Extensive Form Games with Incomplete Information. Microeconomics II Signaling. Signaling Examples. Signaling Games

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1 Extnsiv Fom Gams ith Incomplt Inomation Micoconomics II Signaling vnt Koçksn Koç Univsity o impotant classs o xtnsiv o gams ith incomplt inomation Signaling Scning oth a to play gams ith to stags On play has pivat inomation Signaling Gams: Inomd play movs ist aantis Education Scning Gams: Uninomd play movs ist Insuanc company os contacts Pic discimination oth a spcial cass o Gams ith Obsvd Actions Sinc thy hav to stags pct aysian quilibium is quivalnt to squntial quilibium vnt Koçksn (Koç Univsity) Signaling / 3 vnt Koçksn (Koç Univsity) Signaling / 3 Signaling Exampls Usd-ca dal o do you signal quality o you ca? Issu a aanty An MA dg o do you signal you ability to pospctiv mploys? Gt an MA Entpnu sking inanc You hav a high tun pojct. o do you gt inancd? Rtain som quity Stock puchass Otn sult in an incas in th pic o th stock Manag knos th inancial halth o th company A puchas announcmnt signals that th cunt pic is lo imit picing to dt nty o pic signals lo cost vnt Koçksn (Koç Univsity) Signaling 3 / 3 Signaling Gams o plays Play is snd has pivat inomation θ Θ movs ist by choosing an action a A Play is civ Θ is singlton obsvs a but not θ movs by choosing an action a A Natu s pobability distibution: p (Θ) Payos: Fo all a A,a A and θ Θ Statgis: β(a θ),β(a a) lis: µ(θ a) ui(a,a,θ), i =, β : Θ (A) β : A (A) vnt Koçksn (Koç Univsity) Signaling 4 / 3

2 Expctd Payos Sinc plays may play andomizd statgis and play has incomplt inomation us xpctd payos Givn statgis and blis (β,β,µ) Expctd payo o play o typ θ i sh plays a U(a,β(a a),θ) = a β(a a)u(a,a,θ) Expctd payo o play at a i h plays a U(a,a,µ) = θ µ(θ a)u(a,a,θ) Pct aysian Equilibium Dinition A pct aysian quilibium is a collction o statgis and blis (β,β,µ) that satisis Squntial Rationality: Statgis maximiz xpctd payos givn blis β(a θ) > 0 implis a agmax a β(a a) > 0 implis a agmax a U(a,β(a a),θ) U(a,a,µ) ays Rul: I th is a θ such that β(a θ) > 0 µ(θ a) = β(a θ )p(θ ) θ β(a θ)p(θ) vnt Koçksn (Koç Univsity) Signaling 5 / 3 vnt Koçksn (Koç Univsity) Signaling 6 / 3 An Exampl: o uich Pct aysian Equilibia o o uich Gam 3,, 0. Natu o classs o possibl pu statgy quilibia, 3, Spaating Equilibia: dint typs choos dint actions Pooling Equilibia: both typs choos th sam action Θ = {,} A = {,}, A = {,} p() = 0.,p() = 0.9 vnt Koçksn (Koç Univsity) Signaling 7 / 3 vnt Koçksn (Koç Univsity) Signaling 8 / 3

3 Natu 3, 0, 3, 0, Natu, 3,, 3, 0 0 Spaating Equilibia β( ) =,β( ) = ays ul (R) µ( ) =,µ( ) = Squntial ationality (SR) o β( ) =,β( ) = ut SR o β( ) = 0, contadiction No such PE β( ) =,β( ) = ays ul (R) µ( ) =,µ( ) = Squntial ationality (SR) o β( ) =,β( ) = ut SR o β( ) = 0, contadiction No such PE vnt Koçksn (Koç Univsity) Signaling 9 / 3 vnt Koçksn (Koç Univsity) Signaling 0 / 3 3,, 3,, Natu Natu, 3,, 3, Pooling Equilibia β( ) = β( ) = ays ul (R) µ( ) = 0.,µ( ) = Squntial ationality (SR) o β( ) =,β( ) =? ut SR o play typ β( ) = SR o µ( ) / h olloing is a class o PE β( ) = β( ) =,β( ) =,β( ) = µ( ) = 0.,µ( ) / Pooling Equilibia β( ) = β( ) = ays ul (R) µ( ) = 0.,µ( ) = Squntial ationality (SR) o β( ) =,β( ) =? ut SR o play typ β( ) = SR o µ( ) / h olloing is a class o PE β( ) = β( ) =,β( ) =,β( ) = µ( ) = 0.,µ( ) / vnt Koçksn (Koç Univsity) Signaling / 3 vnt Koçksn (Koç Univsity) Signaling / 3

4 Intuitiv Cition 3,, Play s blis at a not plausibl hy ould play dviat and dink b i h is typ In quilibium h is gtting 0 ighst h can gt by dinking b is Play o typ has potntially somthing to gain In quilibium h gts I h can convinc play that h is typ h could gt 0 Play should put zo pobability on typ at ut thn h ould play olloing, upstting th quilibium Natu vnt Koçksn (Koç Univsity) Signaling 3 / 3 0., 3, Intuitiv Cition Cho and Kps (987) has omalizd this intuition and calld it intuitiv cition ak an quilibium An action a is undominatd o play at a i th xists a bli und hich a is a bst spons to a st payo o play olloing a is th maximum payo that sh can gt hn play plays an undominatd action An action a is quilibium dominatd o typ θ i h bst payo olloing a is stictly small than h quilibium payo Play s blis a asonabl at a i it givs positiv pobability only to thos typs o hom a is not quilibium dominatd h quilibium ails intuitiv cition i th xists a typ and action a o hom quilibium payo is small than th payo to a, givn that play bst sponds to a und asonabl blis vnt Koçksn (Koç Univsity) Signaling 4 / 3 Intuitiv Cition Mo omally: Fo any µ (Θ) and a A lt R(µ,a) = agmaxu(a,a,µ) a A b th st o pu statgy bst sponss to a givn that blis a µ. Also, o any non-mpty ˆΘ Θ R(ˆΘ,a) = R(µ,a) µ:supp(µ(. a)) ˆΘ b th st o pu statgy bst sponss to a givn that blis giv positiv pobability only to typs in ˆΘ. St R(,a) = R(Θ,a) vnt Koçksn (Koç Univsity) Signaling 5 / 3 Intuitiv Cition Fo any assssmnt (β,µ) and any a A lt J(β,µ,a) = {θ Θ : U(β,µ θ) > max a R(Θ,a) U(a,a,µ θ)} In oth ods, J(β,µ,a) is th st o typs o hom payo und (β,µ) is stictly btt than playing a as long as play plays an undominatd action. Dinition A pct aysian quilibium (β,µ) o a signaling gam ails th intuitiv cition i o som a A th xists θ Θ such that U(β,µ θ ) < min a R(Θ\J(β,µ,a),a) U(a,a,µ θ ) An quilibium ails th intuitiv cition i th is an action a and a typ o hom th payo to a is btt than th quilibium payo givn that play bst sponds to a, sticting his blis to thos typs o hom a is not dominatd by th quilibium payo. vnt Koçksn (Koç Univsity) Signaling 6 / 3

5 Intuitiv Cition h pooling quilibium in hich both typs at quich ails intuitiv cition U(β,µ ) = 0,U(β,µ ) = Not that has ho U(β,µ ) = 0 > = U(β,µ ) = < 0 = max a R(Θ,) U(,a,µ ) max a R(Θ,) U(,a,µ ) J(β,µ,) = {}, R(Θ \ J(β,µ,),) = {} U(β,µ ) = < 0 = min a R(Θ\J(β,µ,),) U(,a,µ ) An Application: Spnc s Modl o Education A ok (play ) has poductivity (valu poducd p unit o tim) qual to o, ith > > 0 h ok knos his poductivity but th im (play ) only knos that th popotion o high poductivity oks is p > 0. Fo any bli that th im may hold about th ok s poductivity, th valu o th ok to th im is givn by th xpctd poductivity. assum that th im os a ag that is qual to th xpctd poductivity could modl this by considing a labo makt in hich ims compt o th ok by oing ags In such a modl quilibium ag ould indd b th xpctd poductivity as long as ims hav common blis. h ok chooss a lvl o ducation 0 h im obsvs and maks a ag o Payo unction o th ok is u(,,θ) = θ, θ =, vnt Koçksn (Koç Univsity) Signaling 7 / 3 vnt Koçksn (Koç Univsity) Signaling 8 / 3 PE o Spnc s Modl o Education θ: quilibium ducation choic o ok ith typ θ µ(θ ): im s bli (pobability) that poductivity o th ok is θ i h chooss amount o ducation. quilibium ag schdul = µ( ) +( µ( )) () Spaating Equilibia: ays ul implis that µ( ) = and µ( ) = () implis =, = SR o ok implis = 0: Suppos > 0. hn, u(,,) = < u(0,(0),) by (), and hnc choosing = 0 ould b a poitabl dviation. SR implis that o all 0 and vnt Koçksn (Koç Univsity) Signaling 9 / 3 In paticula qui and hich is quivalnt to ( ) ( ). ays ul dos not apply to blis at any / {,}. h olloing bli and ag spciication is on o many possibl that suppot a PE in hich [( ),( )] { { µ( ) =,, = 0, <,, < Poposition An ducation poil (,) is pat o a pu statgy spaating PE i = 0 and [( ),( )]. Not that ths quilibia a Pato ankd. h bst quilibium is th on ith = ( ). As ill s shotly this is th only quilibium that satisis Intuitiv Cition. vnt Koçksn (Koç Univsity) Signaling 0 / 3

6 A Spaating Equilibium u = Pooling Equilibia: = = ays ul implis µ( ) = p ho, ( ) = p +( p) E[θ] Again nd, o any 0 u = E[θ] E[θ] () (3) Not that () implis E[θ] Substituting o E[θ], this implis p( ). vnt Koçksn (Koç Univsity) Signaling / 3 vnt Koçksn (Koç Univsity) Signaling / 3 Pooling Equilibia A Pooling Equilibium u = E[θ] can suppot any such by th olloing bli and ag spciication { { p, µ( ) = 0, <, = E[θ],, < Poposition An ducation poil (,) is pat o a pu statgy pooling PE i = p( ). E[θ] u = E[θ] vnt Koçksn (Koç Univsity) Signaling 3 / 3 vnt Koçksn (Koç Univsity) Signaling 4 / 3

7 u = Intuitiv Cition in Spnc s Modl Poposition A pu statgy PE satisis intuitiv cition i = 0, = ( ). Poo ill ist sho that all pooling quilibia ail intuitiv cition. t quilibium ducation b. > implis that (viy) th xists an > such that p +( p) > p +( p) < h lt hand sids a quilibium payos has th ight hand sids a th maximum payo that ach typ could gt by playing givn that th im plays a bst spons to som blis, in this cas µ( ) =. Poo (cont d) ho, is quilibium dominatd o and not o. In ou pvious notation J(β,µ, ) = {}. Onc stict th im s bst spons to blis µ( ) =, th minimum payo that typ can gt is bigg than th quilibium payo and hnc th quilibium ails intuitiv cition. Dviation to is poitabl o th high typ E[θ] u = E[θ] Figu: Pooling Equilibium u = E[θ] vnt Koçksn (Koç Univsity) Signaling 5 / 3 vnt Koçksn (Koç Univsity) Signaling 6 / 3 Intuitiv Cition Poo (cont d) No tak a spaating quilibium in hich = 0, > ( ) and lt (( ),). hav > < hich implis that is quilibium dominatd only o. Givn that µ( ) =, playing ould bing at last / to typ, hich is stictly btt than th quilibium payo. ho, all spaating quilibia in hich > ( ) ail intuitiv cition. Dviation to is poitabl o th high typ Figu: Spaating Equilibium u = vnt Koçksn (Koç Univsity) Signaling 7 / 3 vnt Koçksn (Koç Univsity) Signaling 8 / 3

8 u = Poo (cont d) t us viy that spaating quilibia in hich = 0, = ( ) satisy intuitiv cition. I > ( ), thn ( ) > > and hnc J(β,µ, ) = {,}. ut thn th quilibium payo o typ θ is at last as lag as th minimum that h could gt hn th im s blis a not stictd: /θ. I, on th oth hand, < ( ), thn ( ) < and < and hnc J(β,µ, ) =. Again th quilibium payo o typ θ is at last as lag as th minimum that h could gt hn th im s blis a not stictd. vnt Koçksn (Koç Univsity) Signaling 9 / 3 Intuitiv Cition st spaating quilibium is th only on that satisis intuitiv cition u = Figu: st Spaating Equilibium vnt Koçksn (Koç Univsity) Signaling 30 / 3 la Poptis o Equilibia la Poptis o Equilibia hav just sn that although th a multipl quilibia only th bst spaating quilibium satisis th Intuitiv Cition. t us compa th oks la in this quilibium ith complt inomation and no-signaling quilibia. Complt Inomation: In this cas =, =, hich a also th payos o th lo and high typs, spctivly. No Signaling: I th is incomplt inomation but oks do not hav signaling oppotunitis, thn ims ill o = p +( p), hich is qual to th payo o both typs. ho, th lo typ is btt o has th high typ is os o compad to th complt inomation quilibium. 3 st Spaating Equilibium: h payo o lo typ is qual to, has th payo o th high typ is ( ). ho, th lo typ has th sam payo as th complt inomation quilibium and is os o compad to th no-signaling cas. h high typ is alays os o compad to th complt inomation cas. I p < /, h is btt o compad to th no-signaling cas and os o i p > /. vnt Koçksn (Koç Univsity) Signaling 3 / 3 vnt Koçksn (Koç Univsity) Signaling 3 / 3