Cheap talk, neologisms, and bargaining

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CHEAP

TALK,

NEOLOGISMS,

AND BARGAINING

Joseph Farrell

Robert Gibbons

No. 500

July

1988

massachusetts

institute

of

technology

50 memorial

drive

Cambridge, mass.

02139

CHEAP

TALK,

NEOLOGISMS,

AND

BARGAINING

Joseph Farrell

Robert Gibbons

«u?

_^^^

Cheap

Talk,

Neologisms,

and

Bargaining*

by

Joseph Farrell and

Roben

Gibbons

Berkeley

MIT

and

NBER

July 1988

Abstract

Thispaper

shows

that in

some

bargaininggames, cheap talkmust matter: the unique

neologism-proofequilibrium involvesinformative talk.

A

simpletradeoff

makes

both equilibriumtalkand neologisms credible: by saying thathe is interested intrading, the

buyer encourages the seller tocontinue tobargain butreceives poorer termsoftradeiftrade occurs. Alltj^pes of bothparties are (weakly) betteroffinthe neologism-proofequilibrium than in theequilibrium withouttalk. Thus,cheap talk beforebargainingwill happen andis

socially desirable.

* Financial support through

NSF

grants IRI-8712238 (Farrell) and

xxx-88xxxxx

1

1. Introduction

Many

game-theoreticmodelsofbargainingunderasymmetric information arebased

on

Spence's (1973) signalingmodel: a bargainerincurs costs (t)q)ically through delay) to

signal thathe is not desperate totrade and so shouldreceive favorable terms oftrade iftrade occurs.! Outside bargainingtheory, Crawford andSobel (1982) have

shown

that

communicationis possible withoutcostlysignals: costlessmessages (cheap talk) can be

credible iftliere is

some

common

interestbetween theparties.

Farrcll and Gibbons (fonhcoming) and

Matthews

and Postlewaite (forthcoming)

show

thatcheap talkcan matterina specificbargaininggame, a doubleauction. Farrell and Gibbons

show

thateach bargainerfacesa tradeoffthatcan

make

his talk credible: by

expressing an interest in trading,onebargainerencourages theother tocontinueto bargain

but receivespoorer terms oftradeiftradeoccurs; inequilibrium, only buyers (sellers) with high (low) reservation prices are willing toexpress such an interest in trading. Mattheu's

and Postlewaite

show

that talk can coordinate the bargainerson differentdouble-auction

equilibria (eachofwhich

would

have beenan equilibrium without talk)depending on the bargainers'

announced

reser\'ationprices, and that such pre-play

communication

dramatically enlargestheset of equilibriumoutcomesofa doubleauction. Both papers

show

that talkcanmatterbyconstructingequilibria thatcould notbe equilibria withouttalk.

In thispaper

we

show

that in

some

bargaining

games

cheaptalk notonlycan but

mustmatter,in the sensethat theuniqueneologism-proof equilibrium (see Farrell

(forthcoming)) couldnotbe an equilibrium withouttalk. In thisequilibrium,cheap talkis

credible forthe

same

reason

we

identifiedinourearlierpaper: a bargainercan use talkto

encouragehis opponenttocontinue tobargain, but willreceivepoorerterms oftrade if

trade occurs. This tradeoff leads to

two

results inthe

game

we

analyze: (1) buyers with

highreservation prices prefertoreveal themselves throughtalkrather than to

masquerade

^Thelargeliteratureonthissubject beginswithFudenberg andTirole(1983)and Sobcl andTakaliashi

2

asbuyers wdth

low

reservationprices, and (2) buyers with high reservation pricespreferto

reveal themselvesratherthan topool.

The

first oftheseresults impliesthatan informative cheap-talk equilibrium exists, andthesecondestablishesthatequilibriain

which

talkis

absent or uninformativearenot neologism-proof.

We

alsoconsiderthe welfare implicationsofcheap talkbefore bargaining. In the

game we

analyze, alltypesof both partiesare(weakly) better offin theneologism-proof

equilibrium thanin theequilibriumwithouttalk. Thus,cheaptalk before bargaining not onlywill happen butis socially desirable. This contrastswith ourfindings fora double

auction that

some

types of bothparties

do

better inan equilibrium with informative talk, but

some

types

do

worse, andtheex-ante expectedgains fromtrade arelowerwith talk than without.

The

body

ofthepaperisorganized as follows. In Section2

we

developa variation

ondiefamiliartake-it-or-leave-itbargaining model.

Our model

may

be of independent

interest asananalysis oftradeinidiosyncratic goods, but

we

havedeveloped itmainly to

provide an unclutteredsetting forouranalysisofcheaptalk. In Section 3

we

characterize the neologism-proofequilibriainourbargaininggame. Section 4concludes.

2.

The

Model

We

analyze averysimple bargaining

game

in which a seller, atcostc, can

make

a

singletake-it-or-leave-it offerto abuyer.

The

seller's (privately

known)

valuation for the

good

inquestion isdenoted

by

s, the buyer's byb. Iftradeoccursat pricep then the

seller's payoffis _p - s - cand the buyer's is b- _{p. Ifthe}seller

makes

an offer thatthe

buyerrejects then theseller'spayoffis -cand the buyer's iszero. Finally, ifthe seller

makes

no

offerthen each player's payoffis zero.

Both

parties are

risk-neutral.-Perry (1986)showsthattheuniqueperfectBayesianequilibrium ofthisgameis identicaltotheunique

3

The

seller's willingnessto

make

an offerand theprice she

demands

ifshe does so

depend on hervaluationand on herbeliefabout the buyer's valuation.

We

assume

thatitis

common

knowledge

thats andbareindependently

drawn

froma uniform distributionon

[0,1].

We

ask whethercheap talkby the buyer caninfluencetlie seller's beliefaboutb, and

so influence both theseller's decisionto

make

anoffer andthepriceshe

demands

ifshe

makes

an offer.

Our

previous paper

shows

that, in a double auction, the buyerfaces a simple and

intuitive tradeoffthatcan

make

cheap talkmatterin adoubleauction.

To

studythis tradeoff

in ourtake-it-or-leave-itbargaining game, let u([x,y],b)be the expected payofftoabuyer

ofvaluation b

when

the sellerbelievesthat the buyer's valuation is uniformly distributedon

[x,y], an intervalcontained in [0,1]. (Notethatbneed notbe in [x,y] forthis expectation

tobe welldefined.) In orderfortalktobecredible inequilibrium, there

must

exist a valuation b* e(0,1) such thatbuyers with valuationsabove b* prefertohave theseller

believe that theirvaluation is high while thosebelow b* preferhertobelieve that itis low.

(1) u([b*,l],b)

>

u([0,b*],b)

forallbe(bM],

and

(2) u([0,b*],b)

>

u([b*,l].b) forall b e [0,b*).

In the take-it-or-leave-itbargaining game, however, thefact that the sellerhas all the

bargaining

power makes

itimpossible to satisfy both oftheseinequalities, exceptin an

uninteresting way: u([0,b*],b)

=

forall be [0,b*].3 Thus,in this

game,

theonly buyers

willing toadmitto having lowvaluations are those

who

do

not trade following talk.

thesellerhasthe opponuniiytomakethe firstoffer, (ii)thesellerincurs cost ceach limeshemakesan

offer,and(iii)thebuyer incurscost

C

>ceach time hemakesanoffer.

^Formally, u(rb*,l],b*)= becausegiven ihisbelief iheseller will notmake an offer lessthan b*.

Therefore,u([b*,I],b) isarbitrarilyclosetozero forbsufficiently close to(butgreaterthan) b*. Notethat

u([x,y],b) isweaklyincreasing in b,so u([0,b*],b*)=0,else(1) fails forb slightly greater than b*. But

4

In

many

otherbargaininggames,itis nottrue that the lowest-valuation buyerhas an

expected payoffofzero."* Itwill

become

clearbelow,however, thatthe simplicityofthe

take-it-or-leave-itbargaining

game

enables usto derive crispresultsabout

how

talkwill

affect the bargaining. Ratherthanabandon thetake-it-or-leave-it bargaining

game,

therefore,

we

modify its information structureso thatthelowest-type buyerhas a positive

expectedpayoff. In theprocess,

we

developa

model

that

may

be ofindependentinterestas a description ofan importantclassof bargainingproblemsthat has not yet received

much

attention.

We

assume

thatthebuyer does not learnhisexact willingness topay forthe seller's

good

until she defines itprecisely.

Such

imprecision arises

when

the

good

in questionis

complex

or idiosyncratic: ifthesellermust carefully defineits specifications(or iftheseller

must

write a contractcovering

many

contingencies thatmight ariseduringalong-term

relationship with the buyer), thebuyercannotprecisely

know

hisvaluation beforehand.

Our

assumptionis natural giventhat thebuyerandthe sellerarebargaining ratherthan

tradingin a competitivemarket atamarket-determined price:

when

the seller's

good

(or the buyer's need)is idiosyncratic, thereisamatching

problem

aswellas a pricing problem, so

competition disappears and bargaining results.

We

model

the buyer's imprecise

knowledge

ofhis valuation as follows. First,

nature

draws

thevaluations sand b, asabove. Second, the sellerlearns hervaluation, s, and the buyerlearns what

we

will call his type (as distinct

from

his valuation).

The

buyer's

type(t) iseitherhigh (H) or

low

(L),

where

t

=

H

forbe [1/2,1] and t

=

L

forb£ [0,1/2).

Thus,after learninghis type, the buyer's beliefabouthis valuation is uniform

on

[0,1/2) if

t

=

L

and

uniform

on

_[1/2,1] ift

=

H.

"^InChauerjeeand Samuelson's(1983)linearequilibrium inadoubleauction,for instance, ifs is

distributedon [x,y]andbis(indcpendenily)distributedon [w,z], then thebuyerwith valuation

w

earns a

su-icilypositiveexpected payoff provided

w

issufficientlygreaterthan x. Also,ininfinite-horizon games

based onRubinstein's (1982)modelwith discounting, such as ChatterjeeandSamuelson (1987)and

5

Afterlearning histype, thebuyer can send acostlessmessage (cheap talk) to the

seller.

The

message

spaceis unrestricted,but thereis not

much

thatcan besaid. Infact, it

is sufficientto considerthe spaceof equilibrium messages {"H", "L", "pool").^

The

remainder ofthe

game

is a take-it-or-leave-it bargaininggame.

The

seller

chooses whetherto

make

anoffer, and ifsowhatpricetodemand. Ifshe

makes

an offer,

thebuyerlearnshisvaluation, b,and then chooses to accept orreject theoffer. Ifthe seller

does not

make

an offer, the

game

ends, without the buyerlearning b. Tlie payoffsare

exactly asdescribed abovebecause themessagesintroduced herearecostless.

Two

special features ofthis

model

areworth noting. First,cheap talk cannot

communicate

anythingother than thebuyer's tj^e, t: because thebuyerlearns b afterthe

seller

makes

theoffer, there is no scopeforthe buyerto talk about b, and in this

take-it-or-leave-it

game

thereis no role for talkbythesellerabouther valuation, s.^ Second, the

model

has

moving

support: ifthe buyerlearns that t

=

H, say, then he

knows

that bis

certain to exceed 1/2. Relaxingtheseassumptions

would

complicate the analysis but

seems

unlikely toaffect the basic ideaspresented below.

3. Analysis

Our

analysis proceeds by comparing six expected payoffs, analogous to the

expected payoffs u([x,y],b) but based on the buyer's

knowledge

ofhis type but not his

valuation.

These

six expectedpayoffsare denoted U("H", t), U("L", t), and U("pool", t)

-Thisisin conirasltoourearlierpaper,inwhich wcconsidered onlythe restrictedmessagespace {"keen",

"notkeen") andsocouldnoicompleiclycharacterize theequilibriumroleofcheaptalkinadoubleauction.

Also, noie wellthe distinctionbetweentheequilibriummessages ("H", "L", "pool") and the unrestricted

space of possible messages; Farrelldescribesthe largesetofpotentialout-of-cquilibrium messagesinvolved

in the definitionof neologism-proofness.

°Inthistake-it-or-leave-itgame, talkbythe sellerabout her valuation could matteronlybyinducingthe

buyertosend asubsequentcheap-talkmessagethatvarieswiihiheseller's talk. Butnomatterwhather

valuation, thesellerprefers tohavethebuyerrevealmoreratherthanlessbeforeshe hastodecide whether

tomakeanoffer. Thus,independent of hervaluation, thesellerwouldsend themessagethatinducesthe

buyertoreveal asmuch aspossible,so thereisnoreason forthe buyer'stalk to varj' with theseller's: the

6

forte {H,L).

The

firstand secondofthese are theexpected payoffto a buyerof iypt t

given thatthesellerbelieves thatt

=

H

and t

=

L, respectively.

The

thirdis theexpected

payoffto abuyer oftype tgiven that the sellerholdsthe(prior)beliefthateach ofthe

two

types isequally likely^

We

are interestedintwo (perfectBayesian) equilibriaofour

game:

a separating

equilibrium (in which tliebuyer's talkperfectly

communicates

histype tothe seller), and a

pooling equilibrium (in which thebuyer'stalk

communicates

nothing). In each kindof

equilibrium, tliepost-talk bargaining behavioris theuniquesequentially rational

outcome

of

thetake-it-or-leave-it bargaining

game

given theseller's belief.^

A

basic resultin thetheoryof cheap-talk

games

is that apooling (or "babbling")

equihbrium always exists.

We

tlierefore askwhethera separatingequilibrium exists, and

also whetherthe equilibria that exist arereasonable

—

thatis, satisfyarefmement.

Formally,a separatingequilibriumexists if

(3) U("H",

H) >

U("L",

H)

and

(4) U("L", L)

>

U("H", L).

The

refmement

we

use isFarrell's notion ofneologism-proofequilibrium.

(Many

otherrefinements, such as thosedeveloped forsignaling

games

by

Cho

andKreps (1987)

and

Banks

andSobel (1987), have noeffectincheap-talk games.)

We

saythat thepooling equilibriumisnot neologism-proof(and sorejectitas an equilibrium)if

(5)

U("H",

H) >

U("pooI",

H)

and (6) U("H", L)

<

U("pool", L).

'The payofffunction u([x,y],b)isofcourserelated tolliesesixexpectedpayoffs:for instance,U("H", L) -E{u([l/2,l],b)IO<b< 1/2).

'^Note thatthisuniqueness ofscqueniiallyrational posi-talkbargainingbehaviorrulesoutequilibria in

The

ideabehindtheseconditionscan be phrased in terms ofaspeech likethose popularized

by

Cho

and Kreps. Supposethe pooling equilibriumistobe playedand thebuyerlearns

that his typeishigh, t

=

H.

Then

thebuyershoulddeviatefromtheequilibrium

by making

thefollowing speech tothe sellerduringthe cheap-talk phase: "Iclaim

my

typeis high.

Note

thatifyou believethisclaimthen I willbe (strictly)better offthan I

would

have been in equilibrium, by (5). Noie alsothat if

my

t>pe werelow then 1

would

have no (strict)

incentive to getyou to believethisclaim, by (6)." In FarrcH's terminology, (5) and (6)

define a credible neologism bythe high t)'pe; if(5)and (6) hold,

we

say that a credible

neologism by thehigh buyer-type breaks thepoolingequilibrium.

A

credible neologism bythe low buyer-type alsocould break thepooling equilibrium, if

(7) U("L", L)

>

UC'pool", L) and

(8) U("L",

H) <

UC'pool", H).

Proposition 1 below implies that ifa credibleneologism by ilie low buyer-type breaksthe

pooling equilibrium then acredible neologism bythehigh buyer-tj'pe also breaksthe

poolingequilibrium, so

we

ignore neologisms

by

thelow buyer-typein

what

follows.

Ifa separatingequilibrium exists, then

we

also are interested in whetheritis

neologism-proof. Herethe relevantneologism ispooling (the appropriatespeech begins "I

am

notgoingtoreveal

my

type. Notice thatregardless of

my

type, I

have

reason to

make

this deviation...").

The

separating equilibrium is notneologism-proofifand onlyif

(9) UC'pool",

H)

>

U("H",

H)

and

8

Note thatbothinequalities arestrict, by analogy with (5) and (7).

A

basicresult for

two-type cheap-talk

games

is

Lemma

1. Ifa separatingequilibriumexists and thepoolingequilibriumis not

neologism-proofthenthe separating equilibriumis neologism-proof.

Proof. Ifthepooling equilibrium is not neologism-proofthenat leastone of(5) and (7)

must

hold, hence atleast oneof (9) and (10) must fail.

Q.E.D.

We

now

characterize the neologism-proofequilibria ofour model.

The

analysis

rests

on

aseries ofsimple

lemmas

(proofs ofwhich arerelegated to the Appendix).

Lemma

2.

Suppose

the sellerofvaluation s believes that tliebuyer's valuation is uniformly

distributed

on

[x,y].

Then

iftheseller

makes

an offer, theprice

demanded

isp(s|x,y)

=

max{x,(y+s)/2}.

Lemma

3. For all values ofc, U("H", L)

=

andUC'pool", L)

=

0.

Lemma

4. (a) U("H",

H) >

ifand only ifc

<

1/2.

(b) UC'pool",

H)

>

ifand only ifc

<

1/4. (c) U("L", L)

>

ifandonly ifc

<

1/8. (d) U("L",

H)

>

ifand only ifc

<

1/8.

Lemma

5. Forc e (0,1/2), U("H",

H) >

U("pool", H).

Combining

these

Lemmas

(andadding a smallargument given intheAppendix)

Proposition 1.

(a) Ifc

>

1/2 thentheunique equilibrium ispoolinganditis neologism-proof.

(b) Ifc £ (0,1/2) then acredible neologism bythehigh buyer-t>'pebreaks the

poolingequilibrium. Funher, thereexists a criticalcostlevel c* e (0,1/8) such tliat

when

c e (c*,l/2) a separating equilibriumexists and is neologism-proof, but

when

ce (0,c*) a

separating equilibriumdoes notexist(and so theredoes notexista neologism-proof

equilibrium).

Two

featuresoftliis resultare worth noting. First, by

Lemma

4(b), ifc

>

1/4 then

the .sellerdoes not

make

an offerin thepooling equiHbrium, no matterwhat hervaluation.

Thus, by Proposidon 1(a), thepooling equilibrium is neologism-proofonly ifnothing

happens!

And

second, assumingce (c*,l/2) sothatthe separaung equilibriumexists, both

partiesare betteroffin the separating than in the pooling equiHbrium:

Lemma

5 establishes

this forthehigh buyer-type, while

Lemmas

3 and4(c) doso for the low, and theseller is

betteroff(no matterwhat hervaluation) simply becauseshe has

more

information.9 Tlie

realimportance of Proposition 1,however, is thatit leads to our

main

result:

Proposition 2. Forc E (c*,l/2), the post-talkbargaining behavior in the unique

neologism-proofequilibrium could not be equilibrium behavior withouttalk.

The

proofofthis Propositionis simple: the unique equilibrium withouttalkis pooling; the unique neologism-proofequilibriumfor ce (c*,l/2) is separating; and the post-talk

bargaining behaviorinthese twoequilibria differs in

many

obvious ways.

We

now

turn to

ItISofcourse not generally irucin gamesthathaving moreinformationmakesa playerbetteroff. Inour

bargaininggame, however,the buyer'soptimalactiondependsonlyonthepricethesellerdemandsandon

b,andsoismdcpendeniofwhatthebuyerthinks thesellerknows. In essence,thisreducesthegametoa

1

a detaileddiscussionofthese differences inpost-talk bargaining behaviorin orderto

interpret theeffectsof cheap talk inour model.

Thereare

two

ways

in whichpost-talkbargaining behaviorcan differ

among

equilibria: inthe seller's decision to

make

an offer, orin thepriceshe

demands

(or both).

Our

main

resultsinvolve both ofthese differences: the high buyer-type is willingto paya

higherpriceiftradeoccurs in orderto increasethe probability that trade will occur, and the

low buyer-typestrictlyprefers not to

make

this tradeoff.

In

some

ofthe equilibriaconsidered in Proposition 2, however, simpler (and,

we

think, less interesting) arguments suffice to supportthe equilibrium. In

some

cases, for instance, the

low

buyer-type is indifferentabouthis message: (4)holds with equality.

Also, in

some

cases theprice the seller

demands

ifshe

makes

an offeris the

same

in the separatingand poolingequilibria, so theequilibria differonly in theprobabilir\' that the

seller

makes

an offer.

We

now

identifytheequilibriadescribed in Proposition2 that are

relativelyuninteresting for eitherofthese reasons; thisleaves us with tiieequilibria that

precisely

embody

theintuition

we

havedescribed.

Considerfirstthepoolingequilibrium.

Lemma

2implies thatifthe sellerwith

valuation s

makes

an offerthentheprice she

demands

is p(s|0,l)

=

(l+s)/2.

Straightforwardcalculation then

shows

that (assumingc

<

1/4) thesellerswith valuations

s

<

1 - 2^/c

make

offers; asnoted above, wb.enc

>

1/4 no seller-type

makes

an offer.

Now

considertheseparating equilibrium.

When

c e (1/8,1/2),

more

seller-tj^es

make

offers in theseparatingequihbrium than inthepoolingequilibrium, but those types

that

make

offersin both equilibria

make

the

same

offerineach equilibrium.^0

_fhe

factthat

more

seller-types

make

offersin theseparating equilibrium is

most

apparent

when

^^Itisonly byaccidentthatthe price thatasellerwith valuationsdemandsin the pooling equilibrium,

p(s|0,l)=(l+s)/2,isthesameas the price thatshedemandsintheseparatingequilibrium whenthebuyer

claims tobethehigh type, p(s|l/2,l)=(l-i-s)/2. Ifthehighbuyer-i>'peconsistedofbuyerswith valuations

b £[b',1] forsomeb' > 1/2,thenthepriceasellerwith valuationsdemandsin the separating equilibrium

when thebuyerclaimstobethe high typewouldbep(s|b',l)=max(b',(l-t-s)/2),whichdiffersfrom the

1 1

ce(1/4,1/2): theseller

makes

noofferinthepoolingequilibrium, but in theseparating

equilibrium, with probability 1/2 the buyerannounces that his typeis high, in

which

case

theseller updates herbeliefabout hisvaluationb, and, if s is low,

makes

an offer.

An

analogous argument applies

when

c e(1/8,1/4), exceptthatin thiscase ver)'-low-valuation

sellers

make

offers in bothequilibria.

Consequently,

when

c £ (1/8,1/2), saying "H" strictly dominates saying "pool"or

"L" forthe high buyer-typeand weakJydominates these alteniaiivcs forthe low

buyer-type. Forthese values ofc, therefore, the existence ofa separatingequilibrium depends

criticallyon the

weak

inequahtyin (4).

A

more

interesting equilibriumexists

when

talknot

only determines which seller-types

make

offers but alsowhatprices they

demand.

This occurs

when

ce (c*,l/8), as thefollowingtable

makes

clear.

1 Price

Demanded

by 1 HighestValuation 1 1 Sellerof Valuation s 1 Seller to

Make

an Offer 1

Pooling 1

p(s|0,l)=-^

1 s*(cr'pool")= 1-2%^ 1 Equilibrium 1 1 1 Separating 1 1 1 EquiUbrium 1

p(s|l/2,l)=^

1 s*(c|"H")

=

1-V2c 1 following

"H"

I.I

1 Separating 1 1 1 Equilibrium 1 p(sI0,l/2)

=

^^

1 s*(cr'L")

=

^""^ ' following "L" 1 1 1

Thus, a low-valuation seller, forinstance,

makes

offers in both thepooling and separating

equilibriabut

demands

a different pricein thelatterbecause thebuyer's talkhas influenced her beliefabout the buyer's valuation, b.

12

4. Conclusion

The

firstresultofouranalysisisthat the pooling equilibrium(theonlyequilibrium

in

which

talkdoesnot matter) is neologism-proof onlyifthe costof

making

an offeris so

highthat

no

offeris

made

bya sellerofanyvaluation.

We

can rephrasethis

more

strikingly: restrictingattention to neologism-proofequilibria, ifan offer (notto mention

trade) is observed, then talk must haveplayed a role.

The

force behind this result is the natural and credible message availableto the high-type buyer:

when

t

=

H, thebuyer

encourages the seller to

make

anoffer, at

some

cost in tlieterms oftrade iftrade occurs,

and thisbreaks thepoolingequilibrium.

We

expectanalogous cheap-talkmessages tobe

crediblein

many

other bargaininggames, sothatpoolingequilibria in these

games

often

will notbe neologism-proof;

we

willpursue thisidea in future work.

Our

second result is that

when

c E (c*,l/2), theseparating equilibrium not only

existsbut is theunique neologism-proofequilibrium. In this equilibrium, talk affects

whether and at

what

terms trade takes place.

Combined

with ourfirst result, this

shows

that in

some

bargaining

games

(i.e., for

some

values ofc) talk not onlycan matter but

must

matter,inthesensethattheonlyneologism-proofequilibrium could notbe an equilibrium

without talL

We

findthis resultparticularly appealing

when

c e (c*,l/8), in

which

case both buyer-types havea strictincentive to reveal themselvesin a separatingequilibrium

and both buyer-types havea strictincentive toreveal themselvesratherthan topool.

Our

final resultis that, in the

game

we

analyze, alltypes ofboth parties are

(weakly) betteroffin the neologism-proofequilibrium than in theequilibrium withouttalk.

1 3

APPENDIX

Proof of

Lemma

2.

The

seller'sproblemis tochoosep to solve

max

s F(p)

+

_p [1 - F(p)],

P

where

F(p)

=

(p - x)/(y - x) forp e [x,y], is zero forp

<

x, andis one forp

>

y.

The

first-order condition }delds p

=

(y+s)/2, whichis a global maximizerprovided tliat F(p) is

strictlypositive atthisprice. Q.E.D.

Proofof

Lemma

3.

By

Lemma

2, the

minimum

price

demanded

comes

from tlie

zero-valuation sellerand so is

p(0|x,y)

=

max

{x,y/2).

Thus, inU("H", L),

where

the seller believes thatb is uniformly distributed on [1/2,1], the

minimum

price is 1/2, andin U("pool", L), wherethesellerbelieves that bis uniformly

distributed on [0,1], the

minimum

price also is 1/2. In neithercasedoes the low-type buyerreceive an acceptableoffer. Q.E.D.

Proofof

Lemma

4.

Each

caseis solved by computing thevalue ofcsuch that the

zero-valuationselleris indifferent about

making

an offer.

(a)In U("H", H), the zero-valuation sellerbelieves thatb is uniformlydistributed

on [1/2,1] and so

demands

p(0|l/2,l)

=

1/2ifshe

makes

an offer. This offer is accepted

withprobability one andso earns a profitfor the zero-valuation seller of 1/2. Thus, ifc

=

1/2,the zero-valuation selleris indifferentabout

making

anoffer.

(b) In UC'pool", H), the zero-valuation seller believes that b is uniformly

distributedon [0,1] and so

demands

p(0|0,1)

=

1/2 ifshe

makes

an offer. This offeris

accepted with probability 1/2andso earns aprofitfor the zero-valuation sellerof 1/4.

14

(c) In U("L",L), thezerovaluation sellerbelieves tliatb isuniformly distributed on

[0,1/2] and so

demands

p(0|0, 1/2)

=

1/4ifshe

makes

anoffer. This offer is acceptedwith

probability 1/2 and so earns aprofit for the zero-valuation sellerof 1/8. Thus, ifc

=

1/8,

thezero-valuation selleris indifferentabout

making

anoffer.

(d) In U("L",

H)

the zero-valuationsellerbeUevesthat bis uniformlydistributed

on

[0,1/2], as in (c). Tliis belief leads to the

same

sellerbehavior as in (c).

The demand

p(0|0,l/2) = 1/4 is accepted with probabilityone by the high-typebuyer, but the zero-valuationsellerdoesnot expect lliis behaviorfrom the buyerand so is indifferent about

making

an offer

when

c

=

1/8. Q.E.D.

Proofof

Lemma

5.

The

result follows from

Lemmas

4(a) and 4(b) for c e (1/4,1/2). TTie

argument forc e(0,1/4) is as follows.

By

Lemma

2, theprice

demanded

by

the sellerof valuation s is identical in U("H",

H)

and in U("pool", H):

p(s|l/2,l)

=

p(s|0,l)=^.

The

only difference

between

U("H",

H)

and U("pool",

H)

is that, for any given value ofc,

strictly

more

sellers chooseto

make

an offerin theformer, so the high-type buyeris better

off. Formally, simple computations

show

that in U("H",

H)

the highest-valuation sellerto

make

an offeris s*(cl"H")

=

1-V2c, whilein U("pool",

H)

it is s*(c|"pool")

=

l-2Vc.

The

formeris strictly larger. Q.E.D.

Proofof Proposition 1.

(a)

The

separating equihbrium does not existbecause U("H",

H)

=

0, by

Lemma

4(a), so (3) fails. Similarly, the pooling equilibrium is neologism-proof because

by

15

(b)

By

Lemma

5, (5) holds, and by

Lemma

3, (6) holds. Thus, a credible

neologism by thehighbuyer-type breaks thepoolingequilibrium. Therefore,

by

Lemma

1

,

ifthe separating equilibriumexists then itisneologism-proof.

Suppose

first that c e (1/8,1/2).

By

Lemmas

4(a) and 4(d), (3) holds.

By Lemmas

3 and 4(c), (4) holds weakly.

Hence

a separatingequilibriumexists.

Now

suppose that c e (0,1/8).

By

Lemmas

3 and 4(c), (4) holds strictly. It thus

remains to

show

that (3) holds ifand onlyifc E (c*,l/8), for

some

c* E (0,1/8). Using the

values ofp(s|0,l/2), s*(c|"L"), p(s|l/2,l), and s*(c|"H") given in thetext, calculations

show

that

,,,.,. „ „, (l-4x)(3+4x)

U(

L

,

H) =

rr and

where

x

=

Vc72. Note thatifc £ (0,1/8) then x e (0,1/4). Thus, U("H",

H) >

U("L",

H)

if

and onlyif

f(x)

= 32x3

.

_96x2

.

_{24x +}

₅

_<

_0.

Note

that f(l/4)

<

and f(0)

>

0. (Both ofthese areobvious

from

firstprinciples: consider

c

=

1/8 in

Lemmas

4(a) and4(d) andc

=

0, respectively.) Finally, notethat f(x) is

decreasing forx e (0,1/4), sothere exists a unique x* £ (0,1/4) that solves f(x)

=

0.

1 6

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