Günther Lang
Universidade Novade Lisboa
Faculdade de Economia
rst version: April2000
this version: December2000
Abstract
Weanalyzeasimplemodelofbilateralbargainingunder
asymmet-ricinformationwhere theseller ofanobjectcannotsimplysay"no"
by default to a buyer who is supposed to make a take-it-or-leave-it
oer. Rather,hemustacquirethisoptionbeforetheactualbargaining
processbegins. Thischoiceis observableto thebuyer,andhence,the
seller'sprebargainingactionmightsignalprivateinformation. We
de-velop acomplete characterization of Perfect BayesianEquilibriumin
pureand(strictly)mixed strategiesforthisgame. Thenthemodelis
comparedtoastandardbargainingsettingin termsof therealization
ofwelfareenhancingpropertyrightchanges.
KEYWORDS:Bargaining,Signalling.
JEL-Classi cation: D23,D82,L14
Author'saddress: GüntherLang,UniversidadeNovadeLisboa,Faculdadede
E cient tradeamong economic agents requiresa well-established
property-rights system that protects initial endowments as well as produced goods.
Without such an institution, incentives to exert eort are distorted as the
total(marginal)gainsfromindividualperformancedonotfallexclusivelyto
theindividualhimself. Itisfrequentlyarguedthattheprotectionofproperty
rightshaspublic goodcharacter,and indeed, policeprotectionand criminal
investigationareprovided on acollective base.
However,morebasicinstrumentsofprotectionagainstburglaryandtheft
areleft to theindividualor local level: door locksfrom family homes to
in-dustrialcomplexesare nancedbytheirowners,electronicvideosurveillance
systems and private security in condominiums and factories are also
orga-nizedonanindividualbase, rms withinternet presencespendconsiderable
amountsof money intheelectronic protection ofsensitivedata thatisonly
for internal use. Clearly, one can argue that these instruments are rather
goods of private bene t, or at most public goods on arestricted local level.
Nevertheless, theremay well existsigni cantscale eects inorderto justify
joint provisionof these goods andservices.
On the other hand, the magnitude of investment undertaken by an
in-dividual party in the protection of its propertyrights in a certain object
may reveal (or conceal) signi cant information as to the valuation of the
objectbythisparty. Expensesmadeonprotectionmaythereforebeofvalue
intrade relationships under asymmetric information asa signalling device:
high valuation types of sellers could use this instrument for the purpose
of distinguishing themselves from low valuationtypes, inducing potentially
higher bids from buyers who are supposed to make a ( rst) oer. Hence,
welfareenhancing exchanges in propertyrights may be promoted as
com-paredtoasettingwherescaleeectswouldjustifyjointsupplyofprotective
measures.
We will formally analyze the eect of introducing a cost for having the
option to say "no" into a traditional model of bilateral bargaining where a
buyermakesatake-it-or-leave-it proposalto aseller(seeFudenbergand
Ti-role(1983),and(1991),Ch. 10,Sec. 2). Thefollowingsectiondescribesthe
model, subsequently allequilibria of thegame inpureand mixed strategies
areexamined.
The Model
We consider a traditional one-shotsingle-oer bargaininggame with
asym-metric information, with the extension that the seller can only veto the
transferof the objectin hispossessionifhe has made aninitial investment
invest-(bargaining) game.
Weassumethatforbuyer(A)andseller(B)therearetwopossibletypes,
respectively: v2fv;vgforA,witha-prioridistributionp(v);p(v)+p(v)=1;
w2fw;wgforB,witha-prioridistributionq(w);q(w)+q(w)=1. Thetypes
for both of the players stem from an independent draw. We will examine
thetwo relevant cases: the no-gap case with 0 w <v <w < v, and the
gap-case with0w<wv <v. Concerning the amount k necessaryfor
thesellertoinvestinorderto beabletodeclinethebuyer'soer,weassume
thatitis xedwith0w<k <w.
The timingof the gameis asfollows:
(i) Naturedecides independently,according to thedistributions p(v)and
q(w),thetypesofboth playersandexclusivelyrevealstoeach ofthem
hisowntype.
(ii) SellerBdecideswhetheror not to investk. Ifhe doesnotinvestthen
theobjectchangeshandswitha payoof0for thesellerandv forthe
buyer. Otherwise, thegamecontinues withstep(iii).
(iii) BuyerA makesanoer c to B.
(iv) Seller B decideswhether or not to accept this oer. If he does, then
theobjectchangeshands,withpayos ofcforBandv cfor A.Ifhe
does not, Bstays withthe objectand a payoof w, whereas A earns
0.
Onecouldequallywellimagineasettingwheretheseller rstinvestsand
then makes an oer to the buyer. This problem then is only of unilateral
asymetric information and trivialto solve: the sellerinvests ihis maximal
expected payo over all his bids exceeds the cost k. In thegame analyzed
here,however,atruesignallingproblemarises,inthesensethatB'sdecision
to investmayreveal orhide information abouthis type.
WedenotebyÆ
k
(w)2f0;1gB'sbehavorialstrategywhenaskedwhether
to invest (1) or not (0). This decision may be conditioned on B's type.
Analoguously, we denote by c(v) A's bid-strategy. By a(cjw) 2 f0;1g we
designate B's acceptance (1) or refusal (0) of A's oer. This strategy as
well maydependon the history ofthe game revealed to B.The equilibrium
concept applied is that of perfect Bayesian equilibrium (see Fudenbergand
Tirole (1991),Ch. 8).
Asapreliminaryresult,itisclear thatB'stypewalwaysinvests in
equi-libriumbecausehisvaluationoftheobjectexceedsthecostoftheinvestment,
k<w. However, type w mayinvest aswell, although hisvaluationis lower
than costs: in doing sohe avoids revealing his type, which tendentially
in-creasestheexpectedoermadebythebuyerascomparedto thecasewhere
well outmatch joint supply, we assume an alternative setting withn sellers
(owners) where public provision causes costsof G, independently of n, and
whereG=n<k(soscaleeectsaresupposedtohaveturnedpublicprovision
into the less costly alternative). This public system is mandatory for all
sellers,withequalcostsharing,andonceitisimplementedsellersandbuyers
playthe standard gamewith thesellers' default optionto refuse an oer.
No-Gap Case: w < v < w < v
Inwhat follows wewill rst analyzeexistenceof equilibriuminpure
strate-gies. Thenwe lookat equilibria in(strictly) mixedstrategies.
Theorem 1 (SeparatingEquilibrium) Separating equilibria doexist i
ww~ :=
k p(v)w
p(v)
: (1)
For all w w,~ the following strategies constitute a separating equilibrium:
Æ k (w)=0;Æ k (w)=1,c(v)=w, c(v)=w, and a(cjw)= 1 for cw 0 for c<w (2)
forallcandw,withbeliefsatout-of-equilibrium-pathinformationsetschosen
arbitrarily.
Proof: Out-of-equilibrium-path information sets can only be found at the
stagewhereBhastodecideacceptanceorrefusalofA'soer. There,aswell
asat information sets which areon theequilibrium path at this stage,it is
best for B to accept an oer thatmatches at least w,independently of B's
beliefs about A's type;hencea() asgiven aboveis optimal.
Given that Æ
k
=1 isobserved,Bayes' rule requiresA to believe inB'stype
w. A's type v then just oers w, knowing that this oer will be accepted.
A's type v could induce w's acceptance only by incurring a loss. Since
w w~ < w, he avoids just this. Therefore, c(v) = w and c(v) = w,
respectively,areoptimalforthebeliefsimpliedbytheequilibriumstrategies.
B'stype walways invests: Withoutdoing sohis payo wouldbe zerosince
hecouldnotrecuse A'sproposal,but having investedoerstheopportunity
tosaynoandguarantees himapayoofw k >0. w w~ (asgivenin(1))
impliesthatp(v)w+p(v)w k0,andsow nds itoptimalnot to invest.
Onthe other hand,if (1)is not ful lled, i.e. w>w,~ thenw strictly prefers
to investfor anyw>0 since his reservationpayois wanyway. Hence, (1)
Relationship(1)is equivalent top(v)
w w
. Theinterpretationfor the
existenceofaseparating equilibriumisthatsincetheprobabilityofmeeting
A'stypevisrelativelylowandthistypeistheonlywhopotentiallyoers
w B'stypew doesnot ndit worthwhile to invest.
Theorem 2 (Pooling Equilibrium) Apooling equilibrium does exist i
ww~:= k p(v)w p(v) (3) and ww^ :=v v w q(w) : (4)
The following strategies constitute this equilibrium: Æ
k
(w) = 1;Æ
k
(w) = 1,
c(v)=w, c(v)=w, anda(cjw) as in (2). Beliefs at out-of-equilibrium-path
informationsets can be chosen arbitrarily.
Proof: Out-of-equilibrium-path information sets canbe found atthestage
where B has to decide acceptance or refusal of A's oer. By the same
ar-gument as in the proof of Theorem 1, a() is optimal o as well as on the
equilibriumpath.
Wehavec(v)=w,sinceoeringwwouldleaveA'stypevwithacertainloss.
Oeringw,however, attractsB'stype w,and onlythistype. Thisevent
oc-curswithprobabilityq(w),leaving typev thereforewithan expectedpayo
ofq(w)(v w)>0.
A's type v oersc(v) =w i v w q(w)(v w), which isequivalent to
(4). Otherwise he wouldoer only w, inwhich case both typesof A would
bidonly thelowvaluation. Then, however, w would not invest. Hence, (4)
isnecessaryfor theexistenceof apoolingequilibrium.
B'stypewinvests,bythesame(obvious)argumentasinTheorem1. Whether
w invests, given that (4) is ful lled, depends: he invests i p(v)(w k)+
p(v)(w k)0,which isequivalent to (3).
Hence(3)and (4)arenecessaryand su cient for theexistenceof apooling
equilibrium.
In a pooling equilibrium, A's type v must nd it worthwhile to bid w.
This is the case if the probability of meeting B's type w is not too high;
indeed (4) is equivalent to q(w) v w
v w
. B's type w then invests aswell if
theprobabilityofmeetingA'stypev issu ciently high,whichisequivalent
to (3).
Existence ofeithertypeofequilibriumthereforedependsonthe
relation-shipbetween w^ andw:~
^
ww~ : Separating equilibria do exist i w w.~ Pooling equilibria do
existi ww.~ For w=w~ separating and poolingequilibria coexist
i w w.^ For w~ <w < w^ neither pooling nor separating equilibria
doexist: a)IfA'stypevwouldoerw,thenB'stypewwhere induced
toinvest aswell. However,w'sfrequency istoohighinorder to make
itworthwhile for v to do so, and b) the frequency of A's type v, who
ina separating equilibrium is supposed to oer w, istoo high for B's
typew not to invest.
In the following we consider all cases of equilibria in (strictly) mixed
strategies that exist inaddition to thepure-strategy equilibria already
dis-cussed.
Theorem 3 (Mixed-Strategy Equilibria) In theno-gapcasethefol
low-ingequilibriain (strictly) mixed strategies doexist:
(i) For
w<w~ :=
k p(v)w
p(v)
(5)
the only mixed-strategy equilibria consist of the (pure) strategies from
the separating equilibrium of Theorem 1, with the exception that A's
type v mixes hisbid c(v) on the interval [0;w].~
(ii) For
w=w~ :=
k p(v)w
p(v)
(6)
the following strategies constitute the set of all hybridequilibria:
(prob(Æ k (w) =1) = ;prob (Æ k (w) =0) =1 );Æ k (w) =1, c(v) =w
(for =0anymixed strategyon [0;w]~ asin(i)),c(v)=w,anda(cjw)
as in (2), with min(
;1), where
> 0 is the (unique) solution
to q(w) q(w)+q(w) (v w)=v w: (7) (iii) For w>w~ := k p(v)w p(v) (8) then:
a) Forq(w)(v w)=v w(i.e. w=w),^ thepoolingequilibriumfrom
Theorem 2 survives with the modi cation that A's type v mixes:
(prob(c(v) = w) = ;prob(c(v) = w) = 1 ), with
,
where
2(0;1) is the (unique)solution to
p(v)(w k)+p(v)[
(w k)+(1
constitute theset of all hybrid equilibria: (prob(Æ k (w) = 1) = ;prob (Æ k (w) = 0) = 1 );Æ k (w) = 1, c(v) =w, (prob (c(v) =w) = ;prob (c(v) = w) =1 ), and a(cjw) as in (2), with and
as de ned in (ii) and (iii) a),
respectively.
Inallcases, beliefsatout-of-equilibrium-path informationsets can be chosen
arbitrarily.
Proof: (i) From Theorem 1 we know that B'stype w nds it optimal not
to invest for any c(v) w.~ Since w~ <w, B'stype w rejects such an oer
anyway. Mixing onthe interval [0;w]~ thenhasthesame eect. The
mixed-strategy equilibriumispayo-equivalent tothe separatingequilibrium.
It remains to showthat there are no other mixed-strategy equilibria. First
of all, in order to guarantee existence of a best response it is required to
assume that, on the equilibrium path, both types of B accept A's oer if
indierent between acceptingand rejecting. Then (5)implies thatw would
not investeven ifA'stypev oered w,and,on theother hand,ifhe invests
thenA's type v would never oer more than w. Therefore, wneverinvests
inequilibrium,andsoonlytheseparatingequilibriumfromTheorem1,with
mixing asjustdescribed,survives.
(ii) Given that (6) is satis ed, B's type w is indierent between investing
and not investing, provided that c(v) = w and c(v) = w; and therefore
he may mix. The lhs of (7) gives the expected payo of A's type v if he
bids w, whereby the rst term expresses the Bayesian updated probability
oftypewgiven thistype'smixedstrategy. Therhsof(7)istypev'scertain
payoifbidding w. There triviallyexistsaunique solution
to (7)thatis
positive butnot necessarilybelowor equalto one. IfB'stypew mixeswith
probability min(
;1) then the lhsof (7)is lowerthan or equal to the
rhs, and so it is optimal for type v to oer c(v) =w. If A's type v mixes
on[0;w]~ then =0isoptimalforB'stypew,andmoreover,ifanystrategy
from [0;w)~ is played with strictly positive probability then only = 0 is
optimal.
(iii)a)andb)Giventhat(8)holds,B'stypewisindierentbetweeninvesting
andnotinvestingif(9)issatis ed,andhestrictly prefersto investifthelhs
of(9) is strictly positive.
=
p(v)(w k)+p(v)(w k)
p(v)(w w )
solves(9) andlies inthe
interval(0;1),whereas <
impliesthatthelhsisstrictlypositive. Hence,
ifB'stype winvest withprobabilityone,thenq(w)(v w)=v w implies
thatA'stypevisindierentbetweenbiddingwandw. Ifq(w)(v w)>v w
thenthe
thatsolves(7) guarantees thisindierence.
With the same argument as in Theorems 1 and 2, beliefs at the
ratingnorpoolingequilibriadoexist,atleastequilibriuminmixedstrategies
can be guaranteedbypart(iii) b) ofthetheorem.
Finally,wecomparetheexpectede ciencylossesduetoassymetric
infor-mationinthepresentmodeltothoseinthejointprovisionscenariowiththe
guaranteed option to say "no" after the mandatory perhead share G=n is
paid(andafterwhichsellerandbuyerplaythestandardmodel). Ine ciency
inboth models arises incase thata bene cialtrade between A and Bdoes
notoccur. Fortheno-gapcase,inequilibrium,tradebetweenB'stypewand
anyofA'stypesalwaysisbene cialandinfacttakesplace. However, inthe
standardmodel,B'stypewcannotrealizethebene cialtradewithA'stype
v ifthe probability of B's type w is too high, in particular ifq(w) > v w
v w .
Inthe modeldiscussed here,this caseisconsistent only withtheseparating
equilibrium and the mixed-strategy equilibria (i) and (ii). In any of these
equilibria,allwelfareincreasing transfersinproperty-rightsdotake place
although not always voluntarily but social cost consist of B's type w's
and/orw's investment k. Therefore, expected e ciency losses are q(w)k in
theseparating equilibrium, aswell asinthemixed-strategy equilibrium(i),
and they amount to [q(w) +q(w)]k in case of mixed-strategy equilibrium
(ii). Inthe standard modelthey areG=n+p(v)q(w)(v w). Hence, having
investment in property-rights protection as a signalling device pays o i
k < G
nq(w )
+p(v)(v w) inthe rst two cases, and k <
G=n+p(v)q(w)(v w )
q(w) +q(w) in
the last case. On theother hand, ifq(w) < v w
v w
, all e cient trades in the
standard model take place, inthe modeldiscussed here, however, there are
always social costsintermsofat leasttype w'sinvestment k.
Gap Case: w < w v < v
Theorem 4 (Pure-Strategy Equilibria) In the gap case
a) no separating equilibria do exist,and
b) a pooling equilibrium does exist i
(i) ww(v),^ where ^ w(v):=v v w q(w) ; (10) or
(ii) (w(v)^ w<w(v)^ andww~ (w~ given by (1))).
The following strategies constitute this equilibrium: Æ
k
(w) = 1;Æ
k
(w) = 1,
c(v) = w (case (i)), c(v) =w (case (ii)), c(v) =w, and a(cjw) as in (2).
and w(v)^ < w. Note that w(v)^ < w(v).^ Then, if w < w(v),^ none of A's
typesoersw. Hence, B'stypew doesnot invest.
Ifw(v)^ w<w(v),^ thenc(v)=w andc(v)=w. Alsotype w then nds it
optimalto invest i ww.~
If w w(v),^ then both types of A bid the high valuation w. In this case,
both typesof B ndit optimalto invest.
By the same argument as above, beliefs at out-of-equilibrium information
sets where B has to decide acceptance or refusal are irrelevant, and a() is
optimal.
Also inthe gap case we now investigate equilibrium in (strictly) mixed
strategies.
Theorem 5 (Mixed-Strategy Equilibria) In the gap case the following
equilibria in (strictly) mixed strategies doexist:
(i) For w=w(v):^ Æ k (w)=1, Æ k (w)=1, prob (c(v)=w)= ;prob(c(v)= w)=1 ), with min( ;1), c(v)=w,and a(cjw) as in (2). (ii) For w(v)^ <w<w(v):^ a) Forw=w:~ (prob(Æ k (w)=1)= ;prob(Æ k (w)=0)=1 ),with 1> (v), c(v)=w, c(v)=w, anda(cjw) as in (2). b) For w < w:~ (prob (Æ k (w) = 1) = (v);prob(Æ k (w) = 0) = 1 (v)),(prob(c(v)=w)= ;prob (c(v)=w)=1 ),c(v)=w, anda(cjw) as in (2). (iii) For w=w(v):^ a) For w >w:~ Æ k (w) =1, Æ k (w)=1, c(v) =w, (prob(c(v)=w) = ;prob (c(v)=w)=1 ),with , and a(cjw) as in(2). b) Forw=w:~ (prob(Æ k (w)=1)= ;prob(Æ k (w)=0)=1 ),with 1> (v), c(v)=w, c(v)=w, anda(cjw) as in (2). c) For w < w:~ (prob (Æ k (w) = 1) = (v);prob(Æ k (w) = 0) = 1 (v)),(prob(c(v)=w)= ;prob (c(v)=w)=1 ),c(v)=w, anda(cjw) as in (2). iv) For w<w(v)^ <w(v):^ a) For w > w:~ (prob (Æ k (w) = 1) = (v);prob(Æ k (w) = 0) = 1 (v)),c(v)=w,(prob (c(v)=w)= ;prob (c(v)=w)=1 ), anda(cjw) as in (2). b) Forw=w:~ (prob(Æ k (w)=1)= ;prob(Æ k (w)=0)=1 ),with (v) (v), c(v)=w, c(v)=w, and a(cjw) asin (2).
c) For w < w:~ (prob (Æ k (w) = 1) = (v);prob(Æ k (w) = 0) = 1 (v)),(prob(c(v)=w)= ;prob (c(v)=w)=1 ),c(v)=w, anda(cjw) as in (2). Thereby,
(v) is the (unique)solution to
(v)q(w) (v)q(w)+q(w) (v w)=v w; (11) as given in (9), and
is the (unique)solution to
p(v)[
(w k)+(1
)(w k)]+p(v)(w k)=0: (12)
Inallcases, beliefsatout-of-equilibrium-path informationsets can be chosen
arbitrarily.
Proof: GivenA'stypev2fv;vg,heisindierentbetweenoeringwandw
i(11)holds. There isaunique solutionto (11), and <
(v)( >
(v))
impliesthat type v strictly prefers bidding w (w). Also,
(v) <
(v), i.e.
A'shigh-valuationtypeviswillingtooerwatahigherprobabilityoffacing
B'stypew ascompared to hislow-valuation typev.
asgiven in(9) marksB'stype w'sindierence between investing or not,
given that A's type v oersw and type v mixes on fw;wg. Consequently,
typew strictly preferstoinvest(notto invest) i < ( > ). Analoguously,
as given in (12) marks B'stype w's indierence between
investingornot, given thatA'stypev oerswandtype v mixesonfw;wg.
Notethat
isstrictlypositiveanditmaybegreaterthanone. Typewthen
strictlyprefersto invest (notto invest) i <min(
;1) ( >min(
,1)).
Therefore, if w > w(v),^ both types of A oer w even if both types of B
invest, and sothereonly existsthe pooling equilibriumfromTheorem 4.
Withthis knowledgewe can now prove claims(i) to (iv).
(i)w=w(v)^ impliesthatA'stypevoerswandtypevisindierentbetween
vandv,giventhatB'stypespool. Iftypev thenmixeswith min(
;1)
itisoptimalforB'stypewto invest. Onthe otherhand,ifthelatterwould
invest with probability lower than one, then both of A's types should oer
w,inducing himto invest withprobabilityone.
(ii) a) Given B's type w's mixing with 1 >
(v), using (11) we see
that c(v) =w and c(v) =w are indeed optimal. Since w = w,~ B'stype w
thenindeedisindierent. Onthe otherhand,ifthelatterwouldinvestwith
probabilitylowerthan
(v),thenbothofA'stypesshouldoerw,inducing
himto invest withprobabilityone.
(ii)b) Given B'stype w's mixing with
(v), c(v)=w is uniquely optimal,
whereasA'stypevisindierent. Giventhatthelattermixeswith
,which
isindeedlowerthanone becauseofw<w,~ B'stypewisindierentbetween
investing or not. If type w mixed with <
would oer w, inducing pooling. If he used > (v), then c(v) = w and
c(v)=wwereoptimal,inducing type wnot to invest because of w<w.~
(iii)a)GiventhatbothtypesofBpool,w=w(v)^ impliesthattypevstrictly
prefersoeringw,whereastypevisindierent. By(9),B'stypewthen nds
it optimal to invest for
. If type w did not invest with probability
one, then the optimal strategies for A's types would satisfy c(v) w and
c(v)=w,inducing him toinvest becauseof w>w.~
(iii)b)For1>
(v),c(v)=wandc(v)=wareoptimal. Sincew=w,~
type wis indierent between investing and not doing so. =1 impliesthe
pure-strategy pooling equilibrium from Theorem 4. <
(v) wouldimply
c(v)=c(v)=w,making ituniquely optimalfor type wto invest.
(iii) c) Given w's mixing with
(v), type v is indierent, whereas type v
strictlyprefersoeringw. Theformer'smixingwith
thenleavesB'stype
windierent whethertoinvestornot. For <
(v),bothtypesofAwould
oer w, inducing pooling of B's types. >
(v) then makes c(v) = w
uniquely optimal,and because ofw<w,~ type w wouldnot invest.
(iv)a)Givenw'smixing with
(v),c(v)=wisunique optimalchoiceof v,
whereas v is indierent. Hence, he may mix with
. This, in turn, leaves
type w indierent between investing and not investing. <
(v) implies
c(v)= w or w and c(v) =w, making type w invest because of w> w.~ On
the other hand, >
(v) implies c(v) = c(v) =w, sothat type w would
not invest.
(iv) b) For 2 [
(v);
(v)], c(v) = w and c(v) = w are optimal, and
because of w =w,~ B's type w then is indierent between investing or not.
<
(v) implies c(v) = c(v) = w, making it worthwhile for type w to
invest. >
(v) induces c(v) = c(v) = w, which makes type w not to
invest.
(iv) c) Given that B's type w mixes with
(v), A's type v is indierent
between oeringw andw,whereas type v strictlyprefersbidding w. Given
theformer'smixingwith
,B'stypewisjustindierent. <
(v)implies
c(v)=c(v)=w,which makes itworthwhile for typew to invest. >
(v)
implies c(v) =w and c(v) w, which makes it unattractive for type w to
investbecause ofw<w.~
Finally, a(cjw) as in (2) is optimal for both of B's types, independently of
beliefs heldaboutA's type.
Also inthegap-casewe examine underwhich conditions theinvestment
in property rights protection as a signalling device can improve welfare as
compared to the joint supply/ standard model. In the latter, bene cial
trade now may not take place between w and either v or v, depending on
whetherq(w)>h(v):= v w
v w
,i.e. onwhetherB'stypew'sprobabilityistoo
in the standard model, causing social costs of G=n+q(w)[p(v)(v w)+
p(v)(v w)]. Inoursignallingmodel,thiscaseisconsistentonlywith
mixed-strategyequilibriumiv). Giventhecostofinvestment k,pluscostsinterms
ofnon-realizedbene cialtrades,insub-cases a) c)of iv)thepossibilityof
signallingprovideshigher welfare ascompared to thestandard model i
k< 8 > > > > > > < > > > > > > : q(w)p(v)(1 )(v w)+G=n (v)q(w)+q(w) sub-case a) q(w)p(v)(v w)+G=n q(w)+q(w) sub-case b) q(w)[p(v)(v w)+p(v)(v w)(1 )]+G=n (v)q(w)+q(w) sub-case c) (13)
h(v)<q(w )<h(v): Inthiscase,thestandardmodelonlyprecludestrade
between wand v,causing socialcosts ofp(v)q(w)(v w)+G=n. This
con-stellationisconsistentonlywithpoolingequilibrium(ii)andmixed-strategy
equilibrium (ii). In the rst case, also the signalling model precludes trade
between w and v, but it causes the cost of investment k. Hence, welfare as
compared to the standard modelis lower since k >G=n. The same applies
formixed-strategyequilibrium(ii)sub-casea). Insub-caseb),thereistrade
between w and v with probability 1
; hence the signalling framework
provideshigher welfare ik <
q(w)p(v)(1 )(v w )+G=n (v)q(w)+q(w) .
q(w )<h(v)<h(v): Alle cient tradestakeplaceinthestandardmodel.
Thiscase correspondsto pooling equilibrium(i),where theyarerealized as
well, at social costs of k, however. Hence, since k > G=n, the signalling
modeldisplays lowersocial e ciency.
Conclusion
We have analyzed a simple model of bargaining, modi ed by the feature
thatthe optionto declineanoermustbeacquiredinadvance. Inthe
sym-metric information case, only highvaluation sellers would buy this option.
However, if the sellers' type is privateinformation, then also lowvaluation
sellersmay undergo this costly investment inorder to avoid revealing their
type. In the nogap case, this just happens if the ex-ante probability of
thelowvaluationtype sellerissu cientlylow, andthatofahighvaluation
typebuyerissu cientlyhigh. Inthegapcase,aseparatingequilibriumdoes
notexistat allbecause bothtypesofbuyersaresupposedto makethesame
(high)oer. Ontheotherhand,alsoapoolingequilibriummayfailtoexistif
equilibriuminmixedstrategies can always be guaranteed.
Comparisonwiththejointsupply/standardbargainingmodelshowsthat
welfare may well increase under some circumstances if one introduces the
relatively more costly instrument of individual propertyrights defence. In
fact, if the ex-ante probability of the lowvaluation seller is too high, then
in the standard model, both types of buyers tend to make the same low
oer, inpeding bene cial trades between highvaluation sellers and buyers.
Nevertheless, ifcostsof signalling,i.e. of buyingtheoptionto say"no",are
nottoohigh,thenthisdevicewillindeedguaranteethatalle cientchanges
inpropertyrights dotakeplace.
References
[1] Fudenberg, D. and Tirole, J. (1983): Sequential bargaining with
incompleteinformation, Review of Economic Studies,50,221247.
[2] Fudenberg, D. and Tirole, J. (1991): Game Theory, MIT Press,
Cambridge (Mass.).
[3] Myerson, R. and Satterthwaite, A. (1983): E cient mechanisms
