Signaling. Economics of Information and Contracts Signaling and Informed Principal Problem. Signaling Games. Expected Payoffs

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1 Signaling Economics o Inomation and Contacts Signaling and Inomed Pincipal Poblem event Koçkesen Koç Univesity Sceening: agent is inomed and pincipal uninomed Pincipal ties to educe inomation ent o agent Signaling: pincipal is the inomed paty Pincipal may signal pivate inomation though contact oe o action choice beoe contacting Pioneeing aticle: Spence, M. (973) Wokes bette inomed about thei poductivity They may use education as a signal This woks i education is costlie o lowe poductivity wokes Simila models in copoate inance: eland,. E. and D.. Pyle (977) Company owne may signal value o im by etaining equity Myes, S. C. and N. S. Majlu (984) New equity oes may educe stock pice event Koçkesen (Koç Univesity) Signaling / event Koçkesen (Koç Univesity) Signaling / Signaling Games Two playes Playe is sende has pivate inomation θ Θ moves ist by choosing an action a A Playe is eceive obseves a but not θ moves by choosing an action a A Natue s pobability distibution: p (Θ) Payos: Fo all a A,a A and θ Θ Stategies: β(a θ),β(a a) ui(a,a,θ), i =, Expected Payos Since playes may play andomized stategies and playe has incomplete inomation we use expected payos Given stategies and belies (β,β,µ) Expected payo o playe o type θ i she plays a U(a,β(a a),θ) = a β(a a)u(a,a,θ) Expected payo o playe ate a i he plays a U(a,a,µ) = θ µ(θ a)u(a,a,θ) β : Θ (A) β : A (A) Belies: µ(θ a) event Koçkesen (Koç Univesity) Signaling 3 / event Koçkesen (Koç Univesity) Signaling 4 /

2 Peect Bayesian Equilibium Deinition A peect Bayesian equilibium is a collection o stategies and belies (β,β,µ) that satisies. Sequential Rationality: Stategies maximize expected payos given belies. β(a θ) > 0 implies An Example: Bee o uiche 3, B 0. W Natue, a agmax U(a,β(a a),θ) a, T. β(a a) > 0 implies a agmax a U(a,a,µ). Bayes Rule: I thee is a θ such that β(a θ) > 0 µ(θ a) = β(a θ )p(θ ) θ β(a θ)p(θ) event Koçkesen (Koç Univesity) Signaling 5 / B Θ = {W,T} A = {B,}, A = {,} p(w) = 0.,p(T) = 0.9 event Koçkesen (Koç Univesity) Signaling 6 / Peect Bayesian Equilibia o Bee o uiche Game 3,, B 0 0. W Natue Two classes o possible pue stategy equilibia. Sepaating Equilibia: dieent types choose dieent actions. Pooling Equilibia: both types choose the same action, 0 T B Sepaating Equilibia. β( T) =,β(b W) = Bayes ule (BR) µ(t ) =,µ(w B) = Sequential ationality (SR) o β( ) =,β( B) = But SR o β(b W) = 0, No such PBE event Koçkesen (Koç Univesity) Signaling 7 / event Koçkesen (Koç Univesity) Signaling 8 /

3 Natue 3,, 3, 0 B, 0. B 0. W 0. W Natue, T, T B 0. β(b T) =,β( W) = Bayes ule (BR) µ(t B) =,µ(w ) = Sequential ationality (SR) o β( ) =,β( B) = But SR o β( W) = 0, No such PBE event Koçkesen (Koç Univesity) Signaling 9 / Pooling Equilibia. β(b T) = β(b W) = Bayes ule (BR) µ(w B) = 0.,µ(W ) = ee Sequential ationality (SR) o β( B) =,β( ) =? But SR o playe type W β( ) = SR o µ(w ) / The ollowing is a class o PBE B β(b T) = β(b W) =,β( B) =,β( ) = µ(w B) = 0.,µ(W ) / event Koçkesen (Koç Univesity) Signaling 0 / 3, B W, Intuitive Citeion 3, 0.5 B 0., Natue 0. W, T Natue Pooling Equilibia. β( T) = β( W) = Bayes ule (BR) µ(w ) = 0.,µ(W B) = ee Sequential ationality (SR) o β( ) =,β( B) =? But SR o playe type T β( B) = SR o µ(w B) / The ollowing is a class o PBE B β( T) = β( W) =,β( ) =,β( B) = µ(w ) = 0.,µ(W B) / event Koçkesen (Koç Univesity) Signaling /, B Playe s belies ate B ae not plausible Why would playe deviate and dink bee i he is type W In equilibium he is getting 0 ighest he can get by dinking bee is Playe o type T has potentially something to gain In equilibium he gets I he can convince playe that he is type T he could get 0 Playe should put zeo pobability on type W ate B But then he would play ollowing B, upsetting the equilibium T event Koçkesen (Koç Univesity) Signaling /

4 Intuitive Citeion Cho and Keps (987) has omalized this intuition and called it intuitive citeion Take an equilibium An action a is undominated o playe ate a i thee exists a belie unde which a is a best esponse to a Best payo o playe ollowing a is the maximum payo that she can get when playe plays an undominated action An action a is equilibium dominated o type θ i he best payo ollowing a is stictly smalle than he equilibium payo Playe s belies ae easonable ate a i it gives positive pobability only to those types o whom a is not equilibium dominated The equilibium ails intuitive citeion i thee exists a type and action a o whom equilibium payo is smalle than the payo to a, given that playe best esponds to a unde easonable belies Intuitive Citeion Moe omally: Fo any µ (Θ) and a A let BR(µ,a) = agmaxu(a,a,µ) a A pue stategy best esponses to a given that belies ae µ Fo any non-empty ˆΘ Θ BR(ˆΘ,a) = BR(µ,a) µ:supp(µ(. a)) ˆΘ pue stategy best esponses to a given that belies give positive pobability only to types in ˆΘ Set BR(,a) = BR(Θ,a) event Koçkesen (Koç Univesity) Signaling 3 / event Koçkesen (Koç Univesity) Signaling 4 / Intuitive Citeion Fo any assessment (β,µ) and any a A let J(β,µ,a) = {θ Θ : U(β,µ θ) > max a BR(Θ,a) U(a,a,µ θ)} Intuitive Citeion: Bee-uiche The pooling equilibium in which both types eat quiche ails intuitive citeion U(β,µ W) = 0,U(β,µ T) = set o types o whom payo unde (β,µ) is stictly bette than playing a as long as playe plays an undominated action. Deinition A peect Bayesian equilibium (β,µ) o a signaling game ails the intuitive citeion i o some a A thee exists θ Θ such that U(β,µ θ ) < min a BR(Θ\J(β,µ,a),a) U(a,a,µ θ ) An equilibium ails the intuitive citeion i thee is an action a and a type o whom the payo to a is bette than the equilibium payo given that playe best esponds to a, esticting his belies to those types o whom a is not dominated by the equilibium payo. event Koçkesen (Koç Univesity) Signaling 5 / Note that wheeas Theeoe and U(β,µ W) = 0 > = U(β,µ T) = < 0 = max a BR(Θ,B) U(B,a,µ W) max a BR(Θ,B) U(B,a,µ T) J(β,µ,B) = {W}, BR(Θ\ J(β,µ,B),B) = {} U(β,µ T) = < 0 = min a BR(Θ\J(β,µ,B),B) U(B,a,µ T) event Koçkesen (Koç Univesity) Signaling 6 /

5 An Application: Spence s Model o Education A woke (playe ) has poductivity (value poduced pe unit o time) equal to o, with > > 0 The woke knows his poductivity but the im (playe ) only knows that the popotion o high poductivity wokes is p > 0. Fo any belie that the im may hold about the woke s poductivity, the value o the woke to the im is given by the expected poductivity. We assume that the im oes a wage w that is equal to the expected poductivity We could model this by consideing a labo maket in which ims compete o the woke by oeing wages In such a model equilibium wage would indeed be the expected poductivity as long as ims have common belies. The woke chooses a level o education e 0 The im obseves e and makes a wage oe w Payo unction o the woke is u(e,w,θ) = w e θ, θ =, PBE o Spence s Model o Education eθ: equilibium education choice o woke with type θ µ(θ e): im s belie (pobability) that poductivity o the woke is θ i he chooses e amount o education. equilibium wage schedule w(e) = µ( e) +( µ( e)) () event Koçkesen (Koç Univesity) Signaling 7 / event Koçkesen (Koç Univesity) Signaling 8 / Sepaating Equilibia: e e Bayes ule implies that µ( e) = and µ( e) = () implies w(e) =,w(e) = SR o woke implies e = 0, since the wost that she can get by choosing e = 0 is (by equation ()). SR implies that o all e 0 w(e) e and e w(e) e Sepaating Equilibia In paticula we equie which is equivalent to e and e ( ) e ( ) These ae the incentive compatibility constaints Bayes ule does not apply to belies ate any e / {e,e}. The ollowing belie and wage speciication is one o many possible µ( e) = {, e e, w(e) = 0, e < e {, e e, e < e event Koçkesen (Koç Univesity) Signaling 9 / event Koçkesen (Koç Univesity) Signaling 0 /

6 Sepaating Equilibia A Sepaating Equilibium w u = Poposition An education poile (e,e) is pat o a pue stategy sepaating PBE i e = 0 and e [( ),( )]. These equilibia ae Paeto anked The best one is with e = ( ) It also is the only one that satisies Intuitive Citeion u = e w(e) e e e event Koçkesen (Koç Univesity) Signaling / event Koçkesen (Koç Univesity) Signaling / Pooling Equilibia: e = e = e Bayes ule implies µ( e ) = p Theeoe, w(e ) = p +( p) E[θ] Again we need, o any e 0 () and w(0) imply o E[θ] e w(e) e E[θ] e w(e) e E[θ] e e p( ) () (3) Pooling Equilibia The ollowing suppots any such e Poposition µ( e) = { { p, e e 0, e < e, w(e) = E[θ], e e, e < e An education poile (e,e) is pat o a pue stategy pooling PBE i e = e p( ). Eicient pooling equilibium has e = e = 0 event Koçkesen (Koç Univesity) Signaling 3 / event Koçkesen (Koç Univesity) Signaling 4 /

7 A Pooling Equilibium w u = E[θ] e Intuitive Citeion in Spence s Model Poposition A pue stategy PBE satisies intuitive citeion i e = 0,e = ( ). E[θ] e e u = E[θ] e w(e) Poo We will ist show that all pooling equilibia ail intuitive citeion. et equilibium education be e. > implies that (veiy) thee exists an e > e such that p +( p) e e > p +( p) e e < The let hand sides ae equilibium payos wheeas the ight hand sides ae the maximum payo that each type could get by playing e given that the im plays a best esponse to some belies, in this case µ( e ) =. event Koçkesen (Koç Univesity) Signaling 5 / event Koçkesen (Koç Univesity) Signaling 6 / Poo (cont d) Theeoe, e is equilibium dominated o and not o. In ou pevious notation J(β,µ,e ) = {}. Once we estict the im s best esponse to belies µ( e ) =, the minimum payo that type can get is bigge than the equilibium payo and hence the equilibium ails intuitive citeion. Now take a sepaating equilibium in which e = 0,e > ( ) and let e (( ),e). We have e > e < e which implies that e is equilibium dominated only o. Given that µ( e ) =, playing e would bing at least e / to type, which is stictly bette than the equilibium payo. Theeoe, all sepaating equilibia in which e > ( ) ail intuitive citeion. event Koçkesen (Koç Univesity) Signaling 7 / Poo (cont d) et us veiy that sepaating equilibia in which e = 0,e = ( ) satisy intuitive citeion. I e > ( ), then ( ) > e > e and hence J(β,µ,e ) = {,}. But then the equilibium payo o type θ is at least as lage as the minimum that he could get when the im s belies ae not esticted: e /θ. I, on the othe hand, e < ( ), then ( ) < e e and < and hence J(β,µ,e ) =. Again the equilibium payo o type θ is at least as lage as the minimum that he could get when the im s belies ae not esticted. event Koçkesen (Koç Univesity) Signaling 8 /

8 ed u = E[θ] e w(e) e ed e u = u = e w(e) e e u = u = e w(e) Intuitive Citeion Intuitive Citeion In both cases deviation to ed is poitable o the high type Best sepaating equilibium is the only one that satisies intuitive citeion w u = E[θ] e w w E[θ] e Figue : Pooling Equilibium e Figue : Sepaating Equilibium e Figue 3: Best Sepaating Equilibium e event Koçkesen (Koç Univesity) Signaling 9 / event Koçkesen (Koç Univesity) Signaling 30 / Welae Popeties o Equilibia Thee ae multiple equilibia but only the best sepaating equilibium satisies the Intuitive Citeion Woke s welae unde dieent scenaios. Complete Inomation: No education o anybody and w =,w = : best!. No Signaling: Thee is incomplete inomation but wokes cannot signal w = p +( p) and no education low type is bette o, high type is wose o compaed to complete inomation same outcome as eicient pooling equilibium 3. Best Sepaating Equilibium: U = and U = ( ) ow type has the same payo as complete inomation and is wose o compaed to no-signaling case igh type is wose o compaed to the complete inomation case igh type compaed to no-signaling case p < / bette o p > / wose o Inomed Pincipal Poblem Stage I Pincipal (woke) oes a contact: (wi,ei)i=, Stage II Agent (im) accepts o ejects I eject, both get zeo I accept, go to Stage III Stage III Pincipal type i chooses ei and gets wi, i =, Analyzed (in a geneal setting) by Maskin, E. and J. Tiole (99) They showed that i p < /, then best sepaating equilibium is the unique PBE event Koçkesen (Koç Univesity) Signaling 3 / event Koçkesen (Koç Univesity) Signaling 3 /

9 An Application: Advese Selection and Signaling in Copoate Finance A isk neutal entepeneu has no unds to inance a poject costing I Poject yields R i successul and 0 i ailue Poject could be two types igh quality: pobability o success is p ow quality: pobability o success is 0 < p < p Entepeneu obseves the type o the poject endes believe that the poject is igh quality with pobability q ow quality with pobability q endes ae isk neutal, maket o unds is competitive, and isk-ee inteest ate is zeo In equilibium thei expected payo is zeo Entepeneu has limited liability: In case o ailue she pays back 0 Two scenaios. pr > I > pr: only high type is ceditwothy. pr > pr > I: both types ae ceditwothy event Koçkesen (Koç Univesity) Signaling 33 / Peect Inomation Entepeneu obseves quality and oes contact Feasible contacts: pay the lende D in case o success and 0 in case o ailue endes accept only i thei expected payo is non-negative Suppose that lendes also obseve the quality igh type oes epayment D such that and obtains expected payo pd = I p(r D) = pr I > 0 The best ow type can do is to oe D such that pd = I in which case he payo is pr I I pr > I, she obtains inancing, othewise she does not event Koçkesen (Koç Univesity) Signaling 34 / Asymmetic Inomation endes do not obseve quality Sepaating Equilibia: D D? µ( D) =,µ( D) = I both types oes ae accepted, zeo poit condition implies D = I/p > D = I/p Theeoe, ow type mimics the igh type Thee is no sepaating equilibium in which ow type is denied lending eithe: p(r D) = p(r I/p) > 0 i.e., ow type again mimics Thee is no sepaating equilibium Asymmetic Inomation: Pooling Equilibia D = D = D endes expected payo is pd I, whee p = qp +( q)p Two possibilities: ending Equilibium No lending Equilibium ending Equilibium: It must be that pr I Both types ae ceditwothy o q q whee I/p > D = I/p > I/p (q p +( q )p)r I = 0 igh type is hut; ow type beneits om asymmetic inomation ende may make losses on the ow type (coss-subsidization) event Koçkesen (Koç Univesity) Signaling 35 / event Koçkesen (Koç Univesity) Signaling 36 /

10 Asymmetic Inomation: Pooling Equilibia Asymmetic Inomation No ending Equilibium Implies pr < I: Othewise entepeneu could oe R ε, ε > 0 small wost belie is p lende accepts it I q < q whee (q p +( q )p)r I = 0 µ( D) = q o all D and D accepted i D R is such an equilibium maket beakdown: even the wothy boowes ae denied cedit Note that we can wite pr I as [ p p ( q) p and deine an index o advese selection p p χ = ( q) p ] pr I The ight type s pledgable income is discounted by the pesence o ow types event Koçkesen (Koç Univesity) Signaling 37 / event Koçkesen (Koç Univesity) Signaling 38 / Signaling: Equity Oeing and Negative Stock Pice Reaction It is well documented that stock pices decline upon announcement o new equity issue Asquith and Mullins (986) Masulis and Kowa (986) Model based on Myes, S. C. and N. S. Majlu (984) Signaling: Equity Oeing Entepeneu aleady owns the poject; without uthe investment yields R with pobabilities p o p Entepeneu knows the pobability Investos put pobability q on p, q on p Since pob. o success is p Assets in place undevalued o igh type ovevalued o ow type Entepeneu initially owns all shaes Both types can incease pob. o success by τ by investing I τr > I Investing in eicient o both types I must be aised om investos by issuing new shaes Dilutes he owneship ess costly i assets in place ae ovevalued event Koçkesen (Koç Univesity) Signaling 39 / event Koçkesen (Koç Univesity) Signaling 40 /

11 Equity Oeing: Pooling Equilibium Equity Oeing: Pooling Equilibium Entepeneu decides whethe to announce equity oe (E) o not (N) Must oe a stake D such that (p+τ)d = I Without equity oeing, igh type gets pr Theeoe, we need o (p +τ)(r D) pr (4) τr p +τ p+τ I This condition can be witten as τr I χτ I χτ whee χτ is the index o advese selection This is an eicient outcome Both types undetake investment No negative stock pice eaction ( q)(p p) χτ = p +τ event Koçkesen (Koç Univesity) Signaling 4 / event Koçkesen (Koç Univesity) Signaling 4 / Sepaating Equilibium ow oes, igh does not Belies o investos: µ( E) = µ( N) = Implies igh type must pee not to oe o (p +τ)d = I (5) (p +τ)(r D) pr (6) τr p +τ p +τ I Check that ow type pees to oe D event Koçkesen (Koç Univesity) Signaling 43 / Negative Stock Pice Reaction Assume that investment oppotunity is peectly anticipated by the capital maket Pe-announcement value o total shaes: Post-announcement value (5) and (6) imply V0 = q[pr] +( q)[(p +τ)r I] V = (p +τ)r I p +τ pr [(p +τ)r I] > (p +τ)r I p +τ which implies V < V0 This is negative stock pice eaction ighe the volume o equity oe I and lowe the value o poject τ, highe the negative pice eaction event Koçkesen (Koç Univesity) Signaling 44 /