a) Does this game have any separating perfect Bayesian equilibrium? Show youranalysis and, if there is such an equilibrium, report it.

(b) Does this game have any pooling perfect Bayesian equilibrium? Show youranalysis and, if there is such an equilibrium, report it.



8. Consider a contracting game between a firm (player 1) and a consumer (player 2).First the firm chooses what type of contract (Good or Bad) to offer the consumer,then the consumer decides how thoroughly to read it (Read or Not), and finallythe consumer decides whether to accept the contract (Accept or Don’t). If theconsumer selects R, meaning that he exerts effort to read the contract, then theconsumer pays a reading cost of 1 unit and learns the firm’s choice (G or B). Ifthe consumer selects N, meaning that he does not exert effort to read the contract,then he must decide whether to accept it without observing whether the contractis G or B. The G contract yields a value of 5 to the firm and 5 to the consumer,whereas the B contract yields a value of 8 to the firm and −4 to the consumer. Theextensive form of this game is shown below.

(a) In this game, is there a pure-strategy subgame perfect Nash equilibrium in whichG is offered by player 1?

(b) Is there a mixed-strategy subgame perfect Nash equilibrium in which G is offeredand accepted with positive probability? If so, in equilibrium what is the probabilitythat player 1 selects G and what is the probability that player 2 selects R?

9. Consider a first-price, sealed-bid auction in which there are two bidders (players 1and 2) vying for one object. Let vi be the valuation of player i, for i = 1, 2.Each player’s valuation of the object is either 0 (the Low type) or 10 (the Hightype). Nature selects these with equal probabilities and chooses the valuationsof the two players independently. Each player knows his/her own valuation butdoes not observe the valuation of the other player (knowing only that it is 10 withprobability 1/2 and 0 with probability 1/2). After nature selects the valuations,the players simultaneously make bids b1, b2 ≥ 0. The object is given to the playerwho bids the higher amount; in the case of equal bids, the winner is determinedrandomly (with equal probabilities). If player i wins the auction then he/she gets apayoff of vi − bi. If player i loses then he/she gets 0.

This auction game has a Bayesian Nash equilibrium in which the High typeof each player selects his/her bid randomly according to a continuous probabilitydistribution over [0, b], for some number b. Let the function p(b) represent thisprobability distribution in the sense that, for any b, p(b) is the probability that theHigh type bids less than b. Calculate the Bayesian Nash equilibrium and report theLow type’s bid, b, and the function p.



Your name: _________________________________

Economics 109 Final Exam Answer SheetFall 2007, Prof. Watson

_____________________________________________________________________________________

1. (a) The set of rationalizable strategy profiles:

(b) Name the Nash equilibrium/equilibria:

(c) A mixed strategy equilibrium in which X and Y are played? Circle one: YES NO_____________________________________________________________________________________

2. x = y = _____________________________________________________________________________________

3. (a) The strategy profile derived by backward induction:

(b) Number of proper subgames:

_____________________________________________________________________________________

4. Analysis:

Condition on *:

1 2 3 4 5 6 7 8 9 Total

7 5 6 7 8 8 8 8 7 64



5. (a) Analysis:

Second-period priceif no one purchasedin the first period: p =

(b) Analysis:

Optimal first-periodprice: p1* =

_____________________________________________________________________________________

6. (a) Player 1’s fourth-period offer: m4 =

(b) Player 2’s third-period offer: m3 =

(c) Player 2’s second-period offer: m2 =

(a) Player 1’s first-period offer: m1 =



7. (a) Analysis of separating PBE (clearly show whether there is an equilibrium and report all conditions):

(b) Analysis of pooling PBE (clearly show whether there is an equilibrium and report all conditions):

_____________________________________________________________________________________

8. Analysis:



8. (continued)

(a) Is there a pure strategy equilibrium in which G is offered? Circle one: YES NOExplain:

(b) Is there a mixed strategy equilibrium in which G is offered? Circle one: YES NOExplain:

Probability thatplayer 1 chooses G:

Probability thatplayer 2 selects R:

_____________________________________________________________________________________

9. Analysis:

Low type’s bid:

b& =

Function p(b) =



Sample Exam Questions, Jesse Bull 2008

1. An island has 2 lakes and 20 fishermen. Each fisherman can fish on only one lake.The current institution is that a fisherman gets to keep the average number of fishcaught from the lake on which he chose to fish. On lake 1 the total number of fishcaught is given by F1(L1) = 10L1 − (L1)

2/2, where L1 is the number of fishermen onlake 1. For lake 2 the relationship is F2(L2) = 5L2.

(a) Under this institution, what is the total number of fish caught?

(b) The chief of the island asks his economist whether this arrangement isefficient (that is, whether the equilibrium allocation of fishermen to lakesmaximizes the number of fish caught). What is the answer to the chief’squestion? What is the efficient number of fishermen on each lake?

(c) The chief decides to require a fishing license for lake 1 which wouldrequire each fisherman who decides to fish on lake 1 to pay the chief xfish. If it is to bring about the efficient allocation of fishermen to lakes,what should x be?

2. Each firm in a duopoly can produce any positive quantity of output by payingonly a fixed cost of f . Let qi denote the output of firm i, and let Q denote the totaloutput. The inverse demand function for the market is p = 10 −Q.

(a) Find firm 2’s best response function.

(b) Suppose that the firms interact as in the Stackelberg model with firm 1choosing its quantity and then firm 2 choosing its quantity (after observingfirm 1’s choice). Assume that f = 9. Find the equilibrium quantities andprofits.

(c) Suppose that the fixed cost f is the result of an operating fee thatmust be paid to the government in order to produce output. At whatlevel would each firm argue that f should be set, assuming that theyinteract as in part (b)?

3. Player 1 (the hider) and player 2 (the seeker) play the following game. Thereare four boxes (all turned upside down so that the contents of each cannot be seen)arranged in a straight line (say left to right) with an equal distance, say x, betweenconsecutive boxes. For convenience, the boxes are labeled 1, 2, 3, and 4. The boxesare arranged in ascending order from left to right. Player 1 is given a $100 bill (by

1



the administrator of the game) to hide under one of the four boxes. Player 2 doesnot observe where player 1 hides the $100 bill. Once player 1 has hidden the $100 billunder one of the boxes, player 2 must look under one (and only one) of the boxes. Ifthe money is under the box under which player 2 looks, he gets to keep the $100. Ifit is not, player 1 gets to keep the $100.

(a) Describe the Nash equilibrium.

(b) Suppose that the game is modified as follows. Now player 1 choosesthe box in which to hide the money, and then the administrator of thegame places the boxes a large distance apart. (Suppose x is equal to thelength of a city block.) Player 2 is now required to begin a distance ofx to the left of box 1, and must walk to the box that he wishes to lookunder. Player 2 really does not like exercise and, given the warm weather,suffers disutility of $10 for each distance x that he walks. (So, for example,looking under box 1 costs player 2 $10, and looking under box 2 costs him$20, and so on.) What is the Nash equilibrium of this new game?

(c) Suppose that the game is as in part (b), but player 2 begins at box 2.Will player 2’s equilibrium strategy change? Why?

4. Ashley is negotiating an employment contract with a prospective employer, the LaJolla YMCA. The contract specifies two things: (1) Ashley’s job description (surfinginstructor or tennis instructor) and (2) Ashley’s salary t. Ashley is better at teachingsurfing and enjoys it more. If Ashley works as a surfing instructor for the YMCA,then Ashley’s payoff is t − 2000 and the YMCA’s payoff is 102, 000 − t. If Ashleyworks as a tennis instructor, then her payoff is t − 6000 and the YMCA’s payoff is10, 000−t. If Ashley and the YMCA fail to reach an agreement, then the YMCA getszero and Ashley obtains w. In other words, the default outcome of this negotiationproblem leads to the payoff vector (w, 0), where Ashley’s payoff is listed first. Thevalue w is due to Ashley’s outside opportunity, which is to work as a writer for theSan Diego Union Tribune.

Solve this bargaining problem using the Nash bargaining solution, under the as-sumption that Ashley’s bargaining power is πA and the YMCA’s bargaining power isπY . Describe the joint decision that is made (the job description and salary).

5. Suppose that the owner of a factory (F) has hired a contractor (C) to improve thefactory machinery. A properly performed job by the contractor allows the productionline to operate at a faster rate (with certainty) that is more efficient. For simplicity,assume that there are only two possible rates when the improvements have beenmade properly. These are the faster rate and the old rate at which the inefficientmachinery operated. Further, assume that if improperly improved, the machinery

2



can operate only at the slower rate. Denote the “high efficiency” state by H, andthe “low efficiency” state by L. Assume that the contractor possesses no evidence ineither state. The factory owner potentially possesses two pieces of evidence dL anddH , which are just the running of the machinery at the possible speeds. In state Hshe possesses both dL and dH , but in state L she only possesses dL. The state andexisting evidence are common knowledge between the players.

Suppose that improperly improving the machinery yields the contractor a costsavings of 4, but that having the more efficient machinery yields the factory a gainof 100. (Note that any initial payments between the parties have already been made,and are not modeled here.) Clearly, it is efficient for the improvements to be made.Following either production decision (by the contractor) the players have scope forsettlement. Assume their bargaining weights are 1/2, 1/2. If no settlement is reached,they go to court. When they go to court the factory owner can present evidence thatis in her possession. It is costly for the factory owner to present (or produce) evidence.The factory owner can present dL in either state at a cost of 16, while productionof dH in state H costs her 4. Producing no evidence (∅), which is possible in eitherstate, costs the factory zero. Costs are additive, so producing both dL and dH , whenavailable, costs the factory 20.

The court’s action (a transfer between the players) is based on the evidence pre-sented. It is common knowledge that the court’s mapping of evidence to transfers isas follows.

dL ⇒ 13 from C to FdH ⇒ 2 from C to F∅ ⇒ –4 from C to F (4 from F to C)dL and dH ⇒ 15 from C to F.

(a) Model this game by drawing the extensive form.

(b) What happens in each state if they go to court?

(c) What is the equilibrium of this game? Is this efficient?

(d) What does this imply about frivolous law suits by the factory? Doesthis change if the contractor bears a cost of 5 to go to court?

(e) If evidence production is costless and the same transfer schedule is inplace (and no cost of going to court for the contractor), what will happen?Is this efficient?

6. Consider the following interaction between a buyer and seller who have agreedto a contract. The seller chooses whether to perform (P) or to not perform (N).The buyer observes the seller’s production decision (P or N). Following the seller’sproduction decision, the buyer either has evidence available or does not. If the sellerperformed, the buyer has the evidence with probability .2. (With probability .8 the

3



buyer does not have the evidence, when the seller has performed.) If the seller did notperform, the buyer has the evidence with probability .5. Following the realization ofthe buyer’s evidence outcome, which is private information to the buyer, the playerssimultaneously and independently decide whether to go to court (C) or not to go tocourt (N), and the buyer also decides whether to present her evidence if it exists.Going to court costs each player $4. When available, the buyer’s evidence is costlessto disclose. If the buyer wins at court, the court requires that the seller pay $10 tothe buyer. If the buyer loses, the court imposes no transfer. The court outcome isdecided as follows. The buyer wins, regardless of whether she presents evidence, ifshe goes to court and the seller does not. The seller wins if she goes to court and thebuyer does not. If both go to court, the buyer must present the evidence in order towin. Naturally, when no one goes to court, the court takes no action.

Describe the equilibrium. Is this reasonable? Explain. What is the seller’s ex-pected payoff in the litigation game when she has performed and when she has not?If the seller’s immediate payoff in the productive interaction is $4 higher when shedoes not perform than when she performs, does the contract give her the incentiveto perform?

7. Consider the following infinitely-repeated game with discount factor δ. Eachperiod a principal and an agent contract for the agent to produce a good of quality q,which costs the agent c(q), where c′(q) > 0 and c′′(q) > 0. If the good of the agreedupon quality q is delivered, the principal is to pay s to the agent. The principal’spayoff from the transaction is q − s, and the agent’s is s − c(q). Both the principaland agent are risk neutral. The value of each player’s outside option is zero.

Suppose that if the contract is breached, the players go to court. With probabilityv ∈ (0, 1) the court observes both the level of q that is produced and whether s hasbeen paid. If the court observes these, it will enforce the contract. In each period,the players agree on q and s, then the principal must write a contract, and then theyplay the production/trade game. The principal incurs a cost k of writing the contracteach period. (It is incurred each period.) If it enforces the contract, the court imposesexpectations damages—it requires the breaching player to pay the other an amountthat gives the non-breaching player what she expected to receive under the contract.(If the agent does not produce the good of the agreed upon quality q, she must payan amount that gives the principal q− s. If the principal breaches, say by not payingor partially paying, she must pay an amount that gives s to the agent.)

Suppose that the players agree to the same value of q and s for each period. Underwhat conditions are these played in a subgame perfect equilibrium? (Use modifiedtrigger strategies.) Explain.

8. A buyer and seller can potentially trade a single good of quality i ∈ R+. Thegood’s quality is determined by the seller’s level of investment, which is made prior

4



to trade. An investment level of i by the seller costs the seller i. Let φ : R+ → R+

describe the buyer’s, and ψ : R+ → R+ the seller’s value from consuming a good ofquality i. If they do not trade, the seller can consume the good and receive a ψ(i).The function φ is strictly increasing and concave, whereas ψ is weakly increasing andconcave. Both functions φ and ψ are twice differentiable, where limi→0 φ

′(i) = ∞and limi→∞ φ′(i) = 0 to ensure interior solutions. Finally, φ(0) > ψ(0) = 0 andφ′(i) > ψ′(i) for all levels of investment. The trade surplus φ(i)−ψ(i) is, thus, alwaysstrictly positive and strictly increasing in the investment.

The buyer is motivated by his value of the good φ(i) less what he pays to theseller. The seller can be of two possible types, which the buyer does not observe.With probability q, the seller is “greedy,” which means that the seller bears a largecost K (assume K > γ[φ(i) − ψ(i)] for all i) if she sells the good to the buyer ata price that gives her less than γ ∈ (0, 1) of the trade surplus. That is, given aninvestment level i, the greedy seller will not agree to trade if she does not receivepayment from the buyer that is at least γ[φ(i) − ψ(i)] + ψ(i). With probability(1 − q), the seller is “accommodating,” which means she will agree to a price thatgives her any non-negative share of the trade surplus. That is, given an investmentlevel i, the accommodating seller will agree to trade as long as she receives a price atleast as large as ψ(i).

(a) Describe the efficient level of investment.

(b) Suppose that i is verifiable and the buyer can offer a contract to theseller, which she accepts or rejects, before the seller chooses i. When canthe efficient level of investment be implemented? What contract inducesthe efficient level of investment?

Suppose now that the buyer and seller cannot contract prior to the seller’s in-vestment decision. Instead, after the investment has been made, the buyer makes atake it-or-leave-it offer to the seller. The offer p is a fraction of the trade surplus,given i, that the seller is to receive. So, if the buyer’s offer is accepted, his pay-off is (1 − p)[φ(i) − ψ(i)]. If the greedy seller accepts an offer p ≥ γ, her payoff isp[φ(i) − ψ(i)] + ψ(i) − i. If the greedy seller accepts an offer p < γ, her payoff isp[φ(i)− ψ(i)] + ψ(i)− i−K < 0. The accommodating seller’s payoff from acceptingany offer p ≥ 0 is p[φ(i) − ψ(i)] + ψ(i) − i.

(c) Under what conditions can you find a perfect Bayesian equilibriumof this game that induces a level of investment i∗ such that the seller’spayoff from trade is greater than her payoff from consuming the goodherself? Even if you cannot describe the conditions, describe what suchan equilibrium would look like.

(d) Under what conditions can you find a perfect Bayesian equilibriumthat induces the efficient level of investment ie? Even if you cannot de-scribe the conditions, describe the intuition.

5



(e) Would it be possible to induce the efficient level of investment if,instead, we focused on sequential equilibria? Explain.

(f) Suppose that, prior to the seller’s investment decision, the buyer andseller can make a non-binding agreement. Does this help attain the effi-cient level of investment?

9. Suppose that two players who may choose to form a firm interact as follows. Firstthe players each choose whether to make an investment (which are made simultane-ously and independently). If player i invests, it costs her 3. If she does not invest, itcosts her nothing. The investment decisions become common knowledge. Then theplayers decide whether to form a firm and, if they decide to form a firm, how to dividethe profit from the firm. If both have invested, the firm’s profit will be 16. If oneor both has not invested, the firm’s profit will be 12. Assume that the player’s willdivide the surplus from forming a firm according to the standard bargaining solutionwith equal bargaining weights. Let di(0) denote the value of player i’s disagreementpayoff when she has not invested, and let di(3) denote the value of her disagreementpayoff when she has invested. These are symmetric.

(a) What is the efficient outcome?

(b) Describe conditions on di(0) and di(3) such that, in equilibrium, bothplayers invest. Show that this is an equilibrium.

(c) In light of your answers to parts (a) and (b), briefly provide someintuition for your answers in relation to the “hold-up” problem.

10. Two risk-neutral players bid for a single object in an auction. Each player caninvest so as to increase her valuation of the object. (The timing of this and the typeof auction is described below.) The players’ investment decisions are simultaneousand independent. Let xi denote player i’s level of investment, which costs her xi.

(a) Suppose player i’s valuation of the object is equal to x2i . The players

invest first, and their investment levels are then publicly announced beforethe good is auctioned in an English auction (ascending oral bids). Describean equilibrium of this game.

(b) Suppose player i’s valuation of the object is equal to x2i . The object is

first auctioned in a second-price, sealed-bid auction, and then the winnermay make her investment. Describe an equilibrium of this game.

(c) Compare the expected revenue under (a) and under (b).

(d) Instead assume that each player i’s valuation is given by x2i +wi, where

wi ∼ u[1, 2]. Players invest before a second-price, sealed-bid auction. Now

6



they do not observe each others’ investments or valuations (that is xi

and wi are private information). Describe an equilibrium. What is theexpected revenue for the seller?

11. Consider a Bertrand duopoly in which two firms simultaneously and indepen-dently select prices. The demand curve is given by Q = 100−p, where p is the lowestprice charged by the firms. Consumer demand is divided equally between the firmswho charge the lowest price. Suppose that each firm’s cost function is c(q) = 20q.

Suppose that this interaction is infinitely repeated and firms observe each other’schoice of price. Assume that the firms have equal discount factors given by δ. Describeall equilibrium prices that can occur in equilibrium (as a function of δ), and showthat these are, in fact, equilibrium behavior.

12. Consider the following social choice problem. There are two players (1 and 2)who know the state, which is either a or b. There is an uninformed social planner(or external enforcer). (You could think of this as a parent and two children or afirm and two workers, etc.) The social planner would like to impose public action xin state a, and impose public action y in state b (call this a social choice function).These are “real” or productive actions, and are the only actions the social plannercan take. (You could think of these decisions as being which child gets which toy, orwhich worker is assigned which task.)

The social planner does not know the state, but can base his decision (aboutwhether to impose x or y) on the players simultaneously and independently announc-ing the state to him. That is, each player can announce “a” (claiming the state is a)or “b” (claiming the state is b).

The task of the social planner is to design a game form that the players will playby making their announcements. We assume that he would like for both players totruthfully announce the state. The idea is that the game form will specify a publicaction (x or y) as a result of the announcements that players make. So considergame forms that have x imposed when both players have announced “a” and have yimposed when both players have announced “b.”

Players’ preferences over x and y are specified below. Your task is to try to designa game form (a mechanism) that induces, in equilibrium, players to truthfully namethe state and implements x in state a and y in state b.

(a) Suppose that player’s preferences are represented by the followingutility functions (which do not depend on the state).

u1(x) = 10

u1(y) = 5

u2(x) = 5

7



u2(y) = 10

Is there a mechanism (as described above) that implements the desiredsocial choice function?

(b) Now suppose that player’s preferences are represented by the followingutility functions (which do not depend on the state).

u1(x) = 5

u1(y) = 10

u2(x) = 5

u2(y) = 10

Is there a mechanism (as described above) that implements the desiredsocial choice function?

(c) Suppose players have the same preferences as in part (a). Now sup-pose that in state b player 2 possesses a document (evidence) that shedoes not possess in state a. (Player 1 possesses no documents in eitherstate.) Suppose that, in addition to making an announcement of the state,player 2 can also disclose her evidence (when she possesses it). Assumethat evidence disclosure and announcements must occur simulateouslyand independently as before. How does this change your answer to part(a)?

(d) Consider the scenario in part (c), but now assume that in state bplayer 1 possesses a document (evidence) that she does not possess instate a (and player 2 possesses no documents in either state). How doesthis change your answer to part (c)?

(e) Briefly discuss these in relation to “signaling” models.

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