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1 Dynamic Global Games of Regime Change: Learning, Mulipliciy and iming of Aacks Supplemenary Maerial George-Marios Angeleos MI and NBER Chrisian Hellwig UCLA Alessandro Pavan Norhwesern Universiy Ocober 2006 Absrac his supplemenary documen conains a formal analysis of some of he eensions briefly discussed in Secion 5 of he published version. Secion A1 considers he game in which agens receive signals abou he size of pas aacks. Secion A2 considers he game wih observable shocks o he fundamenals. Secion A3 considers he varian in which agens observe he shocks wih a one-period lag. Secion A4 considers he game wih shor-lived agens in which he fundamenals follow a random walk. Finally, Secion A5 collecs he proofs of he formal resuls conained in his documen. A1. Signals abou pas aacks For some applicaions, i migh be naural o assume ha agens collec informaion eiher privae or public no only abou he underlying fundamenals bu also abou he size of pas aacks. o capure his possibiliy, we eend he game wih public news eamined in Secion 5.1 as follows. In every period 2, agens receive privae and public signals abou he size of he aack in he previous period. hese signals are, respecively, X i = SA 1, ξ i and Z = SA 1, ε, where ξ i is idiosyncraic noise, ε is common noise, and S : [0, 1] R R. o preserve Normaliy of he informaion srucure, we adop a specificaion similar o ha in Dasgupa 2002: Φ 1 A + υ if A 0, 1, ξ i N 0, 1/γ, ε N 0, 1/γ z, and S A, υ = υ oherwise. 1

2 he noises ξ i and ε guaranee ha, even if Aθ is monoonic, he fundamenals θ never become common cerainy among he agens. 1 Since in any equilibrium of he game, agens play in period 1 as in he saic benchmark, he size of aack in period 1 is given by A 1 θ = Φ β 1 1 θ, where 1 = ˆ 1. his implies ha in period 2, he signals he agens receive abou A 1 are also addiive signals abou θ : X i2 = β 1 1 θ + ξ i2 and Z 2 = β 1 1 θ + ε 2. he poserior beliefs abou θ condiional on 2, z 2, X 2, Z 2 are hen Normal wih mean β 2 β 2 +α α 2 β 2 +α 2 z 2 and precision β 2 + α 2 where } 2 = β 1 β η 2 β β 1γ2 β X2 β1, } z 2 = α 1 α 2 z 1 + ηz 1 α 2 z 1 + β 1γ2 z α Z2 β1, β 2 = β 1 + η 2 + β 1 γ 2 and α 2 = α 1 + η z 2 + β 1 γ z 2, wih 1, z 1, β 1 and α 1 defined as in he previous secions. ha is, 2 and z 2 are suffi cien saisics for 2, X 2 and z 2, Z 2 wih respec o θ. If he agens sraegies in period 2 are monoonic in 2, X 2, hen he size of aack and hence he regime oucome in ha period are decreasing in θ, which in urn implies ha he agens sraegies in period 2 are necessarily a hreshold sraegy in he saisic 2. A similar argumen applies o every 2 : in any monoone equilibrium, he poserior beliefs abou θ condiional on, z, X, Z are Normal wih mean β β +α + α β +α z and precision β + α, where } = β 1 β 1 + η β β γ 1 β 1 1 X, β 1 } z = α 1 α z 1 + ηz β α z γ z 1 α 1 1 Z, β 1 β = β 1 + η β 1 γ and α = α 1 + η z β 1 γ z ; where 1 1 is an indicaor funcion ha akes value 1 if A 1 0, 1 and 0 oherwise, and 1 is he hreshold played in period 1. I follows ha he condiions in Proposiion 3 coninue o characerize he enire se of monoone equilibria he only difference is ha he saisics and z are now endogenous, as defined above, and ha he hresholds and θ are now funcions, no only of z, bu also of Z. he mulipliciy resul of heorem 2 hus eends direcly o his environmen. Similarly, he srucure of dynamics remains he same as in he game wih public news, ecep for he propery ha an unsuccessful aack does no necessarily reduce he incenives for furher aacks. his is because an unsuccessful aack now also generaes new privae and public signals, which in some cases may offse he impac of he knowledge ha he regime survived pas aacks. o see his, 1 We assume ha hese signals are uninformaive when A = 0 or A = 1 o avoid he possibiliy ha agens can deec collecive deviaions. Since agens are infiniesimal, his would no affec equilibrium oucomes, bu would require us o specify ou-of-equilibrium beliefs. 2

3 consider he case where all signals are privae γ > 0, η 0, γ z = η z = 0, in which case he only novel effec is ha an unsuccessful aack leads o an endogenous increase in β. A furher aack is hen possible only if his increase is large enough, like in he benchmark game. On he oher hand, when he endogenous signal is public γ z > 0 = γ, a new aack becomes possible if his signal is low enough, like in he case wih eogenous public news. Signals abou he size of pas aacks can hus subsiue for he eogenous arrival of privae and public informaion and lead o snow-balling effecs where new aacks become possible immediaely afer unsuccessful ones. A2. Observable shocks Consider he game wih observable shocks described in Secion 5.3 of he paper. he characerizaion of monoone equilibria was compleed here. Here we prove ha essenially all equilibria of he benchmark game Γ0 can be approimaed by equilibria of he game wih observable shocks Γ, for small enough his resul was discussed a he end of Secion 5.3 wihou proof. As in he case wih unobservable shocks heorem 3 in he paper, we rule ou knife-edge equilibria where U is angen o he horizonal ais. Bu, unlike ha case, convergence is esablished in probabiliy, for he equilibrium hresholds here are funcions of he sequences of observable shocks. Proposiion A1 rue for all < ε, : For any ε > 0 and <, here eiss ε, > 0 such ha he following is For any equilibrium, θ } of Γ 0, for which θ / arg ma θ U θ, θ 1, β, α, z for all 2,..., }, here eiss an equilibrium, θ } of Γ such ha θ Pr θ ε 1,..., } 1 ε. A3. Shocks observable wih lag In his secion, we discuss a varian of he game wih shocks in which agens observe he shocks wih a one-period lag. his varian was briefly discussed a he end of Secion 5.3. he game srucure is he same as in he model wih fully observable shocks Secion 5.3, ecep ha becomes known only a he end of period. he propery ha he conemporaneous shock is unobservable inroduces an addiional source of uncerainy abou he regime oucome in he curren period and may even reinroduce he lower-dominance region. A he same ime, he propery ha he shock is revealed a he end of he period ensures ha he learning induced by he knowledge ha he regime survived pas aacks coninues o ake he simple and sharp form of a runcaion in he suppor of he agens beliefs abou θ, as in he case wih fully observable shocks. 3

4 Equilibrium characerizaion, mulipliciy and dynamics. Monoone equilibria are now characerized by sequences 1, θ } such ha agens aack in period if and only if 1 and he saus quo survives period if and only if θ > θ ; noe ha sraegies in period are coningen only on 1 since is no observed a he ime agens choose wheher or no o aack, bu he regime oucome sill depends on, since direcly affecs he size of aack necessary for regime change. o compue he epeced ne payoff from aacking, we need o adjus he condiional probabiliy of regime change as follows. For a given hreshold rule, regime change occurs in period when he fundamenals are θ if and only if θ + Φ β θ, or equivalenly θ; [ Φ β θ θ ] /. Condiional on θ, he probabiliy of regime change in period is herefore given by p θ; Pr θ; = F θ;. he updaing of poserior beliefs, on he oher hand, is he same as in he game wih fully observable shocks. Le θ, Ω be implicily defined by θ +Ω = Φ β θ. Ne, consider any sequence of hreshold rules 1 } and define he sequence θ } recursively by θ = ma θ 1 1, θ 1, }, wih θ0 = and 0 = 0. When agens follow he sraegy associaed wih 1 }, poserior beliefs over θ in period are again characerized by runcaed normal disribuions wih runcaion a θ 1 1. Le hen Ψ θ, θ 1 denoe he c.d.f. of an agen s poserior abou θ condiional on having saisic and on believing ha θ > θ 1 ; his is simply α+β β Φ α+β α + α+β z θ 1 Ψ θ, θ 1 = α+β β Φ α+β α if θ > θ 1 + α+β z θ 1 0 if θ θ 1 which is eacly he same as in he benchmark model. Ne, le v,, θ 1 denoe an agen s epeced ne payoff from aacking in period when he has suffi cien saisic R, all oher agens follow monoone sraegies in ha period wih hreshold R, and he agen believes ha θ > θ 1 ; his is given by Finally, define V v V,, θ + 1 = F, θ 1 θ; dψ θ, θ 1 c. lim + v,, θ 1 if = + v,, θ 1 if R lim v,, θ 1 if = is he analogue of he funcion U in he benchmark model: i represens he ne payoff from aacking in period for he marginal agen wih hreshold.. 4

5 Since v is coninuous in, and θ 1, V is coninuous in and θ 1 for all R. Moreover, since v is bounded and monoone decreasing in, for any given, V, θ 1 is well-defined a = ±. We hus have he following equilibrium characerizaion. Proposiion A2 a } is a monoone equilibrium of Γ if and only if here eiss a sequence 1, θ } such ha: i for all, a = 1 if < 1 and a = 0 if > 1. ii for = 1, 1 R solves V 1 1, = 0; and θ 1 1 = θ 1, 1. iii for all 2, eiher 1 = and V 1, θ 1 1 0, or 1 R solves V 1, θ 1 1 = 0; and θ = ma θ 1 1, θ 1 },. An equilibrium always eiss. he equilibrium characerizaion is hus similar o ha wih observable shocks; one only has o adjus he agens epeced payoff from aacking o ake ino accoun he uncerainy abou he regime oucome inroduced by unobservable conemporaneous shocks. As 0, he impac of shocks on regime oucomes vanishes, hus ensuring a similar convergence resul as he one we esablished in he previous secion for he case wih observable shocks. Proposiion A3 rue for all < ε, : For any ε > 0 and <, here eiss ε, > 0 such ha he following is For any equilibrium, θ } of Γ 0 for which θ / arg ma θ U θ, θ 1, β, α, z for all 2,..., }, here eiss an equilibrium 1, θ } of Γ such ha θ Pr θ ε 1,..., } 1 ε. A4. Changing fundamenals wih shor-lived agens In Secion 5.5 we inroduced and briefly analyzed a game wih shor-lived agens where he fundamenals summarized by he criical size of aack necessary for regime change follow a random walk. Here we prove ha Proposiion 5 and heorem 3, which we esablished for he case wih longlived agens and unobservable shocks, apply also o his game. o keep he analysis self-conained, we firs briefly revisi he descripion of he game and he characerizaion of beliefs and payoffs ha is in Secion 5.5. he game. A regime change occurs in period if and only if A h, where h follows a Gaussian random walk: h 1 = θ Nz, 1/α and h = h 1 + for 2, wih N0, 1, i.i.d. across ime and independen of θ. Once he saus quo is abandoned, he game is over. As long as he saus quo is in place, a new cohor of agens replaces he old one in each period; each cohor is of measure 1 and lives eacly one period. Agens who are born in period mus choose wheher or no o aack he saus quo, afer receiving privae signals i = h + ξ i, where ξ i N 0, 1/β 5

6 is i.i.d. across agens and independen of h s for any s. Payoffs are as in he benchmark model: he ne payoff from aacking in period is 1 c if he saus quo is abandoned in ha period and c oherwise, while he payoff from no aacking is zero. Equilibrium characerizaion, mulipliciy and dynamics. Le Ψ h, 1 denoe he c.d.f. of he common poserior in period abou h, when agens in earlier cohors aacked in periods τ 1 if and only if τ < τ. When earlier cohors followed such sraegies, he saus quo survived period τ if and only if h τ > θ τ τ, where θ τ τ is he soluion o Φ β τ τ h τ = h τ. herefore, for 2, Ψ h ; 1 is recursively defined by + Ψ h ; 1 θ 1 1 Φ h h 1 dψ 1 h 1 ; 2 = 1 Ψ 1 θ 1 1 ; 2 A1 wih Ψ 1 h 1 = Φ α h 1 z. abou h, by Bayes rule, Ne, le Ψ h ; 1 denoe he c.d.f. of privae poserior Ψ h ; 1 = β φ β h dψ h ; 1 β φ β h dψ h ; 1. h + A2 he epeced ne payoff from aacking in period for an agen wih signal is hus given by v1 ; 1 = Ψ 1 θ1 1 c for = 1 and v ; = Ψ θ ; 1 c for 2. Finally, define he payoff of he marginal agen by V lim + v ; if = + v ; if R lim v ; if = A3 he following hen provides he algorihm for characerizing monoone equilibria. Proposiion A4 For any > 0, a } is a monoone equilibrium for Γ if and only if here eiss a sequence } such ha: i for all, a = 1 if < and a = 0 if >, ii for = 1, 1 R and V 1 1 = 0, iii for any 2, eiher = and V 0, or R and V = 0. An equilibrium eiss for any > 0. 2 Finally, he ne resul esablishes ha essenially any equilibrium of he benchmark game can be approimaed by an equilibrium of he random-walk game for small enough. 2 Given a sequence of hresholds } characerizing a monoone equilibrium, he sequence of hresholds h } characerizing he associaed regime oucomes is simply given by h = θ for any 1. 6

7 Proposiion A5 For any ε > 0 and any <, here eiss ε, > 0 such ha he following is rue for all < ε, : For any equilibrium } of Γ 0 such ha / arg ma V 0 1, for all 2,..., }, here eiss an equilibrium } of Γ such ha, for all, eiher < ε, or = and < 1/ε. A5. Proofs claim: Proof of Proposiion A1. o esablish Proposiion A1, we firs prove he following weaker Resul A1.a For any ε > 0, any <, and any sequence θ } ha is par of an equilibrium of Γ 0 and such ha θ / arg ma θ U θ, θ 1, β, α, z for all, here eiss a ˆ = ˆ ε,, θ } > 0 such ha, whenever ˆ, here eiss an equilibrium θ } of Γ such ha θ Pr θ ε, 1,..., } 1 ε. A4 Given Resul A1.a, he sronger resul in he proposiion hen follows by leing ε, be he minimum of ˆ ε,, θ } across all differen sequences θ } ha can be par of an equilibrium of Γ 0 ; ha ε, > 0 is ensured by he fac ha he se of such sequences is finie for any finie <. o prove Resul A1.a, we proceed in four seps, using an argumen based on inducion: sep 1 shows ha he resul holds for = 1; sep 2 provides a suffi cien condiion for he resul o hold for condiional on holding for 1; seps 3 and 4 prove ha his condiion is saisfied boh for he case where θ = θ 1 sep 3 and for he case where θ > θ 1 sep 4. o simplify noaion, le Ω, and for any 1 and any θ, θ 1, Ω such ha θ θ 1 and Ω [ θ, 1 θ ], define V θ, θ 1, Ω U θ + Ω, θ 1 + Ω, β, α, z + Ω. Furhermore, for any ε > 0, <, and θ } R, le } θ } B ε, θ R : θ θ } ε, = 1,...,. Sep 1. By Proposiions 1 and 4, he unique firs-period equilibrium hreshold θ1 of Γ 0 saisfies V 1 θ1,, 0 = 0, while he also unique firs-period equilibrium hreshold θ 1 1 of Γ saisfies V 1 θ 1 1,, 1 = 0. Moreover, since U θ,, β1, α, z is coninuous and sricly decreasing in boh θ and z, V 1 θ,, Ω is also coninuous and sricly decreasing in boh θ and Ω. From he definiion of V and of θ1, we hus have ha V 1 θ1 ε,, 0 > 0 > V 1 θ1 ε,, ε. I follows ha here eiss Ω 0, ε such ha V 1 θ 1 ε,, Ω = 0, implying ha θ1 Ω/ = θ1 ε. Likewise, V 1 θ1 + ε,, 0 < 0 < V 1 θ1 + ε,, ε and hence here eiss Ω ε, 0 such ha V 1 θ1 + ε,, Ω = 0, implying ha θ 1 Ω/ = θ 1 + ε. Since V 1 θ1,, Ω is coninuous and sricly decreasing in boh θ 1 and Ω, θ 1 1 is coninuous and decreasing in 1. Hence θ 1 1 [θ 1 ε, θ 1 + ε] if and only if 1 [ Ω/, Ω/ ]. here 7

8 hus eiss an equilibrium of Γ for which θ1 1 θ1 ε whenever 1 [ Ω/, Ω/ ] and herefore Pr θ 1 1 θ1 ε = Pr 1 [ Ω/, Ω/ ]. Since Ω < 0 < Ω, Pr 1 [ ] Ω/, Ω/ is decreasing [ in and converges o 1 as 0. I follows ha here eiss ˆ > 0 such ha Pr 1 Ω/ˆ, Ω/ˆ ] = 1 ε and Pr θ 1 1 θ1 ε 1 ε for all ˆ, which proves he claim for = 1. Sep 2. Suppose Resul A1.a holds for 1, wih 2. his means ha for any sequence θ } 1 ha is par of an equilibrium of Γ0 and any ε 1 0, ε, here eiss a ˆ 1 = ˆ ε 1, 1, θ } 1 such ha, for any ˆ 1, here eiss an equilibrium θ } of Γ such ha Pr θ } 1 B ε 1, 1 θ } 1 1 ε 1. A5 Now suppose furher ha we are able o prove ha he following is rue. Resul A1.b For any ε > 0 and any sequence θ } ha is par of an equilibrium of Γ0, here eiss an ε 1 0, ε and a ˆ ˆ 1 such ha for any 0, ˆ, here eiss an equilibrium of Γ ha saisfies A5 and such ha, for any 1 for which θ 1 1 θ 1 ε 1, Pr θ ε 1 1 ε + ε 1. θ If Resul A1.b is rue, hen, θ Pr θ ε Bu hen, Pr θ } Pr = Pr θ θ B ε, } 1 Pr } 1 θ 1 ε 1 1 ε + ε 1 > 1 ε, θ } 1 B ε 1, 1 θ } B ε 1, 1 B ε 1, 1 θ ε implying ha Resul A1.a holds also for. θ } 1 θ } 1 and θ } 1 θ 1 ε + ε 1. A6 θ ε θ } 1 B ε 1, 1 θ } 1 o complee he proof of Resul A1.a, i hus suffi ces o show ha Resul A1.b holds. We do so by proving he following: Resul A1.c here eis scalars ε 1 0, ε, Ω < 0 < Ω, > 0, and a funcion ˆθ : R 2 R such ha he following hold: i for any θ 1 [ θ 1 ε 1, θ 1 + ε 1] and any Ω [Ω, Ω], eiher ˆθ Ω, θ 1 = θ 1 Ω, 8

9 or ˆθ Ω, θ 1 > θ 1 and V ˆθ Ω, θ 1, θ 1, Ω = 0; ii for any θ 1 [ θ 1 ε 1, θ 1 + ε ] 1 and any <, Pr ˆθ, θ 1 θ ε ε + ε 1. 1 We prove Resul A1.c in he ne wo seps, disinguishing he case where θ = θ 1 sep 3 and where θ > θ 1 sep 4. Resul A1.b hen follows from Resul A1.c by leing ˆ = min, ˆ ε 1, 1, θ } 1 } and θ } be he equilibrium of Γ whose sequence of hresholds coincides wih ha of he equilibrium ha saisfies A5 for 1 ogeher wih = ˆθ, θ 1 for any 1 such ha θ 1 [ θ 1 ε 1, θ 1 + ε ] 1. θ Sep 3. Suppose ha θ = θ 1, and pick any ε 1 0, ε such ha θ 1 ε 1 > 0. hen, for any θ 1 [ θ 1 ε 1, θ 1 + ε 1], le ˆθ Ω, θ 1 = θ 1 if Ω θ 1 and oherwise le ˆθ Ω, θ 1 be he highes soluion o V ˆθ, θ 1, Ω = 0. Clearly, ˆθ Ω, θ 1 saisfies par i of Resul A1.c for any. o see when par ii is also saisfied, noe ha, for any θ 1 [ θ 1 ε 1, θ 1 + ε ] 1, Pr ˆθ, θ 1 θ ε Pr ˆθ, θ 1 = θ 1 = Pr θ 1 Pr θ 1 ε 1 /. Since θ 1 θ 1 ε 1 > 0, Pr θ 1 ε 1 / is sricly decreasing in and converges o 1 as 0. I follows ha here eiss > 0 such ha Pr θ 1 ε 1 / = 1 ε + ε 1, implying par ii is saisfied for all. Hence, Resul A1.c is saisfied for he case θ = θ 1. Sep 4. Ne assume ha θ > θ 1, in which case θ solves V θ, θ 1, 0 = 0. Suppose furher ha V θ, θ 1, Ω is sricly decreasing in θ in a neighborhood of θ, θ 1, Ω = θ, θ 1, 0 An analogous argumen applies if V is sricly increasing in such a neighborhood, whereas he case ha V is locally non-monoonic is ruled ou by he non-angency assumpion. hen, by he Implici Funcion heorem, here eiss ε 0, ε], Ω < 0 < Ω and a funcion ˆθ : R 2 R such ha V ˆθ Ω, θ 1, θ 1, Ω = 0 for any Ω, θ 1 [Ω, Ω ] [θ 1 ε, θ 1 +ε ]. Clearly, he funcion ˆθ Ω, θ 1 saisfies par i of Resul A1.c by consrucion. o see when i also saisfies par ii, noe ha, by he coninuiy of V, ˆθ is also coninuous, and hence here eis ε 1, ε 2 0, ε ], Ω [Ω, 0, and Ω 0, Ω ], such ha ˆθ Ω, θ 1 [θ ε 2, θ + ε 2] whenever Ω, θ 1 [Ω, Ω] [θ 1 ε 1, θ 1 + ε 1]. Since ε 2 ε ε, i follows ha, whenever θ 1 [θ 1 ε 1, θ 1 + ε 1], Pr ˆθ, θ 1 θ ε Pr ˆθ, θ 1 θ ε2 Pr [ Ω/, Ω/ ]. Since in urn Pr [ Ω/, Ω/ ] is decreasing in and converges o 1 as 0, here eiss > 0 such ha Pr [ Ω/, Ω/ ] 1 ε + ε 1 for all, which esablishes par ii of Resul A1.c. 9

10 Proof of Proposiion A2. he resul follows from eacly he same argumens as he proof of Proposiion 5 in he main e, afer adjusing he noaion for beliefs. Noe ha, unlike in he case of Proposiion 5, here here is no need o prove convergence of beliefs: he belief updaing induced by any given monoone sraegy is idenical o ha in he benchmark model. Proof of Proposiion A3. As in he proof of Proposiion A1, i suffi ces o prove he weaker claim in Resul A1.a. For his purpose, Sep 1 below firs esablishes poinwise convergence of V, θ 1 o V 0, θ 1 U θ, θ 1, β, α, z as 0, where θ θ, 0. Seps 2-5 hen use his propery o prove he resul wih an inducion argumen similar o he one in he proof of Proposiion A1. Sep 1. he proof ha V converges poinwise o V 0 as 0 is similar o Sep 1 in he proof of heorem 3 in he paper; i is acually simplified by he fac ha he equilibrium updaing of beliefs here is idenical o ha in he benchmark model and hence follows direcly from he convergence of regime oucomes. Indeed, for any 1 any R and any θ θ, 1 if θ lim 0 p θ; = p 0 θ, θ; 0 if θ > θ. his immediaely implies ha, for any, any R, and any θ 1 R, lim V, θ 1 = lim p θ; dψ θ, θ 1 c = Ψ θ, θ 1 c = U θ, θ 1, β, α, z V 0 Sep 2. Here we show ha Resul A1.a holds for = 1., θ 1. Fi ε > 0. In period 1, he game in which shocks are observable wih a lag is isomorphic o he game in which shocks are never observable. herefore, for any η > 0, sep 2 of heorem 3 in he paper implies immediaely ha here eiss η > 0, such ha for all η here eiss an equilibrium 1, θ } of Γ such ha 1 1 η. Moreover, since θ1 1 = θ 1 1, 1, define Ω 1 and Ω 1 implicily by θ 1 1, Ω = θ1 ε and θ 1 1, Ω = θ1 + ε. herefore, Pr θ1 1 [θ1 ε, θ 1 + ε] = Pr 1 [ Ω ] 1 /, Ω 1 /. Clearly, Ω 1 > 0 > Ω 1, and by coninuiy, here eiss η 1 0, ε] such ha Ω 1 > 0 > Ω 1 for any 1 [ 1 η 1, 1 + η 1]. Since Pr 1 [ Ω 1 /, Ω 1 / ] is decreasing in and converges o 1 as 0, here eiss > 0 such ha Pr 1 [ Ω 1 /, Ω 1 / ] 1 ε for all. We conclude ha Resul A1.a holds for = 1 wih ˆε, 1 = min, η 1 }. Sep 3. Along he same lines as in sep 2 in he proof of Proposiion A1, we now esablish a suffi cien condiion for Resul A1.a o hold for periods when i holds for 1 periods. In paricular, fi an ε > 0, an ε 1 0, ε, a 2, and a sequence θ } ha is par of an equilibrium of Γ 0, and suppose ha here eiss a ˆ 1 = ˆ ε 1, 1, θ } 1 > 0 such ha, 10

11 whenever ˆ 1, here eiss an equilibrium 1, θ } of Γ ha saisfies he resul for 1 and ε 1. Suppose furher ha we are able o prove he following: Resul A2.c here eis scalars ε 1 0, ε, > 0, such ha, for any <, here eiss a funcion ˆ : R R ha saisfied he following: i for any θ 1 [ θ 1 ε 1, θ 1 + ε 1], eiher ˆ θ 1 = and V, θ 1 0, or ˆ θ 1 > and V ˆ θ 1, θ 1 = 0; ii for any θ 1 [ θ 1 ε 1, θ 1 + ε ] 1, Pr ˆθ, θ 1 θ ε 1 ε + ε 1, where ˆθ is defined by ˆθ Ω, θ 1 ma θ ˆ θ 1, Ω, θ 1 }. hen, for any < minˆ 1, }, here eiss an equilibrium of Γ ha saisfies he resul for 1 and ε 1 and for which 1 = ˆ θ 1 1 and θ = ˆθ, θ 1 1 when 1 is such ha θ 1 1 [ θ 1 ε 1, θ 1 + ε 1]. Bu hen, by he same argumen as in Sep 2 of he proof of Proposiion A1, his equilibrium saisfies Pr θ } B ε, θ } 1 ε, proving ha he resul holds for wih ˆ = minˆ 1, }. In he ne wo seps, we hus prove Resul A2.c, disinguishing again beween he case where θ = θ 1 Sep 4 and he case where θ > θ 1 Sep 5. Sep 4. Suppose ha θ = θ 1, and fi ε > 0. For all, +,, θ 1 = F θ; dψ θ, θ 1 c v = + F θ/ dψ θ, θ 1 c F θ 1 / c, and herefore V, θ 1 F θ 1 / c. Now, selec ε 1 0, ε and 1 > 0 such ha θ 1 > ε 1 and F θ 1 ε 1 / 1 c 0. Whenever 1 and θ 1 θ 1 ε 1,, θ 1 F θ 1 / c F θ 1 ε 1 / 1 c 0. V herefore, whenever 1, ˆ θ 1 = saisfies par i of Resul A2.c, in which case } θ ˆ θ 1, Ω = Ω and hence ˆθ Ω, θ 1 = ma Ω, θ 1. o check ha par ii is also saisfied, noice ha Ω > θ 1 implies ˆθ Ω, θ 1 = θ 1, and hence ˆθ Ω, θ 1 θ ε 1 < ε. herefore, Pr ˆθ, θ 1 θ < ε Pr > θ 1 = 1 F θ 1 / 1 F θ 1 ε 1 / for any θ 1 [ θ 1 ε 1, θ 1 + ε 1]. Since θ 1 > ε 1, here eiss 2 > 0, such ha 1 F θ 1 ε 1 / 1 ε + ε1 for all 2. Hence, Resul A2.c is saisfied whenever min 1, 2 }. 11

12 Sep 5. Suppose now ha θ > θ 1, and fi ε > 0 and ε 0, ε] such ha θ 1 + ε < θ ε. Suppose furher ha V 0 is locally decreasing in a = and fi η 1 > 0 such ha V 0 η, θ 1 > 0 > V 0 + η, θ 1 for all η η1 An analogous argumen applies if V 0 is locally increasing, while angency is ruled ou by assumpion. >From he poinwise convergence of V o V 0, for any η 0, η 1 ], here eiss 1 η > 0, such ha, whenever 1 η, V η, θ 1 > 0 > V + η, θ 1. By coninuiy wih respec o θ 1, here also eiss ε 1 η 0, ε, such ha V η, θ 1 1 > 2 V η, θ 1 1 > 0 > 2 V + η, θ 1 > V + η, θ 1 for all θ 1 such ha θ 1 θ 1 ε 1 η and all 1 η. herefore, whenever 1 η, here eiss ˆ θ 1 [ η, + η] such ha V ˆ θ 1, θ 1 = 0, in which case par i of Resul A2.c is saisfied for ε 1 η and 1 η, for any η η 1. o check when par ii is also saisfied, noe ha θ, Ω [θ ε, θ + ε ] if and only if Ω [Ω θ + ε,, Ω θ ε, ], where Ω θ, Φ β θ θ. Whenever θ ˆ θ 1, Ω [θ ε, θ + ε ], θ ˆ θ 1, Ω > θ 1 and herefore ˆθ Ω, θ 1 = ma θ } ˆ θ 1, Ω, θ 1 = θ ˆ θ 1, Ω. Moreover, θ, Ω [θ ε, θ + ε] for all [ η, + η], and herefore ˆθ Ω, θ 1 = θ ˆ θ 1, Ω [θ ε, θ + ε], whenever Ω [Ω θ + ε, + η, Ω θ ε, η]. We conclude ha Pr ˆθ, θ 1 θ ε Pr ˆθ, θ 1 θ ε Pr [ Ω θ + ε, + η, Ω θ ε, η ] for all θ 1 such ha θ 1 θ 1 ε 1 η. Since Ω θ, = 0 and Ω θ ε, > 0 > Ω θ + ε,, we have Ω θ ε, η 2 > 0 > Ω θ + ε, + η 2 for some η 2 0, η 1 ], and here eiss 2 η 2 > 0 such ha, for all 2 η 2, Pr [Ω θ + ε, + η 2 /, Ω θ ε, η 2 /] 1 ε + ε 1 η. herefore, par ii of Resul A2.c is saisfied wih ε 1 = ε 1 η 2 and = min 1 η 2, 2 η 2 }. Proof of Proposiion A4. Since firs-period beliefs are idenical o hose in he benchmark game, V 1 1 = V for all 1 R, and herefore 1 = ˆ 1 and h 1 = ˆθ 1, where ˆ 1, ˆθ 1 denoe he firs-period equilibrium hresholds of he benchmark game. he res of he proof hen follows from he same argumens as he proof of Proposiion 5 and Lemma A2. In paricular, o see ha 1, + = c < 0 for all 1 R 1 which rules ou equilibria in which = +, noice V ha for = +, and for any > 1, Ψ h, 1 = Ψ 1, 1 Ψ Ψ 1, , 1 φ ; β h φ β 1 dψ h 1

13 as, φ β h φ whenever h β 1 > 1, and herefore lim Ψ 1, 1 = 0. Proof of Proposiion A5. Below we esablish ha, as 0, beliefs and hence payoffs in he game wih shor-lived agens converge poinwise o hose in he benchmark model. Given he converge of payoffs, he resul hen follows from he same argumens as in Seps 2-4 in he proof of heorem 3. Poinwise convergence of poseriors and payoffs. Consider firs beliefs. Le Ψ 0 h ; 1 denoe he period- common poserior abou h in he benchmark model, and Ψ h ; 1 he period- common poserior abou h in he game wih changing fundamenals and shor-lived agens. he former are simply given by he runcaed Normals, Ψ 0 h ; 1 = 1 Φ α z h Φ αz θ 1, while he laer are defined by A1 Recall ha θ 1 minθ : θ Φ β τ τ θ τ } = ma τ θτ τ }. By Bayes rule, he corresponding privae poseriors saisfy Ψ 0 h ; 1 = h β φ β h dψ 0 h ; 1 + β φ β h dψ 0 h ; 1 for he benchmark model, and similarly replacing Ψ 0 wih Ψ for he game wih for he game wih changing fundamenals Clearly, he above definiions and condiions apply o 2; similar ones hold for = 1. o prove poinwise convergence of privae poseriors, i hus suffi ces o prove poinwise convergence of he common poseriors. We esablish his by inducion. Clearly, since period 1 is idenical in he wo games, Ψ 1 h 1 = 1 Φ α z h 1 = Ψ 0 1 h 1 for any h 1. Ne, consider any 2 and suppose ha poinwise convergence holds a 1. By he inducion hypohesis, lim 0 Ψ 1 h ; 2 = Ψ 0 1 h ; 2 = 0 if h θ 2 2 Φ α z h 1 Φ α z θ 2 2 > 0 if h > θ 2 2 for all h and 2. Using he above ogeher he fac ha lim 0 Φ h h 1 / = 1 whenever h 1 < h and lim 0 Φ h h 1 / = 0 whenever h 1 > h, condiion A1 gives 0 if h θ 1 1 lim 0 Ψ h ; 1 = h θ 1 1 dψ0 1 h 1 ; 2 1 Ψ 0 1 θ 1 1 ; 2 if h > θ 1 1 = Ψ 0 h, 1, 13

14 for all h and 1, which proves he poinwise converge of poseriors in period. Ne, consider payoffs. In he benchmark model, firs-period payoffs saisfy whereas for any 2, V = U θ1 1,, β 1, α, z = Ψ 0 1 θ1 1 1 c 1 R, V 0 1, = U θ, θ 1 1, β, α, z = Ψ 0 θ ; 1 c R, 1 R 1. In he game wih changing fundamenals, firs-period beliefs are idenical o hose in he benchmark game, and herefore V1 1 = Ψ 1 θ1 1 1 c = Ψ 0 1 θ1 1 1 c = V R. For 2, payoffs in he game wih changing fundamenals saisfy V 1, = Ψ θ ; 1 c R, 1 R 1, 2. he poinwise convergence of beliefs hus implies ha lim V 1, = lim Ψ 0 0 θ ; 1 c = V 0 1, R, 1 R 1, 2. Noe ha convergence of beliefs and payoffs may fail a =, bu, as in he case of heorem 3, his does no affec he resul. 14