Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he exper and he decision maker, U e (a,, b) and U d (a, ), are assumed o saisfy he following condiions: ) U e ( ) and U d ( ) are wice coninuously differeniable, 2) U( ) i <, 3) U(a, i ) = for some a R, 4) U2( ) i >, i = e, d, 5) U d (a, ) = U e (a,, ) for all (a, ), and 6) U3( ) e > everywhere. Also define s argmax a(r, s) = U d (a, )d, if r < s, a r a d (r), if r = s. The emergence of off-equilibrium beliefs in he amaeur model, which canno be eliminaed by having all messages used, poses a challenge in he equilibrium characerizaion. A full characerizaion requires consideraion of all possible off-equilibrium beliefs, which represen a very large se. In he following, I illusrae he issues and provide a specificaion of beliefs ha, coupled wih a mild condiion on he exper s payoff, ensures he exisence of pariional equilibria. I firs show ha, for whaever beliefs afer false advice, he exper never fully reveals his informaion. Inuiively, if he exper reveals, he amaeur s own informaion will become useless, implying ha he amaeur will respond in exacly he same way as does he novice. Since a biased exper does no fully reveal o he novice, he also will no do so o he amaeur. Proposiion 5. There exiss no separaing equilibrium in he amaeur model. Proof. If he exper in he amaeur model fully reveals his informaion, for all Θ, induces a d () on all inerval ypes. Accordingly, a ype- exper s payoff is U e (a d (),, b). Suppose here exiss a fully separaing equilibrium. For any η >, we can find a < η ha induces a d ( ) in he equilibrium. Suppose deviaes by sending m ha in he equilibrium is reserved for = ɛ, where ɛ > is such ha a d ( ) < a d ( ɛ) < a e (, b). The coninuiy of he payoff funcions and ha U d (a, ) U e (a,, ) guaranee ha such an ɛ exiss. Upon receiving m, all inerval ypes wih / (, ɛ) will ake acion a d ( ɛ), which, given he choice of ɛ and ha U( ) e <, is sricly preferred over a d ( ) by. To he remaining ypes wih (, ɛ), all being low-inerval ypes, m is a false advice, and hey ake acion under off-equilibrium beliefs ψ( l ). For any such beliefs, given ha U2( ) d >, he se of induced acions aken by hese inerval ypes is bounded by a d () and a d ( ɛ). Since U( ) e <, we
have U e (a d (),, b) U e (a,, b) for all a [a d (), a d ( ɛ)]. Thus, he payoff for o send m is, regardless of he specificaion of ψ( l ), bounded below by (A.) U e (a d ( ɛ),, b) ɛ U e (a d (),, b) ɛ U e (a d ( ɛ),, b). Subracing s equilibrium payoff from (A.) gives: (A.2) [U e (a d ( ɛ),, b) U e (a d ( ),, b)] ɛ ɛ [U e (a d ( ɛ),, b) U e (a d ( ),, b)]. [U e (a d (),, b) U e (a d ( ),, b)] We can choose η sufficienly small such ha, for any ɛ ha saisfies he above crierion of choosing he deviaing message, he firs and he hird posiive erms in (A.2) dominae he second negaive erms for some < η. The sric incenive for some o deviae poses a conradicion o he exisence of fully separaing equilibrium. In he CS model, when he indifference condiion is saisfied under a pariional sraegy of he exper (p.7-8) where N 2, i follows ha he following incenive-compaibiliy condiions are saisfied: ) every boundary ype i will (weakly) prefer sending messages in M i over any oher messages; and 2) in he inerior of every I i he inerior ypes will (sricly) prefer he same. The exper s sraegy hus consiues an informaive pariional equilibrium. In he amaeur model, even if he indifference condiion holds under such sraegy, in which (A.3) V e (M i, i, b) = V e (M i, i, b), i =,..., N, = and N =, since he off-equilibrium beliefs may generae a benefi of lying under false advice he incenivecompaibiliy condiions is no necessarily saisfied. Before illusraing how benefi of lying may arise, I pause o characerize he induced acions of he amaeur under a pariional sraegy. I disinguish beween wo ypes of induced acions, effecively induced and ineffecively induced. An acion a is effecively induced by m if he amaeur updaes her beliefs µ( m, s ) using Bayes rule and here exiss Θ such ha, in her maximizaion problem of which a is he soluion, µ( m, s ) φ( s ). Lemma 4. A high-inerval ype h akes effecively induced acions if and only if she receives. subsiuing advice: her hreshold I i, i =,..., N, and he exper reveals ha I j, i < j N; or 2. complemenary advice: her hreshold I i, i =,..., N, and he exper reveals ha I i, and she akes ineffecively induced acions if and only if she receives Wih a sligh abuse of noaions, I use V e (M i, i, b) o sand for V e (m, i, b) for all m M i. 2
3. redundan advice: her hreshold I N, and he exper reveals ha I N ; or 4. false advice: her hreshold I i, i = 2,..., N, and exper reveals ha I k, k < i. A low-inerval ype l akes effecively induced acions if and only if she receives 5. subsiuing advice: her hreshold I i, i = 2,..., N, and he exper reveals ha I k, k < i; or 6. complemenary advice: her hreshold I i, i = 2,..., N, and he exper reveals ha I i, and she akes ineffecively induced acions if and only if she receives 7. redundan advice: her hreshold I, and he exper reveals ha I ; 8. false advice: her hreshold I i, i =,..., N, and he exper reveals ha I j, i < j N. Proof. I prove he cases for high-inerval ypes; he cases for low-inerval ypes are similar. Consider firs Condiions and 2. Suppose h wih I i, i =,..., N, receives m indicaing ha I j. For j > i, h s updaed beliefs under Bayes rule are µ( m, h ) = /( j j ) for ( j, j ] and zero elsewhere. For j = i, h s updaed beliefs are µ( m, h ) = /( i ) for [, i ] and zero elsewhere. In boh cases, here exiss [, ] such ha µ( m, h ) φ( h ), and he resuling acions are effecively induced. This proves he sufficiencies. The necessiies is proved by conraposiive. Suppose h wih I i, i =,..., N, receives m indicaing ha I j, j < i. Noe ha hen Θ σ (m) h = ; Bayes rule canno be applied, and he resuling acion canno be effecively induced. Finally, suppose h wih I N receives m indicaing ha I N. Her updaed beliefs are µ( m, h ) = /( ) for [, ] and zero elsewhere. This is equivalen o φ( h ) for all [, ], and he resuling acion canno be effecively induced. Since here are only wo ypes of acions, effecively induced and ineffecively induced, and hey are muually exclusive, he sufficiencies (necessiies) in Condiions 3 and 4 for ineffecively induced acions follow from he above necessiies (sufficiencies) for effecively induced acions. Using Lemma 4, he profile of acions effecively induced on h, I i, i =,..., N, is { a(, i ), for m M i (complemenary advice), ρ(m, h ) = a( j, j ), for m M j, i < j N (subsiuing advice); and ha induced on l, I i, i = 2,..., N is { a( i, ), for m M i (complemenary advice), ρ(m, l ) = a( k, k ), for m M k, k < i (subsiuing advice). The profile of acions ineffecively induced by redundan advice is ρ(m, h ) = a(, ) and ρ(m, l ) = a(, ). The profile of acions induced by false advice depends on he off-equilibrium beliefs. To 3
illusrae he benefi of lying, suppose he indifference condiion holds wih N = 3 in a proposed equilibrium. Consider boundary ype (Figure 3). 2 If is he rue sae, all high-inerval ypes will have and all low-inerval ypes >. Suppose sends m M, indicaing ha [, ]. Then, all high-inerval ypes will ake acion a(, ), [, ], and all lowinerval ypes will ake acion a(, ) (he second line in Figure 3). Given ha saisfies he indifference condiion, he will be indifferen beween inducing hese acions and hose induced by m M 2, which are a(, 2 ) and a(, ), (, 2 ] (no shown in he figure). Deviaion high-inerval ypes hresholds low-inerval ypes hresholds acions induced: a( 2, ) a? a( 2, ) Proposed Equilibrium high-inerval ypes hresholds low-inerval ypes hresholds acions induced: a(, ) a(, ) 2 M M 3 Figure : Incenives for Deviaions Now, suppose lies by sending m M 3, indicaing ha ( 2, ]. All high-inerval ypes will ake a( 2, ) and all low-inerval ypes wih ( 2, ] will ake a( 2, ) (he firs line in Figure 3). These inerval ypes canno deec he lie and are effecively induced o ake hese acions. And o hese acions are less favorable han hose induced by M (or M 2 ). To he low-inerval ypes wih (, 2 ], he advice can be deeced as false. Wihou any resricion, one can come up wih off-equilibrium beliefs so ha he ineffecively induced acions aken by hese low-inerval ypes will be closer o s ideal han is a(, ), he effecively induced acion aken by hese ypes in he proposed equilibrium. This creaes a benefi of lying, which is absen in he CS model. I is conceivable, especially in equilibria wih more seps, ha such benefi of lying may ouweigh he cos of inducing less favorable acions. Wha equilibria may emerge in siuaions of his sor require ad hoc and deailed specificaions of beliefs. The following proposiion saes, however, ha here is a se of off-equilibrium beliefs ha, ogeher wih a mild resricion on he exper s payoff, ensures he sufficiency of he indifference condiion for he exisence of pariional equilibria. Denoe ψ o be he se of off-equilibrium beliefs of all inerval ypes: ψ = {,s} T {l,h} ψ( s). Proposiion 6. There exiss a se of off-equilibrium beliefs ψ such ha, provided U2( ) e is sufficienly large, he boundary ypes { i } N i= ha saisfy (A.3) always consiue an equilibrium. 2 There could be a benefi of lying for boundary ypes whenever N 3. However, for he inerior ypes, such benefi also arises for N = 2. Thus, he indifference condiion is no always sufficien even for wo-sep equilibria. Indeed, in he CS model, incenive compaibiliy for he inerior ypes is a consequence of ha for he boundary ypes. As will be discussed below, his is also no rue in he amaeur model. 4
Proof. I firs consruc ψ and sae he cases of he exper s payoff V e (m,, b) under ψ. I hen show ha, if ψ = ψ and U e 2( ) is sufficienly large, hen (A.3) is sufficien for he following o always hold: for all [ i, i ], (A.4) V e (M i,, b) = max V e (M j,, b), i, j =,..., N. j The se of off-equilibrium beliefs ψ is consruced as follows. Suppose here exiss a monoone soluion, {,..., N } (, ), o (A.3). If a high-inerval ype h wih ( i, i ], i =,..., N, receives a false advice, her beliefs are ha is disribued on [, i ] wih densiy /( i ) and zero elsewhere; if a low-inerval ype l wih ( i, i ] receives a false advice, her beliefs are ha is disribued on ( i, ) wih densiy /( i ) and zero elsewhere. Then, when [ i, i ] sends m M i under he pariional sraegy and deviaes from i by sending m M g, g i, he profile of his expeced payoff will be (A.5) V e (m,, b) = k i U e (a( k, k ),, b) i r=k i r r U e (a(, r ),, b) U e (a(, i ),, b) U e (a( k, k ),, b), U e (a( i, i ),, b) i U e (a(, i ),, b) i i U e (a( i, ),, b) U e (a( i, i ),, b), U e (a( j, j ),, b) i U e (a( i, ),, b) j r r=i j r U e (a( r, ),, b) U e (a( j, j ),, b), if m M k, k < i, if m M i, if m M j, i < j N. Using he second and he hird cases in (A.5), he expeced payoff for i o send m M i and 5
m M i are, respecively, (A.6) (A.7) i i U e (a( i, i ), i, b) U e (a( i, i ), i, b) i i i U e (a(, i ), i, b) i U e (a( i, i ), i, b) U e (a( i, ), i, b) U e (a( i, i ), i, b). i i Thus, he indifference condiion (A.3) becomes he following second-order difference equaion: (A.8) V ( i, i, i, b) = i [U e (a( i, i ), i, b) U e (a( i, i ), i, b)] i i [U e (a( i, i ), i, b) U e (a(, i ), i, b)] i i [U e (a( i, ), i, b) U e (a( i, i ), i, b)] i [U e (a( i, i ), i, b) U e (a( i, i ), i, b)] =, i =,..., N, =, N =. Suppose here is a sricly increasing pariion,,..., i, ha saisfies (A.8). Tha U e ( ) <, a(, ) is sricly increasing in is argumens, and he coninuiy of V ( i, i,, b) in ensure ha here exiss a unique i > i ha saisfies (A.8). Turning o incenive compaibiliy, I begin by showing ha (A.4) holds for i, i =,..., N, ha saisfy (A.3). If N = 2, here exiss no oher se of messages ha i can send, and (A.4) is saisfied vacuously. So, consider N 3. Suppose i sends message m M in, 2 n N i. Then, from he hird case in (A.5) his expeced payoff is (A.9) i U e (a( in, in ), i, b) in in 2 r=i in U e (a( in, ), i, b) r r U e (a( r, ), i, b) in U e (a( in, in ), i, b). Subracing (A.9) from (A.7), we have D 3 = i [U e (a( i, i ), i, b) U e (a( in, in ), i, b)] i i in 2 r r=i r in [U e (a( i, ), i, b) U e (a( i, ), i, b)] [U e (a( i, i ), i, b) U e (a( r, ), i, b)] in [U e (a( i, i ), i, b) U e (a( in, ), i, b)] [U e (a( i, i ), i, b) U e (a( in, in ), i, b)]. in 6
Noe ha (A.8) implies ha he exper s ideal acion a e ( i, b) (a( i, i ), a( i, i )). Since a( in, in ) > a( i, i ), a( in, ) > a( i, i ) for ( in, in ), and a( j, ) > a( i, i ) for ( j, j ), j = i,..., i n 2, given U e ( ) < and he maximum of U e (a, i, b) is achieved for a (a( i, i ), a( i, i )), he firs, hird, fourh and fifh erms are posiive. Also, he second erm vanishes. Thus, D 3 >. Nex, suppose i sends m M i η, η i. From he firs case in (A.5), his expeced payoff is (A.) i η U e (a( i η, i η ), i, b) i r=i η r i η r U e (a(, r ), i, b) i η U e (a(, i η ), i, b) i U e (a( i η, i η ), i, b). Subracing (A.) from (A.6), we have D 4 = i η [U e (a( i, i ), i, b) U e (a( i η, i η ), i, b)] i η i η [U e (a( i, i ), i, b) U e (a(, i η ), i, b)] i 2 r=i η i r r [U e (a( i, i ), i, b) U e (a(, r ), i, b)] i [U e (a(, i ), i, b) U e (a(, i ), i, b)] i [U e (a( i, i ), i, b) U e (a( i η, i η ), i, b)]. Similar o he above, since a( i, i ) > a( i η, i η ), a( i, i ) > a(, i η ) for ( i η, i η ), and a( i, i ) > a(, j ), for ( j, j ), j = i η,..., i 2, he firs, second, hird and fifh erms are posiive, while he fourh erm vanishes. Thus, D 4 >. Tha D 3 > and D 4 > imply ha (A.4) holds for i, i =,..., N. I show nex ha given (A.8) and for sufficienly large U e 2( ), all ( i, i ) prefer sending m M i over m M i, and all ( i, i ) prefer sending m M i over m M i, i =,..., N, so ha (A.4) holds for all inerior. Consider an arbirary ( i, i ). From he hird case in (A.5), his expeced payoff from sending m M i is (A.) U e (a( i, i ), i, b) i i U e (a( i, ), i, b) U e (a( i, ), i, b) U e (a( i, i ), i, b). i i 7
Subracing his expeced payoff from sending m M i in (A.5) from (A.), we have (A.2) D 5 = i [U e (a( i, i ),, b) U e (a( i, i ),, b)] i [U e (a( i, i ),, b) U e (a(, i ),, b)] i i i [U e (a( i, ),, b) U e (a( i, ),, b)] [U e (a( i, ),, b) U e (a( i, i ),, b)] i [U e (a( i, i ),, b) U e (a( i, i ),, b)]. Differeniaing D 5 wih respec o gives D 5 = i [U e (a( i, i ),, b) U e (a( i, i ),, b)] [U e (a( i, i),, b) U e (a(, i ),, b)] i i i [U e (a( i, ),, b) U e (a( i, i ),, b)] [U e (a( i, i),, b) U e (a( i, i ),, b)] i [U e (a( i, i ),, b) U e (a(, i ),, b)]. Since a( i, i ) > a( i, i ), a( i, i ) > a(, i ) for ( i, ), and a( i, ) > a( i, i ) for ( i, i ), U e 2( ) > implies ha he firs four erms are posiive and he las erm is negaive; when decreases from i, here are negaive effecs on D 5 from he firs four erm and a posiive effec from he las erm. However, for a sufficienly large U e 2( ) a, he negaive effecs ouweigh he posiive. A sufficienly large U e 2( ) hen ensures, given (A.8), D 5 for. Consider nex an arbirary ( i, i ). From he firs case in (A.5), his expeced payoff from sending m M i is (A.3) i U e (a( i, i ),, b) i i U e (a(, i ),, b) i U e (a(, i ),, b) U e (a( i, i ),, b). 8
Subracing (A.3) from he expeced payoff from sending m M i in (A.5), we have (A.4) D 6 = i [U e (a( i, i ),, b) U e (a( i, i ),, b)] i i [U e (a( i, i ),, b) U e (a(, i ),, b)] i [U e (a(, i ),, b) U e (a(, i ),, b)] i [U e (a( i, ),, b) U e (a( i, i ),, b)] i [U e (a( i, i ),, b) U e (a( i, i ),, b)]. Differeniaing D 6 wih respec o gives D 6 = i [U e (a( i, i ),, b) U e (a( i, i ),, b)] i [U e (a( i, i),, b) U e (a(, i ),, b)] i i [U e (a( i, ),, b) U e (a( i, i ),, b)] [U e (a( i, i),, b) U e (a( i, i ),, b)] i [U e (a( i, ),, b) U e (a( i, i ),, b)]. Since a( i, i ) > a( i, i ), a( i, i ) > a(, i ) for ( i, ), and a( i, ) > a( i, i ) for ( i, i ), U e 2( ) > implies ha he firs four erms are posiive. While he las erm is negaive, similar o he above, for a sufficienly large U e 2( ) a, he posiive effecs on D 6 from he firs four erms ouweigh he negaive effec from he las erm; (A.8) hen implies D 6 a. When U e 2( ) is sufficienly large for all inerior ypes, (A.4) holds for all of hem. The off-equilibrium beliefs specified in he proof ha for ( i, i ], h and l believe ha is uniformly disribued on, respecively, [, i ] and ( i, ) are sufficien for he incenive compaibiliy condiion o hold for he boundary ypes. However, since he inerior ypes induce a se of acions no induced by he boundary ypes, a sufficienly large U e 2( ) is called ino he picure o ensure ha incenive compaibiliy also holds for hem. Figure 4 illusraes he raionale wih an example of wo-sep equilibrium. Given ha he indifference condiion holds, he boundary ype s expeced payoff from he profile of acions a(, ) and a(, ), [, ], is he same from ha from a(, ) and a(, ), (, ] (he wo upper lines). Consider he acions induced when he inerior ype sends messages in M and M 2. If we compare he profile of acions in he lower pair of lines wih hose in he upper pair, we can see ha hey are he same excep for (, ]. While gives no 9
boundary ype M M 2 inerior ype M a(, ) a(, ) a(, ) a(, ) M 2 a(, ) a(, ) a(, ) a(, ) a(, ) a(, ) Figure 2: Acions Induced by Boundary and Inerior Types false advice when he sends messages in eiher M and M 2, here is one when sends m M 2. The specificaion of ψ, which allows incenive compaibiliy o hold for, (ineffecively) induce he acion a(, ) (wih aserisk) for he inerior ype if he sends messages in M 2, which is he same as he acion effecively induced by m M. If we could fix he profile of acions, ha U e 2( ) > would have guaraneed ha < sricly prefers o send messages in M over M 2. However, when he exper s ype changes, he profile of acions also changes, and, insofar as he acions aken by (, ] are concerned, is indifferen beween M and M 2. Thus, we have o ensure ha, overall, prefers M enough for / (, ] so ha even wih he indifference for (, ] he incenive compaibiliy sill holds. For his, a sufficienly large U e 2( ) is required. A large U e 2( ) means ha he ideal acion of a higher is sufficienly higher han ha of a lower. This addiional resricion is nohing bu a srenghening of he already exising soring condiion.