Optimal delegation and information transmission under limited awareness

Aus e , Sa ah; Pa oni, Nicola
A icle
Op imal delega ion and in o ma ion ansmission unde
limi ed awa eness
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Aus e , Sa ah; Pa oni, Nicola (2024) : Op imal delega ion and in o ma ion
ansmission unde limi ed awa eness, Theo e ical Economics, ISSN 1555-7561, The Econome ic
Socie y, New Ha en, CT, Vol. 19, Iss. 1, pp. 245-284,
h ps://doi.o g/10.3982/TE5117
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/296459
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Theo e ical Economics 19 (2024), 245–284 1555-7561/20240245
Op imal delega ion and in o ma ion ansmission unde
limi ed awa eness
Sa ah Aus e
Depa men o Economics, Uni e si y o Bonn
Nicola Pa o n i
Depa men o Economics, Bocconi Uni e si y
We s udy he delega ion p oblem be ween a p incipal and an agen , who no only
has be e in o ma ion abou he pe o mance o he a ailable ac ions bu also
supe io awa eness o he se o ac ions ha a e ac ually easible. We p o ide con-
di ions unde which he agen inds i op imal o lea e he p incipal unawa e o
ele an op ions. By doing so, he agen inc eases he p incipal’s cos o dis o ing
he agen ’s choices and inc eases he p incipal’s willingness o g an him highe
in o ma ion en s. We u he show ha he p incipal may use he op ion o ene-
go ia ion as a ool o implemen ac ions ha a e no desc ibable o he a he con-
ac ing s age. I he agen enego ia es, his p oposal signals in o ma ion abou
he payo s a e. Due o he limi ed awa eness, he p incipal makes a coa se in e -
ence om he agen ’s ecommenda ions and, as a esul , accep s a la ge numbe
o he agen ’s p oposals, which ul ima ely bene i s bo h.
Keywo ds. Unawa eness, op imal delega ion, enego ia ion.
JEL classi ica ion. D82, D83, D86.
1. In oduc ion
In many si ua ions, economic agen s delega e decisions o expe s whose p e e ences
may no be pe ec ly aligned wi h hei own. Public and p i a e o ganiza ions use p o-
cu emen manage s o pu chase p oduc s and se ices o he asks a hand; co po a e
headqua e s ely on di ision manage s wi h supe io in o ma ion abou he p o i abil-
i y o new p ojec s; small in es o s seek ad ice om inancial expe s wi h a be e un-
de s anding o he isks and e u ns o he a ailable po olios. O en imes, he in o med
Sa ah Aus e : [email p o ec ed]
Nicola Pa oni: [email p o ec ed]
The au ho s hank Sandeep Baliga, Pie paolo Ba igalli, Syl ain Chassang, Wou e Dessein, An onio Gua -
ino, Yingni Guo, Johannes Hö ne , Na in Ka ik, Ellio Lipnowski, Alessand o Pa an, and Yu al Salan o
e y help ul commen s. The p esen d a bene i ed om se e al commen s ecei ed by he semina pa -
icipan s a he Toulouse School o Economics, Collegio Ca lo Albe o, Uni e si y College London, EIEF in
Rome, Bonn Uni e si y and a he Max Planck Ins i u e, a he Uni e si y o B is ol, he Uni e si y o Ca di ,
a he Bocconi and Ca olica uni e si ies in Milan, a he Uni e si y o Ve ona, Uni e si y o Malaga, Ox-
o d Uni e si y, S ockholm School o Economics, and he Cowles Con e ence on Economic Theo y. Nicola
Pa oni acknowledges inancial suppo om he MIUR-PRIN g an 20157NH5TP. Sa ah Aus e acknowl-
edges unding om he Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion) unde
Ge many’s Excellence S a egy—EXC 2126/1—390838866 and CRC TR 224 (P ojec B02).
©2024 The Au ho s. Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5117
246 Aus e and Pa oni Theo e ical Economics 19 (2024)
pa y no only has a be e unde s anding o wha he mos sui able ac ion is bu also o
he op ions ha a e ac ually a ailable. P ocu emen manage s ha e supe io awa eness
o he easible p oduc s and po en ial supplie s in he ma ke whe e hey ope a e; di i-
sion manage s ha e a be e unde s anding o he p ojec s hey could pu sue; inancial
expe s a e amilia wi h mo e inancial ins umen s han e ail in es o s, e c.
This pape p oposes a amewo k o s udy he implica ions o such asymme y by
inco po a ing unawa eness in o a canonical delega ion model. Mo e speci ically, we
conside he p oblem o a p incipal (she) who needs o ake an ac ion and delega es he
ask o an agen (he). The agen ecei es p i a e in o ma ion abou he payo s o each
a ailable ac ion, and he p incipal’s p oblem is o de e mine a se o ac ions om which
he agen can choose. We depa om he adi ional amewo k o op imal delega ion
by conside ing a si ua ion whe e he p incipal is unawa e o some easible ac ions and
whe e his limi s he language wi h which she can w i e a con ac : he p incipal can
only pe mi ac ions in he delega ion se i she can name hese ac ions explici ly; hence,
i she is awa e o hem. Be o e he delega ion s age and be o e ecei ing p i a e in o -
ma ion, he agen can expand he p incipal’s awa eness by e ealing addi ional ac ions
and he eby en ich he se o easible con ac s o he p incipal.
We a e in e es ed in he ques ion i and how he agen dis o s he p incipal’s awa e-
ness o inc ease his own en . We add ess his ques ion in an en i onmen wi h a con-
inuum o payo s a es, a con inuum o easible ac ions, and an agen who p e e s a
highe ac ion han he p incipal in each s a e. Gi en he awa eness, he p incipal’s op-
imal delega ion se sol es he usual adeo be ween minimizing dis o ions and limi -
ing he agen ’s in o ma ion en . Since he agen has an upwa d bias, op imal delega ion
en ails ha he p incipal limi s he agen ’s choice om abo e. An op imal delega ion
se hus has a h eshold abo e, which no ac ion is pe mi ed. How high his h esh-
old is depends on he p incipal’s awa eness se . We iden i y condi ions unde which
he agen op imally lea es he p incipal unawa e o an in e al o ac ions a ound he
op imal uppe h eshold unde ull awa eness. By choosing he bounds o he in e al
app op ia ely, he agen makes i op imal o he p incipal—who s ill ca es abou he
agen ’s in o ma ion— o pe mi an ac ion abo e he ull awa eness cap, and hence an
ac ion ha would be p ecluded i he p incipal was ully awa e.
An impo an assump ion o ou model is ha he p incipal canno speci y ac ions in
he delega ion se o which she is unawa e. An in ospec i e p incipal migh , howe e ,
ask he sel whe he he e a e o he con ac s ha can imp o e on he op imal delega-
ion se wi hou gi ing he agen he lexibili y o aking unknown, po en ially ha m ul
op ions. One such possibili y o he p incipal is o o ego ull commi men and add
a con ac ual clause ha allows he delega ion se o be adjus ed when new op ions o
con ingencies come o ligh . Adding such a clause and he eby allowing o ex pos ene-
go ia ion is indeed consis en wi h he p incipal’s sophis ica ion and language. In he
second pa o he pape , we s udy he implica ions o he use o such con ac s. The
agen is hen allowed o p opose addi ional ac ions o he p incipal a e he ini ial dele-
ga ion se is ag eed upon and he obse es he s a e. Subsequen ly, he p incipal decides
whe he o pe mi a new ac ion o whe he o main ain he o iginal delega ion se .
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 247
Since he disclosu e o addi ional ac ions is made a e he agen ecei es p i a e in-
o ma ion, he p incipal and agen play a signaling game a he enego ia ion s age. We
cha ac e ize he se o p oposals he p incipal is willing o accep a e he ini ial ag ee-
men is signed. Fixing he agen ’s ini ial disclosu e and ocusing on he equilib ium wi h
he maximal se o accep able p oposals, we p o ide condi ions unde which adding a
enego ia ion clause o he ini ial delega ion se always bene i s he p incipal.Thedown-
side o enego ia ion is ha i limi s he agen ’s disclosu e incen i es in he con ac ing
phase. Indeed, we show ha , depending on he p incipal’s ini ial awa eness, he agen
can signi ican ly inc ease his lexibili y by ini ially holding back some a ailable ac ions
and ailo ing addi ional disclosu es o he in o ma ion he ecei es. F om a modele ’s
iewpoin , he agen ’s s a egy in his equilib ium is ully e ealing. The unawa e p inci-
pal, howe e , canno compa e he agen ’s p oposal a a gi en s a e o he ac ions which
he agen would ha e p oposed in a di e en s a e, and hence in e s s ic ly less in o -
ma ion om he agen ’s ecommenda ion.
An applica ion o ou se ing is p ocu emen delega ion wi hin an o ganiza ion.
Many o ganiza ions ha e p ocu emen manage s in cha ge o pu chasing p oduc s and
se ices acco ding o he o ganiza ion’s cu en needs. This choice is o en limi ed ia
p e-speci ied lis s o app o ed p oduc s o endo s. A possible conce n a ionalizing
such es ic ions is ha he p ocu emen manage can use his disc e ion o u he pe -
sonal goals, such as ca ee enhancemen , minimiza ion o wo kload, pe sonal en ich-
men , e c. (Roge son (1994)). We can hink o ou agen as he p ocu emen manage
and in e p e he agen ’s ac ions as he di e en p oduc s (o supplie s) a ailable in he
ma ke . The s a e cap u es he cha ac e is ics o he ask a hand, de e mining he p o-
cu emen manage ’s “ideal” p oduc . The p incipal can be in e p e ed as a high-le el
manage wi h limi ed awa eness o he a ailable p oduc s.
Ou esul s sugges ha a biased p ocu emen manage ypically bene i s om hid-
ing ce ain p oduc s o supplie s. In pa icula , unde he iden i ied condi ions, he p o-
cu emen manage has incen i es o p opose p oduc s wi h ela i ely ex eme cha ac-
e is ics o app o al bu wan s o hide hose wi h mo e mode a e a ibu es. The mo e
p onounced he awa eness asymme y, he la ge he scope o he p ocu emen man-
age o dis o he p ocu emen decision h ough s a egic disclosu e. Finally, i he
p ocu emen manage has he possibili y o seek app o al o addi ional op ions a e
obse ing he cha ac e is ics o he ask, he can subs an ially expand his lexibili y by
ailo ing he disclosu e o he ci cums ances.
A e he li e a u e e iew, he pape is o ganized as ollows. Sec ion 2p esen s he
delega ion model wi h limi ed awa eness. In Sec ion 3, we analyze he agen ’s op imal
disclosu e and he esul ing delega ion se . Sec ion 4analyzes he game wi h enego ia-
ion and Sec ion 5concludes.
1.1 Rela ed li e a u e
The pape makes bo h applied and heo e ical con ibu ions. I in oduces unawa eness
o he canonical delega ion p oblem and shows how he agen can dis o he p inci-
pal’s delega ion choice h ough s a egic disclosu e. The analysis builds on he li e a-
u e on op imal delega ion. Holms öm (1980) i s de ines he delega ion p oblem and
248 Aus e and Pa oni Theo e ical Economics 19 (2024)
p o ides condi ions o he exis ence o i s solu ion. Following he seminal pape , he
li e a u e was u he de eloped by Melumad and Shibano (1991), Szalay (2005), Ma i-
mo and Semeno (2006), Alonso and Ma ouschek (2008), Ko áˇ
c and Mylo ano (2009),
A ms ong and Vicke s (2010), Amado and Bagwell (2013), and Halac and Ya ed (2020),
among o he s. None o hem conside limi ed awa eness in his amewo k.
The pape is also ela ed o he smalle li e a u e ha applies unawa eness o games
in gene al and con ac ing p oblems in pa icula . In con as o ou se ing, mos o
he exis ing wo k conside s con ac ing p oblems whe e con ingen ans e s a e easi-
ble and whe e he agen has limi ed awa eness, while he p incipal is ully awa e (Von
Thadden and Zhao (2012), Zhao (2011), Filiz-Ozbay (2012), Aus e (2013)). One excep-
ion is F ance ich and Schippe (2020), which s udies a sc eening model whe e he p in-
cipal is unawa e o ce ain cos ypes (bu has ull awa eness o e ac ions) and he agen
decides which ypes o disclose.
In Aus e and Pa oni (2022), we conside a inance applica ion o ou model, in-
e p e ing he agen as a inancial expe and he p incipal as an in es o wi h limi ed
awa eness abou he a ailable inancial p oduc s. We collec sel - epo ed da a om
cus ome s in he I alian e ail in es men sec o and ind suppo o he key p edic-
ions o he model: he menus o e ed o less knowledgeable in es o s con ain ewe
p oduc s, which a e pe cei ed o be mo e ex eme.1Also, Lei and Zhao (2021)s udya i-
nancial ma ke applica ion o ou model bu ocus on he unawa eness o con ingencies
(na u e’s mo es) a he han playe s’ ac ions.
On he heo e ical side, he s udy o he disclosu e p oblem e eals how he agen ’s
in o ma ion en s depend on he se o easible ac ions—o he p incipal’s pe cep ion
he eo —in delega ion se ings. This ques ion is ela ed o a ecen li e a u e looking a
he de e minan s o agency en s in models wi h ull awa eness, ini ia ed by Roesle and
Szen es (2017). Wi h he second pa o he pape , we also con ibu e o he li e a u e on
incomple e con ac s and un o eseen con ingencies (G ossman and Ha (1986); Ha
and Moo e (1988)) by demons a ing he alue o ex pos enego ia ion in se ings wi h
pa ial awa eness. P e ious pape s ha s udy he in e ac ion be ween limi ed awa eness
and he possibili y o enego ia ion a e Ti ole (2009)andPie mon (2017), albei in a he
di e en se ings.
2. En i onmen
The e is a p incipal and an agen . The agen has access o an in e al o ac ions YA=
[ymin,ymax]. The p incipal’s and agen ’s payo s depend on he ac ion ha is chosen and
an unknown payo pa ame e θ, which can be p i a ely obse ed by he agen . Le =
[0, 1]be he se o payo s a es and le F(θ)deno e he cumula i e dis ibu ion unc ion
on , assumed o be wice di e en iable on he suppo . The p incipal and he agen
1The da a we collec ed consis s o app oxima ely 1400 in es o s epo ing on hei expe ience in he
I alian e ail in es men sec o . We eg ess bo h he numbe o o e ed p oduc s and a measu e o pe cei ed
“ex emeness” in he menu on an index o knowledge, which is based on a numbe o ques ions elici ing
he in es o ’s backg ound knowledge. See Aus e and Pa oni (2022) o de ails.

Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 249
ha e wice con inuously di e en iable u ili y unc ions2
UP(θ,y),UA(θ,y).
Fixing θ,Ui o i=P,Ais assumed o be s ic ly conca e in ywi h an in e io maximum
on YA. The p incipal’s and agen ’s condi ionally p e e ed ac ions a e desc ibed by he
unc ions
yP(θ):=a g max
y∈YAUP(θ,y),yA(θ):=a g max
y∈YAUA(θ,y).
We assume Ui
θy >0, which implies ha yP(·),yA(·)a e s ic ly inc easing unc ions. Fu -
he mo e, we assume ha condi ional on he payo pa ame e θ, he agen p e e s a
highe ac ion han he p incipal: o all θ,yP(θ)<yA(θ).
Awa eness Le Ydeno e he se o closed subse s o [ymin,ymax]. The p incipal is awa e
o a subse o a ailable ac ions, deno ed by YP∈Y. Hence, unawa eness in ou ame-
wo k does no ake he o m o un o eseen con ingencies bu conce ns he se o a ail-
able ac ions. Apa om he assump ion ha YPis closed, we impose no u he s uc-
u e on he p incipal’s ini ial awa eness se . Be o e he p incipal con ac s wi h he agen
and he agen obse es θ, he agen can make he p incipal awa e o addi ional ac ions
by e ealing a closed se X∈Y. The p incipal ully unde s ands he op ions ha a e e-
ealed o he and acco dingly upda es he awa eness o he union o wha e e she knew
ini ially and wha he agen e eals.3
Delega ion Gi en he upda ed awa eness, he p incipal o e s a con ac o he agen .
We ule ou mone a y ans e s and assume ha he agen ’s pa icipa ion cons ain is
always sa is ied. The con ac ing p oblem o he p incipal hen educes o he decision
o e he se o ac ions om which he agen can choose once he obse es he payo
pa ame e θ.4Ou subs an ial assump ion is ha he p incipal’s unawa eness es ic s
he language wi h which she can w i e a con ac . In pa icula , we assume ha he
p incipal can only e e o ac ions in he con ac , which she can name explici ly. The
la ge he p incipal’s awa eness se , he iche he se o con ac s she can w i e.
Gi en he p incipal’s upda ed awa eness se , she hen has wo na u al op ions: he
p incipal can ei he name he ac ions she allows he agen o ake o she can name he
ac ions she explici ly o bids. Unde ull awa eness, hese wo op ions a e clea ly equi a-
len . Wi h unawa eness, on he o he hand, speci ying only he o bidden ac ions lea es
2No e ha he p incipal does no ha e ull access o he payo unc ion UPbu jus o a payo unc ion
es ic ed o he domain o ac ions o which she is awa e.
3Assuming ha he agen discloses ac ions be o e ecei ing p i a e in o ma ion a oids signaling e ec s
in he baseline model. Tha is, a e any disclosu e by he agen , he p incipal’s belie s abou he payo s a e
a e desc ibed by he p io F. This is consis en wi h “Re e se Bayesianism” (see Ka ni and Vie ø (2013)),
which pos ula es ha ela i e belie s on e en s o which he decision make was p e iously awa e do no
change when he awa eness g ows.
4The s anda d delega ion p oblem is equi alen o a mechanism design p oblem when he p incipal
es ic s he sel o de e minis ic alloca ions (see Alonso and Ma ouschek (2008) and Ko áˇ
c and Mylo ano
(2009)). Fo mally, he p incipal commi s o a mechanism ha speci ies an ac ion as a unc ion o he agen ’s
message.
250 Aus e and Pa oni Theo e ical Economics 19 (2024)
he p incipal ulne able o he agen aking ac ions ha he p incipal does no an ici-
pa e. We will discuss his case and o he op ions in Sec ion 3.2 and concen a e now on
he case whe e he p incipal speci ies he ac ions which she pe mi s. Since he p incipal
canno speci y ac ions o which she is unawa e, he p incipal’s delega ion se is hen a
subse o he awa eness se . We es ic a en ion o closed delega ion se s. The iming
o he game can be summa ized as ollows:
1. The agen e eals a se o ac ions X∈Yand he p incipal upda es he awa eness
o Y=YP∪X.
2. Gi en awa eness se Y, he p incipal chooses a delega ion se D∈Ysuch ha
D⊆Y.
3. The agen obse es θand chooses an ac ion om se D.
4. Payo s a e ealized.
The game be ween he p incipal and he agen can be o mally ep esen ed by a am-
ily o pa ially o de ed subjec i e game ees. Such amily includes he modele ’s iew o
he objec i ely easible pa hs o play, bu also he easible pa hs o play as subjec i ely
iewed by some playe s, o as he ame o mind a ibu ed o a playe by o he playe s
o by he same playe a a la e s age o he game (Hei e z, Meie , and Schippe (2021)).
5As a solu ion concep , we use a s ong e sion o Pe ec Bayesian Nash Equilib ium
(PBE), which implies subgame pe ec ion, adap ed o gene alized ex ensi e- o m games
wi h unawa eness (e.g., see Halpe n and Rêgo (2014)andFeinbe g (2021)).
Rema k. The e is an al e na i e eading o ou model as one o limi ed au ho i y. We
can hink o a si ua ion whe e he agen , a he han disclosing easible ac ions o he
p incipal, ac ually enables he p incipal o pu sue hem. The agen hus decides on he
se o ac ions he makes a ailable o he p incipal and, as be o e, he p incipal delega es
some subse o hose ac ions o he agen . By deciding which ac ions o make a ailable,
he agen is gi en commi men powe no o ake ce ain ac ions. Since such commi -
men limi s he p incipal’s choice o e easible con ac s, we a e ul ima ely aced wi h a
double delega ion game be ween he agen and he p incipal.
3. Equilib ium analysis
We will wo k backwa d and s a he analysis by conside ing he las s age o he game.
Gi en a delega ion se Dand obse ed payo s a e θ, he agen ’s bes esponse o he
5In a wo king pape e sion o his pape , a ailable on ou websi es, we p o ide a mo e ex ensi e de-
sc ip ion o he amily o game ees ep esen ing he gene alized game wi h unawa eness associa ed o
ou delega ion model acco ding o he app oach p oposed by Hei e z, Meie , and Schippe (2013). We
also desc ibe he se o ou comes ha sa is y a p uden e sion o ex ensi e- o m a ionalizabili y and we
show ha whene e we es ic o pu e s a egies and assume he ie-b eaking ules we adop below o be
commonly known, he PBE ou come we ob ain is also he sole a ionalizable ou come o he gene alized
game.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 251
las s age o he game is de ined by
BRA(θ,D):=a g max
y∈DUA(θ,y).(1)
When he agen is indi e en be ween wo ac ions, le y∗(θ,D):=min BRA(θ,D)be he
selec ion ha akes he smalles alue (indi e ence is b oken in a o o he p incipal).6
Delega ion s age Tu ning o he p incipal’s delega ion choice, we i s de ine he p in-
cipal’s alue o delega ion se D∈Ygi en y∗:
VP(D):=1
0
UPθ,y∗(θ,D)dF(θ).(2)
The e a e ypically ac ions ha he p incipal could pe mi bu he agen will no imple-
men . Wi hou loss o gene ali y, we will es ic a en ion o delega ion se s Dsuch ha
o any y∈D, he e is some s a e θ∈[0, 1]such ha y∗(θ,D)=y.Le D(Y)be he se o
delega ion se s in {D∈Y:D⊆Y} ha sa is y his equi emen . Fo each awa eness se
Y∈Y, he p incipal’s op imal delega ion se sol es he p oblem
max
D∈D(Y)VP(D).(3)
Theo em 1 in Holms öm (1980) gua an ees exis ence o each closed Y(see also P opo-
si ion 12 in Appendix A.8). I p oblem (3) has mul iple solu ions, we assume ha he
p incipal chooses he agen -p e e ed se . Fo each Y,wedeno ebyD∗(·)such selec-
ion om he se o maximize s. Fu he mo e, we assume ha in he case whe e he
p incipal is ully awa e, delega ion is aluable. A su icien condi ion o aluable del-
ega ion is y∗
0>y
A(0),whe ey∗
0∈a gmaxyVP({y}). This equi es he bias o be no oo
la ge and implies ha he p incipal p e e s he delega ion se [yA(0),y∗
0] o he single on
{y∗
0}(see also Alonso and Ma ouschek (2008, Co olla y 2).
Disclosu e s age In he i s s age o he game, he agen chooses an awa eness se
Y∈Y. Since he agen canno make he p incipal unawa e o ac ions ha he p incipal
al eady knows, he induced awa eness se mus con ain he p incipal’s ini ial awa eness
se YP. The smalle YP, he la ge he collec ion o awa eness se s om which he agen
can choose. An op imal awa eness se Y∗sol es he p oblem
max
Y∈Y1
0
UAθ,y∗θ,D∗(Y)dF(θ)such ha YP⊆Y.(4)
Since di e en awa eness se s migh induce he same delega ion se , he solu ion o
p oblem (4) is again ypically no unique. O cou se, his ype o mul iplici y does no
a ec he ou come. We assume ha when wo solu ions o p oblem (4)a enes ed, he
agen discloses he la ge se . This assump ion allows us o dis inguish he ac ions ha
emain undisclosed o s a egic easons om hose ha a e edundan . Le Y∗deno e
he se o all solu ions o (4) sa is ying his equi emen .
6Such selec ion is well-de ined, since he BRAco espondence is nonemp y and uppe hemicon inuous
(see Holms öm (1980)). In addi ion, o each closed D, hese o θ’s o which BRA(·,D)is no a single on
is a mos coun able, and hence o F-measu e ze o (see Lemma 11 in Appendix A.8).
252 Aus e and Pa oni Theo e ical Economics 19 (2024)
Ou model is a sequen ial mo e game wi h in ini e ac ions. This makes equilib ium
exis ence a non i ial issue. In Appendix A.8, P oposi ion 13, we show ha a solu ion o
p oblem (4) exis s. Hence, he e is an equilib ium whe e he agen discloses a se Y∈Y∗,
he p incipal delega es se D∗(Y)and, a e obse ing he s a e ealiza ion θ, he agen
akes ac ion y∗(θ,D∗(Y)).
Equilib ium disclosu e The cen al ques ion o his pape is whe he he agen dis o s
he p incipal’s delega ion choice in his a o by lea ing he p incipal unawa e o some
easible ac ions. Due o he con lic o in e es be ween he p incipal and he agen , a
ully awa e p incipal will no ind i op imal o pe mi he agen his p e e ed ac ion in
e e y payo s a e. Indeed, since he agen is upwa d biased, he p incipal can always
imp o e on ull delega ion by excluding an in e al o high ac ions, o cing he agen o
high ealiza ions o θ o ake an ac ion close o he p incipal’s condi ionally p e e ed
ac ion.
Following his a gumen , we de ine ˆ
y:=maxD∗(YA)<y
A(1)as he highes ac-
ion, which he p incipal pe mi s unde he op imal delega ion se in he ull awa eness
benchma k. The ollowing p oposi ion shows condi ions unde which unawa eness o
ˆ
yis su icien o ensu e ha he agen bene i s om he p incipal’s limi ed awa eness.7
To his end, le s:=(yA)−1deno e he in e se o yA, implici ly de ined by he i s -o de
condi ion UA
y(s(y),y)=0. Since he agen ’s u ili y unc ion is wice con inuously di e -
en iable, he unc ion s(·)is di e en iable (see Lemma 8in he Appendix).
Suppose now he uppe h eshold ˆ
yis a limi poin o D∗(YA). I mus hensa is y
he ollowing op imali y condi ions:
1
s(ˆ
y)
UP
y(θ,ˆ
y)dF(θ)=0(5)
−UP
ys(ˆ
y),ˆ
y s(ˆ
y)s(ˆ
y)+1
s(ˆ
y)
UP
yy(θ,ˆ
y)dF(θ)≤0. (6)
The i s -o de condi ion (5) says ha , condi ioning on he e en θ≥s(ˆ
y),ac ionˆ
yis
op imal o he p incipal in expec a ion. The second-o de condi ion (6) is necessa y
o ˆ
y o cons i u e a local maximize . Fo he ollowing esul , we will main ain ha his
condi ion holds s ic ly.
P oposi ion 1. Assume ha he ull awa eness p oblem (3) has a unique maximize
D∗(YA)and ha he uppe h eshold ˆ
yis a limi poin o D∗(YA),wi h(6)holdingas
s ic inequali y. Then
ˆ
y/∈YP=⇒ YA/∈Y∗.
P oposi ion 1shows ha , unde he s a ed condi ions, i he p incipal is ini ially un-
awa e o he highes ac ion in he op imal delega ion se unde ull awa eness, hen he
agen inds i p o i able o hide some o he easible ac ions om he p incipal. To p o e
he esul , we conside a simple pe u ba ion o he ull awa eness se . The pe u ba ion
en ails ha he p incipal emains unawa e o an in e al (y−,y+)o ac ions a ound he
7Since YPis a closed se , unawa eness o ˆ
yimplies ha he p incipal is unawa e o an in e al a ound ˆ
y.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 259
The es ic ion o uling-in con ac s a ises na u ally i one iews he p incipal’s
p oblem as designing a di ec mechanism. Unde his in e p e a ion, he p incipal com-
mi s o a mapping om messages o ac ions, whe e unawa eness imposes es ic ions
on he image o such mappings. In pa icula , he p incipal canno commi o an ac ion
ha she does no know o exis . I migh be in e es ing o conside mo e complex ela-
ionships be ween he p incipal’s awa eness and implemen able ac ion p o iles h ough
indi ec mechanisms. Fo ins ance, he p incipal could a emp o pe mi addi ional ac-
ions h ough an indi ec desc ip ion o hose. Whe he his imp o es he p incipal’s
wel a e o ac ually hu s he depends on he de ails o he model, o which he p incipal
is unawa e. Any a e sion o such unknown possibili ies migh in ac call o a desc ip-
ion o ac ions as speci ic as possible gi en he p incipal’s language.
An in ospec i e p incipal—one who is awa e o he unawa eness—migh , howe e ,
wonde whe he he e a e any con ac s ha can imp o e on he op imal delega ion se
wi hou gi ing he agen blanke app o al o ake unknown ac ions. One such possi-
bili y is o add a con ac ual clause speci ying ha he ini ial con ac can be adjus ed
when new op ions come o ligh and pa ies mu ually ag ee (see also Pie mon (2017)).
Speci ying a con ac ual clause o his o m would no ely on he p incipal’s abili y o
desc ibe ac ions ou side he awa eness and would hedge he agains he possibili y o
he agen aking ha m ul ac ions wi hou he consen . In he spi i o he incomple e
con ac li e a u e, one could hen iew he ini ial delega ion se as a p elimina y ag ee-
men ha can be enego ia ed when new, mu ually bene icial op ions appea . We ex-
plo e his possibili y in he ollowing sec ion. In con as o he incomple e con ac s
li e a u e, we will main ain he assump ion ha he p incipal has ull commi men , so
enego ia ion is in ac ully a oidable. We ask ins ead whe he unde limi ed awa e-
ness he p incipal can ac ually bene i om olun a ily gi ing up some o ha commi -
men .
4. Ex pos enego ia ion
Suppose ha , a he han ully commi ing o he ini ial delega ion se , he p incipal
p oposes a con ac ha ixes a se o pe mi ed ac ions bu allows o an adjus men
when new op ions appea . We hus conside con ac s unde which he agen can ene-
go ia e wi h he p incipal o e ac ions ha we e no disclosed (o simply no pe mi ed)
in he ini ial s age o he game. C ucially, we allow he agen o p opose such ac ions
a e he obse es he payo s a e, he eby signaling in o ma ion. In pa icula , upon
ecei ing a p oposal o a new ac ion, he p incipal in e s ha he agen makes such
a p oposal only i aking he new ac ion bene i s him. Howe e —due o he p incipal’s
limi ed awa eness—she canno concei e o al e na i e ac ions he agen could ha e dis-
closed ins ead, and hence canno lea n om pa icula ac ions no being p oposed. This
asymme y a ises as a consequence o he p incipal’s limi ed awa eness and plays a c u-
cial ole in he esul s ha ollow.
Themodi iedgamehas wophases, he con ac ing phase and he enego ia ion
phase. The con ac ing phase is he same as be o e: he agen discloses a se o ac ions
and he p incipal de e mines a delega ion se . In he enego ia ion phase, he agen i s

260 Aus e and Pa oni Theo e ical Economics 19 (2024)
obse es he payo s a e θand hen decides be ween wo op ions. Ei he he picks an
ac ion om he delega ion se o he p oposes a di e en ac ion o he p incipal, who
can hen accep he p oposal o keep he o iginal delega ion se .
S a egies and belie s While we e u n o he con ac ing phase a he end o he sec-
ion, ou main ocus will lie on he enego ia ion phase. To his end, we ix he p incipal’s
in e im awa eness se Y∈Yand a delega ion se D⊆Yas p imi i es. The se Yis in e -
p e ed as he p incipal’s upda ed awa eness a e he con ac ing phase and he se Das
he co esponding delega ion se . Nex , we de ine he s a egies o he p incipal and he
agen . The agen ’s possible mo es a e ei he “no new p oposal” (le us call i N) o sin-
gle on p oposals x∈YA. The se o possible p oposals o he agen is hus X:=N∪YA
and he agen ’s s a egy is a map x:[0, 1]→X om he possible ealiza ions o θ o a ec-
ommenda ion. Upon ecei ing a new p oposal, he p incipal needs o decide whe he
o accep o ejec i . He s a egy is a mapping ρ:X→{0, 1},whe eρ(x)=0 means ha
he p incipal ejec s he agen ’s p oposal and keeps he o iginal delega ion se D, while
ρ(x)=1 means ha he p incipal accep s he agen ’s p oposal and he implemen ed
ac ion is x. To accoun o he ac ha a e “no new p oposal” he o iginal delega ion
se mus be kep , we se ρ(N)=0. Whene e he e is no new p oposal o he p oposal is
ejec ed, he agen chooses an ac ion om he o iginal se D. The agen ’s op imal choice
in his case is desc ibed by y∗(θ,D), as in oduced in Sec ion 3. We will ake his pa o
he agen ’s s a egy as gi en.
We es ic s a egies xand ρ o be uppe semicon inuous unc ions.12 In ui i ely,
his amoun s o assuming ha , in case o indi e ence, he agen b eaks ies in a o o
p oposing a new ac ion, and he p incipal b eaks ies in a o o allowing new p oposals.
I is easy o see ha his assump ion gene a es equilib ium se s o pe mi ed ac ions ha
a e closed. We u he concen a e on ou comes in pu e s a egies. Finally, we deno e
o each x∈X he se o concei able p oposals unde he p incipal’s upda ed awa eness
Y∪{x}by Xx:=Y∪{x}∪{N}.
De ini ion 1. Fix an awa eness se Y∈Yand a delega ion se D⊆Y.Thes a egy
p o ile (x∗,ρ∗), oge he wi h a belie unc ion μ∗(·|x)∈([0, 1]) o each x∈Xand a
collec ion o s a egy pe cep ions (x∗
x)x∈X,wi hx∗
x:[0, 1]→Xx, cons i u es a PBE o he
enego ia ion game i and only i he ollowing condi ions hold:
1. P incipal op imali y: o all x∈X N,
ρ∗(x)∈a g max
ρ∈{0,1}ρEμ∗(·|x)UP(θ,x)+(1−ρ)Eμ∗(·|x)UPθ,y∗(θ,D);
2. Agen op imali y: o all θ∈[0, 1],
x∗(θ)∈a gmax
x∈Xρ∗(x)UA(θ,x)+1−ρ∗(x)UAθ,y∗(θ,D);
12To de ine uppe semicon inui y o he agen , associa e a nega i e numbe o Nin he codomain o his
s a egy.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 261
3. Consis ency o belie s: o all x∈X,μ∗(·|x)is consis en wi h he pe cei ed s a egy
x∗
x(·),whe e
x∗
x(θ)∈a g max
x∈Xx
ρ∗xUAθ,x+1−ρ∗xUAθ,y∗(θ,D). (14)
In pa icula , le ing ∗(x)deno e he p eimage o x o he unc ion x∗
x(·),μ∗(·|x)is
de i ed ia Bayes ule whene e ∗(x)dF(θ)>0. I ∗(x)dF(θ)=0bu ∗(x)=∅,
hen μ∗(·|x)is an a bi a y dis ibu ion wi h suppo ∗(x). Finally, i ∗(x)=∅,
hen μ∗(·|x)is un es ic ed;
The main no el y in De ini ion 1is he awa eness-adap ed consis ency condi ion,
which assu es ha he p incipal’s belie s a e cohe en wi h he agen playing op imally
in he game as pe cei ed h ough he p incipal’s awa eness. To o malize his equi e-
men , we need o accoun o he ac ha he p incipal’s pe cep ion o he agen ’s se o
easible s a egies depends on he p incipal’s upda ed awa eness se , and hence on he
agen ’s ealized p oposal. Indeed, each p oposal x∈Xinduces a di e en subjec i e
game in he mind o he p incipal. We hus de ine o each x∈X, a pe cei ed s a egy
x∗
x, which maps he s a e θ∈[0, 1] o a easible ecommenda ion x∈Xxin he p in-
cipal’s subjec i e game. Condi ion (14) hen equi es ha in his game, s a egy x∗
xis
op imal agains he p incipal’s equilib ium s a egy ρ∗. The key implica ion o he con-
sis ency condi ion is ha a e any change o awa eness ollowing he agen ’s p oposal,
he p incipal’s s a egy and he belie s a e pa o an equilib ium in he esul ing subjec-
i e game.13
Accep able p oposals We now ask which p oposals he p incipal is willing o accep in
a enego ia ion equilib ium. To answe he ques ion, we es ic a en ion o ini ial del-
ega ion se s D ha sol e he p incipal’s delega ion p oblem (3): D=D∗(Y).Gi en he
p incipal’s awa eness in he con ac ing phase, delega ion se D∗(Y)is indeed op imal,
since any ac ion in Y, which he p incipal plans o pe mi in he enego ia ion phase,
can di ec ly be included in he delega ion se . Gi en his es ic ion, we can show ha
in any equilib ium o he enego ia ion game, he p incipal pe mi s an agen ’s p oposal
x/∈D∗(Y)only i she would ha e p e e ed o add he ac ion o he ini ial delega ion se
D∗(Y)a he con ac ing s age. The se o ac ions sa is ying his equi emen is de ined
by
A(Y):=x∈YA:VPD∗(Y)∪{x}≥VPD∗(Y),
whe e VPis he p incipal’s alue in he “ ull commi men delega ion” benchma k.
P oposi ion 5. Fix an awa eness se Y∈Yand a delega ion se D=D∗(Y).
(i) In any equilib ium (x∗,ρ∗,(μ∗(·|x),x∗
x)x∈X),i ρ∗(x)=1, henx∈A(Y).
(ii) The e is an equilib ium such ha ρ∗(x)=1 o all x∈A(Y).
13We could impose he consis ency condi ion only on belie s ollowing on-pa h p oposals. None o ou
esul s would be a ec ed. While his is ob ious o P oposi ions 5(ii), 6, and 7, also P oposi ion 5(i) emains
alid, as he a gumen p o ing i does no ely on he speci ica ion o o -pa h belie s (see Appendix A.5).
262 Aus e and Pa oni Theo e ical Economics 19 (2024)
P oposi ion 5cha ac e izes he se o p oposals ha can be accep ed by he p incipal
in equilib ium. By de ini ion o D∗(Y), hese A(Y)does no include any ac ions ha
belong o Yo he han hose al eady in he delega ion se D∗(Y). Hence, in equilib ium
we ha e ρ∗(x)=0 o allx∈Y D∗(Y). This means ha , gi en delega ion se D∗(Y), he
agen can only gain om enego ia ion i he discloses new ac ions o which he p inci-
pal was p e iously unawa e. Conside hen an equilib ium whe e p oposal x/∈Yis ac-
cep ed in he enego ia ion phase. By he consis ency condi ion, he p incipal’s belie s
a e his p oposal ha e suppo
∗(x)=θ∈[0, 1]:UA(θ,x)≥max
y∈D∗(Y)UA(θ,y). (15)
Acco ding o he p incipal’s awa eness, xis he only new ac ion ha he agen can p o-
pose. The p incipal hus belie es ha he agen p oposes xwhene e he p e e s i o e
his bes al e na i e in D∗(Y). The ques ion is hen whe he condi ional on he agen
p e e ing xo e he ac ions belonging o D∗(Y), he p incipal p e e s xas well. The
answe o his ques ion is yes i and only i he p incipal would ha e p e e ed o add x
o he delega ion se D∗(Y), i.e., i and only i x∈A(Y). This is because adding an ac ion
o he delega ion se changes he ou come only in hose s a es whe e he agen p e e s
he ac ion o he al e na i es in he delega ion se . In he enego ia ion phase, he same
conside a ion applies.
The bene i o pa ial commi men The p e ious esul demons a es ha by adding a
enego ia ion op ion o he op imal delega ion se he p incipal can keep some lexi-
bili y o implemen addi ional ac ions should he awa eness g ow while gene a ing he
same ou come as unde ull commi men in case he awa eness emains unchanged.
We show nex ha , i Assump ion 1holds, pa ial commi men indeed domina es ull
commi men .
P oposi ion 6. Le Assump ion 1be sa is ied. Fix an awa eness se Y∈Y, a delega ion
se D=D∗(Y)and conside a enego ia ion equilib ium (x∗,ρ∗,(μ∗(·|x),x∗
x)x∈X)whe e
ρ∗(x)=1 o all x∈A(Y)and ρ∗(x)=0o he wise. The p incipal’s expec ed payo in his
equilib ium is VP(A(Y)) and sa is ies
VPA(Y)≥VPD∗(Y). (16)
P oposi ion 6shows ha , ocusing on he equilib ium whe e he se o accep ed
p oposals is maximal, he p incipal bene i s om pa ially o going he commi men
i Assump ion 1is sa is ied. In equilib ium, he se o implemen able ac ions o he
agen is A(Y)and he p incipal’s equilib ium payo (as iewed om he pe spec i e o
a ully awa e ou side obse e ) is gi en by VP(D∗(Y)∪A(Y)). By Assump ion 1, hese
A(Y)includes all ac ions in [yA(0),maxD∗(Y)]. In ui i ely, his means ha he agen
can “close po en ial gaps” o he o iginal delega ion se D∗(Y) h ough enego ia ion.
By he same assump ion, con exi ying he se no only bene i s he agen bu also he
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 263
p incipal. The se A(Y)may also include ac ions s ic ly highe han max D∗(Y).In he
p oo , we show ha hei inclusion bene i s he p incipal as well.14
In o ma ion ansmission A s iking ea u e o he desc ibed equilib ium is ha he im-
plemen ed ac ion is s ic ly inc easing in he s a e o all θsuch ha yA(θ)≤maxD∗(Y),
e en when he ini ial delega ion se D∗(Y)has gaps. This would no be possible unde
ull awa eness: in any candida e equilib ium whe e ypes pe ec ly sepa a e hemsel es
h ough hei announcemen , he ully awa e p incipal lea ns he payo s a e and has
incen i es o de ia e o a s ic ly lowe ac ion, a leas in some s a es. In he case o lim-
i ed awa eness, howe e , he p incipal canno con empla e he agen ’s mo es o which
she emains unawa e and his limi s he ex en o which she in e s in o ma ion om he
agen ’s ecommenda ion.
In pa icula , i he ealized alue is θand he agen p oposes an ac ion yA(θ)/∈Y
such ha yA(θ)∈[yA(0),max D∗(Y)], he subjec i e game ee ha ep esen s he p in-
cipal’s ame o mind a e upda ing does no include mo es o he agen in ol ing a
p oposal jus below o abo e yA(θ). As a consequence, he p incipal canno concei e
o he ac ha she would ha e pe mi ed such ac ions i he agen had p oposed hem
ins ead. In he p incipal’s subjec i e game ollowing p oposal yA(θ), he e is an equi-
lib ium whe e he agen e eals yA(θ)in all s a es whe e he agen p e e s yA(θ)o e
he ac ions in he ini ial delega ion se . Each o he agen ’s equilib ium p oposals is hus
pe cei ed o be consis en wi h an in e al o s a es and hese in e als o e lap; ha
is, he p incipal’s in o ma ion can no longe be ep esen ed by a pa i ion o he s a e
space in o pai wise disjoin se s. The disc epancy be ween he agen ’s ue s a egy and
he p incipal’s pe cep ion o i is exac ly wha allows o a con inuum o on-pa h p o-
posals. Some imes he p incipal’s coa se in e ence leads he o accep p oposals ha
she should ejec , e.g., when he agen p oposes an ac ion close o he lowe bounda y
o any po en ial gap in D∗(Y). In expec a ion, howe e , she gains om he addi ional
lexibili y ha she g an s in equilib ium.
Disclosu e in he con ac ing phase While o a ixed awa eness se enego ia ion un-
ambiguously bene i s he p incipal, he p ospec o being able o enego ia e a e he
a i al o in o ma ion a ec s he agen ’s disclosu e incen i es in he con ac ing phase.
The agen ’s ini ial disclosu e de e mines he p incipal’s delega ion se and wi h ha he
se o p oposals he p incipal is willing o accep in he enego ia ion phase. His goal is
o maximize he inal se o pe mi ed ac ions, whe he pe mission is gi en in he con-
ac ing phase o in he enego ia ion phase. Focusing again on he case whe e, upon
inducing awa eness Y, he agen expec s he p incipal o delega e D∗(Y)and accep
any addi ional p oposal in A(Y), he agen ’s op imal disclosu e in he con ac ing phase
sol es he p oblem
max
Y1
0
UAθ,y∗θ,A(Y)dF(θ)(17)
14To see why Assump ion 1is needed, suppose i is no sa is ied. Then we canno ule ou he possibil-
i y o ha e an awa eness Y,adelega ionse D∗(Y), and wo ac ions y,y/∈D∗(Y)such ha VP(D∗(Y)∪
{y}),VP(D∗(Y)∪{y})≥VP(D∗(Y)) and VP(D∗(Y)∪{y,y})<VP(D∗(Y)). The p incipal may hus be
wo se o a e enego ia ion.
264 Aus e and Pa oni Theo e ical Economics 19 (2024)
subjec o YP⊆Y⊆YA. When Assump ion 1is sa is ied, his p oblem has a simple
solu ion.
P oposi ion 7. Le Assump ion 1be sa is ied. A solu ion o p oblem (17)exis sandis
gi en by he awa eness se Y ha sol es
max
YP⊆Y⊆YAmaxD∗(Y).
The p oposi ion shows ha he agen ’s highes equilib ium payo is a ained by ha -
ing he agen disclose a se o ac ions in he con ac ing phase ha maximizes he uppe
h eshold o he p incipal’s esul ing delega ion se . Since h ough enego ia ion, he
agen is able o implemen all ac ions below he uppe h eshold, his maximizes he
agen ’s lexibili y in equilib ium, and hus his equilib ium payo .
Fo conc e eness, conside case (ii) o P oposi ion 2, whe e he p incipal’s alue
VP([yA(0),y]) is single-peaked in yand he agen ’s op imal disclosu e policy in oduces
a single gap a ound ˆ
y.Le (ˆ
y−¯
1,ˆ
y+¯
2)be he la ges easible awa eness gap such ha
he co esponding delega ion se is D∗(Y)=[ymin,ˆ
y−¯
1]∪{ˆ
y+¯
2}. The highes ac ion
he p incipal is willing o delega e in he con ac ing phase is hus
max
YP⊆Y⊆YAmaxD∗(Y)=ˆ
y+¯
2.
Gi en awa eness se Y=[ymin,ˆ
y−¯
1]∩[ˆ
y+¯
2,ymax]and delega ion se D∗(Y)=
[ymin,ˆ
y−¯
1]∪{ˆ
y+¯
2}, he e is an equilib ium in he enego ia ion phase whe e he
agen can implemen any ac ion in he in e al [yA(0),ˆ
y+¯
2]. I he ealized s a e
θis such ha yA(θ)∈D∗(Y), he agen does no enego ia e and akes his p e e ed
ac ion yA(θ). I he ealized θis such ha yA(θ)>ˆ
y+¯
2, he agen does no ene-
go ia e ei he , because condi ioning on he e en ha he agen p e e s some ac ion
x>ˆ
y+¯
2o e ˆ
y+¯
2, he p incipal s ic ly p e e s ˆ
y+¯
2. I ins ead he s a e θis such
ha yA(θ)∈(ˆ
y−¯
1,ˆ
y+¯
2), he agen enego ia es and p oposes his p e e ed ac ion
yA(θ). The p incipal is unawa e o o he ac ions in he in e al (ˆ
y−¯
1,ˆ
y+¯
2),and
hus only in e s ha he agen p e e s he p oposed ac ion o all o he ac ions in D∗(Y).
Condi ioning on his in o ma ion, she p e e s xas well.
The model wi h enego ia ion highligh s an impo an aspec conce ning he dy-
namics o unawa eness. Much like in o ma ion, awa eness is no e e sible. This means
ha i a playe becomes awa e o an ac ion oday, he emains awa e o ha ac ion in
he u u e (simila ly o ou comes, e en s, e c.). Hence, he mo e a playe e eals a an
ea ly s age o he game, he smalle he collec ion o he opponen ’s awa eness se s om
which he can choose la e on. When he e is unce ain y abou he u u e, his c ea es in-
cen i es o hide easible ac ions om he o he playe un il he la e s ages o he game.
In ou en i onmen , his p inciple is e lec ed in he ac ha he agen e eals ewe
ac ions in he con ac ing phase when enego ia ing is possible han when i is no . In
he case discussed abo e, he op imal awa eness gap in he con ac ing phase is maxi-
mal when enego ia ing is possible. No ice ha , e en wi hou enego ia ion, he agen
could implemen any single ac ion below ˆ
y+¯
2by e ealing he “ igh ” se o ac ions.

Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 265
He canno , howe e , implemen all ac ions below ˆ
y+¯
2because some ac ions c owd
ou o he s. The agen has o make a choice based on he expec ed alue o he easi-
ble awa eness se s and he esul ing delega ion se s. When enego ia ion is possible,
ins ead, he agen can condi ion he p incipal’s awa eness on he ealiza ion o θ.
Ex an e wel a e Since he possibili y o enego ia e limi s he agen ’s disclosu e incen-
i es in he con ac ing phase, he anking o he p incipal’s expec ed payo be ween
he wo cases, wi h and wi hou enego ia ion, depends on he p incipal’s ini ial le el o
awa eness. To illus a e his, conside he ollowing example based on he model ana-
lyzed in Sec ion 3.1.
Example. Assume UP(y,θ)=−(y−(θ−β))2,UA=−(y−θ)2.Le θbe uni o mly dis-
ibu ed on [0, 1]and assume β<1/2, so ha delega ion is aluable. The op imal ull
awa eness cap in his example is ˆ
y=1−2β, while he pa ame e cha ac e izing he
agen ’s uncons ained solu ion o he disclosu e p oblem is ∗=2(√2−1)β.Deno eby
¯
(YP) he ac ion in he p incipal’s ini ial awa eness se closes o ˆ
y, whe e o ease o
no a ion, we will supp ess he a gumen YP.
A calcula ion shows ha unde his speci ica ion, he p incipal’s expec ed payo in
he case wi h enego ia ion is highe han in he case wi hou i i and only i
¯

β≤√2−1
23(4√2−5)−1≈1.21. (18)
♦
In wo ds, accoun ing o he agen ’s ini ial disclosu e incen i es, he p incipal is be -
e o wi h enego ia ion i and only i ¯
is su icien ly small wi h espec o he agen ’s
bias β. To unde s and his p ope y, no e i s ha i ¯
≤∗, he agen ’s op imal disclo-
su e in he con ac ing phase is no a ec ed by he possibili y o enego ia e la e : in
ei he case, he op imal awa eness gap o he agen is (ˆ
y−¯
,ˆ
y+¯
)and he esul ing
delega ion se is D∗(Y∗)=[0, ˆ
y−¯
]∪{ˆ
y+¯
}. Gi en ha he p incipal p e e s o ha e he
gap in D∗(Y∗)closed, she is s ic ly be e o when enego ia ion is allowed. Mo eo e ,
he la ge he bias β, he la ge ∗, and hence he gain om enego ia ion. I ins ead
¯
>
∗, he possibili y o enego ia e gi es he agen incen i es o lea e he p incipal
unawa e o mo e ac ions in he con ac ing phase han in he case o ull commi men ,
hus a adeo a ises. The p incipal’s expec ed payo in he case o pu e delega ion is
now VP([0, ˆ
y−∗]∪{ˆ
y+∗})(independen o ¯
), while he expec ed payo in he case o
enego ia ion is VP([0, ˆ
y+¯
]).15 The anking o hese wo payo s depends on how high
ˆ
y+¯
is, i.e., how much addi ional lexibili y he agen gains in he case o enego ia ion.
Condi ion (18)isequi alen oVP([0, ˆ
y+¯
]) ≥VP([0, ˆ
y−∗]∪{ˆ
y+∗}).
15The closed- o m exp essions a e:
VP0, ˆ
y−∗∪ˆ
y+∗=−β2+8
3(3−2√2)β3
VP[0, ˆ
y+¯
]=−
(1−2β+¯
)β2+1
3(¯
−β)3−β3.
266 Aus e and Pa oni Theo e ical Economics 19 (2024)
5. Conclusion
This pape o mula es a lexible delega ion model wi h limi ed awa eness and de i es
se e al p ope ies o he op imal solu ion. The solu ion shows ha by lea ing he p inci-
pal unawa e o mode a e op ions, he agen makes i op imal o he p incipal o pe mi
ac ions close o his own p e e ences. As a gued in he In oduc ion, ou amewo k has
in e es ing implica ions o applica ions. We howe e belie e ha a key componen o
he con ibu ion is o p o ide a leas h ee gene al insigh s ha apply o games wi h a
p incipal-agen s uc u e whe e he agen has supe io awa eness o e easible ac ions.
Fi s , he pape illus a es ha limi ed awa eness can impose na u al cons ain s on
he language o con ac s and ha such limi s may be exploi ed by he con ac ing pa y
wi h supe io awa eness. This p inciple is no es ic ed o delega ion p oblems bu ap-
plies o o he con ac ing p oblems. A p incipal acing a p i a ely in o med agen mus
esol e a adeo be ween exploi ing he agen ’s p i a e in o ma ion and limi ing he
agen ’s in o ma ion en s. The dis o ions sol ing his adeo a e op imal o he p in-
cipal bu no o he agen . By manipula ing he p incipal’s awa eness se , and hence
he se o easible con ac s, he agen can inc ease he p incipal’s cos o such dis o -
ions, he eby inc easing he p incipal’s willingness o g an he agen highe in o ma-
ion en s. The uncons ained solu ion o he agen ’s disclosu e p oblem de e mines he
maximal in o ma ion en s he can ge by modi ying he se o easible ac ions.
Second, he pape shows how in con ac ing si ua ions wi h limi ed awa eness, he
op ion o enego ia ion may be used as a ool o implemen ou comes ha a e no de-
sc ibable a he con ac ing s age, wi hou gi ing he o he pa y blanke app o al o
unknown ac ions. While he exis ing li e a u e la gely ocuses on he cos s caused by
he impossibili y o a oiding ex pos enego ia ions in he p esence o ex an e speci ic in-
es men s, we hus see enego ia ion as an oppo uni y o he p incipal o imp o e he
ou come. The downside o ex pos enego ia ion o he p incipal in ou se ing is he
educ ion in he agen ’s incen i es o disclose ac ions ex an e, gi ing ise o an in e es -
ing adeo . A cle e design o he enego ia ion p ocess may shi his adeo u he
in a o o enego ia ion (see Aghion, Dewa ipon , and Rey (1994)andHa and Moo e
(2004)). An in iguing gene al ques ion is indeed wha a designe wi h limi ed awa eness
can achie e wi h mechanisms ha a e exp essible in he language. The cu en pape
may be iewed as a s ep in ha di ec ion.
Thi d, ou modi ica ion o he game wi h enego ia ion exempli ies how unawa e-
ness changes he ways in which agen s in e in o ma ion. I a playe is unawa e o he
se o possible signals and only becomes awa e o he signal s/he obse es, he playe
canno in e in o ma ion om he ac ha a di e en signal did no ealize. This asym-
me y gi es ise o nons anda d in o ma ion s uc u es, and hence o a he di e en
equilib ium ou comes wi h espec o he ull awa eness benchma k.
Appendix A
A.1 P oo o P oposi ion 1
Le :(YA)2→[0, 1]be a symme ic unc ion, indica ing he s a e a which he agen
is indi e en be ween any wo ac ions yand y.I is speci ied as ollows. Fo y=y,se
(y,y)=s(y)( ecall ha s(·)is he in e se o yA(·)). Fo y<y
, (y,y)is de ined by
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 267
–i UA(θ,y)<UA(θ,y) o all θ∈[0, 1], hen (y,y)=0;
–i UA(θ,y)>UA(θ,y) o all θ∈[0, 1], hen (y,y)=1;
–o he wise (y,y)is such ha
UA y,y,y=UA y,y,y. (19)
Due o he single-c ossing condi ion, he solu ion o (19) is unique. Fo y>y
, (y,y)
is pinned down by he symme y condi ion (y,y)= (y,y). The ollowing lemma links
he slope o swi h a pa ial de i a i e o .
Lemma 8. Conside y0such ha s(y0)∈(0, 1), hen
lim
y→y0
d (y,y0)
dy=1
2s(y0)
P oo .Fo hecasewhe e is de e mined by (19), we apply he implici unc ion heo-
em o de i e
d (y,y0)
dy=UA
y (y,y0),y
UA
θ (y,y0),y0−UA
θ (y,y0),y
Taking he limi , we ha e
lim
y→y0
d (y,y0)
dy=lim
y→y0
UA
y (y,y0),y
UA
θ (y,y0),y0−UA
θ (y,y0),y
=lim
y→y0
UA
θy  (y,y0),yd (y,y0)
dy+UA
yy (y,y0),y
UA
θθ (y,y0),y0−UA
θθ (y,y0),yd (y,y0)
dy−UA
θy  (y,y0),y
=
UA
θy s(y0),y0lim
y→y0
d (y,y0)
dy+UA
yys(y0),y0
−UA
θy s(y0),y0
whe e he second equali y ollows om L’Hôspi al’s ule. Also, ecall (y0,y0)=s(y0).We
can sol e he abo e equali y o limy→y0
d (y,y0)
dyand ob ain
lim
y→y0
d (y,y0)
dy=−1
2·UA
yys(y0),y0
UA
y0s(y0),y0
F om UA
y(s(y),y)=0, we de i e ia he implici unc ion heo em:
s(y)=−UA
yys(y),y
UA
θy s(y),y
which is well-de ined gi en ou assump ion ha UA(θ,y)is in C2and UA
θy >0. The wo
esul s oge he es ablish he claim.
268 Aus e and Pa oni Theo e ical Economics 19 (2024)
De ine ¯
D(y):=D∗(YA)∩[yA(0),y]as he se ob ained by capping he op imal dele-
ga ion se unde ull awa eness a y.
Lemma 9. I (6) holds as a s ic inequali y, he e exis s some y<ˆ
ysuch ha o all y∈
(y,ˆ
y)∩D∗(YA),
VP¯
D(y)<VP¯
D(y)∪{ˆ
y}.
P oo . We de ine he di e ence be ween he p incipal’s expec ed payo s when adding
ac ion ˆ
y o a delega ion se whose highes ac ion is y<ˆ
y. Gi en he s ic mono onici y
o he agen ’s p e e ed ac ion in θ, adding ac ion ˆ
y o a delega ion se Dwi h maxD=
y<ˆ
yonly changes he ou come in he s a es whe e he agen op imally swi ches om y
o ˆ
y. The se o s a es whe e his happens is ( (y,ˆ
y),1
], so he payo di e ence is

W(y):=1
(y,ˆ
y)
UP(θ,ˆ
y)dF(θ)−1
(y,ˆ
y)
UP(θ,y)dF(θ)
No e ha o all y∈(y,ˆ
y)∩D∗(YA),weha e
VP¯
D(y)∪{ˆ
y}−VP¯
D(y)=
W(y)
We calcula e he i s de i a i e o 
W(·)and e alua e i a ˆ
y:

W(y)=−1
(y,ˆ
y)
UP
y(θ,y)dF(θ)−UP (y,ˆ
y),ˆ
y−UP (y,ˆ
y),y  (y,ˆ
y)d (y,ˆ
y)
dy

W(ˆ
y)=−1
s(ˆ
y)
UP
y(θ,ˆ
y)dF(θ)
By (5), he abo e e m is equal o ze o. We mus he e o e conside he second de i a i e:

W(y)=−1
(y,ˆ
y)
UP
yy(θ,y)dF(θ)+2UP
y (y,ˆ
y),y  (y,ˆ
y)d (y,ˆ
y)
dy
−UP
θ (y,ˆ
y),ˆ
y−UP
θ (y,ˆ
y),y  (y,ˆ
y)d (y,ˆ
y)
dy 2
−UP (y,ˆ
y),ˆ
y−UP (y,ˆ
y),y  (y,ˆ
y)d (y,ˆ
y)
dy +  (y,ˆ
y)d2 (y,ˆ
y)
dy2

W(ˆ
y)=−1
s(ˆ
y)
UP
yy(θ,ˆ
y)dF(θ)+2UP
ys(ˆ
y),ˆ
y s(ˆ
y)d (y,ˆ
y)
dy y=ˆ
y
Since, by Lemma 8,weha ed (y,ˆ
y)
dy |y=ˆ
y=1
2s(ˆ
y), condi ion (6) holding as a s ic inequal-
i y implies 
W(ˆ
y)>0. Remembe ing 
W(ˆ
y)=0, he e is hen an in e al o y o he
le o ˆ
y,whe e
W(y)<0. Wi h 
W(ˆ
y)=0, his p ope y implies, in u n, ha he e is
some y<ˆ
ysuch ha 
W(y)>0 o ally∈(y,ˆ
y). Hence, o all y∈(y,ˆ
y)∩D∗(YA),we
ha e VP(¯
D(y)∪{ˆ
y})−VP(¯
D(y)) =
W(y)>0.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 275
is s ic ly conca e wi h he in e io solu ion cha ac e ized by (11). The condi ions o ap-
plying he implici unc ion heo em a e again sa is ied, hence he e is a unc ion ∗(β)
desc ibing he uncons ained solu ion o he agen ha sol es he i s -o de condi ion
U(∗(β);β)=0, which becomes an iden i y when seen as a unc ion o β,and
∗(β)=−Uβ∗(β);β
U∗(β);β.
To p o e he s a emen o he p oposi ion, we mus hen show Uβ(∗(β);β)>0. Di -
e en ia ing he exp ession o he i s -o de condi ion (11) wi h espec o βkeeping ∗
ixed, a e some ea angemen , deli e s
Uβ∗(β);β=−ˆ
y(β)1+Fˆ
y(β)−∗(β)−2Fˆ
y(β).
Gi en ˆ
y(β)<0(see(24)),wea edonei 1+F(ˆ
y(β)−∗(β))−2F(ˆ
y(β)) >0, o equi a-
len ly
21−Fˆ
y(β)>1−Fˆ
y(β)−∗(β).
Using ˆ
y(β)=E[θ−β|θ≥ˆ
y(β)], he i s -o de condi ion (11) can be w i en as
1−Fˆ
y(β)−∗(β)Eθ|θ≥ˆ
y(β)−∗(β)−ˆ
y(β)−∗(β)
=21−Fˆ
y(β)β. (25)
Since ˆ
y(β)−∗(β)is s ic ly smalle han ˆ
y(β), he ollowing condi ion holds:
Eθ−β|θ≥ˆ
y(β)−∗(β)−ˆ
y(β)−∗(β)>0.
Equi alen ly, we can w i e
Eθ|θ≥ˆ
y(β)−∗(β)−ˆ
y(β)−∗(β)>β.
Gi en his inequali y, (25) equi es2
(1−F(ˆ
y(β))) >1−F(ˆ
y(β)−∗(β)), as desi ed.
A.5 P oo o P oposi ion 5
P oo . We begin wi h pa (i). Conside a enego ia ion equilib ium and le ˆ
X:={x∈
Y:ρ∗(x)=1}be he se o ac ions in he p incipal’s ini ial awa eness se which a e al-
lowed in equilib ium. By uppe semicon inui y o ρ∗, hisse ˆ
Xis closed. Since also
D∗(Y)is closed, he se D∗(Y)∪ˆ
Xis closed as well. We i s wan o show ha
VPD∗(Y)∪ˆ
X≥VPD∗(Y).
Since he agen can always gua an ee choices in D∗(Y)by p oposing N,agen ’sop-
imali y in he eyes o he p incipal implies x∗
x(θ)=y∗(θ,D∗(Y)∪ˆ
X) o all x∈ˆ
Xand
θ∈x∈ˆ
X∗(x), wi h he exclusion o poin s whe e he agen is indi e en be ween wo
ac ions in D∗(Y)∪ˆ
X. Fo each x∈D∗(Y)∪ˆ
X,we husha e
cl∗(x)=θ∈[0, 1]:UA(θ,x)≥max
y∈D∗(Y)∪ˆ
X
UA(θ,y)

276 Aus e and Pa oni Theo e ical Economics 19 (2024)
Using he ac ha o allx∈ˆ
Xand all θ∈∗(x),weha ex=y∗(θ,D∗(Y)∪ˆ
X)and ha
o each B⊆∗(x)we ha e ∗(x)μ∗(B|x) (θ)dθ=BdF(θ),17 he p incipal’s op imal-
i y condi ion yields
∗(x)∗(x)
UP(θ,x)dμ∗(θ|x)dFθ≥∗(x)∗(x)
UPθ,y∗θ,D∗(Y)dμ∗(θ|x)dFθ
⇐⇒ ∗(x)
UPθ,y∗θ|D∗(Y)∪XdF(θ)≥∗(x)
UPθ,y∗θ,D∗(Y)dF(θ)
o all x∈ˆ
X.
Nex , o θ/∈x∈ˆ
X∗(x),weha ex∗
x(θ)=No ρ∗(x∗
x(θ)) =0. In ei he case, he
agen will ake y∗(θ,D∗(Y)).We u he ha ey∗(θ,D∗(Y)) =y∗(θ,D∗(Y)∪ˆ
X) o all
θ/∈x∈ˆ
X∗(x), since he agen is ee o ake any ac ion in D∗(Y)∪ˆ
X.
Se ing C(ˆ
X):=[0, 1] (x∈ˆ
X∗(x)), we can now in eg a e o e ˆ
Xand ob ain
C(ˆ
X)
UPθ,y∗θ,D∗(Y)∪ˆ
XdF(θ)+ˆ
X∗(x)
UPθ,y∗θ,D∗(Y)∪ˆ
XdF(θ)dx
≥C(ˆ
X)
UPθ,y∗θ,D∗(Y)dF(θ)+ˆ
X∗(x)
UPθ,y∗θ,D∗(Y)dF(θ)dx
o equi alen ly
VPD∗(Y)∪ˆ
X≥VPD∗(Y).
Recall ha D∗(Y)∪ˆ
X⊆Yis a closed se . Since D∗(Y)is he la ges closed op imal
awa eness se wi h espec o Y ha includes ac ions ha will ac ually be aken by he
agen unde some con ingency, his inequali y yields a con adic ion unless D∗(Y)∪ˆ
X=
D∗(Y).
Ha ing shown ha he p incipal only accep s addi ional ac ions in he enego ia-
ion phase i hey do no belong o Y, conside p oposal x∈YA Ysuch ha ρ∗(x)=
1. Pe cei ed agen op imali y hen equi es x∗
x(θ)=x o all θsuch ha UA(θ,x)>
maxy∈D∗(Y)UA(θ,y). P incipal op imali y in u n equi es ha condi ioning on he e en
UA(θ,x)>U
A(θ,y∗(θ,D∗(Y))), he p incipal p e e s xo e y∗(θ,D∗(Y)) in expec a-
ion. This is he case only i x∈A(Y).
To show pa (ii), we need o cons uc an equilib ium o he enego ia ion game
whe e p oposal xis accep ed by he p incipal whene e x∈A(Y).To hisend,we
se ρ∗(x)=1 o allx∈A(Y)and ρ∗(x)=0 o he wise. Recall ha ρ∗(N)=0. Fo he
agen , we se x∗(θ)=a gmaxx∈A(Y)UA(θ,x)i maxx∈A(Y)U(θ,x)≥maxy∈D∗(Y)UA(θ,y)
and x∗(θ)=No he wise.18 Simila ly, o he agen ’s s a egy as pe cei ed by he p in-
cipal when ecei ing p oposal x,wese x∗
x(θ)=xi U(θ,x)≥maxy∈D∗(Y)UA(θ,y)and
17No e ha mono onici y o he agen ’s op imal policy y∗in θimplies ha ∗(x)is ei he o posi i e
measu e o a single on.
18Recall in case o indi e ence he agen b akes ies in a o o he p incipal.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 277
x∗
x(θ)=No he wise. The p incipal belie s sys em μ∗is de ined as ollows. Fo all x∈X,
μ∗(B|x)=B
dF(θ
∗(x)
dF(θ)∀B⊆∗(x),
whene e ∗(x)=(x∗
x)−1(x)is o posi i e measu e, and μ∗({θ}|x)=1i ∗(x)={θ}.I
can be checked di ec ly ha his s a egy and belie p o ile sa is y p incipal op imali y,
agen op imali y, and consis ency o belie s, and hus cons i u e a PBE o he enego ia-
ion game, as speci ied in De ini ion 1.
A.6 P oo o P oposi ion 6
P oo . No e ha , unde Assump ion 1, o anyx∈(minD∗(Y),maxD∗(Y)),weha e
VP(D∗(Y)∪{x})≥VP(D∗(Y)),soxbelongs o he se o accep able p oposals. The same
is ue o all x<minD∗(Y), since condi ioning on he ac ha he (upwa d biased)
agen p e e s xo e minD∗(Y), he p incipal p e e s xas well. The se o implemen able
ac ion A(Y) hus includes all ac ion in [yA(0),max D∗(Y)]. Unde Assump ion 1,we
clea ly ha e VP([yA(0),maxD∗(Y)]) ≥VP(D∗(Y)).I max D∗(Y)=max A(Y), hiscon-
cludes he a gumen . Fo he o he case, le ¯
y:=maxA(Y)assume ¯
y>max D∗(Y).By
de ini ion o A,weha eVP(D∗(Y)∪{¯
y})≥VP(D∗(Y)). Mono onici y o he agen ’s
ac ion in θ hen implies
VPyA(0),maxD∗(Y)∪{¯
y}≥VPyA(0),maxD∗(Y)
Bu gi en ha pe mi ing ac ion ¯
yweakly inc eases he p incipal’s expec ed payo , pe -
mi ing any addi ional ac ion in (max D∗(Y),¯
y)bene i s he p incipal as well (again by
Assump ion 1), so (16) is sa is ied.
A.7 P oo o P oposi ion 7
P oo . We s a by showing he exis ence o a solu ion o maxYD∗(Y). Recall ha
BRP(·)deno es he p incipal’s solu ion co espondence o p oblem (3). Le us hen
de ine
ˇ
y(Y):=sup
D∈BRP(Y)
maxD.
Since any D∈BRP(Y)is compac , he las max is well-de ined. F om he p oo o P opo-
si ion 12, we know ha BRPis uppe hemicon inuous. I we show ha he unc ion
m(D):=maxDis con inuous in D, he gene alized e sion o he maximum heo em
(e.g., Theo em 17.30 in Alip an is and Bo de (2006)) implies ha he max exis s o each
Yand ˇ
y(Y)is uppe semicon inuous in Y. This in u n implies ha he ollowing objec
is well-de ined:
¯
y∗:=max
YP⊆Y⊆YAˇ
y(Y).
278 Aus e and Pa oni Theo e ical Economics 19 (2024)
Lemma 10. The unc ion m(D):=max Dis con inuous wi h espec o he Hausdo me -
ic.
P oo . Recall again ha we a e wo king wi h me ic spaces. Take a sequence Dn→HD.
Conside now he sequence o eal numbe s dn:=max Dn∀n. We need o show ha he
sequence con e ges o d:=maxD∈R. Since Dncon e ges, i is Cauchy. We wan o
show ha also dnis Cauchy. Fo any δ,le Nδbe such ha dH(Dn,Dm)≤δ∀n,m≥
Nδ.Now,i dn=dm he e is no hing o p o e. Suppose dn= dm. Then, wi hou loss o
gene ali y, assume dn>d
m.Weha e
|dn−dm|=|max Dn−max Dm|= in
y∈Dm|dn−y|≤dH(Dn,Dm)≤δ.
Gi en ha δis gene ic, dnis Cauchy, and since Ris comple e, he sequence dnmus con-
e ge. Le d∗be he con e ging poin o he sequence. Again, wi hou loss o gene ali y
assume d∗>d. Bu hen, by he de ini ion o con e gence, i mus be ha o Nla ge
enough, dn>d o all n≥N. Deli e ing
dH(D,Dn)≥in
y∈D|dn−y|=|dn−d|>0∀n≥N,
which con adic s he ac ha Dncon e ges o D.Hence,i mus be ha d∗=d.
Nex , we wan o show ha o each Y,ˇ
y(Y)is equal o max D∗(Y). Suppose his
is no ue. Then he e exis s an awa eness se ˜
Ysuch ha ˇ
y(˜
Y)>maxD∗(˜
Y)and a
delega ion se ˜
D∈BRP(˜
Y)such ha max ˜
D>maxD∗(˜
Y). Since i is ne e op imal o
cu ail he agen ’s lexibili y om below and Assump ion 1holds, we ha e
min ˜
D=minD∗(˜
Y)=y∗(0, ˜
Y)
Gi en min ˜
D=minD∗(˜
Y)≤maxD∗(˜
Y)<max ˜
D, Assump ion 1implies VP(˜
D∪D∗(˜
Y)) ≥
VP(˜
D), and hence (˜
D∪D∗(˜
Y)) ∈BRP(˜
Y). Bu since D∗selec s he agen -p e e ed dele-
ga ion se om BRP(˜
Y)and since D∗(˜
Y)⊆(˜
D∪D∗(˜
Y)),wemus ha eD∗(Y)=˜
D,and
hence maxD∗(Y)=max ˜
D, a con adic ion. Combining hese esul s, we ha e shown
ha
sup
D∈BRP(Y)
maxD=maxD∗(Y),
and hence ¯
y∗=maxYP⊆Y⊆YAmaxD∗(Y).
Le ¯
Y∗be an awa eness se ha maximizes max D∗(Y)o e Ysubjec o YP⊆Y⊆
YA. We now wan o show ha ¯
Y∗sol es (17). Suppose no . Since
A¯
Y∗=yA(0),¯
y∗
he e mus hen exis an awa eness se Ysuch ha max(A(Y)) >¯
y∗. By he de ini ion
o ¯
y∗, he e is no awa eness se Ysuch ha YP⊆Y⊆YAand ¯
y∗∈D∗(Y).Hence,
he emus beap oposalx> ¯
y∗ ha he p incipal accep s in he enego ia ion phase.
Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 279
By P oposi ion 5, his equi esx∈A(Y), o equi alen ly
VPD∗(Y)∪{x}≥VPD∗(Y).
This in u n implies x∈D∗(Y∪{x}), and hence ¯
y∗≥x, a con adic ion.
A.8 Exis ence esul s
Conside he ollowing p ope ies o ou se up:
(a) The se Y⊆YA=[ymin,ymax]is a compac subse o he comple e and sepa able
me ic space (R,|·|).
(b) Recall Ydeno es he se o closed subse s o [ymin,ymax],and
ˆ
D(Y):={D∈Y:D⊆Y}.
Then ˆ
D(Y)is a closed subse o 2[Y]wi h espec o he Hausdo -me ic
dHD,D=maxsup
y∈D
in
y∈Dy−y,sup
y∈D
in
y∈Dy−y,
and hence compac in he opology gene a ed by he Hausdo me ic dH(see
poin 3 o Theo em 3.85 in Alip an is and Bo de (2006)).
(c) Recall ha UAand UPa e con inuous and uni o mly bounded on hei domains
[0, 1]×[ymin,ymax],andFadmi s a densi y.
(d) Recall ha UA
θy >0, UA
y(θ,y)>0, UA
yy(θ,y)<0. Hence, i we ix a closed se D⊆
YAand an open in e al O⊆YA D, he e is a mos one alue o θsuch ha
yA(θ)∈Oand BRA(θ,D)is no single- alued.
(e) The se (ymin,ymax ) Dis an open se , and hence i can be uniquely de ined as a
coun able union o disjoin open in e als (e.g., Theo em 6, p. 51, in Kolmogo o
and Fomin (1975)).
Recall he agen chooses acco ding o BRA(θ,D):=a g maxy∈DUA(y,θ).No e ha by
con inui y BRA(θ,D)is nonemp y o each θ, since D⊆YAis compac om (a) and UA
is con inuous om (c). In addi ion, since he easibili y se Dchanges con inuously wi h
Din he Hausdo no m (and UAis con inuous in (D,y,θ)), he maximum heo em
implies ha BRAis an uppe hemicon inuous co espondence when seen as a unc ion
o (D,θ). In addi ion, combining (d) wi h (e), and no icing ha o all θsuch ha yA(θ)∈
Dwe ha e BRA(θ,D)={yA(θ)},weconclude ha o eachD he se o θ’s o which
BRA(θ,D)is no single- alued is coun able. Since Fadmi s a densi y, he se o alues
o which he agen is indi e en is hen o F-measu e ze o. In summa y, we ha e he
ollowing.
Lemma 11. BRAis a nonemp y uppe hemicon inuous co espondence in (θ,D).Mo e-
o e , o each D, he se A(D):={θ∈[0, 1]|BRA(θ,D)is no a single on}has measu e
ze o acco ding o F.
280 Aus e and Pa oni Theo e ical Economics 19 (2024)
Recall as well, we deno ed wi h y∗ he selec ion ha esol es ies in a o o he p in-
cipal. Due o Lemma 11, o each awa eness se Y∈Y, he p incipal op imally selec s a
delega ion se D⊆Y o sol e
max
D∈ˆ
D(Y)
VP(D)whe e VP(D)=1
0
UPθ,y∗(θ,D)dF(θ). (26)
F om Lemma 4 in Holms öm (1980), VPis uppe semicon inuous in D o each closed
Y⊆YA(whe e dis ances in Da e de ined acco ding o he Hausdo -me ic). Since
acco ding o his me ic he easibili y se ˆ
D(Y)is compac , we ha e he ollowing.
P oposi ion 12. An op imal solu ion o he p incipal’s p oblem in ˆ
D(Y)—and hence in
D(Y)—exis s. In addi ion, VPis con inuous in D.
The exis ence o an op imal solu ion in D(Y)is gua an eed by he ac ha o any
solu ion o (26)in ˆ
D(Y)we ob ain a solu ion in D(Y)by elimina ing ac ions ha he
agen does no ake in equilib ium. The con inui y o Vis implied by he ac ha when-
e e an uppe hemicon inuous co espondence is single- alued i is con inuous. Hence,
he second pa o Lemma 11 implies ha any selec ion h om BRAwill ha e discon i-
nui ies a a se o poin s ha ha e p obabili y ze o acco ding o F.Tha is,
VP(D)=1
0
UPθ,y∗(θ,D)dF(θ)=1
0
UPθ,h(θ,D)dF(θ),
and he la e a ies con inuously wi h Dby he con inui y p ope y o he selec ion h
and he con inui y o UPsumma ized in (c).
Deno e by BRP(Y) he solu ion co espondence o he p incipal’s p oblem. I we
can show ha he co espondence om Y o D(Y)is con inuous, hanks o p ope -
ies (a)–(e), we can apply he heo em o he maximum o show ha BRP(Y)is uppe
hemicon inuous.
Now ecall, we indica e wi h D∗(Y) he selec ion om BRP(Y) ha esol es ies in
a o o he agen . The p oblem o he agen a he ini ial disclosu e s age sol es
max
Y∈YVA(Y)whe e VA(Y)=1
0
UAθ,y∗θ,D∗(Y)dF(θ)such ha
YP⊆Y⊆YA.
I we can show ha he alue VA(Y)is uppe semicon inuous by he p ope y o he
selec ion D∗, we ha e a solu ion. Hence, le us show he ollowing esul .
P oposi ion 13. BRP(Y)is uppe hemicon inuous and he p oblem o he agen has a
leas one solu ion.
P oo . As a gued abo e, he e a e wo c ucial s eps. Fi s , he uppe semicon inui y o
VAand hen he con inui y o he co espondence D(Y).

Theo e ical Economics 19 (2024) Op imal delega ion and in o ma ion ansmission 281
Lemma 14. The unc ion VAis uppe semicon inuous in Yunde he Hausdo me ic.
P oo . Recall ha D∗has he ollowing p ope y:
D∗(Y)=a g max
D∈BRP(Y)1
0
UAθ,y∗(θ,D)dF(θ).
To show uppe semicon inui y, ake a con e ging sequence Yn→H¯
Yand suppose he e
is a sequence (in he eal numbe s) Vncon e ging o ¯
V(in he |·|me ic), whe e o
each n,
Vn=VA(Yn)=1
0
UAθ,y∗θ,D∗(Yn)dF(θ).
We need o show ha ¯
V≤VA(¯
Y). To simpli y no a ion, le ˆ
Dn=D∗(Yn).Wehence
ha easequence ˆ
Dnsuch ha ˆ
Vn=1
0UA(θ,y∗(θ,ˆ
Dn))dF(θ)→¯
V. Since Yis compac ,
he e is a con e ging subsequence, ˆ
Dn→Hˆ
D,and ecall ha ˆ
Vn→¯
V.No enow ha
he unc ion T(D):=1
0UA(θ,y∗(θ,D))dF(θ)is con inuous in Dbased on he same
a gumen s as in he p oo o P oposi ion 12. I mus hence be he case ha T(ˆ
Dn)→
T(ˆ
D). This implies ha ¯
V=1
0UA(θ,y∗(θ,ˆ
D))dF(θ). Now, ecall ha BRPis uppe
hemicon inuous, i.e., i has a closed g aph G . We ha e shown ha ˆ
Dis he limi o a
sequence Dnsuch ha ˆ
Dn∈BRP(Yn) o all n, i.e., (ˆ
Dn,Yn)∈G o all n. The limi poin
mus be in he g aph as well: (ˆ
D,¯
Y)∈G . This is equi alen o saying ha ˆ
D∈BRP(¯
Y),
and hence, om he de ini ion o D∗, we ha e he desi ed inequali y
¯
V=1
0
UAθ,y∗(θ,ˆ
D)dF(θ)≤1
0
UAθ,y∗θ,D∗(¯
Y)dF(θ).
Lemma 15. The co espondence mapping each se Y om he me ic space (Y,dH) o
ˆ
D(Y)is bo h uppe and lowe hemicon inuous
P oo . Fi s o all, no e ha om ˆ
D(Y),weha e
D∈ˆ
D(Y)⇐⇒ D⊆Y.
Since we a e wo king wi h me ic spaces (and hence i s coun able opological spaces),
we can p o e ou s a emen using sequences. (i) Uppe hemicon inui y: ake any Y∈Y
and a gene ic con e ging sequence Yn→HY. Now, ake a sequence Dnsuch ha Dn⊆
Yn o all n. We wan o show ha he e is a subsequence Dnscon e ging o D⊆Y.The
exis ence o a con e ging sequence is implied by he compac ness o he space. So, le
Dbe such a poin . We need o show ha D⊆Y. This is implied by he con e gence
condi ion dH(Dn,D)→0 in he Hausdo me ic. Suppose D⊃Y.I mus hencebe
ha dH(D,Y)=ε>0, i.e., he dis ance be ween he wo se s is posi i e. Now, since
bo h sequences con e ge, o each δ he e is a Nδsuch ha o all n≥Nδwe ha e bo h
dH(Dn,D)≤δand dH(Yn,Y)≤δ. This indica es ha Dncanno be smalle han he
maximal educ ion o Dcompa ible wi h he dis ance and Ycanno be la ge han he
maximal ex ension o Ycompa ible wi h he dis ance. Such educ ions and ex ensions
282 Aus e and Pa oni Theo e ical Economics 19 (2024)
can be made a bi a ily small, so i we ha e D⊃Y,wemus ha e o Nla ge enough
DN⊃YN, which is a con adic ion.
(ii) Lowe hemicon inui y: ake any D⊆Yand a con e ging sequence Yn→HY.We
need o show ha he e is a sequence Dn→HDsuch ha Dn⊆Yn o all n.I D=Y,
we can ake he sequence Dn=Yn. Al e na i ely, suppose D⊂Y. Since Yncon e ges o
Y, o Nsu icien ly la ge, we ha e D⊂Yn o all n≥N. Hence, conside he ollowing
sequence: Dn=Yn o n<Nand Dn=D o n≥N.
This concludes he p oo .
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