- Email: [email protected]
Price as a signal of quality
European Journal of Political Economy 9 (I
ri
Accepted for publrcation July i992
This paper discusses non-cooperative equiiibrium notions that have :ecen n applied in the analysis of signalling games. As an illustration, we analyse the role of as sigxils when consumers are incompletely informed about product quality before purchase. We investigate the incentives of a high-quaiit~ k-m to separate from a lowquaiiiy firm tiirough its pricing srraregy. Starting from the set of sequential equilibria we show that, when stronger restrictions on out-ofequilibrium inferences of the type suggested by Cho and Kreps (1983) are imposed, a unique szporoting equilibrium outcome remains. ??x paper offers a critical assessment of the formal game theore!:ca.l refinements needed to reach this conclusion, and argues that in the class of signaiiing games considered here other outcomes involving pooiing ment atten!ron besause of their economic content.
The objective o f this paper is to discuss the merit of various noncooperative equilibrium concepts that have recently been applied in the TIll___l_-L_-AI.’ discussion, we wi!! set up a -z_-IBZ_- -Kc rl&r&in2 io SqpalZLr!g gmies. r CJ I~!w5’,zz;c:s :m ;n+nrqr+;o~ simple model of goods m~tp+ . ..&S_W.C I..C”LCVC. 1 , where agents on the demand side are incompletely informed about the quality of the good before purchase. To facilitate matters, we assume that the consumers are served by a monepotist firm, whose exogenous product quality is unknown to the potential customers. Hen&, &he firm is privately informed, and this symmetric info mation structure will embody the adverse selection prable needed to set Correspondence to: Per Baker Overgaard, fnstitute of Economics, University of Aarhus, IX-8 Aarhus C, Jktmark. *This paper is a revised version of cit. B in my Ph.D.-thesis [ a%99t)I. Comments from S. Albzk, F. Forges, hl. Hviid. I..-F. Mertens, J.-P. Ponssard, E. va my thesis adviser C. !YAspremont, and two anonymous referees have improved the exposition ese share any responsibility for and are greatly appreciated. Of course, n Academy is grat&lly a~~~ow~~ errors. Financial support from the Danish work by Mzilat finishing the final draft, my attention was alert $-b, &rc &se?‘” ralakr in Fujiwara and Post!ewaite fl992) that pre spirit to section 4 of this paper. 01762680/93/SCt6.00 (%, 199~Elsevier
Science Publishers B.V. Ail rights reserved
P.B. Orwgaard.
484
Price as a signal of quulity
the market interaction as a signailing game. Adverse selection arises since the various types of the monopolist firm associated with dlryerent quality levels may have differing incentives to reveal or conceal their priv??- information. o~~quc~iti~, we study a ‘classical’ signaliing prch;._~- , t”:t:,,t? tk i~itiriiid
~~~t~atesthe game by sending a signai, :-: &it:. 3 t&e utiinntormed party end optimally given ihz i&;r~ukti~n contained in the signal. ative of the tradition that grew from Akerlof’s nce’s ( 1973) signalling-model, and it addresses at of Mllgrom 2nd RGkiiS i i986). it SS w~i~~kn~w~that signalling games, in which any particular equilibrium ts only a few information sets, give rise to an uncomforting of Nash equilibria, and it is on the selection among these that r wiii focus. As a tarting point, the analysis shows that the notions reps and Wilson (1982)] and (itcrated) ciiminastrategies bring us some way, and sometimes ail the way, the mult~~l~c~typroblem. But, we also demonstrate that ere these concepts do not yield a unique equilibrium outcome. In such cases appeal is often made to more refined selection c~~te~~, such as those suggested by Kreps (1984), Cho and Kreps (1987), s and Sobel (1987), Cho and Sobei (1990) and Kohlberg and Mertens rty
e &nw u.._ .. th7t . ..i. !hcna refinements do, indeed, have sig;;ificairi uutting power ve elimination of dominated strategies, and in the game generailv give rise to a unique equilibrium outcome which is ut, and thii is the main motivation for the paper, we present an fif th\nr~ 9ttrm-tivenocc argumen! wkir~ .. ;..-L. s1uest!on$?UL..U11. .%,.%.30 v, cnnc~pis in the game *SC%== c~~~~dercd, since ii terms of economic rationale they do not always choose the most sensible outcome. Instead, this paper provides a rationale for an alternative outcome which may be separating or pooling. The paper is organized as follows. In section 2 we first outline a simpie e of goods market interaction with asymmetric quality information. r, sequential equilibria are defined and the elimination of weakly dol~~in~ted strategies are presented as a further restriction 06 out-of~~~~~b~un~ beliefs. Section 3 considers separating and pooling equilibria in turn and characterizes the set of undominafed sequential equilibria. In this tion the spirit and methodology of the qna3ysis is closely related to that of agweii and Ramey (1988, 199Q).I[n section 4 we first introduce the further refinements suggested by Cho and Mrcps I: 1987) and show how they lead to a unrque r:zfined A*ryLa 1 l+brium outcome. Then we turn to a critical assessment of these extant formal refinements of the equilibrium concept, and we conclude by p?esent~~jg 8~ alternative selection criterion that leads to a different refined outccme in the gcnle PE\~P;IJ~~~A C*.‘*U)“UVI*U. Proofs are ccntained in the
P.R.
in
Ooergaurd.
Price
us a signal
of qualiry
485
;hfz
section we pro-k!e 3n ihstrativ &ar. We claim, however, that this exampie is representative in the . ..W.U.U. Hence, it shouid bet that ?he qualitative arguments presented a of the model used, but are of releva problems that have been analysed in a signalling ap~ii%!
litpratfllrC
AC _____r?ntaA ---.-, 2kF?VP xge :LI”Leb*I mc.Jnl thn L. n ‘i_ CLlk intnraet;a-....LLILb-iad*ll%_.:tug D&1ci.
with asymmetric information about quality. The trade good, by which is meant that potential customers are u ver:,y ‘f the quality of the good before purchase. concluded will the true quality of the good be revealed t assume that ex-post trades among consumers are i further, we focus on a case where the price avaiiable to the tirm. We thus exciu advertising, research budgets, etc., may se rule out repeat sales and set up the market ~~ter~~t~~~~ as a For examples of similar models that focus on adv~rt~si the reader is referred to Milgrom and R ch. 2), which also contain discussions o games. To set the scene for the Consider a one-shot market number of potential custome Since we are not interested in the m quality, we assume that, prior to the quality level qE (LH), where O< L tion fully described by pr(H)= p@. common knowledge. Hence, pa is t that Nature chooses a hig potential customers obr erve beliefs as a function 3f t Let us assume
that
of
486
P.B. Overguard,
Price us a signal
ofquality
To further simplify the analysis we assume unit costs to be strictly increasing in quality. i.e. vE.< uH. Let Bqap denote the unique maximizer of ~~(P,ejl,p). Thus, PL.0 is the full information maximizer fot the low-quality t-iPH.l is the full informatIon maximizer for tbs. high~q~a~~tyfi To define equilibria, we formalize the market interactions tween the firm ers as an extensive form ame with the f&s initially, Nature chooses a pr(H) ==#. Then, the firm o chooses a price 0 E R +. t&e Paststage the fir te beliefs to p(P) and tever demand is forthcoming at the Turning to the definition of equilibrium, we note at this point that is restricted to pure strategy quilibria.” Further, ~que~tial and ayesian equilib~a c~i~c~d~ in t s case where each player only has one move. Now, the c~~~ect~~uf P,, P,,p( P)) forms a ~q~~~t~~~ cqtiirli if: (i) The strategy of each ty
of the firm is sequentially rational.
i.e. b& types uf the firm optimize in their choice of price, P4, given the liefs (hence, demand strategies) of the consumers. hc system of consumer befiefs, p(P), is &yes-consistent, i.e. (a) if PH P:_ then p(B,)- 1 and P(f’t)=O, (b) if P, = PI. then p( PH)= p”, and PM,P, then the consistency re uiremcni d43esnot constrain beliefs, (c) ny p(P) t5[O, 13 is consistent. (ii) is simply meant that consumer beliefs are updated rule w~never it applies. (a) and (b) are selfexplanatory. ness of beliefs 0iT the uilibrium of this mode!, and this is the y equilibrium gives a positive ows that any other price in the ~~t-of~uilib~um play. The model adds no
set of outcomes can
suppwted by sequential ~~~~~~~~ alv in
In the interaction
and only if max R(P. q*g~j5 mia x( P. q. 67). P P In our model, demand x(P,p) is strictly increasing in p; and it fo!!ows that ---;i;G”p ff(*D’, $, p) sp ;r(p, q, 1) anG z(_P.q+I. is S*rr;\QCtng :‘SC-.I_~“‘Zi:D in r”. ES,,i iciiic, min, n( P. q. p) = n(P, q,O). We therefore say that P’ is weakly dominated .^sr ore generally, we say that a strategy P’ type q by P if aU”,q, l)$g#P,q,Q). is weakly dominated for type q if and only if
~(CY,q,l)jmaxn(P,q,O)~~(P,.,,q,O), P
i.e. a price P’ is weak11 dominated for some type. if the profits induced by the strategy under the most favourabte maximum profits attainable under the te denote a( Pgwo, q, 0) by 9, and call this the security level smce the type q firm is aiways able to obtain at lea consumer beiiefs. ith this definition of wea ihe notion of an undominated if the associated system of beliefs satisfies t (iii) Weakly dominated strategic weakly dominated for type L
P.B. Overgaard,
488
Price as a signal of quality
repgczt‘:.Q strgegfec, ‘,hg,t are dominated for one type but not the other. If a strategy is weakly dominatecl for both types, then arbitrary beliefs pi [O, 1) can still be specified. We will use requirement (iii) in the next section, but let us remark here that an iitirtilu’ocequilibrium selection criterion follows naturally from (iii), since one round of elimination of dominated strategies may reveal further strategies that are dominated given the first-round eliminations, see e.g. Kreps (19234).We will briefly return to such an iterative criterion when we discuss pooling equilibria in the next section. N&e
that
this r&&ion
onjy has
pnwer
w&h
uilibria
In this section we first briefly characterize the set ‘of separating sequential equilibria, show that these exist, and show that a unique outcome arises from undominated separating equilibria. Having shown this uniqueness result for separating equilibria, we turn to pooling equi!ibrfa and characterize the set of outcomes that remains when weakly dominated strategies are removed iteratively.
3. i. Sepuraring equiiihia
In this simple model, a separating equilibrium outcome is fully described by a pair of prices (P;, P h’); where PL f PE1 that allows consumers to infer
the iv,j=
quaiit,y
with
cer&ntjt_
From
&yes
f&cws
p{P,j=O
a-f! Wi’-_
1.
We claim that in any separating equilibrium, the low-quality firm chooses i.e. the fuli information price. We see this by noting that for any &=Pt.s, other price, P, deviating to the full information price increases profits since L 0) < R(PL.0. L, 0) VP # PI_0. Turning to the behaviour of type H, we exploit the weakness of ayes-consistency by setting o(P) =OVP# PH, i.e., at any price different from ted equilibrium price of type H the consumers infer q= & with This system of beliefs is the least favourable from the point of view of the firm and will therefore support the largest set of sequential equilibria. e note that this system of beliefs is consistent with requirement (ii). ii piic.z B, F-0: is be part of’ a separating equilibrium it must ‘be ~n~~o~tab~e for the type L firm to masquerade as a type hi relative to e full information price- --a_ SA be Let us define the set Ls __ recognized. _ L, l)ssLf. I.e., LS-constitutes the prices that type L rvould r?ot deviate to; every if ccmsumerb i If firm to choose PM m should not have an ven the “b&e& piPj=
O?y?~!Jt,\.(f~). We define the set p-d”as {PIz(P,H,l)~sHj. Hence, with P, E tp” the type H firm has no incentive to deviate. With these definitions at hand, we see that necessary con&tions for a separating sequential equilibrium are P, = P,,, and PH E Ls A I?“. Such strategies are sustained by belief systems that specify p[Fj =0 V~PER+\{ a,] and p(P,)= 1. Hence, the conditions are also surTicient.Below we prove that L” n H” is non-empty, which implies that separating sequential equilibria always exist in this pnode!. In fact, Ls n .q” wr,li!! **~n;~~lL Lo nm :-*---s-l S~Y.w~n~~ vti ~EI III~GI val, iiiiuA we conchide, unsurprisingly, that a multiplicity of ssquential equilibrium outcomes exists in this model. To remedy this multiplicity problem, we invoke condition (iii) from section 2. and prove that a unique undominated separating sequential equilibrium outcome can be sustained. To this end, we introduce a further piece of notation. Define for any real number 4 the following profit expression:
w~q)=w-ti4)w,
1,:
’
IYe generAze the assumptions about costs by assuming that u is strictly increasing in 4, and v(L)= uL, u(H)=v,. Thus, we assume for a moment that any quality qE R is possible. Hence, tr(Pl4) is the profit to a hypothetical firm with quality 4 [and unit costs v(q)] when consumers believe q=H with certainty. Further, let P,(q) be the price that uniquely maximizes ~r(P/q), and assume that P,(q) is continuous in q. For later reference, we show in the ap-pndix tRat r”,(q) is strictiy increasing in q. Note that P,(H) = Pii. 1qi.e. the
full information price 0 f type M as defined earlier. Now, denote by a(P,(q)l L) the profit to the type L firm from selecting the optimum of the hypothetical type 4 firm, r”,(q), i.e., the optimum for type 4 when consumers infer 4 = H. We can write this as
By inspecting this expression, we conclude that there exists a 4> L sufficiently high so that w(P&j 1L)<~c(P~,~, L,O)ZI:~. Thm when 4 is sullTiciently large, tke type L firm prefers its full information price to P n(q) even if p(Pl(4))= 1. By continuity of x( *) and U,(s) there exists a unique qualify, 4> L, such that @P,(g) 1L)=st Finally, to make the signalling problem interesting we assume P,, 1$ b;‘,i.e. ;*a of type M is not in the set of prices i31efull information optimizing pp.,, that are dominated for tppe L. if, instead, P “, 1 E L’, then the type %fc:ly choose its full inrormation optimizer, PH.1 H, since type 8, has no incentive to mimic thi separating sequential equilibrium in I’,f, I E problem is mute.
P.B. Overgaard.
F
?,O
Price as a signal qf quality
FHo F,($
P
1
Fig. I
Now, we are able to state the main result on separating equi!ihria: There exists a unique undominated separaiing ij > H. come where PIa= P L,I! and P, = P :(/i) $r ~17~*1110 TlwmrP$P2 I_ d
..__
‘
_.._
equilibrium
out-
Coroiiary 1. PH = PI(q) > P,. ,, i.e. the separating equilibriu_m price of the type H firrat is distorted upwards relative to the full irlformation price for that typePrasf.
See appendix.
There results are illustrated in fig. 1.
The previous subsection made clear that, in the interesting case where P,, , $ L”, the price of the high-quahty firm is distorted upwards. Hence, signalhng the ccrrcct ty.pe is costly when quality is high. We may therefore wonder whether pooiing eq-uiiibria exist that survive the invocation of a domination criterion of the type suggested in section 2. We will show that the answer is afirmative when the prior consumer beliefs, pe, are sufiicicntly Favourable from the point of view of the firm. y definition, pa-S-vvalSle equifibria sham the characteristic that both types of the firm charge the same price, i.e. Pt = P, = P*. It follows that consumers infer no new information from the equilibrium play, and Bayes-consistency requires p( P*) = pO. To characterize the set of undominated pooling equilibria, let us recali that the iast subsection showed that the type M firm can ensure a profit of , i) in the separating equilibrium. M/e use this result to refine the hat type H should receive at least us therefore define the set I-J*@*) as PL=Bff=
P
t in a p
with
PO) is a
f5Saiy coGi$tiOfa
e further that H*(p’)#@ su~~tj~n
showed
that
L to stay in the
for p0
cf e use this to state a I as P* Al.* where
type
L is assured
ed pooling equilibrium ?* E L*(p’) CT prove in the appendix that P* EL*(P’) n II* is also a sufkient condition, and that N*~Q’) c Z..*(,-I~)_ This a!!nws ur to stata the following result: Theorem I. Any PE H*(p’) ~~~~~i~ri~ oumme.
can be susfained as an undominated pooling
Cor&.w_~ 2. For p” suflcienfly large W*(p’) #@: i.e. undnminntpd pan@ equilibria exist when the consumer priors are su&ienSly favourable.
Pmg. TO m
_
See ai;pndix. that
the
at
of
undominatd
pooling
equiI&&m
~~;~~~~s
is
non-empty for p” suE.kntly large, we recall that the set H*(p’) is defined by the inequality ~(f, H,p’)h n(Pi(4)9 H, 1). Thus, the set of pooling prices is non-empty if p” hpmia, where
492
P.B. Ouergaard.
Price as a signal $qucdirg
the firm, the incentive for type H to separate is too weak, and pooling equilibria exist which survive the iterated elimination of dominated strategies. Combining with theorem 1, we conclude from section 2 that restricting behefs according to the iterative domination criterion reduces the original set of sequential equilibria to a smaller set, consisting of multiple pooling equilibria for values of p0 in [pmin,1) and the unique undominwted cepara?*.*A-1P.Y.a..Y..ICIIC WVI-3 to &iG _,_:_:_F.:,...r&momarr* ing eouilibriam, Thus, aSthou@ th ic p,-!l#f*+r”illm nate sequential equilibria, the basic problem of multipiicity remains. In .“W f’~>:! a given set of particular, two different types of equi!ibria U_ a¶meter values. Hence, the model prediction of the price distortions is unclear. Problems of this kind has prosmpted a search for further refinements of equilibrium concepts applied in signalling games, most notably by Kreps (1984), Cho and Kreps i i987), Banks and Sobel (19873, and Cho and Sobeil (1990). These papers draw on the general treatment of strategic stability by Kohlberg and Mertens (1986). Below, we use the notions of Cho and Kreps a5 an illustration and assess hCW ihey work in the present context. In the remainder of the paper we wih assume p” E [Pin, I), and we will not attempt to dispute that separation is the unique outcome when this assumption is not satisfied. At a more general level it may be argued that even the (iterated) domination argument requires too much even from sophisticated players since the common knowledge and coordination assumptions are unduly strong. Below we consider further refinements and require even more of the piayers, and it foliows that the tentative objections raised will apply a fortiori if iterated dominance is considered strong. The particular refinement of Cho and Kreps (1987) wi!! i!lustrate the genera! -idea 4n the papers listed above. Let us fix a sequential equilibrium with prices (I’,,p#). and with equilibrium profits to type y given by n(ti,, y, p( p,J). Consider some alternative price P” # gL, pa. A deviation to P” is said to be equilibrium hmitmte$ for type q if
The ‘intuitive’ restriction suggested by Cho and Mreps is that the consumers should not believe that the firm will experiment with an equilibrium dominated deviation since, even if beliefs were favourabte following such a deviation, the 5rm is assured of a higher Layoff in eqnihbrium. Hence, a sequentia: equilibrium i P,, P,, p(P); :-13 iiidj .~;Jr-t~~~i~r~in~ if, in addition to (I) and (ii), the following is satisfied: (iv) Elimination of equilibrium dominated strategies; i.e. p(P)= 1 equilibrium dominated for type L(W) but rlnt foF :yp~ Eii,j.
(6)
if
P
is
to satisfy (iv), the m ltiplicity problem is
tc that the se
brium satisfies (iv).
P.B. Overgaard,
Price u; cz signal
n(P,L,p)
I
L.-B P LJJ
:
:
.
P* Pd P,($
ofquality
493
= n(P”,L,pO)
3 P
Fig. 3
““” ~GQ ~,,,~, rcca]i that p,(e) is defined Hence, self-enforcing equilibria exist. To cpp by W,(W, I)=& and it follows that any equilibrium undominated deviation by type M from PI(G) to PO, say, would be mimicked by type L if p(P’)= 1 (see fig. 2). Next, we state the following result.
Proof: See appendix. The proof of Theorem 3 demonstrates that at every proposed pooling equilibrium price P* E H*(p*) there exists an out-of-equilibrium price P” such that P’ is equilibrium dominated for type J#_,but not for type H. Hence, condition (iv) requires p(P ‘; = 1, and type H obtains z(Pd, H, 1) > II( P*, H: p”) ~kh destabilizes the equil,ibrium.’ This is ilhsstratcd in fig. 3. In the following we want to argue that, while the suggestions of Cho and Kreps (1987) summarized in condition (iv) certainly have the valuable characteristic that a unique equilibrium outcome is picked out, it is not clear that the arguments oehind the se!ection procedure arc persuasive.’ “vt’eshall argue that condition (iv) rules out equilibrium outcomes that can be rationalized in a sensible way, and we shall given an example of such an outcome. Before doing ~3, we wili make aV,Sz., rfimp observations concerning the refined selection criteria used so far. First, recall that a multiplicity of undominated outcomes exists, of which one is fully revealing and a continuum is non-revealing. Secondly, among the undominated equilibrium outcomes, the separating outcome is the least .mU .. that ‘refined’ cquiiibrium pr&rrcd by b&z types of the firm. It follrtws ‘As pointed out in note 1, we resfrict attention to pi;re strategy equilibria. If ihe consumer prior is suficientiy high. a third class of hybrid equilibria exists. These arise ii the players are allowed !ti mix between a separating and a pooliing price. But. WI eW@k::: G: &e proud ol -Gkh ii are elmMated if condi$ion (iv) is iir:posed. Theorem 3 GA!!&i.+v ihai & L&k1 rqu2:.,; ‘We should note that we are certainly not the firs! to qu4sz ihe logic Mind !hc su tians of Cho and Kreps. In a diflerent context, vi~n Damze (1?U. QYXldix e) has a c Cho-#reps re$nenleai. qUeStiOnedthe Ft3Wf’kQ khiti t%
P.B.
Ouergaard, Price as a signal
ofquality
Fig. 4
selection picks the worst undominated equilibrium outcome from the point of view of the informed party.4 i’hirdly, we observe that it is ‘as if’ consumer beliefs are used strategically as a ‘threat’ against the firm; e.g. the type W firm can be forced to choose PI(G) only because it accepts that lower prices will be taken by consumers to indicate that it was played by a type L tfrm. Finally, there exist p0 E [pmi”, 1) and PE II*(#) such that (9 (RP”)>~ (~I.,O,G;, (ii) (RP*)>~ (PI(Q), 1 I- and (iii) iP,~“j>~ (P,.,, 0) with probability atzi:ity ;P*
(I -p”)
and (P,(i), 11 with prob-
where >,. denotes performance relation for player i and C denotes consumers. I.e., undominated pooling equilibria exist for _JZ* suficientiy iarge that a:__ ~rp ntpt’prtp~ hrt hn#)? nC *ha k-r IL_ y.-s”e..‘~~~ “j -_-%-%iitxpum ‘_~~.=a~ =L=: ‘EEL_’firm --. --_ ag& .. =_.- fhp .___ pp,.n~::me~c _____+A r-siz !&a,-~ \_‘__ __ the C.._ _;Gi expected utility/consumer surplus is higher in the pooling equilibrium than in the separating eqtdibtium!, With these observations in mind we turn to consider the situation 1 illustrated in fig. 4. Assume &at pnor co2 sumec beliefs are given by oo3 and ihilt the c;msuxmers expect Che unique undomir-*+ .-d separating equilibrium to kp “1 Ar9..PI( ySerJsu. I.e., consumers expect with probability p* to observe P,(G), in which case beiiefs wiii ‘be updated to p(F,i@= !, and with probability (! --a@) to observe P &,*, in which case beliefs are updated to p(P&=O. Fdote that prior bePp-cc--L a high pi&tliiiQ 1613PLLWII to the type H firm; the type that consumers prefer. The question we wish to address in the following is: HOW should consumers react if they observe P*? First of all, the consumers are surprised by P* since it is a zero-probabiliiy event when the separating equilibrium is expected. We now want to argue that the consumers should also be confused. If the type H firm ‘deviates’ G-c-, P I Ab i: 18. QaG;V&f+& s.v_ ir; we, c,l~ ex-p&& qy&i&;ing i."asa r. l\y4Jr 3irir3e6 ii ZiCS’. ‘This is true for the criterion deveao by CSro and Kreps, as well as thse thst z&et b&&s even further. such as Universal Divinity [Banks and Sobel (1987)] and a suitable version
of Strategic Stability CVoLlkp - ~m.,rg and Mertens ! 198Ajl_*
and 5h(P”, t, pf P*,) > Te(P,.,. L,
Now, suppose p0 >p*
and assume for a moment
consider P* to be compieiziy ~M~~a!ive,
t
i.e. p{ P*) =: p
IriP+,H.pO)>II(P,(?).H,l)
and KCP*,L,pO)rR(PL.o,L,O).
So, if p( P*9=p” is indeed the ccnsumet inference, both types of the tirm will prefer to play P* to P,(g) and PLSo, respectively. Menue, if consumers interpret P* as unfnfcrrmaiive, kfi+k _W.** :yq
cf tt;: fi;7;; hZVC &ii iiiiCi& ;si deviating from the separating choices. This reinforces that the consumers .z .__ _ -’ -1 _i:- _I.__ -
if the firm takes the pessimistic ‘beiicfs a; our-of-equiiibiium pomrs as given will the m-iii stay at the spec~riett separating choice. 3IlUUlU
ut
LUIIIU3cW
alltt
UUSGrViii&
-
i?
Qdy
-_ _--. -*here u(p,q) ¬es ;hc @~fi~cmer p&y& izicssgfgd QE some van Neumann-Morgenstern utility scale. Define p*+ by jminip”jj3P,p”(u(P,H)-u(P,(~,H))
l(i-pO)(u(P,.,,Lf--ujP,,Lii). Thus, the inequality above states that the exp2cred consumer payoff associated with the price P* and posterior beliefs p” dominates t payoff in the unique separating equilitrium. Hence, even if the completely uninformative, it is preferred by consumers if the priQ,r of type W, p”, is s~~~ie~t~y high. As proved earlier, the only admissible in (iv) is p(P*j=O. -With the above remarks i to suggest an alternative to the reaso other outcomes can be rationalized
of carrying out arbitrarily comp!ex of the model i&ted above, and est that ~~ns~rn~rs can rely on a move to a ~~~e~~-du~~~~~~~e ~i~~bri~rn.If such pessimism 1s sittlation emerges To c;t?Lz thir;,
LO) is implied by z(P,H,p’)z that ~(P,~.p~)~~(P~.~, 1). Hence, am defin the set of prices, for given beliefs, which by all! to the begat ~~u~~~brjurnoutcome. By definition, the is a eon-~rn~ty closed subset of N
set P
0
> ZZZ
P
*_
P
min
and
when ~~&j=rnax(p*,p**{. Note that any message in by the type N firm will tentially be duplicated by the type L firm, en the Dekfs, since S*(p’) are preferred by both to the noted_ this rB=iarCWJ-*c ._ .. .. . .._“.. that consumers !&““:A -L-WE_ fter observing at least som2 mssagcs Iii S*(p’]. nt of view
of the type H firm and look at prices in the nt that p(P)=p’ for any PEP(#), and
is certainly the case for p” close to 1. Now, n_pCrand sequential rationa!ity Since the type L firm to follow the proposed deviation bv type
p(P) = p” VP E S*(p’).
criterion (where A stands Car alternativk): th
typesof the firm choose a p&q F; ihat
price ilial saiisfks criterion A, which consumers can cieariy ,,,..:_ ill . . .L_L.&PJLIWUL\b &Wx..lAtGl1IQIifl an*.3 UrG--I--_ y1KPt3.But, so far the story have nix compietely specified beliefs foiiowing
P.B. Orergaard. Price as a signai oJpquailty
497
PT -I
P,!g!---
I
P
H.1
i
I
?!-I
a
I I
1
p”
Fig. 5
~p~~*(po)\,.p~_
szstafn
---* s--. ,--!--- -:-‘---I ~JILKGU uui oy crirerion A 2: a pooiing equilibrium, we must specify p(P), VP E 2*(j3’)\, f Pcj, such that ir(P, i, p(3)) 5 R(P~.~,L,O). Criterion A picks out, among the undn*;=e+e’ “I‘IIIIt&L~” ~~~l;nn ta”“UUC) W~PPC r_‘___ in -1; S*fpO), tit2 p&~&f-Chiiiiidi;ilg uuicome for type H. Above. the reader lmay have wondered why we took the point of view of the R fhm ____ typo1 II . . . . .., and suggest that this is quite arbitrary. To this we note that, in models of this type, it is existence of a t9.w SjP;- L firm that causes the efficiency problem. At any point in the game the type L firm is deciding whether or not to mimic the type H firm, hence, deciding whether to accent a+~prlllilihrin~ nrni’;la by :yi;e H= 1lJl-r z- i’ksp,J&, _-‘-q~“‘Y..1. pr.“,.IY ynnnc.+ v,vyvos *r ?&.! --.WC r-f-htzltt:IL l!iLS virv_ p ihe
To
the
&)p’ce
l
emphasis should be on the decision of type II, and at P, nOthe
AWse
dkct
or. profits to type li of the possible existence of a low-quality type is minimized. Since the type L firm and the consumers realize this, the price ? H.pObecomes focal, and the consumers should attack, a lo=* probabihty to type H after a message P # PH,Po.The reader can easily verify for himself that
the assumption P H,pO~S*(~o) is immaterial for the qualitative rest&s, and that a maximizer under criterion A always exists since S*(p’) is compact. Thus: the alternative criterion picks out a unique point from the set of undominatcd sequential equihbrium outcomes. “VVewant to compare this outcome with the separating outcome resulting from the selection procedure of Cho and Kreps ( 198 i). A questionable feature of the Cho-Kreps outcome is its indc-$ndence of the consumer prior, p”, and this causes a puzzling discontinuity at p@= !. To
e see this, assum_ethat consumer N-A-I -. CDE (cl. 3:. ad :et )J‘ Iv13 iii3 @?X Sy p” = 1 Ep+ 0. From the analysis above, it is clear that as &pdecrecases the outcome chosen according to reqwirement (iv) is unchanged, i.e. P, = Pi(q) W s(O,1). Now. set EP~0, which imp!ies that the consaumers discard the ~oss~b~i~t~df Tfa ,Bs;c crkc.a *l-w sa04:K.a.l‘-rcat11mC” t* .U.l l;lil .*.‘“.a.aU..V... .nfr\rpracsl;nra tyIe L altog&!sr, 1.2., ,mp 11. CIllJ Q.c&L9& &elk ‘srl1ya.h ..” .W&U... optimizer of type pi is P, = P,, , . Hence, the self-enforcing equilibrium is discontinuous at J-I’= 1 as iiiustrated in fig.. 5. der criterion A, ty? Compare this with the alternative
II picks, and type L mimics, the pri
498
P,=
at-g max lr{F, ii’,
p”i
PES'(P"I
for all JIO~~= ax{p*. p** i. This is illustrated in fig. 6. This alternative also &iiitj & disiontiiiuitjr iji the priCt2futict n of type N. But, the di~o~t~nuity appears at i, which is a point where the reto-rankg of ik xyalaiing and pooling equilibria changes. PM+,O~S*@“f for p” sufficiently close to 1, and it follows that the equilibrium selection criterion suggested here h= particular economic appeal when the a priori probability of a low-quality type is small, since the type W firm and consumers suffer only vanishing payoff losses from the existence of a low-quality type as p” approaches 1. Thus, the introduction of a low-quality firm with a small prior probabiiity has a nealirsible effect on the expected payoffs of the high-quality firm and _ _ On the contrary, if we insist on the consumers when criterion A i-II bmptoyed. e r,_\ "eQi~~~~Z?TSZf
a."e."f
i:Yj*
thc
ty;<
j;
z~m
2nd
ihe
c~g~~pme~s
su-ff&-
z
non-frivi&
is=
even from an infinitesimal prior probability of a !ow quality. The aim of this section has nui ‘been to shut that criterion A is generaliy more sensible or intuitive than &= Chn-Kreps criterion. A sin& refinement of the sequential equilibrium concept seems too much to hope for, and our purpose ;,-_,,c,.%r- *t-r +.?,, L+S,i” “&III\91I,\.*nrc ,,*a, ai2 &Xiiaiiv~ selection proCedure can be LMK9 LiGtiII.-
cf'gwcffs
_
iesds to outcomes which involve pooiisg of dik:cM typ. hether a discontinuity at p”= 1 is considered to be serious depends on the situation. we want the model to dcscrik~?and hence o he interpretation of seemmgly trivial changes in the model. It may well the case thai a
model rn which p” = different kom a model in which /Pm 1, since they may describe very different phenomena, and as such we .the urn outcomes to differ. In t&is interTpretation, .__!_usmg illustrates the familiar point that adding or deleting ith ‘zero’ prior probability ay ultimately change the outcome of a gan-W6 1
---I:
is
sr%ni~~nliy
in signaIfing games. Two variants of signatli ted& in the industrial organization literature. variants are characterized by the space of actions available to the uninform uninfmned
plaver to vatv cnntinrbnllclv - - ‘--‘J piah the signa! a~! the pos:;:io;
beliefs [see ais*- the quai&signailing modei of Milgrom and and the Spence-model discussed in Kreps !lQW) and 6110 -___ -_ ~11.4 Roughly, the conclusion from such mohels is that a t-nique separating equilibrium outcome survives refinements of the Cho-Xreps ty general results of Cho and Sobel (1990)]. A feature of the separati is that it is the Iesp_stpreferred by the informed player in the set of undominated equilibrium outcomes. Therefore. we con&de that our discussion of equilibrium selection appli-ya +ro ihis class of games as a whole.’ In the second variant, repro sented by the limit-pricing mode1 of Roberts ( 1982) a-r&d,pariicuiariy, the version of Bagwell and the best respo:rse of the uninformed player maps the posterior beliefs onto a two-element set (viz. {Enter, Not enter)). 4s a con result can be expected from ~ti!ikium refinements .4 typical result is that a uniq outcomes remain, Often, see e.g.
equilibria admiis a unique Pareto-optimal
equili pi ayers tlI&;Fk *.*I**. is pOO/ifig_Furthermore _-__-“, the “god
type receives iis fuii-
‘in the Spence-model the payoff to the rl”iEifGzizd player is the same in an to the zero-profit c~=r.diti~n.So, the critique of the ~~arati~g e~~i~~~~~rn payoff-domination arguments for the uninformed player, but will rely on an ar pay&s d :k aark% types of t”neirr;Nlil~ I”iaypr.
VW) -
(A.ij
and
these to obtain
and substitute L into (A.!) to obtain
9. 1).-(P,(q2)-v,).~P,(q2), reasing
ve
l)>O.
in 9 for q>
t. Now, the continuity of ue tj> L such that n(P,(q3 i L)= we must have
&&i*8m ..m..3 c&a&o -._ _ 0. ierminate the e.g. make a ‘mpra that a rational t
W firm in our
. .
since, by definlt:oR,
if)” maxrm,,e, . ;s
c.
and we conclude x(P,(&, l)>x(P,, v(@> oHj into (A.2) which impiies
.RiD
,‘,a
,I,,
LP
l? 1,
*et “,I
PS
A.4
.
Then
1) or P,(q)< P,,. Now, substitute q (i.e.
ut, this is a contradiction since P,(q) maximizes x(R)i) by assumption. Thus, we must have R(P~, L, 1)= sL. Thirdly, we prove that P 1(i) is the unique maximizer of 7t(P,H, i j subject to the condition 7t(P,L,1)=sL. Pick any P#P,(q’) such that 7t(P,f., l)=sL. Then
and (P-U,jX(Y,
-
_.
I)-(P,Iq)-uL)x(JJ,(q),
1)=-O.
(A.3)
Add these to obtain
tp1(4)-~,)xtp,t(63,I)-w-b&w,
1).
Add (A.3) to get
Since this is positive, we conclude that P,!$j is the unique maximizer r.f x(P,W, 1) given n(P,& f)=sL. Finalty, to show genera! existence of sequential separating must picve i’ n ii”+@. Hence, all that remains is to s the de~R~t~o~of 4 ensu is the maximizer of ?t(
P.B. Owrgaard, Price as a signal
502
ofquality
Add these to obtain
which implies x(P H,O,Q)<:x(PI(q),1). But then (since G>H)
and since x((P,,,, ~)>x(P~,~,O) we infer
.
But, this is a contradiction since P,(i) is the maximizer of TE(PI4). We conclude P,(g) E If” and separating equilibria exist. This ends the proof of Theorem 1. ?roof of Theorem 2
We first prove that H*(p”) c I,*(#‘), and then we show that any price in H*(p*) can be sustained by specifying admissible beliefs. First, we claim that any price in H*(pO) is also in L*(p*). Suppose not, A rannll Frfirn *ha rc#v-J “L ,-J Th.7W-W.Pm 1 tll\nt dD Ia P lb-J Than 3E:u .kdwcz.. II”‘,. L1.1, ydbl”I 1 Il’bVLWII‘ a CIlQL rqr 1\y,, U) .p-.i . i..w..
uw%-~,)XC~I~~,U-w--i;,W,pOb=G and (~-~,)x~~,vO)-(P*(~)-~“)x(P~(~);
I)@.
(A-4)
Adding these gives (FM--oL)(jrQ,
which implies x(P:(#, 1) >x(P,#).
But then, using (A.4), we conclude
And, since x(P, 1)> .x(P,PO),we obtain
ique maximizer of z( P I&.
We
P.B. Ooergaard, Price a:; a signal of quality
503
any P$L*(pO), p(P)= I, (2) for any P E L”(p”)\I-P(pO)* p(P) = 0, (3) for P= P* EH*(~O), p(P) = PO,and e.g. (4) for any P E H*(pO)\{P* >, p(P) =o. (1)
fOi
First of all, a potential candidate for a self-enforcing pooling price must be in H*(p’). Otherwise, the equilibrium is trivially broken. Now, fix any pooling price P* in II*( By continuity of B,(q), there exists a $ > L such that (‘4.5)
Any P* satisfying (AS) must be less than Pr(qr)_ Hence x(,P*,~~)> .JD i,l\ l\ Xl L... _..I-.,A-%5(1 r\y J, IJ. lYUW,tva1us1e (P*-
4fw*~P"!
-(P*(41)-uH)X(Pl(41),
1).
Subtracting (A.5) we get (VL
Ezixe,
-~~rs~c~(~*,P"~-xcP,~ql),
iiic iype
H trrnr siriciiy
w
wiih ixTi& pjF,(q’jj=
i to P
with beliefs p(P *) = p’. We conclude that there exist prices in a close neighbourhood of P1(ql) which are equ;r;h*;lllUrlUm dominated for type L but not for type H. Thus, imposing condition (iv) destabilizes P*. Referams Akerlof, G., 1970, The market for ‘lemons’: Quality uncertainty and the market mechanism, Quarterly Journal oi Economics 84. no. 3,488-%O. Albaek, S. and P.R. Overgaardi, 1991, Manufacturer-retailer relations when the wholesa!e price signals demand intensity, The Academy of Marketing Science, International Conference Series 5, 167-371. AlbaPk, S. and P.R. Overgaard, 1993. IJpstream pricing and advertising signal downstream demand, Journal of Economics and Management Strategy !, no. 4.677-698. Bagwell, K., 1987, Introductory price as a signal of cost in a model of repeat business, Review of CA . . ~onuiiiic . c-..A:-3I”urls .‘9; II=_,:3, 35384. Bagwell, K. and G. Ramey, 1988, Advertising and limit pricing, Rand Journal of Economics 19, no. I, 59-‘31. Ragtell, K. and G. Ramey, 1990, Advertising and pricing to deter or accommodate entry when demand is unknown, International Journal of Industrial Organization 8. no. !,93-“113. ina .+_-__--. -7mes. Econometrica 55, no. 3, Ranks, J.S. and J. Sobel, 1987, Equilibrium selection in signal..., 647-H. Binmore, K., 1987, Remodeled rational players, Mimeo. Cho, L-K., 1987, A rellnement of sequential equilibrium, Econometrica 55, no. 6, 1347-1389. Cho, I.-K. and D.M. Kreps, 1987, Signaling games and stable equilibria, Economics 102, no. 2, l79-221.
’
L-K. and J. Sobel, 1990, Strategic stability and uniqueneslr in s~~~~~~~~ Economic Theory 50. no. 2. 381-413. Kohlberg, E. and J.-F. Mertens. 1986. On the strategic stab~~~~yof no. 5, 1003-1037. nu. 758 ~~!~~f~~~~ Kreps, D.M.. 1984. Signahng games and stable ~~~~~bria, University. Graduate School of Business, Stanford, CA). Kreps, D.M;, P. Miigrom; 3. Roberts and R Wilson, 19R2. repeated prisoners’ dilemma, Journal of Economic Theo Kreps, D.M. and R. Wilson, 1982, Sequential ~~i~ib~~a, EI; Mailath, G.J., M. Okuno-Fujiwara and A. ~o~~~ew~~~c,1 signalling games, Mimeo. Milgrom, P. and J. Roberts, 1986, Price Gad advertising si~~~~s of Political Economy 94. no. 4. 796821. Overgaard. P.B.. 1991, Product quality uncertainty: Strate ati product markets with adverse selection and adverse P Universitt Cathoiique de Louvain. Louvain-la-Neuvet. Spence, A.M., 1973, Job market signaiing, Quarterly Journal of Economics 87. no, 3. t 35-374~ van Damme, E.C., 1987. Stable e+ilibria and forward induction, DP no. 12$ ( Bonn, Department of Economics.
Chc,
ri
Accepted for publrcation July i992
This paper discusses non-cooperative equiiibrium notions that have :ecen n applied in the analysis of signalling games. As an illustration, we analyse the role of as sigxils when consumers are incompletely informed about product quality before purchase. We investigate the incentives of a high-quaiit~ k-m to separate from a lowquaiiiy firm tiirough its pricing srraregy. Starting from the set of sequential equilibria we show that, when stronger restrictions on out-ofequilibrium inferences of the type suggested by Cho and Kreps (1983) are imposed, a unique szporoting equilibrium outcome remains. ??x paper offers a critical assessment of the formal game theore!:ca.l refinements needed to reach this conclusion, and argues that in the class of signaiiing games considered here other outcomes involving pooiing ment atten!ron besause of their economic content.
The objective o f this paper is to discuss the merit of various noncooperative equilibrium concepts that have recently been applied in the TIll___l_-L_-AI.’ discussion, we wi!! set up a -z_-IBZ_- -Kc rl&r&in2 io SqpalZLr!g gmies. r CJ I~!w5’,zz;c:s :m ;n+nrqr+;o~ simple model of goods m~tp+ . ..&S_W.C I..C”LCVC. 1 , where agents on the demand side are incompletely informed about the quality of the good before purchase. To facilitate matters, we assume that the consumers are served by a monepotist firm, whose exogenous product quality is unknown to the potential customers. Hen&, &he firm is privately informed, and this symmetric info mation structure will embody the adverse selection prable needed to set Correspondence to: Per Baker Overgaard, fnstitute of Economics, University of Aarhus, IX-8 Aarhus C, Jktmark. *This paper is a revised version of cit. B in my Ph.D.-thesis [ a%99t)I. Comments from S. Albzk, F. Forges, hl. Hviid. I..-F. Mertens, J.-P. Ponssard, E. va my thesis adviser C. !YAspremont, and two anonymous referees have improved the exposition ese share any responsibility for and are greatly appreciated. Of course, n Academy is grat&lly a~~~ow~~ errors. Financial support from the Danish work by Mzilat finishing the final draft, my attention was alert $-b, &rc &se?‘” ralakr in Fujiwara and Post!ewaite fl992) that pre spirit to section 4 of this paper. 01762680/93/SCt6.00 (%, 199~Elsevier
Science Publishers B.V. Ail rights reserved
P.B. Orwgaard.
484
Price as a signal of quulity
the market interaction as a signailing game. Adverse selection arises since the various types of the monopolist firm associated with dlryerent quality levels may have differing incentives to reveal or conceal their priv??- information. o~~quc~iti~, we study a ‘classical’ signaliing prch;._~- , t”:t:,,t? tk i~itiriiid
~~~t~atesthe game by sending a signai, :-: &it:. 3 t&e utiinntormed party end optimally given ihz i&;r~ukti~n contained in the signal. ative of the tradition that grew from Akerlof’s nce’s ( 1973) signalling-model, and it addresses at of Mllgrom 2nd RGkiiS i i986). it SS w~i~~kn~w~that signalling games, in which any particular equilibrium ts only a few information sets, give rise to an uncomforting of Nash equilibria, and it is on the selection among these that r wiii focus. As a tarting point, the analysis shows that the notions reps and Wilson (1982)] and (itcrated) ciiminastrategies bring us some way, and sometimes ail the way, the mult~~l~c~typroblem. But, we also demonstrate that ere these concepts do not yield a unique equilibrium outcome. In such cases appeal is often made to more refined selection c~~te~~, such as those suggested by Kreps (1984), Cho and Kreps (1987), s and Sobel (1987), Cho and Sobei (1990) and Kohlberg and Mertens rty
e &nw u.._ .. th7t . ..i. !hcna refinements do, indeed, have sig;;ificairi uutting power ve elimination of dominated strategies, and in the game generailv give rise to a unique equilibrium outcome which is ut, and thii is the main motivation for the paper, we present an fif th\nr~ 9ttrm-tivenocc argumen! wkir~ .. ;..-L. s1uest!on$?UL..U11. .%,.%.30 v, cnnc~pis in the game *SC%== c~~~~dercd, since ii terms of economic rationale they do not always choose the most sensible outcome. Instead, this paper provides a rationale for an alternative outcome which may be separating or pooling. The paper is organized as follows. In section 2 we first outline a simpie e of goods market interaction with asymmetric quality information. r, sequential equilibria are defined and the elimination of weakly dol~~in~ted strategies are presented as a further restriction 06 out-of~~~~~b~un~ beliefs. Section 3 considers separating and pooling equilibria in turn and characterizes the set of undominafed sequential equilibria. In this tion the spirit and methodology of the qna3ysis is closely related to that of agweii and Ramey (1988, 199Q).I[n section 4 we first introduce the further refinements suggested by Cho and Mrcps I: 1987) and show how they lead to a unrque r:zfined A*ryLa 1 l+brium outcome. Then we turn to a critical assessment of these extant formal refinements of the equilibrium concept, and we conclude by p?esent~~jg 8~ alternative selection criterion that leads to a different refined outccme in the gcnle PE\~P;IJ~~~A C*.‘*U)“UVI*U. Proofs are ccntained in the
P.R.
in
Ooergaurd.
Price
us a signal
of qualiry
485
;hfz
section we pro-k!e 3n ihstrativ &ar. We claim, however, that this exampie is representative in the . ..W.U.U. Hence, it shouid bet that ?he qualitative arguments presented a of the model used, but are of releva problems that have been analysed in a signalling ap~ii%!
litpratfllrC
AC _____r?ntaA ---.-, 2kF?VP xge :LI”Leb*I mc.Jnl thn L. n ‘i_ CLlk intnraet;a-....LLILb-iad*ll%_.:tug D&1ci.
with asymmetric information about quality. The trade good, by which is meant that potential customers are u ver:,y ‘f the quality of the good before purchase. concluded will the true quality of the good be revealed t assume that ex-post trades among consumers are i further, we focus on a case where the price avaiiable to the tirm. We thus exciu advertising, research budgets, etc., may se rule out repeat sales and set up the market ~~ter~~t~~~~ as a For examples of similar models that focus on adv~rt~si the reader is referred to Milgrom and R ch. 2), which also contain discussions o games. To set the scene for the Consider a one-shot market number of potential custome Since we are not interested in the m quality, we assume that, prior to the quality level qE (LH), where O< L tion fully described by pr(H)= p@. common knowledge. Hence, pa is t that Nature chooses a hig potential customers obr erve beliefs as a function 3f t Let us assume
that
of
486
P.B. Overguard,
Price us a signal
ofquality
To further simplify the analysis we assume unit costs to be strictly increasing in quality. i.e. vE.< uH. Let Bqap denote the unique maximizer of ~~(P,ejl,p). Thus, PL.0 is the full information maximizer fot the low-quality t-iPH.l is the full informatIon maximizer for tbs. high~q~a~~tyfi To define equilibria, we formalize the market interactions tween the firm ers as an extensive form ame with the f&s initially, Nature chooses a pr(H) ==#. Then, the firm o chooses a price 0 E R +. t&e Paststage the fir te beliefs to p(P) and tever demand is forthcoming at the Turning to the definition of equilibrium, we note at this point that is restricted to pure strategy quilibria.” Further, ~que~tial and ayesian equilib~a c~i~c~d~ in t s case where each player only has one move. Now, the c~~~ect~~uf P,, P,,p( P)) forms a ~q~~~t~~~ cqtiirli if: (i) The strategy of each ty
of the firm is sequentially rational.
i.e. b& types uf the firm optimize in their choice of price, P4, given the liefs (hence, demand strategies) of the consumers. hc system of consumer befiefs, p(P), is &yes-consistent, i.e. (a) if PH P:_ then p(B,)- 1 and P(f’t)=O, (b) if P, = PI. then p( PH)= p”, and PM,P, then the consistency re uiremcni d43esnot constrain beliefs, (c) ny p(P) t5[O, 13 is consistent. (ii) is simply meant that consumer beliefs are updated rule w~never it applies. (a) and (b) are selfexplanatory. ness of beliefs 0iT the uilibrium of this mode!, and this is the y equilibrium gives a positive ows that any other price in the ~~t-of~uilib~um play. The model adds no
set of outcomes can
suppwted by sequential ~~~~~~~~ alv in
In the interaction
and only if max R(P. q*g~j5 mia x( P. q. 67). P P In our model, demand x(P,p) is strictly increasing in p; and it fo!!ows that ---;i;G”p ff(*D’, $, p) sp ;r(p, q, 1) anG z(_P.q+I. is S*rr;\QCtng :‘SC-.I_~“‘Zi:D in r”. ES,,i iciiic, min, n( P. q. p) = n(P, q,O). We therefore say that P’ is weakly dominated .^sr ore generally, we say that a strategy P’ type q by P if aU”,q, l)$g#P,q,Q). is weakly dominated for type q if and only if
~(CY,q,l)jmaxn(P,q,O)~~(P,.,,q,O), P
i.e. a price P’ is weak11 dominated for some type. if the profits induced by the strategy under the most favourabte maximum profits attainable under the te denote a( Pgwo, q, 0) by 9, and call this the security level smce the type q firm is aiways able to obtain at lea consumer beiiefs. ith this definition of wea ihe notion of an undominated if the associated system of beliefs satisfies t (iii) Weakly dominated strategic weakly dominated for type L
P.B. Overgaard,
488
Price as a signal of quality
repgczt‘:.Q strgegfec, ‘,hg,t are dominated for one type but not the other. If a strategy is weakly dominatecl for both types, then arbitrary beliefs pi [O, 1) can still be specified. We will use requirement (iii) in the next section, but let us remark here that an iitirtilu’ocequilibrium selection criterion follows naturally from (iii), since one round of elimination of dominated strategies may reveal further strategies that are dominated given the first-round eliminations, see e.g. Kreps (19234).We will briefly return to such an iterative criterion when we discuss pooling equilibria in the next section. N&e
that
this r&&ion
onjy has
pnwer
w&h
uilibria
In this section we first briefly characterize the set ‘of separating sequential equilibria, show that these exist, and show that a unique outcome arises from undominated separating equilibria. Having shown this uniqueness result for separating equilibria, we turn to pooling equi!ibrfa and characterize the set of outcomes that remains when weakly dominated strategies are removed iteratively.
3. i. Sepuraring equiiihia
In this simple model, a separating equilibrium outcome is fully described by a pair of prices (P;, P h’); where PL f PE1 that allows consumers to infer
the iv,j=
quaiit,y
with
cer&ntjt_
From
&yes
f&cws
p{P,j=O
a-f! Wi’-_
1.
We claim that in any separating equilibrium, the low-quality firm chooses i.e. the fuli information price. We see this by noting that for any &=Pt.s, other price, P, deviating to the full information price increases profits since L 0) < R(PL.0. L, 0) VP # PI_0. Turning to the behaviour of type H, we exploit the weakness of ayes-consistency by setting o(P) =OVP# PH, i.e., at any price different from ted equilibrium price of type H the consumers infer q= & with This system of beliefs is the least favourable from the point of view of the firm and will therefore support the largest set of sequential equilibria. e note that this system of beliefs is consistent with requirement (ii). ii piic.z B, F-0: is be part of’ a separating equilibrium it must ‘be ~n~~o~tab~e for the type L firm to masquerade as a type hi relative to e full information price- --a_ SA be Let us define the set Ls __ recognized. _ L, l)ssLf. I.e., LS-constitutes the prices that type L rvould r?ot deviate to; every if ccmsumerb i If firm to choose PM m should not have an ven the “b&e& piPj=
O?y?~!Jt,\.(f~). We define the set p-d”as {PIz(P,H,l)~sHj. Hence, with P, E tp” the type H firm has no incentive to deviate. With these definitions at hand, we see that necessary con&tions for a separating sequential equilibrium are P, = P,,, and PH E Ls A I?“. Such strategies are sustained by belief systems that specify p[Fj =0 V~PER+\{ a,] and p(P,)= 1. Hence, the conditions are also surTicient.Below we prove that L” n H” is non-empty, which implies that separating sequential equilibria always exist in this pnode!. In fact, Ls n .q” wr,li!! **~n;~~lL Lo nm :-*---s-l S~Y.w~n~~ vti ~EI III~GI val, iiiiuA we conchide, unsurprisingly, that a multiplicity of ssquential equilibrium outcomes exists in this model. To remedy this multiplicity problem, we invoke condition (iii) from section 2. and prove that a unique undominated separating sequential equilibrium outcome can be sustained. To this end, we introduce a further piece of notation. Define for any real number 4 the following profit expression:
w~q)=w-ti4)w,
1,:
’
IYe generAze the assumptions about costs by assuming that u is strictly increasing in 4, and v(L)= uL, u(H)=v,. Thus, we assume for a moment that any quality qE R is possible. Hence, tr(Pl4) is the profit to a hypothetical firm with quality 4 [and unit costs v(q)] when consumers believe q=H with certainty. Further, let P,(q) be the price that uniquely maximizes ~r(P/q), and assume that P,(q) is continuous in q. For later reference, we show in the ap-pndix tRat r”,(q) is strictiy increasing in q. Note that P,(H) = Pii. 1qi.e. the
full information price 0 f type M as defined earlier. Now, denote by a(P,(q)l L) the profit to the type L firm from selecting the optimum of the hypothetical type 4 firm, r”,(q), i.e., the optimum for type 4 when consumers infer 4 = H. We can write this as
By inspecting this expression, we conclude that there exists a 4> L sufficiently high so that w(P&j 1L)<~c(P~,~, L,O)ZI:~. Thm when 4 is sullTiciently large, tke type L firm prefers its full information price to P n(q) even if p(Pl(4))= 1. By continuity of x( *) and U,(s) there exists a unique qualify, 4> L, such that @P,(g) 1L)=st Finally, to make the signalling problem interesting we assume P,, 1$ b;‘,i.e. ;*a of type M is not in the set of prices i31efull information optimizing pp.,, that are dominated for tppe L. if, instead, P “, 1 E L’, then the type %fc:ly choose its full inrormation optimizer, PH.1 H, since type 8, has no incentive to mimic thi separating sequential equilibrium in I’,f, I E problem is mute.
P.B. Overgaard.
F
?,O
Price as a signal qf quality
FHo F,($
P
1
Fig. I
Now, we are able to state the main result on separating equi!ihria: There exists a unique undominated separaiing ij > H. come where PIa= P L,I! and P, = P :(/i) $r ~17~*1110 TlwmrP$P2 I_ d
..__
‘
_.._
equilibrium
out-
Coroiiary 1. PH = PI(q) > P,. ,, i.e. the separating equilibriu_m price of the type H firrat is distorted upwards relative to the full irlformation price for that typePrasf.
See appendix.
There results are illustrated in fig. 1.
The previous subsection made clear that, in the interesting case where P,, , $ L”, the price of the high-quahty firm is distorted upwards. Hence, signalhng the ccrrcct ty.pe is costly when quality is high. We may therefore wonder whether pooiing eq-uiiibria exist that survive the invocation of a domination criterion of the type suggested in section 2. We will show that the answer is afirmative when the prior consumer beliefs, pe, are sufiicicntly Favourable from the point of view of the firm. y definition, pa-S-vvalSle equifibria sham the characteristic that both types of the firm charge the same price, i.e. Pt = P, = P*. It follows that consumers infer no new information from the equilibrium play, and Bayes-consistency requires p( P*) = pO. To characterize the set of undominated pooling equilibria, let us recali that the iast subsection showed that the type M firm can ensure a profit of , i) in the separating equilibrium. M/e use this result to refine the hat type H should receive at least us therefore define the set I-J*@*) as PL=Bff=
P
t in a p
with
PO) is a
f5Saiy coGi$tiOfa
e further that H*(p’)#@ su~~tj~n
showed
that
L to stay in the
for p0
cf e use this to state a I as P* Al.* where
type
L is assured
ed pooling equilibrium ?* E L*(p’) CT prove in the appendix that P* EL*(P’) n II* is also a sufkient condition, and that N*~Q’) c Z..*(,-I~)_ This a!!nws ur to stata the following result: Theorem I. Any PE H*(p’) ~~~~~i~ri~ oumme.
can be susfained as an undominated pooling
Cor&.w_~ 2. For p” suflcienfly large W*(p’) #@: i.e. undnminntpd pan@ equilibria exist when the consumer priors are su&ienSly favourable.
Pmg. TO m
_
See ai;pndix. that
the
at
of
undominatd
pooling
equiI&&m
~~;~~~~s
is
non-empty for p” suE.kntly large, we recall that the set H*(p’) is defined by the inequality ~(f, H,p’)h n(Pi(4)9 H, 1). Thus, the set of pooling prices is non-empty if p” hpmia, where
492
P.B. Ouergaard.
Price as a signal $qucdirg
the firm, the incentive for type H to separate is too weak, and pooling equilibria exist which survive the iterated elimination of dominated strategies. Combining with theorem 1, we conclude from section 2 that restricting behefs according to the iterative domination criterion reduces the original set of sequential equilibria to a smaller set, consisting of multiple pooling equilibria for values of p0 in [pmin,1) and the unique undominwted cepara?*.*A-1P.Y.a..Y..ICIIC WVI-3 to &iG _,_:_:_F.:,...r&momarr* ing eouilibriam, Thus, aSthou@ th ic p,-!l#f*+r”illm nate sequential equilibria, the basic problem of multipiicity remains. In .“W f’~>:! a given set of particular, two different types of equi!ibria U_ a¶meter values. Hence, the model prediction of the price distortions is unclear. Problems of this kind has prosmpted a search for further refinements of equilibrium concepts applied in signalling games, most notably by Kreps (1984), Cho and Kreps i i987), Banks and Sobel (19873, and Cho and Sobeil (1990). These papers draw on the general treatment of strategic stability by Kohlberg and Mertens (1986). Below, we use the notions of Cho and Kreps a5 an illustration and assess hCW ihey work in the present context. In the remainder of the paper we wih assume p” E [Pin, I), and we will not attempt to dispute that separation is the unique outcome when this assumption is not satisfied. At a more general level it may be argued that even the (iterated) domination argument requires too much even from sophisticated players since the common knowledge and coordination assumptions are unduly strong. Below we consider further refinements and require even more of the piayers, and it foliows that the tentative objections raised will apply a fortiori if iterated dominance is considered strong. The particular refinement of Cho and Kreps (1987) wi!! i!lustrate the genera! -idea 4n the papers listed above. Let us fix a sequential equilibrium with prices (I’,,p#). and with equilibrium profits to type y given by n(ti,, y, p( p,J). Consider some alternative price P” # gL, pa. A deviation to P” is said to be equilibrium hmitmte$ for type q if
The ‘intuitive’ restriction suggested by Cho and Mreps is that the consumers should not believe that the firm will experiment with an equilibrium dominated deviation since, even if beliefs were favourabte following such a deviation, the 5rm is assured of a higher Layoff in eqnihbrium. Hence, a sequentia: equilibrium i P,, P,, p(P); :-13 iiidj .~;Jr-t~~~i~r~in~ if, in addition to (I) and (ii), the following is satisfied: (iv) Elimination of equilibrium dominated strategies; i.e. p(P)= 1 equilibrium dominated for type L(W) but rlnt foF :yp~ Eii,j.
(6)
if
P
is
to satisfy (iv), the m ltiplicity problem is
tc that the se
brium satisfies (iv).
P.B. Overgaard,
Price u; cz signal
n(P,L,p)
I
L.-B P LJJ
:
:
.
P* Pd P,($
ofquality
493
= n(P”,L,pO)
3 P
Fig. 3
““” ~GQ ~,,,~, rcca]i that p,(e) is defined Hence, self-enforcing equilibria exist. To cpp by W,(W, I)=& and it follows that any equilibrium undominated deviation by type M from PI(G) to PO, say, would be mimicked by type L if p(P’)= 1 (see fig. 2). Next, we state the following result.
Proof: See appendix. The proof of Theorem 3 demonstrates that at every proposed pooling equilibrium price P* E H*(p*) there exists an out-of-equilibrium price P” such that P’ is equilibrium dominated for type J#_,but not for type H. Hence, condition (iv) requires p(P ‘; = 1, and type H obtains z(Pd, H, 1) > II( P*, H: p”) ~kh destabilizes the equil,ibrium.’ This is ilhsstratcd in fig. 3. In the following we want to argue that, while the suggestions of Cho and Kreps (1987) summarized in condition (iv) certainly have the valuable characteristic that a unique equilibrium outcome is picked out, it is not clear that the arguments oehind the se!ection procedure arc persuasive.’ “vt’eshall argue that condition (iv) rules out equilibrium outcomes that can be rationalized in a sensible way, and we shall given an example of such an outcome. Before doing ~3, we wili make aV,Sz., rfimp observations concerning the refined selection criteria used so far. First, recall that a multiplicity of undominated outcomes exists, of which one is fully revealing and a continuum is non-revealing. Secondly, among the undominated equilibrium outcomes, the separating outcome is the least .mU .. that ‘refined’ cquiiibrium pr&rrcd by b&z types of the firm. It follrtws ‘As pointed out in note 1, we resfrict attention to pi;re strategy equilibria. If ihe consumer prior is suficientiy high. a third class of hybrid equilibria exists. These arise ii the players are allowed !ti mix between a separating and a pooliing price. But. WI eW@k::: G: &e proud ol -Gkh ii are elmMated if condi$ion (iv) is iir:posed. Theorem 3 GA!!&i.+v ihai & L&k1 rqu2:.,; ‘We should note that we are certainly not the firs! to qu4sz ihe logic Mind !hc su tians of Cho and Kreps. In a diflerent context, vi~n Damze (1?U. QYXldix e) has a c Cho-#reps re$nenleai. qUeStiOnedthe Ft3Wf’kQ khiti t%
P.B.
Ouergaard, Price as a signal
ofquality
Fig. 4
selection picks the worst undominated equilibrium outcome from the point of view of the informed party.4 i’hirdly, we observe that it is ‘as if’ consumer beliefs are used strategically as a ‘threat’ against the firm; e.g. the type W firm can be forced to choose PI(G) only because it accepts that lower prices will be taken by consumers to indicate that it was played by a type L tfrm. Finally, there exist p0 E [pmi”, 1) and PE II*(#) such that (9 (RP”)>~ (~I.,O,G;, (ii) (RP*)>~ (PI(Q), 1 I- and (iii) iP,~“j>~ (P,.,, 0) with probability atzi:ity ;P*
(I -p”)
and (P,(i), 11 with prob-
where >,. denotes performance relation for player i and C denotes consumers. I.e., undominated pooling equilibria exist for _JZ* suficientiy iarge that a:__ ~rp ntpt’prtp~ hrt hn#)? nC *ha k-r IL_ y.-s”e..‘~~~ “j -_-%-%iitxpum ‘_~~.=a~ =L=: ‘EEL_’firm --. --_ ag& .. =_.- fhp .___ pp,.n~::me~c _____+A r-siz !&a,-~ \_‘__ __ the C.._ _;Gi expected utility/consumer surplus is higher in the pooling equilibrium than in the separating eqtdibtium!, With these observations in mind we turn to consider the situation 1 illustrated in fig. 4. Assume &at pnor co2 sumec beliefs are given by oo3 and ihilt the c;msuxmers expect Che unique undomir-*+ .-d separating equilibrium to kp “1 Ar9..PI( ySerJsu. I.e., consumers expect with probability p* to observe P,(G), in which case beiiefs wiii ‘be updated to p(F,i@= !, and with probability (! --a@) to observe P &,*, in which case beliefs are updated to p(P&=O. Fdote that prior bePp-cc--L a high pi&tliiiQ 1613PLLWII to the type H firm; the type that consumers prefer. The question we wish to address in the following is: HOW should consumers react if they observe P*? First of all, the consumers are surprised by P* since it is a zero-probabiliiy event when the separating equilibrium is expected. We now want to argue that the consumers should also be confused. If the type H firm ‘deviates’ G-c-, P I Ab i: 18. QaG;V&f+& s.v_ ir; we, c,l~ ex-p&& qy&i&;ing i."asa r. l\y4Jr 3irir3e6 ii ZiCS’. ‘This is true for the criterion deveao by CSro and Kreps, as well as thse thst z&et b&&s even further. such as Universal Divinity [Banks and Sobel (1987)] and a suitable version
of Strategic Stability CVoLlkp - ~m.,rg and Mertens ! 198Ajl_*
and 5h(P”, t, pf P*,) > Te(P,.,. L,
Now, suppose p0 >p*
and assume for a moment
consider P* to be compieiziy ~M~~a!ive,
t
i.e. p{ P*) =: p
IriP+,H.pO)>II(P,(?).H,l)
and KCP*,L,pO)rR(PL.o,L,O).
So, if p( P*9=p” is indeed the ccnsumet inference, both types of the tirm will prefer to play P* to P,(g) and PLSo, respectively. Menue, if consumers interpret P* as unfnfcrrmaiive, kfi+k _W.** :yq
cf tt;: fi;7;; hZVC &ii iiiiCi& ;si deviating from the separating choices. This reinforces that the consumers .z .__ _ -’ -1 _i:- _I.__ -
if the firm takes the pessimistic ‘beiicfs a; our-of-equiiibiium pomrs as given will the m-iii stay at the spec~riett separating choice. 3IlUUlU
ut
LUIIIU3cW
alltt
UUSGrViii&
-
i?
Qdy
-_ _--. -*here u(p,q) ¬es ;hc @~fi~cmer p&y& izicssgfgd QE some van Neumann-Morgenstern utility scale. Define p*+ by jminip”jj3P,p”(u(P,H)-u(P,(~,H))
l(i-pO)(u(P,.,,Lf--ujP,,Lii). Thus, the inequality above states that the exp2cred consumer payoff associated with the price P* and posterior beliefs p” dominates t payoff in the unique separating equilitrium. Hence, even if the completely uninformative, it is preferred by consumers if the priQ,r of type W, p”, is s~~~ie~t~y high. As proved earlier, the only admissible in (iv) is p(P*j=O. -With the above remarks i to suggest an alternative to the reaso other outcomes can be rationalized
of carrying out arbitrarily comp!ex of the model i&ted above, and est that ~~ns~rn~rs can rely on a move to a ~~~e~~-du~~~~~~~e ~i~~bri~rn.If such pessimism 1s sittlation emerges To c;t?Lz thir;,
LO) is implied by z(P,H,p’)z that ~(P,~.p~)~~(P~.~, 1). Hence, am defin the set of prices, for given beliefs, which by all! to the begat ~~u~~~brjurnoutcome. By definition, the is a eon-~rn~ty closed subset of N
set P
0
> ZZZ
P
*_
P
min
and
when ~~&j=rnax(p*,p**{. Note that any message in by the type N firm will tentially be duplicated by the type L firm, en the Dekfs, since S*(p’) are preferred by both to the noted_ this rB=iarCWJ-*c ._ .. .. . .._“.. that consumers !&““:A -L-WE_ fter observing at least som2 mssagcs Iii S*(p’]. nt of view
of the type H firm and look at prices in the nt that p(P)=p’ for any PEP(#), and
is certainly the case for p” close to 1. Now, n_pCrand sequential rationa!ity Since the type L firm to follow the proposed deviation bv type
p(P) = p” VP E S*(p’).
criterion (where A stands Car alternativk): th
typesof the firm choose a p&q F; ihat
price ilial saiisfks criterion A, which consumers can cieariy ,,,..:_ ill . . .L_L.&PJLIWUL\b &Wx..lAtGl1IQIifl an*.3 UrG--I--_ y1KPt3.But, so far the story have nix compietely specified beliefs foiiowing
P.B. Orergaard. Price as a signai oJpquailty
497
PT -I
P,!g!---
I
P
H.1
i
I
?!-I
a
I I
1
p”
Fig. 5
~p~~*(po)\,.p~_
szstafn
---* s--. ,--!--- -:-‘---I ~JILKGU uui oy crirerion A 2: a pooiing equilibrium, we must specify p(P), VP E 2*(j3’)\, f Pcj, such that ir(P, i, p(3)) 5 R(P~.~,L,O). Criterion A picks out, among the undn*;=e+e’ “I‘IIIIt&L~” ~~~l;nn ta”“UUC) W~PPC r_‘___ in -1; S*fpO), tit2 p&~&f-Chiiiiidi;ilg uuicome for type H. Above. the reader lmay have wondered why we took the point of view of the R fhm ____ typo1 II . . . . .., and suggest that this is quite arbitrary. To this we note that, in models of this type, it is existence of a t9.w SjP;- L firm that causes the efficiency problem. At any point in the game the type L firm is deciding whether or not to mimic the type H firm, hence, deciding whether to accent a+~prlllilihrin~ nrni’;la by :yi;e H= 1lJl-r z- i’ksp,J&, _-‘-q~“‘Y..1. pr.“,.IY ynnnc.+ v,vyvos *r ?&.! --.WC r-f-htzltt:IL l!iLS virv_ p ihe
To
the
&)p’ce
l
emphasis should be on the decision of type II, and at P, nOthe
AWse
dkct
or. profits to type li of the possible existence of a low-quality type is minimized. Since the type L firm and the consumers realize this, the price ? H.pObecomes focal, and the consumers should attack, a lo=* probabihty to type H after a message P # PH,Po.The reader can easily verify for himself that
the assumption P H,pO~S*(~o) is immaterial for the qualitative rest&s, and that a maximizer under criterion A always exists since S*(p’) is compact. Thus: the alternative criterion picks out a unique point from the set of undominatcd sequential equihbrium outcomes. “VVewant to compare this outcome with the separating outcome resulting from the selection procedure of Cho and Kreps ( 198 i). A questionable feature of the Cho-Kreps outcome is its indc-$ndence of the consumer prior, p”, and this causes a puzzling discontinuity at p@= !. To
e see this, assum_ethat consumer N-A-I -. CDE (cl. 3:. ad :et )J‘ Iv13 iii3 @?X Sy p” = 1 Ep+ 0. From the analysis above, it is clear that as &pdecrecases the outcome chosen according to reqwirement (iv) is unchanged, i.e. P, = Pi(q) W s(O,1). Now. set EP~0, which imp!ies that the consaumers discard the ~oss~b~i~t~df Tfa ,Bs;c crkc.a *l-w sa04:K.a.l‘-rcat11mC” t* .U.l l;lil .*.‘“.a.aU..V... .nfr\rpracsl;nra tyIe L altog&!sr, 1.2., ,mp 11. CIllJ Q.c&L9& &elk ‘srl1ya.h ..” .W&U... optimizer of type pi is P, = P,, , . Hence, the self-enforcing equilibrium is discontinuous at J-I’= 1 as iiiustrated in fig.. 5. der criterion A, ty? Compare this with the alternative
II picks, and type L mimics, the pri
498
P,=
at-g max lr{F, ii’,
p”i
PES'(P"I
for all JIO~~= ax{p*. p** i. This is illustrated in fig. 6. This alternative also &iiitj & disiontiiiuitjr iji the priCt2futict n of type N. But, the di~o~t~nuity appears at i, which is a point where the reto-rankg of ik xyalaiing and pooling equilibria changes. PM+,O~S*@“f for p” sufficiently close to 1, and it follows that the equilibrium selection criterion suggested here h= particular economic appeal when the a priori probability of a low-quality type is small, since the type W firm and consumers suffer only vanishing payoff losses from the existence of a low-quality type as p” approaches 1. Thus, the introduction of a low-quality firm with a small prior probabiiity has a nealirsible effect on the expected payoffs of the high-quality firm and _ _ On the contrary, if we insist on the consumers when criterion A i-II bmptoyed. e r,_\ "eQi~~~~Z?TSZf
a."e."f
i:Yj*
thc
ty;<
j;
z~m
2nd
ihe
c~g~~pme~s
su-ff&-
z
non-frivi&
is=
even from an infinitesimal prior probability of a !ow quality. The aim of this section has nui ‘been to shut that criterion A is generaliy more sensible or intuitive than &= Chn-Kreps criterion. A sin& refinement of the sequential equilibrium concept seems too much to hope for, and our purpose ;,-_,,c,.%r- *t-r +.?,, L+S,i” “&III\91I,\.*nrc ,,*a, ai2 &Xiiaiiv~ selection proCedure can be LMK9 LiGtiII.-
cf'gwcffs
_
iesds to outcomes which involve pooiisg of dik:cM typ. hether a discontinuity at p”= 1 is considered to be serious depends on the situation. we want the model to dcscrik~?and hence o he interpretation of seemmgly trivial changes in the model. It may well the case thai a
model rn which p” = different kom a model in which /Pm 1, since they may describe very different phenomena, and as such we .the urn outcomes to differ. In t&is interTpretation, .__!_usmg illustrates the familiar point that adding or deleting ith ‘zero’ prior probability ay ultimately change the outcome of a gan-W6 1
---I:
is
sr%ni~~nliy
in signaIfing games. Two variants of signatli ted& in the industrial organization literature. variants are characterized by the space of actions available to the uninform uninfmned
plaver to vatv cnntinrbnllclv - - ‘--‘J piah the signa! a~! the pos:;:io;
beliefs [see ais*- the quai&signailing modei of Milgrom and and the Spence-model discussed in Kreps !lQW) and 6110 -___ -_ ~11.4 Roughly, the conclusion from such mohels is that a t-nique separating equilibrium outcome survives refinements of the Cho-Xreps ty general results of Cho and Sobel (1990)]. A feature of the separati is that it is the Iesp_stpreferred by the informed player in the set of undominated equilibrium outcomes. Therefore. we con&de that our discussion of equilibrium selection appli-ya +ro ihis class of games as a whole.’ In the second variant, repro sented by the limit-pricing mode1 of Roberts ( 1982) a-r&d,pariicuiariy, the version of Bagwell and the best respo:rse of the uninformed player maps the posterior beliefs onto a two-element set (viz. {Enter, Not enter)). 4s a con result can be expected from ~ti!ikium refinements .4 typical result is that a uniq outcomes remain, Often, see e.g.
equilibria admiis a unique Pareto-optimal
equili pi ayers tlI&;Fk *.*I**. is pOO/ifig_Furthermore _-__-“, the “god
type receives iis fuii-
‘in the Spence-model the payoff to the rl”iEifGzizd player is the same in an to the zero-profit c~=r.diti~n.So, the critique of the ~~arati~g e~~i~~~~~rn payoff-domination arguments for the uninformed player, but will rely on an ar pay&s d :k aark% types of t”neirr;Nlil~ I”iaypr.
VW) -
(A.ij
and
these to obtain
and substitute L into (A.!) to obtain
9. 1).-(P,(q2)-v,).~P,(q2), reasing
ve
l)>O.
in 9 for q>
t. Now, the continuity of ue tj> L such that n(P,(q3 i L)= we must have
&&i*8m ..m..3 c&a&o -._ _ 0. ierminate the e.g. make a ‘mpra that a rational t
W firm in our
. .
since, by definlt:oR,
if)” maxrm,,e, . ;s
c.
and we conclude x(P,(&, l)>x(P,, v(@> oHj into (A.2) which impiies
.RiD
,‘,a
,I,,
LP
l? 1,
*et “,I
PS
A.4
.
Then
1) or P,(q)< P,,. Now, substitute q (i.e.
ut, this is a contradiction since P,(q) maximizes x(R)i) by assumption. Thus, we must have R(P~, L, 1)= sL. Thirdly, we prove that P 1(i) is the unique maximizer of 7t(P,H, i j subject to the condition 7t(P,L,1)=sL. Pick any P#P,(q’) such that 7t(P,f., l)=sL. Then
and (P-U,jX(Y,
-
_.
I)-(P,Iq)-uL)x(JJ,(q),
1)=-O.
(A.3)
Add these to obtain
tp1(4)-~,)xtp,t(63,I)-w-b&w,
1).
Add (A.3) to get
Since this is positive, we conclude that P,!$j is the unique maximizer r.f x(P,W, 1) given n(P,& f)=sL. Finalty, to show genera! existence of sequential separating must picve i’ n ii”+@. Hence, all that remains is to s the de~R~t~o~of 4 ensu is the maximizer of ?t(
P.B. Owrgaard, Price as a signal
502
ofquality
Add these to obtain
which implies x(P H,O,Q)<:x(PI(q),1). But then (since G>H)
and since x((P,,,, ~)>x(P~,~,O) we infer
.
But, this is a contradiction since P,(i) is the maximizer of TE(PI4). We conclude P,(g) E If” and separating equilibria exist. This ends the proof of Theorem 1. ?roof of Theorem 2
We first prove that H*(p”) c I,*(#‘), and then we show that any price in H*(p*) can be sustained by specifying admissible beliefs. First, we claim that any price in H*(pO) is also in L*(p*). Suppose not, A rannll Frfirn *ha rc#v-J “L ,-J Th.7W-W.Pm 1 tll\nt dD Ia P lb-J Than 3E:u .kdwcz.. II”‘,. L1.1, ydbl”I 1 Il’bVLWII‘ a CIlQL rqr 1\y,, U) .p-.i . i..w..
uw%-~,)XC~I~~,U-w--i;,W,pOb=G and (~-~,)x~~,vO)-(P*(~)-~“)x(P~(~);
I)@.
(A-4)
Adding these gives (FM--oL)(jr
which implies x(P:(#, 1) >x(P,#).
But then, using (A.4), we conclude
And, since x(P, 1)> .x(P,PO),we obtain
ique maximizer of z( P I&.
We
P.B. Ooergaard, Price a:; a signal of quality
503
any P$L*(pO), p(P)= I, (2) for any P E L”(p”)\I-P(pO)* p(P) = 0, (3) for P= P* EH*(~O), p(P) = PO,and e.g. (4) for any P E H*(pO)\{P* >, p(P) =o. (1)
fOi
First of all, a potential candidate for a self-enforcing pooling price must be in H*(p’). Otherwise, the equilibrium is trivially broken. Now, fix any pooling price P* in II*( By continuity of B,(q), there exists a $ > L such that (‘4.5)
Any P* satisfying (AS) must be less than Pr(qr)_ Hence x(,P*,~~)> .JD i,l\ l\ Xl L... _..I-.,A-%5(1 r\y J, IJ. lYUW,tva1us1e (P*-
4fw*~P"!
-(P*(41)-uH)X(Pl(41),
1).
Subtracting (A.5) we get (VL
Ezixe,
-~~rs~c~(~*,P"~-xcP,~ql),
iiic iype
H trrnr siriciiy
w
wiih ixTi& pjF,(q’jj=
i to P
with beliefs p(P *) = p’. We conclude that there exist prices in a close neighbourhood of P1(ql) which are equ;r;h*;lllUrlUm dominated for type L but not for type H. Thus, imposing condition (iv) destabilizes P*. Referams Akerlof, G., 1970, The market for ‘lemons’: Quality uncertainty and the market mechanism, Quarterly Journal oi Economics 84. no. 3,488-%O. Albaek, S. and P.R. Overgaardi, 1991, Manufacturer-retailer relations when the wholesa!e price signals demand intensity, The Academy of Marketing Science, International Conference Series 5, 167-371. AlbaPk, S. and P.R. Overgaard, 1993. IJpstream pricing and advertising signal downstream demand, Journal of Economics and Management Strategy !, no. 4.677-698. Bagwell, K., 1987, Introductory price as a signal of cost in a model of repeat business, Review of CA . . ~onuiiiic . c-..A:-3I”urls .‘9; II=_,:3, 35384. Bagwell, K. and G. Ramey, 1988, Advertising and limit pricing, Rand Journal of Economics 19, no. I, 59-‘31. Ragtell, K. and G. Ramey, 1990, Advertising and pricing to deter or accommodate entry when demand is unknown, International Journal of Industrial Organization 8. no. !,93-“113. ina .+_-__--. -7mes. Econometrica 55, no. 3, Ranks, J.S. and J. Sobel, 1987, Equilibrium selection in signal..., 647-H. Binmore, K., 1987, Remodeled rational players, Mimeo. Cho, L-K., 1987, A rellnement of sequential equilibrium, Econometrica 55, no. 6, 1347-1389. Cho, I.-K. and D.M. Kreps, 1987, Signaling games and stable equilibria, Economics 102, no. 2, l79-221.
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L-K. and J. Sobel, 1990, Strategic stability and uniqueneslr in s~~~~~~~~ Economic Theory 50. no. 2. 381-413. Kohlberg, E. and J.-F. Mertens. 1986. On the strategic stab~~~~yof no. 5, 1003-1037. nu. 758 ~~!~~f~~~~ Kreps, D.M.. 1984. Signahng games and stable ~~~~~bria, University. Graduate School of Business, Stanford, CA). Kreps, D.M;, P. Miigrom; 3. Roberts and R Wilson, 19R2. repeated prisoners’ dilemma, Journal of Economic Theo Kreps, D.M. and R. Wilson, 1982, Sequential ~~i~ib~~a, EI; Mailath, G.J., M. Okuno-Fujiwara and A. ~o~~~ew~~~c,1 signalling games, Mimeo. Milgrom, P. and J. Roberts, 1986, Price Gad advertising si~~~~s of Political Economy 94. no. 4. 796821. Overgaard. P.B.. 1991, Product quality uncertainty: Strate ati product markets with adverse selection and adverse P Universitt Cathoiique de Louvain. Louvain-la-Neuvet. Spence, A.M., 1973, Job market signaiing, Quarterly Journal of Economics 87. no, 3. t 35-374~ van Damme, E.C., 1987. Stable e+ilibria and forward induction, DP no. 12$ ( Bonn, Department of Economics.
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