SUPPLEMENT TO PERSUASION OF A PRIVATELY INFORMED RECEIVER (Econometrica, Vol. 85, No. 6, November 2017, )

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1 Economeica Supplemenay Maeial SULEMENT TO ERSUASION OF A RIVATELY INFORMED RECEIVER Economeica, Vol. 85, No. 6, Novembe 2017, ANTON KOLOTILIN School of Economics, UNSW Business School TYMOFIY MYLOVANOV Depamen of Economics, Univesiy of isbugh and Kyiv School of Economics ANDRIY ZAECHELNYUK School of Economics and Finance, Univesiy of S Andews MING LI Depamen of Economics, Concodia Univesiy and CIREQ AENDIX A: MULTILE ACTIONS IN THIS AENDIX, we allow he eceive o make a choice among muliple acions. We chaaceize he implemenable eceive s ineim uiliies and show ha he sende can geneally implemen a sicly lage se of he eceive s ineim uiliies by pesuasion mechanisms han by expeimens. We also fomulae he sende s opimizaion poblem and show ha he sende can achieve a sicly highe expeced uiliy by pesuasion mechanisms han by expeimens. A.1. efeences Le A ={0 1 n} be a finie se of acions available o he eceive. The sae ω =[0 1] and he eceive s ype R =[0 1] ae independen and have disibuions F and G. We coninue o assume ha he eceive s uiliy is linea in he sae fo evey ype and evey acion. I is convenien o define he eceive s and sende s uiliies, uω a and vω a, ecusively by he uiliy diffeence beween each wo consecuive acions. Fo each a {1 n}, uω a uω a 1 = b a ω x a vω a vω a 1 = z a + ρ uω a uω a 1 and he uiliies fom acion a = 0 ae nomalized o zeo, uω 0 = vω 0 = 0fo all ω and all. Fo each a {1 n}, he eceive s and sende s uiliies can be expessed as a a uω a = b i ω b i x i Anon Koloilin: akoloilin@gmail.com Tymofiy Mylovanov: mylovanov@gmail.com Andiy Zapechelnyuk: azapech@gmail.com Ming Li: ming.li@concodia.ca 2017 The Economeic Sociey DOI: /ECTA13251

2 2 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI a vω a = z i + ρuω a We assume ha b a > 0foall and all a {1 n}. This assumpion means ha evey ype pefes highe acions in highe saes. Noe ha x a is he cuoff sae a which he eceive of ype is indiffeen beween wo consecuive acions a 1anda. Define x 0 = and x n+1 =. Denoe by x a he cuoff uncaed o he uni ineval, x a = max { 0 min { 1 x a }} We assume ha he cuoffs ae odeed on [0 1] such ha x 1 x 2 x n fo all R Thus, ype opimally chooses acion a on he ineval of saes x a x a+1. 1 A.2. Expeimens Because he eceive s uiliy is linea in he sae fo evey ype and evey acion, evey expeimen σ can be equivalenly descibed by he pobabiliy ha he poseio mean sae is below a given value x, H σ x = σx ω dfω In fac, as in Blackwell 1951, Rohschild and Sigliz 1970, andgenzkow and Kamenica 2016, i is convenien o descibe an expeimen by a convex funcion C σ : R R defined as C σ x = 1 Hσ m dm x Obseve ha by 9, fo evey expeimen σ,wehavec σ = U σ fo all,wheeu σ is he eceive s ineim uiliy unde σ in he poblem of Secion 2, wih wo acions and uω a = aω. Hence, by Theoem 1, he se of all C σ is equal o C ={C : C C C and C is convex} whee C and C coespond o he full and no disclosue expeimens, Cx = x 1 Fm dm Cx = max { E[ω] x 0 } 1 This assumpion ensues ha he acions ha can be opimal fo ype ae consecuive. If acions a 1 and a + 1 ae opimal fo ype unde saes ω and ω, hen hee mus be a sae beween ω and ω whee acion a is opimal. This assumpion simplifies he exposiion. Relaxing his assumpion poses no difficuly: i will only equie us, fo each ype, o omi fom he analysis he acions ha ae neve opimal fo his ype.

3 ERSUASION OF A RIVATELY INFORMED RECEIVER 3 A.3. Implemenable Ineim Uiliies The expeced uiliy of ype unde expeimen σ is equal o U σ = max a A um a dh σ m fo all R ROOSITION 1: U is implemenable by an expeimen if and only if hee exiss C C such ha U= n b a C x a fo all R 12 ROOF: The eceive s ineim uiliy unde expeimen σ is U σ = n a=0 xa+1 x a um a dh σ m = n x a b a m x a dh σ m whee we used um 0 = 0. Fo each a {1 n}, inegaion by pas yields x a b a m x a dh σ m = b a m x a 1 H σ m x a + b a 1 Hσ m dm = b a C σ xa x a Summing up he above ove a {1 n} yields 12. I followsfomsecion A.2 ha he se of he eceive s ineim uiliies implemenable by expeimens is equal o he se of funcions ha saisfy 12 fo evey C C. Q.E.D. A pesuasion mechanism can be descibed by a possibly, infinie menu of expeimens, Σ. The eceive of ype chooses one expeimen fom he menu and hen obseves messages only fom his expeimen. Obviously, he eceive chooses he expeimen ha maximizes his expeced uiliy, U Σ = max U σ σ Σ fo all R By oposiion 1, i is immediae ha he eceive s ineim uiliy U is implemenable if and only if hee exiss a menu C Σ C such ha { n U= max b a C x a } fo all R. C C Σ Theoem 1 shows ha he sende can implemen he same se of eceive s ineim uiliies by expeimens as by pesuasion mechanisms. Wih moe han wo acions, howeve, he sende can geneally implemen a sicly lage se of ineim uiliies by pesuasion mechanisms han by expeimens, as shown in Example 2.

4 4 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI EXAMLE 2: Le A ={0 1 2}, F admi a sicly posiive densiy, and uω a be coninuous in fo all ω and all a. Fuhemoe, suppose ha hee exis wo ypes <, such ha fo all, 0 <x 1 <x 1 < x 1 <x 2 <x 2 < x 2 < 1 Conside a pesuasion mechanism consising of he menu of wo expeimens epesened by paiions { },whee and ae he fis-bes paiions fo ypes and, = {[ 0 x 1 [ x 1 x 2 [ x 2 1 ]} = {[ 0 x 1 [ x 1 x 2 [ x 2 1 ]} Types and choose, especively, and and ge hei maximum possible uiliies U and U. By he coninuiy of uω a in, hee exiss a ype who is indiffeen beween choosing and. By his indiffeence, whee, fo each a {1 2}, L a = L 1 + L 2 = L 1 + L 2 xa x a u ω a u ω a 1 dfω denoes he uiliy loss of ype fom using cuoff x a ahe han his fis-bes cuoff x a o decide beween acions a 1anda. Analogously, fo each a {1 2}, wedefine L a. Figue 4 illusaes his example. The hee blue lines depic he uiliy of he eceive of ype fom aking acion a, uω a, fo each a = The kinked solid blue line is he uiliy of ype fom aking he opimal acion, max a {0 1 2} uω a.infigue4, he loss of ype fom expeimen elaive o he fis bes, L 1 + L 2, is he oal aea of he wo shaded iangles assuming ha ω is unifomly disibued. Similaly, he loss of ype fom expeimen elaive o he fis bes, L 1 + L 2, is he oal aea of he wo hached iangles. Fo ype, hese shaded and hached aeas ae equal, so ype is indiffeen beween he wo expeimens. An expeimen ha gives he maximum possible uiliies U and U o ypes and mus a leas communicae he common efinemen of paiions and. Theefoe, he uiliy of ype unde such an expeimen is a leas U min { L 1 L 1 } min { L 2 L 2 } which is sicly lage han his uiliy unde he pesuasion mechanism, U L 1 L 2 unless L 1 = L 1 and L 2 = L 2. In Figue 4, foype, he loss L 1 lef hached iangle is smalle han he loss L 1 lef shaded iangle. Similaly, he loss L 2 igh shaded iangle is smalle han he loss L 2 igh hached iangle. Hence, he oal loss is smalle unde he expeimen ha is he coases common efinemen of and he aea of he smalle shaded and hached iangles han unde eihe expeimen o expeimen.

5 ERSUASION OF A RIVATELY INFORMED RECEIVER 5 FIGURE 4. The uiliy of ype in Example 1. A.4. Sende s oblem In his secion, we impose he following addiional assumpions. Fo each a {0 1 n}, funcionx a is sicly inceasing, and is ange conains. Le funcion a x be he invese of funcion x a. Moeove, fo each a {0 1 n}, funcionz a is diffeeniable, and funcion G a x is wice diffeeniable. Fo a given expeimen σ, he sende s expeced uiliy condiional on he eceive s ype being is 2 n a Hσ V σ = ρu + z i xa+1 H σ xa We now expess he sende s expeced uiliy as a funcion of C σ, and all he model paamees ae summaized in funcion I, he same way as in Lemma 2 in Secion 4. LEMMA 3: Fo evey expeimen σ, V σ dg = K + whee K is a consan independen of σ and n d Ix= z a a x d dx dx G a x R C σ xix dx + ρ a x b a a x d dx G a x 2 Fo each whee H σ x a+1 is disconinuous, his fomula assumes ha ype beaks he indiffeence in favo of acion a if he poseio mean sae is x a+1. This assumpion is innocuous because G admis a densiy, and hee ae a mos counably many disconinuiies of H σ.

6 6 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI ROOF: Wehave n a Hσ z i xa+1 H σ xa = n z a 1 H σ xa = n z a C σ xa wheeweusedx n+1 = hence, H σ x n+1 = 1 and 1 H σ x = C σ x. By oposiion 1, we, hus, obain n a Hσ V= ρu + z i xa H σ xa+1 = ρu n z a C σ xa = n ρba C σ xa z a C σ xa Fix a {1 n} and define he vaiable x = x a. Hence, = a x. Using his vaiable change, we have ρba C σ xa z a C σ xa dg R = ˆK a + ρ a x b a a x C σ x z a a x C x σ dg a x whee ˆK a is a consan independen of σ because =[0 1] [x a 0 x a 1], and,foall σ and all x/, wehavec σ x = max{0 E[ω] x}. Now we inegae by pas z a a x C x dg σ a x = K a C σ x d z a a x d dx dx G a x dx = K a Cx d z a a x d dx dx G a x dx whee K a is a consan independen of σ because, fo all σ, wehavec σ 0 = E[ω] and C σ 1 = 0. Thus, we obain ρba C σ xa z a C σ xa dg R = K a + ρ a x b a a x dg a x + d z a a x d dx dx dx G a x C σ x dx whee K a = ˆK a + K a. Summing he above ove a {1 n}, weobaink + C σx Ixdx, wheek = K a a,andi is defined in Lemma 3. Q.E.D. The sende s opimal expeimen is descibed by a funcion C C ha solves CxIx dx max C C The soluions o his poblem ae chaaceized by Theoem 2.

7 ERSUASION OF A RIVATELY INFORMED RECEIVER 7 As shown in Secion A.3, when he eceive has moe han wo acions, he sende can implemen a sicly lage se of eceive s ineim uiliies by pesuasion mechanisms han by expeimens. We now show ha he se of implemenable ineim acions is also sicly lage unde pesuasion mechanisms. Theefoe, even if he sende caes only abou he eceive s acion, and no his uiliy, ρ = 0foall, he sende can achieve a sicly lage expeced uiliy unde pesuasion mechanisms. EXAMLE 2 Coninued: In addiion, le hee exis x 2 x 1 x 2 such ha E[ω ω x 2 ]=x 2 and E[ω ω<x 2 ] <x 1. An expeimen ha maximizes he pobabiliy of acion a = 2foype mus send message x 2 if and only if ω [x 1]. Unde any such expeimen, ype 2 akes acion a = 2ifandonlyifω [x 1], because fo 2 ω<x 2, his expeimen mus geneae messages disinc fom x 2 and, hus, below x, which is in un below x 2 2. Conside now a pesuasion mechanism consising of he menu of wo expeimens epesened by he following paiions: = {[ 0 x ε 2 \ [ x 1 x 1 [ x 1 x 1 [ x ε 1]} 2 = {[ 0 x ε 2 [ x ε [ 2 x 2 x 1]} 2 whee ε>0is sufficienly small. Type sicly pefes o because, fo a sufficienly small ε, he benefi of aking acion a = 1 ahe han a = 0 on [x 1 x 1 exceeds he cos of aking acion a = 2 ahe han a = 1 on [x ε 2 x 2.Type is indiffeen beween and, because unde boh paiions he weakly pefes o ake acion a = 0 on [0 x ε and acion a = 2 1on[x 2 ε 1]. Theefoe, unde his mechanism, ype akes acion a = 2ifandonlyifω [x 1], 2 buype akes acion a = 2ifandonlyif ω [x ε 1]. As shown above, hese pobabiliies of a = 2 2foypes and canno be achieved by any expeimen. Finally, an opimal pesuasion mechanism need no be an expeimen. Suppose ha he sende caes only abou acion a = 2, ha is, ρ = z 0 = z 1 = 0andz 2 = 1 fo all. Also suppose ha he suppo of G conains only and wih being likely enough, so ha he sende s opimal expeimen maximizes he pobabiliy of acion a = 2 fo ype. The pesuasion mechanism consuced above gives a sicly lage expeced uiliy o he sende han any expeimen. AENDIX B: NONLINEAR UTILITIES In his appendix, we allow he sende s and eceive s uiliies o be nonlinea in he sae. We chaaceize condiions unde which pesuasion mechanisms ae equivalen o expeimens, and show, in paicula, ha cuoff mechanisms ae equivalen o expeimens. We also show ha he equivalence of implemenaion by pesuasion mechanisms and by expeimens geneally fails. B.1. efeences As in Secion 2, he eceive has wo acions, A ={0 1}, he se of saes is =[0 1], and he se of eceive s ypes is R =[0 1]. The eceive s uiliy, howeve, is uω a = auω

8 8 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI whee uω is diffeeniable, sicly inceasing in ω, and sicly deceasing in. We also nomalize he uiliy such ha, fo each ω, The sende s uiliy is uω ω = 0 vω a = avω whee vω is diffeeniable. Sae ω and ype ae independen and have disibuions F and G. 3 B.2. Chaaceizaion of Expeimens We sa wih he chaaceizaion of pesuasion mechanisms ha ae equivalen o expeimens. ROOSITION 2: An incenive-compaible pesuasion mechanism π is equivalen o an expeimen if and only if πω is noninceasing in fo evey ω. Inuiively, because fo each expeimen σ, he disibuion σ ω of condiional on each sae ω is nondeceasing in, eachπω [1 σ ω 1 σ ω] see 6 is noninceasing in. ROOF: Conside a mechanism π ha is equivalen o an expeimen σ. Sinceσ ω is a disibuion funcion of condiional on ω, i is nondeceasing in fo each ω. Then, by 6, πω is noninceasing in fo each ω. Convesely, le π ω be noninceasing in fo all ω. Fo evey ω and, define σ ω = 1 π + ω,wheeπ + ω denoes he igh limi of π ω a. Since π + ω [0 1] is noninceasing and igh-coninuous in, he funcion σ ω is a disibuion, which descibes he disibuion of messages fo evey given sae ω. Thus,σ is an expeimen. I emains o veify ha he consuced expeimen is diec and induces he same acion by he eceive as mechanism π, ha is, when he expeimen sends a message, hen ype is indiffeen beween he wo acions. Fo all, U π = = uω πω dfω= uω πω + dfω= uω + πω + dfω uω 1 σ ω dfω= U σ whee he fis equaliy holds by he definiion of U π, he second by he absolue coninuiy of U π Theoem 1 of Milgom and Segal 2002, he hid by he coninuiy of u in, he fouh by he definiion of σ, and he las by he definiion of U σ fo diec expeimens. 3 Noe ha if ω and ae coelaed, he analysis below applies if we impose sic monooniciy on funcion ũω = uω g ω/g ahe han on u, wheeg and g ω denoe, especively, he maginal densiy of and he condiional densiy of fo a given ω. This is because he eceive s ineim uiliy unde a mechanism π can be wien as U= ũω πω dfω.

9 ERSUASION OF A RIVATELY INFORMED RECEIVER 9 Thee exis lef and igh deivaives of U π fo all Theoem 3 of Milgom and Segal 2002 ha saisfy U uω π + = π + ωdfω U π = uω π ωdfω Since U π = U σ and σ ω = 1 π + ωfo all, wehave U uω σ + = 1 σ ω dfω U uω σ = 1 σ ω dfω showing ha ype is indiffeen beween he wo acions upon eceiving message. Q.E.D. B.3. Binay Sae Hee we apply oposiion 2 o show ha if hee ae only wo saes in he suppo of he pio F, hen evey incenive-compaible mechanism is equivalen o an expeimen. COROLLARY 2: Le he suppo of F consis of wo saes. Then evey incenive-compaible mechanism π is equivalen o an expeimen. ROOF: Conside F whose suppo consiss of wo saes, wihou loss of genealiy, {0 1}, andleπ be an incenive-compaible pesuasion mechanism. By oposiion 2, i is sufficien o show ha π is noninceasing in fo all 0 1. Incenive compaibiliy implies ha fo all ˆ 0 1, uω πω πω ω = ω=0 1 Rewiing 13 wice, wih = 2 1 and = 1 2, yields he inequaliies u0 2 u1 2 δ δ u0 1 u1 1 δ whee δ 2 1 ω= πω 2 πω 1 ω = 1. Because u0 <0andu1 >0 fo = 1 2, he monooniciy of u in implies ha 0 < u0 2 u1 2 u0 1 u1 1 fo Combining 14and15 gives πω 2 πω 1 if 2 1 fo each ω = 0 1. Q.E.D. Noe ha if F has a wo-poin suppo, hen he eceive s uiliy is linea in he sae wihou loss of genealiy, and, hence, Theoem 1 applies. Howeve, Coollay 2 makes a songe saemen, because i asses ha evey incenive-compaible mechanism is equivalen o an expeimen, no jus implemens he same eceive s ineim uiliy.

10 10 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI B.4. Beyond Binay Sae Suppose now ha he suppo of he pio F consiss of hee saes ω 1 <ω 2 <ω 3 and le f i = ω i >0foi = When hee ae a leas hee saes and he uiliy of he eceive is nonlinea in any ansfomaion of he sae, hen he poseio disibuion of he sae induced by an expeimen can no longe be paameeized by a one-dimensional vaiable such as he poseio mean sae in he case of linea uiliies, he poseio pobabiliy of one of he saes in he case of binay-valued sae, and he cuoff value in he case of cuoff mechanisms. As a consequence, he ineim acion q and, hence, he sende s ineim uiliy V ae no longe pinned down by he eceive s ineim uiliy U. ROOSITION 3: Le π 1 and π 2 be wo mechanisms ha ae disinc fo each ω 1 ω 3 bu implemen he same diffeeniable eceive s ineim uiliy U. Then, he ineim acion q ishesamefoπ 1 and π 2 if and only if hee exis funcions b, c, and d such ha uω = c + bdω fo each ω {ω 1 ω 2 ω 3 } ω 1 ω 3. ROOF: Foall ω 1 ω 3 and j = 1 2, we have U= U = 3 uω i π j ω i f i 3 uω i π j ω i f i whee he fis line holds by he definiion of U, and he second line by he incenive compaibiliy of π. The expeced acion, q πj = 3 π jω i f i, is he same acoss j = 1 2 fo each if and only if he vecos uω, uω,and1 ae linealy dependen fo each. Tha is, fo each ω {ω 1 ω 2 ω 3 } ω 1 ω 3, hee exis funcions γ and μ such ha uω The soluion of diffeenial equaion 16isgivenby uω = e ω1 μx dx ηω + γxe ω 1 + μuω = γ 16 xω1 μy dy dx whee funcion ηω saisfies he iniial nomalizaion condiion uω ω = 0. This complees he poof wih b, c,anddω given by b c dω = e ω1 μx dx xω1 μy γxe dy dx e ω1 μx dx ηω ω 1 Q.E.D. When he eceive s uiliy is nonlinea, he sende can implemen a sicly lage se of he eceive s ineim acions by pesuasion mechanisms han by expeimens. Theefoe, he sende can achieve a sicly highe expeced uiliy by pesuasion mechanisms, even if he uiliy v is sae-independen.

11 ERSUASION OF A RIVATELY INFORMED RECEIVER 11 FIGURE 5. The uiliies of ypes blue and black in Example 2. EXAMLE 3: Le hee be hee saes, ω 1 <ω 2 <ω 3, and wo ypes of he eceive, >.Denoeu = uω i i and u i = uω i fo i {1 2 3}. Assume u < 0 2 <u. 3 Moeove, assume u 1 /u 1 >u 2 /u, which means ha poin ω 2 1 u 1 lies above he dashed line in Figue 5. Finally, assume ha he pobabiliy masses f 1 f 2 f 3 on he saes saisfy f 1 u + f 1 3u < 0andf 3 2u + f 2 3u < 0. 3 Le Σ be he se of all expeimens ha maximize he pobabiliy of acion fo ype. I is easy o check ha any σ Σ induces ype o ac wih pobabiliy 1 if ω = ω 3,wih pobabiliy f 3 u /f 3 2u if ω = ω 2 2, and wih pobabiliy 0 if ω = ω 1. Obseve ha by he monooniciy of u in, each message of σ Σ ha induces ype o ac also induces ype < o ac. Moeove, by he definiion of Σ, each message of σ Σ ha induces ype nooaccanbesenonlyinsaesω 1 o ω 2 whee he uiliy of ype is negaive by assumpion, u 1 <u < 0, so ype 2 does no ac eihe. Thus, fo each expeimen σ unde which ype acs wih pobabiliy f 3 1 u 3 /u i.e., σ 2 Σ, ype acs wih he same pobabiliy as ype. We now consuc a pesuasion mechanism ha also maximizes he pobabiliy ha ype acs, bu induces ype o ac wih a diffeen pobabiliy. Le Σ be he se of all expeimens σ ha induce ype o ac wih pobabiliy 1 if ω = ω 3, wih pobabiliy 0 if ω = ω 2, and wih pobabiliy f 3 u /f 3 1u if ω = ω 1 1. Conside a pesuasion mechanism ha consiss of a menu of wo expeimens {σ σ } wih σ Σ and σ Σ.Noice ha ype is indiffeen beween σ and σ as he obains zeo expeced uiliy in eihe case. Howeve, ype sicly pefes σ o σ, because u 1 /u 1 >u 2 /u 2 by assumpion. Theefoe, unde his pesuasion mechanism, ype acs wih pobabiliy f 3 1 u 3 /u, 2 bu ype acs wih diffeen pobabiliy f 3 1 u 3 /u. 1 Finally, when he eceive s uiliy is nonlinea, he se of eceive s ineim uiliies implemenable by pesuasion mechanisms as compaed o expeimens can be sicly lage.

12 12 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI EXAMLE 3 Coninued: In addiion, le be such ha u 1 /u 1 <u 2 /u 2,whee u i = uω i fo i {1 2 3}. I is easy o check ha Σ and Σ ae he ses of expeimens σ and σ ha maximize he uiliy of ypes and, especively, subjec o he consain ha ype ges uiliy U = 0. Because Σ and Σ do no inesec, no expeimen can achieve he ineim uiliy induced by a pesuasion mechanism ha consiss of he menu of wo expeimens σ Σ and σ Σ. AENDIX C: LINEAR UTILITIES This appendix exends ou main esuls o he class of uiliy funcions ha ae linea in he sae. Specifically, we conside he model defined in Secion 2, wih he modificaion ha he uiliies ae linea in he sae and ae abiay funcions of he eceive s ype. Le he eceive s and sende s uiliies be nomalized o zeo if he eceive does no ac, a = 0, and be linea in he sae if he eceive acs, a = 1, uω a = a bω vω a = a cω + d whee = b c d R 4 denoes he eceive s ype. The ype has disibuion G ha admis a diffeeniable densiy g, which is sicly posiive on a compac se in R 4 and zeo eveywhee else. The sae ω =[0 1] is independen of and has disibuion F. Le H σ be he disibuion of he poseio mean induced by an expeimen σ. Asin Secion A.2, i is convenien o descibe σ by C σ = 1 Hσ m dm ROOSITION 4: Fo each expeimen σ, he eceive s ineim uiliy is U σ = b C σ + min{0 b} E[ω] 17 Thee exis K R and I : R R such ha, fo each σ, he sende s expeced uiliy is V σ = K + C σ I d 18 R oposiion 4 allows us o exend Theoems 1 and 2 o his seing. Recall fom Secion A.2 ha he se of all C σ is equal o C ={C : C C C and C is convex} whee C and C coespond o he full and no disclosue expeimens. Each pesuasion mechanism can be descibed by a possibly, infinie menu of expeimens, Σ, which he eceive chooses fom. By 17, fo a given menu Σ, he eceive s ineim uiliy is max U σ = b σ Σ max σ Σ C σ + min{0 b} E[ω] Noice ha max σ Σ C σ is he uppe envelope of convex funcions C σ C and hence i is in C. Theefoe, by oposiion 4, any implemenable pai of he sende s and eceive s

13 ERSUASION OF A RIVATELY INFORMED RECEIVER 13 expeced uiliies is implemenable by an expeimen. Moeove, he sende s poblem can be expessed as max CId C C R and Theoem 2 holds wih U eplaced by C. ROOF OF ROOSITION 4: Fixaype = b c d R 4 and evaluae U σ = 1 0 max { 0 bm } dh σ m Clealy, if b = 0, hen U σ = 0. We now conside wo cases, b>0andb<0. Case 1: b>0. Given a poseio mean m, he eceive acs if and only if <m.by inegaion by pas, he eceive s ineim uiliy is U σ = 1 bm dh σ m = bc σ Again, by inegaion by pas, he sende s ineim uiliy is V σ = 1 cm + d dhσ m = cc σ dc σ Case 2: b<0. Given a poseio mean m, he eceive acs if and only if m. By inegaion by pas, he eceive s ineim uiliy is U σ = 0 bm dhm = bc + b E[ω] Again, by inegaion by pas, he sende s ineim uiliy is V σ = 0 cm + d dhm = cc + dc + d + c E[ω] We, hus, obain 17. We now show ha he sende s expeced uiliy is given by 18. Le gb c d be he densiy of b c d condiional on, andleg be he maginal densiy of.define c + = cgb c d 1 b>0 db c d and c = cgb c d 1 b<0 db c d Similaly, define d + and d. Fix and ake he expecaion wih espec o b c d on he se of b>0: V σ b c d 1 b>0 gb c d db c d = c + C σ d + C σ b c d

14 14 KOLOTILIN, MYLOVANOV, ZAECHELNYUK, AND LI Now, inegaing wih espec o, V σ b c d 1 b>0 dgb c d = c+ C σ d + C σ g d b c d Similaly, fo b<0, b c d = V σ b c d 1 b<0 gb c d db c d c + g + d d [ d+ g ] C σ d = c C σ + d C σ + d + c E[ω] Now, inegaing wih espec o, V σ b c d 1 b<0 dgb c d = c C σ d C σ g d + K b c d whee = K = d + c E[ω] g 1 b<0 d c g + d d [ d g ] C σ d + K isaconsanindependenofc σ. Since he measue of ypes wih b = 0 is zeo, we obain V σ dg = V σ 1 b>0 + 1 b<0 dg = IC σ d + K whee I= c + c g + d d [ d+ d g ] Q.E.D. REFERENCES BLACKWELL, D. 1951: Compaison of Expeimens, in oceedings of he Second Bekeley Symposium on Mahemaical Saisics and obabiliy, Vol. 1, [2] GENTZKOW, M.,AND E. KAMENICA 2016: A Rohschild-Sigliz Appoach o Bayesian esuasion, Ameican Economic Review, apes & oceedings, 106, [2] MILGROM,., AND I. SEGAL 2002: Envelope Theoems fo Abiay Choice Ses, Economeica, 70, [8,9] ROTHSCHILD, M., AND J. STIGLITZ 1970: Inceasing Risk: I. A Definiion, Jounal of Economic Theoy, 2, [2] Co-edio Joel Sobel handled his manuscip. Manuscip eceived 23 Febuay, 2015; final vesion acceped 26 June, 2017; available online 28 June, 2017.