Imperfect Signaling and the Local Credibility Test

Transcription

1 Imperfect Signaling and the Local Credibility Tet Hongbin Cai, John Riley and Lixin Ye* Abtract In thi paper we tudy equilibrium refinement in ignaling model. We propoe a Local Credibility Tet (LCT) which i omewhat tronger than Cho and Krep Intuitive criterion but weaker than the trong Intuitive Criterion of Groman and Perry. Allowing deviation by a pool of nearby type, the LCT give conitent olution for any poitive correlation between the ignal ender true type (e.g., ignaling cot) and the value to the ignal receiver (e.g., marginal product). It alo avoid ruling out reaonable pooling equilibria when eparating equilibria do not make ene. We identify condition for the LCT to be atified in equilibrium for both the finite type cae and the continuou type cae, and demontrate that the reult are identical a we take the limit of the finite type cae. We then apply the characterization reult to the Spence education ignaling model and the Milgrom and Robert advertiing ignaling model. Intuitively, the condition for a eparating equilibrium to urvive our LCT tet require that a meaure of ignaling effectivene i ufficiently high for every type and that the type ditribution i not tilted upward too much. *UCLA, UCLA, and Ohio State Univerity. We would like to thank In-Koo Cho, David Cooper, Maimo Morelli, Jame Peck, and eminar participant at Arizona Univerity, Illinoi Workhop on Economic Theory, Ohio State Univerity, Penn State Univerity, Rutger Univerity, UC Riveride, UC Santa Barbara, and Cae Wetern Reerve Univerity, for helpful comment and uggetion. All remaining error are our own.

2 . Introduction Since the eminal work of Cho and Krep (987), variou refinement concept have been propoed to rank different equilibria in ignaling game in term of their reaonablene. However, the miion i till far from being completed. In many application, there i a poitive yet imperfect correlation between the ignal ender true type (e.g., ignaling cot) and the ignal receiver expected value (which then determine her repone), ee Riley (00, 00). Conider a ituation in which two of the ender type have a ame ignaling cot but quite different value to the receiver. If thee two type do not oberve their value to the receiver, they are effectively the ame type with the expected value, o the exiting refinement concept, uch a the Cho and Krep Intuitive Criterion, apply in the uual way. However, if thee two type do oberve their different value to the receiver, then the Intuitive Criterion i unable to rank equilibria. The reaon i that if one of the two type like a deviation, the other alo like it. Thi i highly unatifactory becaue the two cae are obervationally equivalent. The reaon for the inconitent olution in the above example i that the exiting refinement concept focu on deviation by a ingle type only and do not conider deviation by a pool of type. Groman and Perry (986a,b), in a bargaining context, propoe an equilibrium refinement concept trengthening the Cho-Krep Intuitive criterion to allow pooling deviation. In thi paper, in a general ignaling model, we weaken lightly the Groman-Perry Strong Intuitive Criterion, and propoe a Local Credibility Tet (LCT) in which a poible deviation i interpreted a coming from one or more type whoe equilibrium action are nearby. We conider only local pooling deviation, firt becaue they eem to u more natural, econd becaue they have much of the power of global pooling deviation, and third becaue they are more eaily analyzed. Moreover, the Local Credibility Tet doe not alway rule out pooling equilibria in favor of eparating equilibria. We will argue that in ome ituation eparating equilibria eem unreaonable while pooling equilibria can be rather appealing. By allowing pooling deviation, the LCT avoid ruling out pooling equilibria in uch ituation. Conider a imple two type education-ignaling model, in which the high type mut take a quite cotly ignal (e.g., everal year of unproductive education) to eparate from the low type. Now uppoe there i only one low type agent in every 5 million high

3 type agent. In uch a ituation eparation eem highly unreaonable, becaue without taking the cotly ignaling action an agent hould not be perceived much differently from being the high type. By the LCT, it i eay to ee that in any eparating equilibrium a pooling deviation to ome ufficiently low cot level of the ignal i profitable to both type, o no eparating equilibrium atifie the LCT. The thrut of our analyi i to derive condition under which there exit equilibria atifying the LCT. We tudy a family of continuou type model of which the Spence education ignaling and the Milgrom and Robert advertiing ignaling are both member. We begin by formulating the concept of the LCT for the dicrete type model firt, ince the intuition i eaier to preent. Then we conider a dicretization of the continuou type model, and take the limit a the dicretization become finer. We characterize condition under which the Pareto dominant eparating equilibrium in our general ignaling model atifie the LCT. The required condition are intuitive. A long a a meaure of ignaling effectivene i ufficiently high for every type and the type ditribution i not tilted upward too much, the eparating equilibrium can urvive our LCT tet. In the continuou type cae, the et of equilibrium ignal i dene o that out-ofequilibrium ignal can be only found outide the interval of equilibrium ignal. However, thinking of the continuou type cae a the limiting cae of the dicrete type cae with many cloe type, it i natural to generalize the concept of the LCT to the continuou type. An equilibrium urvive the LCT if no deviation-perception pair i credible in the following ene: for any poible deviation ignal (on- or off-equilibrium), if it i interpreted a from type of a mall neighborhood of the immediate equilibrium type, it i profitable for the type in thi neighborhood to deviate to thi ignal, but unprofitable for type out of thi neighborhood to do o. Another way of thinking about thi credibility tet in the continuou type cae i the following. If, for an on-equilibrium ignal, there i uch a deviation-perception pair, then thoe nearby type can credibly deviate to the particular on-equilibrium ignal by throwing away ε amount of money. We derive condition under which the LCT i atified in equilibrium in the continuou type cae. The condition are exactly the ame a in the limiting dicrete type cae. Thi i atifactory, becaue model with continuou type and model with finitely

4 many type are theoretical tool for analyzing the ame kind of real world problem. Put differently, it would be highly unatifactory if an equilibrium refinement concept applie to one cae but not the other, or give different anwer for the two cae. The paper i tructured a follow. The next ection ue imple example to illutrate the baic idea of the LCT. Then Section 3 preent the general ignaling model. In Section 4, we formulate the concept of the LCT for the dicrete type cae. Then we derive condition under which the LCT i atified by the Pareto dominant eparating equilibrium in a dicretized continuou type model a the dicretization become finer. Section 5 generalize the formulation of the LCT to the continuou type cae, and how that the condition for the LCT are exactly the ame a in the dicrete type cae. We dicu an iue of robutne in Section 6. Concluding remark are in Section 7.. Example In thi ection we ue imple example to illutrate the baic idea of the Local Credibility Tet. A conultant ( i, v j) ha a ignaling cot i and a marginal product of v j, where < <... < n and v < v <... < vm. He can ignal at level z at a cot of Cz (, ). If paid a wage w hi payoff i U( z, w, ) = w C( z, ). In a competitive labor market for i conultant, hi wage will be hi marginal product perceived by the market. Activity z i C a potential ignal becaue the marginal cot of ignaling, (, zi ) i a decreaing z function of i. The et of ignaling cot type S = {,..., n }, and the et of poible productivity level i V = { v,..., v m }. The probability of each type, π ( i, vj), i common knowledge. i i If π (, v ) = 0 for all i j, the model reduce to the uual Spence model in which the negative i j correlation between ignaling cot and value to receiver i perfect. While we aume π (, v ) > 0 for all i and j, the analyi applie quite generally. 3 i j

5 Initially we aume that each conultant oberve hi own ignaling cot type but not hi productivity. We define v ( ) = Ev { }, and aume that type with lower i ignaling cot have higher expected productivity, that i, v ( ) < v ( ) <... < v ( n ). There i a continuum of eparating Nah equilibria in thi game. A Nah eparating equilibrium with three ignaling cot type i depicted below. Each curve i an indifference curve for ome ignaling cot type. A le heavy curve indicate a lower ignaling cot. Note that the optimal choice for each type i (indicated by a haded dot) i trictly preferred over the choice of the other type. Such an equilibrium fail the Intuitive Criterion firt propoed by Cho and Krep (987). i v ( 3) I I I 3 v ( ) v ( ) z = ẑ z 0 z 3 z Fig: -: Separating Nah Equilibria To ee thi, uppoe an individual chooe the ignal ẑ and argue that he i type. I thi credible? If the individual i believed, hi wage will be bid up to earn the ame wage a in the Nah equilibrium but incur a lower ignaling cot. v ( ) o he A noted by Cho and Krep (987), with more than two type, it i neceary to modify their original Intuitive Criterion or it loe much of it power. For the modified Intuitive Criterion the quetion i whether any particular type i uniquely able to benefit from ome out-of-equilibrium ignal if the ignal receiver correctly infer the ignaler type. 4

6 However (,( zv ˆ )) i trictly wore than ( z, v( )) for type and trictly wore than ( z3, v( 3)) for type 3. Thu the claim i indeed credible. Similar argument rule out any Nah equilibrium where different ignaling cot type are pooled. Thu the only equilibrium that atifie the Intuitive Criterion i the Pareto dominant eparating equilibrium (i.e., the Riley outcome) in which each local upward contraint i binding. Next uppoe that each conultant know both hi ignaling cot and hi marginal product. Again conider the Nah Equilibrium depicted above. Suppoe in thi equilibrium three different type are pooled at each ignal level. Conider the three type (, v), (, v), (, v 3) pooled at z. Suppoe a conultant chooe ẑ and claim to be type (, v 3). I thi credible? If the claim i believed, the conultant wage will rie from v ( ) to v 3 thu the conultant i indeed better off. But any offer that make type (, v 3) better off alo make type (, v) and (, v ) better off, ince they have the ame ignaling cot. Thu there i no credible claim that type (, v 3) alone can make. A imilar argument hold for each of the other type. Thu any Nah eparating equilibrium atifie the Intuitive Criterion. An almot identical argument etablihe that any Nah Equilibrium with (partial) pooling atifie the Intuitive Criterion a well. 3 Since all the type with the ame ignaling cot are obervationally equivalent, it eem to u that any argument for ranking the equilibria in the firt model (productivity unknown) hould alo be applicable to the econd model a well. There i a imple modification to the Intuitive Criterion that achieve thi goal. A conultant take an outof-equilibrium action ẑ and argue that he i one of the type who, in the Nah Equilibrium would have choen z. Thi being the cae, applying the Baye Rule, hi 3 It can be verified that the Cho and Sobel (990) refinement concept of divinity, which i built on the idea of tability of Kohlberg and Merten (986) and can be conidered a a logic offpring of the Intuitive Criterion, doe not have power either in the above example. Ramey (996) extend the Cho and Sobel divinity concept to the cae of a continuum of type. Like the Intuitive Criterion, divinity face the ame problem of ditinguihing type (, v ), (, v ), (, v ) to interpret a poible deviation, while thee 3 type have the ame incentive to deviate. Riley (00) dicue in greater detail thee and other refinement concept. 5

7 expected marginal product i v ( ). Then once again, the unique Nah Equilibrium atifying the modified Intuitive Criterion i the Pareto Dominant eparating equilibrium. We now argue that for ome parameter value, thi equilibrium defie common ene. Suppoe that S = N = o that there are four type of agent. For type the cot of ignaling i cz ( ) = 000z and mean marginal product i 0. For type the ignaling cot i c ( z) = 999z and mean marginal product i 000. The lower type mut be indifferent between (0, v ( )) and the choice of type, that i ( z, v( )). Therefore, U = 0 = z and o z =. The payoff for type i therefore U = v( ) c ( z ) = z =. Suppoe that only in 000 conultant i of type. Then the unconditional mean marginal product i 999. Thu the low ignaling cot type have an income which i approximately one thouandth of the wage in the Nah pooling equilibrium! We believe a better criterion for ranking equilibria hould not rule out pooling equilibria in uch circumtance. We now introduce a further imple modification of the Intuitive Criterion that achieve thi outcome. Local Credibility Tet: Suppoe that an out-of-equilibrium ignal ẑ i oberved and that z i the larget Nah Equilibrium ignal le than ẑ and z + i the mallet Nah Equilibrium ignal greater than ẑ, if they exit. Let Ŝ be the ubet of ignaling cot type chooing z or z + with poitive probability. For each S Sˆ, define v( S) = E{ v S}. Then the equilibrium pae the Local Credibility Tet (LCT) if there i no S uch that (, zvs ˆ ( )) i trictly preferred over the Nah Equilibrium outcome if and only if S. Note that if ẑ i maller (greater) than all equilibrium ignal, then z ( z + ) doe not exit and z + ( z ) i the mallet (larget) equilibrium ignal. By the above definition, Ŝ i the ubet of type chooing z + ( z ). Alo note that by conidering a ubet of Ŝ 6

8 to be the ingleton et of z + or z, the definition of the Local Credibility Tet allow deviation by ingle type. It follow that only eparating equilibria can pa the Local Credibility Tet. The idea of the Local Credibility Tet i weaker than the Strong Intuitive Criterion (SIC) propoed by Groman and Perry (986a,b). For any out-of-equilibrium ignal ẑ, their criterion conider any ubet of type a a potential deviating pool. An equilibrium fail the SIC if ẑ i credible for one ubet of type. Here we retrict attention to local deviation. Thi make the analyi more tractable and, we believe, more plauible. For the imple conulting example above, there are only two level of the ignal in the efficient eparating Nah Equilibrium. Conider any convex combination zˆ = ( λ) z + λz = λz = λ. Then S ˆ = {, }. Conider S = Sˆ. The average productivity of thee two type i 999. For ẑ = λ to be a credible deviation by S, both type mut trictly prefer ( z ˆ,999). Note that U ( z,999) = λ and U ( zˆ,999) = λ, ˆ and the equilibrium payoff are 0 and. Thu, for all λ = zˆ < 998 / 999, both type are indeed better off chooing the out-of-equilibrium ẑ. The efficient eparating Nah Equilibrium thu fail the LCT. In fact no equilibrium pae the LCT, o the criterion fail to rank the different Nah Equilibria. We now how how the LCT can be applied when there are many type and, in the limit, a continuum of type. Let the et of ignaling cot type be S = {,..., n }, with probabilitie { f,..., f n } where f i =. Suppoe type i ha a ignaling cot i i+ i i i cz ( ) Cz (, ) =,where = δ > 0. We alo aume that the expected marginal product of thoe of type i have an expected marginal product of v ( i ) = i. We eek condition under which the Pareto dominant eparating equilibrium pae the LCT. In thi equilibrium, the local upward contraint are binding. Therefore, a depicted below, thoe with ignaling cot type i are indifferent between 7

9 ( z, ) and ( z, ). (The indifference curve i labeled I.) We contruct the ignal i i i i level ẑ and z i + a follow. Chooe z o that thoe with ignal type i are indifferent between ( zi, i) and ( z, i + i + ). In the Figure below, thee are the point Ci and C. i w (, zw ˆ ˆ) Ii I i I i + i + C i i i C i C i ( z, ) i i zi z z i + z Fig. -: Applying the LCT with many type Then chooe z i + o that thoe with ignal type i + are indifferent between ( z, + i + ) and ( zi+, i+ ). In Figure., thee are the point C and C i +. We will i argue that type i mut be indifferent between Ci and C i + a depicted. That i, C i + i the efficient eparating contract for thoe with ignaling cot type i +. A type j conultant i indifferent between ( zw, ) and ( z, w ) if and only if cz ( ) cz ( ) w = w, j j that i, if cz ( ) cz ( ) = ( w w) (.) j 8

10 i By contruction, a type i conultant i indifferent between ( zi, i) and ( z, z + z i + ). Appealing to (.), cz ( ) cz ( ) = ( + ) = ( ). (.) i i i i+ i i i+ i Alo, a type i + conultant i indifferent between ( z, z + z ) and ( z, ). Again appealing to (.), i i+ i+ i+ cz ( ) cz ( ) = ( ) = ( ). (.3) i+ i+ i+ i i+ i+ i+ i Adding equation (.) and (.3) and noting that, by hypothei, = +, it follow that cz ( ) cz ( ) = ( ). i+ i i i+ i 9 i i i+ Appealing, finally, to (.), it follow that type i i indeed indifferent between ( zi, i) and ( zi+, i+ ). Thu the point C i + i the efficient eparating Nah Equilibrium contract for type i +. Note that for any wage ŵ above i + i +, there i a ignal ẑ between z and z i + that i trictly preferred by thoe with ignaling cot type i and i + but i not preferred by type i (or lower type.) If fi < f i +, the expected marginal product of fii+ fi+ i+ thee two type, exceed i + i + and, we can chooe ŵ to be equal to f + f i i+ the expected marginal product. Then if type i and i + chooe the out-of-equilibrium ignal ẑ, they can expect to be paid ŵ. The out-of-equilibrium ignal i therefore credible and o the eparating equilibrium fail the LCT. Converely if fi > f i +, the expected marginal product of thee two type i le than i + i + the out-ofequilibrium ignal ẑ by the two type i not credible. If thi hold for all i, that i fi > fi+, i =,..., n, the efficient eparating equilibrium pae the LCT. Note that thi condition i independent of the tep ize δ between ignaling cot type. Treating the continuum of type a the limit a δ 0, it follow that the Pareto

11 dominant eparating equilibrium pae the LCT if the denity function f ( ) i everywhere decreaing. 3. A General Signaling Model We conider the following ignaling environment. A player, the ender, ha private information denoted byθ Θ n R. The ender (multidimenional) private information affect her expected payoff through an aggregated variable called her true type (e.g., marginal cot of ignaling). Let (): i Θ R, S = [, ] be the true type of the ignal ender. The ender chooe an action (ignal), y Y, where Y = [ y, y] i the et of feaible ignal. Let v(): i Θ R, v V = [ v, v] be the value of the ender product or ervice to each receiver. We aume that a receiver payoff i linear in v o that it i only the perceived expected value ˆv which determine a receiver repone. We can then write the ender payoff a a function U(, vˆ, y ) of hi true type, the reponder expected value and the ender ignal. When the ender private information θ i i multidimenional, it i natural that the ender and the receiver may have imilar preference overθ, but place different weight on it different dimenion. In other word, the ender true type and the receiver value v will be correlated, but not perfectly correlated. Let z(): i S Y be a trictly monotone ignaling function that fully reveal the true type of the ender. Accordingly, U() = U(,(), v z()) i the utility of the ender of true type in a eparating equilibrium with the ignaling chedule z. () We maintain the following tandard aumption: (a) U(,, vˆ y) i third order differentiable in all it element; (b) U ˆ (,, v y ) > 0; (c) The ingle croing condition hold: 0

12 dvˆ U U U U U = = < 0 dy U U U Under thee aumption, the tandard reult in the literature (Riley, 979; Mailath, 987) how that a eparating equilibrium of the ignaling game atifie the following differential equation: U(, v(), z()) z () = (3.) U (, v(), z()) 3 In many ignaling model, the ignal i cotly o that U3 (, v(), z()) < 0for all. In thi paper we make thi aumption, and dicu what happen when thi aumption i relaxed. Naturally, in a eparating equilibrium the lowet type gain nothing to chooe any cotly ignal, o hould jut chooe the leat cotly ignal y. The ignaling chedule given by (3.) and the initial condition ((), v y ) give rie to the unique Pareto dominant eparating equilibrium of the game. Thi formulation of the ignaling problem i quite general and include many ignaling model tudied in the exiting literature a pecial cae. We illutrate thi with two example. 4 Example : The Spence education ignaling model In the Spence education ignaling model, a worker know her own peronal characteritic related to productivity θ Θ (e.g., technical knowledge, dicipline, learning ability, work habit, etc). Thee characteritic can be ummarized into a ingle dimenional variable: productivity, denoted by. Then a worker expected payoff i Uy (,, ˆ ) = ˆ Cy (, ), where i her productivity unknown to firm, ŝ i her productivity perceived by firm and hence i alo the wage offered to him by competing firm, and y i the education ignal the worker can chooe. It i typically aumed that for 4 Many other application fit into the general ignaling model. In Appendix B, we how that with ome variable tranformation, the reerve price ignaling model of Cai, Riley and Ye (003) atifie all the aumption. We then apply our characterization reult to check when the Pareto dominant eparating equilibrium in that model atifie LCT.

13 all ( y,, ) (i) C (, y ) < 0; (ii) C (, y ) > 0; and (iii) C (, y ) < 0. It can be eaily verified that the ingle croing and all other condition are atified. By the tandard reult, a eparating equilibrium atifie U (,, z()) U(,, z ()) C(, z ()) () = = 3 z In the Pareto dominant eparating equilibrium, the lowet type chooe the minimum education level y. Example : An advertiing ignaling model Thi example i adopted from Milgrom and Robert (986). A monopolitic firm can produce a good with a contant marginal cot c. It ell to a unit ma of conumer. The firm know it product quality, denoted by. Among the conumer, α < are informed about. The ret α of conumer are uninformed about, and their belief i given by the ditribution G ( ) on [., ] For product of quality, the conumer invere demand function i p = bq, where b i a poitive parameter and q i the quantity. So, given price p, the demand of an informed conumer i q = ( p) b, and I that of a uninformed conumer i q = ( ˆ p) b, where ŝ i hi perception of. The U total demand i then q= αq + ( α) q. I U Suppoe the firm pend z on advertiing, which lead to a perception of ŝ by the uninformed conumer. By chooing an optimal price given ŝ, the firm maximum [ α+ ( α) ˆ c] profit i U =. Conider a poible eparating ignaling chedule z. ( ) 4b The firm payoff function i [ α+ ( α) ˆ c] U(,, ˆ z) = z 4b It can be checked that thi example fit the general model a well. A eparating equilibrium atifie U (,, z()) ( α)( c) U (,, z()) b () = = 3 z

14 In the Pareto dominant eparating equilibrium, the lowet type chooe zero advertiing. 4. Finite Type Cae In thi ection we explore condition under which an equilibrium atifie the LCT in the dicrete type cae. Specifically, the ender ha N true type, denoted by = < < < N =. Let G( ) = prob.{ } be the probability ditribution of the ender true type. Let ŝ i i be the receiver perception of the ender true type. The ender expected payoff iu(,, vˆ y ), which atifie all the aumption made in Section. Since it i not central to what follow, we will focu on the ender true type intead of her private information θ and ue the reduced form payoff function U(,, vˆ y ). Let zi = z( i), where z = z < z < < zn = z, be an equilibrium ignaling chedule. Conider any ignal yˆ Y, yˆ z, for all i. i (L): No credible Deviation by the lowet type When ŷ < z, ( yv ˆ, ( )) i a credible deviation ifu(, v( ), z) < U(, v( ), yˆ ). (L): No credible Deviation by the highet type When ŷ > z, ( yv ˆ, ( )) i a credible deviation if U(, v( ), z) < U(, v( ), yˆ ). (L3): No local deviation by a ingle type When z ( i ) < yˆ < z ( i+ ), ( yv ˆ, ( i )) i a credible deviation if U( i, v( i), y ˆ) > U(, v( ), z ), and, U(, v( ), yˆ ) < U(, v( ), z ), j i. i i i (L4): No Local Pooling Deviation j i j i j Define v ( i, i+ ) = Ev { = i or = i+ }. When z ( i) < yˆ < z ( i + ), ( yv ˆ, ( i, i + )) i a credible deviation if U( j, v( i, i+ ), yˆ ) > U( j, v( j), zj), j = i and j = i+ and U( j, v( i, i+ ), yˆ ) < U( j, v( j), zj), j i or i+. By the definition of the Local Credibility Tet given in Section, if there doe not exit any kind of credible out-of-equilibrium ignal-perception, then the equilibrium {(, z )} urvive the LCT. 3 i i

15 We ay that an equilibrium atifie a requirement among (L)-(L4) if there i no uch credible deviation decribed in the requirement. Requirement (L), (L) and (L4) are all about deviation of one ingle type, which are the local condition of the Cho-Krep Intuitive Criterion. When N =, they are exactly the Cho-Krep Intuitive Criterion. Requirement (L3) i the local condition of the Groman and Perry criterion, allowing a deviation to be interpreted a from the two nearby type. Propoition : If an equilibrium atifie the LCT, it i the Pareto dominant eparating equilibrium (the Riley outcome). Proof: See the Appendix. By Propoition, to check when an equilibrium atifie the LCT, we only need to check when the Pareto dominant eparating equilibrium atifie (L3). Conider the Pareto dominating eparating equilibrium depicted in Figure 4. below. U n U n U n + n + v n z n y z n + Figure 4. 4

16 For any interior point n with n, define ( vy, ) be the intercept of indifference curve Un and U n + depicted in Figure. Then U( n, v, y) = U( n, n, zn) U( n+, v, y) = U( n+, n+, zn+ ) U( n, n, zn) = U( n, n+, zn+ ) (4.) Let v be the average type of n and n +, that i, v [ G ( ) G ( )] + [ G ( ) G ( )] = G ( ) G ( ) n n n n+ n n+ n+ n (4.) For the lower boundary point =, define v a the olution to the following equation: U(, v, z) = U(,, z). The point v i depicted in Figure 4. below. The U U v = z = z z Figure 4. G ( ) + [ G( ) G( )] The expected type between and i v =. G ( ) 5

17 Propoition : The Pareto dominating eparating equilibrium atifie the LCT if and only if for all n, v v. Proof: Note that only point in the hadowed area in Figure 4. and 4. are preferred by both type of and + to their repective equilibrium point. If v v, then there i n n no ignal between z and z to which both type of + and + are willing to deviate if n n the receiver ha the perception of v. So the LCT i atified. On the other hand, if the LCT i atified, then it mut be the cae that v v. Otherwie, it i clear from Figure 4. and 4. that there are ignal that both type of and + are willing to deviate to but other type do not. Contradiction. Q.E.D n n n n The idea of Propoition i imple. For both interior point and boundary point, v i the minimum perception by the receiver uch that the two type n and n + can find a common profitable deviation that i not attractive to any other type. The receiver correct perception about the pool of n and n + i v. If v < v for all n, then there i no pair of type can find a local pooled deviation uch that under the correct perception by the receiver, it i profitable only to them but not any other type. If that i the cae, the LCT i atified by the Pareto dominant eparating equilibrium. Propoition can be ued to check the exitence of equilibrium atifying the LCT in any dicrete type model. Let u return to the continuou type ignaling model introduced in Section in which the type pace i S = [, ] with a mooth ditribution function G. ( ) We tudy the following dicrete type verion where =, = + δ, =, and δ = i+ i N N k. We let N = for k, that i, a k increae by one, each interval i divided into two even one. We are intereted in the limit cae when k, or δ 0, a an approximation of the continuou type cae. For implicity, we make the following technical aumption: B: U = 0 for all (,, y ˆ ); 6

18 B: U = 0 for all (,, y ˆ ). Condition B and the ingle croing condition together imply thatu 3 > 0. Note that the Spence education ignaling model atifie both condition, but the advertiing ignaling model doe not. In the next Section, we will how that even though our reult do not directly apply to that example, our method can be eaily applied to it to get imilar characterization reult. k Conider interior type firt. For any k and n {,3,... }, fix n =. A k increae, let n and be the nearet type to, o n+ = n δ( k) and = n + + δ( k). Let v(, δ ) = v = E[ = n or n+ ], and vδ (, ) be the olution to (4.) Lemma : When δ 0, (i) v(, δ ) ; (ii) v (, δ ) 0.5 ; (iii) G"( ) v(, δ ) =. G'( ) Proof: See the Appendix. Lemma : Suppoe condition B and B hold. When δ 0, (i) v (, δ ) ; (ii) v (, δ ) 0.5 ; (iii) UU 3U33+ UUU UU 3 3U3 UU UU 3 3 v(, δ ) =. 4UU Proof: See the Appendix. 3 3 For the boundary type, let vδ (, ) be the olution to U( + δ, v(, δ ), z) = + δ + δ, where U(,, z ) z i uch that U(,, z) = U(, + δ, z). Let v(, δ ) be G ( ) + ( + δ )[ G ( + δ ) G ( )]. Note that if G ( ) = 0, then v(, δ ) = + δ for all δ, G ( + δ ) which mean that v (, δ ) < v (, δ ) and hence the condition of Propoition i violated. However, in the dicrete type cae, G ( ) = 0 mean that hould not be conidered at all. Thu, we uppoe G ( ) > 0. We have 7

19 Lemma 3: Suppoe condition B and B hold. When δ 0, (i) v(, δ ), v (, δ ) 0, and G'( ) v(, δ ) ; (ii) v (, δ ), G () v (, δ ) 0, and v U (,, z) U (,, z) δ 3. (, ) Proof: See the Appendix. 3 From Propoition and Lemma -3, we have our main characterization reult for the limiting dicrete type cae. Theorem : Suppoe condition B and B hold. When δ 0, the Pareto dominant eparating equilibrium of the dicrete type model atifie the LCT if (i) for any (, ], UU U + UUU + UU U UU + 4 UU G ( ) > G ( ) UU 3 3 (4.3) U3(,, z()) G () (ii) G ( ) > 0 and >. (4.4) U (,, z()) G() 3 The intuition for Theorem i eay to undertand. Conider the example depicted in Figure 3.3. Beide the type ditribution already dicued, another angel to ee the problem of non-exitence i how the difference curve differ acro type. Since the marginal rate of ubtitution between z and t i imilar for the different type in that example, the indifference map are imilar and o indifference curve are cloe together. A a reult, both type are better off deviating to ẑ if the receiver believe that both may be chooing to deviate, thu violating the requirement of no credible deviation. However, if the marginal rate of ubtitution decline ufficiently rapidly with type, the indifference curve I in Figure 3.3 will be everywhere abovet. Now only type t i better off deviating to ẑ if the receiver till think that both type may be deviating, making the deviation-perception pair not credible. Intuitively, the rate at which the marginal rate of ubtitution decline with i a meaure of ignaling effectivene. Thu Theorem ugget that when ignaling 8

20 effectivene i ufficiently large, the eparating equilibrium will urvive the LCT. Thi intuition i reflected in condition (4.3) and (4.4). Note that the lope of the indifference U3(,, z()) map i given by MRS(,, z()) =, and by Aumption B, U (,, z()) MRS(,,()) v z v v= U = U 3 U3 U. The LHS of (4.4) can be rewritten a U U 3, which meaure how rapidly the MRS decline with at (normalized at the level of MRS itelf). Figuratively, when the LHS i greater, the indifference curve I i far from * I in Figure 3.3, hence there will be no credible deviation with the perception at t. The RHS of (4.4) i the denity function of, normalized by the probability ma at. Clearly, the maller thi ratio, the maller the perception of a pool involving. Figuratively, when the expected value of t for both type, t, i lower in Figure 3.3, there will be no credible deviation. The intuition for (4.3) i imilar, though le tranparent. The lat term of the LHS U 3 (over the denominator),, ha exactly the ame interpretation: a meaure of how U 3 rapidly the MRS decline with. From Figure 4.., the critical value of v depend on how rapidly the curve Un increae with v and how lowly the curve U n + increae U 3 with z. Note that MRS(,, v z()) = by Aumption B and v U U U + U U MRS(,, z) = z ( U) v=. The firt and third term in (4.3) can be rewritten a U ( U U U U ) + U U U U U U U U MRS z MRS v = = + UU U U MRS MRS Figuratively, when the curve Un i more traight-up (large MRS v ) and the curve U + i more flat (mall MRS n z ), there will be no credible deviation with the perception at v (again, the wrong notation). The RHS of (4.3) i the concavity of the 9

21 ditribution function of, G, ( ) normalized by it denity function. Intuitively, the more concave Gi ( ) (i.e., the maller G i), the more probability ma on maller in any et of type, thu the maller the expected value of any et of type. Conequently, the maller G i, the le likely a deviation i credible. In ummary, Theorem ay that the Pareto dominant eparating equilibrium will atify the LCT if ignal are effective in ditinguihing type (i.e., MRS declining fat with type) and the type ditribution i not too tilted upward. 5. Continuou Type Cae In thi ection we derive condition under which an equilibrium atifie the LCT in the continuou type cae. We how that the reult here are exactly the ame a thoe in the limiting dicrete type cae tudied in the preceding ection. A in the dicrete type cae, we can focu on eparating equilibria. Conider any eparating equilibrium z (): S [, zz] Y. Conider any ignal ŷ Y. (L): When ŷ < z, ( y ˆ, ) i a credible deviation if U(,, z) < U(,, yˆ ). (L): When ŷ > z, ( y ˆ, ) i a credible deviation if U(,, z) < U(,, yˆ ). (L3c): When yˆ [ z, z], let z ( ˆ 0) = yand conider a mall neighborhood of 0, S0 S. Let ˆ = E[ S0]. If () i Uy (,, ˆ ˆ) > Uz (,, ()),forall intso. ( ii) U(,, ˆ yˆ) < U(,, z()),forall So. then the ignal-perception ( y ˆ, ˆ) i credible. By the definition of the Local Credibility Tet given in Section, if there doe not exit any credible ignal-perception, then the eparating equilibrium z () urvive the LCT. Requirement (L) and (L) are exactly the ame a in the dicrete type cae. It i eay to ee that in the continuou type cae the only equilibrium that atifie (L) and (L) i the Pareto dominant eparating equilibrium. Therefore, to check whether there exit an equilibrium atifying the LCT, we only need to check whether the unique Pareto dominant eparating equilibrium atifie Requirement (L3c). 0

22 Requirement (L3c) i the counterpart of (L3) in the dicrete cae. In the continuou cae, any ignal yˆ [ z, z] i an on-equilibrium ignal, o requirement (L4) in the dicrete cae doe not apply at all. However, thinking of the continuou type cae a the limit of the dicrete type cae with many very cloe type, it i eay to undertand how an on-equilibrium ignal can be alternatively interpreted a a deviation ignal and how to check credibility of uch deviation. Conider any on-equilibrium ignal ŷ, and uppoe the type of ender for thi ignal in thi eparating equilibrium i 0. Suppoe in a dicrete type verion of the model, 0 ( n, n + ) for ome n. Suppoe the nearby type S0 = ( n, n + ) all deviate to thi ignal, and thi i correctly perceived by the receiver and o the perception of the average type i ŝ. Requirement (L3c) ay that if given the perception ŝ, all the deviating type can gain relative to their equilibrium payoff while all other type cannot, then thoe nearby type can credibly deviate to the particular onequilibrium ignal. If a eparating equilibrium doe not allow any uch credible deviation, then it atifie the LCT. A mentioned in the Introduction, another way of thinking about on-equilibrium deviation i a follow. If for an on-equilibrium ignal there i uch a deviation-perception pair ( y ˆ, ˆ) a decribed in (L3c), then thoe nearby type can credibly deviate to ŷ by throwing away a mall amount of money. Since other type will not gain by mimicking, throwing away money by type in S 0 can ignal to the receiver that they are the type deviating to ŷ. Since the continuou type cae i viewed a an approximation of the cae of many cloe dicrete type, we only need to check whether there are any credible interior deviation and any credible boundary deviation for very mall interval, in the ene that will be made precie below. Conider interior deviation firt. For any two type, uppoe thoe in the interval [, ] pool at a certain ignal y. Let v(, ) be the expected type of thi pool: ' zdg( z) v(, ) = G ( ) G ( ). Let v (, ) [, ] and y (, ) Y be a olution to U(, v, y) = U(,, z( )) U(,, v y) = U(,, z()) (5.)

23 The point ( y (, ), v (, )) i depicted below in Figure 5.. Given thi ignal-perception pair, all thoe type in (, ) prefer the pool to their eparating equilibrium payoff. * I * I ( yv, (, )) ( z ( ), ) z () ( z ( ), ) z Fig. 5.: Pool of type in [, ] Similar to Propoition, it can be argued that the type in [, ] cannot find a credible deviation defined in (L3c), if and only if the ignal-perception pair of ( y (, ), v (, )) i not credible. Therefore, the eparating equilibrium z ( ) atifie (L3c), if and only if for any [, ) and >, v (, ) < v (, ) a. Note that for any [, ), v(,) = v(,) =. Furthermore, we have Lemma 4: For any [, ), (i) v (,) = / ; (ii) G''( ) v(,) =. 6 G'( ) Proof: See the Appendix. Lemma 5: Suppoe condition B and B hold. For any uch that U 3 (,, z ()) 0, (i) v (,) = ; (ii) UU 3U33 + UUU UU 3 3U3 UU UU 3 3 v(,) = UU Proof: See the Appendix. 3 3

24 We now conider the boundary deviation of the following kind. For any type >, uppoe thoe in the interval [, ] all chooe z = z(), the equilibrium ignal by. Let v(, ) be the expected type of thi pool. Suppoe G ( ) > 0, we have Let v (, ) [, ] be a olution to ' G () + zdgz () v(, ) = (5.) G ( ) U(, v, z) = U(,, z( ')) = U ( ) (5.3) In order for the eparating equilibrium characterized by z ( ) to atify the LCT, it mut be that for cloe to, any uch ignal-perception pair of ( zv, (, )) i not credible. That i, v(, ) < v(, ) a. Note that v(,) = v(,) =. * Lemma 6: Suppoe condition B and B hold. (i) Suppoe G ( ) > 0. Then, v (,) = 0, G'( ) v(,) = G () ; (ii) v (,) = 0, and U3(,,()) z v(,) = U (,, z()). Proof: See the Appendix. 3 Theorem below follow immediately from Lemma 4 to 6. Theorem : Suppoe condition B and B hold. The Pareto dominant eparating equilibrium z () in the continuou type model atifie the LCT if (i) for any (, ], UU U + UUU + UU U UU + 4 UU G ( ) > G ( ) UU 3 3 (5.4) U3(,, z()) G () (ii) G ( ) > 0 and >. (5.5) U (,, z()) G() 3 Therefore, Theorem and how that the concept of the LCT can be applied to the continuou type model exactly a in the dicrete type model. While the continuou 3

25 type model can be eaier to work analytically in term of charactering the eparating equilibrium, it hould be viewed a an approximation of the ituation with many cloe dicrete type. Our poition i that it hould be ubject to the ame crutiny of credibility a dicrete type model, even though ignal are literally out-of-equilibrium in the continuou type model. The equivalence of Theorem and demontrate that thi approach i valid. We want to make everal remark on the condition that G ( ) > 0. Note that when G ( ) = 0, it can be hown that a, v (, ) 0.5. By Lemma 6, in the neighborhood of, v(, ) > v( ), hence there i alway a credible boundary deviation at the lowet ignal y = z(). Thu, the condition G ( ) > 0 i neceary for the LCT. Thi condition i completely natural in the dicrete type cae. Viewing the continuou type cae a an approximation of the dicrete type cae, we can think of thi condition a requiring that moothing out the probability mae of all other type except for type. Another way of jutifying G ( ) > 0 i a follow. Imagine that for ome participation cot not modeled here, there are ome type lightly lower than who do not actively participate in the market. A can be een from Figure 4., while thoe type do not find the ignal-perception pair (, z ) attractive, they may well find the ignalperception pair (, z ˆ ) profitable and thu would like to join the pooling deviation. Thu, when we conider a poible boundary deviation, the exitence of thee type lightly lower than will affect the perception of the receiver a if there were an equivalently probability ma on. Below we apply our reult to the example introduced above. Example (continued): The Spence education ignaling model Since Uy (,, ˆ ) = ˆ Cy (, ), the aumption B and B are both atified. In the common formulation of the model, U3(,, v y) = C(, y) < 0for all ( v,, y ). U (,, v y) = C (, y) and U33(,, v y) = C(, y), condition (5.4) become Since 3 4

26 C "( ) C C C > G C C C C C G'( ) where all function are evaluated at (,()) z. If the above condition hold, then the U(,,()) z eparating equilibrium given by z () = = and goe through U(,, z ()) C(, z ()) 3 z () = 0doe not allow credible interior deviation. Condition (5.5) i imply that G ( ) > 0 and C(, y) G ( ) >. When thee condition hold, then the Pareto dominant C (, y) G( ) eparating equilibrium atifie the LCT. In a imply cae with Cy (, ) = ( A β y ), where β > 0 meaure how fat marginal cot of education y decreae in type and A> β i a poitive contant, the intuition of the condition i very clear. The LCT will be atified if the ignal effectivene meaured by the marginal rate of ubtitution C C = β ( A β ) i large and the type ditribution i not tilted upward too much. Clearly, the greater β i, the more likely the LCT i atified. Example (continued): The advertiing ignaling model From the firm payoff function, we haveu = α( α) / b and U = α b. Since thi model doe not atify the aumption B and B, we ( ) / cannot apply Theorem directly. However, uing the ame method, it i not difficult to derive condition under which the LCT i atified. We have the following reult. Propoition 3: In the advertiing ignaling model, the Pareto eparating equilibrium atifie the LCT if (i) G () < 0for >, and (ii) G> ( ) 0 and α /( c) > G ( ) / G( ). Proof: See the Appendix. 5

27 Becaue of the quadratic function form of the firm payoff function and other pecial feature of thi model (e.g., many cro derivative are zero), the meaure of ignal effectivene i contant and equal zero. So for the condition of no credible interior deviation, it all depend on the type ditribution. Propoition 3 alo ay that for the boundary deviation not to be credible, the proportion of informed conumer mut be relatively large. Thi i intuitive. Since informed conumer will not be fooled by deviation in advertiing, more of informed conumer reduce the incentive to deviate and make the eparating equilibrium more likely to atify the LCT. A pecial cae commonly tudied in application i when the type i uniformly ditributed. ThenG = 0 for >. It can be hown that a ', all higher order partial derivative of v (, ) and v(, ) with repect to are zero, which implie that in the limit, v (, ) = v (, ). So thi i really a knife edge cae. By our definition, however, the LCT i not atified. 6. Interior Optimum for the Lowet Type So far we have aumed that U3 (,, z()) < 0for all. When the ignal i partially productive, thi aumption may not hold. Then the perfect information optimal choice for the lowet type, z * ( ) arg max U(,, y), may be an interior optimal y Y * olution, which implie that U (,, ( )) 0 3 z =. In thi cae, Theorem and do not hold any more. To illutrate the problem, conider the following example. Example 3: The price ignaling model Thi example i alo adopted from Milgrom and Robert (986). The environment i identical to that in the advertiing ignaling model, except that the firm ue price rather than advertiing to ignal it quality. Suppoe the firm chooe price p, which lead to a perception of ŝ by the uninformed conumer. Then the firm profit i π = ( p c)( αq + ( α) q ). Conider a poible eparating ignaling chedule p(). The firm payoff function i I U 6

28 π (,, p ˆ ) = ( p c)( α+ ( α) ˆ p)/ b It can be checked that thi model atifie the tandard aumption and aumption B and B. By the tandard reult, a eparating equilibrium atifie U (,, z()) ( α)( p c) U (,, z()) p c () = = 3 z However, for the lowet type, the full information optimal price i choen o that U 3 (,, p( )) = 0, which give * p = 0.5( + c). To check interior deviation, for >, condition (5.4) become: p()( + 3 α) αc ( + α ) G () > ( p c ) G () (6.) For cloe to, it can be checked that the nominator on the LHS i negative, hence the LHS goe to negative infinity. Thi how that the Pareto dominant eparating equilibrium fail the LCT around the lower boundary point. The above lower endpoint problem can be overcome if we modify the model o that there i a ignificant fixed cot to enter the market. If the fixed cot i greater than the maximum profit the lowet type can get, the lowet type and all the type lower than a certain threhold will tay out of the market. Then for the lowet type that doe enter the market, it cannot chooe it complete information optimal price. Otherwie, ome of the lower type would find it profitable entering the market mimicking it. With thi type of truncation of the type ditribution by entry cot, it i again the cae that U 3 < 0 o Theorem applie. 7. Concluding Remark Except in the pecial cae of perfect correlation, tandard refinement (Intuitive Criterion, Divinity, Stability) are not applicable. We argue that to have any bite at all, a refinement i needed in which the ignal receiver take into account the way ender type are ditributed. We then propoe a Local Credibility Tet which i milder than, but in the pirit of the Groman-Perry Criterion. For a cla of model which include the 7

29 baic Spence model and the advertiing ignaling model, we provide condition under which the Pareto dominant eparating equilibrium atifie the LCT. Thee condition are the more likely to be met, (a) the le rapidly the denity increae or the more rapidly the denity decreae with type, and (b) the more rapidly the marginal cot of ignaling decreae with type. What i the right equilibrium when our condition are not met? Thi i a challenging quetion for which we have no atifactory anwer. A can be een from the imple two-type model depicted in Figure 3.3, our LCT tet and the even tronger Groman and Perry SIC how that the Pareto dominant eparating equilibrium i not reaonable. But at the ame time, all other equilibria are killed a well. We conjecture that pooling or partial pooling mut be a part of any more complete analyi of ignaling. To make the point a tarkly a poible, let p be the probability that a type i low. With p = 0, the high type doe not have to ignal at all. A long a p i poitive, the high type ha to take cotly ignaling to eparate herelf from the low type. Thu the eparating equilibrium ha an extreme dicontinuity at p = 0. When p i cloe to zero, the pooling outcome eem more reaonable than the highly inefficient eparating equilibrium. That i, reaonable out-of-equilibrium belief do not necearily lead to reaonable outcome. 8

30 Appendix A: Proof Proof of Propoition : When N =, the propoition i obviouly true by the tandard argument of the Cho-Krep Intuitive Criterion. Suppoe N >. We firt how that the only eparating equilibrium atifying the LCT mut be the Riley outcome, and then argue that no pooling equilibrium atifie the LCT. For a eparating equilibrium, * requirement (L) implie that z ( ) = y( ), where y * ( ) maximize U(,, y ) ; otherwie, if * z ( ) > y( ), then the out-of-equilibrium ignal-perception pair * ( y ( ), ) i credible. From z = z = y, the Pareto dominant eparating * ( ) ( ) equilibrium can be characterized by the following condition U(,, z( )) = U(,, z( )) i i i i i+ i+ For a different eparating equilibrium, there i at leat one k uch that U( k, k, z( k)) > U( k, k+, z( k+ )). That i, type k + ue an exceive ignal z ( k + ) uch that type k trictly prefer her equilibrium ignal z ( k ) to mimicking type k +. By Requirement (L4), the logic of the Cho-Krep Intuitive Criterion for the two type cae implie that k+ < N. Requirement (L4) alo implie that U(,, z( )) = U(,, z( )). Otherwie, uppoe k+ k+ k+ k+ k+ k+ U(,, z( )) < U(,, z( )). Then for mall ε > 0, conider a deviation k+ k+ k+ k+ k+ k+ z ( t+ ) ε. We have (i) U( k+, k+, z( k+ )) < U( k+, k+, z( k+ ) ε), (ii) U(,, z( )) > U(,, z( ) ε) ; and (iii) k k k k k+ k+ U(,, z( ) ε) < U(,, z( )). By the ingle croing condition, part (iii) k+ k+ k+ k+ k+ k+ alo hold for type higher than k +. So z ( t+ ) ε i a credible deviation for type k +, violating (L4). However, U( k+, k+, z( k+ )) = U( k+, k+, z( k+ )) implie that U(,, z( )) > U(,, z( )). Iteration implie that for k+ k+ k+ k+ k+ k+ U(,, z( )) > U(,, z( )). But then it follow from the logic of the Cho- N N N N N N Krep Intuitive Criterion for the two type cae that Requirement (L4) i violated. 9

31 Therefore, if a eparating equilibrium atifie Requirement (L), (L) and (L4), it mut be the Pareto dominant eparating equilibrium. To rule out pooling and partial pooling equilibria, note that, by the tandard argument of the Cho-Krep Intuitive Criterion, Requirement (L) implie that the highet type cannot pool. Let k be the highet type that i pooled at z with a perception of k ˆk U(, ˆ, z ) = U(,, z( )). < k by the receiver. By the previou argument, k k k k k+ k+ Then it i eay to verify that a deviation by type k to zˆk ε, where z i uch that ˆk U(, ˆ, z ) = U(,, zˆ ), i credible. Thu, an equilibrium atifying the LCT cannot k k k k k k involve pooling. Q.E.D. Proof of Lemma : Uing n =, n = δ, n+ = + δ, we rewrite (4.) a follow: v (, )( G ( ) G ( )) [ G ( ) G ( )] [ G ( ) G ( )] δ n+ n = n + n+ n+ Differentiating with repect to δ on both ide, we have v (, δ )[ G( ) G( )] + v(, δ )[ G ( ) + G ( )] = G ( ) + G ( ) + G( ) G( ) n+ n n+ n n n+ n+ n+ v (, δ )[ G( ) G( )] + v (, δ)[ G ( ) + G ( )] + v(, δ)[ G ( ) G ( )] n+ n n+ n n+ n = G ( ) + G ( ) + G ( ) n n+ n+ n+ v (, δ )[ G ( ) G ( )] + 3 v (, δ )[ G ( ) + G ( )] n+ n n+ n + 3 v (, δ)[ G ( ) G ( )] + v(, δ)[ G ( ) + G ( )] n+ n n+ n = G ( ) + G ( ) + 3 G ( ) n n+ n+ n+ G"( ) Letting δ 0 we obtain: v(, δ) ; v(, δ) /; v(, δ). Q.E.D G'( ) Proof of Lemma : Firt by the continuity of U we have v (,0) Differentiating (4.) with repect toδ, we have = from (4.). 30