Reserve Price Signaling Hongbin Cai, John Riley and Lixin Ye*

Transcription

1 Reerve Price Signaling by Hongbin Cai, John Riley and Lixin Ye Sepember, 00 UCLA, UCLA, and Ohio Sae Univeriy

2 . Inroducion In he andard privae value aucion model, he eller profi maximizing raegy i o e a reerve price above her own valuaion. In o doing, addiional revenue i capured whenever here i only one buyer wih a valuaion much higher han her own. Moreover, hi reerve price i independen of he number of bidder. Thu excep when he number of bidder i very mall, he probabiliy ha here i only one high valuaion bidder i low. Thu he exra profi reuling from eablihing an opimal reerve price i low. In hi paper we ak wheher he reerve price can have a much more cenral role in an aucion. We conider an environmen in which he eller ha privae informaion abou he objec characeriic ha are valued by he buyer and he eller. If direc verificaion of he eller informaion i cole, eller wih ufficienly high valuaion have an incenive o reveal heir informaion ince buyer will hen be willing o bid more for he objec. Since he ame argumen hold for hoe wih he mo valued produc characeriic in any pool, hee ype alway have an incenive o reveal and o he only Nah equilibrium i full revelaion of he eller privae informaion. Bu if direc revelaion i ufficienly coly, eller wih high value characeriic have an incenive o ry o ignal hi o buyer by eing a high reervaion price. In ecion we decribe he model and how ha i i he eller markup over her own valuaion ha i he key variable, raher han he reerve price. Then, in ecion we characerize he eparaing Nah equilibrium. Since he model i formulaed o ha i belong o he family of creening If he minimum valuaion of buyer i ufficienly higher han he eller valuaion, hi can never be he cae. The eller hen ha no incenive o e a reerve price above he buyer minimum valuaion.

3 model examined by Riley (979), andard reul in he lieraure imply ha hi eparaing equilibrium i he unique equilibrium which aifie he Cho-Krep Inuiive Crierion. In ecion 4 we argue ha hi crierion can only be applied in knife-edge cae and inead conider a Srong Inuiive Crierion. For imple finie ype example i i eay o ee ha he crierion i aified if and only if he probabiliy of more favorable ype i no oo high. Finally, in ecion 5 we conider a family of coninuou model of which he Spence ignaling model and he reerve price ignaling model are boh member. We obain neceary condiion for he Srong Inuiive Crierion o be aified and argue ha here are plauible parameer value for each model for which he neceary condiion hold. However i i alo eay o find parameer value for which no Nah equilibrium aifie he crierion. We eablih ha a neceary condiion for he crierion o be aified i ha here i a ricly poiive ocial co of ignaling for all ype. Many of he model in he lieraure (or a lea heir coninuou varian) have he propery ha he marginal ocial co for he lowe ype i zero. While our reul are no direcly applicable, we conjecure ha no Nah equilibrium will aify he Srong Inuiive Crierion in hee model a well.. The Model We conider a econd-price aucion in which n buyer bid for a ingle, indiviible objec. Buyer i valuaion for he objec Vi = + Xi where i a common value componen and X i i a privae value componen. The common value componen i oberved only by he eller, and he privae value componen, X i, only by buyer i. Ex ane, i diribued wih c.d.f. G() and uppor T = [, ] and he X i are i.i.d.wih c.d.f F() and uppor

4 [ xx., ] Le f( x ) denoe he deniy funcion of F(), and g () denoe he deniy funcion of G (.). We aume ha and X i are independen, i =,,..., n, and f( x ) > 0 for all x [ xx, ] and g () > 0 for all [, ]. Throughou we will aume he following regulariy condiion A: Diminihing marginal revenue: Fx ( ) J( x) = x i ricly increaing in x. f( x) The eller move fir by announcing a reerve price. The buyer hen ubmi ealed bid. If here i no ale, he eller valuaion for he objec i α (0,]. α, where Conider an aucion in which i i believed ha he eller ype S T. Define θ = E { S}. Then a buyer of ype xi ha a valuaion θ + xi. Suppoe ha he eller acual ype i and he e a reerve price of r. By he well-known reul of he andard privae independen value aucion, all buyer bid heir valuaion θ + xi, reuling in he following aucion oucome. If θ + x() < r, he objec i no old. If θ + x() > r and x() θ + < r, he objec i old a he reerve price. If θ + x() > r, he iem i old a he price θ + x(). The eller expeced payoff i herefore x U = αf ( r θ) + rf [ ( r θ) F ( r θ)] + ( θ + xdf ) ( x) () () () () r θ x = αf ( r θ) + θ( F ( r θ)) + ( r θ)( F ( r θ) F ( r θ)) + xdf ( x) (.) () () () () () r θ where () () n F ( x) = Pr{ X x} = F ( x) and F x = X x = F x + n F x F x. n ( ) Pr{ } ( ) ( ( )) ( ) () () () By he Revenue Equivalence Theorem (Myeron 98, Riley and Samuelon 98), all our reul coninue o hold if he objec i old in an open acending price aucion.

5 Define m r θ. Then we can rewrie he expeced eller payoff a follow. U (, θ, m) = αf ( m) + θ( F ( m)) + Bm ( ) (.) () () where x Bm ( ) mf ( ( m) F ( m)) + xdf ( x) (.) () () () m Since i will be ueful below we noe ha B ( m) = F ( m) F ( m) mf ( m) = F ( mjm ) ( ) (.4) () () () () Differeniaing he eller payoff by θ, Uθ (, θ, m) = F ( m). (.5) Thu U (, θ, m) > 0for m ( xx, )and U (, θ, x) = 0. θ Alo, () θ Um (, θ, m ) = ( α θ ) F ( m ) + B ( m ) () Since = ( α θ J( m)) F ( m) (.6) U m i increaing in and U θ i independen of i follow ha Thu he ingle croing propery hold. () dv U m = 0 dm < (.7) U U v Define M(, θ ) o be he opimal choice of m o ype if he percepion i θ. By (.6) and Aumpion A, if U (, θ, m) 0, hen U (, θ, m ) < 0 for all m > m. Thu he opimal choice i unique: m m M(, θ ) = x, α θ < J( x) J α θ α θ > J x ( ), ( ) (.8) 4

6 Full Informaion If were direcly obervable by buyer, he percepion her markup o maximize Um (,, ) ha i o chooe θ =, o he eller would chooe m () = M (,). Wih α = he opimal markup i independen of, oherwie i i ricly decreaing. The inuiion i clear. Wih α < he bigger i he bigger i he gap beween he eller valuaion and reerve price and hu he greaer i he lo if he iem remain unold. Hence he eller opimal raegy i o mark he reerve price up le. In he uniform cae wih uppor [ xx,, ] J( m) = m ( x x), and we have m () = Max{ x, ( x x ( α))} Thi i he cae depiced in Figure. under he aumpion ha x < ( ( ) ) x x α i hu m ( ) = ( x x ( α) ). i i The indifference curve I for a ype eller ju ouche he line θ = a m ( ). Tha i, given he correc percepion, he eller opimal markup i m ( ). Similarly he indifference curve I for a ype eller ouche he line θ = a m ( ). Aymmeric Informaion Wih aymmeric informaion, he ype eller mu chooe a markup ha i incenive compaible. Thu i canno be in he preferred e for a ype (he haded region in Figure..) The Pareo efficien eparaing choice of markup i herefore m ( ). 5

7 θ, m () I I I m ( ) m ( ) m ( ) m Figure.: Separaing Equilibrium. Characerizaion of he eparaing Nah Equilibrium In hi ecion we fir conider he cae in which J( x) < ( α ). Then he full informaion choice of markup i m () = J ( ( α)). (.) To implify noaion le m be he choice of he lowe eller ype, ha i m= M(, ) (.) Le Tm ( ) be a oluion o he following differenial equaion: 6

8 Um(,, m) ( m) = (.) U (,, m) θ The family of oluion o hi differenial equaion on he domain over which Tm ( ) i increaing i depiced in Figure.. m () = Tmk (, ) = Tmk (, ) = Tmk (, ) z m Figure.: Soluion o he differenial equaion Following argumen paralleling hoe in Riley (979), here i a unique oluion hrough he lower endpoin ( m ( ), ). Suppoe ha he buyer percepion i ha θ = Tm ( ). A eller of ype hu chooe m o maximize UTm (, ( ), m ). Differeniaing by m, d UTm (, ( ), m ) = U θ (, Tm ( ), mt ) ( m ) + U m(, Tm ( ), m ) dm Um(, Tm ( ), m) = Uθ (, Tm ( ), m)[ T ( m) + ] U (, Tm ( ), m ) θ 7

9 Um( T( m), T( m), m) Um(, Tm ( ), m) = Uθ (, Tm ( ), m)[ + ] U ( T ( m ), T ( m ), m ) U (, Tm ( ), m ) Thu, ince he ingle croing propery (.7) hold, we have incenive compaibiliy. θ Thi implie ha in equilibrium, he ype eller chooe a reerve price m (), which i a oluion o he following differenial equaion: d Um(,, m) = (.4) dm U (,, m) Subiuing (.5) and (.6) ino he above equaion and implifying, we obain θ d f() ( m)[ J( m) + ( α)] = dm F ( m) () θ = f ( m)[ Jm ( ) J( m ()] () F ( m) () (.5) Conider a oluion o he differenial equaion (.5) uch ha ype ha her opimal (full informaion) ignal. Tha i, he oluion goe hrough he poin Le m= m() = M (,), hen we have he following characerizaion reul: ( m ( ), ). Propoiion : The oluion o (.5) wih boundary condiion m () characerize he unique eparaing equilibrium. = m To compue he equilibrium payoff for differen ype of eller, define Then by he Envelope Theorem, we have u () = MaxU(, Tm ( ), m) m 8

10 u () = u ( ) + α F ( d ) = U(,, m) + α F ( d ) () () (.6) From (.5) d ( F() ( m)) ( α) f() ( m ) = f() ( mjm ) ( ) dm Muliplying boh ide by ( F() ( m)) α, d α α [( F() ( m)) m ( )] = f() ( m)( F() ( m)) J( m). dm Inegraing we obain: m α () () () m α ( F ( m)) m ( ) = f ( y)( F ( y)) J( y) dy. Therefore, m ( α) α m ( ) = ( F() ( m)) f() ( y)( F() ( y)) Jydy ( ) + (.7) m In he pecial cae when X i uniform wih uppor [0,] and α =, we can inegrae analyically o obain where m =. + m m m ( ) = 4( m m) + ln( ) ln( ) + m m Comparaive Saic We have hown ha for any α (0,], here i a unique eparaing equilibrium aociaed wih he boundary condiion ( m, ), which can be characerized by he ordinary differenial equaion (.5). 9

11 Wrie / bmκ (,, ) a he righ hand ide of equaion (.5), where κ i a parameer in he model (e.g. α, n, ec.). Then (.5) can be re-wrien a dm d = bm (,, κ ) (.8) Lemma : Suppoe (i) bmκ (,, )/ κ < 0 for all ( m,) and (ii) for any κ > κ, m( κˆ ) m( κ), where m( κˆ ) and m( κ) are defined by ( m( κ)) = bm ( ( κ),, κ) = 0 and ( m( κˆ)) = bm ( ( κˆ),, κˆ) = 0. Then he oluion o equaion (.5), mκ (, ) i decreaing in κ, or mκ (, )/ κ < 0 for all >. m b Proof: Differeniaing (.8) by κ we have = < 0 κ κ by (i). So mκ (, )/ κ i decreaing in. By condiion (ii) we have mκ (, )/ κ < 0 for all >. Q.E.D. Propoiion : In he eparaing equilibrium, for every reerve price r () = + m () i higher for larger α or larger n. Proof: See he Appendix. >, he markup and hence he Inuiively, when he eller ha a larger reervaion value (larger α ), ignaling co from higher reerve price are maller becaue he eller payoff i α in he even of no ale. Similarly, where here are a larger number of bidder (larger n ), he ignaling co from higher reerve price are maller becaue he probabiliy of no ale i lower. A ignaling co go down, reerve price will be higher in equilibrium. The model can be readily exended o iuaion where he relaive imporance of common value and privae value componen in bidder valuaion can ake on any 0

12 arbirary value. Tha i, le Vi = + β Xi where β (0, ). Clearly β = in he baic model. A before le m= r θ bu now call m = m/ β he relaive markup. The inereing queion in hi cae i how he relaive markup change a β change. Le θ = θ / β and = / β, hen he expeced revenue o he eller wih relaive ype, relaive percepion θ and relaive markup m i a follow: U (, θ, m ) = αf ( m/ β) + ( m+ θ)( F ( m/ β) F ( m/ β)) + ( θ + βxdf ) ( x) () () () () m / β = βα [ F ( m ) + ( m + θ )( F ( m ) F ( m )) + ( θ + xdf ) ( x)] = βu (, θ, m ) where U (.,.,.) i defined by (.). () () () () m So he problem can be viewed a a normalizaion from our baic model. Thu all he analyi follow a before immediaely. In he cae of full informaion ( i known o he buyer), we have x x (.9) m () = x, ( α) < J( x) J ( ( α)), ( α) > J( x) (.0) Clearly, he full informaion relaive markup m () i increaing in. Given, a β goe o infiniy, m goe o max{ xj, (0)}. Tha i, when he privae value componen dominae, he opimal (relaive) reerve price largely depend upon he diribuion of he privae value x. On he oher hand, a β goe o zero, m goe o x and o go o zero. Then he reerve price converge o, ha i, he opimal reerve price depend on he common value componen. m mu

13 In he cae of aymmeric informaion, he differenial equaion (.5) (wih he normalized variable) characerize he eparaing equilibrium. In he equilibrium, m () i increaing in. So for any given, he relaive reerve markup m i decreaing in β. A he privae value componen become more imporan, he relaive markup decreae. The inuiion i ha a β increae, ignaling co from higher reerve price increae becaue he opporuniy co from no ale increae. Higher ignaling co implie ha he relaive markup will be lower in equilibrium. Ouide Cerificaion We now conider iuaion where in addiion o ignaling hrough reerve price, he eller can credibly reveal o he bidder hrough an ouide cerificaion agency a a fixed co of c > 0. The queion i when he eller i willing o pay for uch a ervice. Le u UM Um () = (,, (,)) = (,, ()) be ype eller expeced revenue under he full informaion eup, and le u () = Um (,, ()) be ype eller expeced revenue in he eparaing equilibrium (a before). Alo le W() = u () u (). Immediaely, W () = 0 and W () 0 for all by he definiion of M(, θ ). To furher implify noaion, le m M m m = (,) and = (). By (.6), dw du du = d d d = ( α) F ( m ) αf ( m) () () = + F() ( m) ( α)( F() ( m) F() ( m )) Since m m, we have dw / d > 0. Clearly he eller i willing o pay for he cerificaion ervice if W() c. The following reul i immediae.

14 Propoiion : For any c > 0, here exi a cuoff ype uch ha for all he eller hire he ouide cerificaion agency; for all [, ), he eller ignal hrough reerve price r (). [,], 4. Choice of Refinemen Conider he following modificaion of he baic model of ecion. There are hree ype of eller τ { τ, τ, τ}. The eller i ype τ i wih probabiliy f i. Type τ ha common value componen and reervaion value componen and reervaion value α. Type τ ha common value α. Type τ ha common value componen and reervaion value α, where α = α. From (.), he payoff and he common value componen for he hree ype are a follow. U( τ, θ, m) = αf ( m) + θ( F ( m)) + Bm ( ), τ = () () U( τ, θ, m) = αf ( m) + θ( F ( m)) + Bm ( ), τ = () () U( τ, θ, m) = α F ( m) + θ( F ( m)) + Bm ( ) () () = αf ( m) + θ( F ( m)) + Bm ( ), τ = () () Thu ype τ ha he ame common value componen a ype τ bu he ame ignaling co a ype τ. Define S = { τ, τ} and = E { τ S}. Conider he ignal percepion K couple {( m, ),( m, )} in Figure 4.. A depiced ype τ ricly prefer ( m, ), K while he oher ype ricly prefer ( m, ). Thu he incenive compaibiliy condiion K hold and {( m, ),( m, )} i a Nah equilibrium.

15 θ, I I m C m K m m Figure 4.: Separaing Equilibrium Now conider he andard refinemen propoed by Cho and Krep (987). Inuiive Crierion: Suppoe ha when ome ype make an ou of equilibrium choice m, her ype i correcly perceived and, a a reul, ype i beer off. If no oher ype T i beer off mimicking ype, he percepion of he buyer i credible. Cho and Krep howed ha for he andard ignaling model he Pareo efficien eparaing Nah equilibrium i he only equilibrium for which here i no credible ou-ofequilibrium ignal. For he knife-edge cae wih f = 0 and hence =, i i eay o ee ha Pareo inferior eparaing equilibria canno urvive hi credibiliy e. Conider 4

16 he eparaing Nah equilibrium depiced in Figure 4. and he ou-of-equilibrium ignal C m made by ype τ. If buyer correcly perceived ha i i ype τ who ignal, ype C K i ricly beer off, U( τ,, m ) > U( τ,, m ). Moreover, a depiced in Figure 4., C C U( τ,, m ) < U( τ,, m). Thu ype τ ha no incenive o mimic and o ( m, ) i indeed credible. Wih hree ype, however, he Inuiive Crierion ha no power. For if here i C ome pair ( m, θ ) ha i ricly preferred by ype τ, i will be preferred by ype τ a well. Thu here i a coninuum of eparaing Nah equilibria ha urvive he credibiliy e. Thi i highly unaifacory. Given our aumpion, he andard model wih wo ype and he -ype model are obervaionally equivalen. Thu i i naural o eek a refinemen ha elec he ame ube of Nah equilibria in each cae. In our imple example i i clear how hi can be done. Suppoe buyer oberve he ou-of-equilibrium ignal C m. Knowing ype τ and τ have idenical preference, he percepion i ha he expeced common value componen i. Given hi percepion, C U( τ, m, ) > U( τ, m, ), i=,. i K i C However U( τ, m, ) < U( τ, m, ) o ype τ ha no incenive o mimic. Thu he Pareo dominaed eparaing equilibrium fail our renghened Inuiive Crierion (formally defined below). The Srong Inuiive Crierion wa fir inroduced by Groman and Perry (986a, b). Since we will be uing i o conider he ignaling model 5

17 wih a coninuum of ype, we modify heir crierion o ha i can be applied in hi cae a well. Srong Inuiive Crierion: Le T = [, ] be he e of ype, le Y = [ y, y] be he e of poible ignal, and le z() be a ricly increaing mapping from T inoy. Thi i a eparaing Nah ignaling equilibrium if, for all T, Uz (,, () Uz (,, ( )). Suppoe ha here i ome ignal ŷ, a ube of ype S T, percepion θ = E { S} and ε > 0 uch ha () i U (, θ, yˆ ) U (, θ(), z ()) + ε,forall S ( ii) U (, θ, yˆ ) > U (, θ(), z ()) + ε,forome S ( iii) U (, θ, yˆ ) < U (, θ(), z ()), forall S Then he ou-of-equilibrium ignal-percepion ( yˆ, θ ) i credible. An equilibrium aifying he SIC if here doe no exi uch a credible ou-ofequilibrium ignal-percepion ( yˆ, θ ). Noe in he above definiion of credibiliy, all ype who chooe he ou-of-equilibrium ignal have a payoff ha rie by a lea ε. Thu each uch ype can ignal by pending ε. Then he ou-of-equilibrium ignal can be a choice of ŷ ha i in he e Z = {(), z T}. Uing he hree ype example, we now how ha here may be credible ou-of equilibrium ignal for all Nah equilibria. Conider Figure 4.. The Pareo dominan Wih a finie ype pace and coninuou ignal pace Y, he e of ignal ha are no eleced in a eparaing equilibrium i dene in Y. Thu here i no need o ignal by hrowing money away. 6

18 S eparaing equilibrium i {( m, ), ( m, )}. Define = E{ T}. A depiced, if an ou of equilibrium ignal ˆm i oberved and i i perceived ha i could be from any of he hree ype o ha he expeced common value componen i, all hree ype are beer off. Thu he parially eparaing equilibrium doe no aify he Srong Inuiive Crierion. I i eay o ee ha a Nah equilibrium in which all hree ype are pooled canno aify he Srong Inuiive Crierion eiher. θ, I S I m ˆm S m m Figure 4.: Separaing Equilibrium Therefore, whenever he SIC i applied, i i neceary o characerize condiion under which here exi an equilibrium aifying he refinemen. We will do o for our reerve price ignaling model a well a in a more general model in he nex ecion. 7

19 There we will make a number of implifying aumpion. Thi will allow u o apply he following impler crierion o e wheher an ou-of-equilibrium ignal i credible. Propoiion 4: Suppoe ha he ingle-croing propery hold and ha U (, θ, y) = 0. Le z (), T be a Nah ignaling equilibrium. Suppoe ha here i ome ignal ŷ, S T and percepion θ ˆ < θ E { S} uch ha () i U (, θˆ, yˆ ) U (, θ(), z ()), for all S ( ii) U (, θˆ, yˆ ) > U (, θ(), z ()), forome S ( iii) U (, θˆ, yˆ ) < U (, θ(), z ()), forall S Then he ou-of-equilibrium ignal-percepion ( yˆ, θ ) i credible. Proof: Given he ingle croing propery, if here i uch a ube S, i mu be an inerval [, ]. Thi i depiced in Figure 4. below. θ, I I. ( y, θ ˆ) ( z ( ), ) z () ( z ( ), ) z Figure 4.: Type [, ] pooling 8

20 Suppoe ha < < <. For (i) and (iii) o hold i mu be he cae ha U (, θ( ˆ ), z ( )) = U (, θ, yˆ ) and U (, θ( ˆ ), z ( )) = U (, θ, yˆ ). Then, by he ingle croing propery, (ii) hold for all (, ). Since θ ˆ < θ = E { [, ]}, we have U (, θ( ), z ( )) = U (, θ, yˆ) [ U (, θ, yˆ) U (, θˆ, yˆ)] i i i i i i θ = U ( i, θ, yˆ) U ( i, θ, yd ˆ) θ, i =,. (4.) θ θˆ Define θ ε() = U (, θ, yd ˆ) θ. By hypohei, θ U (, θ, y) = 0, hence we may wrie θˆ ε() = εˆ. From (4.), U (, θ( ), z ( )) = U (, θ, yˆ ) εˆ, i =,. i i i i By ingle croing, for (, ) U (, θ(), z ()) < U (, θˆ, yˆ) = U (, θ, yˆ) [ U (, θ, yˆ) U (, θˆ, yˆ)] Thu θ = U (, θ, yˆ) U (, θ, yd ˆ) θ = U (, θ, yˆ) εˆ θ U (, θ(), z ()) < U (, θ, yˆ ) εˆ,if (, ). Alo by ingle croing, for [, ] U (, θ(), z ()) > U (, θˆ, yˆ) = U (, θ, yˆ) [ U (, θ, yˆ) U (, θˆ, yˆ)] θˆ θ = U (, θ, yˆ) U (, θ, yd ˆ) θ = U (, θ, yˆ) εˆ θ Thu condiion (i) (iii) of he Srong Inuiive Crierion are aified. θˆ 9

21 The proof when S = [, ]or S = [, ] i almo idenical. Q.E.D. 5. Condiion for a Nah equilibrium o aify he SIC To explore condiion for a eparaing equilibrium o aify he SIC, we fir digre o conider a general ignaling model ha include he reerve price ignaling model a a pecial cae. Then we apply he condiion derived for he general model o he reerve price ignaling model, a well a o he well-known Spence educaion ignaling model. A General Signaling Model To begin, le u fix he noaion. A before, [, ] i ill he rue ype of he ignal ender, and θ i he ype of he ender ha i perceived by he ignal receiver(). A ignal choen by he ender i denoed by y Y, where Y i he e of feaible ignal. Le z () be a ricly monoone ignaling chedule ha fully reveal he rue ype of he ender. U (, θ, y) i he uiliy of he ender of rue ype of who end ou a ignal of y and i perceived o be ype θ. Accordingly, U () = Uz (,, ()) i he uiliy of he ender of rue ype in he eparaing equilibrium z (). We make he andard aumpion ha (a) U (, θ, y) i hird order differeniable in all i elemen; (b) U (,, y) 0 θ > ; and (c) he ingle croing condiion hold: dθ U UU U U = = < 0 dy U U U Under hee aumpion, he following are andard reul from he ignaling lieraure: (i) From he IC condiion, U(,, z) + U(,, z) z'( ) = 0, o when 0

22 U (,, z ()) 0, U(,, z ()) z () = ; and (ii) By he envelope heorem, U (,, z ()) U () = U (,, z ()). We alo make he following aumpion: A: U = 0 for all (, θ, y) ; A: U = 0 for all (, θ, y). Aumpion A implie U(, θ, y) = U(, θ, y) for any,, θ, y. I alo follow from A ha U = 0, hence for any,, θ, y, U(, θ, y) = U(, θ, y). Example : The reerve price ignaling model In he model udied in previou ecion, he eller expeced payoff can be expreed a U (, θ, y) = αy+ θ( y) + H( y), where we adop he ranformaion y = F () ( m) [0,] and H ( y ) = Bmy ( ( )). Uing (.4), we can ee ha H ( y) = B ( m) y ( m) = F ( mjm ) ( ) F ( m) = Jmy ( ( )) H ( y) = J ( m) F ( m) () () () Wih hi ranformaion, he derivaive of U (, θ, y) are U (, θ, y) = αy, U (, θ, y) = y, U (, θ, y) = α θ + H ( y) U (, θ, y) = U (, θ, y) = 0, U (, θ, y) = α U (, θ, y) = 0, U (, θ, y) =, U (, θ, y) = H ( y) I i eay o check ha he andard aumpion and A and A are all aified. By he andard reul, when U (,, z ()) = H (()) z ( α ) 0, we have

23 U (,, z ()) z () U (,, z ()) ( α) H (()) z () = = z U (,, z ()) = U (,, z ()) = α z () Example : The educaion ignaling model In a common formulaion of he Spence educaion ignaling model, a worker expeced payoff i U (, θ, y) = θ Cy (, ), where i he worker produciviy unknown o firm, θ i he worker produciviy perceived by he firm and hence i alo he wage offered o her by compeing firm, and y i he educaion ignal he worker can chooe. I i ypically aumed ha for all (, y ) (i) C (, y ) < 0; (ii) C (, y ) > 0 ; and (iii) C (, y ) < 0. The derivaive of U (, θ, y ) are U (, θ, y) = C (, y), U (, θ, y) =, U (, θ, y) = C (, y) U (, θ, y) = C (, y), U (, θ, y) = 0, U (, θ, y) = C (, y) U (, θ, y) = U (, θ, y) = 0, U (, θ, y) = C (, y) I i eay o ee ha he andard aumpion and A and A are all aified. By he andard reul, and ince by aumpion U(, θ, y) = C(, y) 0 for all (, vy,, ) we have U(,, z ()) z'( ) = = U (,, z ()) C (, z ()) ' = = U (,, z ()) U (,, z ()) C (, z ())

24 Local Condiion for No Credible Inerior Deviaion We now derive neceary condiion for he eparaing equilibrium characerized by z () o aify he SIC in he general model. For any wo ype, uppoe hoe in he inerval [,] pool a a cerain ignal y. Le v (,) be he expeced ype of hi pool, ha i, v (,) = [ G () G ()] θdg( θ), where G(.) i he c.d.f. of. Le v (,) [,] and y (,) Y be a oluion o Uvy (,, ) = Uz (,, ()) = U () Uvy (,, ) = Uz (,, ( )) = U () (5.) The poin ((,), y v (,)) i depiced below in Figure 5.. Given hi ignal-percepion couple, all hoe ype in (,) prefer he pool o heir eparaing equilibrium oucome. θ I I (,(,)) yv ((), z ) ((), z ) z Figure 5.: Pool of ype in [,] From Propoiion 4, in order for he eparaing equilibrium characerized by z () o aify he SIC, i mu be ha for any [, ) and any >, v (,) < v (,). Noe

25 ha for any [, ), v (,) = v (, ) =. Thu, if v(,) v(,) < 0 for any >, hen v (,) < v (,). Lemma : For any [, ), (i) v (, ) = /; (ii) G''( ) v(, ) = ; (iii) if G() i 6 G'( ) concave, v (,) < / for all >. Proof: See he Appendix. Lemma : For any uch ha U (,, ( )) 0 z, (i) under Aumpion A, v (, ) 0.5 = ; (ii) under Aumpion A-A, U 4U + U U U v z z (,) = + [ + ] '( ) + [ '( )] 6 U U U U where all funcion are evaluaed a ( z,,()). Proof: See he Appendix. The propoiion below follow immediaely from Lemma -. Propoiion 5: For any uch ha U (,, ( )) 0 z, he eparaing equilibrium characerized by z () aifie he SIC locally above if U 4 + U G "( ) U U U U G'( ) + [ + ] z'( ) + [ z'( )] > (5.) where =. z () U ( z,, ( )) U ( z,, ( )) 4

26 Proof: We know ha v (,) v (,) = 0. From Lemma and 4, we have v (, ) v (, ) = 0. If he condiion of he propoiion i aified, hen for a neighborhood around, i mu be v(,) < v(,), and hence v (,) < v (,). Q.E.D. We now provide ome inuiion for hi reul. Conider he marginal rae of ubiuion MRS(, θ, z) a θ = and he marginal rae of ubiuion MRS(, θ, z) a θ =, where >. If he raio of hee marginal rae of ubiuion i cloe o, (o ha he relaive difference in ignaling co i mall), he gain o eparaion for ype i mall. Thu a pool i more likely o be profiable. Converely, if he raio of hee marginal rae of ubiuion i much le han, he gain o eparaion i larger and pool become le aracive. Thu he rae a which he logarihm of he marginal rae of ubiuion decline wih i a meaure of ignaling effecivene. Given aumpion A and A, ln MRS(,, z) = ln[ U (,, z)] ln U (,, z). U ln MRS(,, z) = + U U. Thu he larger i eiher U or U, he greaer i he ignaling effecivene. Looking a Propoiion 5, whenever eiher of hee econd derivaive i ufficienly large, he SIC i aified. Propoiion 5 give a ufficien condiion for z () o aify he SIC locally. From he proof, i i clear ha a ufficien condiion for i o aify he SIC globally, ha i, v (,) < v (,) for any >, i ha v(,) < v(,) for any >. However, i i no illuminaing o pell ou hi condiion, a boh expreion of v(,) and v(,) are 5

27 quie cumberome. By par (iii) of Lemma, when G() i concave, i uffice for z () o aify he SIC if v (,) > 0. Thi can be ueful in ome applicaion. Noe alo ha Propoiion 5 applie only o hoe uch ha U (,, ( )) 0 z. If a eparaing equilibrium z () ha he propery ha U (,, ( )) 0 z for all [, ], hen we only need o check (5.) o ee if z () aifie he SIC (locally). If for ome uch ha U (,, ( )) 0 z =, hen Propoiion 5 doe no apply. In hi cae, z'( ) goe o infiniy. From he expreion of v (,), a, we can ee ha if U ( z,, ( )) > 0, hen (5.) i aified a (and i neighborhood); if U (,, ( )) 0 z <, hen (5.) i violaed a (and i neighborhood). In he laer cae, he eparaing equilibrium z () doe no aify he SIC. Below we apply Propoiion 5 o he wo example inroduced above. Example (coninued): The reerve price ignaling model We now analyze when he eparaing equilibrium characerized in Propoiion aifie he SIC. Fir conider he cae in which in (.8), ( α ) < J( x) o ha he eparaing equilibrium goe hrough m ( ) = M(, ) = x a. Since U ( z,, ( )) = ( α ) Jm ( ( )) i decreaing in, U ( z,, ( )) < ( α ) J ( x ) < 0 for all [, ]. In hi model, ince U(,, vy) = U(, vy, ) = 0, (5.) become 4α + H (()) z z () G () > ( α) H (()) z G () 6

28 where z () z () =. When hi condiion hold, he eparaing equilibrium ( α) H (()) z characerized in Propoiion aifie he SIC a lea locally. For he cae in which in (.8), ( α ) J( x), hen U (,, z ()) = 0. Since U( vy,, ) = H"( y) < 0, he eparaing equilibrium characerized in Propoiion violae he SIC around. Example (coninued): The educaion ignaling model In he common formulaion of he model, U(, vy, ) = C(, y) < 0 for all (, vy., ) From he derivaive of Uvy (,, ) derived before, i i eay o ee ha U(,, vy) = C(, y) and U(,, vy) = C(, y). Condiion (5.) become C "( ) C C C > G C C C C C G'( ) where all funcion are evaluaed a (,()) z. If he above condiion hold, hen he eparaing equilibrium ha i given by U(,, z ()) z'( ) = = and goe U (,, z ()) C (, z ()) hrough z () = 0 aifie he SIC a lea locally. Global Condiion for No Credible Inerior Deviaion So far we have focued on local condiion for he SIC. In a more rucured model, i may be poible o obain global condiion for he SIC. Conider he following model in which he ender uiliy aifie 7

29 A4: Semi-eparable uiliy, ha i, U (, θ, y) = a( θ) + bcy ()(), where a(), a (), b(), b () > 0. Lemma 4: If Q() i increaing and ricly convex (concave), hen Q'( d ) ( + ) > ()0 < Q ( ) ( ) Q. Proof: Define Q'( d ) = q (, ) Q and ( ) Q ( ) ( ) = +. [ q (, ) ][ Q ( ) Q ( )] = ( Q ) '( d ) = ( )[ Q '( ) Q'( )] d which i greaer (le) han zero if Q() i increaing and ricly convex (concave). Q.E.D. Propoiion 6: Suppoe he ender uiliy funcion aifie A4. The SIC i aified for all poible pool of ype who ignal if (i) a () G (), (ii) ' a () G () '' b () b () a"( ) > 0, [, ] b () b () a'( ) If neiher inequaliy hold for ome ype who ignal, he SIC i violaed. Remark: For he imple Spence labor marke model U (, θ, y) = θ cy ( )/ γ, (i) and (ii) hold if G() i concave and γ >. 8

30 Proof: If A4 hold, U = a( θ) + bcz ()(). We begin by ranforming variable and defining = a (), ˆ = a( θ ). Then we can rewrie he uiliy funcion a follow. Uy (, ˆ, ) = ˆ+ bcy ( ) ( ), where b() ba ( ()). Le G ˆ () be he c.d.f. of. Tha i, G ˆ( ) Pr{ a () } Pr{ a ()} Ga ( = = = ()). Le zbe () a eparaing equilibrium chedule. The fir order condiion for incenive compaibiliy i hen + bc () ( z ( )) z'( ) = 0 d Define cz (()) = c (()) z z '( ). From he IC condiion d d cz (()) d = b(). Inegraing over he inerval [, ], d ( ( )) ( ( )) = = + + ( ) b() b( ) b( ) d b() (5.) cz cz d d Now conider any pool [, ], who chooe a common ignal y (, ). The expecaion of he ender ype for hi pool i ˆ v(, ) = G'( d ) G ˆ( ) G ˆ. If ( ) v(, ) v (, ) for any <, where v (, ) i defined by he indifference condiion, hen z () aifie he SIC globally. Since ˆ( ) ( ()) G = Ga, G ˆ '( ) = G'() a'( ). Taking he logarihm and differeniaing by, Gˆ "( ) G"( ) a"( ) = ˆ G'( ) a'( ) G'() a'( ) 9

31 Thu, by Lemma 4, v(, ) + if G ˆ () i concave, or G"( ) a () 0, ha i, G'( ) a () if condiion (i) hold. The indifference condiion (5.) become U ( ) = + bcz ( )(( )) = v+ b( ) c( y). From he fir equaliy and from he econd, U ( ) = + b( ) cz ( ( )) = v+ b( ) cy ( ) v + ( ( )) = + cy ( ) b ( ) ( ) cz b v + ( ( )) = + c( y) b ( ) ( ) cz b Subracing he econd expreion from he fir, Subiuing from (5.) ( + ) v= + cz (( )) + cz (( )) b ( ) b( ) b ( ) b( ) d ( + ) v = ( ) d b ( ) b ( ). d b() Hence v (, ) = d ( ) d d b () d ( ) d d b() 0

32 if b() b() i increaing o i b(). I follow from Lemma 4 ha v (, ) > ( + ) i convex. Le k () = =. Differeniaing and aking logarihm, b() ba ( ()) ln k () = ln b() ln b () ln a(). Then k () () () () () b b a a = k () b () b () a () Thu, b() i convex if k"( ) > 0, or he righ hand ide above i greaer han zero, which i condiion (ii). In um, if condiion (i) and (ii) hold, hen v (, ) > ( + ) v(, ). On he oher hand, if boh condiion fail, hen G ˆ () i ricly convex and / b () i ricly concave. Hence v(, ) > ( + ) and v (, ) < ( + ), in which cae he SIC i clearly violaed. Q.E.D. Condiion for No Credible Exerior Deviaion Finally, for he eparaing equilibrium characerized by z () o aify he SIC in he general model, we need o conider he following kind of deviaion, in addiion o he kind of inerior deviaion repreened by (5.). For any ype >, uppoe hoe in he inerval [, ] all chooe he ignal y = z( ), he equilibrium ignal by. Le

33 v(, ) be he expeced ype of hi pool, ha i, Le v ( ) [, ] be a oluion o v(, ) [ G ( ) G( )] dg() =. U(, v, y) = U (,, z ( )) = U ( ) (5.4) In order for he eparaing equilibrium characerized by z () o aify he SIC, i mu be ha for any >, v(, ) < v ( ). Noe ha v(, ) = v ( ) =. Thu i i ufficien if v(, ) v'( ) < 0 for any >. From (5.4), oal differeniaing give U(,, z ( )) U(, v, y) v'( ) = U (, vy, ) A a, v and z ( ) y. Therefore, v'( ) 0. However, i i eay o how ha, v(, ) 0.5. So a lea in he neighborhood of, v(, ) > v ( ), hu violaing he SIC. The analyi above poin ou a lower endpoin problem for z () o aify he SIC in he general model. However, hi problem can be overcome if we modify he model o ha given he ignaling chedule z (), he ender of ome lowe ype doe no acively paricipae in he marke becaue of ome fixed enry co. Le u ay ha now he lowe ype who paricipae in he marke i ay ou of he marke. Now uppoe for ome chooe he ignal y heir expeced ype a higher han = z ( ), he equilibrium ignal by >, and he ender of all ype > hoe in he inerval [, ] all <. If he ignal receiver perceive and hu pay hem accordingly, hen all hoe lowe ype < will now find i profiable o paricipae in he marke and join he pool

34 of [, ]. Bu when he (probabiliy or populaion) ma of ype [, ] i large, he correc percepion of he expanded pool for hoe who chooe ignal y = z ( ) will be maller han, making i unaracive for ype of [, ] o deviae. Le u conider he wo pecific example udied earlier. In he aucion model, FB uppoe he eller need o pay a fixed co of C o run he aucion. Le Uy (,, ()) be he uiliy of he ender of ype who chooe her opimal acion FB y () = F ( m ()) in () he cae of full informaion, where m () = M (,) from (.8) i he opimal markup. Define o be he oluion o FB FB FB FB Uy (,, ()) = αy () + [ y ()] + H( y ()) = C + α FB FB FB FB FB Since duy (,, ()) d = U(,, y ()) + U(,, y ()) = αy () + y () > α for FB allα <, o Uy (,, ()) α i ricly increaing in. Hence here i a unique oluion if C i ufficienly large. By monooniciy, he ender of ype maller han be willing o pay he aucion co, while he ender of ype greaer han will no will pay he aucion co and chooe reerve price according o z () aring from y ( ). When he aucion co C i ufficienly large, he ma of he ype excluded FB [, ] will be ufficienly large o ha he lower endpoin problem doe no arie. Working wih he runcaed diribuion on [, ], we can check he neceary (local) condiion for z () o aify he SIC dicued before. In paricular, i i required ha a he new aring poin, ( ) J( x) α <.

35 The analyi of he andard educaion ignaling model i very imilar. Suppoe he worker ha an ouide opporuniy of w >. Then he worker of ype maller han w will no be willing o paricipae in hi marke, ince he highe payoff he can ge in a eparaing equilibrium i < w. By monooniciy, he worker of ype greaer han w will paricipae in hi labor marke and chooe educaion according o z () aring FB from y ( w ) = 0. When he ouide opporuniy w i ufficienly large, he ma of he ype excluded [, w ] will be ufficienly large o ha he lower endpoin problem doe no arie. Working wih he runcaed diribuion on [ w, ], we can check he neceary (local) condiion for z () o aify he SIC dicued before. 4

36 Appendix Propoiion : In he eparaing equilibrium, for every reerve price r () = + m () i higher for larger α or larger n. Proof: Wrie he righ hand ide of (.8) a d f() ( m)[ J ( m) + ( α)] = =. dm bm (,, α) F ( m) () >, he markup and hence he Obviouly, for any α, he iniial poin i ( =, m= m( α)) where, from (.8), m( α ) i increaing wih α. Moreover bmα (,, ) i increaing wih α. Thu he condiion of Lemma hold wih κ = α. reerve price i higher for larger α. Therefore, for every >, he markup and hence he Now wrie he righ hand ide of (.8) a bmn (,, ). Fir he iniial poin ( m, ) i independen of n. We wan o how ha bmn (,, ) i increaing in n, which i equivalen o howing ha ϕ ( mn, ) i decreaing in n where We have ϕ ( mn, ) = ln f ( m) ln( F ( m)) = ln n+ ( n )ln f( m) ln( F ( m)). () () ϕ( mn.) F ( m)ln Fm ( ) = + ln Fm ( ) n n n F ( m) n F ( m) + nln Fm ( ) =. n n( F ( m)) n n Le δ ( mn, ) = F ( m) + nln Fm ( ). For any n, δ ( mn, ) = 0a m= x. Furhermore, for all m < x, n 5

37 n nf( m) δ ( mn, ) = nf ( m) f( m) + m Fm ( ) nf ( m) ( n = F ( m )) > 0. Fm ( ) So i mu be ha δ ( mn, ) < 0 for all m< x and for all n. Therefore ϕ ( mn, ) i decreaing in n. Q.E.D. Lemma : For any [, ), (i) v (, ) = /; (ii) G''( ) v(, ) = ; (iii) if G() i 6 G'( ) concave, v (,) < / for all >. Proof: By definiion, v (,) = [ G () G ()] θdg( θ). Muliplying boh ide by G () G () and hen differeniaing by, Differeniaing by again, Seing v (,)( G () G ()) + vg (,) '( ) = G'( ). v (,)( G () G ()) + v (,) G'( ) + vg (,) ''( ) = G'( ) + G''( ). (6.) =, i follow immediaely ha v (, ) = /. Differeniaing (6.) by again, v(,)( G () G ()) + v(,) G'( ) + v(,) G''( ) + vg (,) '''( ). = G"( ) + G"'( ) Since v (, ) = and v (, ) = /, eing = we obain v G"( ) (, ) = 6 G'( ) 6

38 Suppoe G() i concave. Then v (, ) < 0 o ha v (,) i a decreaing funcion of over ome inerval. To how ha v (,) < / for all >, uppoe oherwie. Le be he malle value of uch ha v(, ) = /. Then in ome lef neighborhood of, v (,) i an increaing funcion of, o v(, ) 0. However, appealing o (6.), v (, )( G ( ) G ()) = ( v (, )) G"( ) < 0. Conradicion. Thu v (,) < / for all >. Q.E.D. Lemma : For any uch ha U (,, ( )) 0 z, (i) under Aumpion A, v (, ) 0.5 = ; (ii) under Aumpion A-A, U 4U + U U U v z z (,) = + [ + ] '( ) + [ '( )] 6 U U U U where all funcion are evaluaed a ( z,,()). Proof: Toal differeniaing (6.) give U (, vyd, ) + U (, vydv, ) + U (, vydy, ) = U (,, z ()) d U ( vyd,, ) + U ( vydv,, ) + U ( vydy,, ) = U ( z,, ( )) d Solving he equaion, we have where ; dv = d+ d dy = d+ d = U (, vyu, ) ( vy,, ) U ( vyu,, ) (, vy, ) = U ( vyu,, )[ (,, z ()) U (, vy, )], = U (, vy, )[ U ( z,, ( )) U ( vy,, )] = U ( vy,, )[ U (,, z ()) U (, vy, )], = U (, vyu, )[ ( z,, ( )) U (, vy, )] Under Aump ion A, we have 7

39 dy U (, vy, ) U(,, z ()) = = d U (,, vy) U (, vy, ) (6.) Fix any, a, i mu be ha v, z () z (), and y z (). For he impliciy of noaion, wrie v () = v(,) and y () = y(,). Applying he I Hopial rule, a, we ge dy d U (, vy, ) + U(, vyv, ) () + U(, vyx, ) () U(,, z ()) U(,, z ()) U(,, z ()) z () = lim U ( vyv,, ) () + U ( vyy,, ) () U (, vyv, ) () U (, vyx, ) () U (, vy, ) U =lim dy = z () d (, v, y) y () U (,, z ()) z () U (, vy, ) dy Hence a, 0.5 z'( ) d a long a z'( ) = U( z,, ( )) U( z,, ( )) i defined a, or U (,, ( )) 0 z a. Since dv dy U ( vy,, ) dy = = = d d U ( vy,, ) d (6.) we have dv U( vy,, ) dy U( z,, ( )) dy = lim = lim d U ( vy,, ) d U ( z,, ( )) d U ( z,, ( )) = U ( z,, ( )) = 0.5 z'( ) 0.5 for any uch ha U (,, ( )) 0 z. Thi prove par (i). For par (ii), fir noe ha from z'( ) = U( z,, ( )) U( z,, ( )), 8

40 ''( ) U '( ) U + U + = U z U U z z From (6.), and by Aumpion A, we have '( ) dy U (, vy, ) + U(, vy, ) U(,, z ()) U(,, z ()) z'( ) d y = d (,, ) (,, ) d U vy U vy U vy U vy U vy dy dy dy (,, ) (,, ) (,, ) d d U ( vy,, ) U (, vy, ) d U (, vy, ) U (,, z ()) U(, vy, ) U( vy,, ) dy = + U ( vy,, ) U (,, vy) U ( vy,, ) U (, vy, ) d dy U(, vy, ) U(,, z ()) z'( ) + d U (,, vy) U (, vy, ) (6.4) Le Li (,) be he ih erm on he righ hand ide of he above equaion. For any uch ha U (,, ( )) 0 z, i can be checked ha Therefore, lim L U ( z,, ( )) z'( ) = U ( z,, ( )) U ( z,, ( )) lim L = [ z'( )] 4 U( z,, ( )) d y U ( z,, ( ))[ z'( )] lim L = z () d U ( z,, ( )) d y U ( z,, ( )) U ( z,, ( )) z'( ) 6 = z () + z'( ) d U ( z,, ( )) From (6.), and uing Aumpion A, we have 9

41 dv U( vy,, ) d y U( vy,, ) U( vyu,, ) ( vy,, ) dy = d U( vy,, ) d U( vy,, ) U( vy,, ) d (6.5) dv A, we know ha 0.5 d and dy U( vy,, ) 0.5 Uz '( ) = 0.5U. So, d dv U ( z,, ( )) d y U ( z,, ( )) U ( z,, ( )) z'( ) 0.5 z'( ) d U ( z,, ( )) d U ( z,, ( )) = d y d U '( ) = U z z z'( ) U 0.5 '( ) U + U + Uz'( ) U + 0.5U = [ ] + U 6 U [ z'( )] 0.5 z'( ) z'( ) U U z'( ) 0.5 z '( ) U U + U + U z'( ) U U z'( ) U U z'( ) = + U 6 U U U 4U+ U U U = + [ + ] z'( ) + [ z'( )] 6 U U U U Thi prove par (ii). Q.E.D. 40

42 Reference Cho, In-Koo and Krep, David M. (987), Signaling Game and Sable Equilibria, Quarerly Journal of Economic, 0, 79-. Groman, Sanford and Perry, Moy (986a), Sequenial bargaining under Aymmeric Informaion, Journal of Economic Theory, 9, Groman, Sanford J. and Perry, Moy (986b), Perfec Sequenial Equilibrium, Journal of Economic Theory, 9, Myeron, Roger B. (98), Opimal Aucion Deign, Mahemaic of Operaion Reearch, 6, Riley, John (975), Compeiive Signaling, Journal of Economic Theory, 0, Riley, John G. (979), Informaional Equilibrium, Economerica, 47, -59. Riley, John G. (00), Silver Signal: 5 year of Screening and Signaling, Journal of Economic Lieraure, 9, Riley, John G. Weak and Srong Signal (00) Scandinavian Journal of Economic, 04, -6. Riley, John G. and William F. Samuelon, (98), Opimal Aucion, American Economic Review, 7, 8-9. Rohchild, Michael and Sigliz, Joeph (976), Equilibrium in Compeiive Inurance Marke: An Eay on he Economic of Imperfec Informaion, Quarerly Journal of Economic, 90, Spence, A. Michael (97), Job Marke Signaling, Quarerly Journal of Economic, 87,