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Game Theory for Applied

Economists

Robert Gibbons

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Princeton University Press Princeton, N ew Jersey

Copyright© 1992 by Princeton University Press Published by Princeton University Press, 41 Williarn Street, Princeton, New Jersey 08540

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Gibbons, R. 1958- Garne theory for applied econornists I Robert Gibbons. p. crn. Includes bibliographical references and index. ISBN 0-691-04308-6 (CL) ISBN ISBN 0-691-00395-5 (PB)

1. Garne theory. 2. Econornics, Mathernatical. 3. Economics-Mathernatical Models. I. Title. HB144.G49 1992 330'.01'5193-dc20 92- CIP

This book was cornposed with Ib-Tp< by Archetype Publishing Inc., P.O. Box 6567, Charnpaign, IL 61821.

Princeton University Press books are printed on acid-free paper and rneet the guidelines for perrnanence and durability of the Cornrnittee on Production Guide- lines for Book Longevity of the Council on Library Resources.

Printed in the United States

10 9 8 7 6 5 4 3

Outside of the United States and Canada, this book is available through Harvester Wheatsheaf under the title A Primer in Game Theory.

for Margaret

viii

CONTENTS

2.2.A Theory: Subgame Perfeetion.... 2.2.B Bank Runs .............. 2.2.C Tariffs and Imperfect International Competition 2.2.D Tournaments............. Repeated Games ............... 2.3.A Theory: Two-Stage Repeated Games 2.3.B Theory: Infinitely Repeated Games. 2.3.C Collusion between Cournot Duopolists 2.3.D Efficiency Wages .......... 2.3.E Time-Consistent Monetary Policy Dynamic Games of Complete but Imperfect Information ........... 2.4.A Extensive-Form Representation of Games 2.4.B Subgame-Perfeet Nash Equilibrium Further Reading Problems. References

. 112 . 115 . 115 . 122 . 129 . 130 . 138

3 Static Games of Incomplete Information 143 3.1 Theory: Static Bayesian Gamesand Bayesian Nash Equilibrium........................ 144 3.1.A An Example: Cournot Competition under Asymmetrie Information ........ 3.1.B Normal-Form Representation of Static Bayesian Games............. 3.1.C Definition of Bayesian Nash Equilibrium 3.2 Applications............ 3.2.A Mixed Strategies Revisited 3.2.B An Auetion..... 3.2.C A Double Auction. 3.3 The Revelation Principle 3.4 Further Reading 3.5 Problems. 3.6 References

4 Dynamic Games of Incomplete Information 173 4.1 Introduction to Perfeet Bayesian Equilibrium.. 175 4.2 Signaling Games.................. 183 4.2.A Perfeet Bayesian Equilibrium in Signaling Games...................... 183

Contents

4.2.B Job-Market Signaling............. 4.2.C Corporate Investment and Capital Structure 4.2.D Monetary Policy ......... 4.3 Other Applications of Perfeet Bayesian Equilibrium................ 4.3.A Cheap-Talk Games.......... 4.3.B Sequential Bargaining under Asymmetrie Information .............. 4.3.C Reputation in the Finitely Repeated Prisoners' Dilemma.......... 4.4 Refinements of Perfeet Bayesian Equilibrium. 4.5 Further Reading 4.6 Problems. 4.7 References

Index

ix

Preface

Game theory is the study of multiperson decision problems. Such problems arise frequently in economics. As is widely appreciated, for example, oligopolies present multiperson problems - each firm must consider what the others will do. But many other ap- plications of game theory arise in fields of economics other than industrial organization. At the micro level, models of trading processes (such as bargaining and auction models) involve game theory. At an intermediate level of aggregation, labor and finan- cial economics include game-theoretic models of the behavior of a firm in its input markets (rather than its output market, as in an oligopoly). There also are multiperson problems within a firm: many workers may vie for one promotion; several divisions may compete for the corporation' s investment capital. Finally, at a high level of aggregation, international economics includes models in which countries compete (or collude) in choosing tariffs and other trade policies, and macroeconomics includes models in which the monetary authority and wage or price setters interact strategically to determine the effects of monetary policy. This book is designed to introduce game theory to those who will later construct (or at least consume) game-theoretic models in applied fields within economics. The exposition emphasizes the economic applications of the theory at least as much as the pure theory itself, for three reasons. First, the applications help teach the theory; formal arguments about abstract games also ap- pear but play a lesser role. Second, the applications illustrate the process of model building- the process of translating an infor- mal description of a multiperson decision situation into a formal, game-theoretic problern to be analyzed. Third, the variety of ap- plications shows that similar issues arise in different areas of eco- nomics, and that the same game-theoretic tools can be applied in

Game Theory for Applied Economists

Chapter 1

Static Games of Complete

Information

In this chapter we consider games of the following simple form: first the players simultaneously choose actions; then the players receive payoffs that depend on the combination of actions just cho- sen. Within the dass of such static (or simultaneous-move) games,

we restriet attention to games of complete information. That is, each

player's payoff function (the function that determines the player's payoff from the combination of actions chosen by the players) is common knowledge among all the players. We consider dynamic (or sequential-move) games in Chapters 2 and 4, and games of incomplete information (games in which some player is uncertain about another player's payoff function-as in an auction where each bidder's willingness to pay for the good being sold is un- known to the other bidders) in Chapters 3 and 4. In Section 1.1 we take a first pass at the two basic issues in game theory: how to describe a game and how to solve the re- sulting game-theoretic problem. We develop the tools we will use in analyzing static games of complete information, and also the foundations of the theory we will use to analyze richer games in

later chapters. We define the normal-form representation of a game

and the notion of a strictly dominated strategy. We show that some

games can be solved by applying the idea that rational players do not play strictly dominated strategies, but also that in other games this approach produces a very imprecise prediction about the play of the game (sometimes as imprecise as "anything could

4 STATIC GAMES OF COMPLETE INFORMATION

strategy s; is a member of the set of strategies 5;.) Let (st, ... , sn)

denote a combination of strategies, one for each player, and let

u; denote player i's payoff function: u;(s1, ... , Sn) is the payoff_ to

player i if the players choose the strategies (s1, ... , Sn)· Collectmg

all of this information together, we have:

Definition The normal-form representation of an n-player game spec-

ifies the players' strategy spaces 5 1 , ... , Sn and their payoff functions

u1, ... , Un. We denote this game by G = {S1, ... , Sn; Ut, ... , Un}·

Although we stated that in a normal-form game the players choose their strategies simultaneously, this does not imply that the parties necessarily act simultaneously: it suffices that each choose his or her action without knowledge of the others' choices, as would be the case here if the prisoners reached decisions at ar- bitrary times while in their separate cells. Furthermore, althoug_h in this chapter we use normal-form games to represent only stahc games in which the players all move without knowing the other players' choices, we will see in Chapter 2 that normal-form repre- sentations can be given for sequential-move games, but also that

an alternative-the extensive-form representation of the game-is

often a more convenient framework for analyzing dynamic issues.

l.l.B Iterated Elimination of Strictly Dominated

Strategies

Having described one way to represent a game, we ~ow take a first pass at describing how to solve a game-theorehc problem. We start with the Prisoners' Dilemma because it is easy to solve, using only the idea that a rational player will not play a strictly dominated strategy. In the Prisoners' Dilemma, if one suspect is going to play Fink, then the other would prefer to play Fink and so be in jail for six months rather than play Mum and so be in jail for nine months. Similarly, if one suspect is going to play Mum, then the other would prefer to play Fink and so be released immediately ~ather than play Mum and so be in jail for one month. Thus, for pnsoner i, playing Mum is dominated by playing Fi~k-for_ each strat~gy

that prisoner j could choose, the payoff to pnsoner 1 from playmg

Mum is less than the payoff to i from playing Fink. (The same

would be true in any bi-matrix in which the payoffs 0, -1, -6,

Basic Theory 5

and -9 above were replaced with payoffs T, R, P, and 5, respec-

tively, provided that T > R > P > 5 so as to capture the ideas

of temptation, reward, punishment, and sucker payoffs.) More generally:

Definition In the normal-form game G = {S1, ... , Sn; u1, ... , Un}, Iet

sj and sj' be feasible strategies for player i (i.e., sj and sj' are members of

5;). Strategy sj is strictly dominated by strategy sj' if for each feasible

combination of the other players' strategies, i's payoff from playing sj is

strictly less than i's payoff from playing sj':

for each (s1, ... , s;_ t, si+ll ... , Sn) that can be constructed from the other

players' strategy spaces S1, ... , S;-1, Si+ll ... , Sn.

Rational players do not play strictly dominated strategies, be- cause there is no belief that a player could hold (about the strate- gies the other players will choose) such that it would be optimal to play such a strategy.^1 Thus, in the Prisoners' Dilemma, a ratio- nal player will choose Fink, so (Fink, Fink) will be the outcome reached by two rational players, even though (Fink, Fink) results in worse payoffs for both players than would (Mum, Mum). Be- cause the Prisoners' Dilemma has many applications (including the arms race and the free-rider problern in the provision of pub- lic goods), we will return to variants of the game in Chapters 2 and 4. For now, we focus instead on whether the idea that rational players do not play strictly dominated strategies can lead to the solution of other games. Consider the abstract game in Figure 1.1.1.2 Player 1 has two strategies and player 2 has three: S1 = {Up, Down} and 5 2 = {Left, Middle, Right }. For player 1, neither Up nor Down is strictly (^1) A complementary question is also of interest: if there is no belief that player i could hold (about the strategies the other players will choose) such that it would be optimal to play the strategy Sj, can we conclude that there must be another strategy that strictly dominates Si? The answer is "yes," provided that we adopt appropriate definitions of "belief" and "another strategy," both of which involve the idea of mixed strategies to be introduced in Section 1.3.A. (^2) Most of this book considers economic applications rather than abstract exam- ples, both because the applications are of interest in their own right and because, for many readers, the applications are often a useful way to explain the under- lying theory. When introducing some of the basic theoretical ideas, however, we will sometimes resort to abstract examples that have no natural economic interpretation.

6 STATIC^ GAMES^ OF^ COMPLETE^ INFORMATION

Player 1

Up Down

Left 1, 0,

Player 2 Middle 1, 2 0,

Figure 1.1.1.

Right 0, 1 2,

dominated: Up is better than Down if 2 plays Left (because 1 > 0),

but Down is better than Up if 2 plays Right (because 2 > 0). For

player 2, however, Right is strictly dominated by Middle (because

2 > 1 and 1 > 0), so a rational player 2 will not play Right.

Thus, if player 1 knows that player 2 is rational then player 1 can eliminate Right from player 2's strategy space. That is, if player 1 knows that player 2 is rational then player 1 can play the game

in Figure 1.1.1 as if it were the game in Figure 1.1.2.

Player 2 Left Middle

Player 1

Up 1,0^ 1, Down 0,3^ 0,

Figure 1.1.2.

In Figure 1.1.2, Down is now strictly dominated by Up for player 1, so if player 1 is rational (and player 1 knows that player 2 is rational, so that the game in Figure 1.1.2 applies) then player 1 will not play Down. Thus, if player 2 knows that player 1 is ra-

tional, and player 2 knows that player 1 knows that player 2 is

rational (so that player 2 knows that Figure 1.1.2 applies), then player 2 can eliminate Down from player 1's strategy space, leav- ing the game in Figure 1.1.3. But now Left is strictly dominated by Middle for player 2, leaving (Up, Middle) as the outcome of the game.

This process is called iterated elimination of strictly dominated

strategies. Although it is based on the appealing idea that ratio-

nal players do not play strictly dominated strategies, the process has two drawbacks. First, each step requires a further assumption

l

Basic Theory 7

Player 2 Left Middle Player (^1) Up I 1,0 1, 2

Figure 1.1.3.

about what the players know about each other's rationality. If we want to be able to apply the process for an arbitrary number

of steps, we need to assume that it is common knowledge that the

players are rational. That is, we need to assume not only that all the players are rational, but also that all the players know that all the players are rational, and that all the players know that all the

players know that all the players are rational, and so on, ad in-

finitum. (See Aumann [1976] for the formal definition of common

knowledge.) The second drawback of iterated elimination of strictly domi- nated strategies is that the process often produces a very impre- cise prediction about the play of the game. Consider the game in Figure 1.1.4, for example. In this game there are no strictly dom- inated strategies tobe eliminated. (Smce we have not motivated this game in the slightest, it may appear arbitrary, or even patho- logical. See the case of three or more firms in the Cournot model in Section 1.2.A for an economic application in the same spirit.) Since all the strategies in the game survive iterated elimination of strictly dominated strategies, the process produces no prediction whatsoever about the play of the game.

T

M

B

L c 0,4 4, 4,0 0, 3,5 3,

Figure 1.1.4.

R

We turn next to Nash equilibrium-a solution concept that produces much tighter predictions in a very broad dass of games. We show that Nash equilibrium is a stronger solution concept

10 STATIC^ GAMES^ OF^ COMPLETE^ INFORMATION

T

M

B

L c

0.1: 1:, 1:,0 0,1: 3,5 3,

Figure 1.1.5.

R

Q,Q

Figure 1.1.1. These strategy pairs are the unique Nash equilibria of these games. 4 We next address the relation between Nash equilibrium and iterated elimination of strictly dominated strategies. Recall that the Nash equilibrium strategies in the Prisoners' Dilemma and Figure 1.1.1-(Fink, Fink) and (Up, Middle), respectively-are the only strategies that survive iterated elimination of strictly domi- nated strategies. This result can be generalized: if iterated elimina- tion of strictly dominated strategies eliminates all but the strategies ~1 ,... ,s~ ), then these strategies are the unique Nash equilibrium of the game. (See Appendix l.l.C for a proof of this claim.) Since it- erated elimination of strictly dominated strategies frequently does not eliminate all but a single combination of strategies, however, it is of more interest that Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies,

in the following sense. If the strategies (s]' ,... , s~) are a Nash equi-

librium then they survive iterated elimination of strictly domi- nated strategies (again, see the Appendix for a proof), but there can be strategies that survive iterated elimination of strictly dom- inated strategies but arenot part of any Nash equilibrium. To see the latter, recall that in Figure 1.1.4 Nash equilibrium gives the unique prediction (B, R), whereas iterated elimination of strictly dominated strategies gives the maximally imprecise prediction: no strategies are eliminated; anything could happen. Having shown that Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies, we must now ask whether Nash equilibrium is too strong a So- lution concept. That is, can we be sure that a Nash equilibrium

(^4) This statement is correct even if we do not restriet attention to pure-strategy Nash equilibrium, because no mixed-strategy Nash equilibria exist in these three games. See Problem 1.10.

Basic Theory 11

exists? Nash (1950) showed that in any finite game (i.e., agame in which the number of players n and the strategy sets S 1 , ... , S 11 are all finite) there exists at least one Nash equilibrium. (This equi- librium may involve mixed strategies, which we will discuss in Section 1.3.A; see Section 1.3.B for a precise statement of Nash's Theorem.) Cournot (1838) proposed the same notion of equilib- rium in the context of a particular model of duopoly and demon- strated (by construction) that an equilibrium exists in that model; see Section 1.2.A. In every application analyzed in this book, we will follow Cournot's lead: we will demonstrate that a Nash (or stronger) equilibrium exists by constructing one. In some of the theoretical sections, however, we will rely on Nash's Theorem (or its analog for stronger equilibrium concepts) and simply assert that an equilibrium exists.

We conclude this section with another dassie example- The

Battle of the Sexes. This example shows that a game can have mul-

tiple Nash equilibria, and also will be useful in the discussions of mixed strategies in Sections 1.3.B and 3.2.A. In the traditional ex- position of the game (which, it will be clear, dates from the 1950s), a man and a wo man are trying to decide on an evening' s enter- tainment; we analyze a gender-neutral version of the game. While at separate workplaces, Pat and Chris must choose to attend either the opera or a prize fight. Both players would rather spend the evening tagether than apart, but Pat would rather they be tagether at the prize fight while Chris would rather they be tagether at the opera, as represented in the accompanying bi-matrix.

Pat Opera Fight

Chris

Opera 2,1 0, Fight (^) 0,0 1,

The Battle of the Sexes

Both (Opera, Opera) and (Fight, Fight) are Nash equilibria. We argued above that if game theory is to provide a unique solution to a game then the solution must be a Nash equilibrium. This argument ignores the possibility of games in which game theory does not provide a unique solution. We also argued that

12 STATIC^ GAMES^ OF^ COMPLETE^ INFORMATION

if a convention is to develop about how to play a given game, then the strategies prescribed by the convention must be a Nash equilibrium, but this argument similarly ignores the possibility of games for which a convention will not develop. In some games with multiple Nash equilibria one equilibrium stands out as the compelling solution to the game. (Much of the theory in later chapters is an effort to identify such a compelling equilibrium in different classes of games.) Thus, the existence of multiple Nash equilibria is not a problern in and of itself. In the Battle of the Sexes, however, (Opera, Opera) and (Fight, Fight) seem equally compelling, which suggests that there may be games for which game theory does not provide a unique solution and no convention will develop. 5 In such games, Nash equilibrium loses much of its appeal as a prediction of play.

Appendix l.l.C

This appendix contains proofs of the following two Propositions, which were stated informally in Section 1.1.C. Skipping these proofs will not substantially hamper one's understanding of later material. For readers not accustomed to manipulating formal def- initions and constructing proofs, however, mastering these proofs will be a valuable exercise.

Proposition A In the n-player normal-form game G = { S1, ... , Sn;

u 1 , ..• , u 11 }, if iterated elimination of strictly dominated strategies elimi-

nates all but the strategies (si, ... , s~), then these strategies are the unique

Nash equilibrium of the game.

Proposition B In the n-player normal-form game G = { S1, ... , Sn;

u 1 , ... , u 11 }, if the strategies (si, ... , s~) are a Nash equilibrium, then they

survive iterated elimination of strictly dominated strategies.

(^5) In Section 1.3.8 we describe a third Nash equilibrium of the Battle of the Sexes (involving mixed strategies). Unlike (Opera, Opera) and (Fight, Fight), this third equilibrium has symmetric payoffs, as one might expect from the unique solution to a symmetric game; on the other hand, the third equilibrium is also inefficient, which may work against its development as a convention. Whatever one's judgment about the Nash equilibria in the Battle of the Sexes, however, the broader point remains: there may be games in which game theory does not provide a unique solution and no convention will develop.

Basic Theory 13

Since Proposition B is simpler to prove, we begin with it, to warm up. The argument is by contradiction. That is, we will as- sume that one of the strategies in a Nash equilibrium is eliminated by iterated elimination of strictly dominated strategies, and then we will show that a contradiction would result if this assumption were true, thereby proving that the assumption must be false. Suppose that the strategies (sj, ... , s~) are a Nash equilibrium

of the normal-form game G = {St, ... , Sn; Ut, ... , Un}, but suppose

also that (perhaps after some strategies other than (sj, ... , s~) have

been eliminated) sj is the first of the strategies (sj, ... , s~) to be

eliminated for being strictly dominated. Then there mustexist a strategy s;' that has not yet been eliminated from S; that strictly

dominates sj. Adapting (OS), we have

u;(s1, ... ,s;-t,si,s;+t, ... ,sn)

< u;(s1, ... ,s;-t,s;',s;+t, ... ,sn) (1.1.1)

for each (s1, ... , s;-1, Si+1, ... , sn) that can be constructed from the strategies that have not yet been eliminated from the other players'

strategy spaces. Since sj is the first of the equilibrium strategies to

be eliminated, the other players' equilibrium strategies have not yet been eliminated, so one of the implications of (1.1.1) is

But (1.1.2) is contradicted by (NE): sj must be a best response to

(sj, ... , sj_ 1 , sj+ 1 , .•• , s~), so there cannot exist a strategy s;' that

strictly dominates sj. This contradiction completes the proof.

Having proved Proposition B, we have already proved part of Proposition A: all we need to show is that if iterated elimination

of dominated strategies eliminates all but the strategies (sj, ... , s~)

then these strategies are a Nash equilibrium; by Proposition B, any other Nash equilibria would also have survived, so this equilib- rium must be unique. We assume that G is finite. The argument is again by contradiction. Suppose that iterated elimination of dominated strategies eliminates all but the strategies (sj, ... , s~) but these strategies are not a Nash equilibrium. Then there must exist some player i and some feasible strategy s; in S; such that (NE) fails, but s; must have been strictly dominated by some other strategy s;' at some stage of the process. The formal

16 STATIC GAMES OF COMPLETE INFORMATION

solve for equilibrium. We assume that the firm's payoff is simply its profit. Thus, the payoff u;(s;,sj) in a general two-player game in normal form can be written here as 7

n;(q;, qj) = q;[P(q; + qj)- c] = q;[a- (q; + qj)- c].

Recall from the previous section that in a two-player game in nor-

mal form, the strategy pair (sj, si) is a Nash equilibrium if, for

each player i, u;(si, sj) 2: u;(s;, sj) (NE)

for every feasible strategy s; in S;. Equivalently, for each player i, si must solve the optimization problern

max u;(s;,sj). S;ES;

In the Cournot duopoly model, the analogaus statement is that

the quantity pair (qj, qi) is a Nash equilibrium if, for each firm i,

qi solves

max n;(q;, qj) = max q;[a- (q; + qj)- c]. o::;q;<oo o::;q;<oo

Assuming qj < a- c (as will be shown tobe true), the first-order

condition for firm i's optimization problern is both necessary and sufficient; it yields q; = ~(a- qj- c). (1.2.1)

Thus, if the quantity pair (qi, qi) is tobe a Nash equilibrium, the firms' quantity choices must satisfy

qt^ *^ = 2.1 ( a- q2- *^ c)

and

q2^ ^ = 1 (- a^ - ql - c .)

2

(^7) Note that we have changed the notation slightly by writing u;(s;,s;) rather than u;(s 1 , s 2 ). Both expressions represent the payoff to player i as a function of the strategies chosen by all the players. We will use these expressions (and their n-player analogs) interchangeably.

;

Applications 17

Solving this pair of equations yields

* * a- c

ql = q2 = -3-,

which is indeed less than a - c, as assumed.

The intuition behind this equilibrium is simple. Each firm would of course like to be a monopalist in this market, in which

case it would choose q; to maximize 1r;(q;, 0)-it would produce

the monopoly quantity qm = (a - c)/2 and earn the monopoly

profit 7r;(qm, 0) = (a- c) 2 /4. Given that there are two firms, aggre-

gate profits for the duopoly would be maximized by setting the aggregate quantity q1 + q2 equal to the monopoly quantity qm, as

would occur if q; = qm/2 for each i, for example. The problern

with this arrangement is that each firm has an incentive to devi- ate: because the monopoly quantity is low, the associated price

P(qm) is high, and at this price each firm would like to increase its

quantity, in spite of the fact that such an increase in production drives down the market-clearing price. (To see this formally, use

(1.2.1) to checkthat qm/2 is not firm 2's best response to the choice

of qm/2 by firm 1.) In the Cournot equilibrium, in contrast, the ag-

gregate quantity is higher, so the associated price is lower, so the temptation to increase output is reduced-reduced by just enough that each firm is just deterred from increasing its output by the realization that the market-clearing price will fall. See Problem 1.

for an analysis of how the presence of n oligopolists affects this

equilibrium trade-off between the temptation to increase output and the reluctance to reduce the market-clearing price. Rather than solving for the Nash equilibrium in the Cournot game algebraically, one could instead proceed graphically, as fol- lows. Equation (1.2.1) gives firm i's best response to firm j's

equilibrium strategy, qj. Analogaus reasoning leads to firm 2's

best response to an arbitrary strategy by firm 1 and firm 1's best

response to an arbitrary strategy by firm 2. Assuming that firm l's

strategy satisfies q 1 < a- c, firm 2's best response is

likewise, if q2 < a- c then firm 1's best response is

Rt(q2) =

2

(a- q2- c).

.,.•~,..,..,_,., ,.,.. .... --·-.., ... ,.,~··· -:>. '"" 't ; · H :; 'l idT:·· ~.- r• ·, •. ·

18 STATIC^ GAMES^ OF^ COMPLETE^ INFORMATION

(0, a- c)

(O,(a-c)/2)

((a-c)/2,0) (a-c,O) ql

Figure 1.2.1.

As shown in Figure 1.2.1, these two best-response functions inter- sect only once, at the equilibrium quantity p~i~ (~j, qi~· A third way to solve for this Nash eqmhbr.mm 1s to apply the process of iterated elimination ~f strictly.dommated str~t.egies. This process yields a unique soluhon-wh1ch, by Proposition A

in Appendix l.l.C, must be the Nash equilibrium (qj, qi). The

complete process requires an infinite nu~ber of s.teps, ~ach of which eliminates a fraction of the quanhhes remammg m each firm' s strategy space; we discuss only the first two steps. Firs~, the

monopoly quantity qm = (a- c)/2 strictly dominates any h1gher

quantity. That is, for any x > 0, 7r;(qm,qj) > 7r;(qm +x,qj) for all

qj ?: 0. To see this, note that if Q = qm + x + qj < a, then

a- c[a- c ]

7r;(qm, qj) = -2- --2- - qj

and

7r;(qm+x,qj)= [a;c +x] [a;c -x-qj] =7r;(qm,qj)-x(x+qj),

and if Q = qm + x + qj ?: a, then P(Q) = 0, so producing a smaller

Applications 19

quantity raises profit. Second, given that quantities exceeding qm

have been eliminated, the quantity (a- c)/4 strictly dominates

any lower quantity. That is, for any x between zero and (a- c)/4,

1r;[(a- c)j4,qj] > 1r;[(a- c)/4- x,qj] for all qj between zero and

(a- c)/2. To see this, note that

7r,. (a- 4^ c^ )%·) _- ~ 4 [3(a - 4 c)^ _^ q, ·]

and

7r; a- -4--^ c^ x,qj )

[

a- -4- c - X J [3(a- 4 c) (^) + X - qj J

After these two steps, the quantities remaining in each firm's strategy space are those in the interval between (a - c)/4 and

(a- c)/2. Repeating these arguments leads to ever-smaller inter-

vals of remaining quantities. In the limit, these intervals converge

to the single point qj = (a- c)/3.

lterated elimination of strictly dominated strategies can also be described graphically, by using the observation (from footnote 1; see also the discussion in Section 1.3.A) that a strategy is strictly dominated if and only if there is no belief about the other players' choices for which the strategy is a best response. Since there are only two firms in this model, we can restate this observation as:

a quantity q; is strictly dominated if and only if there is no belief

about qj suchthat q; is firm i's best response. We again discuss only the first two steps of the iterative process. First, it is never a best response for firm i to produce more than the monopoly quantity, qm = (a-c)/2. To see this, consider firm 2's best-response function, for example: in Figure 1.2.1, R2(q1) equals q 111 when q1 = 0 and

declines as q1 increases. Thus, for any qj ?: 0, if firm i believes

that firm j will choose qj, then firm i's best response is less than or

equal to q 111 ; there is no qj such that firm i's best response exceeds q 111 • Second, given this upper bound on firm j's quantity, we can derive a lower bound on firm i's best response: if qj :::; (a - c)/2,

then R;(qj) ?: (a - c)/4, as shown for firm 2's best response in

Figure 1.2.2. 8 (^8) These two argurnents are slightly incornplete because we have not analyzed

22 STATIC GAMES OF COMPLETE INFORMATION

are again two players. This time, however, the strategies available

to each firm are the different prices it might charge, rather than

the different quantities it might produce. We will assume that negative prices are not feasible but that any nonnegative price can be charged-there is no restriction to prices denominated in pen- nies, for instance. Thus, each firm' s strategy space can again be represented as Si = [0, oo ), the nonnegative real numbers, and a

typical strategy Si is now a price choice, Pi 2: 0.

We will again assume that the payoff function for each firm is

just its profit. The profit to firm i when it chooses the price Pi and

its rival chooses the price Pj is

Thus, the price pair (pj, Pi) is a Nash equilibrium if, foreachfirm i,

pj solves

max 1fi(Pi, pj) = max [a- Pi+ bpj][pi- c].

o:::;p;<= o:::;p;<=

The solution to firm i's optimization problern is

Therefore, if the price pair (pj, pi) is tobe a Nash equilibrium, the

firms' price choices must satisfy

Pi = ~ ( a + bpi + c)

and

Pi = ~ ( a + bp} + c).

Solving this pair of equations yields

• • a + c P1 = P2 = 2 _ b.

1.2.C Final-Offer Arbitration

Many public-sector workers are forbidden to strike; instead, wage disputes are settled by binding arbitration. (Major league base-

. ball may be a higher-profile example than the public sector but is substantially less important economically.) Many other disputes, including medical malpractice cases and claims by shareholders against their stockbrokers, also involve arbitration. The two ma- jor forms of arbitration are conventional and final-offer arbitration. In final-offer arbitration, the two sides make wage offers and then the arbitrator picks one of the offers as the settlement. In con- ventional arbitration, in contrast, the arbitrator is free to impose any wage as the settlement. We now derive the N ash equilib- rium wage offers in a model of final-offer arbitration developed by Farber (1980).^9 Suppose the parties to the dispute are a firm and a union and the dispute concerns wages. Let the timing of the game be as follows. First, the firm and the union simultaneously make offers, denoted by wr and Wu, respectively. Second, the arbitrator chooses one of the two offers as the settlement. (As in many so-called static games, this is really a dynamic game of the kindtobe discussed in Chapter 2, but here we reduce it to a static game between the firm and the union by making assumptions about the arbitrator's behavior in the second stage.) Assurne that the arbitrator has an ideal settlement she would like to impose, denoted by x. Assurne further that, after observing the parties' offers, Wf and Wu, the arbitrator simply chooses the offer that is closer to x: provided

that wf < Wu (as is intuitive, and will be shown to be true), the

arbitrator chooses wf if x < (wf + wu)/2 and chooses Wu if x > (wf + wu)/2; see Figure 1.2.3. (lt willbeimmaterial what happens if x = (wf + wu)/2. Suppose the arbitrator flips a coin.) The arbitrator knows x but the parties do not. The parties

believe that x is randomly distributed according to a cumulative

probability distribution denoted by F(x), with associated prob-

ability density function denoted by f(x).1° Given our specifi-

cation of the arbitrator's behavior, if the offers are wf and Wu

(^9) This application involves some basic concepts in probability: a cumulative probability distribution, a probability density function, and an expected value. Terse definitions are given as needed; for more detail, consult any introductory probability text. (^10) That is, the probability that x is less than an arbitrary value x* is denoted F(x*), and the derivative of this probability with respect to, x* is denoted f(x*). Since F(x*) is a probability, we have 0 :::; F(x*) :::; 1 for any x*. Furthermore, if x** > x* then F(x**) 2': F(x*), so f(x*) 2': 0 for every x*.

24 STATIC GAMES OF COMPLETE INFORMATION

w 1 chosen wu chosen

X

Figure 1.2.3.

then the parties believe that the probabilities Prob{ wf chosen} and Prob{ Wu chosen} can be expressed as

Prob{ w Wf+Wu}^ (wt+Wu)

1 chosen}^ =^ Prob^ x^ <^2 =^ F^ 2

and

Prob{wu chosen} = 1- F Wf^ +wu) 2

Thus, the expected wage settlement is

wf · Prob{ Wf chosen} + Wu · Prob{ Wu chosen}

= Wf. F ( Wf ; Wu) + Wu. [ (^) 1 _ F ( Wf ; Wu)].

We assume that the firm wants to minimize the expected wage settlement imposed by the arbitrator and the union wants to max- imize it.

Applications 25

If the pair of offers (wj, w~) is tobe a Nash equilibrium of the game between the firm and the union, wj must solve 11

~ Wf. F w^ f +w*) 2 u + w~.

and w~ must solve

mufux wj · F wj^ +^ Wu)^ [^ (wj^ +^ Wu)] 2

• Wu · 1 - F 2

Thus, the wage-offer pair (wj, w~) must solve the first-order con- ditions for these optimization problems,

( w*^ -^ w*)^. -^1 f^ (^ w*f^ +^ w*)u^ =^ F^ (^ w*f^ +^ w*u^ )

u f 2 2 2

and

( w~ - wj). 21 f (^ w*f + 2 w*u^ )^ = [^ 1 - F (^ w*f + 2 w*u^ ) ].

(We defer considering whether these first-order conditions are suf- ficient.) Since the left-hand sides of these first-order conditions are equal, the right-hand sides must also be equal, which implies that

wj + w~) = :!_. F 2 2' (1.2.2)

that is, the average of the offers must equal the median of the arbitrator's preferred settlement. Substituting (1.2.2) into either of the first-order conditions then yields

1 w- w = · u f (w+w)'

f ~

that is, the gap between the offers must equal the reciprocal of the value of the density function at the median of the arbitrator's preferred Settlement. (^11) In formulating the firm's and the union's optimization problems, we have assumed that the firm's offer is less than the union's offer. It is Straightforward to show that this inequality must hold in equilibrium.