Benke , Jean-Michel; Ma ysko a, Ludmila; S a ko , Ego
Wo king Pape
S a egic a ibu e lea ning
Discussion Pape s, No. 24-11
P o ided in Coope a ion wi h:
Depa men o Economics, Uni e si y o Be n
Sugges ed Ci a ion: Benke , Jean-Michel; Ma ysko a, Ludmila; S a ko , Ego (2024) : S a egic
a ibu e lea ning, Discussion Pape s, No. 24-11, Uni e si y o Be n, Depa men o Economics, Be n
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S a egic A ibu e Lea ning
Jean-Michel Benke , Ludmila Ma ysko a,
Ego S a ko
24-11
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DISCUSSION PAPERS
S a egic A ibu e Lea ning
*
Jean-Michel Benke , Ludmila Ma ysko ´a, and Ego S a ko
Decembe 13, 2024
Abs ac
A esea che alloca es a budge o in o ma i e es s ac oss mul iple unknown a -
ibu es o in luence a decision-make . We de i e he esea che ’s equilib ium lea n-
ing s a egy by sol ing an auxilia y single-playe p oblem. The a ibu e weigh s in
his p oblem depend on how much he esea che and he decision-make disag ee.
I he esea che expec s an excessi e esponse o new in o ma ion, she o goes lea n-
ing al oge he . In an o ganiza ional con ex , we show ha a manage a o s mo e
di e se analys s as he hie a chical dis ance g ows. In ano he applica ion, we show
how an app op ia ely opposed ad iso can cons ain a disc imina o y poli ician, and
iden i y he wel a e-inequali y Pa e o on ie o esea che s.
Keywo ds: A ibu es, In o ma ion acquisi ion, Gaussian dis ibu ion, S a egic lea ning
JEL classi ica ion: D72, D81, D83
*
Fo aluable commen s, we hank: A jada Ba dhi, Na han Hanca , Toomas Hinnosaa , Alessand o
Ispano, Igo Le ina, An oine Loepe , Ma c M¨olle , Nick Ne ze , F ancisco Poggi, Johannes Schneide ,
A min Schmu zle , Jakub S eine , Pe e No man Sø ensen, Dezs¨o Szalay, and he audiences a he
Uni e si ies o Alican e, Be n, Copenhagen, Mad id Ca los III, Mannheim, Zu ich, and CERGE-EI, as
well as he No dic Theo y Mee ings, he Annual Mee ing o Eu opean Associa ion o Young Economis s,
he Lisbon Mee ings in Game Theo y and Applica ions, he Ba celona Summe Fo um, EEA-ESEM,
and he Annual Mee ing o he V S. Ma ysko ´a g a e ully acknowledges unding om he Gene ali a
Valenciana (P ome eo/2021/073) and om he g an PID2022-142356NB-I00 inanced by MICIU/AEI
/10.13039/501100011033 and FEDER, UE.
Benke : Depa men o Economics, Uni e si y o Be n. Ma ysko ´a: Depa men o Economics,
Uni e si y o Alican e. S a ko : Depa men o Economics, Uni e si y o Copenhagen. Email: jean-
michel.benke @unibe.ch, ludmila.ma ysko [email p o ec ed], ego .s a k[email p o ec ed].
1
1 In oduc ion
Decision-making unde unce ain y o en in ol es mul iple dimensions o po en ially
unequal impo ance. Fo example, when designing a new p oduc , a company may
conside he p oduc ’s unce ain ecep ion ac oss a ious ma ke segmen s, some o
which may be mo e impo an han o he s. Simila ly, a poli ician’s op imal policy may
need o add ess he di e se and unce ain needs o dis inc social g oups, wi h some
g oups po en ially ca ying mo e weigh han o he s do.
In bo h examples, decision-make s mus lea n abou he a ious dimensions, o a -
ibu es, in luencing hei decisions bu o en lack he esou ces o a ho ough in es-
iga ion. Consequen ly, a specialized esea che is o en in cha ge o he lea ning ask.
Howe e , he esea che may p io i ize a ibu es di e en ly om he decision-make
and s a egically p ocu e in o ma ion abou he unknown a ibu es o in luence he
inal decision.
In his pape , we explo e ques ions pe aining o such s a egic conce ns in lea ning
abou complex decisions. Fi s , wha kinds o biases may a ise when mul iple a ibu es
a e a play, and how do hey in e ac ? Fo ins ance, in an o ganiza ional applica ion, we
explo e wo o hogonal biases—hie a chical dis ance and di e si y—and analyze when
a manage (decision-make ) p e e s di e se analys s ( esea che s) as a unc ion o hi-
e a chical dis ance. Second, how can biases mi iga e socially undesi able decisions?
Fo ins ance, in he con ex o poli ical disc imina ion, we show how an app op ia ely
opposed ad iso ( esea che ) can cons ain a disc imina o y poli ician whose decision
a ec s well-being o a ious social g oups (a ibu es). Mo e b oadly, how does p e -
e ence misalignmen in a mul ia ibu e en i onmen a ec lea ning, when esea che s
shape no only he ex en o lea ning bu also i s di ec ion?
To add ess hese ques ions, we de elop a amewo k ha cap u es he s a egic in e -
ac ion be ween a esea che , who lea ns abou an unknown s a e o he wo ld, and a
decision-make , who ac s based on he esul ing in o ma ion. C ucially, we assume ha
he s a e o he wo ld consis s o mul iple a ibu es and cap u e he playe ’s p e e -
ence misalignmen by allowing o di e ing weigh s on he a ibu es. As such, ou
main con ibu ion is a ac able model o s udying how p e e ence misalignmen a ec s
mul i-a ibu e lea ning and decision-making.
2
In Sec ion 2, we o mally in oduce ou amewo k. In ou model, he a ibu es a e
independen ly and no mally dis ibu ed and join ly de e mine he s a e o he wo ld and
hus he playe s’ p e e ed decisions. The s a e can be impe ec ly lea ned by alloca ing
a gi en budge o es s ac oss di e en a ibu es. The mo e es s a e alloca ed o an
a ibu e, he mo e in o ma i e a signal abou i becomes. Playe s aim o minimize he
quad a ic loss be ween he inal decision and hei bliss poin , which is a weigh ed sum
o a ibu es.1
We begin by analyzing a help ul benchma k in Sec ion 3: a si ua ion in which a single
playe con ols bo h he decision and he lea ning, whe e he op imal lea ning s a egy
is he p ima y ocus o ou analysis. Theo em 1 shows ha —absen s a egic mo i es—
he playe chooses he es alloca ion which achie es he highes educ ion in esidual
unce ain y ega ding he op imal decision. In pa icula , op imal lea ning e lec s bo h
he a ibu e’s ele ance o he agen and i s p io unce ain y. Pu di e en ly, he
mo e impo an and mo e unce ain an a ibu e is, he mo e es s a e alloca ed o i .
Mo eo e , he budge size de e mines how many a ibu es he agen lea ns abou .
In Sec ion 4, we e u n o he p ima y amewo k, in which a esea che con ols he
lea ning and a decision-make makes he decision. Ou main esul (Theo em 2) shows
ha he esea che ’s equilib ium es alloca ion hen coincides wi h he single-playe
solu ion wi h auxilia y weigh s, which a e di e en om he ac ual weigh s o ei he
playe . We show ha hese auxilia y weigh s dec ease as he misalignmen be ween play-
e s g ows. I he esea che alues an a ibu e less (o mo e) han he decision-make
does, he iews he decision-make ’s eac ion o any in o ma ion abou ha a ibu e as
excessi e (o insu icien ). When he misalignmen esul s in o e eac ion, i can lead
he esea che o abs ain om lea ning abou he a ibu e en i ely. No ably, when all
o he esea che ’s weigh s a e su icien ly low ela i e o he decision-make ’s weigh s,
he esea che en i ely o goes lea ning—con as ing wi h he single-playe case, whe e
lea ning gene ically akes place. The ac ha he esea che ’s equilib ium es allo-
ca ion coincides wi h he single-playe solu ion wi h auxilia y weigh s cla i ies how he
s a egic mo i e in luences he esea che ’s lea ning, educing i s impac o a shi in a -
ibu e weigh s. Thus, by compa ing hese auxilia y weigh s o he esea che ’s o iginal
weigh s, we can pinpoin how misalignmen d i es adjus men s in he esea che ’s lea n-
1Ou model is a s a ic a ian o he amewo k in Liang, Mu, and Sy gkanis (2022), who s udy a
single agen ’s dynamic lea ning unde a ibu e co ela ion.
3
ing s a egy—insigh ha would be ha de o achie e i he s a egic solu ion de ia ed
undamen ally om he single-playe case.
We explo e wo economical applica ions, each wi h a ailo ed pa ame iza ion o playe
misalignmen . These applica ions demons a e he e sa ili y o ou analy ical ools
in s udying misalignmen and yield new insigh s wi hin each speci ic con ex . In ou
i s applica ion, we examine how biases ac oss laye s o hie a chy in luence lea ning
in o ganiza ions. He e, an analys ( esea che ) ga he s in o ma ion, and a manage
(decision-make ) ac s on i . While he manage i ially p e e s an analys whose p e -
e ences ma ch he s, we examine he p e e ed analys when ull alignmen is un easible.
We pa ame ize misalignmen by decomposing he analys ’s p e e ences in o sensi i i y
(how closely he playe s align in absolu e e ms) and dis o ion (how much he playe s
disag ee in ela i e e ms). P oposi ion 2 shows ha he manage may p e e ana-
lys s wi h high dis o ion when he analys s ha e low sensi i i y. An analys wi h low
sensi i i y iews he manage ’s eac ion o any in o ma ion as excessi e and hus may
comple ely o go lea ning. In such cases, he manage hen p e e s an analys who is
e y dis o ed owa d some a ibu e, as hen he manage ’s eac ion o in o ma ion
abou ha a ibu e is no longe iewed as excessi e, esul ing in some lea ning. As
any lea ning is be e o he manage han no lea ning, she p e e s a s ongly dis o ed
analys o e one wi h li le o no dis o ion. Assuming ha he mo e hie a chical he
o ganiza ion, he bigge he di e ence in he sensi i i y be ween he manage s and he
analys s, ou esul s sugges balancing di e si y and uni o mi y ( e lec ed in dis o ion)
in o ganiza ions as a unc ion o hie a chical dis ance.
In ou second applica ion, we explo e he issue o disc imina ion. A poli ician (decision-
make ) chooses a policy o mee he unce ain needs o wo a p io i iden ical social g oups
(a ibu es) bu a o s one g oup o e he o he . Wi h an unchecked poli ician—who
con ols bo h decision-making and lea ning—such a o i ism esul s in inequali y and
educes u ili a ian wel a e compa ed o he si ua ion wi hou a o i ism. We exam-
ine how delega ing lea ning o an ad iso ( esea che ) wi h di e en p e e ences can
mi iga e hese nega i e e ec s. P oposi ion 4 demons a es ha he e exis s a wel a e-
inequali y Pa e o on ie o po en ial ad iso s ha s ic ly domina es unchecked dis-
c imina ion. A one end o his on ie is an impa ial ad iso , who—gi en he poli i-
cian’s a o i ism—maximizes wel a e, bu lea es he g oups wi h unequal ou comes
(Lemma 3). A he o he end o he on ie is an ad iso who is sui ably pa ial owa ds
4
he disad an aged g oup. Such an ad iso lea ns p ecisely enough abou his g oup o
coun e ac he poli ician’s a o i ism, and hus elimina es inequali y when appoin ed
(Lemma 4). Howe e , i comes a a cos o wel a e, as he poli ician hen elies less on
he ad iso ’s in o ma ion, and in o ma ion is hus “was ed.” No ably, he le el o ad-
iso ’s pa iali y equi ed o elimina e inequali y does no inc ease wi h he poli ician’s
a o i ism bu exhibi s a non-mono one ela ionship.
In Sec ion 7, we ou line wo ex ensions de eloped in de ail in he online appendix. The
i s examines mul iple esea che s in he con ex o media ma ke s, showing ha while
compe i ion be ween media ou le s ( esea che s) leads o pola iza ion in equilib ium,
his ou come is ac ually bene icial o a o e (decision-make ). The second ex ension
add esses unce ain y abou he decision-make ’s p e e ences. We s udy his wi hin a
dual-sel model, whe e a single agen may be ei he sophis ica ed o na¨ı e abou po en ial
changes in he p e e ences be ween he lea ning and he decision-making s ages, inding
ha na¨ı e e can, in ac , be ad an ageous.
We conclude in Sec ion 8 by discussing ou esul s and p ospec i e a enues o u u e
esea ch.
Rela ed li e a u e. Ou wo k is closely ela ed o Ba dhi (2024), who examines
a s a egic mul i-a ibu e lea ning p oblem whe e a p ojec ’s payo is composed o
co ela ed a ibu es, weigh ed di e en ly by a decision-make and a esea che . The
esea che selec s which a ibu es o sample (and pe ec ly lea n), gi en ha he can
sample a limi ed numbe . Ba dhi (2024) p ima ily ocuses on he ole o co ela ion
be ween a ibu es. In con as , we aim o p o ide a ac able model o s udy p e e ence
misalignmen . Fi s , we ocus on independen a ibu es o sepa a e misalignmen e ec s
om co ela ion. Second, we conside noisy lea ning, whe e he choice in ol es no only
which a ibu es o lea n abou bu also how much o lea n abou each. Ou analysis
demons a es ha wi h as ew as wo a ibu es, noisy lea ning e ec i ely cap u es he
complexi y o bias in e ac ions.2
Ki ne a (2023), building on Tamu a (2018), conside s a ela ed mul idimensional se -
ing, whe e he decision-make can acqui e addi ional cos ly in o ma ion on op o wha
2Mo e b oadly, ou amewo k is ela ed o he Gaussian-sampling li e a u e such as Ba dhi and
Bobko a (2023), Callande (2011), and Ca nehl and Schneide (2024).
5
is p o ided by he esea che . The need o in luence he decision-make ’s lea ning s a -
egy (and no only he decision) hen shapes he esea che ’s lea ning s a egy. The
esea che may hen p o ide pa ial in o ma ion abou he a ibu e on which he play-
e s’ p e e ences a e misaligned in o de o di e he decision-make ’s lea ning away
om i .3In con as , we do no allow o independen lea ning by he decision-make ,
c ea ing a dis inc s a egic en i onmen and shu ing down he “a en ion di e sion”
channel.
Ou amewo k builds on Liang e al. (2022), adap ing hei dynamic model o mul iple
a ibu es o a s a ic o m. Fu he mo e, while hey s udy a single agen ’s dynamic
lea ning unde a ibu e co ela ion, we ocus on s a egic lea ning and abs ac om
co ela ion o highligh he ole o p e e ence misalignmen be ween playe s. No ably,
he e is a key pa allel in ou indings. Bo h pape s show ha a “g eedy” lea ning
s a egy, which achie es he highes educ ion in esidual unce ain y and is op imal in
a single-playe model wi hou co ela ion, emains op imal in he p esence o co ela ion
(Liang e al., 2022) and in he p esence o s a egic mo i es (ou pape ), espec i ely.
Ou i s applica ion, on di e si y in o ganiza ion, connec s o he li e a u e on delega ed
expe ise, whe e a decision-make delega es lea ning abou he s a e o in e es o a
biased esea che .4We explo e he e ec s o di e en biases be ween he esea che
and he decision-make , which is ela ed o s udies by Ball and Gao (2024), Che and
Ka ik (2009), and Ilino e al. (2022). These pape s show ha delega ing lea ning o
a esea che wi h some p e e ence misalignmen can be op imal in single-dimensional
se ings, as misalignmen encou ages he acquisi ion o mo e cos ly in o ma ion. In
con as , we explo e a mul idimensional se ing, whe e he esea che decides which
a ibu es o lea n abou and o wha ex en . Ou indings in Sec ion 6.1 e eal ha
highe misalignmen on a ce ain ype o bias can be bene icial, as i may p e en he
b eakdown o lea ning caused by misalignmen on ano he , o hogonal, ype o bias.
Ou second applica ion, on disc imina ion in policymaking, ela es o Fosge au e al.
(2023) and Echenique and Li (2023), who explo e how decision-make s’ s a egic lea n-
3Simila insigh s—namely, ha when he ecei e can acqui e in o ma ion, he sende ’s choices a e
p ima ily d i en by a desi e o in luence he lea ning p ocess a he han he inal decision—ha e also
been ob ained by Ma eenko and S a ko (2023).
4Pionee ed by Demski and Sapping on (1987), his li e a u e has seen enewed in e es , o ins ance,
see Deimen and Szalay (2019) and Lindbeck and Weibull (2020).
6
ing choices ein o ce disc imina ion.5They show ha employe s’ disc imina o y be-
lie s shape job candida es’ incen i es o in es in hei skills, po en ially leading o
a sel -sus aining disc imina o y equilib ium. In con as , we ocus on coun e ing dis-
c imina o y endencies by sepa a ing lea ning om decision-making and delega ing i
o independen ad iso s. Fu he , ou esul s in Sec ion 6.2 complemen Liang e al.
(2024). They s udy he ai ness-accu acy Pa e o on ie in he con ex o an in e ac ion
be ween an egali a ian esea che and a u ili a ian decision-make , whe e he esea che
may coa sen o ban in o ma ion abou ce ain a ibu es. In con as , we examine he
wel a e-inequali y on ie in a se ing whe e he decision-make is inhe en ly un ai bu
can be pai ed wi h a esea che holding di e en p e e ences, who can lexibly alloca e
in o ma i e es s ac oss di e en a ibu es.
2 Model
2.1 Se up
Playe s. The e a e wo playe s, a decision-make (D, she) and a esea che (R, he).
Fi s , he esea che chooses he lea ning s a egy o examine payo - ele an a ibu es
o he s a e o he wo ld by choosing how o alloca e a budge o es s T. Second, he
decision-make obse es he esul s o hese es s and makes a decision.
A ibu es. The e is an unknown s a e o he wo ld ˜
θ= (˜
θ1,...,˜
θK) consis ing o
K≥2 a ibu es.6All a ibu es ˜
θk o k= 1, . . . , K a e join ly no mally dis ibu ed
wi h commonly known p io means µ0
k∈Rand p io a iances Σ0
k>0; ha is, ˜
θk∼
Nµ0
k,Σ0
k.Mo eo e , he a ibu es a e independen : ˜
θl⊥˜
θj o l=j.
Ac ions. The decision-make chooses a decision d∈R. The esea che , gi en an
exogenous budge o es s T > 0, chooses a es alloca ion τ∈ T := {τ∈RK
+:
τ1+. . . +τK≤T}, whe e τkis he amoun o es s alloca ed o lea n abou a ibu e
5See Onuchic (2024) o a ecen o e iew o heo ies o disc imina ion.
6We deno e a andom a iable and i s ealiza ion by ˜xand x, espec i ely.
7
Figu e 1. The esea che ’s solu ion: Auxilia y single-playe weigh s
02αR
k
ˆαk
αD
k
αR
k
αR
k
Panel (a)
0
ˆαk
αR
k
αD
k/2αD
k
αD
k
45◦
Panel (b)
No es: The igu es depic ˆαkas a unc ion o he decision-make ’s weigh , αD
k, in panel (a), and as a
unc ion o he esea che ’s weigh , αR
k, in panel (b), keeping he weigh o he o he playe cons an .
how misalignmen d i es adjus men s in he esea che ’s lea ning s a egy—insigh ha
would be ha de o achie e i he s a egic solu ion de ia ed undamen ally om he
single-playe case.
F om he esea che ’s pe spec i e, he ex an e ma ginal alue o lea ning abou a ibu e
˜
θkis p opo ional o he pa ame e λk, which—unlike in he single-playe case—can
be nega i e. Theo em 2 explains he link be ween he auxilia y weigh s ˆαkand he
pa ame e s λk. No e ha λk(and hus ˆαk) is dec easing in he misalignmen be ween he
playe s’ weigh s ∆k. When ∆k>0, he esea che iews he decision-make ’s eac ion o
in o ma ion abou ˜
θkas ei he excessi e (αD
k> αR
k) o insu icien (αD
k< αR
k), he eby
educing he esea che ’s incen i es o lea n abou ha a ibu e. I he eac ion is oo
s ong (αD
k≥2αR
k) o oo weak (αD
k= 0), he esea che a oids lea ning abou ha
a ibu e en i ely. I λk≤0 o all k, he esea che abs ains om lea ning, as any
in o ma ion would lead o undesi able o e eac ion (o no eac ion) by he decision-
make , making he s a us quo decision he esea che ’s p e e ed ou come.
When playe s disag ee on he weigh o an a ibu e, he auxilia y weigh di e s om
he weigh o ei he playe . Figu e 1 illus a es how ˆαkdepends on αR
kand αD
k, keeping
he weigh o he o he playe ixed. Panel (a) shows ha ˆαkalways sa is ies ˆαk≤
αR
k: he misalignmen e ec i ely educes he impo ance he esea che assigns o he
a ibu e. Ne e heless, he ex en and di ec ion o any dis o ion in he equilib ium
es alloca ion—compa ed o he esea che ’s non-s a egic op imum— hen depend on
how he a ios o he auxilia y weigh s di e om he a ios o he esea che ’s weigh s.
14
5 Equi alen payo speci ica ions
This sec ion in oduces al e na i e payo s uc u es ha esul in he same equilib ium
es alloca ion as he baseline model. The objec i e o p esen ing hese amewo ks is
o o e al e na i es ha may be be e ailo ed o pa icula economic applica ions,
hus demons a ing he adap abili y o ou model and he analy ical ools de ailed in
Sec ions 3 and 4.
Baseline model. In oduced in Sec ion 2, he baseline model ea u es a decision-
make aking a single decision, d∈R. Gi en a decision dand a ealized s a e θ=
(θ1, . . . , θK), he u ili y o playe i=R, D is
ui(d, θ) = −(d−bi(θ))2=− d−X
k
αi
kθk!2
.(15)
F amewo k A. In his amewo k, he decision-make also akes a single decision,
d∈R. Gi en a decision dand a ealized s a e θ= (θ1, . . . , θK), he u ili y o playe
i=R, D is
ui
A(d, θ) = −X
k
αi
k(d−θk)2,(16)
whe e Pkαi
k= 1 o bo h playe s.
F amewo k B. This amewo k ea u es he decision-make simul aneously aking K
dis inc decisions d1, . . . , dK∈R. Gi en decisions d= (d1, . . . , dK) and a ealized s a e
θ= (θ1, . . . , θK), he u ili y o playe i=R, D is
ui
B(d, θ) = −X
k
(dk−αi
kθk)2.(17)
In all h ee models: (i) playe s aim o minimize a loss gi en by a quad a ic dis ance,
and (ii) he weigh s αide e mine he ela i e impo ance o di e en a ibu es. Be-
yond hese simila i ies, he amewo ks di e in how he playe s agg ega e losses ac oss
a ibu es and decisions. Despi e hese di e ences, he ollowing p oposi ion (p o ed in
15
Online Appendix C.1) shows ha F amewo ks A and B a e equi alen o he baseline
model in e ms o he equilib ium es alloca ion.
P oposi ion 1. Gi en he decision-make ’s weigh s αD, he esea che ’s weigh s αR,
he es budge T, and he p io dis ibu ion o he s a e ˜
θ, he esea che ’s equilib ium
es alloca ions in he baseline model, F amewo k A, and F amewo k B a e iden ical.
The displayed lexibili y allows us o s udy a ich se o economic p oblems. Fo ins ance,
we use F amewo k A o s udy disc imina ion (Sec ion 6.2) and media pola iza ion (On-
line Appendix B.1). To illus a e an applica ion o F amewo k B, conside a po olio
choice p oblem, whe e an in es o makes decisions dkon how much o in es in each
asse k= 1, . . . , K. He e, ˜
θk ep esen s he unce ain u u e e u n on asse k. The
in es o e alua es he u u e e u ns objec i ely (αD
k= 1 o all k), while an ad iso
may ha e an incen i e o s ee he in es o owa ds some asse s and away om o he s:
αR
k= 1 o some k.
6 Applica ions
In his sec ion, we illus a e how ou amewo k can be used o s udy p e e ence mis-
alignmen in wo applica ions, each wi h a ailo ed pa ame iza ion o he misalignmen .
Fi s , we adop an o ganiza ional se ing o showcase how biases ac oss hie a chical lay-
e s impac lea ning in o ganiza ions. Nex , we analyze a policy-making model wi h a
disc imina o y poli ician and explo e how in oducing independen esea che s o in o m
policy decisions can mi iga e nega i e e ec s o disc imina ion on wel a e and inequali y.
In he applica ions, we adop he simples speci ica ion, a wo-a ibu e case, which is
al eady su icien ly ich o demons a e he ension be ween he playe s a ising in he
mul i-a ibu e con ex . Th oughou , we assume ha bo h weigh s o he decision-make
a e s ic ly posi i e, αD
k>0 o k= 1,2.
6.1 Di e si y in o ganiza ions
In his applica ion, we conside a s ylized model o an o ganiza ion, comp ising a man-
age (decision-make ) and an analys ( esea che ). This se up e lec s an o ganiza ional
16
Figu e 2. The bias decomposi ion o an analys
αR
2
αR
1
β
γ
γ > 0
αD
No es: The dashed line ep esen s di e en αR alues whe e γis ixed a a posi i e alue and a ious
alues o βa e conside ed.
hie a chy whe e he manage holds au ho i y o e he analys and akes decisions based
on he analys ’s in o ma ion. Gi en a di e se pool o analys s wi h a ying weigh s,
he manage selec s one o p o ide in o ma ion. The key ques ion is which analys he
manage will choose.
Fi s , i is immedia e ha he manage ’s ideal choice would be an analys whose weigh s
ma ch he s. Howe e , ou ocus is on iden i ying he ype o analys he manage would
p e e when, due o an inhe en hie a chical s uc u e, pe ec alignmen is unachie able.
To analyze his ques ion, we begin by es ablishing a sui able pa ame iza ion o he
misalignmen be ween he playe s’ weigh s.
We ix he manage ’s weigh ec o , αD= (αD
1, αD
2), and de ine an o hogonal ec o
¯αD:= (−αD
2, αD
1). Nex , we exp ess he analys ’s weigh ec o αRas a linea combi-
na ion o he manage ’s weigh ec o and i s o hogonal coun e pa as
αR(β, γ) := βαD+γ¯αD(18)
o some β∈R+and γ∈Γ(β) := h−βαD
2
αD
1
, β αD
1
αD
2i.13 He e, β ep esen s he absolu e bias,
e e ed o as sensi i i y ( o new in o ma ion), and γ ep esen s he ela i e bias, e e ed
o as dis o ion. We say ha he analys becomes mo e sensi i e when βinc eases.14
13The cons ain γ∈Γ(β) ollows om he equi emen ha αR
k≥0 o bo h k= 1,2. As no ed in
Foo no e 8, his es ic ion is wi hou loss and only o ease o exposi ion.
14No e ha when β= 1, he analys has he same sensi i i y as he manage .
17
Figu e 3. Equilib ium es alloca ion
Panel (a)
αR
2
αR
1
αD
1
2
αD
2
2
β
no es ing
o β≤1/2
manage ’s i s
bes o β > 1/2
αD
Panel (b)
αR
2
αR
1
αD
1
2
αD
2
2
αD
L1
L2L12
L0
γ
τ∗
1↓, τ∗
2↑
γ
No es: Panel (a) depic s he undis o ed analys ’s equilib ium es alloca ion: no es ing i β≤1/2,
and manage ’s i s bes i β > 1/2. Panel (b) de ails he equilib ium es alloca ion o di e en (β, γ)
pai s: he analys lea ns (i) only abou a ibu e ˜
θ1i αR∈L1; (ii) only abou a ibu e ˜
θ2i αR∈L2;
(iii) abou bo h a ibu es i αR∈L12; (i ) abou no a ibu e i αR∈L0. The black hick lines
cap u es analys s wi h inc easing dis o ion owa ds a ibu e ˜
θ2 om an undis o ed analys (dashed
line) o ixed sensi i i y a β < 1/2 (line close o he o igin) and β > 1/2 (line u he away om he
o igin).
We u he say he analys becomes mo e dis o ed when γ≥0 inc eases o γ≤0
dec eases. Speci ically, when γ > 0 (γ < 0), he analys exhibi s bias owa ds a ibu e
˜
θ2(a ibu e ˜
θ1). When γ= 0, we call he analys undis o ed. Figu e 2 isualizes
his bias decomposi ion.15 In he con ex o ou o ganiza ional applica ion, we posi
ha analys s who a e sepa a ed by mo e laye s o hie a chy om he manage a e less
sensi i e.
Fi s , we examine how he manage ’s payo changes wi h he analys ’s dis o ion γ.
P oposi ion 2 below shows ha he manage is wo se o as he analys becomes mo e
dis o ed, p o ided he analys is su icien ly sensi i e (high β). Howe e , he con e se
is ue when he analys is oo insensi i e (low β).
P oposi ion 2. Fix he analys ’s le el o sensi i i y β∈R+.
(i) I β≤1/2 ( he analys is oo insensi i e), hen he manage ’s expec ed equilib ium
15Al e na i e decomposi ions o he analys ’s weigh s in e ms o “absolu e” and “ ela i e” bias exis .
Fo ins ance, one could p esen bo h αRand αDin pola coo dina es and le βbe he di e ence in
dis ances om he o igin and γbe he di e ence in angles. The insigh s om ou P oposi ions 2 and 3
would ex end o such al e na i e ep esen a ions.
18
payo weakly inc eases as he analys becomes mo e dis o ed.
(ii) I β > 1/2 ( he analys is su icien ly sensi i e), hen he manage ’s expec ed
equilib ium payo weakly dec eases as he analys becomes mo e dis o ed.
The manage and an undis o ed analys sha e he same weigh a ios, so he analys ’s
equilib ium lea ning s a egy aligns wi h he manage ’s i s -bes solu ion, p o ided he
analys ’s sensi i i y is high enough o mo i a e lea ning (β > 1/2). In his case, i he
analys ’s dis o ion inc eases, his equilib ium lea ning s a egy de ia es u he om
he manage ’s i s bes , making he manage wo se o . Con e sely, when β≤1/2,
an undis o ed analys abs ains om lea ning, pe cei ing he manage ’s eac ion o
any in o ma ion as excessi e. Then, i he analys ’s dis o ion inc eases, his p e e ences
align mo e closely wi h he manage ’s on one pa icula a ibu e, po en ially p omp ing
him o lea n abou i . While he added dis o ion does inc ease misalignmen on he
o he a ibu e, i does no nega i ely impac lea ning since he ini ial misalignmen
was al eady su icien o de e he analys om lea ning abou i . O e all, since any
acqui ed in o ma ion is p e e able o none, he manage ul ima ely bene i s.
The esul s, which a e illus a ed in Figu e 3, highligh he impo ance o bo h di e si y
and uni o mi y wi hin o ganiza ions. Ou indings sugges a s a egic app oach o o ga-
niza ional s uc u e i employees lowe down he hie a chy a e less engaged and hus less
sensi i e o new in o ma ion han he manage s highe up a e. In o ganiza ions wi h
many laye s o hie a chy and hus po en ially signi ican di e ences in engagemen , i is
bene icial o ha e subs an ial di e si y a he lowe le els, as indica ed by ela i ely high
deg ees o dis o ion |γ|compa ed o he manage s. Con e sely, in smalle , less hie a -
chical o ganiza ions, a mo e uni o m wo k o ce wi h minimal dis o ion is p e e able as
he a ia ion in engagemen le els among manage s and analys s is less p oblema ic in
hese se ings.
The abo e unde sco es he nuanced ole o dis o ion in shaping he analys ’s align-
men wi h he manage ’s objec i es. Howe e , when we shi ocus om dis o ion o
sensi i i y, he e ec s become mo e s aigh o wa d.
P oposi ion 3. Fix he analys ’s dis o ion γ∈R. Then, he manage ’s expec ed
equilib ium payo is weakly inc easing in he analys ’s sensi i i y β.
The manage always p e e s a mo e sensi i e analys , as his leads o an equilib ium
19
es alloca ion ha aligns mo e closely wi h he i s -bes solu ion. In o he wo ds, he
manage seeks highly engaged employees who a e sensi i e o new in o ma ion ele an
o he o ganiza ion. To p o e his esul , we show ha o any dis o ion γ, he equilib-
ium es alloca ion τ∗inc easingly aligns wi h he manage ’s i s bes as βinc eases,
e ec i ely neu alizing he impac o dis o ion.
6.2 Disc imina ion, wel a e and inequali y
In his applica ion, we explo e a scena io whe e a poli ician decides on a policy a ec ing
wo social g oups and a o s one g oup. We in es iga e how appoin ing an ad iso
wi h di e en p e e ences, who s a egically cu a es he in o ma ion p o ided o he
poli ician, can mi iga e he nega i e impac o his ype o disc imina ion on u ili a ian
wel a e and inequali y.
Suppose he e a e wo social g oups k= 1,2. Each g oup khas an unknown op imal
policy ˜
θk∼ N(0,1), and so he wo g oups a e a p io i iden ical. A poli ician (decision-
make ) decides on a common policy, d∈R. The u ili y o g oup kis gi en by uk(d, θk) =
−(d−θk)2. A budge T= 1 o es s is a ailable o inqui e abou he op imal policies
o bo h g oups. An ad iso ( esea che ) chooses a es alloca ion τ= (τ1, τ2)∈ T. The
lea ning p ocess and no a ion a e he same as in he baseline model.
The poli ician and he ad iso a e u ili a ian wi h pa icula weigh s. The poli ician
ca es abou bo h g oups bu a o s g oup k= 1, he payo being
uD(d, θ1, θ2;δ) = 1 + δ
2
|{z}
αD
1(δ):=
u1(d, θ1) + 1−δ
2
|{z}
αD
2(δ):=
u2(d, θ2)
o a gi en le el o disc imina ion δ∈(0,1). On he o he hand, he ad iso (possibly)
a o s g oup k= 2, his payo being
uR(d, θ1, θ2;p) = 1−p
2
|{z}
αR
1(p):=
u1(d, θ1) + 1 + p
2
|{z}
αR
2(p):=
u2(d, θ2)
o a gi en le el o pa iali y p∈[0,1). When p= 0, indica ing he ad iso ca es equally
abou bo h g oups, we call he ad iso impa ial.
20
Ou ocus in his applica ion is on wel a e W(p, δ), de ined as he sum o ex an e equi-
lib ium expec ed payo s o bo h g oups (and is hence u ili a ian wi h equal weigh s),
and inequali y I(p, δ), de ined as hei di e ence (and is hence an egali a ian measu e):
W(p, δ) := Ehu1˜
d∗(τ∗),˜
θ1i+Ehu2˜
d∗(τ∗),˜
θ2i,(19)
I(p, δ) := Ehu1˜
d∗(τ∗),˜
θ1i−Ehu2˜
d∗(τ∗),˜
θ2i.(20)
In he exp essions abo e, he igh -hand side depends on pand δ ia he ad iso ’s choice
o es alloca ion τ∗and he poli ician’s policy choice ˜
d∗(τ∗), which can be mo e ully
desc ibed as ˜
d∗(τ∗(p, δ), δ). When I(p, δ) = 0, we say he e is equali y in equilib ium.16
As a benchma k, we conside he case o unchecked disc imina ion, when he poli ician
con ols bo h lea ning and decision-making. He e, wel a e and inequali y a e de ined
analogously o equa ions (19) and (20), wi h he poli ician selec ing he op imal es
alloca ion ins ead o an ad iso doing so. In his con ex , inc eased disc imina ion
nega i ely impac s bo h wel a e and inequali y: wel a e declines while inequali y ises.
We aim o unde s and how appoin ing an ad iso can mi iga e hese ad e se e ec s.
We analyze a scena io whe e wel a e and inequali y a e in luenced solely h ough he
ad iso ’s ole in he lea ning s age, while he poli ician e ains ull con ol o e decision-
making.
Fi s , we ask which ad iso maximizes wel a e. Since bo h wel a e and he p e e ences
o he impa ial ad iso a e u ili a ian wi h equal weigh s, i immedia ely ollows ha
appoin ing an impa ial ad iso maximizes wel a e, ega dless o he poli ician’s le el
o disc imina ion. In con as , appoin ing a pa ial ad iso o allowing unchecked dis-
c imina ion yields lowe wel a e. The nex lemma desc ibes he (wel a e-maximizing)
lea ning s a egy o he impa ial ad iso .
Lemma 3. Fo e e y le el o disc imina ion δ, wel a e is maximized by appoin ing he
impa ial ad iso (p= 0). Mo eo e , he impa ial ad iso ’s equilib ium es alloca ion
τ∗(0, δ) = (1/2,1/2) is independen o δ.
As he le el o disc imina ion δinc eases, wo e ec s eme ge. F om he ad iso ’s pe -
spec i e, in o ma ion abou g oup 1 is inc easingly o e emphasized in he poli ician’s
16This model i s F amewo k A in Sec ion 5. By P oposi ion 1, Theo em 2 can be used o ind he
equilib ium es alloca ion.
21
decision, while in o ma ion abou g oup 2 is inc easingly unde used. Each e ec inde-
penden ly educes he ad iso ’s incen i e o lea n abou he espec i e g oup. Howe e ,
o he impa ial ad iso , hese e ec s a e equal in magni ude and cancel each o he
ou . This leads him o be un esponsi e o changes in δand always choose an equal es
alloca ion. Consequen ly, as δinc eases, inequali y inc eases unde an impa ial ad iso .
The abo e aises he ques ion o whe he a pa ial ad iso can coun e balance he poli i-
cian’s disc imina ion and es o e equali y. I so, does he equi ed le el o pa iali y p
inc ease wi h he le el o disc imina ion δ? The nex lemma add esses hese ques ions.
Lemma 4. Fo e e y poli ician’s le el o disc imina ion δ, he e exis s a unique ad iso ’s
le el o pa iali y ˆp(δ)>0 ha ensu es equali y in equilib ium: I(ˆp(δ), δ) = 0. The
unc ion ˆp(δ) is con inuous and non-mono one: he e exis s a unique δ∈(0,1) such
ha ˆp(δ) is s ic ly inc easing o δ < δ and s ic ly dec easing o δ > δ.
As no ed ea lie , inc eases in δ educe he ad iso ’s incen i e o lea n abou bo h g oups,
as in o ma ion abou g oup 1 is inc easingly o e emphasized in he poli ician’s decision,
while in o ma ion abou g oup 2 is inc easingly unde used. Fo an ad iso wi h pa iali y
p > 0, he o me e ec domina es he la e , p omp ing him o lea n mo e abou g oup
2 as δinc eases. Mo eo e , he dispa i y be ween he wo e ec s is g ea e o ad iso s
wi h highe pa iali y p, as hey p io i ize g oup 2 o e g oup 1 mo e. As such, ad iso s
wi h highe pa iali y adjus hei lea ning s a egies mo e sha ply in esponse o changes
in δ han do hose wi h lowe pa iali y.
On he o he hand, as he le el o disc imina ion δinc eases, es o ing equali y e-
qui es mo e was ed in o ma ion: he ad iso mus lea n less abou g oup 1 and mo e
abou g oup 2, e en hough he poli ician becomes less esponsi e o in o ma ion abou
g oup 2. Hence, he poli ician’s decision inc easingly elies on he p io belie s—whe e
he e is no disag eemen be ween g oups— a he han he ad iso ’s in o ma ion. As he
amoun o “e ec i ely” u ilized in o ma ion diminishes wi h highe δ, he adjus men s
equi ed in he lea ning s a egy o es o e equali y become p og essi ely smalle .
In summa y, wo e ec s a e a play: ad iso s wi h highe pa iali y p espond mo e
s ongly o changes in δ, while he deg ee o adjus men in es s o achie e equali y
diminishes wi h highe δ. These dynamics hen lead o he non-mono onici y esul in
Lemma 4. Addi ionally, since es o ing equali y in ol es was ing mo e in o ma ion as δ
22
inc eases, i esul s in a wel a e loss. Indeed, wel a e W(ˆp(δ), δ) dec eases wi h δ.
As we ha e seen, wel a e is maximized wi h an impa ial ad iso , bu hen inequali y
inc eases wi h δ. Con e sely, wi h pa ial ad iso s who es o e equali y, wel a e hen de-
c eases wi h δ. Hence, wel a e maximiza ion and inequali y minimiza ion a e misaligned
objec i es, as cap u ed in he ollowing p oposi ion.
P oposi ion 4. Fo any le el o disc imina ion δ, a wel a e-inequali y Pa e o on ie
is o med by ad iso s ha ing pa iali y le els p∈[0,ˆp(δ)], whe e bo h wel a e and
inequali y s ic ly dec ease in p. Mo eo e , wel a e wi h he equali y- es o ing ad iso
ˆp(δ) is s ic ly highe han unde unchecked disc imina ion.
P oposi ion 4 illus a es a ade-o be ween wel a e and inequali y. Reducing inequali y
equi es was ing mo e in o ma ion, achie ed by ad iso s wi h highe le els o pa iali y,
while inc easing wel a e equi es using in o ma ion mo e e icien ly, achie ed by less
pa ial ad iso s. This dynamic gi es ise o he Pa e o on ie o med by ad iso s
anging om impa ial o hose es o ing equali y. Mo eo e , any ad iso along his
on ie unambiguously imp o es bo h wel a e and inequali y compa ed o unchecked
disc imina ion. No ably, e en an ad iso who es o es equali y esul s in highe wel a e
han does unchecked disc imina ion. This is because wel a e is conca e in es alloca ion,
and he lea ning s a egy o an unchecked poli ician is hea ily skewed owa ds lea ning
abou he needs o g oup 1, whe eas an “equalizing” ad iso lea ns mo e e enly abou
bo h g oups (e en hough his lea ning is skewed owa ds g oup 2).
7 Ex ensions
In his sec ion, we ou line wo ex ensions, p esen ed in mo e de ail in Online Appendix
B. The i s examines media pola iza ion by in oducing wo esea che s in o he model,
ep esen ing media ou le s compe ing o in luence a o e (decision-make ). Each ou le
and he o e seek implemen a ion o hei p e e ed policy mix on wo policy issues. In
he model, i s , he o e alloca es he a en ion be ween he wo ou le s. Second, he
ou le s simul aneously decide how much co e age o de o e o he wo policy issues. The
combined a en ion and co e age p o ide he o e wi h signals abou he wo policy
issues. Finally, he o e cas s he ballo . We p o ide condi ions unde which media
23
and (iii) co (˜
θk,˜
θj) = 0 o all k=jsince he a ibu es a e independen .
A.2.2 P oo o Theo em 2
Recall ha ψD(τ) = σ2,D
0−ˆσ2,D(τ) is he a iance o he decision-make ’s decision,
whe e ˆσ2,D(τ) is gi en by (8). Then, om Lemma 2 and d opping e ms ha a e
cons an in τ, he esea che ’s alue is gi en by
Vs (τ) = 2 co ˜
d∗(τ),˜
bR+ ˜σ2,D(τ) = 2 X
k
αD
kαR
kΣ0
kτkˆ
Σk(τk) + X
kαD
k2ˆ
Σk(τk)
=X
k2αD
kαR
kΣ0
kτk+αD
k2Σ0
k
1 + τkΣ0
k
subjec o he non-nega i i y and he budge cons ain s. The pa ial de i a i es o
Vs (τ) wi h espec o τk≥0 a e
∂V s (τ)
∂τk
=λkΣ0
k
1 + τkΣ0
k2
=λk1
Σ0
k
+τk−2
(30)
whe e we deno e λk:= αD
k2αR
k−αD
k o e e y k.
No e ha when λk≤0, he unc ion Vs (τ) is (weakly) dec easing in τk. Then he
solu ion dic a es τ∗
k= 0 o such k.21 Hence, he maximize s o unc ion Vs (τ) a e
he same as he maximize s o an adjus ed unc ion Vs ,adj(τ) := Vs (τ) whe e we se
αD
k=αR
k= 0 o all kwi h λk≤0 (and hence Vs ,adj(τ) is independen o τk o such
k).
Fo e e y k, deno e ˆαk=pmax{0, λk}and, w.l.o.g., le us elabel he a ibu es such
ha ˆα1Σ0
1≥. . . ≥ˆαKΣ0
K. By no ing ha (i) he unc ion Vs ,adj(τ) is conca e, (ii)
o e e y k, he pa ial de i a i e ∂V s ,adj (τ)
∂τkis he same as he pa ial de i a i e ∂V (τ)
∂τk
o a single-playe objec i e unc ion (21) in which we se αk= ˆαk o all k, and (iii)
he cons ain s a e he same o bo h ( he adjus ed s a egic and he single-playe )
maximiza ion p oblems, we ob ain he s a emen .
21When λk≤0 o all kand he e exis s a leas one jsuch ha λj= 0, we use he assump ion ha
in case o indi e ence he esea che abs ains om lea ning, hus yielding τ∗= (0,0,...,0). O he wise,
se ing τ∗
k= 0 whene e λk≤0 is uniquely op imal: (i) ei he he e exis s a leas one jwi h λj>0
(and hus he unc ion Vs (τ) is s ic ly inc easing in τj), o (ii) λk<0 o all k(and hus he unc ion
Vs (τ) is s ic ly dec easing in e e y τk).
30
A.3 P oo s o Sec ion 6.1: Di e si y in o ganiza ions
We i s s a e ou lemmas (p o ed in Online Appendix C.2) ha we use o p o e bo h
P oposi ions 2 and 3. Lemma 5 and Lemma 6 cha ac e ize he equilib ium lea ning
s a egy τ∗(β, γ) as a unc ion o γwhen β≤1/2 and β > 1/2, espec i ely. Lemma 7
es ablishes ha he decision-make ’s in e im expec ed payo is s ic ly inc easing and
conca e in (τ1, τ2). Lemma 8 de i es he esea che ’s equilib ium es alloca ion o
γ= 0.
Lemma 5. Le αD
1≥αD
2>0 and assume he analys ’s weigh ec o αRis gi en by
decomposi ion (18). Fo any β∈[0,1/2] he e exis unique γ1(β), γ2(β)∈Γ(β) such
ha γ1(β)≤0≤γ2(β) and:
τ∗(β, γ) =
(0, T) i γ > γ2(β),
(0,0) i γ∈[γ1(β), γ2(β)],
(T, 0) i γ < γ1(β).
(31)
Fu he , he e exis β1, β2∈Rwi h 0 < β2≤β1<1/2 such ha γ1(β)∈in (Γ(β)) i
and only i β∈(β1,1/2], and γ2(β)∈in (Γ(β)) i and only i β∈(β2,1/2].
Lemma 6. Fix he manage ’s weigh ec o αD= (αD
1, αD
2) wi h αD
k>0 o bo h
k= 1,2. Le he analys ’s weigh ec o αR(β, γ) be gi en by he decomposi ion (18).
Fix β > 1/2. Then he e exis γI(β), γII(β)∈in (Γ(β)) wi h γI(β)< γII(β) such ha
he analys ’s equilib ium es alloca ion is
τ∗(β, γ) =
(T, 0) i γ≤γI(β)
(˜τ1(β, γ),˜τ2(β, γ)) i γ∈(γI(β), γII(β))
(0, T) i γ≥γII(β)
(32)
whe e
˜τ1(β, γ) := ˆα1(β, γ)Σ0
1−ˆα2(β, γ)Σ0
2
Σ0
1Σ0
2(ˆα1(β, γ) + ˆα2(β, γ)) +ˆα1(β, γ)
ˆα1(β, γ) + ˆα2(β, γ)T
˜τ2(β, γ) :=T−˜τ1(β, γ)
(33)
31
and
ˆα1(β, γ) := max n(2β−1) αD
12−2γαD
1αD
2,0o
ˆα2(β, γ) := max n(2β−1) αD
22+ 2γαD
1αD
2,0o(34)
Mo eo e , ˜τ1(β, γ) (˜τ2(β, γ)) is s ic ly dec easing (s ic ly inc easing) in γ o γ∈
(γI(β), γII(β)).
Lemma 7. The decision-make ’s in e im expec ed equilib ium payo is a s ic ly in-
c easing and s ic ly conca e unc ion o (τ1, τ2).
Lemma 8. Fix he es budge T > 0 and he decision-make ’s weigh ec o αD∈R2
++.
Suppose γ= 0 ( he agen is no dis o ed).
(i) I β≤1/2 ( he agen is oo insensi i e), hen he agen op imally chooses no
es ing: τ∗(β, 0) = (0,0).
(ii) I β > 1/2 ( he agen is su icien ly sensi i e), hen he agen op imally chooses
he decision-make ’s mos p e e ed es alloca ion: τ∗(β, 0) = τ∗(1,0).
A.3.1 P oo o P oposi ion 2
Suppose he p emise o P oposi ion 2 holds. P oposi ion 2(i) is a di ec implica ion o
Lemma 5. The manage ’s in e im expec ed payo is s ic ly inc easing in τ1and τ2
by Lemma 7. Hence, he manage always s ic ly p e e s any posi i e es alloca ion
τ= (0,0) o abs aining om lea ning.
Now ake β > 1/2. Lemma 8 shows ha (gi en he cons ain τ∈ T) he man-
age ’s in e im expec ed payo VD(τ) is maximized when he agen ’s ela i e bias is
γ= 0. Fu he mo e, since VD(τ) is s ic ly inc easing in τ1and τ2, he cons ain
τ1+τ2≤Tmus be binding a he manage ’s mos p e e ed es alloca ion. Fu he -
mo e, Lemma 6 shows when β > 1/2, he agen ’s equilib ium es alloca ion τ∗(β, γ)
is such ha τ∗
1(β, γ) and τ∗
2(β, γ) a e weakly dec easing and weakly inc easing in γ,
espec i ely, and τ∗
1(β, γ) + τ∗
2(β, γ) = T. The claim o P oposi ion 2(ii) hen ollows
om he s ic conca i y o VD(τ) (shown in Lemma 7).
32
A.3.2 P oo o P oposi ion 3
Suppose γ= 0. Then, Lemma 8 shows ha he esea che ’s lea ning s a egy is o
no lea n o β≤1/2 and hen o implemen he decision-make ’s p e e ed lea ning
s a egy. As he decision-make ’s in e im expec ed payo VD(τ) is s ic ly inc easing
and conca e in τ1and τ2(Lemma 7), his implies ha VD(τ) is weakly inc easing in β
o γ= 0.
Suppose om his poin onwa ds ha γ > 0 (case γ < 0 is comple ely analogous). Then,
o he esea che ’s weigh s o be weakly posi i e, we mus ha e β≥β:= γαD
2
αD
1
. The
equilib ium es alloca ion is gi en by Theo ems 1 and 2, whe e we can exp ess ˆαk o
k= 1,2 as (34),22 and so
λ1= (2β−1)(αD
1)2−2γαD
1αD
2,
λ2= (2β−1)(αD
2)2+ 2γαD
1αD
2.
No e ha λk o k= 1,2 is s ic ly inc easing in β. The e o e, he e exis β1and β2such
ha ˆαk= 0 o β≤βk, and ˆαkis s ic ly posi i e and s ic ly inc easing o β > βk.
These alues a e gi en by he espec i e oo s o λk= 0:23
β1:= 1
2+γαD
2
αD
1
, β2:= 1
2−γαD
1
αD
2
.
Obse e ha β2<1/2< β1and β < β1. Theo ems 1 and 2 hen imply ha i β > β2,
hen ∃k:τ∗
k(β, γ)>0 and τ∗
1(β, γ) + τ∗
2(β, γ) = T(since a leas one o he auxilia y
weigh s ˆαkis hen s ic ly posi i e). The e o e, we can conclude ha i β > β2, hen
τ∗(β, γ) = (0, T) o all β∈[β, β1], and i β < β2, hen
τ∗(β, γ) =
(0,0) o β∈[β, β2],
(0, T) o β∈(β2, β1].
As VD(τ) is s ic ly inc easing in τ∗
2, i ollows ha VD(τ) is weakly inc easing in βin
ei he o he wo cases.
Conside now he case when β > β1, so we ha e ˆα1,ˆα2>0. Theo ems 1 and 2imply
22While Lemma 6 is only s a ed o β > 1/2, exp ession (34) is well-de ined o all β > β.
23No e ha hese βka e di e en om he ones de ined in he p oo o Lemma 5.
33
ha he equilib ium es alloca ion is gi en by
τ∗
1= max (0,min (ˆα1
ˆα2Σ0
1−Σ0
2
Σ0
1Σ0
2(ˆα1
ˆα2+ 1) +
ˆα1
ˆα2
ˆα1
ˆα2+ 1T, T)),
τ∗
2=T−τ∗
1.
(35)
No e ha he exp ession abo e (and, hence, VD(τ)) only depends on β h ough ˆα1
ˆα2.
Obse e ha lim
β→+∞
ˆα1
ˆα2=αD
1
αD
2
, so ha lim
β→+∞τ∗(γ, β) = τ∗(αD), so he equilib ium es
alloca ion con e ges o he decision-make ’s op imal (payo -maximizing) es alloca ion
as β→ ∞. Fu he , ˆα1
ˆα2is s ic ly inc easing in βin he case conside ed:
∂
∂β
ˆα1
ˆα2
=1
ˆα2
∂ˆα1
∂β −ˆα1
ˆα2
2
∂ˆα2
∂β =ˆα1
ˆα2 αD
1
ˆα12
−αD
2
ˆα22!>0,
whe e he inequali y ollows om ˆαk=√λkand
αD
1
αD
22
>ˆα1
ˆα22
=(2β−1)(αD
1)2−2γαD
1αD
2
(2β−1)(αD
2)2+ 2γαD
1αD
2
=(αD
1)2−2γ
2β−1αD
1αD
2
(αD
2)2+2γ
2β−1αD
1αD
2
.
Toge he wi h he limi esul abo e, his implies ha ˆα1
ˆα2and τ∗(γ, β)mono onically
con e ge o αD
1
αD
2
and τ∗(αD), espec i ely, as β→ ∞. As VD(τ) is conca e in τ(see
Lemma 7) and maximized by τ∗(αD), i mus be weakly inc easing in β∈(β1,+∞).
A.4 P oo s o Sec ion 6.2: Disc imina ion, wel a e and inequali y
A.4.1 P oo o Lemma 3
Fo he i s pa o he s a emen , obse e ha he maximiza ion p oblem o an impa -
ial ad iso is
max
τ∈T
1
2Ehu1˜
d∗(τ),˜
θ1+u2˜
d∗(τ),˜
θ2i.
By he de ini ion o wel a e, an impa ial ad iso hus chooses he wel a e-maximizing
es alloca ion in equilib ium, since he objec i es a e scaled e sions o each o he .
Le us u n o he second pa o lemma. Fix δ∈(0,1). Using Theo em 2, we can sol e
34
he disc imina ion model as a solu ion o a single-agen model wi h weigh s
ˆα1(p, δ) :=
1
2p(1 + δ)(1 −2p−δ) i p < 1−δ
2
0 o he wise
(36)
ˆα2(p, δ) :=
1
2p(1 −δ)(1 + 2p+δ) i p > −1+δ
2
0 o he wise
(37)
Le p= 0. Then ˆα1(0, δ) = ˆα2(0, δ) = 1
2√1−δ2. The equilib ium es alloca ion
depends on he weigh s only h ough hei a ios ˆα1(0,δ)
ˆα1(0,δ)+ˆα2(0,δ)=1
2and ˆα2(0,δ)
ˆα1(0,δ)+ˆα2(0,δ)=
1
2. These a ios a e independen o he le el o disc imina ion δ, which comple es he
p oo .
A.4.2 P oo o Lemma 4
We i s de ine se e al new no ions. Gi en he poli ician’s disc imina ion δ∈(0,1) and
he ad iso ’s pa iali y p≥0, le τ∗(p, δ)=(τ∗
1(p, δ), τ∗
2(p, δ)) deno e he espec i e
equilib ium es alloca ion. No e i holds τ∗
1(p, δ) + τ∗
2(p, δ) = 1, i.e., he budge is
always exhaus ed in equilib ium.24
Gi en a pa icula es alloca ion τand equilib ium decision s a egy, le Vk(τ, δ) deno e
he ex an e expec ed u ili y o g oup k, i.e.,
Vk(τ, δ) = Eh−(˜
d∗D(τ;δ)−˜
θk)2i.
Unde he addi ional cons ain ha he budge is exhaus ed, and so he es alloca ion
sa is ies τ1= 1 −τ2, le us de ine he di e ence be ween he expec ed u ili ies o g oup
1 om g oup 2 as
∆(τ2, δ) := V1((1 −τ2, τ2), δ)−V2((1 −τ2, τ2), δ).(38)
No e ha inequali y is hen gi en by I(p, δ) = |∆(τ∗
2(p, δ), δ)|.
24This esul ollows om he assump ion ha he sums o he weigh s o each playe a e equal:
PkαR
k(p) = PkαD
k(δ). Thus, i canno simul aneously hold αR
k(p)≤αD
k o bo h k(which is necessa y
and su icien o ˆα1(p, δ) = ˆα2(p, δ) = 0), and he e is hus always lea ning in equilib ium.
35
We now s a e wo in e media y esul s (p o ed in Online Appendix C.3), used o p o e
Lemma 4.
Lemma 9. Fo each δ∈(0,1), he e exis s a h eshold pa iali y le el paux(δ)∈0,1−δ
2
such ha he equilib ium lea ning s a egy sa is ies τ∗
2(p, δ) = 1 o ad iso s wi h p≥
paux(δ) and
τ∗
2(p, δ) = τaux
2(p, δ) := 2ˆα2(p, δ)−ˆα1(p, δ)
ˆα1(p, δ) + ˆα2(p, δ) o ad iso s wi h p∈[0, paux(δ)],
whe e ˆα1(p, δ) and ˆα2(p, δ) a e gi en by equa ions (36) and (37).
Lemma 10. Fo each δ he e exis s a unique alue o ˆτ2(δ)∈(1/2,1) o which i holds
∆(ˆτ2(δ); δ) = 0.
Gi en he poli ician’s le el o disc imina ion δand he equilib ium decision s a egy, he
alue ˆτ2(δ) is he amoun o es s alloca ed o lea n abou g oup 2 ha gua an ees ha
he expec ed u ili ies o bo h social g oups a e he same. F om he p oo o Lemma 10,
we ob ain
∆(τ2, δ) = (1 + δ)(1 −τ2)
1+1−τ2−(1 −δ)τ2
1 + τ2
.
Sol ing ∆(ˆτ2(δ), δ) = 0 o ˆτ2(δ)≥0, we ob ain
ˆτ2(δ) = −(1 −δ) + √1+3δ2
2δ.(39)
F om Lemma 9, he equilib ium es alloca ion sa is ies: τ∗
2(p, δ)=1/2 a p= 0,
τ∗
2(p, δ) = 1 a p≥paux(δ), τ∗
2(p, δ) is con inuous on p∈[0,1] and i is s ic ly in-
c easing in p∈[0, paux(δ)). Since ˆτ2(δ)∈(1/2,1), we hus ha e ha o any le el o
disc imina ion δ∈(0,1) he e exis s a unique pa iali y le el ˆp(δ)∈(0, paux(δ)) such
ha ˆτ2(δ) = τ∗
2(ˆp(δ), δ). Sol ing his equa ion, we ob ain
ˆp(δ) = δ1
√1+3δ2−1
2,
36
which is con inuous on δ∈(0,1). Taking he i s de i a i e, we ge
ˆp′(δ) = 1
(1 + 3δ2)3/2−1
2
>0 i δ < δ
= 0 i δ=δ
<0 i δ > δ
whe e δ:= 1
31−21/2+ 22/3.
A.4.3 P oo o P oposi ion 4
The i s pa o he s a emen ollows di ec ly om he ollowing Lemma 11, which is
p o ed in Online Appendix C.3.
Lemma 11. Wel a e W(p, δ) and inequali y I(p, δ) a e bo h (i) con inuous in p; and (ii)
s ic ly dec easing on p∈(0,ˆp(δ)), whe e ˆp(δ) es o es equali y. Fu he mo e, wel a e
s ic ly dec eases and inequali y s ic ly inc eases on p∈(ˆp(δ), paux(δ)), whe e paux(δ)
is cha ac e ized in Lemma 9; and hey a e bo h cons an on p≥paux(δ).
Hence, he ad iso wi h ˆp(δ) s ic ly imp o es bo h wel a e and inequali y ou comes
compa ed o any ad iso wi h p > ˆp(δ), and hence he ad iso s wi h p > ˆp(δ) do no
cons i u e he Pa e o on ie .
Fo he second pa o he s a emen , we build on he ollowing Lemma 12, which is
p o ed in Online Appendix C.3. Fo a gi en δ, le (1 −¯τ2(δ),¯τ2(δ)) deno e he op imal
lea ning s a egy o a poli ician in he unchecked disc imina ion, and le (1−ˆτ2(δ),ˆτ2(δ)),
whe e ˆτ2(δ) is gi en by equa ion (39), deno e he equilib ium es alloca ion chosen by
he ad iso ˆp(δ).
Lemma 12. Fix δ. Then i holds
|τ2(δ)−1/2|>|ˆτ2(δ)−1/2|.
37
Le
ω(τ2, δ) := V1((1 −τ2, τ2), δ) + V2((1 −τ2, τ2), δ)
=−3−δ2+1/2(1 + δ2) + (1 + δ)(1 −τ2)
2−τ2
+1/2(1 + δ2) + (1 −δ)τ2
1 + τ2
deno e he sum o he expec ed u ili ies o he wo g oups as a unc ion o es alloca ion
τ= (τ1, τ2) unde he cons ain ha he budge is ully used, τ1= 1 −τ2, and whe e
Vk(τ, δ) is gi en by equa ion (C.3.2). Since he budge is exhaus ed in equilib ium (as
shown in he p oo o Lemma 4), wel a e is hus gi en by W(p, δ) = ω(τ∗
2(p, δ), δ).
Obse e he ollowing symme y p ope y: ω(τ2, δ) = ω(1 −τ2, δ). Mo eo e , o any
δ∈(0,1), he unc ion ω(τ2, δ) is s ic ly conca e and maximized a τ2= 1/2, when he
es budge is spli equally be ween he wo g oups. The e o e, he unc ion ω(τ2, δ) is
s ic ly dec easing in |τ2−1/2|. Lemma 12 hen implies ha wel a e wi h an ad iso
ˆp(δ) is s ic ly highe han wel a e unde unchecked disc imina ion.
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39
We i s cha ac e ize ˜
θk(s), he o e ’s pos e io belie abou θkgi en some signal ealiza-
ions sA
k, sB
k. I is immedia e o e i y di ec ly om no mal p.d. .s ha i ˜
X∼ N(µ, σ2
X),
˜ε∼ N(0, σ2
ε) and Y=X+ε, hen ˜
X|Y∼ N ˆµ, ˆσ2, whe e
ˆµ=
1
σ2
X
1
σ2
X
+1
σ2
ε
µ+
1
σ2
ε
1
σ2
X
+1
σ2
ε
Y, ˆσ2=1
1
σ2
X
+1
σ2
ε
.
This di ec ly implies cha ac e iza ion (4)–(6) in Sec ion 2.2. In he con ex o his p oo ,
his gi es i s ha ˜
θk|sA
k∼ N(ˆµk,A,ˆ
Σk,A) wi h
ˆµk,A =
1
Σ0
k
1
Σ0
k
+ AqA
k
µ0
k+ AqA
k
1
Σ0
k
+ AqA
k
sA
k,ˆ
Σk,A =1
1
Σ0
k
+ AqA
k
,
and hen, subsequen ly, ha (˜
θk|sA
k)|sB
k∼ N(ˆµk,ˆ
Σk), whe e
ˆµk=
1
ˆ
Σk,A
1
ˆ
Σk,A + BqB
k
ˆµk,A + BqB
k
1
ˆ
Σk,A + BqB
k
sB
k=
1
Σ0
k
µ0
k+ AqA
ksA
k+ BqB
ksB
k
1
Σ0
k
+ AqA
k+ BqB
k
=
1
Σ0
k
1
Σ0
k
+τk
µ0
k+1
1
Σ0
k
+τk AqA
ksA
k+ BqB
ksB
k,(B.1.2)
ˆ
Σk=1
1
ˆ
Σk,A + BqB
k
=1
1
Σ0
k
+ AqA
k+ BqB
k
=1
1
Σ0
k
+τk
.(B.1.3)
Compa ing (B.1.3) wi h i s coun e pa (6) in he baseline model, we no e ha hey
coincide. Nex , compa e (B.1.2) wi h i s coun e pa (5) in he baseline model. In o de
o conclude ha he dis ibu ion o pos e io s is he same as in he baseline model, we
need o show ha he andom a iable de ined as
˜s
k:= 1
τk AqA
k˜sA
k+ BqB
k˜sB
k=˜
θk+1
τk AqA
k˜εA
k+ BqB
k˜εB
k
has he same dis ibu ion as ˜sk om he baseline model. No e ha ˜s
k|θk∼ N θk,1
τk
since E[˜εm
k] = 0 and a mqm
k
τk˜εm
k=1
τk, and hen ˜s
k∼ N µ0
k,Σ0
k+1
τk. Fo a
gi en τ, hese coincide wi h he dis ibu ions o ˜sk|θkand ˜sk om he baseline model,
espec i ely. The e o e, he dis ibu ion o ˆµkabo e coincides wi h ha o (4), so he
dis ibu ion o ˜
θk(s
k) in he media model coincides wi h he dis ibu ion o ˜
θk(sk) in he
baseline model when s
k=sk. Toge he , he wo la e ac s imply ha he dis ibu ion
o ˜
θk(˜s
k) in he media model coincides wi h he dis ibu ion o ˜
θk(˜sk) in he baseline
6
model.
Ou nex lemma below show ha all playe s’ espec i e ex an e expec ed u ili ies only
depend on ( , qA, qB) h ough τ=τ( , qA, qB). Speci ically, de ine playe i’s alue unc-
ion as
Vi(τ) := E"−
2
X
k=1
αi
kd∗(˜s , τ)−˜
θk(˜s , τ)2#(B.1.4)
o i=A, B, V , whe e d∗(˜s , τ) is he o e ’s equilib ium decision s a egy gi en τ=
τ( , qA, qB), and s = (s
1, s
2) is a ec o o ic i ious “agg ega e” signals in oduced in
he p oo o Lemma 13.
Lemma 14. In he media model, gi en ( , qA, qB), he expec a ions o all playe s’ u il-
i ies (B.1.1) a e gi en by alues (B.1.4) and only depend on τ , qA, qB:
Ehuid∗˜s, , qA, qB,˜
θ˜s, , qA, qBi=Viτ , qA, qB
P oo . Rew i ing he expec ed u ili y ( he le -hand side o he equali y in he lemma)
using de ini ion (B.1.1), we ge
E"−
2
X
k=1
αi
kd∗˜s, , qA, qB−˜
θk˜s, , qA, qB2#.(B.1.5)
The o e ’s equilib ium decision o any signal ealiza ion sis gi en by
d∗s, , qA, qB=
2
X
k=1
αV
kEh˜
θks, , qA, qBi.
By Lemma 13 we know ha ˜
θk˜s, , qA, qB∼˜
θk˜s , τ( he wo ha e he same dis-
ibu ions), whe e ˜s = (˜s
1,˜s
2) wi h ˜s
k=1
τk AqA
k˜sA
k+ BqB
k˜sB
k. This implies ha
d∗˜s, , qA, qB∼d∗(˜s , τ). Then we can w i e (B.1.5) as
E"−
2
X
k=1
αi
kd∗(˜s , τ)−˜
θk˜s , τ2#,
which is exac ly (B.1.4), he de ini ion o Vi(τ).
We now make a s a emen abou he media model equilib ia in cases no co e ed in
7
P oposi ion B.1.1, and p o e bo h s a emen s simul aneously.
P oposi ion B.1.2. In each o he cases lis ed below, all equilib ia a e payo -equi alen
o he ollowing:
1. i αA
1> αV
1> αB
1and Tis no la ge enough: bo h media ou le s choose he same
ex eme co e age, ei he qA=qB= (1,0), o qA=qB= (0,1), and he o e
achie es he bes in o ma ion;
2. i αV
1≥αA
1: he o e only ollows ou le A, = (T, 0), and ou le Aachie es i s
bes in o ma ion;32
3. i αB
1≥αV
1: he o e only ollows ou le B, = (0, T), and ou le Bachie es i s
bes in o ma ion.
P oo o P oposi ions B.1.1 and B.1.2. Lemmas 13 and 14 imply ha he media se ing
wi h ou signals s= (sA
1, sA
2, sB
1, sB
2) can, wi hou loss, be eplaced by he baseline
model wi h ic i ious agg ega e signals s = (s
1, s
2), which ha e he same dis ibu ion
condi ional on agg ega e a en ion alloca ion τ , qA, qBas he signals in ou main
model o Sec ion 2. Fo b e i y, we o en d op he a gumen s o he agg ega e a en ion
alloca ion and w i e i as simply τ.
P oceeding by backwa ds induc ion, we i s no e ha he o e ’s equilib ium decision
d∗s , τis analogous o he single-playe and s a egic e sions o ou model, see he
p oo o Lemma 14. In he emainde o his p oo , we assume he o e ollows his
equilib ium decision s a egy d∗s , τ.
P oceeding u he , Lemma 14 implies ha he ou le s’ p oblem o choosing co e age
qmand he o e ’s p oblem o choosing a en ion alloca ion educe o he p oblem
o maximizing alue unc ion (B.1.4) o e τ , qA, qB. Fix some and conside he
ou le s’ p oblem. Le TV( ) := 1+ 2deno e he o al amoun o a en ion de o ed
o he wo ou le s by he o e . Le τi∗( ) := a g maxτ∈R2
+Vi(τ) s. . τ1+τ2≤TV( )
o all i=A, B, V . No e ha playe i’s bes in o ma ion, by de ini ion, is gi en by
max ∈R τi∗( ). Fo bo h m=A, B,τm∗( ) is hen desc ibed by Lemma 2 and Theo em
2 wi h T=TV( ). By Lemma 2, i m’s expec ed payo (12) is weakly inc easing in
32We de ine ou le m’s bes in o ma ion by analogy wi h he o e ’s bes in o ma ion in he ex , i.e.,
( , qA, qB) mus join ly maximize m’s expec ed payo gi en he o e ’s equilib ium decision s a egy.
8
τk, i is also s ic ly conca e in τk.33 F om Theo ems 1 and 2, we ob ain a closed- o m
solu ion o τi∗( ):
τi
1∗( ) = max 0,min ˆαi
1Σ0
1−ˆαi
2Σ0
2
Σ0
1Σ0
2(ˆαi
1+ ˆαi
2)+ˆαi
1
ˆαi
1+ ˆαi
2
TV( ), TV( ),
τi
2∗( ) = TV( )−τi
1∗( ),
(B.1.6)
whe e ˆαi
k:= qmax 0,(αi
k)2−(αi
k−αV
k)2 o all k= 1,2 and i=A, B, V (no e
ha ˆαV
k=αV
k, so o i=V, (B.1.6) coincides wi h he solu ion in Theo em 1). No e
ha τi
1∗( ) is weakly inc easing in αi
1(subjec o he αi
1+αi
2= 1 cons ain ). This
cha ac e iza ion implies ha since αi
1+αi
2= 1 o all i=A, B, V , he auxilia y weigh s
ˆαm
kcanno simul aneously be ze o o bo h k= 1,2 o any m=A, B, so τm
1∗( ) +
τm
2∗( ) = TV( ) o all ( he media wan no a en ion was ed). We no e ha subjec o
his cons ain , alue Vi(τ1, TV( )−τ1) is ei he s ic ly mono one, o s ic ly conca e
in τ1 o all i=A, B, V .
The cha ac e iza ion in Theo em 2 also implies ha gi en , i ˆαm
1Σ0
1>ˆαm
2Σ0
2and
TV( )≤ˆαm
1
ˆαm
2Σ0
2−1
Σ0
1
(o o all TV( ) i ˆα2= 0), hen τm∗( ) = (TV( ),0), and ice
e sa: i ˆαm
2Σ0
2>ˆαm
1Σ0
1and TV( )≤ˆαm
2
ˆαm
1Σ0
1−1
Σ0
2
(o o all TV( ) i ˆα1= 0), hen
τm∗( ) = (0, TV( )). I none o hese condi ions apply, hen τm∗( ) is in e io (and
s ic ly mono one in αm
1subjec o he αm
1+αm
2= 1 cons ain ). The e o e, i we le
ˆ
T:= max 0,min
m∈{A,B}ˆαm
1
ˆαm
2Σ0
2−1
Σ0
1,min
m∈{A,B}ˆαm
2
ˆαm
1Σ0
1−1
Σ0
2,
hen whene e TV( )≤ˆ
T, bo h media ou le s op imally p e e he same ex eme co e -
age: τA
1∗( ) = τB
1∗( )∈ {0, TV( )}. No e ha i ˆαA
1Σ0
1>ˆαA
2Σ0
2and ˆαB
1Σ0
1<ˆαB
2Σ0
2, hen
ˆ
T= 0, and such a p e e ence p o ile (τA
1∗( ) = τB
1∗( )∈ {0, TV( )}) canno a ise. Fu -
he , i TV( )>ˆ
T, hen τm∗( ) is in e io o a leas one ou le m, and since αA
1> αB
1
(and αA
2< αB
2), we hen ha e τA
1∗( )> τB
1∗( ).
Nex , we claim ha i τ=τm∗( ) o some ou le min equilib ium, hen mis ei he
pola ized (i.e., qm∈ {((1,0),(0,1)}), o igno ed ( m= 0). To see his, p oceed by
con adic ion and conside a s a egy p o ile , qA, qBsuch ha τk , qA, qB< τm
k∗( ),
qm
k<1, and m>0 o some k= 1,2 and m=A, B. Then inc easing qm
k(while
33This is mos e iden om (30) in he p oo o Theo em 2: one can eadily see by analogy ha bo h
mono onici y and conca i y in his p oblem depend on whe he λi
k:= αV
k2αm
k−αV
k≷0.
9
dec easing he o he weigh qm
−k) b ings he agg ega e es alloca ion τclose o m’s
op imum τm∗( ), which s ic ly inc eases m’s expec ed payo (12).34 This is a p o i able
de ia ion o ou le m, and he o iginal s a egy p o ile hus canno be an equilib ium.
The same is ue o he case when τk> τm
k∗( ) and qm
k>0 and m>0 o some k= 1,2,
m=A, B.
The logic abo e also implies ha as αA
1> αB
1, he e is no equilib ium wi h τsuch
ha τ1> τA
1∗( ) o τB
1∗( )> τ1, since in he o me case bo h ou le s mwan o
dec ease qm
1and inc ease qm
2(and a leas one has he scope o such a de ia ion), and
in he la e hey wan o do he opposi e. The e o e, in equilib ium, he agg ega e
a en ion alloca ion τ=τ ∗, qA∗, qB∗mus be such ha τ1∈τB
1∗( ∗), τA
1∗( ∗)and
τ2=TV( ∗)−τ1∈τA
2∗( ∗), τB
2∗( ∗).
I emains o conside he o e ’s a en ion alloca ion p oblem. Since τm∗( ) only depend
on h ough TV( ), he o e ’s p oblem can be spli in o i s choosing he o al amoun
o a en ion o de o e o media, TV, and hen choosing how o spli i by way o choosing
τsuch ha τ1∈τB
1∗(TV), τA
1∗(TV)and τ2=TV−τ1. No e ha in he second s ep,
all agg ega e alloca ions in his in e al a e a ailable o he o e o a gi en TV. By
se ing m=TV o some mand −m= 0, she can induce τ=τm∗(TV). Any alloca ion
τin he in e io o he in e al can be achie ed by se ing A=τ1and B=τ2, in which
case he unique bes esponse o ou le Ais qA= (1,0), and he unique bes esponse
o ou le Bis qB= (0,1), as a gued abo e. We shall hus e e o τ ha sa is y hese
equi emen s as easible gi en some TV.
The o e ne e wan s any a en ion was ed, since he payo is s ic ly inc easing in τk
o bo h k. F om (B.1.6) we can see ha d
dTVτm
k∗(TV)∈[0,1] o all m, k, hence o any
TV, any τa ha is easible gi en TV, and any ε > 0, he e exis s τb ha is easible gi en
TV+εand is such ha τb
k≥τa
k o bo h k, wi h a leas one inequali y being s ic .35
The e o e, inc easing TValways o e s a s ic imp o emen he o e , so in equilib ium,
TV( ∗) = T.
We now mo e on o he second s age o he o e ’s p oblem. I T≤ˆ
T, hen τA
1∗(T) =
34 “Close ” he e is used in he sense o he new agg ega e a en ion alloca ion ¯τbeing a con ex
combina ion o he o iginal τand he op imal τm∗( ). The payo unc ion being conca e on he ele an
in e al, and τm∗( ) being i s maximize hen di ec ly imply ha ¯τyields a highe payo han he
o iginal τ.
35A way o see his is o no ice ha an inc ease in TVimplies an inc ease in he s ong se o de o
he in e als o easible in τ1and τ2.
10
τB
1∗(T)∈ {0,1}, so he se o a ailable τis a single on, and he o e is hen indi e en
be ween all a en ion alloca ions. Suppose now ha T > ˆ
T. I αA
1> αV
1> αB
1, hen
τV
1∗(T)∈τB
1∗(T), τA
1∗(T), so he o e can a ain he op imum τV∗(T) by ollowing
he s a egy desc ibed abo e. I is immedia e om (B.1.6) ha in his case he e exis s
ˆ
TV≥ˆ
Tsuch ha τV∗(T) is in e io i and only i T > ˆ
TV. Fo T > ˆ
TV,τV∗(T) is
uniquely op imal o he o e , hence = (τV
1∗(T), τV
2∗(T)), qA= (1,0), and qB= (0,1)
is he unique equilib ium, p o ing P oposi ion B.1.1. Fo T≤ˆ
TV, as a gued abo e, he
o e can s ill achie e τV∗(T), bu he equilib ium may no be unique. This p o es case
1 o P oposi ion B.1.2. In case αV
1/∈αA
1, αB
1, since he o e ’s alue unc ion (11) is
s ic ly conca e in τ, she will op imally choose he a ailable alloca ion closes (in he
sense o Foo no e 34) o τV∗(T). In pa icula , i αB
1≥αV
1, hen τV
1∗(T)≤τB
1∗(T),
so he o e will choose = (0, T), esul ing in τ=τ∗
B(T). And i αV
1≥αA
1, hen
τV
1∗(T)≥τA
1∗(T), and he o e will choose = (T, 0), esul ing in τ=τ∗
A(T). This
p o es pa s 2 and 3 o P oposi ion B.1.2 and comple es his p oo .
Finally, we s a e he ollowing gene al co olla y be o e p o ing i along wi h Co olla y
B.1.1.
Co olla y B.1.2. The o e weakly p e e s he equilib ium when bo h media ou le s
m=A, B a e a ailable o he equilib ium when only one ou le m=A, B is p esen in
he ma ke .
P oo o Co olla ies B.1.1 and B.1.2. In he monopoly case, i only ou le mis a ailable,
hen he o e always lends i he ull a en ion, m=T, since he alue (B.1.4) educes
o (11), which is s ic ly inc easing and s ic ly conca e in τ. The e o e, mcan choose
agg ega e a en ion alloca ion τ eely subjec o τ1+τ2=T. Since αi
1+αi
2= 1 o
all i, Theo em 2 implies ha mdoes no wan o was e any o he o e ’s a en ion.
The unique equilib ium agg ega e a en ion alloca ion is, he e o e, τm∗(T), as de ined
in he p oo o P oposi ions B.1.1 and B.1.2, which is m’s bes in o ma ion.
P oposi ions B.1.1 and B.1.2 abo e shows ha unde media duopoly, he equilib ium
agg ega e a en ion alloca ion τ∗is gi en by τV∗(T) i αA
1> αV
1> αB
1, by τA∗(T)
i αA
1≤αV
1, and by τB∗(T) i αV
1≤αB
1. No e ha in he wo la e cases, i is
he o e ’s p e e ed media ou le ha has he monopoly powe , which p o es Co olla y
B.1.2 o hose cases. In he o me case, τV∗(T) being i s maximize o he o e ’s alue
11
unc ion imply ha he o e p e e s duopoly, concluding he p oo o Co olla y B.1.2.
I τV∗(T)∈τB∗(T), τA∗(T), which is he case i and only i T > ˆ
TV, as de ined in he
p oo o P oposi ions B.1.1 and B.1.2, hen i is also he unique maximize (since he
alue unc ion is s ic ly conca e on ha in e al in ha case), which p o es Co olla y
B.1.1.
B.2 Changing P e e ences
In his second ex ension, we s udy how a (po en ial) change in an agen ’s p e e ences
be ween lea ning and decision-making a ec s his lea ning s a egy. Following he li e -
a u e on ime-inconsis en p e e ences (e.g., O’Donoghue and Rabin, 1999), we conside
sophis ica ed and nai e agen s and in es iga e he impac o such a change in p e e ences
on he lea ning s a egy and on he agen ’s wel a e.
We model his simila ly o ou baseline model by conside ing a single agen acing a
wo-s age p oblem. In he i s s age, he agen acqui es in o ma ion abou unknown
a ibu es. In he second s age, he agen akes a decision based on he acqui ed in o -
ma ion. The agen ’s p e e ence may change be ween he wo s ages. Fo ins ance, his
change may occu due o a change in he agen ’s ci cums ances, such as losing he job o
alling ill. Al e na i ely, he change in p e e ences may be due o he agen ’s sel -con ol
p oblems, such as succumbing o emp a ion.
In he i s s age, when acqui ing in o ma ion, he agen has u ili y unc ion
uR(d, θ) = −d−αR
1θ1−αR
2θ22,(B.2.1)
whe e we assume αR
k>0 o k= 1,2. A e wa ds, he agen ’s p e e ence may change,
which we model as a change in he weigh o a ibu e ˜
θ2occu ing wi h (ex an e)
p obabili y p∈(0,1). In he second s age, when making he decision, he agen has
u ili y unc ion
uD(d, θ) =
−d−αR
1θ1−αR
2θ22wi h p obabili y 1 −p,
−d−αR
1θ1−cαR
2θ22wi h p obabili y p,
12
whe e c > 0 and c= 1.36 The nai e agen belie es he p e e ences will s ay he
same ac oss bo h s ages, while he sophis ica ed agen unde s ands hey migh change.
Th oughou his sec ion, we simpli y he analysis by assuming µ0= (0,0), so ha , ab-
sen any new in o ma ion, he esea che and bo h ypes o decision-make s ag ee on
he op imal decision d∗= 0. Addi ionally, we ix Σ0
1= Σ0
2=T= 1.
As he nai e agen inco ec ly assumes no possibili y o change, he nai ’s lea ning
p oblem coincides wi h ha o a single playe in Sec ion 3, and he equilib ium es
alloca ion ollows om Theo em 1 wi h weigh s αR. In con as , o he sophis ica e, a
s a egic game simila o ha in Sec ion 4 ensues. Howe e , Theo em 2 does no apply
di ec ly, as he decision-make ’s p e e ences a e s ochas ic. By app op ia ely adap ing
Lemma 2 and hen applying he same logic as in he p oo o Theo em 2, we ind ha he
sophis ica e’s equilib ium es alloca ion can ne e heless be exp essed as he solu ion
o a single-playe p oblem wi h auxilia y weigh s37
(ˆα1,ˆα2) := αR
1, αR
2pmax{1−p+p(c(2 −c)),0}.
Thus, he po en ial change in p e e ences educes he weigh pu on a ibu e ˜
θ2, as he
sophis ica e an icipa es he misalignmen be ween his cu en and u u e sel . No ably,
he nex esul shows ha his educ ion may e en induce he sophis ica e o engage in
“s a egic igno ance” (see, e.g., Ca illo and Ma io i, 2000).
P oposi ion B.2.1. The sophis ica e a oids lea ning abou a ibu e ˜
θ2 o any T > 0
i and only i √p(c−1) ≥1.
In he p esence o a po en ial shi in p e e ences, ep esen ed by c, he sophis ica e may
choose s a egic igno ance ega ding a ibu e ˜
θ2. I he change educes he weigh on ˜
θ2
o does no inc ease i oo much, he agen s ill lea ns abou i , bu wi h less in ensi y
(ˆα2< αR
2). In con as , he sophis ica e a oids lea ning abou a ibu e ˜
θ2i he change
is su icien ly high (c > 2) and he p obabili y o change is no oo small. In sho , when
he sophis ica e expec s he u u e sel o o e eac o in o ma ion abou ˜
θ2, she may
op o s a egic igno ance.
36This se ing ex ends he baseline model by allowing o unce ain y abou he decision-make ’s
p e e ences. An al e na i e in e p e a ion is ha he e a e mul iple po en ial decision-make s, and he
esea che is unce ain abou who will make he decision. We ex end ou equilib ium concep o his
se ing in Sec ion B.2.1.
37The o mal s eps a e con ained in he p oo o P oposi ion B.2.1.
13
Ha ing es ablished he di e ences be ween he nai and he sophis ica e’s equilib ium
lea ning s a egies, we wan o unde s and he implica ions o he agen ’s wel a e. In
his con ex , he choice o he app op ia e wel a e c i e ion is unclea . Fo ins ance, i
he change in p e e ences is igge ed by succumbing o emp a ion, he ini ial u ili y
unc ion uRseems mos app op ia e. Howe e , i he change in p e e ences is due o
al e ed ci cums ances, such as illness o pa en hood, hen he changed u ili y unc ion
uDseems mo e app op ia e. We emain agnos ic abou he wel a e c i e ion and conside
bo h al e na i es.
P oposi ion B.2.2. I αR
1/∈(ˆα2/2,2αR
2), he expec ed payo o he nai and he
sophis ica e coincide o bo h wel a e c i e ia. O he wise, i he wel a e c i e ion is he
ini ial u ili y, uR, he wel a e o he sophis ica e exceeds ha o he nai . I he wel a e
c i e ion is he changed u ili y, uD, he nai ’s wel a e exceeds ha o he sophis ica e
o c > 1 and ice e sa o c < 1.
The in e es ing case a ises when a po en ial change in p e e ences in luences equilib-
ium lea ning s a egies. As shown in he appendix, he sophis ica e alloca es (a leas
weakly) mo e es s o a ibu e ˜
θ1 han he nai . An icipa ing a shi in p e e ences,
he sophis ica e educes emphasis on a ibu e ˜
θ2 o a oid misalignmen wi h he u u e
sel ’s choices. When ini ial u ili y is used as he wel a e c i e ion, he sophis ica e’s
expec ed payo is highe because hey op imize lea ning, awa e o possible p e e ence
changes, while he nai makes subop imal choices. Howe e , when he wel a e c i e ion
is he changed u ili y, he nai can ou pe o m he sophis ica e. I c > 1, he nai lea ns
mo e abou ˜
θ2, bene i ing whe he p e e ences change o no : I a change occu s, he
nai has lea ned mo e abou he now mo e impo an a ibu e ˜
θ2 han he sophis i-
ca e (in ex eme case, he sophis ica e engages in s a egic igno ance and lea ns no hing
abou ˜
θ2); i no change occu s, he nai has implemen ed he op imal es alloca ion due
o he nai e e. In con as , he sophis ica e’s “hedging” lea ning s a egy agains he
po en ial change, which unde weighs ˜
θ2, is less e ec i e. Howe e , his is e e sed when
c < 1. Then, he nai ’s lea ning s a egy is s ill op imal when no change occu s bu
(subs an ially) wo se when a change occu s.
The esul in P oposi ion B.2.2 has a pa allel o O’Donoghue and Rabin’s (1999) ind-
ings in hei “doing i now o la e ” amewo k. They ind ha in case o immedia e
cos s he sophis ica e is always be e o han he nai (as in ou case wi h he ini ial
14
u ili y). Con e sely, in case o immedia e g a i ica ion, ei he ype o he agen can be
be e o (as in ou case wi h changed u ili y). In bo h, O’Donoghue and Rabin (1999)
and ou model, he sophis ica e may engage in a o m o o e compensa ion. In ou case
i happens by pu ing less weigh on a ibu e ˜
θ2 o hedge agains p e e ence changes,
which leads o a wo se ou come i espec i e o whe he p e e ences ac ually change.
Summa izing, bo h models highligh ha nai e s a egies can some imes lead o be e
ou comes i u u e p e e ences o ci cums ances alida e hose s a egies. This unde -
sco es he complexi y o modeling changing p e e ences and how di e en assump ions
abou u u e beha io o he eason o change can impac wel a e.
B.2.1 P oo s o he ex ension on changing p e e ences
Equilib ium concep . A weak Pe ec Bayesian Equilib ium o he game wi h po-
en ially changing p e e ences (he eina e e e ed o simply as “equilib ium”) is a uple
(τ∗, d∗
c(s, τ), d∗
u(s, τ),˜
θ(s, τ)) such ha :
1. he esea che ’s es alloca ion s a egy τ∗∈ T maximizes his expec ed payo
gi en he decision-make ’s s a egies d∗
c(s, τ) and d∗
u(s, τ);
2. he decision-make ’s decision s a egies d∗
c(s, τ) : R2×T → Rand d∗
u(s, τ) : R2×
T → Rmaximize he expec ed payo gi en changed and unchanged p e e ences,
espec i ely, and he pos e io belie s ˜
θ(s, τ);
3. he decision-make ’s pos e io belie s ˜
θ(s, τ) : R2× T → ∆(R2) a e ob ained ia
Bayes’ ule gi en he signal ealiza ions sand he esea che ’s choice τ.
P oo o P oposi ion B.2.1. Obse e ha he decision-make ’s equilib ium decision is
gi en by
d∗
u(s, τ) = αR
1ˆµ1(s, τ) + αR
2ˆµ2(s, τ) wi h p obabili y 1 −p,
d∗
c(s, τ) = αR
1ˆµ1(s, τ) + cαR
2ˆµ2(s, τ) wi h p obabili y p.
Thus, o any es alloca ion τ, he unce ain y abou he decision-make ’s p e e ences
ansla es acco dingly in o unce ain y abou he decision-make ’s decision. Hence, we
15
p oo o Lemma 5, i holds ha αR
1(β, γ)≤1/2αD
1i and only i γ≥γ1(β); and
αD
2(β, γ)≤1/2αD
2i and only i γ≤γ2(β). By Theo em 2, he analys ’s equilib ium
es alloca ion sa is ies
τ∗(β, γ) = (τ∗
1(β, γ), τ∗
2(β, γ)) =
(T, 0) i γ≤γ2(β),
(0, T) i γ≥γ1(β).
(C.2.5)
Now suppose γ∈[γ2(β), γ1(β)]. Le ˜τk(β, γ) and ˆαk(β, γ) o k= 1,2 be gi en by equa-
ions (33) and (34). No e ha exp essions in (34) a e well-de ined o γ∈[γ2(β), γ1(β)].
Fu he mo e, we ha e
˜τ1(β, γ) =
1
Σ0
2
+T > T a γ=γ2(β),
−1
Σ0
1
<0 a γ=γ1(β),
since ˆα2(β, γ2(β)) = 0, ˆα1(β, γ2(β)) >0, and ˆα1(β, γ1(β)) = 0, ˆα2(β, γ1(β)) >0. Fu -
he , ˜τ1(β, γ) is con inuous and s ic ly dec easing in γ. To see he s ic mono onici y,
le us ake he pa ial de i a i e o ˜τ1(γ) wi h espec o γ. Fo ease o no a ion, we
supp ess he dependence on βand γ om he igh -hand side o he exp ession. We ge
∂˜τ1(β, γ)
∂γ =
(Σ0
1+ Σ0
2)ˆα2∂ˆα1
∂γ −ˆα1∂ˆα2
∂γ
Σ0
1Σ0
2(ˆα1+ ˆα2)2+ˆα2∂ˆα1
∂γ −ˆα1∂ˆα2
∂γ
(ˆα1+ ˆα2)2<0,
whe e we used ∂ˆα1(β,γ)
∂γ =−αD
2
√ˆα1(β,γ)<0 and ∂ˆα2(γ)
∂γ =αD
1
√ˆα2(β,γ)>0 o de e mine he
sign. By he in e media e alue heo em, he e exis γI(β), γII (β)∈(γ2(β), γ1(β)) such
ha
˜τ1(β, γ) =
Ta γ=γI(β),
0 a γ=γII (β).
(C.2.6)
F om he s ic mono onici y o ˜τ1(β, γ), i u he holds ha
˜τ1(β, γ)
> T i and only i γ < γI(β),
<0 i and only i γ > γII (β).
(C.2.7)
Finally, no e ha ˆαk(β, γ) as gi en by (34) a e equi alen o (13) and (14) gi en de-
22
composi ion (18). By Theo ems 1 and 2, he analys ’s equilib ium es alloca ion o
γ∈[γ2(β), γ1(β)] is
τ∗
1(β, γ) = max {0,min {˜τ1(β, γ), T}} (C.2.8)
τ∗
2(β, γ) = max {0,min {˜τ2(β, γ), T}} (C.2.9)
Combining he esul s (C.2.8), (C.2.9) wi h (C.2.7) and (C.2.5), oge he wi h he s ic
mono onici y esul o ˜τ1(β, γ), we ob ain he claim o Lemma 6.
C.2.3 P oo o Lemma 7
Gi en a es alloca ion τ= (τ1, τ2), he decision-make ’s in e im expec ed payo is
VD(τ) = −αD
12Σ0
1
1 + τ1Σ0
1−αD
22Σ0
2
1 + τ2Σ0
2
.(C.2.10)
The i s , he second, and he c oss de i a es o he decision-make ’s in e im expec ed
payo wi h espec o τk≥0 and τl=τka e ∂V D(τ)
∂τk=αD
kΣ0
k
1+τkΣ0
k2>0, ∂2VD(τ)
∂τ2
k
=
−2αD
k2Σ0
k
1+τkΣ0
k3<0, and ∂2VD(τ)
∂τk∂τl= 0. Hence, VD(τ) is s ic ly inc easing and
s ic ly conca e unc ion.
C.2.4 P oo o Lemma 8
Fix he es budge T > 0 and he decision-make ’s weigh ec o αD∈R2
++. Suppose
γ= 0 ( he agen is no dis o ed).
(i) I β≤1/2 ( he agen is oo insensi i e), hen he agen op imally chooses no
es ing: τ∗(β, 0) = (0,0).
(ii) I β > 1/2 ( he agen is su icien ly sensi i e), hen he agen op imally chooses
he decision-make ’s mos p e e ed es alloca ion: τ∗(β, 0) = τ∗(1,0).
Suppose he p emise o Lemma 8 holds. Pa (i) immedia ely ollows om Lemma 5.
Fo pa (ii) ake β > 1/2 and le ˜τ1(β, γ) and ˜τ2(β, γ) be gi en by (33). A γ= 0, we
23
ge
˜τ1(β, 0) = √2β−1αD
1Σ0
1−√2β−1αD
2Σ0
2
Σ0
1Σ0
2√2β−1αD
1+αD
2+√2β−1αD
1
√2β−1αD
1+αD
2T
=αD
1Σ0
1−αD
2Σ0
2
Σ0
1Σ0
2(αD
1+αD
2)+αD
1
αD
1+αD
2
T
Lemma 6 and Theo em 1 hen imply ha he agen ’s equilib ium es alloca ion coincides
wi h he op imal es alloca ion o a single playe wi h weigh ec o α=αD.
C.3 De ails o he P oo s in Appendix A.4
C.3.1 P oo o Lemma 9
Le us se T= 1 and Σ0
1= Σ0
2= 1. F om Theo ems 1 and 2, he equilib ium es
alloca ion sa is ies τ∗
1(p, δ)=1−τ∗
2(p, δ) and
τ∗
2(p, δ) = min ˆα2(p, δ)−ˆα1(p, δ)
ˆα1(p, δ) + ˆα2(p, δ)+ˆα2(p, δ)
ˆα1(p, δ) + ˆα2(p, δ),1= min2ˆα2(p, δ)−ˆα1(p, δ)
ˆα1(p, δ) + ˆα2(p, δ)
| {z }
τaux
2(p,δ):=
,1
whe e ˆα1(p, δ) and ˆα2(p, δ) a e gi en by equa ions (36) and (37), espec i ely. The
unc ion τaux
2(p, δ) is well-de ined, since ˆα1(p, δ) + ˆα2(p, δ)= 0 o all alues o pand δ.
No e ha τaux
2(p, δ)=1/2 when p= 0 and τaux
2(p, δ) = 2 when p≥1−δ
2. Fu he mo e,
when p∈0,1−δ
2we ha e
∂τaux
2(p, δ)
∂p =2∂ˆα2
∂p −∂ˆα1
∂p (ˆα1+ ˆα2)−(2ˆα2−ˆα1)∂ˆα1
∂p +∂ˆα2
∂p
(ˆα1+ ˆα2)2
=3
4
(1 −δ2)
ˆα1ˆα2(ˆα1+ ˆα2)2>0.(C.3.1)
No e ha τaux
2(p, δ) is con inuous in p o p≥0.39 Fu he mo e, τaux
2(p, δ) is s ic ly
inc easing in pon p∈[0,(1 −δ)/2). A p= 0 we ha e τaux
2(p, δ)=1/2<1 and a
p≥(1 −δ)/2 we ha e τaux
2(p, δ) = 2 >1. Hence, o each alue o disc imina ion δ∈
(0,1) he e exis s a h eshold pa iali y le el o he ad iso paux(δ)∈0,1−δ
2, de ined
implici ly by he equa ion τaux
2(paux(δ), δ) = 1, such ha he equilib ium es ing s a egy
39Since ˆα1(p, δ) and ˆα2(p, δ) a e bo h con inuous in p, he only possible poin o discon inui y o
τaux
2(p, δ) is i ˆα1(p, δ) + ˆα2(p, δ) = 0. Equa ions (36) and (37) imply his ne e happens o all easible
alues o pand δ. Hence, τaux
2(p, δ) is con inuous on p≥0.
24
sa is ies τ∗
2(p, δ) = τaux
2(p, δ) o p∈[0, paux(δ)] and τ∗
2(p, δ) = 1 o p≥paux(δ).
C.3.2 P oo o Lemma 10
The ex an e expec ed u ili y o g oup k,Vk(τ, δ), can be de i ed om he expec ed
payo o a esea che VR(τ) in he baseline model in he p oo o Lemma 2, whe e
we se he decision-make ’s weigh s o αD
1(δ), αD
2(δ) and he esea che ’s weigh s (now
g oup kweigh s) o αR
k= 1 and αR
−k= 0. In o he wo ds, gi en alloca ion τand he
poli ician’s equilib ium s a egy d∗(s, τ;δ), he ex an e expec ed payo o g oup kis he
same as ha o he esea che in he baseline model who solely ca es abou a ibu e ˜
θk
(wi h weigh αR
k= 1) and no abou he o he a ibu e (wi h weigh αR
−k= 0). We ge
Vk(τ, δ) = Eh−(˜
d∗D(τ;δ)−˜
θk)2i
=− D
0(δ)−µ0
k2−σ2,D
0(δ)+Σ0
k+ ˆσ2,D(τ;δ) + 2 co ˜
d∗D(τ;δ),˜
θk
=−αD
1(δ)µ0
1+αD
2(δ)µ0
2−µ0
k2−αD
1(δ)2Σ0
1+αD
2(δ)2Σ0
2+ Σ0
k
+Σ0
k
1 + τkΣ0
kαD
k(δ)2+ 2αD
k(δ)Σ0
kτk+Σ0
−k
1 + τ−kΣ0
−kαD
−k(δ)2
Using µ0
1=µ0
2= 0, Σ0
1= Σ0
2= 1, we ge
Vk(τ, δ) = −αD
1(δ)2+αD
2(δ)2+ 1
+1
1 + τkαD
k(δ)2+ 2αD
k(δ)τk+1
1 + τ−kαD
−k(δ)2.(C.3.2)
The di e ence be ween he expec ed u ili ies o he wo g oups is
V1(τ, δ)−V2(τ, δ) = 2αP
1(δ)τ1
1 + τ1−2αP
2(δ)τ2
1 + τ2
=(1 + δ)τ1
1 + τ1−(1 −δ)τ2
1 + τ2
whe e we used αP
1(δ) = 1
2(1 + δ) and αP
2(δ) = 1
2(1 −δ). The di e ence be ween he
expec ed u ili ies o he wo g oups unde he addi ional cons ain ha τ1+τ2= 1 is
∆(τ2, δ) = V1((1 −τ2, τ2), δ)−V2((1 −τ2, τ2), δ) = (1 + δ)(1 −τ2)
1+1−τ2−(1 −δ)τ2
1 + τ2
.
The unc ion ∆(τ2, δ) is con inuous and s ic ly dec easing in τ2since δ∈(0,1) and
hus ∂∆(τ2,δ)
∂τ2=−(1+δ)
(1+1−τ2)2−(1−δ)
(1+τ2)2<0. Fu he mo e, we ha e ∆(1/2, δ)>0 and
25
∆(1, δ)<0. Hence, o each δ he e exis s a unique alue o ˆτ2(δ)∈(1/2,1) o which
i holds ∆(ˆτ2(δ); δ) = 0.
C.3.3 P oo o Lemma 11
Le
ω(τ2, δ) := V1((1 −τ2, τ2), δ) + V2((1 −τ2, τ2), δ)
=−3−δ2+1/2(1 + δ2) + (1 + δ)(1 −τ2)
2−τ2
+1/2(1 + δ2) + (1 −δ)τ2
1 + τ2
deno e he sum o he expec ed u ili ies o he wo g oups as a unc ion o es allo-
ca ion τ= (τ1, τ2) unde he cons ain ha he budge is ully used, τ1=T−τ2
wi h T= 1, and whe e Vk(τ, δ) is gi en by equa ion (C.3.2). Since he budge is
exhaus ed in equilib ium (as shown in he p oo o Lemma 4), wel a e is hus gi en
by W(p, δ) = ω(τ∗
2(p, δ), δ). The pa ial de i a i e o wel a e wi h espec o p > 0,
whene e di e en iable, is hen
∂W(p, δ)
∂p =∂τ∗
2
∂p
∂ω(τ∗
2, δ)
∂τ2
=∂τ∗
2
∂p 1/2(1 + δ2)−(1 + δ)
(2 −τ∗
2)2
| {z }
<0,since δ<1
+−1/2(1 + δ2) + (1 −δ)
(1 + τ∗
2)2
whe e we supp essed he dependence o τ∗
2(p, δ) on (p, δ) om he RHS o b e i y. No e
ha 1/2(1 + δ2)−(1 + δ)>−1/2(1 + δ2) + (1 −δ)and ha (2 −τ∗
2)2<(1 + τ∗
2)2,
since τ∗
2(p, δ)>1/2 o p > 0. Hence, ∂ω(τ∗
2,δ)
∂τ2<0. We hus ha e sign ∂W(p,δ)
∂p =
−sign ∂τ∗
2
∂p .
Le τaux
2(p, δ), ˆp(δ) and paux(δ) wi h ˆp(δ)< paux(δ), be he a iables de ined in he p oo
o Lemma 4. F om equa ion (C.3.1), i ollows
sign ∂W (p, δ)
∂p =−sign ∂τ∗
2(p, δ)
∂p =
−sign τaux
2(p,δ)
∂p <0p∈(0, paux(δ))
0p>paux(δ)
,
and he de i a i e does no exis a p=paux(δ). We hus ha e ha wel a e W(p, δ) is
(i) con inuous in p(since Vk(τ∗(p, δ), δ) is con inuous in p); (ii) s ic ly dec easing on
p∈(0, paux(δ)); and (iii) cons an on p≥paux(δ), whe e all he ad iso s use he en i e
budge o lea n exclusi ely abou g oup 2. Fu he mo e, he equalizing pa iali y le el
26
ˆp(δ)< paux(δ).
Nex , inequali y is gi en by
I(p, δ) = |∆(τ∗
2(p, δ), δ)|
whe e ∆(τ∗
2(p, δ), δ) is gi en by equa ion (38). As we ha e shown in he p oo o
Lemma 4, he unc ion ∆(τ∗
2(p, δ), δ) is con inuous in τ∗
2(p, δ), s ic ly dec easing in
τ∗
2(p, δ), and is ze o a τ∗
2(ˆp(δ), δ), whe e τ∗
2(ˆp(δ), δ)< T ( he ad iso who es o es
equali y, ˆp(δ), does no use he en i e es budge o lea n exclusi ely abou g oup 2).
As shown abo e, we ha e: he equilib ium alloca ion o es s o g oup 2, τ∗
2(p, δ), is
con inuous in p, s ic ly inc easing on p<paux(δ), and cons an on p≥paux(δ), whe e
he ad iso s use he en i e es budge o lea n exclusi ely abou g oup 2: τ∗
2(p, δ) = T.
Hence, inequali y I(p, δ) is (i) con inuous in p; (ii) s ic ly dec easing on p∈(0,ˆp(δ))
(iii) ze o a p= ˆp(δ); (i ) s ic ly inc easing on p∈(ˆp(δ), paux(δ)); and ( ) cons an on
p≥paux(δ).
The e o e, wel a e W(p, δ) and inequali y I(p, δ) a e bo h (i) con inuous in p; and (ii)
s ic ly dec easing on p∈(0,ˆp(δ)). Fu he mo e, wel a e s ic ly dec eases and inequal-
i y s ic ly inc eases on p∈(ˆp(δ), paux(δ)); and hey a e bo h cons an on p≥paux(δ).
C.3.4 P oo o Lemma 12
By Theo em 1, he poli ician’s op imal lea ning s a egy unde unchecked disc imina ion
yields
¯τ2(δ) =
1−3δ
2i δ < 1
3,
0 i δ≥1
3.
F om he p oo o Lemma 4, equa ion (39), he es alloca ion es o ing equali y ˆτ2(δ) =
−(1−δ)+√1+3δ2
2δ.
Fix δ∈[1/3,1). Then |τ2(δ)−1/2|>|ˆτ2(δ)−1/2|, since ˆτ2(δ)∈(1/2,1), as shown in
27
he p oo o Lemma 4. Nex , ix δ∈(0,1/3). Then we ha e, equi alen ly,
|τ2(δ)−1/2|=1
2−1−3δ
2>−1 + δ+√1+3δ2
2δ−1
2=|ˆτ2(δ)−1/2|
1+3δ2>p1+3δ2
which holds o any δ∈(0,1/3).
28
