Bargaining with Noisy Communication

Vincenzo Denicolo and Paolo G. Garella

Departmen t of Economics { University of Bologna

Piazza Scaravilli 2 { 40126 Bologna { Italy

denicolo@ecn01.economia.unibo.it garella@spbo.unibo.it

December 1996

Abstract

Inthispap erw eshowthatinabargainingsituationthesellerma y

not necessarily w ant to fully exploit comm unication p ossibilities. In

thestandardt wo-p erio d bargaining modelwith one-sided incomplete

information, the seller, who o wns an indivisible goo d, mak es oers

which the buyer can either accept or reject. We ask whether the

sellercanprotfrommanipulatingthecomm unicationmechanismby

sending oers that reach the buyer with probability less than one

(noisy comm unication). Noisy comm unication is a w ay to improve

theseller's second p erio d b eliefs ab out thebuyer'swillingnesstopay

forthegoo dand istherefore a w ayto"buy" commitment. Westudy

thecaseofadiscretedistributionofbuyer'st yp esandshowthatthere

existequilibriawithnoisycomm unicationwhenthereareatleastthree

dierentt yp es ofbuyers.

Keyw ords: Bargaining,Comm unication, Incomplete Information.

1 Introduction

In this pap er we show thatin abargaining situation theseller ma y not

nev ertheless, strategic considerations induce him to refrain from comm

uni-cating tothe buyer with maxim umeectiv eness.

More sp ecically, we consider the Fudenb erg and Tirole (1983) model

of bargaining with one-sided incomplete information, where one seller and

one buyer bargain ov erone indivisible object. There are twop eriods. The

seller's valuation is common knowledge and the buyer's valuation is private

information. In the original model, the seller mak es an oer in p eriod 1,

whic hthe buyercaneitheracceptorreject. Ifthe buyeraccepts,the gameis

over;if he rejects, the seller can mak e another oerin p eriod 2. Again, the

buyercaneitheracceptorrejectthesecondoer;ifherejects, thegameends

and the object isleft unsold. Wealter this basic structureby assumingthat

the messagessentbytheseller ma ynothit thebuyer withaprobabilit ythat

is controlled by the seller. We show that the seller ma y gain from sending

messagesinthe rstp eriodthat arereceiv edbythe buyerwithaprobabilit y

strictly lowerthan 1 (noisy comm unication).

It is not clear to us what could prev ent a seller from in tro ducing some

frictionsin tothe comm unicationpro cess, andthereforewefeelthat aprop er

modeling of the bargaining pro cess should allow for such a strategy. Our

theoretical analysis is consistent with the observation that comm unication

p ossibilities are sometimes v oluntarily underexploited. For instance, some

shops donot p ostprices. Tomak eanexample whic hiscloser tothe subject

of this pap er, our analysis ma y help explain why the owner of a house (in

asmall village, say)ma ysomewhat concealhiswillingness tosell, informing

only a few p eople and letting the information circulate by word of mouth,

instead of resorting to more eectiv e ways of comm unication, ev en if the

cost of more in tense advertising would b e negligible compared to the value

of the transaction. This kind ofb ehavior seemsto b efrequen tinbargaining

situations 1

.

1

Atrst,onema ythinkthatincompleteinformationab outtheseller'svaluationcould

provideanalternativ eexplanation; that is,this b ehaviorma yb e used asa signal.

How-ever,onre ection,whileitisreasonableforthesellertosignalthroughanabnormallyhigh

price,itseemsdiculttounderstandwhyhewouldnotcontactallp otentialbuyers. Thus

incomplete information by itself is not sucient to explain noisy comm unication. Our

analysis,thatassumesthatthesellerhasnoprivateinformation,showsthatitisnot

nec-essaryeither. Whethernoisycomm unicationcanb e optimalundertwo-sidedincomplete

committoaxedsequenceofoers. Asaconsequence,ifthe rst oeris

re-jected, hewill haveanincen tivetoreducethe secondp eriod price,asshown

by Fudenb erg and Tirole (1983). An ticipating this b ehavior, the buyer will

reject some rst p eriod oers that he would otherwise hav e accepted. But

if the seller sends messages that do not hit the buyer with probabilit y one,

his second p eriod b eliefs ab out the buyer's willingness to pay will b e more

optimistic than in the case of deterministic messages. This translates in to a

higher second p eriod price. In short, the seller uses noisy comm unicationto

'buy' some commitmen t. Of course, noisy comm unication is costly as some

opp ortunities for immediate agreemen t are foregone, and showing the

opti-malit yofnoisycomm unicationrequiresprovingthatthe strategic advantage

illustrated ab ov ema youtweighthe cost.

Another in terpretation of the model isthat the seller is a durable-goo ds

monopolist selling to a numb er of consumers whose willingness to pay for

the go o d is distributed according to a known distribution function. In this

context, the counterpart of noisy comm unication is that the seller rations

randomly a fraction of consumers at the rst p eriod. This in terpretation is

dev elop edinacompanionpap er(DenicoloandGarella,1996),wherethecase

ofacontinuousdistributionofconsumers'typ esisanalyzed. Inthispap er,we

fo cus onthe case of adiscretedistribution,whic h allowsa moretransparent

analysis of the comparativ e advantages and costs of noisycomm unication.

Totheb estofourknowledge,theb ehavioranalyzedinthispap erhasnot

b een discussedinthe literature sofar. However,relatedproblems hav eb een

treated in the bargaining literature and in the game-theoretic literature on

comm unication.

Inthebargainingliterature,the basiccomm unicationstructureof

Fuden-b erg and Tirole(1983)has b een c hangedinanumb erof ways. Forinstance,

Mutho o (1994) considers the p ossibilit y that the oeris not irrevo cable. In

hismodel, the sellercan c ho ose towithdrawhisoerafterthe buyer has

ac-ceptedit. AdmatiandPerry(1987)studyamodelwithtwo-sidedincomplete

information where the playerscan c ho ose the time b et ween oers, and use

this instrumen ttosignal theirbargaining strength 2

. The general messageof

these pap ers is that the structure of the bargaining pro cess can b e

manip-ulated by the players in order to gain a strategic advantage. The presen t

2

Noisy comm unication b et ween players has b een explicitly modelled in

the game-theoretic literature. Rubinstein (1989), in a pap er on

common-knowledge where no bargaining problem is in volved, assumes that messages

sent by playershave anexogenously giv enprobabilit yof getting lost.

Com-m unication has also b een studied in the literature on c heap-talk in

sender-receiv ergames(seeForges,1986andFarrell,1993). My erson(1991)discusses

anexampledue toFarrellwhereiftheinformedplayersendsanoisymessage

with a probabilit y1/2 of b eing receiv ed, anequilibrium is reac hedin whic h

b oth playersget a payoexceeding the one theycan get in anyequilibrium

withnoiselesscomm unication. Butobviouslysender-receivergamesarequite

dieren tthan bargaining models.

The rest of the pap er is organized as follo ws. We presen t the model in

section 2. In section3, we provideac haracterizationof equilibria wherethe

seller do es not fully exploit the p ossibilities of comm unication (Equilibria

with Noisy Communication, ENC). In section 4 we analyze the case where

the buyer can b e one of twotyp es,showing that noENC existsin this case.

The three typ escase is studied insections 5 and 6. It is shown that ENC's

existinthethreetyp ecase. Wealsobrie ydiscuss,insection7,theeciency

prop erties of ENC.Section8 concludes.

2 The Model

Assumethatabargainingin volvinganindivisiblego o d canlast fortwo

trad-ing p eriods. The seller's valuation of the go o d is public information, and

is strictly lowerthan any p ossible valuation of the buyer; thus, without any

furtherlossofgeneralit y,itcanb enormalizedtozero. Thebuyer'svaluation,

v; is instead private information. The seller has a probabilit y distribution

F(v) oversome supp ort 

v

0 ;v



,with v>v>0, whic hiscommon knowledge.

Both players are risk neutral. The seller's utilit y is simply equal to the

discounted price  t 01

p

t

( t = 1;2 ) if the object is sold, and zero otherwise;

the buyer's utility frombuying at date t isgiv enby

u= t 01

(v0p

t

); (1)

bilit y1anoerp 3 wherep 3 =argmax p p[10F(p)]. Weshallrefertop 3 asthe

static optimalprice. Tosimplifythe exposition,we assumethat p[10F(p)]

is strictly quasi-concaveso that p 3

is unique :

A teac hp eriodt=1;2,theseller sendsamessagewhic hisanirrev o cable

oer to sell the go o d at a sp ecied price p

t

. The seller c ho oses also the

probabilit y10

t

in (0;1) that the messagehits the p otentialbuyer. There

is no cost of increasing this probabilit y. If the buyer receiv es the message

and acceptsthe oer,theobjectissold. Finally,weassume thatif thebuyer

receiv es the message he can also observe 

t

, but the seller cannot observe

whether the buyerhas receiv edthe message ornot.

It is clear that in the second p eriod the seller has nothing to gain from

c ho osing

t

>0,forthiscouldonlylowertheprobabilit ythatanagreemen tis

reac hedwithoutincreasing the equilibrium price. Without lossof generalit y

therefore we can dene a strategy of the seller as a triplet (p

1 ;;p 2 ; ), where p 1

is the rst p eriod price,  = 

1

is the noise in the rst p eriod message,

and p

2

is the second p eriodoer whic his made with zero noise, conditional

on disagreemen tin the rst p eriod.

In the second p eriod, the buyer's decision problem is trivial and he will

accept the oer if and only if p

2

 v. If the buyer do es not receiv e the

message in the rst p eriod he has no decision to take. Thus, a strategy for

the buyer can b e described simply as arst p eriod reservationprice, b .

One maywonderwhether thebuyer whohas not receiv edanyoercould

tell the seller that the messagewentlost, or whether the buyer who has not

acceptedanoercouldsendanegativ ereplytotheseller. Sincethisb ehavior

will mak enoisycomm unicationineectiv ethus destroyingthe ENC,a seller

who would gain from noisy comm unication, would want a buyer to reply

onlyif heacceptsareceivedoer. Inthis pap er, weassumethat the selleris

empoweredtoc ho ose the comm unicationstructure and thereforewe assume

thatthebuyercannotsendunwantedmessages. Forinstance,thesellercould

use an appropriate electronic device, or instruct a middleman to transmit

only acceptances.

If the seller could commit to a prescribed price sequence, it would b e

optimal to set p 1 = p 2 = p 3

and the solution to the seller's problem would

b etrivial. Weassume,however,that theseller cannotcommittoasequence

of oers. This implies that the second p eriod price will b e set in order to

In a p erfect Bayesian equilibrium, conditional on no agreement in the

rst p eriod,p

2

will b esetsoas tomaximize p

2 Pr (v p 2 ),where Pr(vp 2 )

isthe revisedb eliefthat the buyer'swillingness topayishigherthan p

2

con-ditionalon disagreemen tinthe rst p eriod. On the other hand, the buyer's

reservation price b (v) will satisfy the condition that the buyer is indieren t

b et ween accepting and rejecting a rst p eriod oer p

1

= b (v). Since the

buyer can anticipate the second p eriod price incase of adisagreemen t, b (v)

will satisfy (v0b )=(v0p 2 ), sothat: b (v)=(10)v+p 2 : (2) A rst p eriodoer p 1

willtherefore b e accepted bya buyerwhose valuation

is atleast v 1 = p 1 0p 2 10 ; (3)

andwill b erejectedif thebuyer'svaluationislowerthanv

1

. Thisholdsb oth

with p erfect and noisy comm unication; all that c hanges is the value of p

2

(whic hina p erfect Bayesianequilibrium isanticipatedby the buyer).

3 A Characterization of Equilibria with Noisy

Communication

An equilibrium with noisy comm unication (ENC) is an equilibrium where

the seller sets >0.

In thissectionweprov ethatanyENCnecessarilyen tailsarisingpattern

of named prices. Moreover, the second p eriod price is always equal to the

static optimalprice (i.e. the priceif bargaininglasted one p eriod only).

proposition 1 At any ENC, p

2 >p

1 .

All pro ofs are in the App endix.

The in tuition is that if, contrary tothe Prop osition, p

2 <p

1

and >0,

thenthesellercouldincreaseprotsbydecreasingandincreasingp

1

insuch

a wayas to lea ve Pr(v p

2

) unaltered overthe relevantrange. This would

increase rst p eriod prots, while lea ving expected second p eriod rev enue

unchanged,thus showing that the initial strategy issub-optimal.

corollary 1 Atany ENC, p

2

=p >p

1 .

The reason is that when p

2  p

1

the optimalstrategy tothe buyer is to

setb =v (hecannotgainbywaitingforthesecondp eriodoer)andtherefore

at anyENCthe secondp eriod price will solvethe maximization problem

max

p

(10)p[10F(p)] (4)

providedthe solution isgreater than p

1

: It follo ws immediatelythat p

2 =p 3 if p 3 >p 1 . If instead p 1  p 3

; then it would b e optimalto setp

2 = p

1 ; but

by Prop osition 1 this cannotb e an ENC.

Thiscompletesourgeneralc haracterizationofENC's. Inordertoaddress

existence, we further sp ecialize our model assuming a discrete distribution

function. Werstconsiderthecase oftwotyp esofconsumers andshowthat

therecannot b eequilibriawith >0inthatcase. Then weturn tothe case

of three typ es,where ENC's exist.

4 Two types

Inthissectionweassumethatthebuyer'svaluationisv

H withprobabilit yx H andv L <v H withprobabilit yx L =10x H

. Letting=0byassumption,one

obtains exactly the model analyzed by Fudenb ergand Tirole (1983). They

show that equilibrium takesone of the follo wingforms:

(i) p

1 =v

L

, allbuyers acceptimmediately;

(ii) p 1 =(10)v H +v L and p 2 =v L

,ahigh valuation buyeracceptsthe

rst oer and a lowvaluation buyer acceptsthe second oer;

(iii) p

1 =v

H

, this oer isrejected by ahigh valuation buyer with

proba-bilit yy;ifthe buyerdo es not accept,thesellermak es asecondoerwhic his

again v

H

. Here y is determined so as to mak ethe seller indieren tb et ween

setting p 2 = v H and p 2 = v L

: This requires that the probabilit y that the

buyer b e of typ e H in case the rst p eriod oer is rejected, up dated bythe

Bayesrule, is v L =v H , that is: yx H yx H +(10x H ) = v L v H (5)

Fudenb erg and Tirole's (1983) c haracterization of the equilibrium do es not

in the twotyp e case.

proposition 2 With two types of consumers only therecan be no ENC.

In tuitively,inthe twotyp ecase the results of the previous sectionimply

thatanyENCm ustin volvep

1 =v

L

. Butthentheonlyeect ofsetting>0

istoreducetheprobabilit yof agreemen tinthe rstp eriodwithoutaecting

the seller's incen tives in the second p eriod, b ecause the seller's p osterior

b eliefsab outv will coincidewiththe ex-antedistribution. Asaconsequence,

at equilibrium itm ust b e=0.

5 Three Types: Necessary Conditions for an

ENC

We now assume that the buyer can b e one of three typ es. Let v

H > v M > v L

denotethe valuationsofthe three typ es,whic hare inprop ortions x

H ;x M and x L =10x H 0x M .

In this section we describesituations where ENC'scannot emerge. This

analysis serv estwo purp oses. First, it showscircumstances wherethe

tradi-tionalanalysis, that assumesawaynoisycomm unication,isvalid. Second,it

provides necessarycondition for anENC.

We b egin with auseful lemma.

lemma 1 In the three typecase, setting p

1 =v

L

cannot be part of a ENC.

The in tuition for this result is similar to that b ehind Prop osition 2.

An immediate implication of this lemma, com bined with Prop osition 1 and

Corollary1 is that aENC can exist only ifthe static optimalprice is v

H .

proposition 3 A necessary condition for a ENC is that p 3

=v

H .

That is, it m ust b e v

H x H > v M (x H +x M

) (so that pricing at v

M is

sub optimal in the one-p eriod set-up) and v

H x

H > v

L

(so that pricing at v

L is not optimal). Assume that p 3 = v H

. By Prop osition 1 and Lemma 1 it then follo ws

that the candidate ENC in volvessettingp

1 =v M and p 2 =v H .

1 M 2 H

Nowsupp ose p

1 =v

M

and considerthe seller's problem atthe b eginning

of p eriod 2. If v M x M > v L (x L +x M

), then, irrespectiv e of , it is optimal

to price at v

M

in the second p eriod. This case is in fact equivalent to the

two typ e case considered in the previous section b ecause the presence of

the lowestvaluation typ e do es not alter the seller's incen tivesin the second

p eriod. Lik einthetwotyp ecase,itfollo wsthatnoisycomm unicationcannot

b e optimal. proposition 4 If v M x M >v L (x L +x M

), there cannot exist any ENC.

To show existence of ENC's, hereafter we conne our attention to the

case where the ab ov einequality isrev ersed.

6 Existence of an ENC in the Three Type

Case

InthissectionweshowthatENC'sexistinthethreetyp ecase. Letusdenote

h =v H x H , m= v M (x H +x M ), and ` =v L

. The rest of our analysis will b e

based onthe follo wing restrictions onthe parameters:

A1: h>m>`: A2: v M x M <v L (x L +x M ):

A1 guarantees that the static optimalprice is v

H

, and at the same time

ensures quasi-concavity of the function p[1 0F(p)]. Under A2, the three

typ e case is genuinely dieren tfrom the two typ eone. Indeed, the strategy

of pricing so as to induce only the high valuation typ e to accept the rst

p eriodoerandthensettingp

2 =v

M

withoutnoisycomm unication(namely,

Fudenb erg and Tirole's(1983)equilibriumof typ e(ii) restricted tohighand

medium valuation buyers only)) under A2 is no longer subgame p erfect. It

ma ystill b e optimalto setp

1 soas tosatisfy v H 0p 1 =(v H 0v M ),sothat

a high valuation buyer, b eing indierent b et ween accepting or refusing the

rst p eriod oer, randomizes with an appropriate probabilit y of rejecting.

But this equilibrium has now a cost to the seller b ecause the rev enuefrom

pricing strategy (whic h do es not in volve noisy comm unication) and creates

a comparativ eadvantagefor noisycomm unication.

To pro ceed, we rst describe the subgame p erfectequilibria that donot

in volve noisy comm unication and that can emerge giv en assumptions

A1-A2. Then,wecalculate thecandidate ENC,and thecorresp onding expected

prot to the seller, using the results of section 3. Finally, we compare the

prot under the equilibria without noisy comm unication to the ENC prot

and show that the latter ma yb e the highestfor some parameter values.

6.1 Full comm unication subgame perfect equilibria

The equilibria with full comm unication that can emerge giv en A1-A2 are

described inthe follo wingprop osition.

proposition 5 With three types of buyers, if A1-A2 hold, only three types

of full communication equilibria can emerge, namely:

H) p 1 =p 2 =v H ; M) p 1 =(10)v H +v M and p 2 =v M , L) p 1 =(10)v H +v L and p 2 =v L :

Equilibria H and M involve the use of mixed strategies on the part of the

buyer, with the highest valuation buyer randomizing appropriately between

buying soon or waiting with probabilities

y H =max " v M x M x H (v H 0v M ) ; v L (x M +x L ) x H (v H 0v L ) # : and y M = v L (x L +x M )0v M x M x H (v M 0v L ) of waiting, respectively.

All other p ossible pricing strategies either are not subgame p erfect or

yield lower prots. On the other hand, it can b e shown that eac h one of

The candidate ENC can b e easilyc haracterizedusing the results ofthe

pre-vious sections. It m ust b e p

1 = v M and p 2 = v H

. The necessary amoun t of

noise  in the rst p eriod message is giv en by the condition that the seller

do es not nd it protable to lower the second p eriod price. It is clear that

p

2 = v

M

cannot b e optimal for any  giv en that v

H

is the optimal static

price. Hence, itmustb e x

H v H (x L +x H +x M )v L ;that is  v L x L x H (v H 0v L )0v L x M  :~ (6)

Conditions A1 and A2 imply ~<1.

The corresp onding expectedprot is:

 NC

=(10)m+h (7)

and is thereforelinear in: Thus we can restrictour attention tothe p oints

=~ and =1.

In particular, if m< h the optimal strategy with noisycomm unication

would in volve =1, so that bargaining is eectiv ely dela yedto the second

p eriod. However, setting p

1 = v M and p 2 = v H

with  = 1 cannot b e the

globally optimal strategy. Indeed, the seller could dob etter at the

equilib-rium where the price is equal to v

H

in b oth p eriods with  = 0 and the

high valuation buyer randomizes, lik e in strategy H described in Prop

osi-tion 5 ab ov e. This would yield prots higher than h. Therefore, only the

ENC with  = ~ can dominate the full comm unication strategies and thus

itwill b eour unique candidateENC 3

.The expecteddiscountedprot atthe

candidate ENC istherefore:

 NC =(10)m~ +h:~ (8) Obviously NC is increasing in : 3

This impliesthat <m=his a necessaryconditionfornoisycomm unicationsinceit

guaran teesthat  =~ is sup erior to =1. Actually, even stronger conditionsmustb e

Inordertounderstandtheadvantagesofnoisycomm unication,wenow

com-parethecandidateENCtotheequilibriadescribedinProp osition5. Itisalso

instructiv etocompare these equilibriawith the fullcommitmen toptim um.

Let us b egin with equilibrium L, whic h is the only full comm unication

equilibrium that do es not in volve mixed strategies. Equilibrium L indeed

illustrates the typical Coasian dynamics: an agreemen t is reac hedwith

cer-taintyand in the rst p eriodthe seller is unabletoextract allthe ren tfrom

the high valuation buyer who can wait for the price discount in the second

p eriod. Exp ected prot is:

 L

=(10)h+` (9)

It is obviousthat if playersare impatien t( is close to zero),then the seller

retainsm uchofitsbargainingp ower. Indeed,for =0thesolutioncoincides

with the full commitmen t optim um. As playersget more patien t, however,

expected prot falls,and achievesaminimum of v

L

at =1:

Noisycomm unicationisawaytolimitthescop eforstrategic rejectionby

the highvaluationbuyer b ecausethe p ossibilit ythat thebuyer has notb een

reac hed by the message eliminates the seller's incen tive to cut the second

p eriod price. Thus, at the candidate ENC, we have p

2 = v

H

lik e under

full commitmen t. However, as compared to the full commitment strategy

p 1 = p 2 = v H

that yields the static optimal prot, noisy comm unication

in volvestwo typ es of costs. First, in the rst p eriodthe price is lower than

v

H

whic h in volves a loss of expected rev enue. Second, the agreemen t is

dela yed to the second p eriod with p ositiv e probabilit y. While the former

typ e of cost is independen tof , the latter b ecomes less and less important

asincreases. Asaconsequence,forlowvaluesof,equilibriumLissup erior

to noisy comm unication but when  approaches 1 it can b e easily c hecked

from(8) and (9) that the candidateENC giv eshigher prots:

lemma 2  NC < L for 0  h0(10)m h(1+)0l ; and  NC > L for h0(10)m h(1+)0l  1:

strategies. Equilibrium H mimics the equilibrium with full commitmen t in

that the price stic ks to the static optimal level in b oth p eriods. However,

subgamep erfectionrequiresthatthebuyerrandomizewithasucien tlylarge

probabilit y of rejecting the oer in the rst p eriod (since a high valuation

buyer gets zero surplus, he is indieren t b et ween accepting the rst p eriod

oer or waiting and so can randomize). The cost of subgame p erfection is

therefore giv en by the discounting of a fraction of static optimal rev enue.

Exp ected prot is:

 H = h 10(10)y H i h (10) where y H

isdetermined bythe subgame p erfection constraint (see the pro of

ofProp osition 5intheapp endixfordetails). Clearly,equilibriumHis

equiv-alen ttothe fullcommitmen tstrategyat =1but itsprotabilit ydecreases

as the discount factor  decreases.

The candidate ENC is similar to equilibrium H in that in b oth cases

p

2 =v

H

. Moreover,the candidateENCalso in volvesdela yingtheagreemen t

with a p ositiv e probabilit y. However, noisy comm unication not only leads

to a p ositiv e b elief that the buyer b e of the highest typ e in p eriod 2 but it

alsolowersthe b eliefthatthebuyerhasmediumvaluationb ecauseamedium

valuationbuyerwouldhav eacceptedtherstp eriodoer. Asaconsequence,

the probabilit y1-~ that a high valuation buyer ishit by the message inthe

rst p eriod at the candidate ENC is higher than the probabilit y 1-y H

that

a high valuation buyer accepts the oer in the mixed strategy equilibrium

H. This means that the share of expected rev enue from the high valuation

buyer that is dela yed is lower at the candidate ENC than at equilibrium

H. As a consequence, for low values of  the candidate ENC ma y dominate

equilibrium H as can b e c heckedcomparing (6.3) and (6.5):

lemma 3  NC < H for (10 )m~ 0(10y H )h h(y H 0)~  1; and  NC > H for 0  (10)m~ 0(10y H )h h(y H 0 )~ ;

provided (10 )m > (10y )h: If, instead, (10)m  (10y )h, then

strategyH dominatesthe candidate ENC for all values of  :

Let us nally consider equilibrium M. This in volves b oth the cost of

Coasian dynamics (lik e equilibrium L) and of dela yed rev enue (lik e

equi-libriumH).Indeed, ifA2didnot hold,thehighvaluationbuyercouldaccept

with probabilit y1 the rst p eriod oerwithout upsetting equilibrium M; in

this case, as shown by Prop osition 4, noisy comm unication cannot b e

op-timal. But when A2 holds the p ossibilit y that the buyer's valuation b e v

L

would destroy p erfection of equilibrium M, unless a high valuation buyer

reject the oerwith a sucientlyhigh probabilit y. Exp ected prot is:

 M =(10)y M h+(10y M )m (11) wherey M

istheprobabilit ythatahighvaluationbuyerrejectstherstp eriod

oer. Thehighertheincen tivetocutthesecondp eriodpricetov

L

,thehigher

is y M

, and therefore the higher is the cost in terms of dela yed expected

rev enue. The cost of dela yed rev enue may b e so high that equilibrium M

b ecomes lik elydominated by the candidate ENC.

To summarize, our informal discussion suggests that equilibrium H will

o ccur when  isclose to1,equilibrium Lwill o ccurwhen  isclose to0,and

the candidate ENC can b e optimal for in termediate values of . We now

conrmthis conjecture.

proposition 6 A necessary andsucient condition for an ENC to exist in

the three type case underA1-A2 is

(10)m0(10y H )h h(y H 0) > h0(10)m h(1+)0l : (12)

In fact, if the ab ov einequalit y holds, then there is a non empt yin terval

( ; 

)such that ENC's existif and only if 2( ; 

):

A numerical example will show that condition (6.7) is consistent with

A1-A2, and will also help to illustrate the ab ov ediscussion. Let v

H = 700; v M = 420; and v L =140, and let x H = 2=7; x M =1=7; and x L =4=7. The

optimal static price is then 700 and the corresp onding (full commitmen t)

oer with probabilit y 1=4 so that expected discounted prots to the seller

are  H

=50+150.

Under strategy M, ahigh valuation buyerrandomizes b et ween accepting

the rst p eriod oer or not with probabilit y1/2; total discounted expected

prots are  M

=100+80 :

Strategy L yieldsprots  L

=200060.

Finally,inthecandidateENCthe noiseintherst p eriodoeris=4/7

and total discountedprots are  NC =(540+800)=7. Figure 1 illustrates  NC ; H ; M and  L as functions of . Strategy M is

nev er optimal. The candidate ENC yields the highest discounted prot for

43=61 19=25 (approximately,0: 7050: 76):

7 Eciency Properties of ENC's

Asiswellknown,bargainingequilibriamayb eex-postParetoinecien tasan

agreemen t mayfail too ccur ev enwhen itwould b e in b oth parties in terest,

or itma yb e dela yed.

Can noisy comm unication improve the eciency of the bargaining

pro-cess? Surprisingly, it turns out that it can. However, there are also cases

wherenoisycomm unicationresultsinanev enmoreinecien toutcome than

the standard equilibrium withoutnoisycomm unication.

The example discussedat the end ofthe previous sectionma yhelp

illus-tratingthewelfareconsequencesofnoisycomm unication. Theyobviously

de-p end onwhic htyp eofequilibriumwithoutnoisycomm unicationisdisplaced

by the ENC. In the example, the ENC displaces equilibrium of typ e L for

43=61 <<5=7and it displacesequilibrium of typ eH for5=7 < <19=25.

When the ENC displaces equilibrium H, eciency clearly increases, for

tworeasons. First,thereisap ositiv eprobabilit ythatanagreemen tisreac hed

with a typ e M buyer in the rst p eriod. Second, there is larger probabilit y

thatanagreemen twithatyp eHbuyerisreac hedintherstp eriod. Another

waytoconrmthat expectedso cial welfare(i.e.,the sumofexpectedseller's

and buyer's surpluses) is increased is to note that in the ENC the buyer

obtains a p ositiv e expected surplus, and the seller obtains a larger prot

than inthedisplaced equilibrium H.Thus noisycomm unicationneednot b e

equilibrium L,anagreemen t isalwaysreac hed. Moreov er,abuyer oftyp e H

will alwaysreac hanagreemen tin therst p eriod. Thesetwo eectstend to

mak eequilibriumLmoreecien tthantheENC.However,intheENCthere

is ap ositiv eprobabilit ythat atyp e M buyer will reac han agreemen tinthe

rst p eriod,whic hcannothapp en inequilibrium L. Inour example, one can

easilyv erifythatequilibriumL issup eriortothe ENConeciencygrounds.

Tosumup,wehav eshownthattherearecircumstanceswherenoisy

com-m unication,inadditiontob eing protabletothe seller,also leadstogreater

so cial welfare. In fact, ev enthe buyer ma y gain from an ENC displacing a

typ e-Hequilibrium,inwhic hcasetheENCParetodominatestheequilibrium

without noisycomm unication.

8 Concluding remarks

In this pap er wehav eshown thata seller mayndit protableto in tro duce

some frictions in tothe comm unicationmechanismina bargaining situation.

Thereasonisthatthisallowshimtobuysomecommitmen t,whic hisvaluable

in bargaining.

Our analysisassumesthattheseller mak esalloers, andalso decidesthe

comm unication structure. In the bargaining literature, models have b een

studied wherethe buyer playsamore activ erole, forexample making

coun-teroers. Inthepresen tframew ork,allowingthebuyertoplaymoreactiv ely

wouldlead tothe question ofwhether hecould also gain frommanipulating

the comm unication mec hanism. For instance, the buyer ma y wish to use

noisycomm unicationhimself, orhe maytry tomak eit ineectiv ethe use of

noisy comm unicationon the part of the seller. This op ens man yin teresting

Proof of Proposition1. Supp ose to the contrarythat p 1 >p 2 and > 0 at the equilibrium. If p 1 > p 2

, then the probabilit y of anagreemen t at date 2

is [ 10F(v 1 )]+F(v 1 )0F(p 2 ) where v 1

is giv en by (2.4). Consider now

a strategy where  0 = 0 and p 1 is increased to p 0 1

, with a corresp onding

value v 0

1

, in such a way that [ 10F(v

1 )]+F(v 1 ) = F(v 0 1 ). Clearly, the c hoiceof p 2

inthe second p eriod will not c hange,and hence the probabilit y

of an agreemen t in the second p eriod will b e unaltered. Since the ov erall

probabilit yof agreemen t 10F(p

2

) also remains unchanged,the probabilit y

ofan agreemen tin the rstp eriod isunaected. Butp

1

isincreased, sothat

expected prots mustb e higher. 2

ProofofProposition2. Ourc haracterizationofENC'sinsection3implies

immediately that there can b eno noisycomm unicationif the optimalstatic

price is v

L

. Thus supp ose that v

H x

H > v

L

, so that the optimal static price

is v

H

. Prop osition 1 and Corollary 1 imply that at any ENC it must b e

p 2 =v H and p 1 < v H

. In this context the latter inequality implies p

1 =v

L ,

sinceloweringthepriceb elo wv

L

;aswellasoeringanypricestrictlyb et ween

v

H and v

L

,is clearly sub optimal.

The expected protat the candidate ENC isthen

 NC =(10)v L +v H x H : (13)

Nownotice that when p

1 =v L , setting p 2 =v H

is optimal independen tly of

the value of  b ecause v

H

is by assumption the optimal static price. Since

thec hoiceofisunconstrained and NC

islinearin;atthe optim umeither

=0(implyingthatthereisnonoisycomm unication),or=1(ifv

H x H > v L

). However,setting =1 eectiv elymeans that bargaining isdeferred to

the secondp eriod. Theseller'sexpectedprot withthis"deferring" strategy

is x

H v

H .

But the Fudenb erg and Tirole(1983)strategy (iii) describedin section4

ab ov e dominates this deferring strategy. Indeed, let p

1 = p 2 = v H : A high

valuationconsumerwillthenb eindieren tb et weenacceptingthe rstorthe

probabilities10y and y resp ectively,suchthat yx H v H =(x L +yx H )v L : (14)

This implies that in the second p eriod the seller will still nd it optimal to

oer v

H

. The prot at this equilibrium is (10y)x

H v H +yx H v H >v H x H :

Hence noisy comm unicationis not optimalin this case. 2

Proof of Lemma 1. When p

1 = v

L

, the optimal second p eriod pricing

strategy is independen t of  since the value of  do es not alter the seller's

second p eriod b eliefs. It follo ws that in the case of noisy comm unication

expected prot is linear in , so that either  = 0 or  = 1 at the

opti-m uopti-m. Thusnoisy comm unicationcan o ccur only if it is optimalto deferall

agreemen ts to the second p eriod. However, deferring all agreemen ts to the

secondp eriodisdominated(for >0)bythestrategy ofoering v

H

inb oth

p eriods, with a p ositiv e probabilit yof an agreemen t in the rst p eriod, lik e

in the pro of of Prop osition 2 ab ov e.2

ProofofProposition4. Considerthefollo wingsubgamep erfectequilibria,

whic hcorresp ondtothoseconsideredbyFudenb ergandTirole(1983)intheir

analysis of the twotyp e case:

(i) p 1 =(10)v H +v M ; p 2 =v M ; (ii) p 1 =p 2 =v H :

A t equilibrium (i), a high valuation consumer accepts the rst p eriod

oer and a medium valuation consumer accepts the second p eriod oer. A t

equilibrium (ii),onlyhighvaluationconsumers acceptsthe seller'soersand

they are indieren t b et ween reac hing an agreemen t in the rst or in the

second p eriod. Inthe resulting mixedstrategy equilibrium, ahigh valuation

consumerrandomizes: heacceptstherstp eriodoerwith probabilit y10y

andwaitforthesecondp eriodoerwithprobabilit y y,whereyisdetermined

so as to mak e the seller indieren t b et ween setting p

2 = v H and p 2 = v M : This requires yv H x H =v M (x M +yx H ) or y= x M v M x H (v H 0v M ) (15) wherey 1bycondition v H x H >v M (x H +x M

); whic hm usthold byProp

M M L L M

acceptstherst p eriodoerwithprobabilit y1itisoptimaltopriceatv

M in

the second p eriod (thus aiming at reac hing an agreemen t with the medium

valuation consumer only), equilibrium (i) is clearly subgame p erfect. To

c heck that equilibrium (ii) is also subgame p erfect, it remains to b e shown

that in p eriod2 itis not optimal tosetp

2 =v L , that is: yv H x H  v L (x L +x M +yx H ) (16)

This follo ws easily fromthe condition v

H x H > v M (x H +x M

). Thus (i)and

(ii), withygiv enby(7.3),arecandidate subgamep erfectequilibriaandthey

do notin volvenoisycomm unication.

In the mixed strategy equilibrium (ii) expectedprot is:

 ( ii) = " 10(10) x L v L x H (v H 0v L ) # x H v H : (17)

whereas atequilibrium (i) expected prot is:

 ( i) =(10)v H x H +v L : (18)

By Corollary 2, on the other hand, we know that at any ENC p

1 = v M and p 2 =v H

. It follo ws that atthe candidate ENC expected prot is:

 NC =(10)v M +x H v H : (19)

Nowit iseasy toc heck that:

max [  ( i) ; ( ii) ]> R ; (20)

forall<1andfor all>0. Thismeansthat noisycomm unicationcannot

b e used atequilibrium. 2

Proofof Corollary 2. ByProp osition 2andCorollary1,p

1

mustb elower

thantheoptimalstaticprice,whereasbyLemma1itm ustb ehigherthanv

L .

Now supp ose the optimal static price is v

M . It follo ws that v L < p 1 < v M ,

but this isclearly dominated byp

1 =v

M .2

Proof of Proposition5. Clearly, in the second p eriod the optimal price

oer m ustb e one of fv

H ;v

M ;v

L

M,andequilibriumH,accordingtowhetherthesecondp eriodpriceisv L ;v M or v H

, resp ectiv ely.

Equilibrium H

Forp

2 =v

H

tob e part of asubgame p erfectequilibrium, the rst p eriod

oer cannot b e lower than v

H

. The probabilit y y that a high valuation

buyer defers agreemen t to the second p eriod m ust b e such that pricing at

v

H

in the second p eriod is optimal to the seller. This requires y H v H x H  v M (x M +y H x H ),sothatpricingatv M isdominated,andy H v H x H v L (x L + x M +y H x H ),sothatpricingatv L

isdominatedaswell. Bothconditionsmust

hold and this implies

y H =max[ v M x M x H (v H 0v M ) ; v L (x M +x L ) x H (v H 0v L ) ] :

Exp ected prot is:

 H = h 10(10)y H i h: Equilibrium M Forp 2 = v M

to b e part of a subgame p erfect equilibrium;it is necessary

that p

1 6= v

M

. The reason is as follo ws. If p

1 = p 2 = v M , then a high

valuation buyer will strictly prefer to accept the rst p eriod oer, whereas

a medium valuation buyer will b e indieren tb et ween accepting the rst or

the second p eriod oer. However, ev en if a medium valuation buyer waits

for the second p eriod with probabilit y1, by condition A2 it would then b e

optimal to set p

2 = v

L

. On the other hand, setting p

1

at a lev el such that

a high valuation buyer strictly prefers to wait for the second p eriod oer

implies that in the second p eriod p

2 = v

H

is the optimal price. Thus the

only p ossibilit yis that of setting p

1

atthe highestlev elwhic h mak es a high

valuation buyer willing to accept the rst p eriod oer instead of waiting,

i.e. p 1 = (10)v H +v M

. Being indieren t as to when to accept, a high

valuation buyer atequilibrium randomizes and accepts the rst p eriod oer

with a probabilit y(10y M ) such that: v M (x M +y M x H )=v L (x L +x M +y M x H ) (21)

that is, such that in the second p eriod the seller do es not nd it convenient

y M = v L (x L +x M )0v M x M x H (v M 0v L ) : (22) Note that y M

 0 by A1. It m ust also b e y M

1, that is ` <m , whic h

is satisedbyA1. Exp ected prot is:

 M =(10)(10y M )h+m: (23) Equilibrium L When p 2 = v L

, an agreemen t is always reac hed. Clearly, p

2 = v

L can

b e part of a subgame p erfectequilibrium only if the rst p eriod price takes

one of the follo wing values: a) p

1 = v L ; b) p 1 = (10)v H +v L , c) p 1 = (10)v M +v L . With p 1 = v L

the agreemen t is reac hed immediately and

prot is `. Withp 1 =(10)v M +v L

b oth a high and a medium valuation

buyer will accept the rst p eriod oer and prot is (10)m+`: Finally,

setting p 1 = (10)v H +v L

implies that only a high valuation buyer will

accept the rst p eriod oer and yields prot (10)h+`. By condition

A1, it follo ws immediately that this latter equilibrium dominates the other

ones. 2

Proofof Proposition6. Inequality(6.9) impliesthatthereisanonempt y

in tervalof values of  where NC > L and  NC > H

. Thus we m ustshow

that NC

> M

overatleastasubsetofthisin terval. Toshowthis,itsuces

to compare the values of  NC

and  M

at the upp er b ound of that in terval,

i.e. at p oint   = (10)m~ 0(10y H )h h(y H 0)~ : (24)

A tthat p oint, by construction NC

= H

: Thus,we m ust showthat

(10y H )h+  y H h>(10  )(10y M )h+  m; (25) that is (y M 0y H )h>   h (m0h+h(y M 0y H ) i : (26) Substituting, weget (y H 0)(y~ M 0y H )h 2 > h (10)m~ 0(10y H )h ih (m0h+h(y M 0y H ) i : (27)

(y H 0)(y~ M 0 y H )h 2 > h (10)h~ 0(10y H )h ih (m0h+h(y M 0y H ) i (28)

whic his alwaystrue giv enA1. 2

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Values for  are rep orted on the horizontal axes, values for prots on the

v erticalaxes. The decreasing function is L

, the prot under equilibrium L;

theincreasingfunctionwithhighestvalueequalto200is H

;theprotunder

equilibriumH;theincreasingfunctionwhic hstartsab ov e H

,andterminates

justb elo wit,is NC

, theprotwithnoisycomm unication. Thevalueof M

,

the prot under equilibrium of typ e M do es not exceed that of  NC

for any