Reputation building under uncertain monitoring

Deb, Joyee; Ishii, Yuh a
A icle
Repu a ion building unde unce ain moni o ing
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Deb, Joyee; Ishii, Yuh a (2025) : Repu a ion building unde unce ain moni o ing,
Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 20, Iss. 1, pp.
169-208,
h ps://doi.o g/10.3982/TE4758
This Ve sion is a ailable a :
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Theo e ical Economics 20 (2025), 169–208 1555-7561/20250169
Repu a ion building unde unce ain moni o ing
Joyee Deb
Depa men o Economics, New Yo k Uni e si y
Yuh a Ishii
Depa men o Economics, Pennsyl ania S a e Uni e si y
We s udy he s anda d epu a ion model wi h a long- un (LR) playe acing a se-
quence o sho - un (SR) opponen s, wi h one di e ence: he SR playe s a e un-
ce ain abou he moni o ing s uc u e, while he LR playe knows i . We cons uc
examples whe e he s anda d epu a ion esul b eaks down: E en i he e is a
possibili y ha he LR playe is a commi men ype who always plays he ac ion
o which he wan s o commi , he e exis “bad” equilib ia in which he LR playe
ge s payo s subs an ially lowe han his commi men payo s. In con as , i he e
is he possibili y o dynamic commi men ypes who swi ch be ween “signaling”
ac ions ha help he SR playe s lea n he moni o ing s uc u e and “collec ion”
ac ions ha a e desi able o payo s, ou main heo em shows ha a su icien ly
pa ien LR playe ob ains payo s o a leas he commi men payo s in each s a e
in e e y equilib ium.
Keywo ds. Repu a ion, moni o ing, epea ed games, lea ning.
JEL classi ica ion. C73, L14.
1. In oduc ion
Conside a long- un i m building a epu a ion o p oducing en i onmen ally- iendly
p oduc s. Such a epu a ion is aluable o he i m when consume s ca e abou he
en i onmen al impac o hei pu chases and a e o en willing o pay mo e o g een
p oduc s. Consume s make pu chase decisions based on whe he p oduc s ha e “eco-
iendly” labels, bu a e ypically unsu e o how much o us he labels. Many o hese
labels a e genuine ce i ica ions wi h s ingen s anda ds, bu nume ous o he s ha e
been disc edi ed as being ake. As a esul , on seeing an eco-label, consume s a e un-
ce ain abou i s in o ma ional con en , and may no be con inced abou he p oduc
Joyee Deb: [email p o ec ed]
Yuh a Ishii: [email p o ec ed]
Fo help ul commen s ha signi ican ly imp o ed he pape , we hank he h ee anonymous e e ees, as
well as Dilip Ab eu, Heski Ba -Isaac, Ma in C ipps, Mehme Ekmekci, D ew Fudenbe g, Johannes Hö ne ,
Michihi o Kando i, Ba y Nalebu , Aniko Ö y, Ha y Pei, Andy Sk zypacz, Alex Woli zky, and Jidong Zhou.
We also hank Haoning Chen and Ki i a dhan Singh o excellen esea ch assis ance. Finally, we a e g a e-
ul o semina pa icipan s a B own, Duke, ITAM, Ox o d, Queen Ma y Uni e si y o London, Uni e si y
College London, Uni e si y o Wa wick, Yale, and he SITE Summe Wo kshop 2016 in Dynamic Games,
Con ac s, and Ma ke s.
©2025 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/TE4758
170 Deb and Ishii Theo e ical Economics 20 (2025)
being en i onmen ally iendly.1Bu i consume s do no us p oduc labeling, a i m,
e en a e hones in es men in g een p oduc s and a e unde going eliable labeling,
may ind i di icul o es ablish a posi i e epu a ion and con ince consume s ha i s
p oduc s a e indeed en i onmen ally iendly. This mo i a es he cen al ques ion o he
pape : Can epu a ions be buil in en i onmen s wi h such unce ain y in moni o ing?
To s a , conside epu a ion building in en i onmen s in he absence o such un-
ce ain y. Canonical models o epu a ion (e.g., Fudenbe g and Le ine (1992)) conside
a long- un (LR) agen (a i m) who epea edly in e ac s wi h sho - un (SR) opponen s
(consume s). The e is incomple e in o ma ion abou he i m’s ype: consume s en e -
ain he possibili y ha he i m is o a “commi men ” ype ha is commi ed o playing
a pa icula ac ion in e e y pe iod. E en when he ac ions o he i m a e noisily ob-
se ed, he classical epu a ion esul s a es ha i a su icien ly ich se o commi men
ypes occu s wi h posi i e p obabili y, a pa ien i m can achie e payo s a bi a ily close
o hei S ackelbe g payo o he s age game in e e y equilib ium.2In ui i ely by mim-
icking a commi men ype ha always plays he S ackelbe g ac ion, a LR i m can e en-
ually signal o he consume i s in en ion o play he S ackelbe g ac ion in he u u e
and hus ob ain high payo s in any equilib ium. Impo an ly, his esul emains alid
e en on in oduc ion o o he a bi a y commi men ypes. This in ui ion c i ically e-
lies on he consume ’s abili y o accu a ely in e p e he noisy signals, bu i moni o ing
is unce ain, he epu a ion builde may ind i di icul o signal his in en ions.
To s udy he e ec o unce ain moni o ing, we also conside he canonical model
o a LR i m acing a sequence o SR consume s, bu wi h one key di e ence. A he
beginning o he game, a pe sis en s a e (θ,ω)∈×is ealized, which de e mines
bo h he ype o he i m, ω, and he moni o ing s uc u e, πθ:A1→(Y): a mapping
om ac ions aken by he i m o dis ibu ion o signals, (Y), obse ed by consume s.
We assume ha he i m knows he s a e o he wo ld, bu he consume does no .
We i s show in a simple example ha unce ain moni o ing can cause he adi-
ional epu a ion esul o b eak down: E en i consume s belie e ha he i m may be
a commi men ype ha plays he S ackelbe g ac ion e e y pe iod, he e exis equilib ia
in which e en a pa ien i m ob ains payo s a below i s S ackelbe g payo . Such “bad
equilib ia” a ise due o an iden i ica ion p oblem ha s ems om he unce ain y abou
moni o ing: Good ac ions in one s a e canno be s a is ically dis inguished om a bad
ac ion in a di e en s a e.
Ou simple example wi h such a bad equilib ium leads us o ask wha migh es o e
epu a ion building unde unce ain moni o ing in he ace o such iden i ica ion p ob-
lems. Unde an assump ion ha he ac ion space is su icien ly ich, we cons uc a se o
commi men ypes such ha , i hese ypes occu wi h posi i e p obabili y, a su icien ly
pa ien i m ob ains payo s a bi a ily close o he S ackelbe g payo in all equilib ia,
1The Fede al T ade Commission main ains, “Ve y ew p oduc s, i any, ha e all he a ibu es consume s
seem o pe cei e om such claims, making hese claims nea ly impossible o subs an ia e” (Sou ce: E. Wy-
a , “FTC Issues Guidelines o Eco-F iendly Labels,” New Yo k Times, Oc 1, 2012).
2The S ackelbe g payo is he payo ha he LR playe would ge i he could commi o an ac ion in he
s age game, and he S ackelbe g ac ion is he co esponding commi men ac ion.
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Theo e ical Economics 20 (2025) Repu a ion building 171
e en when he consume s a e unce ain abou he moni o ing en i onmen .3Impo -
an ly, he esul holds independen o he ine de ails o he ype space in ha i emains
alid e en i we include o he a bi a y commi men ypes. The commi men ypes
ha we cons uc a e commi ed o dynamic ( ime-dependen ) s a egies ha swi ch
in ini ely o en be ween signaling ac ions ha help he consume lea n he unknown
moni o ing s a e and collec ion ac ions ha a e desi able o payo s ( he S ackelbe g
ac ion). A key con ibu ion is he cons uc ion o hese dynamic commi men ypes
ha play pe iodic s a egies. As we will discuss la e , such dynamic commi men ypes
a e gene ally necessa y o epu a ion building unde unce ain moni o ing, because
signaling he unknown s a e and S ackelbe g payo collec ion may equi e he use o
di e en ac ions in he s age game.
The p oo o he main esul in ol es es ablishing wo p ope ies, which oge he
imply ha he LR playe can gua an ee payo s close o S ackelbe g payo s in any equi-
lib ium. Fi s , we show ha by mimicking any commi men ype, he LR playe can
ensu e in any equilib ium wi h high p obabili y ha he SR playe s’ p edic ions o he
public signal dis ibu ion a e close o he ue dis ibu ion gene a ed by his commi -
men ype in all bu a ini e numbe o pe iods. This s ep demons a es he classic esul
in he spi i o “me ging o opinions,” à la Blackwell and Dubins (1962), and is p o ed
using s anda d a gumen s om Gossne (2011).4In ou se ing, ensu ing accu a e p e-
dic ions o he public signal dis ibu ion by he SR playe s is no su icien o a epu a-
ion esul due o po en ial iden i ica ion p oblems ac oss s a es. Second, we show ha
by mimicking he app op ia e commi men ype, he LR playe can addi ionally ensu e
ha he SR playe s lea n he s a e a a a e ha is uni o m ac oss all equilib ia. We p o e
his by es ablishing a esul on obus lea ning, which p o ides an easy- o-check su i-
cien condi ion ha gua an ees ha an obse e will lea n he alidi y o an e en a a
uni o m a e ac oss a ich class o lea ning en i onmen s. The condi ion ela es he uni-
o m a e a which Hellinge ans o ms anish ac oss all lea ning en i onmen s in he
class o uni o m lea nabili y o an e en .5To he bes o ou knowledge, he obus lea n-
ing heo em is a no el me hodological con ibu ion, which applies o gene al lea ning
en i onmen s beyond he speci ic epu a ion con ex o his pape .
A key ea u e o he cons uc ed dynamic commi men ypes is ha hey e u n o
he signaling phase in ini ely o en. One migh easonably conjec u e ha he inclusion
o a commi men ype ha begins wi h a su icien ly long phase o signaling ollowed
by a pe manen swi ch o playing he S ackelbe g ac ion o he ue s a e would su ice
o epu a ion building. We show in examples ha his is gene ally no su icien . Also,
while his pape is mo i a ed by en i onmen s wi h unce ain moni o ing, ou model
allows o unce ain y bo h abou moni o ing and abou he payo s o he epu a ion
3We can also in e p e ou model as one ha ep esen s subjec i e unce ain y ha consume s ha e
abou he ac ual moni o ing s uc u e and he beha io o he epu a ion-building i m. We show ha he
i m can indeed e ec i ely es ablish a epu a ion, as long as he consume s assign posi i e p obabili y o
he cons uc ed commi men ypes and he co ec moni o ing s uc u e.
4See also he discussion a e Lemma 3.
5See Sec ion 5.3 o p ecise s a emen s o ou su icien condi ion, as well as To ge sen (1991) and
Mosca ini and Smi h (2002) o illus a ions o o he applica ions o he Hellinge ans o m.
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172 Deb and Ishii Theo e ical Economics 20 (2025)
builde . Finally, ou main esul con inues o hold e en i he signals obse ed by he SR
playe s a e unobse ed by he LR playe .
While he main esul es ablishes a lowe bound on he LR playe ’s equilib ium pay-
o , a na u al ques ion is whe he he LR playe can ob ain payo s much highe han
he S ackelbe g payo . Wi h unce ain moni o ing, a pa ien LR playe may be able o
ob ain payo s ha a e s ic ly highe han he S ackelbe g payo o he ue s a e. The
eason is ha he LR playe may no ind i op imal o signal he ue s a e, bu would
a he block lea ning o a ain payo s ha a e highe han he S ackelbe g payo in he
ue s a e. P o iding a gene al, sha p cha ac e iza ion o an uppe bound on a pa ien
LR playe ’s equilib ium payo s is di icul , as i depends on he speci ic se o commi -
men ypes and he p io dis ibu ion o e ypes.6Ne e heless, we p o ide a join su -
icien condi ion on he moni o ing s uc u e and s age game payo s ha ensu es ha
he lowe bound and he uppe bound coincide: Loosely speaking, hese a e games in
which s a e e ela ion is desi able o he LR playe .
1.1 Rela ed li e a u e
We con ibu e o he li e a u e on epu a ion ha s a ed wi h K eps and Wilson (1982)
and Milg om and Robe s (1982), and includes he canonical models o Fudenbe g and
Le ine (1989,1992), and mo e ecen con ibu ions by Gossne (2011). As a as we know,
his pape is he i s o s udy epu a ion unde unce ain moni o ing.
Aumann, Maschle , and S ea ns (1995)andMe ens, So in, and Zami (2014)s udy
epea ed games wi h unce ain y in bo h payo s and moni o ing, bu ocus on ze o-sum
games. Wiseman (2005), Hö ne and Lo o (2009), and Hö ne , Lo o, and Tomala (2011)
s udy payo unce ain y in non-ze o-sum epea ed games, bu do no allow unce ain y
abou he moni o ing s uc u e. Ou amewo k is closes o Fudenbe g and Yamamo o
(2010), who s udy a epea ed game in which he e is unce ain y abou bo h moni o ing
and payo s. Howe e , Fudenbe g and Yamamo o (2010) ocus on pe ec public ex pos
equilib ium in which playe s play s a egies whose bes esponses a e independen o
any belie abou he s a e. As a esul , in equilib ium, no playe has an incen i e o a ec
he belie s o he opponen s abou he moni o ing s uc u e. We s udy mo e gene al
equilib ia whe e he LR playe may ha e incen i e o a ec he belie s o he SR playe s
abou he moni o ing s uc u e.
The necessi y o dynamic commi men ypes o epu a ion building due o iden i i-
ca ion p oblems is no el. Dynamic commi men ypes also a ise in epu a ion building
agains LR opponen s, as in Aoyagi (1996), Celen ani, Fudenbe g, Le ine, and Pesendo -
e (1996), and E ans and Thomas (1997), because es ablishing a epu a ion o ca ying
ou punishmen s a e ce ain his o ies can be bene icial o he epu a ion builde .7,8
6This is in con as o he p e ious pape s in he li e a u e, whe e he payo uppe bound is gene ally
independen o he ine de ails o he ype space such as he ela i e p obabili ies o commi men ypes.
7A akan and Ekmekci (2011,2015), and Ghosh (2014) also use simila ideas.
8In his li e a u e, some pape s do no equi e he use o dynamic commi men ypes by es ic ing a -
en ion o con lic ing in e es games. See, o example, Schmid (1993) and C ipps, Dekel, and Pesendo e
(2005).
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Theo e ical Economics 20 (2025) Repu a ion building 173
Bu in ou se ing wi h SR playe s, he h ea o punishmen s has no bi e. Dynamic com-
mi men ypes u n ou o s ill be necessa y o esol e a ade-o be ween signaling he
co ec s a e and collec ing he S ackelbe g payo , which a e bo h desi able o he ep-
u a ion builde .
In a ecen pape , Pei (2020) s udies epu a ion wi h in e dependen alues. Pei
(2020) es ic s a en ion o pe ec moni o ing and a ini e numbe o s a iona y com-
mi men ypes, and s udies he condi ions unde which he epea ed game yields a ep-
u a ion esul . In con as , we s udy a model whe e ac ions a e impe ec ly obse ed, bu
he obse ed public signals can po en ially con ey in o ma ion abou he s a e. We sim-
ila ly show ha epu a ion building can b eak down when he ype space only consis s
o s a iona y commi men ypes, and u he cons uc dynamic commi men ypes ha
would es o e a epu a ion esul gi en gene al ype spaces ha con ain hese dynamic
commi men ypes in i s suppo .
Ou nega i e examples demons a e ha epu a ion building may be agile in he
p esence o unce ain y abou moni o ing, because mul iple combina ions o s a e and
ac ion lead o he same dis ibu ion o e obse ed public signals. Iden i ica ion p ob-
lems can also gi e ise o long- un disag eemen s be ween di e en agen s in Acemoglu,
Che nozhuko , and Yildiz (2016), and can esul in con e gence o inco ec belie s in
dynamic games wi h lea ning, as in Fudenbe g and Le ine (1993a,1993b). The no el
ques ion ha we add ess he e is whe he o no such iden i ica ion p oblems can be
ci cum en ed by a pa ien long-li ed playe in a epu a ion se ing.
Finally, ou obus lea ning heo em also ela es o a ecen li e a u e ha s udies
a es o lea ning in decision heo e ic se ings. Mosca ini and Smi h (2002)andMu,
Poma o, S ack, and Tamuz (2021) bo h p o ide exac cha ac e iza ions o he speed
o lea ning in decision heo e ic se ings, ocusing on lea ning en i onmen s whe e he
signals a i e in an independen and iden ically dis ibu ed (i.i.d.) manne condi ional
on he ealized s a e. On he o he hand, ou obus lea ning heo em ocuses only on
a lowe bound on he a e o lea ning, while allowing o signals ha may exhibi a bi-
a y o ms o se ial co ela ion. Ou obus lea ning esul also ela es loosely o ideas
o uni o m lea ning om Vapnik–Che onenkis heo y used, o example, in Al-Najja
(2009)andAl-Najja and Pai (2014). These pape s s udy he uni o m lea ning o a ich
class o e en s gi en any i.i.d. p ocess. The main concep ual dis inc ion o ou obus
lea ning esul is ha we s udy uni o m lea ning o ini ely many e en s, bu allow o
any a bi a y s ochas ic p ocess ha may in ol e a bi a y se ial co ela ions.
2. Model
2.1 No a ion
We i s in oduce some no a ion ha we use h oughou he pape . Gi en a coun able
se X,le (X)deno e he se o all p obabili y measu es on X.Le +(X)be he se o
ull suppo p obabili y measu es on X.Fo anyB⊆X,wele Bcdeno e he comple-
men o B.
Gi en x,x∈Xand some eal numbe λ∈[0, 1],wele λx ⊕(1−λ)x∈(X)de-
no e he p obabili y measu e ha assigns p obabili y λ o xand 1 −λ o x.I ν∈
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174 Deb and Ishii Theo e ical Economics 20 (2025)
(X1×···×Xn), henma gXjνis he ma ginal dis ibu ion o νon Xj:ma gXjν(xj)=
i=jν(xj,x−j). Gi en a p obabili y measu e ν∈(X)and some unc ion g:X→R,
de ine Eν[g(x)] o be he expec a ion o g(x)when xis dis ibu ed acco ding o ν.
Gi en a ini e se Yand a coun able se X, de ine S(Y,X)as he se o all possi-
ble s ochas ic p ocesses o e Y∞wi h s a e space Xas ollows. Fo mally, an elemen
s∈S(Y,X)is a sequence s={s }∞
=0, whe e o each ,s ∈(Y ×X)sa is ies he con-
sis ency condi ion ma gY −1×Xs =s −1. By Kolmogo o ’s ex ension heo em, o any
s∈S(Y,X), he e exis s some s∞∈(Y∞×X)such ha ma gY ×Xs∞=s o all .Fo
any s∈S(Y,X)and any subse C⊆X, we can also de ine sC∈S(Y,X)as he co e-
sponding s ochas ic p ocess condi ional on C:sC=(s (·|C))∞
=0.
We use N o ep esen he se o all na u al numbe s including ze o and le N+:=
N {0}. Finally, we es ablish he con en ion ha bo h in ∅=min∅=∞and sup∅=
max∅=−∞.
2.2 Se ing
A long- un (LR) playe , playe 1, aces a sequence o sho - un (SR) playe 2s. Be o e he
in e ac ion begins, a pai (θ,ω)∈×o a s a e o he wo ld and ype o playe 1 is
d awn independen ly acco ding o he p oduc measu e γ0:=ν0×μ0wi h ν0∈+()
and μ0∈+(). We assume ha is ini e and enume a e :={θ0,,θm−1},bu 
may possibly be coun ably in ini e.9The ealized pai o s a e and ype (θ,ω)is hen
ixed o he en i e y o he game.
In each pe iod =0, 1, 2, , playe s simul aneously choose ac ions om hei e-
spec i e ac ion spaces a
1∈A1and a
2∈A2. We assume A1and A2a e ini e. Le
A=A1×A2.Le Ai:=(Ai)be he se o mixed ac ions o playe iwi h ypical elemen
αi.
In each pe iod ≥0, a e playe s ha e played ac ion p o ile a ∈A, a public signal y
is d awn om a ini e signal space Yacco ding o he p obabili y measu e, ψ(·|a ,θ)∈
(Y). No e impo an ly ha bo h he ac ion p o ile chosen a ime and he s a e o
he wo ld θpo en ially a ec he signal dis ibu ion. The s a e o he wo ld θ ep esen s
he unknown moni o ing s uc u e. Deno e by H :=Y he se o all -pe iod public
his o ies wi h ypical elemen h =(y0,,y −1)and assume by con en ion ha H0:=∅.
Le H:=∞
=0H deno e he se o all public his o ies o he epea ed game.
We assume ha he LR playe obse es he ealized s a e o he wo ld θ∈pe ec ly
so ha his p i a e his o y a ime is o mally a ec o , h
1∈H
1:=×A
1×Y . Mean-
while he SR playe a ime obse es only he public signals up o ime and so his
in o ma ion coincides exac ly wi h he public his o y H
2:=H .
A s a egy o playe iis a map σi:∞
=0H
i→Ai. Deno e he se o s a egies o
playe iby i. Finally, le B1be he se o s a ic s a e-con ingen mixed ac ions o playe
1, B1:={β1:→A1}wi h ypical elemen β1.
9The assump ion o allowing  o be coun ably in ini e is s anda d in he exis ing li e a u e (e.g., Fu-
denbe g and Le ine (1992)) when he S ackelbe g ac ion o he s age game can be mixed. We do no know
whe he ou a gumen s can be ex ended o he se ing whe e ||is coun ably in ini e. We lea e his open
o u u e esea ch.
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Theo e ical Economics 20 (2025) Repu a ion building 175
2.3 Type space
We assume ha =com ∪{ωs},whe ecom is he se o commi men ypes and ωsis
a s a egic ype. Each commi men ype ω∈com is associa ed wi h a s a egy σω
1∈1
such ha ype ωalways plays σω
1. In con as , ype ωs∈is a s a egic ype who chooses
a s a egy σ1∈1 o maximize payo s, which we desc ibe in he nex subsec ion. Thus,
a s a egy p o ile, deno ed σ=((σ1(ω))ω∈,σ2),isa uple o whichσ1(ω)=σω
1 o all
ω∈com.
2.4 Payo s and equilib ium
Any s a egy p o ile σ oge he wi h he p io γinduces a unique s ochas ic p ocess,
(πσ
)∞
=0∈S(Y×A,×) o all . By he Kolmogo o ex ension heo em, he e exis s
some πσ
∞∈(H∞×A∞××)such ha o all ,ma gH ×A ××πσ
∞=πσ
.
To s udy SR playe s’ bes esponses, i will also be use ul o de ine he ollowing be-
lie s o he SR playe s a e obse ing a public signal his o y:
λσ
·|h :=ma gA1×πσ
·|h ∈(A1×),
γσ
·|h :=ma g×πσ
·|h ∈(×),
νσ
·|h :=ma gπσ
·|h ∈(),
μσ
·|h :=ma gπσ
·|h ∈().
Then SR playe s’ expec ed payo s in any pe iod depend on he belie , λ∈(A1×):
u2(a2,λ):=Eλu2(a1,a2,θ)=
a1∈A1,θ∈
u2(a1,a2,θ)λ(a1,θ).
Thus, a s a egy p o ile, σ, yields he expec ed payo o u2(σ2(h ),λσ
(h )) in pe iod
a e he public his o y h .Le B2(λ)deno e he mixed bes esponses o playe 2, i.e.,
B2(λ):=a gmaxα2∈A2u2(α2,λ). Wi h a sligh abuse o no a ion, we w i e B2(α1,θ)=
B2(α1×1θ),whe e1θis he Di ac p obabili y measu e ha assigns p obabili y 1 o θ,
and B2(β1,p)=B2(λβ1,p),whe e o β1∈B1and p∈(),λβ,p(a1,θ)=p(θ)β1(a1|θ).
The payo o he LR s a egic ype, ωs, in s a e θis gi en by
U1(σ1,σ2,θ;δ):=Eπσ
∞(1−δ)
∞

=0
δ u1a
1,a
2,θ|θ,ωs.
Then he ex an e expec ed payo o ype ωsis
U1(σ1,σ2;δ):=Eν0U1(σ1,σ2,θ;δ).
Finally, we can de ine he s a ewise-S ackelbe g payo o he s age game. The S ack-
elbe g payo o playe 1 in s a e θis gi en by
u∗
1(θ):=sup
α1∈A1
in
α2∈B2(α1,θ)u1(α1,α2,θ).
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176 Deb and Ishii Theo e ical Economics 20 (2025)
Fo each ε>0, le Sε
θbe he se o ε-S ackelbe g ac ions in s a e θ, which a e he mixed
ac ions ha app oxima e u∗
1(θ)up o εin θ∈:
Sε
θ:=α1∈A1:in
α2∈B2(α1,θ)u1(α1,α2,θ)>u
∗
1(θ)−ε.
We analogously de ine Sε⊆B1as
Sε:=β1∈B1:β1(θ)∈Sε
θ o all θ∈.
Ou analysis will ocus on Bayes Nash equilib ia; o sho en he exposi ion, subsequen ly
we will e e o Bayes Nash equilib ium simply as equilib ium. We le BNEδdeno e he
se o all equilib ia o he game.10
2.5 In o ma ion s uc u e and key assump ions
We now impose wo key assump ions on he in o ma ion s uc u e, Assump ions 1and
2, which we main ain o he en i e y o he pape . We s a wi h a de ini ion.
De ini ion 1. A signal s uc u e ψsa is ies ac ion iden i ica ion o (α1,θ)∈A1×i ,
o all α2∈A2,
ψ(·|α1,α2,θ)=ψ·|α
1,α2,θ=⇒ α1=α
1.
Le Bid ⊆B1be he se o all β1∈B1such ha (β1(θ),θ)sa is ies ac ion iden i ica ion o
all θ∈.
Assump ion 1. Fo e e y ε>0,Sε∩Bid = ∅.
In wo ds, he abo e assump ion holds i and only i in e e y s a e θ, he e exis s some
ε-S ackelbe g ac ion in s a e θsuch ha his ac ion would be s a is ically iden i ied om
all o he ac ions ega dless o he ac ions played by he SR playe . No e ha his is gen-
e ally a minimal condi ion ha is equi ed o a LR playe o be able o gua an ee S ack-
elbe g payo s in s a e θ, since wi hou i , epu a ion building may be impossible e en
when θis common knowledge.
While he abo e assump ion conce ns s a is ical iden i ica ion o ac ions o a ixed
s a e θ, his is gene ally no su icien o a epu a ion heo em. We u he mo e impose
he ollowing assump ion, which conce ns he s a is ical iden i ica ion o ac ions ac oss
s a es.
Assump ion 2. Fo e e y θ= θ, he eexis someα1∈A1such ha
ψ(·|α1,α2,θ)= ψ·|α
1,α2,θ
o all α
1∈A1and all α2∈A2.
10Ou main heo ems p o ide bounds on payo s ac oss all equilib ia. So hese bounds also apply e en
when es ic ing a en ion o mo e s ingen solu ion concep s such as pe ec Bayes Nash equilib ia.
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Theo e ical Economics 20 (2025) Repu a ion building 183
Figu e 6. The in o ma ion s uc u e.
signaling ac ions ha help he consume lea n he unknown moni o ing s a e and col-
lec ion ac ions ha a e desi able o payo s o he LR playe . Because o he necessi y o
play bo h ypes o ac ions, ou commi men ypes a e nons a iona y, playing a pe iodic
s a egy ha al e na es be ween signaling phases and collec ion phases.23
Finally, as we ha e al eady emphasized, ou epu a ion esul does no depend on
speci ic dis ibu ional assump ions on he ype space. In pa icula , i emains alid
e en i we include o he possibly bad commi men ypes, as ichness o he ype space
(,μ)only equi es he exis ence o ypes ωβ1, while placing no es ic ions on he exis-
ence o absence o o he commi men ypes.
4.3 Necessa y cha ac e is ics o commi men ypes
The commi men ypes, ωβ1, ha e wo key ea u es: (i) They swi ch play be ween sig-
naling and collec ion phases, and (ii) hey do so in ini ely o en. These wo ea u es a e
impo an and in some sense also necessa y o epu a ion building, gi en he possibili y
o iden i ica ion p oblems in he moni o ing s uc u e.
Conside again he s age game om Figu e 2and suppose ha he in o ma ion
s uc u e is now gi en by Figu e 6. To highligh he impo ance o (i), we p o ide an ex-
ample below in which he s a egic LR playe ega dless o his discoun ac o ob ains a
low equilib ium payo in s a e θ=bi all commi men ypes play s a iona y s a egies.
To highligh he impo ance o (ii), we conside ype spaces in which all commi men
ypes play s a egies ha on -load he signaling phases and again cons uc equilib ia
in which LR ge s a payo much below he S ackelbe g payo in s a e b.
4.3.1 S a iona y commi men ypes Conside any a bi a y coun able se ∗o com-
mi men ypes, each o which is associa ed wi h he play o a s a e-con ingen ac ion
β∈B1a all pe iods. Fo each ω∈∗,le βωbe he associa ed s a e-con ingen mixed
ac ion plan o ype ω. No ice ha his ype space con ains only s a iona y commi men
ypes. We now show ha he exis ence o such ypes is gene ally no su icien o epu-
a ion building.
Fo mally, gi en any coun able se o s a iona y commi men ypes, ∗,wecancon-
s uc a se o commi men ypes com ⊇∗and a p obabili y measu e μ∈+(com ∪
{ωs})such ha he e exis s an equilib ium in which he s a egic LR playe ob ains a
payo signi ican ly below he S ackelbe g payo in s a e b.
23A simila epu a ion heo em can be p o ed also wi h s a iona y commi men ypes ha ha e access o
a public andomiza ion de ice. In pa icula , we would need a ich space o s a iona y commi men ypes
ha each signal he s a e wi h di e en p obabili ies. We hank Johannes Hö ne o poin ing his ou .
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184 Deb and Ishii Theo e ical Economics 20 (2025)
To simpli y no a ion, le Ab:={α1(C)≥2/3}.No ice ha Bis a bes esponse o α1
in s a e bi and only i α1∈Ab.Le b:={ω∈∗:βω(b)∈Ab}.
Gi en he in o ma ion s uc u e, ψ, o e e y α1∈Ab, he e exis s a co esponding
bad ac ion, α1, in s a e gsuch ha ψ(·|α1,b)=ψ(·|α1,g). Fo e e y ω∈b,le ωdeno e
a ypewhoplaysβω(b)in s a e gand Din s a e b. Le he ype space consis o
=∗∪{ω:ω∈b}∪ωs.
Claim 1. Suppose ha γ0(ω,b)<2
3γ0(ω,g) o all ω∈b. Then o e e y δ∈(0, 1),i
is a PBE o he LR o always play Dand he SR o always play N. In pa icula , his PBE
yields a payo o 0<u
∗
1(b)=4/3 o he LR in s a e θ=b.
P oo .Le σdeno e he abo e s a egy p o ile and conside he belie , λσ
((C,b)|h ),
ha he SR assigns o he e en (C,b)a a his o y h . By cons uc ion, o any ω∈b,
γσ
((ω,b)|h )=γ0(ω,b)
γ0(ω,g)γσ
((ω,g)|h ) o any h . The e o e,
λσ
(C,b)|h =
ω∈b
γσ
(ω,b)|h βω(C|b)+
ω/∈b
γσ
(ω,b)|h βω(C|b)
<
ω∈b
2
3γσ
(ω,g)|h +
ω/∈b
2
3γσ
(ω,b)|h ≤2
3.
Recall ha i is a bes esponse o play Na a his o y i λσ
((C,b)|h )<2/3. Hence, i is
abes esponse o heSR oplayNa all his o ies. Then i is immedia e ha i is a bes
esponse o e LR o play Da all his o ies.
I ν0(b)=1, as long as he closu e o Ab∩{βω(b):ω∈b}con ains 2/3( hemixed
S ackelbe g ac ion), hen a su icien ly pa ien playe ob ains payo s close o 4/3inany
equilib ium, since a de ia ion o mimicking one o he good commi men ypes in b
gua an ees such a high payo . Now conside he case when ν0(b)=1/2. Conside again
a de ia ion o mimicking a good ype in b. Such a de ia ion no longe gua an ees a
high payo , since he e a e now also bad commi men ypes in {¯ω:ω∈b}in s a e g
ha eplica e exac ly he same dis ibu ion o e public signals as he good commi men
ypes. As a esul , SR playe s a e ne e able o di e en ia e be ween hese ypes, and i
he p io places ela i ely highe weigh on such ypes in s a e g, hen he SR playe s will
ne e become op imis ic abou he e en b×{b}.
4.3.2 Type spaces wi h on -loaded signaling Nex we p esen an example whe e each
commi men ype swi ches be ween signaling and collec ion, bu no in ini ely o en;
i.e., hey can play signaling ac ions o a mos Npe iods and hen swi ch o collec ion
o e e . In such ype spaces, we show ha a epu a ion heo em again does no hold
gene ally.
Again conside he same s age game (Figu e 2) and in o ma ion s uc u e (Figu e 6)
om he p e ious subsec ion. Le ωbbe a bad commi men ype who always plays α∗
1=
2
3C⊕1
3Din s a e gand always plays Din s a e b.No e ha ψ(·|α∗
1,g)=ψ(·|C,b).Le
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Theo e ical Economics 20 (2025) Repu a ion building 185
ω deno e a commi men ype who plays Dun il pe iod (signaling phase) and he e-
a e swi ches o he ac ion C o e e a e (collec ion phase).24 Fo any N∈N+∪{∞},
conside he se o ypes N:={ω : ∈N+, ≤N}∪{ωs,ωb}.25
We now show in he ollowing claim ha wi hou u he dis ibu ional assump ions
on he ype space, epu a ion building canno be gua an eed.
Claim 2. Le N∈N+∪{∞}and ν0(g)=ν0(b)=1/2. Then he e exis s some μ0∈+(N)
such ha o any δ∈(0, 1),i isaPBE o heLR oalwaysplayDand he SR o always
play N. Mo eo e , his PBE yields a payo o 0<4/3=u∗
1(b) o ωs o all discoun ac o s
in s a e b.
P oo . Conside he p obabili y dis ibu ion o e ypes gi en by
μ0ω =κ ε,μ0ωb=ε,μ0ωs=1−
N

τ=0
κτε.
We assume ha κ∈(0, 1/4]and ε∈(0, 1/2), in which case, μ0is a alid p obabili y mea-
su e since N
τ=0κτε<1.
Le σdeno e he abo e s a egy p o ile. Conside he p obabili y, λσ
((C,b)|h ).
Since only ypes {ω1,,ω }play Cin s a e ba such a his o y,
λσ
(C,b)|h =γσ
ω1,,ω −1×{b}|h ,
bu o each τ, he likelihood a io be ween (ωτ,b)and (ωb,g)is gi en by
γσ
ωτ,b|h 
γσ
ωb,g|h =μ0ωτ
μ0ωb
τ

τ=1
ψ(yτ|D,b)
ψ(yτ|α1,g)≤μ0ωτ
μ0ωb3
2τ
=3
2κτ
The e o e,
λσ
(C,g)|h ≤
−1

τ=13
2κτ
γσ
ωb,g|h ≤
∞

τ=13
8τ
<2
3.
Again ecall ha whene e λσ
((C,g)|h )<2
3, he SR has a s ic incen i e o play N.
The e o e, his shows ha i is indeed op imal o he SR o play Na all his o ies. Gi en
his, i is immedia e ha i is a bes esponse o ωs o always play D.
Repu a ion building ails in his example because all signaling is on -loaded by all
commi men ypes. To see he basic idea, conside again he de ia ion o a s a egy o
mimicking ω . The hope unde such a de ia ion o he LR is ha he ini ial pe iods o
signaling would be su icien o con ince he SR playe s ha he s a e is b o a su icien
24Fo exposi ional simplici y, we ocus only on hose commi men ypes who play he pu e S ackelbe g
ac ion in he collec ion phase. The example can be easily ex ended o se ings whe e such ypes play mixed
ac ions in he collec ion phase.
25When N=∞,N={ω : ∈N+}∪{ωs,ωb}.
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186 Deb and Ishii Theo e ical Economics 20 (2025)
deg ee o con idence ha i elimina es iden i ica ion p oblems ac oss s a es. Howe e ,
he claim abo e shows ha his is in easible o he LR. The p oblem is ha unde he
cons uc ed belie , μ0∈+(), he likelihood o (ω ,b)is much smalle han he like-
lihood o (ωb,g), so ha e en a e pe iods o signaling, he SR s ill main ains high
p obabili y on (ωb,g).
Rema k. I ins ead γ0(ω ,b)we e su icien ly la ge ela i e o o e e y , hena epu-
a ion esul would hold in he example. Howe e , as p e iously emphasized, his illus-
a es he dependence o epu a ion building on he ine de ails o he p io dis ibu ion
o e ypes (beyond jus he suppo o he p io dis ibu ion) when signaling is on -
loaded.
Bo h o hese issues highligh ed in he abo e examples a e no longe p oblema ic
gi en he commi men ypes cons uc ed in he main heo em. Fi s , because he com-
mi men ypes en e signaling phases many imes, he e a e no bad ypes in o he s a es
ha can eplica e simila dis ibu ions o e public signals du ing hese signaling phases
o long pe iods o ime. Second, he commi men ypes signal he s a e inde ini ely so
ha a LR playe who mimics such a commi men ype can ensu e e en ual co ec lea n-
ing o he s a e e en i he p obabili y o such a commi men ype is ini ially e y small.
Finally, while he abo e analysis demons a es he necessi y o dynamic commi -
men ypes in pa icula examples, he e a e many speci ic se ings whe e ei he s a ion-
a y commi men ypes o commi men ypes wi h on -loaded signaling su ice.26 An
exac cha ac e iza ion in gene al games o when such simple ypes su ice is beyond he
scope o his pape .27 Despi e his, we emphasize again ha Theo em 1holds as long as
ichness is sa is ied wi hou any es ic ions on wha o he ypes a e o a e no p esen .
5. P o ing Theo em 1
We now e u n o p o e Theo em 1. The o e all s uc u e o he p oo ollows he s an-
da d app oach in he epu a ion li e a u e. We show ha o β1∈Sε, a su icien ly pa-
ien LR playe , by playing he s a egy, σβ1, associa ed wi h ype ωβ1, can ob ain payo s
a leas u∗
1(θ)−ρin any equilib ium.
To show his, we p o e wo key p ope ies ha hold uni o mly ac oss all equilib ia:
Fo e e y ε>0, he e exis s some J( ha can be chosen independen o he choice o
equilib ium) such ha by de ia ing o play σβ1in any equilib ium, he ollowing s a e-
men s hold:
P1. The SR playe s’ p edic ions o he cu en pe iod’s public signal dis ibu ion a e
app oxima ely co ec in all bu Jpe iods wi h p obabili y a leas 1 −ε(see
Lemma 3 o de ails).28
26See he discussion a e Theo em 1and he ema k abo e.
27The main obs acle is he di icul y o explici cons uc ion o equilib ia in gene al epu a ion games.
28By “app oxima ely co ec ,” we mean ha he SR playe s’ p edic ions will be close o he ac ual public
signal dis ibu ion unde σβ1and s a e θ.
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Theo e ical Economics 20 (2025) Repu a ion building 187
P2. The SR playe s’ belie s assign p obabili y a leas 1 −εon he co ec s a e θin all
bu Jpe iods wi h p obabili y a leas 1 −ε(see Lemma 4 o de ails).
P ope y P1 holds in p e ious pape s s udying epu a ion building unde impe ec
moni o ing such as Fudenbe g and Le ine (1986)andGossne (2011), and i s p oo ol-
lows hese s anda d a gumen s. In hose en i onmen s, wi h app op ia e iden i ica ion
assump ions, P1 implies ha wi h high p obabili y, he SR playe s’ bes esponses will be
app oxima ely co ec in all bu J1pe iods. Howe e , his p ope y alone is inadequa e
o a epu a ion heo em in ou en i onmen . E en i he SR playe s’ p edic ions o o-
day’s public signal dis ibu ion is exac ly he same in all pe iods as ha o σβ1in s a e
θ, because o iden i ica ion p oblems ac oss s a es, his does no necessa ily imply ha
he SR playe s’ belie s a e concen a ed on he co ec s a e θ.
P ope y P2 add esses his issue. To p o e i , we p o e a heo em on obus lea ning
(Theo em 2) ha es ablishes a simple- o-check su icien condi ion o ensu e ha an
obse e lea ns he ele an s a e a a a e ha is uni o m ac oss a ich class o gene al
lea ning en i onmen s. We hen apply his heo em o he epu a ion se ing o show
ha SR playe s lea n he s a e θa a a e ha is uni o m ac oss all equilib ia.
5.1 Fo mal de ails o he p oo o Theo em 1
We now p o ide de ails o he p oo o Theo em 1. P oo s no p o ided in he ex can
be ound in he Appendices. We i s ex end he no ion o ε-en opy con i ming bes
esponse o Gossne (2011) o ou amewo k.29 To s a e his, i s ecall he de ini ion
o he Kullback–Leible di e gence o wo p obabili y measu es: Gi en wo p obabili y
measu es P,Q∈(Y),
D(PQ):=
y∈Y
P(y)logP(y)
Q(y).
Recall he basic p ope ies o ela i e en opy ha D(PQ)≥0 o allP,Q∈(Y),and
D(PQ)=0 i and only i P=Q.
De ini ion 3. Le (κ,ε)∈[0, 1]2.Thenα2∈A2is a (κ,ε)-con i ming bes esponse a
(α1,θ)i he e exis s some λ∈(A1×)such ha
(i) α2∈B2(λ)
(ii) D(ψ(·|α1,α2,θ)ψ(·|λ,α2))≤ε(see oo no e 30)30
(iii) ma gλ(θ)≥1−κ.
We le CBRκ,ε(α1,θ)be he se o all (κ,ε)-con i ming bes esponses a (α1,θ).
29Fudenbe g and Le ine (1992) p o ide a simila de ini ion ha uses he no ion o o al a ia ional dis-
ance be ween p obabili y measu es ins ead o Kullback–Leible di e gence.
30We de ine ψ(·|λ,α2):=a1,a2,θψ(·|a1,a2,θ)λ(a1,θ)α2(a2).
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188 Deb and Ishii Theo e ical Economics 20 (2025)
The ollowing lemma mo i a es he de ini ion o (κ,ε)-con i ming bes esponses
and shows ha o εsmall, i sho - un playe s play an (ε,ε)-con i ming bes esponse,
hen he LR playe ob ains payo s close o hose as i he SR playe we e bes esponding
wi h pe ec knowledge o bo h he LR playe ’s ac ion and s a e.
Lemma 1. Fo e e y α1∈A1,
limin
ε→0in
α2∈CBRε,ε(α1,θ)u1(α1,α2,θ)≥in
α2∈B2(α1,θ)u1(α1,α2,θ).
P oo . By Assump ion 1and he p ope y ha D(P|Q)=0 i and only i P=Q,weha e
ha CBR0,0(α1,θ)=B2(α1,θ).Mo eo e ,CBR
ε,ε(α1,θ)is uppe hemi-con inuous wi h
espec o ε, and so he inequali y ollows.
No ice ha a (1, ε)-con i ming bes esponse is essen ially he ex ension o he idea
o ε-en opy con i ming bes esponse in Gossne (2011) o he cu en se ing. Un-
de a (1, ε)-con i ming bes esponse, condi ion (iii) in De ini ion 3is i ially sa is-
ied and so he de ini ion only equi es ha he public signal dis ibu ion associa ed
wi h he belie λ equi ed o sus ain α2as a bes esponse be ε-close in Kullback–
Leible di e gence o he ue dis ibu ion o public signals unde he ac ion p o ile
(α1,α2)and s a e θ.Whenκis small, condi ion (iii) addi ionally equi es ha λin-
deed places la ge p obabili y on he s a e θ. This addi ional equi emen is impo an
in Lemma 1, since gene ally, limin ε→0in α2∈CBR1,ε(α1,θ)u1(α1,α2,θ)may be s ic ly less
han in α2∈B2(α1,θ)u1(α1,α2,θ).
The ollowing lemma cons i u es he key s ep in he p oo o he main heo em,
which shows ha i he LR de ia es o play σβ1in any equilib ium, hen he SR plays
s a egies consis en wi h (ε,ε)-con i ming bes esponses in all bu a ini e numbe o
pe iods wi h e y la ge p obabili y. Fo mally, de ine he ollowing se o his o ies gi en
an equilib ium σand a ype ω∈who plays s a egies ha only depend on H ×31
Mσ,(ω,θ)(J,κ,ε):=h∞∈H∞: :σ2h /∈CBRκ,εσ1ω,h ,θ,θ<J.
These a e he se o public his o ies, h∞,whe e ypeωand he SR playe s oge he
play ac ion p o iles ha a e (κ,ε)-con i ming bes esponses a s a e θin all bu Jpe i-
ods. The ollowing lemma p o ides a lowe bound on he p obabili y o such his o ies
ha applies uni o mly ac oss all equilib ia.
Lemma 2. Suppose ha μ(ωβ1)>0. Then o e e y ε>0, he e exis s some Jsuch ha
in σ∈BNEδπσ,(ωβ1,θ)
∞Mσ,(ωβ1,θ)(J,ε,ε)≥1−2ε.32
31In he analysis, we a e conce ned wi h hese se s only o ypes ωβ1who play s a egies ha only de-
pend on H ×. The e o e, he es ic ion o such ypes is no es ic i e.
32No ice ha in his lemma, we do no necessa ily equi e ha β1∈Sε o some ε>0 small. La e in he
p oo o Theo em 1, when we use his lemma, we will use Lemma 2 o he pa icula case in which β1∈Sε
o ε>0 small o ensu e ha by mimicking ωβ1, he LR can ensu e high payo s.
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Theo e ical Economics 20 (2025) Repu a ion building 189
The e a e wo aspec s o he lemma abo e ha a e wo h emphasis. Fi s is ha he
se o his o ies in Mσ,(ω,θ)(J,ε,ε)ensu es ha playe s play ac ion p o iles consis en
wi h (ε,ε)-con i ming bes esponses in all bu Jpe iods. One could weaken his o an-
alyze he p obabili y o he se o his o ies in Mσ,(ω,θ)(J,1,ε) ha only equi e playe s o
play ac ion p o iles consis en wi h (1, ε)-con i ming bes esponses in all bu Jpe iods
as in Gossne (2011). Indeed, he a gumen s o Fudenbe g and Le ine (1986)andGoss-
ne (2011) imply a uni o m lowe bound on he p obabili y o such his o ies ac oss all
equilib ia. Howe e , his is insu icien o ou epu a ion heo ems since as p e iously
discussed, Lemma 1does no apply o (1, ε)-con i ming bes esponses.
The conclusion o he abo e lemma does no hold o any a bi a y ype ωand holds
only o ypes ωβ1. This is again because he de ini ion o Mσ,(ω,θ)(J,ε,ε) equi es SR
playe s o hold app oxima ely co ec belie s on he s a e θin all bu Jpe iods. In pa ic-
ula , i ωwe e a s a iona y commi men ype, hen πσ,(ω,θ)
∞(Mσ,(ω,θ)(J,ε,ε)|ω,θ)may
ac ually be qui e small o some equilib ia, σ.
We p o e Lemma 2in Sec ion 5.2. Be o e his, we p esen he p oo o Theo em 1,
which is now immedia e.
P oo o Theo em 1. De ine u:=mina∈Aminθ∈u1(a,θ). and choose any θ.Wewill
show ha he e exis s some δ∗<1 such ha whene e δ>δ
∗,U1(σ,θ;δ)>u
∗
1(θ)−ρ o
all σ∈BNEδ. This hen p o es he heo em, since he e a e ini ely many s a es θ∈.
Fi s choose some ε∗>0such ha o allε<ε
∗,
(1−2ε)u∗
1(θ)−ρ
4+2εu >u
∗
1(θ)−ρ.
By assump ion, we can choose β1∈Sρ/8such ha μ(ωβ1)>0. By Lemma 1, he eexis s
some ε∈(0, ε∗)such ha
u1β1(θ),α2,θ>min
α2∈B2(β1(θ),θ)u1β1(θ),α2,θ−ρ
8≥u∗
1(θ)−ρ
4
o all (β1(θ),α2) ha is an (ε,ε)-con i ming bes - esponse a θ, whe e he las inequal-
i y ollows om cons uc ion ha β1∈Sρ/8.
By Lemma 2, he e exis s some Jsuch ha o e e y equilib ium, σ,
πσ,(ωβ1,θ)
∞Mσ,(ωβ1,θ)(J,ε,ε)≥1−2ε.
As a esul , in any equilib ium, σ, by mimicking he s a egy o he commi men ype
ωβ1, he LR playe 1 ob ains a leas he payo
(1−2ε)1−δJu+δJu∗
1(θ)−ρ
4+2εu.
Then we can choose some δ∗<1such ha o allδ>δ
∗,
(1−2ε)1−δJu+δJu∗
1(θ)−ρ
4+2εu >u
∗
1(θ)−ρ.
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190 Deb and Ishii Theo e ical Economics 20 (2025)
5.2 P o ing Lemma 2
We now p o e ou key lemma, which ollows in a s aigh o wa d manne om he ol-
lowing wo lemmas. The comple e p oo o Lemma 2is p o ided in Appendix D.To
simpli y no a ion, gi en C⊆×, de ine
φσ
·|h =ma gYπσ
·|h ,φσ,C
·|h =ma gYπσ
·|h ,C.
In wo ds, φσ
(·|h )is he dis ibu ion o e y ∈Yin pe iod in he equilib ium σ,con-
di ional on he public his o y h .33 Addi ionally, φσ,C
is his dis ibu ion when condi-
ioned on he e en (ω,θ)∈C.
Lemma 3 (Me ging). Suppose ha γ0(ω,θ)>0. Then o e e y ε>0, he e exis s some J1
such ha in e e y equilib ium σ,
πσ,(ω,θ)
∞h∞∈H∞: :Dφσ,(ω,θ)
·|h φσ
·|h >ε
<J
1≥1−ε.
Lemma 4 (Uni o m Lea ning). Suppose μ0(ωβ1)>0. Then o e e y ε>0, he eexis s
some J2such ha o all σ∈BNEδ,
πσ,(ωβ1,θ)
∞h∞∈H∞: :νσ
θ|h <1−ε<J
2≥1−ε.
As in Fudenbe g and Le ine (1986)andGossne (2011), Lemma 3s eng hens he
classical me ging esul s, e.g., Blackwell and Dubins (1962)andKalai and Leh e (1993),
by es ablishing a uni o m uppe bound ac oss all equilib ia on he p obabili y o his-
o ies in which he SR playe ’s p edic ion o oday’s public signal dis ibu ion, φσ
(·|h ),
di e ges subs an ially om he “ ue” public signal dis ibu ion, φσ,(ω,θ)
(·|h ),inmo e
han J1 ime pe iods, when LR plays σ(ω)in s a e θ. The p oo ollows using s an-
da d me ging a gumen s o Gossne (2011), which we include o comple eness in Ap-
pendix C.
To p o e Lemma 4, we show ha in any s a e θ, by playing σβ1(θ), he LR playe can
ensu e ha he SR playe s lea n he s a e θa a a e ha is uni o m ac oss all equilib ia.
Indeed s anda d a gumen s immedia ely imply ha in any equilib ium, SR playe s lea n
he ue s a e θwhene e he LR playe plays σβ1(θ). Howe e , he addi ional uni o mi y
equi emen equi es u he analysis, which we now add ess in Sec ion 5.3.
5.3 A obus lea ning heo em
Conside he ollowing gene al model o lea ning. The e is a ini e signal space Yand a
coun able s a e space .Alea ning en i onmen is some π∈S(Y,) o which π:=
ma gπhas ull suppo on . Recall ha o any B⊆,πB∈S(Y,)deno es he
s ochas ic p ocess condi ional on ξ∈B:πB=(π (·|B))∞
=0. No e ha his allows he
s ochas ic p ocess, πξ, o anyξ∈, o be e y gene al, which may po en ially con ain
a bi a y o ms o se ial co ela ions.
33In ac , φσ
(·|h )is he SR playe s’ subjec i e belie o he pe iod public signal a e obse ing h .
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Theo e ical Economics 20 (2025) Repu a ion building 191
To in e p e , in a lea ning en i onmen , a he beginning o each pe iod =1, 2, ,
an obse e upda es he belie s abou he ue s a e ξ∈acco ding o Bayes’ ule upon
he ealiza ion o a his o y o signals h =(y0,,y −1).Le ρπ
(·|h )∈()deno e he
obse e ’s belie s a e obse ing h . We now desc ibe o mally ou de ini ion o obus
lea ning.
De ini ion 4. Le ξ∗∈B⊆and S∗⊆S(Y,). Then we say ha an obse e S∗-
obus ly lea ns Ba ξ∗i o e e y κ∈(0, 1), he e exis s some Ksuch ha
in
π∈S∗π∞∞

=Kh∞:ρπ
B|h ≥1−κ|ξ∗≥1−κ.
In ui i ely, S∗- obus lea ning equi es an obse e ’s belie s o concen a e on B o -
e e a e pe iod Kwi h high p obabili y o all lea ning en i onmen s in S∗.
Ou main heo em in his sec ion es ablishes a simple su icien condi ion on S∗
ha gua an ees S∗- obus lea ning o Ba ξ∗. To s a e i , we i s need a ew de ini-
ions ha a e well known om he heo y o s a is ical expe imen s. Fi s ix a lea n-
ing en i onmen π∈S(Y,),someξ∗∈,andB⊆. We now de ine he unc ion
Hπ
(·;B,ξ∗):[0, 1]→R, which is also known as he Hellinge ans o m. Fo mally his
unc ion is de ined as
Hπ
z;B,ξ∗:=
h ∈H πB
h zπξ∗
h 1−z=Eπξ∗
πB
h 
πξ∗
h z.
This is he momen gene a ing unc ion o he ( andom) log-likelihood a io a ime ,
log πB
(h )
πξ∗
(h ),whenh is dis ibu ed acco ding o πξ∗
. Towa d ou obus lea ning esul ,
le us also de ine
Hπ
B,ξ∗=in
z∈[0,1]Hπ
z;B,ξ∗∈[0, 1].
Roughly speaking, Hπ
(B,ξ∗)measu es he in o ma i eness o he lea ning en i on-
men a ime wi h espec o lea ning he ela i e likelihoods o B s. ξ∗.No ice ha by
Jensen’s inequali y, Hπ
(B,ξ∗)≤1. In ui i ely, a comple ely unin o ma i e lea ning en i-
onmen a ains his maximal alue o Hπ
(B,ξ∗)=1. On he o he hand, i he suppo s
o πB
and πξ∗
a e disjoin so ha he lea ning en i onmen dis inguishes B om ξ∗pe -
ec ly, hen Hπ
(B,ξ∗)=0. In Appendix A, we lis some addi ional use ul p ope ies o
he Hellinge ans o m.34
In he ollowing heo em, we show ha when he Hellinge ans o ms con e ge
o ze o (in o ma ion con e ges o pe ec in o ma ion) a a as enough a e uni o mly
ac oss all lea ning en i onmen s, π∈S∗, hen he obse e S∗- obus ly lea ns Ba ξ∗.
34See also To ge sen (1991) and Mosca ini and Smi h (2002) o mo e de ails on he Hellinge ans o m.
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192 Deb and Ishii Theo e ical Economics 20 (2025)
Theo em 2. Le S∗⊆S(Y,)and ξ∗∈B⊆. Suppose ha in π∈S∗π(ξ∗)>0and
lim
K→∞ sup
π∈S∗
∞

=K
Hπ
Bc,ξ∗=0.
Then an obse e S∗- obus ly lea ns Ba ξ∗.35
The ollowing co olla y will be use ul: I shows ha i we can gua an ee S∗- obus
lea ning o a ini e collec ion o se s a ξ∗, hen we can also gua an ee S∗- obus lea ning
o he in e sec ion o hese se s a ξ∗.
Co olla y 1. Le ξ∗∈B1,,Bn⊆and S∗⊆S(Y,). Suppose ha in π∈S∗π(ξ∗)>
0and ha o all =1, 2, ,n,
lim
K→∞ sup
π∈S∗
∞

=K
Hπ
Bc
,ξ∗=0.
Then he obse e S∗- obus ly lea ns B1∩B2∩···∩Bna ξ∗.
5.3.1 Uni o m signaling o he s a e in epu a ion building Gi en any equilib ium, σ,
he SR playe s ace a lea ning en i onmen abou he s a e space =×along he
samelinesasinSec ion5.3. O cou se, when we iew an equilib ium, σ, as a lea ning
en i onmen , we can also de ine he app op ia e Hellinge ans o ms. Thus, o any
equilib ium σand any e en A⊆×, we de ine he Hellinge ans o m as
Hσ
z;B,(ω,θ)=
h ∈H πσ,B
h zπσ,(ω,θ)
h 1−z.
We also acco dingly de ine
Hσ
B,(ω,θ)=in
z∈[0,1]Hσ
z;B,(ω,θ).
Th ough a s aigh o wa d compu a ion in Lemma 8in Appendix B, we show ha
o any θ= θ,
lim
K→∞ sup
σ∈BNEδ
∞

=K
Hσ
×θ,ωβ1,θ=0.
By Co olla y 1, he SR playe s BNEδ- obus ly lea n θ=θ×( {θ})=×{θ}a
(ωβ1,θ), which p o es Lemma 4.
35We lea e open he ques ion o whe he his condi ion is also necessa y o S∗- obus lea ning o u u e
esea ch.
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Theo e ical Economics 20 (2025) Repu a ion building 199
Lemma 10. Suppose ha γ0(ω,θ)>0. Then o e e y σ∈BNEδ,
πσ,(ω,θ)
∞h∞∈H∞: :Dφσ,(ω,θ)
·|h φσ
·|h >ε
≥J≤−logγ0(ω,θ)
Jε .
P oo . Fo e e y T, by he chain ule o Kullback–Leible di e gence,
Dma gHTπσ,(ω,θ)
Tma gHTπσ
T=Eπσ,(ω,θ)
TT

=0
Dφσ,(ω,θ)
·|h φσ
·|h 
=Eπσ,(ω,θ)
∞T

=0
Dφσ,(ω,θ)
·|h φσ
·|h .
Mo eo e , D(ma gHTπσ,(ω,θ)
Tma gHTπσ
)≤−log γ0(ω,θ)by he p e ious lemma.
The e o e, by he mono one con e gence heo em,
Eπσ,(ω,θ)
∞∞

=0
Dφσ,(ω,θ)
·|h φσ
·|h ≤−logγ0(ω,θ).
Then by Ma ko ’s inequali y,
πσ,(ω,θ)
∞h∞∈H∞: :Dφσ,(ω,θ)
·|h φσ
·|h >ε
>J
≤πσ,(ω,θ)
∞∞

=0
Dφσ,(ω,θ)
·|h φσ
·|h >Jε

≤−log γ0(ω,θ)
Jε .
The p oo o Lemma 3is now immedia e.
P oo o Lemma 3. Choose J1su icien ly la ge such ha −log γ0(ω,θ)
J1ε<ε.Then
Lemma 3is immedia e om Lemma 10.
Appendix D: P oo o Lemma 2
By Lemmas 3and 4, he eexis Jsuch ha o all σ∈BNEδ,
1−ε≤πσ,(ωβ1,θ)
∞h∞: :Dφσ,(ω,θ)
·|h φσ
·|h >ε
<J,
1−ε≤πσ,(ωβ1,θ)
∞h∞: :νσ
θ|h <1−ε<J.
The e o e, o all σ∈BNEδ,
πσ,(ωβ1,θ)
∞Mσ,(ωβ1,θ)(2J,ε,ε)
≥πσ,(ωβ1,θ)
∞h∞: :Dφσ,(ω,θ)
·|h φσ
·|h >ε
, :νσ
θ|h <1−ε<J
≥1−2ε.
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200 Deb and Ishii Theo e ical Economics 20 (2025)
Appendix E: P o ing Theo em 3
The p oo o Theo em 3uses ideas om Me ens, So in, and Zami (2014)wi hsome
modi ica ions. Le us begin wi h some no a ion. Gi en any p obabili y ec o x∈(),
le xdeno e he Euclidean no m:
x2=
θ∈
x(θ)2.
No e ha i playe 1 plays a s a egy ha induces λ∈(A1×)as he join dis ibu-
ion o e A1×and playe 2 plays a2, hen playe iob ains he expec ed u ili y
ui(a2,λ):=Eλui(a1,a2,θ)=
a1,θ
ui(a1,a2,θ)λ(a1,θ).
We now ex end he de ini ion o a bes esponse o ε-bes esponse:
BRε
2(λ):=a2∈A2:max
a
2∈A2
u2a
2,λ−u2(a2,λ)≤ε.
De ine o any ε≥0,
Wε(λ)=max
a2∈Bε
2(λ)u1(a2,λ).
Finally, gi en λ∈(A1×),le q(·|y,λ)be he induced pos e io belie abou θa e
obse a ion o he signal y:
q(θ|y,λ)=
a1∈A1
λ(a1,θ)ψ(y|a1,θ)

θ∈
a1∈A1
λa1,θψy|a1,θ.
P oposi ion 1. Fo e e y ε>0, he e exis s some ρ>0such ha
E
q(·|y,λ)−ma gλ

2<ρ⇒W0(λ)≤ca V(ma gλ)+ε.
See Appendix H o he p oo .
The ollowing lemma p o ides a uni o m bound (ac oss all equilib ia) on he num-
be o imes whe e he expec ed mo emen (in e ms o ·
2dis ance) in he SR playe s’
belie s is g ea e han ε.
Lemma 11. Fo any σ∈BNEδand any ε>0,
 :Eπσ
∞
νσ
+1h +1−νσ
h 

2≥ε≤1
ε.
P oo . Conside any ime +1:
Eπσ
∞
νσ
+1h +1−ν0

2=Eπσ
∞
νσ
+1h +1

2−ν02≤1.
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Theo e ical Economics 20 (2025) Repu a ion building 201
By he ma ingale p ope y o belie s, πσ
∞-almos su ely, Eπσ
∞[νσ
τ+1(hτ+1)|hτ]=νσ
τ(hτ).
The e o e, i is s aigh o wa d o show ha
1≥Eπσ
∞
νσ
+1h +1

2−ν02]=

τ=0
Eπσ
∞
νσ
τ+1hτ+1−νσ
τhτ

2.
Since heabo eholds o e e y , i implies ha
∞

τ=0
Eπσ
∞
νσ
τ+1hτ+1−νσ
τhτ

2≤1,
which implies he claim.
We can now p o e Theo em 3.
P oo o Theo em 3. We i s p o ide an uppe bound on
Eπσ
∞(1−δ)
∞

=0
δ u1a
1,a
2,θ
ha holds ac oss all σ∈BNEδ. No ice ha he abo e payo is no equal o U1(σ,θ;δ),
since he expec a ion does no condi ion on ωs. Howe e , one can in e p e he payo
abo e as ollows. Fo any equilib ium σ∈BNEδ,le ¯σ1deno e he s a egy, whe e he
LR playe ic i iously d aws some ω∈acco ding o μ0and plays he s a egy in he
equilib ium, σ, associa ed wi h ha ype o he en i e y o he epea ed game.41 Indeed
U1(¯σ1,σ2;δ)co esponds o he payo abo e.
By P oposi ion 1, he e exis s some ρ>0such ha
Eλ
q(·|y,λ)−p

2<ρ⇒W0(λ)≤ca V(ma gλ)+ε/8.
Choose n∈Nsuch ha 1
n(u−u)<ε/8andδ∗such ha o all δ>δ
∗,
1−δnm
ρu+δnm
ρca V(ν)+ε
4<ca V(ν)+ε
2.(2)
Fo any σ∈BNEδ,le
Tσ:= :Eπσ
∞
νσ
+1·|h +1−νσ
·|h 

2≥ρ
n.
Fo all /∈Tσ, by Ma ko ’s inequali y, we ha e
πσ
∞
νσ
+1·|h +1−νσ
·|h 

2≥ρ≤1
n.
41Fo example, i he LR playe d aws a commi men ype ω, hen he LR playe plays σω. I ins ead he
LR playe indeed d aws ωs, hen he LR playe simply plays σ1.
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202 Deb and Ishii Theo e ical Economics 20 (2025)
Figu e 7. Quali y choice.
Thus, a all /∈Tσ,
Eπσ
∞W0νσ
·|h ≤1
n(u−u)+Eπσ
∞ca Vνσ
·|h +ε/8≤ca V(ν0)+ε/4.
The e o e,
U1(¯σ1,σ2;δ)≤(1−δ)
∞

=0
δ Eπσ
∞W0(νσ
·|h 
=(1−δ)
∈Tσ
δ u+
/∈Tσ
δ Eπσ
∞W0νσ
·|h 
≤(1−δ)
∈Tσ
δ u+
/∈Tσ
δ ca V(ν0)+ε/4.
By Lemma 11, o e e y σ∈BNEδ,|Tσ|≤nm/ρ. The e o e, o all δ>δ
∗and any σ∈
BNEδ,
U1(¯σ1,σ2;δ)≤1−δnm/ρu+δnm/ρca V(ν0)+ε/4<ca V(ν0)+ε/2.
Finally, no e ha
U1(¯σ1,σ2;δ)≥1−μ0cU1(σ1,σ2;δ)+μ0cu.
Le χ∗>0besuch ha o allχ<χ
∗,
1
1−χca V(ν0)+ε
2−χu<ca V(ν0)+ε.
Thus, o all δ>δ
∗and μ0(c)<χ
∗,
U1(σ1,σ2;δ)≤1
1−μcca V(ν0)+ε
2−μ0cu<ca V(ν0)+ε.
Appendix F: Example
The ollowing example shows ha he p obabili y o commi men ypes ma e s o he
uppe bound e en when δis close o 1. Conside he quali y choice game wi h he
s age game payo s gi en by Figu e 7. In he epea ed game his s age game is epea -
edly played and all payo s a e common knowledge. No e ha he S ackelbe g payo o
he abo e game is 3/2. Fu he mo e, no e ha Bis a bes esponse o he SR playe in
he s age game i and only i α1(C)≥1/2.
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Theo e ical Economics 20 (2025) Repu a ion building 203
Figu e 8. The in o ma ion s uc u e.
The e a e wo s a es ={1, −1} ha only a ec he signal dis ibu ion o he public
signal. The e a e wo ypes in he game: ={ωc,ωs}. The commi men ype, ωc,in his
game is a ype ha always plays he mixed ac ion, 2
3H⊕1
3L, ega dless o he s a e.42 In
pa icula , we assume ha he p obabili y o each s a e is iden ical and he p obabili y
o he commi men ype is μ∈(0, 1).
The signal space is bina y, Y={¯
y,y}and he in o ma ion s uc u e is gi en by Fig. 8.
No e ha acco ding o his in o ma ion s uc u e, (2
3H⊕1
3L,θ)is s a is ically indis-
inguishable om (L,−θ):ψ(·|2
3H⊕1
3L,θ)=ψ(·|L,−θ). In his example, we ha e he
ollowing obse a ion.
Claim 3. The e exis s μ∗such ha o all μ>μ
∗and any δ∈(0, 1), he e exis s an equi-
lib ium in which he s a egic playe ob ains a payo o 2in bo h s a es.
P oo . Conside he candida e equilib ium s a egy p o ile in which he s a egic LR
playe always plays L. Choose μ∗=3
4. Then we will show ha when μ>μ
∗, his s a egy
p o ile is indeed an equilib ium o any δ∈(0, 1).
Conside he incen i es o he SR playe . To s udy his, we wan o compu e he p ob-
abili y ha he SR playe assigns o ac ion Tgi en he candida e equilib ium s a egy o
he LR playe :
λσ
H|h =2
3μσ
ωc|h =2
3γσ
ωc,1|h +γσ
ωc,−1|h .
Conside he likelihood a io
γσ
ωc,θ|h 
γσ
ωs,−θ|h =γσ
ωc,θ|h0
γσ
ωs,−θ|h0=μ
1−μ.
This hen implies ha o all h ,μ(ωc|h )=μ,μ(ωs|h )=1−μ. Thus, o all h and all
μ>μ
∗,
λσ
H|h =2
3μ>1
2.
This hen implies ha o all h , he SR playe ’s bes esponse is o play L. Fu he mo e,
because he SR playe is playing he same ac ion a all his o ies, he s a egic LR playe ’s
bes esponse is o play Ba all his o ies. Thus, he p oposed s a egy p o ile is indeed
42No e ha his is in eali y no he mixed S ackelbe g ac ion. Howe e , by app op ia ely modi ying he
in o ma ion s uc u e, he same conclusions hold, e en i he commi men ype plays some o he mixed
ac ion in e e y pe iod.
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204 Deb and Ishii Theo e ical Economics 20 (2025)
an equilib ium. Fu he mo e, acco ding o his s a egy p o ile, he s a egic LR playe ’s
payo is 2 in bo h s a es, concluding he p oo .
The abo e discussion shows ha when he commi men ype occu s wi h la ge p ob-
abili y, e en an a bi a ily pa ien s a egic LR playe ob ains a payo s ic ly g ea e
han he S ackelbe g payo in equilib ium. We now examine an uppe bound when he
commi men ype p obabili y is small.
Claim 4. Le ε>0. Then he e exis s some μ∗>0and δ∗<1such ha o all μ<μ
∗and
δ>δ
∗,U1(σ,δ)<3/2+ε o all σ∈BNEδ.
P oo .Conside V(p) o any p∈(). Because he s age game u ili ies a e s a e-
independen , i is s aigh o wa d o show ha
V(p)≤sup
α1∈A1
max
a2∈B2(α1)u1(α1,a2)=3/2,
whe e he equali y ollows om a s aigh o wa d calcula ion. The claim hen ollows
om Theo em 3.
Appendix G: P oo o Co olla y 2
The lowe bound is a consequence o Theo em 1. Le usnowshow heuppe bound.
Choose some ν∈(0, minθ∈ν0(θ)).
Suppose by way o con adic ion ha he e exis s some s a e θ∗∈and some se-
quence δn→1andσn∈BNEδnsuch ha o all n,U1(σn,θ∗;δn)≥u∗
1(θ∗)+ε.ByTheo-
em 3,
ν0θ∗u∗
1θ∗+ε+lim sup
n→∞ 
θ=θ∗
ν0(θ)U1σn,θ;δn<
θ∈
ν0(θ)u∗
1(θ)+νε.
Toge he wi h Theo em 1,weha e

θ=θ∗
ν0(θ)u∗
1(θ)≤lim sup
n→∞ 
θ=θ∗
ν0(θ)U1σn,θ;δn≤
θ=θ∗
ν0(θ)u∗
1(θ)−ν0θ∗−νε,
bu his is a con adic ion.
Appendix H: P o ing P oposi ion 1
Le us i s de ine he se
ˆ
NR(p):=λ∈(A1×):λ(·|θ)θ∈∈NR(p),ma gλ=p,
ˆ
NR :=
p∈()
ˆ
NR(p).
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Theo e ical Economics 20 (2025) Repu a ion building 205
No ice ha V(p)=supλ∈ˆ
NR(p)W0(λ). Analogously, we can de ine o any ε>0,
Vε(p)=sup
λ∈ˆ
NR(p)
Wε(λ).
Finally, de ine also o e e y ε≥0,
ε(a2):=λ∈ˆ
NR : a2∈Bε
2(λ).
We begin wi h some lemmas.
Lemma 12. Le ε>0. Then he e exis s some ρ>0such ha o all λ∈(A1×),
E
q(·|y,λ)−ma gλ

2<ρ⇒in
ˆ
λ∈ˆ
NR
λ−ˆ
λ<ε.
See Lemma V.3.6 in Me ens, So in, and Zami (2014) o he p oo .
Lemma 13. Le ε>0. Then he e exis s some ρ>0such ha o all λ,ˆ
λ∈(A1×),
λ−ˆ
λ<ρ⇒W0(λ)≤Wε(ˆ
λ)+ε.
P oo .Le ε>0. Fi s choose ρ>0 su icien ly small such ha
λ−ˆ
λ<ρ
⇒max
a2∈A2u2(a2,λ)−u2(a2,ˆ
λ)≤ε.
Then he e exis s some ρ∈(0, ρ)such ha λ−ˆ
λ≤ρ=⇒ B0
2(λ)⊆Bε
2(ˆ
λ). The e o e,
whene e λ−ˆ
λ≤ρ,
W0(λ)=max
a2∈B0
2(λ)
u1(a2,λ)≤max
a2∈Bε
2(ˆ
λ)
u1(a2,λ)≤max
a2∈Bε
2(ˆ
λ)
u1(a2,ˆ
λ)+ε=Wε(ˆ
λ)+ε.
Lemma 14. Fo e e y ε>0, he e exis s some ρ>0such ha o all a2∈A2,
λ∈ρ(a2)⇒in
λ∈0(a2)
λ−λ
<ε.
P oo . Suppose o he wise. Then o some a2∈A2and ε>0, he e exis s some se-
quence ρn→0andλn∈ρn(a2)such ha
in
λ∈0(a2)
λn−λ
≥ε.(3)
By Bolzano–Weie s ass, wi hou loss o gene ali y, by eplacing he o iginal sequence
wi h an app op ia e subsequence, we can assume his sequence o be con e gen o
some limi λ. Howe e , no e ha since λn→λand λn∈ρn(a2) o all n,λ∈0(a2).
This con adic s (3).
Lemma 15. Fo e e y ε>0, he e exis s some ρ∗>0such ha o all λ∈ˆ
NR and all ρ<
ρ∗,
Wρ(λ)≤ca V(ma gλ)+ε.
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206 Deb and Ishii Theo e ical Economics 20 (2025)
P oo . Fi s , because ca Vand u1(·,a2)a e Lipschi z con inuous o all a2∈A2, he e
exis s some ε>0 such ha whene e λ−λ<ε
, hen
ca V(ma gλ)−ca Vma gλ,max
a2∈A2u1(a2,λ)−u1a2,λ<ε/2.
By he p e ious lemma, le ρ>0besuch ha o alla2∈A2,
λ∈ρ(a2)⇒in
λ∈0(a2)
λ−λ
<ε
.
Recall ha
Wρ(λ)=max
a2∈Bρ
2(λ)
u1(a2,λ).
Le aρ
2(λ)∈Bρ
2(λ)be he solu ion o he abo e maximiza ion p oblem. Thus, o e e y
λ∈ˆ
NR, λ∈ρ(aρ
2(λ)). The e o e, o all λ∈ˆ
NR, he e exis s some λ(λ)∈0(aρ
2(λ))wi h
λ−λ(λ)≤ε.
Then o any λ∈ˆ
NR,
Wρ(λ)=u1aρ
2(λ),λ≤max
a2∈B0
2(λ(λ))
u1(a2,λ)
≤W0λ(λ)+ε/2
≤ca Vma gλ(λ))+ε/2≤ca V(ma gλ)+ε.
We can now p o e P oposi ion 1.
P oo o P oposi ion 1. By Lemma 15, he e exis s some ρ∗∈(0, ε/3)such ha o
all ˆ
λ∈ˆ
NR,
Wρ∗(ˆ
λ)≤ca V(ma gˆ
λ)+ε/3.
By Lemma 13 and Lipschi z con inui y o ca V, he e exis s some ρ>0such ha

λ−λ
<ρ
⇒W0(λ)≤Wρ∗λ+ρ∗,ca V(ma gλ)−ca Vma gλ<ε/3.
By Lemma 12, he eexis sρ>0such ha o allλ o which E[q(·|y,λ)−ma gλ2]<
ρ, he eexis sˆ
λ(λ)∈ˆ
NR such ha ˆ
λ(λ)−λ<ρ
.
Thus, o any λin which E[q(·|y,λ)−ma gλ2]<ρ,weha e
W0(λ)≤Wρ∗ˆ
λ(λ)+ρ∗≤ca V(ma gˆ
λ(λ)+2ε/3≤ca V(ma g(λ)+ε.
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