Li, Bingbing; Fö s e , Manuel
Wo king Pape
A dynamic model o au ho i y in o ganiza ions
Cen e o Ma hema ical Economics Wo king Pape s, No. 753
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Li, Bingbing; Fö s e , Manuel (2025) : A dynamic model o au ho i y in
o ganiza ions, Cen e o Ma hema ical Economics Wo king Pape s, No. 753, Biele eld Uni e si y,
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753
Sep embe 2025
A dynamic model o au ho i y in
o ganiza ions
Bingbing Li and Manuel Foe s e
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
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A dynamic model o au ho i y in o ganiza ions
Bingbing Li
*
and Manuel Foe s e
Sep embe 29, 2025
Abs ac
In ou p incipal-agen model, he p incipal can epea edly delega e au-
ho i y o an agen wi h unce ain p e e ences o ake he decisions himsel .
The p incipal lea ns he s a e a he end o each pe iod and hen upda es
his belie abou he agen ’s bias based on he decision implemen ed i he
delega ed au ho i y. We demons a e ha equilib ia a e cha ac e ized by
an “imi a ion” in e al o agen ypes (biases) who mimic less biased ypes
in o de o be e ained. In e es ingly, he p incipal gene ally bene i s om
he agen ’s imi a ion compa ed o a benchma k. Fu he mo e, compa a i e
s a ics e eal ha , su p isingly, he p incipal may be wo se o wi h be e
in o ma ion. Finally, an ex ension o ini ely many pe iods shows ha he
imi a ion in e al g adually shi s, such ha agen ypes wi hin he in e al
imi a e less biased ypes.
JEL classi ica ion: D23; D82; D83; D73; C72.
Keywo ds: Delega ion, p e e ence unce ain y, p i a e in o ma ion, dynamic
game, o ganiza ional design.
1 In oduc ion
In a ious o ganiza ional con ex s, decision-make s engage be e -in o med ex-
pe s lowe in he hie a chy o assis ance o ad ice. Fo ins ance, policy-make s
seek in o ma ional suppo om subo dina e bu eauc a s when o mula ing d a
policies. Howe e , bu eauc a s o en ha e a pa isan o ien a ion, which may no
*
Cen e o Ma hema ical Economics, Biele eld Uni e si y, PO Box 10 01 31, 33501 Biele eld,
Ge many. Email: [email p o ec ed].
Cen e o Ma hema ical Economics, Biele eld Uni e si y, PO Box 10 01 31, 33501 Biele eld,
Ge many. Email: [email p o ec ed].
1
align wi h ha o he cu en policy-make and a ec hei beha io .1C ucially,
he policy-make —a leas ini ially— aces unce ain y ega ding hese pa isan
o ien a ions.
When delega ing decision-making au ho i y o subo dina es, he policy-make
he e o e has o ake in o accoun bo h hei expe ise and po en ial con lic s o
in e es . O e ime, he chosen subo dina e may hen e eal he pa isan p e e -
ences h ough he decisions. Ano he impo an aspec he e o e is he dynamic
na u e o he delega ion p ocess (unde p e e ence unce ain y): The policy-make
will no only ge o know he en i onmen in he policy domain bu also lea n he
p e e ences o subo dina es, allowing he o eassess he choices.
The ade-o be ween keeping au ho i y and delega ing i o a specialis wi h
di e en objec i es has been ex ensi ely discussed in he o ganiza ional economics
li e a u e. Ye , he p e e ence con lic be ween p incipal and agen has been con-
side ed common knowledge (e.g., Aghion and Ti ole,1997;Dessein,2002;Alonso
and Ma ouschek,2008). In his pape , we build a heo e ical model in which
a p incipal epea edly decides be ween keeping au ho i y and g an ing decision
igh s o an agen wi h unce ain p e e ences. We demons a e how he p ospec
o epea ed in luence on he decision-making p ocess can discipline expe s wi h
pa isan in e es s o beha e like “good” ones whose objec i es a e oughly in line
wi h he p incipal’s.
In ou model, a p incipal (he) has o ake a decision whose payo depends
on he e ol ing s a e o he wo ld in each o wo pe iods. He can ei he keep
au ho i y and decide himsel o delega e au ho i y o a be e -in o med agen (she)
wi h unce ain p e e ences. A he end o he i s pe iod, he p incipal obse es
he ue s a e o he wo ld, he eby acqui ing pa ial insigh s in o he s a e o
he nex pe iod (i.e., he s a e is co ela ed ac oss pe iods). I he p incipal has
delega ed au ho i y in he i s pe iod, he o ms expec a ions ega ding he agen ’s
ype based on his in o ma ion and he decision implemen ed. A he beginning
o he second pe iod, he p incipal hen decides again whe he o keep au ho i y
o delega e i o a be e -in o med agen —i he delega ed au ho i y in he i s
pe iod, he p incipal may ei he e ain o eplace he cu en agen .
Obse e ha a p incipal who p e e s o keep decision-making au ho i y in he
1In he U.K., bu eauc a s ha e ied o “ us a e his p ocess o B exi because i goes agains
he g ain so undamen ally”, see h ps://www.bbc.com/news/uk-42782637?u m_sou ce, ac-
cessed Augus 8, 2025. Fo simila cases in he U.S., see h ps://www.washing onpos .com/
poli ics/2025/02/07/ ump- esis ance- ede al-wo ke s/?u m_sou ce, accessed Augus
8, 2025. See also B ehm and Ga es (1999), who conclude ha “ he o e whelming e idence [. . . ]
indica es ha he bu eauc a ’s own p e e ences ha e he g ea es e ec on beha io .”
2
i s pe iod will do so also in he second pe iod, as he hen is be e in o med.
We hus ocus on he case when he p incipal delega es decision-making au ho i y
in he i s pe iod. We i s es ablish ha , when he discoun a e is low, he
unique equilib ium is cha ac e ized by an imi a ion in e al. While agen s who
ha e a low p e e ence con lic wi h he p incipal choose hei bliss poin and a e
subsequen ly e ained, agen s wi hin his in e al imi a e an agen o he o me
g oup in o de o be e ained as well. Ha ing an in e media e con lic o in e es
wi h he p incipal, hey ade o an immedia e loss due o a de ia ion om hei
bliss poin o a u u e gain due o being e ained and hus able o choose hei
bliss poin in he nex pe iod. In u n, he p incipal e ains he agen despi e he
imi a ion because he expec ed p e e ence con lic is small enough.
Second, when he discoun a e is high, he alue o e en ion o he agen s
is high, as i allows hem o implemen hei bliss poin in he second pe iod.
In equilib ium, he p incipal will hen e ain he agen upon obse ing po en ial
mimicking only wi h a ce ain p obabili y, and in u n, ewe agen ypes mimic.
We hen show ha he p incipal gene ally bene i s om he agen ’s imi a ion.
Disciplining he agen , i yields a gain om a be e decision in he i s pe iod
compa ed wi h a benchma k in which all agen s choose hei bliss poin s. In he
second pe iod, i hen yields a po en ial loss om e aining a he biased agen s
ha he p incipal would no e ain i in o med abou hei ype. In e es ingly, we
es ablish ha he loss is, gene ally, second o de .
Nex , we examine whe he he p incipal bene i s om a mo e slowly changing
en i onmen , such ha he has be e in o ma ion abou he s a e in he second
pe iod. Ou indings e eal ha , su p isingly, he p incipal may be wo se o wi h
be e in o ma ion. Some ela i ely biased agen s hen cease mimicking a less
biased one and ins ead implemen hei bliss poin in he i s pe iod, which educes
he p incipal’s ins an aneous, and po en ially his o al, u ili y. The eason is ha
imi a ion becomes less a ac i e compa ed o he al e na i e o he p incipal
aking he decision himsel , as he la e becomes be e in o med.
Finally, we ex end he model o ini ely many pe iods. We es ablish ha he
p incipal is less and less willing o e ain he agen upon obse ing po en ial mim-
icking as oppo uni ies o mimic in he u u e anish. In equilib ium, he imi a ion
in e al hus g adually shi s, such ha he agen ypes wi hin he in e al imi a e
less biased ypes.
Rela ed li e a u e. This pape is si ua ed wi hin he li e a u e on he dele-
ga ion o o mal au ho i y going back o he seminal wo k by Aghion and Ti ole
3
(1997). They show ha he p incipal may delega e au ho i y in o de o gi e he
agen be e incen i es o acqui e in o ma ion. Dessein (2002) akes he in o ma-
ion s uc u e as gi en and in es iga es he p incipal’s ade-o be ween keeping
au ho i y and delega ing i o he agen depending on hei p e e ence con lic .
While Dessein shows ha he p incipal p e e s delega ion o e communica ion i
objec i es a e su icien ly aligned, Deimen and Szalay (2019) show he e e se may
hold i he agen has o decide on he amoun o in o ma ion she obse es abou
each o wo s a es. O he con ibu ions ha e ex ended he amewo k o op imal
delega ion mechanisms (Alonso and Ma ouschek,2008), ans e s (K ¨ahme ,2006;
Lim,2012), o bo h (Foe s e and Habe mache ,2025a).2In con as o hese pa-
pe s, we conside unce ain y ega ding he p e e ence con lic and show how he
p ospec o epea ed in luence disciplines expe s wi h pa isan in e es s.
The disciplining e ec o epea ed in e ac ion ela es ou pape o he poli -
ical agency li e a u e, which has shown ha elec ions can se e he pu pose o
disciplining a “bad” incumben poli ician (agen ) o beha e like a “good” one o
ge e-elec ed by o e s ( he p incipal) (Be ganza,2000;Besley and Sma ,2007;
Foe s e and Voss,2022). We show ha he same mechanism applies o o ganiza-
ions when a p incipal aces unce ain y ega ding he agen ’s p e e ences and can
e-alloca e au ho i y depending on he decisions. F om a modeling poin o iew,
ou model di e s om hese pape s in a ious aspec s. Fi s and o emos , we
conside a con inuum o ypes bu abs ac om di e ences in abili y as in Foe -
s e and Voss (2022). This has an in e es ing consequence: Since biased agen s in
Foe s e and Voss (2022) imi a e less able ones, he disciplining e ec may, unlike
in ou model, be nega i e om he p incipal’s poin o iew. Second, we allow o
a co ela ion o he s a e ac oss pe iods, such ha he p incipal may ha e be e
in o ma ion in he second pe iod. I u ns ou ha , su p isingly, mo e in o ma ion
can some imes ha m he p incipal. Thi d, we ex end he model o ini ely many
pe iods and in es iga e co ela ions o he s a e ac oss pe iods. Mo eo e , he
ex ension o ou model o ini ely many pe iods sha es some ea u es wi h Banks
and Duggan (2008). Ins ead o a p incipal-agen model, hey conside a gene al
model o epea ed elec ions in which he challenge in each pe iod is andomly
chosen om he se o o e s, who a e assumed o ha e p i a e policy p e e ences.
Banks and Duggan show ha incumben s choose policy comp omises in o de o
ge e-elec ed— eminiscen o expe s imi a ing less biased ones in o de o be
e ained in ou model.
2Foe s e and Habe mache (2025b) ex end he amewo k o Be and compe i ion be ween
(policy) expe s.
4
In ano he ela ed pape , P ende gas (2007) s udies a p incipal who may hi e
a bu eauc a mo i a ed by own objec i es o exe e o . In an ex ension o
p e e ence unce ain y, he shows ha he wage o e ed by he p incipal may se e
as a selec ion de ice. In con as , we ocus on he alloca ion o decision igh s
wi hin an o ganiza ion and conside selec ion h ough epea ed in e ac ion.
To he bes o ou knowledge, we a e he i s o s udy he epea ed alloca ion
o au ho i y o an agen wi h unce ain p e e ences in an o ganiza ional con ex .
Ou esul s show how he p ospec o epea ed in luence on he decision-making
p ocess can discipline agen s wi h pa isan in e es s o beha e like—and he eby
imi a e—agen s whose objec i es a e oughly in line wi h he p incipal’s.
The es o he pape p oceeds as ollows. In Sec ion 2, we se up he model.
Sec ion 3de i es he pu e and mixed equilib ia and analyzes he bene i s o imi a-
ion. Sec ion 4de i es compa a i e s a ics on he co ela ion o he s a e. Sec ion
5ex ends he model o ini ely many pe iods. Sec ion 6concludes.
2 Model and no a ion
We conside an economy popula ed by many ex an e iden ical agen s and a p in-
cipal. In pe iod = 1 he unknown s a e (o he wo ld) θ1∈Θ = Ris dis ibu ed
acco ding o a commonly known dis ibu ion Fon Θ wi h expec ed alue µand
a iance σ2. In = 2, he s a e θ2∈Θ is pa ially co ela ed wi h θ1a a a e
λ∈[0,1), i.e., θ2=λθ1+ (1 −λ)˜
θ2wi h ˜
θ2and θ1i.i.d.
In each pe iod = 1,2, he p incipal P(he) has o ake a decision y ∈R. In
he i s pe iod, P(who does no obse e he s a e) decides whe he o delega e
au ho i y o e he decision y1 o an agen A1(she), whose ype b1∈B=Ris
he p i a e in o ma ion. We assume ha b1is andomly d awn om a commonly
known dis ibu ion Gwi h con inuous, s ic ly posi i e, and symme ic densi y g
on Rwi h expec ed alue µG= 0 and a iance σ2
G.3I A1is selec ed, she obse es
he ue s a e θ1and hen akes he decision y1. I P e ains he au ho i y, he
akes he decision himsel . A he end o he i s pe iod, Pobse es he s a e θ1
and he ou come y1.
A he beginning o he second pe iod, Pdecides whe he o e ain A1(i he
has delega ed he decision o A1in he i s pe iod), o choose ano he agen , A2,
om he pool, o o ake he decision himsel . A2’s ype b2∈Bis he p i a e
in o ma ion and andomly d awn om G(independen ly o b1).
3We assume symme y o simpli y he exposi ion. Ou esul s a e quali a i ely obus o
non-symme ic dis ibu ions.
5
The u ili y o Pis gi en by
(y1, y2|θ1, θ2) = −
2
X
=1
δ −1(θ −y )2,
whe e 0 < δ < 1 deno es he discoun ac o . Tha is, Phas bliss poin θ in
pe iod = 1,2. Simila ly, he u ili y o an agen o ype b, who may o may no
ha e au ho i y o e he decision in one o bo h pe iods, is
u(y1, y2|b, θ1, θ2) = −
2
X
=1
δ −1(θ +b−y )2,
i.e., he agen o ype bhas bliss poin θ +bin pe iod = 1,2. To summa ize,
he iming o e en s is as ollows:
1. Na u e d aws he s a e θ1and he ype b1o A1.
2. Pdecides whe he o delega e au ho i y o A1.
3a. A1 akes he decision y1i Phas delega ed au ho i y o he .
3b. P akes he decision y1himsel i he keeps au ho i y.
4. Plea ns θ1and y1a he end o pe iod 1.
5. Pdecides whe he o e ain A1, o choose ano he agen A2, o o ake he
decision himsel a he beginning o pe iod 2.
6. Na u e d aws he s a e θ2. I Phas delega ed au ho i y o ano he agen
A2, na u e d aws he ype b2.
7. P(A1o A2) akes he decision y2.
8. Payo s ealize.
The solu ion concep we employ is pe ec Bayesian equilib ium.
3 Equilib ium analysis
We i s de e mine he second-pe iod decision depending on au ho i y and hen
he alloca ion o au ho i y. Second, we de e mine A1’s decision in he i s pe iod.
No e ha we will es ic a en ion o he in e es ing case whe e Pdelega es
au ho i y o A1a he beginning o he i s pe iod. To conse e on no a ion, we
6
do no explici ly de ine he s a egy o P, who decides whe he o e ain A1based
on he obse ed decision ˆy1and he s a e θ1a he end o pe iod 1.
3.1 Second-pe iod decision and alloca ion o au ho i y
I Pkeeps he au ho i y in he second pe iod, he maximizes his one-pe iod ex-
pec ed u ili y gi en he i s -pe iod s a e θ1, and hus chooses
y∗
2,P = a g max
y2
E[−(θ2−y2)2|θ1] = E[θ2|θ1] = λθ1+ (1 −λ)µ. (1)
The esidual a iance is (1 −λ)2σ2. I Phas delega ed au ho i y o agen A2o
ype bin he second pe iod, he la e maximizes he one-pe iod expec ed u ili y,
as e ealing bias hen is i ele an , and hus chooses he bliss-poin decision
y∗
2,A(b, θ2) = a g max
y2
E[−(θ2+b−y2)2|θ2] = θ2+b. (2)
The esidual a iance is ze o.
We now u n o he decision o Pwhe he o delega e au ho i y a he begin-
ning o pe iod 2. Suppose o he momen ha Phas kep au ho i y and hus
aken he decision himsel in he i s pe iod. He will hen con inue o do so in he
second pe iod because he s a e is co ela ed, such ha she has be e in o ma ion
han in he i s pe iod. The e o e, we hence o h assume ha he p io a iance
is la ge enough such ha Pdelega es au ho i y o A1in he i s pe iod:
Assump ion 1. σ2≥σ2
G.
I Phas delega ed au ho i y o A1in he i s pe iod and obse ed he decision
ˆy1and he s a e θ1, he upda es his p io belie abou A1’s ype o he pos e io
˜
G=˜
G(·; ˆy1, θ1).
P oposi ion 1. Suppose ha Phas delega ed au ho i y o A1in he i s pe iod.
I is (weakly) op imal o P o e ain A1a he beginning o pe iod 2 i his pos e io
belie ˜
Gis such ha
E˜
G[b2]≤min{(1 −λ)2σ2, σ2
G}.(3)
O he wise, i is weakly op imal o choose ano he agen om he agen pool i
σ2
G≤(1 −λ)2σ2.(4)
O he wise, i is op imal o P o ake he decision himsel .
7
e ained. The ollowing example illus a es he change in he imi a ion in e al as
λinc eases.
Figu e 2: Imi a ion in e al (¯
b(λ),¯
β(λ)] depending on λin Example 1.
Example 1. Suppose ha b∽N(0,1), i.e., σ2
G= 1,σ2= 4, and δ= 0.6. Fi s ,
no ice ha Assump ion 1holds since σ2
G= 1 <4 = σ2, such ha Palways chooses
one agen in he i s pe iod. Second, we ha e δ≤δ∗ o all λ∈[0,1], such ha a
pu e LCSE always exis s. When λ≤0.5, he condi ion σ2
G≤(1−λ)2σ2is sa is ied,
such ha ¯
b(λ)≈0.3and ¯
β(λ)≈2a e cons an in λ. Howe e when λ > 0.5, he
condi ion σ2
G>(1 −λ)2σ2holds. In his ange, bo h ¯
b(λ)and ¯
β(λ)dec ease as λ
inc eases, ul ima ely eaching 0when λ= 1. In pa icula , he imi a ion in e al
anishes as λ→1, see Figu e 2 o an illus a ion. No e ha al hough ¯
b(λ)may
in p inciple inc ease in λ, his equi es pa icula p io dis ibu ions G.
A e ha ing de e mined he e ec o changes in he co ela ion o he s a e
ac oss pe iods on he imi a ion in e al, we nex analyze he e ec on P’s equilib-
ium payo . Obse e ha by P oposi ion 2, he unique pu e LCSE is cha ac e ized
by he imi a ion in e al. I hus ollows om Lemma 3 ha P’s expec ed u ili y
emains cons an as λinc eases when he co ela ion o he s a e ac oss pe iods is
low. When i is high, howe e , his ex an e expec ed u ili y in equilib ium can be
calcula ed as
−2 Z¯
b
0
(b2+δb2)dG +Z¯
β
¯
b
(¯
b2+δb2)dG +Z∞
¯
β
(b2+δ(1 −λ)2σ2)dG!.
14
Fi s , he agen s b1∈[0,¯
b] choose hei bliss poin s and a e e ained by P. Second,
he agen s b1∈[¯
b, ¯
β] imi a e agen ¯
bin he i s pe iod and choose hei bliss poin s
in he second pe iod. All o he agen s b1>¯
βchoose hei bliss poin s in he i s
pe iod, upon which Pdoes no e ain hem and ins ead akes he decision himsel
in he second pe iod. On he one hand, P’s expec ed payo may dec ease as λ
inc eases because some ela i ely biased agen s who did imi a e b′
1=¯
bbe o e will
no do so anymo e and ins ead choose hei bliss poin s b1>¯
b(¯
β(λ) dec eases
by Lemma 3). On he o he hand, P’s expec ed payo may inc ease when he
gain om a be e decision in he second pe iod, in case he decides by himsel ,
domina es he o me loss.
P oposi ion 5. (i) When λ≤1−qσ2
G
σ2,P’s ex-an e expec ed u ili y is cons an
in λ.
(ii) When λ > 1−qσ2
G
σ2,P’s ex-an e expec ed u ili y may inc ease o dec ease
in λ.
The ollowing example illus a es he change in P’s payo due o changes in
he co ela ion o he s a e.
Example 2. Suppose ha b∽N(0,1), i.e., σ2
G= 1,σ2= 4, and δ= 0.6.
Figu e 3illus a es P’s payo as a unc ion o λ. The payo emains cons an
when λ≤0.5. Fo λ > 0.5, we ha e ha ¯
b(λ)and ¯
β(λ)a e dec easing in λ(c .
Example 1). P’s expec ed payo is dec easing o λ∈(0.5,0.83), and inc easing
o λ∈[0.83,1].
Example 2 e eals a possible coun e in ui i e ou come: Pmay be wo se o in
equilib ium when he en i onmen changes less ac oss pe iods, al hough he hen
is be e in o med. No e ha we can also cons uc such examples in case δ > δ∗
o he mixed equilib ium conside ed in P oposi ion 3.
5 Ex ension o ini e ho izon
So a , we ha e analyzed how he p ospec o in luence in he second pe iod a ec s
A1’s beha io in he i s pe iod. We now ex end he model o ini ely many
pe iods T≥3.
In pe iod = 1 he unknown s a e θ1∈Θ = Ris dis ibu ed acco ding o a
commonly known dis ibu ion Fon Θ wi h expec ed alue µand a iance σ2. In
= 2,3, . . . , T, he s a e θ ∈Θ is pa ially co ela ed wi h he p e ious pe iod a
a a e λ∈[0,1), i.e., θ =λθ −1+ (1 −λ)˜
θ wi h θ1and (˜
θ )T
=2 i.i.d.
15
Figu e 3: Change o P’s expec ed payo in Example 2.
The game o he wise p oceeds as desc ibed in Sec ion 2; in pa icula , Pob-
se es he s a e o he cu en pe iod and he decision implemen ed a he end
o each pe iod. A he beginning o he nex pe iod, P hen decides whe he o
e ain he agen , choose a new agen om he pool, o ake he decision himsel .
We es ic a en ion o Ma ko pe ec Bayesian equilib ium, whe e a he
beginning o each pe iod ,Pmakes he au ho i y delega ion decision based solely
on he s a e and he decision implemen ed in he las pe iod −1 (igno ing p e ious
pe iods −2, −3, . . .). Le
u (ˆy , y +1, . . . , yT|b , θ , θ +1, . . . , θT) = −
T
X
′=
δ ′−1(θ ′+b −y ′)2
deno e he es ic ion o agen b ’s u ili y unc ion o pe iods ′≥ .
De ini ion 3 (Ma ko pe ec Bayesian equilib ium).A s a egy p o ile y∗
is a
Ma ko (pe ec Bayesian) equilib ium o he signaling game in pe iod i , o
some pos e io belie ˜
G=G(·|θ ,ˆy )in pe iod + 1 ha is consis en wi h y∗
and
o each agen ype b ∈Band each s a e θ ∈Θ,
y∗
(b , θ )∈∆a gmax
ˆy
E˜
G[u (ˆy , y∗
+1, . . . , y∗
T|b , θ , θ +1, . . . , θT)|θ ],∀ ≤T−1.
I is s aigh o wa d o ex end he de ini ion o LCSE (De ini ion 2) o Ma ko
pe ec Bayesian equilib ia. Simila ly o he baseline model, we p oceed by back-
wa d induc ion. We es ic a en ion o small enough discoun ac o s δsuch ha
16
we can apply he esul s om P oposi ion 2and le [¯
b, ¯
β] deno e he imi a ion in-
e al in he pu e LCSE o he wo-pe iod game. We es ablish ha he bounda ies
o he imi a ion in e al a e dec easing o e ime and e en ually coincide wi h he
imi a ion in e al o he wo-pe iod game.
P oposi ion 6. (Fini e-ho izon pu e Ma ko LCSE) I δ≤δ∗, hen he e exis s
a unique pu e Ma ko LCSE such ha in each pe iod = 1,2,...T −1,
(i) agen b who go he decision delega ed imi a es ˜
b i b ∈(˜
b ,˜
β ]and chooses
he bliss poin θ +b else;
(ii) P e ains he agen i and only i y ∈[θ , θ +˜
b ];
wi h [˜
bT−1,˜
βT−1] = [¯
b, ¯
β]and ˜
b <˜
b −1<˜
β <˜
β −1 o 2≤ ≤T−1.
In he second bu las pe iod, s a egic incen i es a e iden ical o he wo-
pe iod game unde Ma ko s a egies, so ha we ob ain he imi a ion in e al
[¯
b, ¯
β] as in he baseline model. In ea lie pe iods, Pbene i s mo e om e aining
an agen because he la e may con inue o imi a e in he u u e. This shi s
he lowe bound and as a esul also he uppe bound o he imi a ion in e al o
he igh as we mo e back o he beginning o he game. Hence, P igh ens he
equi emen s o e aining he cu en agen o e ime by equi ing decisions ha
a e less biased.
Example 3. Suppose ha b∽N(0,1), i.e., σ2
G= 1,σ2= 4,δ= 0.6,λ= 0.8,
and T= 15. Figu e 4illus a es he speci ic changes o he imi a ion in e al’s
bounda ies o e ime. No e ha λis la ge enough ha P akes he decision o e e
i he doesn’ e ain he selec ed agen . The bounda ies ˜
b and ˜
β a e dec easing
o e ime un il being equal o ¯
band ¯
β(c . Example 1) a pe iod 14 ( he las
pe iod whe e imi a ion akes place). Obse e also ha he changes a e becoming
inc easingly apid o e ime.
6 Conclusion
In ou model, he p incipal aces a ade-o be ween keeping au ho i y and dele-
ga ing i o an in o med agen wi h unce ain p e e ences, while he selec ed agen
compa es he u u e bene i s o epea ed in luence on he decision-making p ocess
wi h he ins an aneous loss o de ia ing om he bliss poin . We ocus on he case
whe e he p incipal is willing o delega e au ho i y in he i s pe iod. When he
17
Figu e 4: Imi a ion in e al (˜
b ,˜
β ] o e ime in Example 3.
discoun a e is low, he unique LCSE is cha ac e ized by an imi a ion in e al
such ha he agen s wi hin his in e al imi a e he “good” ones whose p e e ences
a e oughly aligned wi h he p incipal. When he discoun a e is high, he e is a
mixed equilib ium in which he p incipal e ains he agen upon obse ing po en-
ial mimicking only wi h a ce ain p obabili y. Compa ed wi h he benchma k in
which all agen s choose hei bliss poin s and e eal hei bias in he i s pe iod,
imi a ion allows he p incipal o ob ain highe ex an e expec ed u ili y.
Fu he mo e, we demons a e ha he p incipal may be wo se o wi h be e
in o ma ion, as he incu s a loss om some ela i ely biased agen s ceasing o mimic
a less biased one in he i s pe iod. In e es ingly, a p incipal may hus ha e
an incen i e o s a egically limi hei access o in o ma ion, he eby a oiding
he ad e se incen i e e ec s ha ull obse abili y can gene a e. In o he wo ds,
choosing o emain pa ially unin o med can se e as a commi men de ice ha
induces mo e a o able beha io om he agen . Finally, we ex end he wo-pe iod
model o ini ely many pe iods and show ha he imi a ion in e al g adually shi s
o e ime.
In he u u e, his pape can be ex ended in he ollowing ways. Fi s , we
can examine how he equilib ium changes when we inco po a e cheap alk and
mone a y ans e s, which, o cou se, complica es he analysis. Second, we can
also design a mechanism ha allows he p incipal o imp o e his payo as he
knows mo e, such as commi ing o he beha io o agen s du ing he au ho iza ion
decision s age in he decision-making p ocess.
18
Acknowledgemen s
We would like o hank Ole Jann, And eas Kleine , Sebas ian E ne , and semina
pa icipan s a Biele eld Uni e si y, he QED Jambo ee 2025, he Lisbon Mee ings
in Game Theo y and Applica ions 2025, and he 14 h Con e ence on Economic
Design 2025 o help ul commen s and discussions. We g a e ully acknowledge
esea ch unding by he Deu sche Fo schungsgemeinscha (DFG) h ough he Re-
sea ch P ojec T aining G oup (RTG 2865)—Coping wi h Unce ain y in Dynamic
Economics (CUDE).
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A Appendix: P oo s
P oo o P oposi ion 1.Suppose Phas selec ed an agen in he i s pe iod.
Gi en pos e io ˜
G, his expec ed u ili y om e aining he selec ed agen is
E˜
G[−(θ2−y∗
2,A(b, θ2))2] = −E˜
G[b2].
The expec ed u ili y o P om choose a new agen om he agen pool is
EG[−(θ2−y∗
2,A(b, θ2))2] = −EG[b2] = −σ2
G.
Finally, he expec ed u ili y o P om aking he decision himsel is −(1 −λ)2σ2,
which p o es he claim.
P oo o Lemma 1.The selec ed agen b2would choose he bliss poin y∗
2,A(b2, θ2) =
θ2+b2in he second pe iod, and is hus ce e is pa ibus weakly be e o when she
is being e ained han when she is no being e ained. Since P e ains he agen
i and only i y1∈[θ1+b, θ1+¯
b], choosing he bliss poin y∗
1,A(b1, θ1) = θ1+b1in
he i s pe iod is op imal i b1∈[b,¯
b].
20
I b1>¯
b(b1< b is analogous), hen he agen has o ake a decision y1∈
[θ1+b, θ1+¯
b] in o de o be e ained. To es ablish he claim, obse e ha
a g min
y1∈[θ1+b,θ1+¯
b](θ1+b1−y1)2=θ1+¯
b.
P oo o Lemma 2.Fix any decision y1o he selec ed agen in he i s pe iod.
Suppose i s ha (1 −λ)2σ2≥σ2
G, such ha by P oposi ion 1Pchooses ano he
agen om he agen pool i he does no e ain he selec ed agen . Using (2), he
alue o e en ion o agen A1 hen is:
R(b1) =E[u(y1, y∗
2,A(b1, θ2)|b1, θ1, θ2)|b1, e ained] −E[u(y1, y∗
2,A(b2, θ2)|b1, θ1, θ2)|b1,no e ained]
=0 −δEG[−(θ2+b1−y∗
2,A(b2, θ2))2|b1]
=δ(b2
1+σ2
G).
Second, suppose ha (1 −λ)2σ2< σ2
G, such ha by P oposi ion 1P akes he
decision himsel i he does no e ain he selec ed agen . Using (1), he alue o
e en ion o agen A1 hen is:
R(b1) =E[u(y1, y∗
2,A(b1, θ2)|b1, θ1, θ2)|b1, e ained] −E[u(y1, y∗
2,P (λ, θ1)|b1, θ1, θ2)|b1,no e ained]
=0 −δEF[−(θ2+b1−y∗
2,P (λ, θ1))2|b1]
=δ(b2
1+ (1 −λ)2σ2).
P oo o P oposi ion 2.Le (1 −λ)2σ2< σ2
G, so ha Pp e e s o ake he
decision himsel in he second pe iod ins ead o choosing ano he agen om he
pool ((1 −λ)2σ2≥σ2
Gis analogous). Suppose ha P e ains he agen i and only
i y1∈[θ1, θ1+¯
b], whe e ¯
b > 0. By Lemma 1, agen s b1∈[0,¯
b] hen choose hei
bliss poin , while agen s b1>¯
b ake he decision y1=θ1+¯
bi hey wan o be
e ained. We i s p o e he ollowing lemma.
Lemma 4. I δ≤δ∗, hen he e exis unique ¯
b=¯
b(δ)>0and ¯
β=¯
β(δ,¯
b)>¯
b
such ha
EG[b2|b∈[¯
b, ¯
β]] = min{σ2
G,(1 −λ)2σ2}.
21
P oo . Fix any ¯
b > 0. We i s p o e ha he e exis s a unique ¯
β > ¯
bsuch ha
NR(¯
β,¯
b) = 0, i.e., agen s b1∈(¯
b, ¯
β] p e e o choose y1=θ1+¯
band a e e ained.
Le b1>¯
b, hen Lemma 1and Lemma 2yield
NR(b1,¯
b) = R(b1)−c(θ1+¯
b|b1, θ1)=(δ−1)b2
1+ 2¯
bb1+δ(1 −λ)2σ2−¯
b2.
which is a conca e quad a ic unc ion. And because
NR(b1,¯
b) = R(b1) = δ((1 −λ)2σ2+b2
1)>0
o b1∈[0,¯
b], he e exis s a unique ¯
β=¯
β(δ,¯
b)>¯
bsuch ha NR(¯
β,¯
b) = 0 by
con inui y.
So he agen s b1≤¯
βp e e o imi a e b′
1=¯
bin o de o be e ained in he
second pe iod, while he agen s b1>¯
βins ead p e e o choose hei bliss poin
(acco ding o Lemma 1).
Second, we p o e ha he e exis s ¯
b > 0 such ha EG[b2|b∈[¯
b, ¯
β]] =
(1 −λ)2σ2i δ≤δ∗. By de ini ion o ¯
βand gi en δ=δ∗, we ha e
NR(b1,0) = (δ∗−1)( ¯
β(δ∗,0))2+δ∗(1 −λ)2σ2= 0
o b1=¯
β(δ∗,0). Then we ge ¯
β(δ∗,0) = qδ∗
1−δ∗(1 −λ)σi δ∗<1 and ¯
β(1,0) =
+∞, whe e ∂¯
β(δ∗,0)/∂δ > 0. By de ini ion o δ∗and he la e ,
EG[b2|b∈[0,¯
β(δ, 0)]] ≤(1 −λ)2σ2i δ≤δ∗.
Nex , no e ha , gi en any ¯
b,¯
β=¯
β(δ,¯
b) sa is ies
(δ−1)¯
β2+ 2¯
b¯
β+δ(1 −λ)2σ2−¯
b2= 0,
which yields
¯
β(δ,¯
b) = ¯
b+pδ¯
b2+δ(1 −δ)(1 −λ)2σ2
1−δ
and hus ∂¯
β(δ,¯
b)/∂¯
b > 0. The e o e, EG[b2|b∈[¯
b, ¯
β(δ,¯
b)]] is inc easing in ¯
b,
which es ablishes he claim. No e ha he e canno be wo wo solu ions ¯
b < ¯
b′,
as hen ¯
β(δ,¯
b)<¯
β(δ,¯
b′) and hus
(1 −λ)2σ2=EG[b2|b∈[¯
b, ¯
β(δ∗,¯
b)]] < EG[b2|b∈[¯
b′,¯
β(δ∗,¯
b′)]] = (1 −λ)2σ2,
which is a con adic ion.
22
We nex es ablish he equilib ium. I δ≤δ∗, hen he conside ed s a egies
cons i u e an equilib ium by Lemma 1and de ini ion o ¯
b > 0 and ¯
β(δ,¯
b)>¯
b. In
pa icula , all agen s b1<¯
bsepa a e a leas cos and he e canno be ano he
equilib ium in which a la ge mass o agen s sepa a es by de ini ion o ¯
b, which
implies ha i is he unique LCSE.
Nex , conside δ > δ∗and suppose ha he e is a closed in e al [b′,¯
b′], b′≤¯
b′,
such ha P e ains he agen upon ˆy1∈[θ1+b′, θ1+¯
b′] and ha he e is no
decision ˆy1> θ1+¯
b′ o which his is he case. Then each agen b1∈[b′,¯
b′]
chooses he bliss poin and is e ained by Lemma 1. Recall ha by de ini ion o
δ∗,EG[b2|b∈[0,¯
β(δ, 0)]] >(1 −λ)2σ2. Since u he ∂¯
β(δ,¯
b)/∂¯
b > 0, we ha e
EG[b2|b∈[¯
b′,¯
β(δ,¯
b′)]] >(1 −λ)2σ2 o any ¯
b′>0,
a con adic ion. Finally, suppose ha Pdoes no e ain any agen . Then all
agen s choose hei bliss poin s and disclose hei ype in he i s pe iod. Bu
hen P e ains he selec ed agen when obse ing y1∈[θ1, θ1+ (1 −λ)σ], as hen
b1∈[0,(1 −λ)σ] and hus b2
1≤(1 −λ)2σ2, a con adic ion. This es ablishes ha
he e is no pu e equilib ium i δ > δ∗, which inishes he p oo .
P oo o P oposi ion 3.I δ > δ∗(in his case, σ2
G>(1−λ)2σ2, see Rema k 1),
we conside he mixed equilib ium ha P e ain A1only when obse ing y1=θ1
wi h a posi i e p obabili y PR, o he wise he ake he decision himsel .
Simila o pu e equilib ium, he ne alue unc ion o agen b1 o imi a e he
neu al one is
NR(b1, δ) = −b2
1+PR·δ((1 −λ)2σ2+b2
1).
Then we can ind he indi e en agen s ¯
β′(δ) = qPR·δ
1−PR·δ(1−λ)σ < qδ
1−δ(1−λ)σ=
¯
β(δ, 0) om NR(¯
β′(δ), δ) = 0. The agen b1∈(0,¯
β′] imi a es he neu al one
because NR(b1, δ)≥0, while he agen b1∈(¯
β′,∞) chooses he bliss poin s ( he
p oo is analogous o P oposi ion 2) gi en PR.
When de e mining he s a egies o A1, he expec ed u ili y o P o e ain A1
equals ha o ake he decision himsel when obse ing y1=θ1, which is
Z¯
β′(δ)
0
−b2dG(b) + Z+∞
¯
β′(δ)
−δ(1 −λ)2σ2dG(b) = −(1 −λ)2σ2.
Then we can ge ¯
β′(δ) om EG[b2|b∈[0,¯
β′]] = (1−λ)2σ2and PR=¯
β′
δ(¯
β′+(1−λ)2σ2)=
¯
β′
R(¯
β′) om NR(¯
β′, δ) = 0, which inishes he p oo .
23
Le
e ain(b −1) deno e he con inua ion payo o Pa pe iod ≥2 i he
e ains A −1condi ional on he bias b −1. Since Pin equilib ium will be indi e en
be ween e aining A2and deciding himsel in subsequen pe iods upon obse ing
ˆy2=θ2+˜
b2, we ob ain
E[ 3
e ain(b2)|b2∈(˜
b2,˜
β2]] = −1−δm−1
1−δ(1 −λ)2σ2.
No e ha
e ain(b′
−1)<
e ain(b′′
−1) i b′
−1> b′′
−1. Le u
e ained(b −1) deno e he
con inua ion payo o A −1wi h bias b −1i she is e ained a pe iod ≥2.
Nex , conside he i s pe iod and suppose ha he indi e en agen b1=˜
β1
is selec ed. Then, in equilib ium, we ha e
δ−δm+1
1−δ((1−λ)2σ2+˜
β2
1) =
(˜
β1−˜
b1)2−δ2u3
e ained(˜
β1), i ˜
β1≤˜
b2<˜
β2
(˜
β1−˜
b1)2+δ(˜
β1−˜
b2)2−δ2u3
e ained(˜
β1), i ˜
b2<˜
β1≤˜
β2
(˜
β1−˜
b1)2+δ2−δm+1
1−δ((1 −λ)2σ2+˜
β2
1), i ˜
β1>˜
β2
because agen ˜
β1is indi e en be ween choosing he bliss poin in he i s pe iod
(le -hand side) and imi a ing he agen ˜
b1( igh -hand side).
I ˜
β1≤˜
b2<˜
β2, we ha e
1−δm
1−δ(1 −λ)2σ2
=EG[b2
1|b1∈[˜
b1,˜
β1]] −δE[ 3
e ain(b2)|b2∈[˜
b1,˜
β1]]
<(1 −λ)2σ2−δE[ 3
e ain(b2)|b2∈[˜
b2,˜
β2]]
=(1 −λ)2σ2+δ1−δm−1
1−δ(1 −λ)2σ2
=1−δm
1−δ(1 −λ)2σ2,
(10)
a con adic ion. The i s equali y in (10) holds since Pis indi e en be ween
making he decisions himsel and e aining A1i he obse es y1=θ1+˜
b1in he i s
pe iod. Suppose P e ains A1in he i s pe iod. In ha case, A1chooses he bliss
poin in he second pe iod and is e ained again in he hi d pe iod because ˜
β1≤˜
b2.
So he expec ed u ili y o P o e ain he agen A1a he beginning o pe iod 2 is
−EG[b2
1|b1∈[˜
b1,˜
β1]] + δE[ 3
e ain(b2)|b2∈[˜
b1,˜
β1]]. The second equali y in (10)
holds because E[ 3
e ain(b2)|b2∈[˜
b2,˜
β2]] = −1−δm−1
1−δ(1 −λ)2σ2. The inal equali y
in (10) holds because o he calcula ion. Nex , we p o e he inequali y in (10)
holds. Ob iously, E[ 3
e ain(b2)|b2∈[˜
b1,˜
β1]] > E[ 3
e ain(b2)|b2∈[˜
b2,˜
β2]], since
30
˜
b1<˜
β1≤˜
b2<˜
β2. Suppose EG[b2
1|b1∈[˜
b1,˜
β1]] >(1 −λ)2σ2, hen we ha e
−1−δm−1
1−δ(1 −λ)2σ2> E[ 3
e ain(b2)|b2∈[˜
b1,˜
β1]]
> E[ 3
e ain(b2)|b2∈[˜
b2,˜
β2]] = −1−δm−1
1−δ(1 −λ)2σ2.
The i s inequali y holds since EG[b2
1|b1∈[˜
b1,˜
β1]] >(1 −λ)2σ2and 1−δm
1−δ(1 −
λ)2σ2=EG[b2
1|b1∈[˜
b1,˜
β1]] −δE[ 3
e ain(b2)|b2∈[˜
b1,˜
β1]]. The e is a con a-
dic ion, so EG[b2
1|b1∈[˜
b1,˜
β1]] ≤(1 −λ)2σ2, such ha he inequali y in (10)
holds.
Simila ly o he h ee-pe iod case, we can exclude o he cases and p o e ha
˜
b2<˜
b1<˜
β2<˜
β1, which es ablishes he claim.
Now, we ha e p o en ha when δis small enough, we ge ˜
b <˜
b −1<˜
β <˜
β −1
(2 ≤ ≤T−1) o any T≥3. Finally, no e ha when δ=δ∗, we ha e
0 = ˜
bT−1<˜
βT−1and 0 <˜
b <˜
β o any 1 ≤ ≤T−2. This implies ha he e
exis s a unique Ma ko LCSE when δ≤δ∗.
Finally, we p o e ha i Pdoesn’ e ain he agen , he e exis s 1 ≤ ∗≤T
such ha he chooses ano he agen om he pool i < ∗, o he akes he decision
himsel i ≥ ∗. Le
pool(b −1) deno e he con inua ion payo o Pa pe iod
≥2 i he doesn’ e ain A −1, hen he equi alen p oo is ha :
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