Robust relational contracts with subjective performance evaluation

Bhaska , V.; Olszewski, Wojciech; Wiseman, Thomas
A icle
Robus ela ional con ac s wi h subjec i e pe o mance
e alua ion
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Bhaska , V.; Olszewski, Wojciech; Wiseman, Thomas (2024) : Robus ela ional
con ac s wi h subjec i e pe o mance e alua ion, Theo e ical Economics, ISSN 1555-7561, The
Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 3, pp. 1027-1055,
h ps://doi.o g/10.3982/TE5911
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/320259
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc/4.0/
Theo e ical Economics 19 (2024), 1027–1055 1555-7561/20241027
Robus ela ional con ac s wi h subjec i e pe o mance
e alua ion
V. Bhaska
Depa men o Economics, Uni e si y o Texas a Aus in and CEPR
Wojciech Olszewski
Depa men o Economics, No hwes e n Uni e si y
Thomas Wiseman
Depa men o Economics, Uni e si y o Texas a Aus in
We s udy a epea ed p incipal–agen model wi h ans e able u ili y, whe e he
p incipal’s e alua ion o he agen ’s pe o mance is subjec i e. Ou ocus is on
equilib ia ha a e obus o he addi ion o small p i a ely obse ed shocks o
he payo s. Exis ing cons uc ions o posi i e-e o equilib ia a e no obus o
such payo shocks. Allowing o simul aneous cheap- alk announcemen s makes
some e o sus ainable in a obus equilib ium, and payo s can be a bi a ily
close o ully e icien ones i playe s a e su icien ly pa ien . In con as o he ex-
is ing li e a u e, ou nea -e icien equilib ia exhibi ealis ic ea u es: he bonus
size is easonable, he h eshold o being paid a bonus is non i ial, and he base
wage need no be nega i e.
Keywo ds. P i a e moni o ing, epea ed games, ela ional con ac s.
JEL classi ica ion. C73, D86.
1. In oduc ion
In many o ganiza ions, he asks ha employees mus pe o m lack an objec i e, con-
ac ible measu e o pe o mance. Thus, pe o mance e alua ion is subjec i e and he
wo ke canno obse e he employe ’s e alua ion o his own pe o mance. The employe
may be unable o commi o an incen i e scheme, and may need o ely on ela ional
con ac s and he subjec i e e alua ion o incen i ize he wo ke , as in Le in (2003),
Fuchs (2007), and Maes i (2012).1The con ac s s udied in he li e a u e ypically spec-
i y ha he wo ke exe s e o , and ha he employe pays a bonus o he wo ke i
and only i he subjec i e e alua ion o he wo ke ’s pe o mance is good. To p o ide
incen i es o he employe o disclose he e alua ion u h ully, she is made indi e en
V. Bh as k a : [email p o ec ed]
Wojciech Olszewski: [email p o ec ed]
Thomas Wiseman: [email p o ec ed]
We a e g a e ul o Ma cin P˛eski, and h ee e e ees o e y use ul commen s.
1Bake , Gibbons, and Mu phy (2002) and Malcomson (2013) examine he ci cums ances unde which
ela ional con ac s a e aluable in an o ganiza ional se ing.
©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/TE5911
1028 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
be ween paying and no paying he bonus. To achie e ha indi e ence, he ela ionship
mus be dissol ed wi h some p obabili y in he e en ha he bonus is no paid. Such an
equilib ium is ine icien , since a p oduc i e ela ionship mus be dissol ed wi h pos-
i i e p obabili y. Fuchs (2007) shows ha e iciency can be enhanced by equi ing he
p incipal o epo on he agen ’s pe o mance only e e y Tpe iods. As in Ab eu, Mil-
g om, and Pea ce (1991), he ex en o ine iciency dec eases wi h T. When bo h playe s
become a bi a ily pa ien , he equilib ium can a ain ull e iciency.
Ou pape begins wi h he obse a ion ha he equilib ia so cons uc ed a e ag-
ile and hey do no su i e i he p incipal is subjec o small payo shocks o he low
e enues. In his case, equilib ia whe e he p incipal is indi e en be ween paying he
bonus o no do no su i e, because he p incipal s ic ly p e e s o pay he bonus when
she lea ns ha low e enues in u u e pe iods a e likely o be high, and s ic ly p e e s
no o pay i when she lea ns ha hey will be low. In consequence, she will condi ion
he bonus paymen on he shock, no on he agen ’s pe o mance. This b eaks he link
be ween he agen ’s e o and bonus paymen s, and des oys his incen i e o p o ide
e o . Mo e p ecisely, an equilib ium is pu i iable i i is he limi o a sequence o equi-
lib ia o a sequence o games wi h payo shocks, as he shocks anish. We show ha o
a la ge class o equilib ia, ei he he agen will ne e exe e o o he equilib ium can-
no be pu i ied. In pa icula , none o he equilib ia p oposed in he li e a u e su i es
when he e a e payo shocks.2
We show ha he e exis posi i e e o equilib ia ha a e obus o small payo
shocks i we allow he p incipal and he agen o make simul aneous cheap- alk an-
nouncemen s a he end o each pe iod. The in ui ion o why cheap alk allows o a
obus equilib ium is as ollows. Conside an equilib ium o he base game whe e he
bonus is paid whene e ou pu is high, he bonus is se a a le el whe e he agen is
indi e en be ween wo king and shi king, and whe e he wo ke ’s expec ed compensa-
ion equals his ou side op ion. I he employe does no pay he bonus, he ela ionship
is e mina ed wi h a p obabili y ha makes he employe indi e en be ween paying
and no paying he bonus. Modi y his cons uc ion as ollows: he wo ke shi ks wi h
a small p obabili y, and wo ke and employe e eal hei p i a e in o ma ion, i.e., he
wo ke announces whe he he exe ed e o o shi ked, and he employe announces
whe he ou pu was high o low. I he announcemen s misma ch (i.e., he wo ke an-
nounces e o and he employe announces low ou pu , o hey announce shi king and
high ou pu , espec i ely), hen he ela ionship is e mina ed wi h an addi ional small
p obabili y. Random e o implies ha he p incipal’s obse a ions o ou pu a e in o -
ma i e, and he addi ional e mina ion p obabili ies gi e he employe s ic incen i es
o epo u h ully. We p o ide s ic incen i es o he wo ke o ell he u h by making
him pay a small ine when epo s misma ch; his base wage is inc eased sligh ly by he
2I could be a gued ha he p incipal may ha e a small p e e ence o being hones , which could gi e
he s ic incen i es o ell he u h ega ding he agen ’s pe o mance. Howe e , i could also be he case
ha he p incipal wan s o pay he bonus i and only i she belie es ha he agen has wo ked. In his case,
in any equilib ium whe e he wo ke always wo ks, he p incipal’s belie s do no depend on he obse ed
signals.
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1029
expec ed alue o ines and he paymen o a pa o he base wage, ne o ines, is de-
e ed o he end o he pe iod.3When he noise in moni o ing is small and he discoun
ac o is abo e a cu o , we can cons uc an equilib ium whe e he agen wo ks wi h
a bi a ily high p obabili y and he e iciency loss is close o ze o. Mo eo e , since bo h
playe s ha e s ic incen i es o u h- elling and he agen ’s andomiza ion is his o y-
independen , he equilib ium is pu i iable.
Fo he case whe e he noise in moni o ing is la ge, we explo e equilib ia whe e an-
nouncemen s a e made only e e y Tpe iods. A di icul y a ises: any equilib ium ha in-
cen i izes u h ul announcemen s wi h penal ies o misma ched epo s equi es he
agen o shi k wi h posi i e p obabili y in e e y pe iod. We p esen wo app oaches o
esol ing ha di icul y. In he i s , he agen andomly picks a single pe iod o he T-
pe iod block in which o shi k. This cons uc ion allows us o pu i y a la ge a ie y o
block equilib ia, including he equilib ia o Fuchs (2007) men ioned in he i s pa a-
g aph. In he second app oach, he agen mus be indi e en be ween always shi king
and always wo king wi hin each T-pe iod block. The block equilib ia o Fuchs (2007),
whe e penal ies a e imposed only i all Tsignals a e bad, do no sa is y ha equi emen .
Ins ead, we in oduce a di e en cons uc ion, bo owing an idea om Ma sushima
(2004), ha does yield he necessa y indi e ence. Bo h app oaches achie e e iciency
in he limi as he playe s become a bi a ily pa ien .
Ou ela ional con ac s based on block s a egies di e quali a i ely om exis ing
cons uc ions in he li e a u e. In Fuchs (2007), he agen ge s a bonus a he end o he
block excep when ou pu is low in e e y pe iod o he block. Thus, he agen is e y
likely o ea n he bonus e en when he shi ks in e e y pe iod. The ma ginal inc ease
in he p obabili y o achie ing he bonus a ge by wo king an addi ional pe iod is also
small. This implies ha he pe -pe iod bonus4 ends o in ini y as he leng h o he block,
T,goes o∞and he base wage goes o −∞. In bo h ou equilib ium cons uc ions, he
h eshold o ea n he bonus can be se a a le el ha he agen is likely o each only
wi h consis en e o , and he pe -pe iod bonus is app oxima ely equal o he cos o e -
o pe pe iod. Consequen ly, he base wage is close o he ou side op ion o he agen .
We p o ide a o mal esul showing ha hese a ac i e ea u es a e a pa o ou sec-
ond equilib ium cons uc ion. We iew hese p ope ies as cap u ing mo e accu a ely
how i ms se , o example, sales a ge s o hei employees. Ano he dis inc i e ea u e
o ou cons uc ion is he use o epo s om he wo ke as well as om he employe .
Many i ms inco po a e ha so o employee sel -e alua ion in hei pe o mance e-
iews.
The small shocks o payo s ha we conside , and equi e obus ness o, would be
p esen in nea ly any economic applica ion. Fo example, an employe ’s indi e ence
be ween paying a bonus o no would be b oken by whe he o no she has a pen and
checkbook handy. Tha agili y mo i a es ou ocus on pu i iable equilib ia: heo e ical
3Since he wo ke ’s expec ed con inua ion alue equals he ou side op ion, he will ha e no incen i e o
make hese paymen s olun a ily; his is he eason o making he employe esponsible o he paymen s.
4The pe -pe iod bonus is τ/T ,whe eτis he bonus ha is paid a he end o he block and Tis he leng h
o he block.
1030 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
p edic ions ha a e no obus o such small de ails a e unlikely o accu a ely desc ibe
eal-wo ld beha io .
On he heo e ical side, he pape demons a es how inco po a ing cheap- alk e-
po s o p i a e signals and incen i izing u h- elling h ough penal ies o disag ee-
men can be used o cons uc pu i iable equilib ia unde p i a e moni o ing. Tha
echnique may be use ul mo e gene ally in cons uc ing equilib ia whe e playe s ha e
s ic incen i es o condi ion on his o y in epea ed games wi h p i a e moni o ing. The
belie - ee cons uc ions o en used in ha se ing ely on playe s’ exac indi e ence be-
ween punishing o no so as o p o ide incen i es. Bhaska , Maila h, and Mo is (2013)
discuss he di icul y o making hose equilib ia pu i iable. Ou echnique also p o ides
an addi ional ole o mixed s a egies in epea ed games: non i ial andomiza ion by
bo h playe s is equi ed so ha each playe ’s p i a e in o ma ion is a p edic o o he
o he ’s announcemen .
1.1 Rela ed li e a u e
The e is a la ge li e a u e on epea ed games wi h p i a e moni o ing ha is ele an
and ha we will no do ull jus ice o. B ie ly, while he ea ly li e a u e used bo h “belie -
based” and “belie - ee” app oaches, mos o he subsequen wo k has buil on elemen s
o he belie - ee equilib ia pionee ed by Piccione (2002) and Ely and Välimäki (2002).
Ma sushima (2004) uses belie - ee app oach o show ha block-s a egy equilib ia can
ensu e asymp o ic e iciency in he epea ed p isone s’ dilemma when he playe s’ p i-
a e signals a e independen . Ely, Hö ne , and Olszewski (2005) gene alize belie - ee
equilib ia o a la ge class o games. Sugaya (2022), building on he cons uc ions o
Hö ne and Olszewski (2006) and Ma sushima (2004), p o es a gene al e sion o he
olk heo em o epea ed p i a e-moni o ing games. In he wo k on ela ional con ac s
(Le in (2003), Fuchs (2007)), he a ailabili y o ans e s makes i easie o achie e he in-
di e ences equi ed o belie - ee cons uc ions wi hou any need o andomiza ion.
The p esen pape di e s om he li e a u e on epea ed games wi h p i a e moni-
o ing in h ee aspec s. Fi s , he s age game conside ed he e has a non i ial ex ensi e
o m, whe eas he olk heo em o Sugaya (2022) ob ains o simul aneous-mo e s age
games. Second, moni o ing o he agen by he p incipal is p i a e, bu he p incipal’s
ac ions a e public. Thi d, and mos impo an , is ou insis ence on equilib ia ha a e
obus o p i a e payo shocks and, hus, a e pu i iable. None heless, we also build on
hese p e ious app oaches. Since he agen shi ks wi h posi i e p obabili y in e e y pe-
iod and his con inua ion s a egy a ies wi h ealized e o , ou cons uc ion is belie -
based in his espec . Howe e , ou equilib ium cons uc ions may also be iewed as
modi ica ions o belie - ee app oaches, since hey equi e he p incipal, i she does no
pay a bonus o he agen , o dissol e he ela ionship wi h some p obabili y so ha he
loss is equal o he bonus. In addi ion, one o ou cons uc ions bo ows an idea om
Ma sushima (2004).
In con as wi h mos o he li e a u e, ou posi i e esul s equi e cheap- alk an-
nouncemen s in addi ion o andomiza ion by he agen . The di e ence a ises due o

Theo e ical Economics 19 (2024) Robus ela ional con ac s 1031
he sequen ial na u e o ou s age game and since he p incipal’s ac ions (bonus pay-
men s) a e public a he han p i a e. The cheap- alk announcemen s in oduce an ele-
men o simul anei y ha allows us o ci cum en he induc ion a gumen s ha unde lie
ou nega i e esul s.
Cheap alk plays an impo an ole in ou analysis. In hei pionee ing wo k on e-
pea ed games wi h p i a e moni o ing, Comp e (1998) and Kando i and Ma sushima
(1998) p o e a olk heo em by using cheap- alk announcemen s o coo dina e beha -
io . When he e a e wo playe s, and signals a e independen condi ional on he ac ion
p o ile, he equilib ia ha hey cons uc ha e a belie - ee la o , in ha each playe
is made indi e en be ween he possible announcemen s. Cheap- alk plays a di e en
ole he e, since playe s a e p o ided s ic incen i es o u h- elling. Fu he mo e, an-
domiza ion by he agen plays an essen ial ole in p o iding s ic incen i es, whe eas
andomiza ion plays no such ole in his ea lie wo k.
The eade migh ask, “Why is i ha we equi e cheap- alk announcemen s in addi-
ion o andomiza ion by he agen ?” P e ious wo k on he epea ed p isone ’s dilemma
wi h p i a e moni o ing has cons uc ed belie -based equilib ia, whe e ini ial andom-
iza ion by bo h playe s, coupled wi h wo-sided p i a e moni o ing, su ices o p o-
ide s ic incen i es in subsequen pe iods (Sekiguchi (1997) and Bhaska and Oba a
(2002)). The di e ence a ises due o he sequen ial na u e o ou s age game and since
he p incipal’s ac ions (bonus paymen s) a e public a he han p i a e. The cheap- alk
announcemen s in oduce an elemen o simul anei y ha allows us o ci cum en he
induc ion a gumen s ha unde lie ou nega i e esul s.
Ou nega i e esul s, ha pu e s a egy equilib ia canno be pu i ied, ha e he ol-
lowing an eceden s. Ma sushima (1991) s udies pu e s a egy equilib ia in epea ed
games wi h condi ionally independen p i a e moni o ing. He imposes he es ic ion
ha a playe ’s s a egy does no condi ion on he his o y o p i a e signals unless he e
is a s ic incen i e o do so, and he inds ha playe s mus play a Nash equilib ium o
he s age game in e e y pe iod. Bhaska , Maila h, and Mo is (2013) show ha in games
whe e only one playe mo es a a ime and whe e moni o ing is public bu subjec o
bounded memo y, only Ma ko pe ec equilib ia can be pu i ied. Ou posi i e esul s
equi e wo ke andomiza ion and communica ion, and a e ela ed o Miyagawa, Miya-
ha a, and Sekiguchi (2008), who s udy epea ed games wi h cos ly moni o ing, whe e
each playe has o be incen i ized o moni o he opponen s. Simila ly, in Rahman
(2012), bo h wo ke and moni o mus andomize hei e o and inspec ion decisions,
espec i ely, so as o incen i ize each o he .
Rahman and Oba a (2010) s udy pa ne ships and assume ha incen i e schemes
mus sa is y budge balance. They show ha a media o can be used o i ually imple-
men he e icien ou come whe e all pa ne s wo k, by making, wi h a small p obabili y,
a sec e ecommenda ion o shi k o a andomly chosen pa ne , and by making ewa ds
con ingen bo h on ou pu and on he ecommenda ion. We no e ha a media o makes
i easie o cons uc nea -e icien equilib ia, since playe s do no ha e o be indi e en
be ween wo king and shi king.
1032 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
2. The basic model
Time is disc e e and he ho izon is in ini e. The e a e wo playe s: he p incipal and he
agen . The p incipal selec s a base wage w, and in each pe iod un il he ela ionship
is e mina ed by ei he playe , he p incipal pays he agen w, and he agen chooses
be ween exe ing e o (E) and shi king (S), wi h e o cos c>0. The esul ing ou pu
y, which is p i a ely obse ed by he p incipal, is s ochas ic and akes alues in he se
{G,B},whe eG>B.5We assume ha P (y=G|E)=pand P (y=G|S)=q, sa is ying
1>p>q>0, so ha ou pu is a noisy signal o he agen ’s e o choice. A e obse ing
y, he p incipal may choose o pay he agen an addi ional bonus. The agen ’s ou side
op ion is ¯
w; we no malize he p incipal’s ou side op ion o 0.
Le ¯
yand ydeno e he expec ed alues o ou pu when Eand Sa e chosen, espec-
i ely. We assume ha ¯
y−c> ¯
w>y
. Thus, i is e icien o he agen o be employed
and o exe e o ( he inequali ies imply ha (p−q)(G−B)>c), bu i he agen shi ks,
hen i is p e e able o dissol e he ela ionship. Bo h playe s a e isk neu al and hey
ace no limi ed liabili y cons ain s. They maximize he discoun ed sum o payo s, wi h
common discoun ac o δ∈(0, 1).
In he in e es o p ecision, le us conside he ollowing s age game  ha is played
in e e y pe iod, condi ional on he ela ionship no ha ing been e mina ed by ei he
playe .
•The agen is paid he base wage wand chooses a∈{E,S}.
•The p incipal obse es y∈{G,B}and decides whe he o no o pay a bonus τ,o e
and abo e he base wage w.
•The p incipal and agen obse e he ealiza ion o a public andomiza ion de ice6
and simul aneously decide whe he o no o e mina e he ela ionship. The ela-
ionship con inues o he nex pe iod i and only i bo h pa ies wan o con inue.
We deno e he game ha is epea ed in ini ely, unless e mina ed by a playe , by ∞.
The undamen al di icul y is ha moni o ing is impe ec and p i a e. The p inci-
pal does no obse e he agen ’s ac ion, and he agen does no obse e he signal y.To
incen i ize e o , he agen ’s bonus paymen s (o his con inua ion alue) mus depend
on he p incipal’s obse a ion o ou pu . Howe e , because his obse a ion is p i a e,
he p incipal’s con inua ion alue mus be independen o he ou pu he p incipal ob-
se es. MacLeod (2003)andLe in(2003) p opose a solu ion ha he p incipal is indi -
e en be ween paying he bonus o no paying i . This indi e ence can be achie ed ia
a public andomiza ion de ice ha dec ees ha he ela ionship (which is p o i able o
he p incipal) be dissol ed wi h some p obabili y whene e he bonus is no paid. In
o he wo ds, a pa o he expec ed su plus om he ela ionship mus be des oyed, be-
cause he agen canno be punished while simul aneously ewa ding he p incipal. As
5Ou analysis ex ends o any ini e signal space Y. The p oo s o ou nega i e esul s hold in ha case.
Fo ou posi i e esul s, i we o de he signals yn om lowes o highes likelihood a io P (yn|E)/P (yn|S),
hen we may ocus on a bina y pa i ion o he signal space, {G,B},whe eG={yn∈Y|n≥¯
n}and B={yn∈
Y|n<¯
n} o some cu o ¯
n.
6Tha is, he ealiza ion o a andom a iable ha is uni o mly dis ibu ed on in e al [0, 1].
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1033
usual, ha su plus des uc ion could also ake he o m o he p incipal bu ning money
o gi ing i o a hi d pa y ha nei he he p incipal no he agen ca es abou .
The equilib ium is ine icien because some su plus is des oyed. Fuchs (2007)
shows ha ine iciency can be mi iga ed i he playe s a e pa ien , by di iding he in-
e ac ion in o blocks o Tpe iods. The bonus is wi hheld only i he agen ails ( ha is,
ou pu is B) in e e y pe iod in he block, and his me hod o le e aging a single bonus
o incen i ize e o in mul iple pe iods educes he loss in su plus. Fuchs (2007)also
shows ha he mos e icien equilib ium o a ixed discoun ac o is an e iciency wage
ype equilib ium whe e he only eedback p o ided by he p incipal is when she e mi-
na es he ela ionship.
3. Pu i iabili y
The majo p oblem wi h he block equilib ium (and also he one-pe iod cons uc ion)
is ha i elies on he p incipal’s indi e ence be ween paying he bonus and no pay-
ing i , and on he b eaking his indi e ence acco ding o he his o y o p i a e signals.
Consequen ly, he equilib ium is agile. In pa icula , i he alue o he ela ionship o
he p incipal is subjec o small shocks ha a e p i a ely obse ed by he p incipal, hen
she will condi ion he bonus paymen on he ealiza ion o hese shocks, no on ou -
pu signals. This p oblem also a ises in he no- eedback e iciency wage cons uc ion:
in any pe iod whe e he p incipal i es he agen wi h posi i e p obabili y, she mus
be indi e en be ween e aining he agen and i ing, and his indi e ence is un enable
wi h shocks o he p incipal’s con inua ion alue.7
To make his a gumen p ecise, we will also s udy he ollowing class o ξ-pe u bed
games ∞(ξ,λ),whe eξand λa e posi i e numbe s, de ined as ollows:
•Be o e he p incipal makes he bonus decision in pe iod , he p incipal p i a ely
obse es a andom shock z +1. The p incipal’s payo om ou pu in pe iod is
gi en by B+ξz and G+ξz , i.e., i depends on he shocks ha we e ealized in
pe iod −1.
We assume ha z ollows a s a iona y au o eg essi e p ocesses o o de 1, ha is,
z =λz −1+(1−λ)˜
z ,whe eλ∈(0, 1)and ˜
z a e independen andom a iables, dis-
ibu ed iden ically and con inuously on a bounded in e al Zwi h mean 0. Fo com-
ple eness, le z0=˜
z0. P i a e shocks display some pe sis ence, al hough i may be a -
bi a ily small. Pe sis ence implies ha he p incipal’s u u e su plus om he ela ion-
ship is s ochas ic, and she has p i a e in o ma ion ega ding i . This ensu es ha one
can ne e make he p incipal indi e en be ween paying he bonus and p obabilis ic
e mina ion.
I he shocks we e independen and iden ically dis ibu ed (i.i.d.), a mino modi ica-
ion o he T-pe iod block-s a egy equilib ium, whe e we allow he p incipal o make
an announcemen , would su ice. The p incipal could announce a he end o a block
whe he he agen had passed o ailed he es , bu bo h he paymen o he bonus
7The same p oblem a ises i he e a e shocks o he p incipal’s ou side op ion.
1034 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
and he s ochas ic e mina ion o he ela ionship could be de e ed o K>1pe iods.
This ensu es ha he p incipal would be indi e en be ween paying and no paying he
bonus, since she has no p i a e in o ma ion abou he payo consequences o he an-
nouncemen . The assump ion ha shocks a e e e so sligh ly pe sis en ensu es ha he
p incipal is ne e indi e en be ween paying and no paying he bonus, no ma e he
iming o hese e en s.
Ou payo shocks ha e bounded suppo , and by aking ξ o be small, hei e ec i e
ange can be made a bi a ily small. None heless, in hei p esence, i is di icul o sus-
ain an equilib ium whe e he wo ke exe s e o on he equilib ium pa h. We will now
p esen wo esul s showing ha no e o can be sus ained in some classes o equilib-
ia, which bo h include he equilib ia s udied in he exis ing li e a u e, i we equi e in
addi ion ha hese equilib ia be pu i iable.
To make his p ecise, he ollowing p elimina ies a e necessa y. In he pe u bed
game, in any pe iod , he agen alone obse es his e o choice a . The p incipal alone
obse es ou pu y and he shock o ou pu z +1. Bo h pa ies obse e he public e en s,
such as he bonus paymen and he ealiza ions o he public andomiza ion de ice,
whichwedeno ebyω ∈. Te mina ion decisions a e also public, bu since he game
ends i one pa y chooses o e mina e, we es ic a en ion o his o ies whe e bo h pa -
ies ha e chosen o con inue he ela ionship o da e. Le  −1deno e he se o possible
public his o ies a he beginning o pe iod .
Le σ=(σ )∞
=1deno e he s a egy o he p incipal. A s a egy o he p incipal p e-
sc ibes a bonus paymen and a i ing decision a all possible his o ies. Thus, σ consis s
o a pai (σ1
,σ2
). The bonus paymen is de e mined by σ1
: −1×{G,B} ×Z→[0, ∞).
This depends on he public his o y o da e, he his o y o obse ed ou pu s, and he
cu en alue o he shock o low e enues, z +1. The exp ession σ2
: −1×[0, ∞)×
{G,B} ×Z×[0, 1]→{F,R}de e mines he p incipal’s i ing/ e en ion decision, which
is based addi ionally on he p incipal’s bonus paymen his pe iod and he ealiza ion o
he public andomiza ion.8
Le ρ=(ρ )∞
=1deno e he s a egy o he agen . A s a egy o he agen p esc ibes
an e o choice and a qui ing decision a all possible his o ies. Thus, ρ consis s
o a pai (ρ1
,ρ2
). The agen ’s e o choice is de e mined by ρ1
: −1×{E,S} −1→
{E,S}. This depends on he public his o y o da e and he agen ’s his o y o e o
choices. The exp ession ρ2
: −1×[0, ∞)×{E,S} ×[0, 1]→{Q,C}de e mines his
qui ing/con inua ion decision, which is based addi ionally on his cu en pe iod e -
o choice, he p incipal’s bonus paymen his pe iod, and he ealiza ion o he public
andomiza ion.
No e ha he agen ’s s a egies a e he same ma hema ical objec s in he o iginal
game ∞andin hepe u bedgame∞(ξ,λ). In u n, he p incipal’s s a egies in hese
wo games a e o mally di e en objec s. So o compa e he equilib ia o he wo games,
we conside expec ed s a egies o he p incipal in he pe u bed game. Mo e p ecisely,
le hi
deno e a gene ic his o y o public e en s and ou pu s obse ed by he p incipal up
8A playe ’s p i a e his o y also includes pas alues o he payo shock, bu since hese a e payo i el-
e an , he p incipal will no condi ion on hese shocks in he pe u bed game; see Bhaska , Maila h, and
Mo is (2013), Lemma 2.
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1041
e o in pe iod is c+ξx ,whe ex ollows a s a iona y au o eg essi e p ocesses o o -
de 1, x =λx −1+(1−λ)
x ,whe eλ∈(0, 1)and 
x a e independen andom a iables,
dis ibu ed iden ically and con inuously on a bounded in e al Xwi h mean ze o, and
( o comple eness) le x0=
x0.Le hebonusτbe chosen so he ype o agen wi h cos
c+ξx∗is indi e en be ween wo king and shi king. The bonus is unchanged o e ime
and, hus, he ype o he agen who is indi e en is also in a ian o e ime. Choose
x∗close o he uppe end o in e al X. This gua an ees ha he p obabili y ha he
agen p e e s shi king o wo king is small, bu bounded away om ze o in e e y pe iod,
when λis su icien ly small. Howe e , since cos s a e au oco ela ed, he p obabili y o
he agen exe ing e o in pe iod will a y o e ime. This does no ma e , since he
p ecise p obabili y ha he agen exe s e o plays no ole in he cons uc ion, as long
as i is s ic ly less han 1. Finally, as ξ→0 and he au oco ela ion anishes, he agen ’s
beha io con e ges o a andomiza ion p obabili y ha is cons an ac oss pe iods.
Fo φclose o 1 and close o 0, he p incipal’s alue o he ela ionship is close
o VP. Thisisless hany−w−c, he highes possible su plus. Some su plus mus be
bu ned because moni o ing is impe ec , and he ela ionship mus be e mina ed wi h
posi i e p obabili y since o he wise she would ha e an incen i e o always claim ha
she obse ed low ou pu . These payo losses a ise e en in he nonpu i iable equilib-
ium o Le in (2003), whe e he wo ke chooses e o o su e, and whe e he e is no
cheap alk. We, he e o e„ ha e he ollowing p oposi ion.
P oposi ion 3. Assume VP>0and δ>δ. Fo e e y >0, he e exis s a pu i iable
equilib ium wi h cheap alk such ha he p incipal’s expec ed payo is a leas VP−.
P oposi ion 3 equi es he discoun ac o δ o be s ic ly g ea e han some δ<1,
since o he wise he e a e no easible alues o αV P ha sa is y he p incipal’s indi e -
ence condi ion (2). In o he wo ds, i he p incipal is oo impa ien , she canno be in-
cen i ized o pay he bonus.
The addi ional ine iciency imposed by he pu i iabili y equi emen is wo old: i s ,
he wo ke mus shi k wi h posi i e p obabili y and, second, he e is an addi ional p ob-
abili y o e mina ion induced by he phase o message exchange. None heless, hese
cos s can be a bi a ily small, since bo h he shi king p obabili y 1 −φand he addi-
ional e mina ion p obabili ies αij can be a bi a ily small.
Finally, as he noise in moni o ing anishes, e iciency is achie able; mo e gene ally,
we ha e he ollowing co olla y.
Co olla y 1. Assume VP>0and δ>δ. Fo e e y >0,i (1−p)/(p−q)is su i-
cien ly close o 0, hen he e exis s a pu i iable equilib ium wi h cheap alk such ha he
p incipal’s expec ed payo is a leas y−w−c−.
5. T-pe iod block equilib ia wi h cheap alk
The equilib ia om he p e ious sec ion achie e e iciency only when he noise in mon-
i o ing anishes. Wi h non anishing noise, Fuchs (2007) shows ha pa ien playe s can

1042 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
app oach e icien ou comes by di iding play in o blocks o Tpe iods. A he end o he
block, he p incipal pays a bonus o he agen unless he ou pu was Bin all Tpe iods.
We ind ha adding cheap alk can deli e a simila e iciency esul , bu in pu i iable
equilib ia; in addi ion, some equilib ia ha we cons uc ha e quali a i ely di e en
(pe haps, mo e ealis ic) ea u es.
As in he one-pe iod case in he p e ious sec ion, we gene a e incen i es o u h-
elling by penalizing misma ched epo s. Tha app oach equi es non i ial andom-
iza ion by he agen . Le us examine he condi ions ha mus be sa is ied in a T-pe iod
cons uc ion. Suppose ha he equilib ium equi es he agen o choose Ewi h high
p obabili y in e e y pe iod o he block. Then he play o Ein any pe iod o he block
mus be incen i ized, meaning ha he p incipal’s epo ing decision mus depend on
whe he signal Go Bis ealized in ha pe iod. Howe e , i he p incipal is o ha e s ic
epo ing incen i es, hen he h pe iod signal mus be in o ma i e o he agen ’s be-
ha io (and, hence, he agen ’s epo ). Tha link is possible only i he agen chooses
bo h Eand Swi h posi i e p obabili y in he h pe iod o he block. In o he wo ds,
bo h Eand Smus be played wi h posi i e p obabili y in each pe iod o he block.
We p esen wo app oaches o cons uc ing ha pe iod-by-pe iod unce ain y in
pu i iable T-pe iod equilib ia. Ou i s app oach builds on Fuchs (2007). The agen
plays Sin a mos one pe iod o he block, bu i could be any pe iod. In he second
app oach, he agen plays Sei he in all Tpe iods o in none. The la e has he ad an-
age ha i also deli e s mo e ealis ic con ac s, since he bonus need no be ex emely
high and he base wage need no be nega i e. Bo h o hose ex eme app oaches may
be un ealis ic in applica ions, bu i is s aigh o wa d o use hem o cons uc in e me-
dia e cases (whe e, o example, he agen may pu in low e o o a week when he is
unexpec edly busy a home).12
5.1 S ic incen i es o u h- elling in block-s a egy equilib ia
In his sec ion we p o ide a gene al cons uc ion o modi ying block-s a egy equilib-
ia o pu i iable equilib ia, wi h a minimal loss in e ms o hei e iciency. Le (σ∗,ρ∗)
be a T-pe iod block equilib ium o he baseline en i onmen ∞wi h no exchange o
messages, wi h he p ope y ha , in equilib ium, he agen exe s e o in e e y pe iod.
In pa icula , (σ∗,ρ∗)speci ies a “ es ” ha maps he p incipal’s obse ed signals o a
bonus and a e mina ion p obabili y. We hen augmen (σ∗,ρ∗) oughly as ollows: he
agen andomly chooses a pe iod Sin which o shi k and he announces Sa he end o
he block. The p incipal simul aneously announces he signal om each o he Tpe i-
ods. Then he announced pe iod Sis “ h own ou ” and he es om (σ∗,ρ∗)is un on
he emaining pe iods. Tha ule, wi h some adjus men s, p ese es he e o incen i es
om (σ∗,ρ∗).13 Finally, o p o ide incen i es o u h- elling, we use small adjus men s
o bonus paymen s.
12In addi ion, i is easie o p o e ha he equilib ia cons uc ed by he second app oach a e pu i iable
when we en ich he model by au oco ela ed shocks o he agen ’s cos o e o .
13These adjus men s a e necessa y and essen ial, because o he wise he p incipal who ob ained signal
Bin all pe iods o a block could no be gi en s ic incen i es o epo ing he signals u h ully, and he
augmen ed equilib ium would no be pu i iable.
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1043
Fo mally, de ine a T-pe iod e iew s a egy p o ile (σ∗,ρ∗)=(τ,χ)o he baseline
game as ollows. As a unc ion o he numbe o Gsignals nin he p incipal’s p i a e
T-pe iod his o y o signals, (σ∗,ρ∗)speci ies whe he o no he agen passes he es ,
χ(n)∈{0, 1}. I he agen passes, hen he p incipal pays him a bonus τ. I he agen ails
he es , hen he ela ionship is e mina ed by ei he playe wi h a p obabili y αso ha
he low alue o he payo loss o he p incipal equals τ.14 The agen ’s s a egy is o play
Ein e e y pe iod. An agen -s ic T-pe iod e iew s a egy equilib ium (σ∗,ρ∗)=(τ,χ)
is one in which he agen has a s ic incen i e o play Ein each pe iod.
Nex , de ine an augmen a ion o (σ∗,ρ∗)=(τ,χ)as ollows.
5.1.1 Ac ions and epo s The ho izon is di ided in o blocks o leng h Teach. The
agen chooses exac ly one pe iod in which she plays Sby a ai lo e y o e he Tpe iods
plus a ic i ious (T+1)s pe iod. In all o he pe iods, he agen plays E. I he ic i ious
pe iod T+1 is selec ed by he lo e y, hen he agen p o ides e o in all Tpe iods.
A he end o each block, he agen epo s o he p incipal he pe iod S ha she chose
o playing S, and he p incipal e eals he signals ha she ob ained in he Tpe iods
o he block. The “signal” o he p incipal in he ic i ious pe iod T+1 is gene a ed by
public andomiza ion. Mo e p ecisely, i he agen plays Sin one o he Tpe iods, hen
he public andomiza ion gene a es in he ic i ious pe iod T+1 a signal in he way he
signal would be gene a ed by he agen ’s ac ion Ein ha pe iod. I S=T+1, hen he
public andomiza ion gene a es a signal in he way he signal would be gene a ed when
he agen played ac ion S. The epo s a e simul aneous and he ealiza ion o public
andomiza ion is obse ed a e he epo s. I is unde s ood ha he agen mus epo
ac ion Ein all bu one o he T+1 pe iods; in he emaining pe iod she mus epo
ac ion S.
5.1.2 Agen ’s e iew Nex , he agen is subjec o he es χ.Thepe iod Sdoes no
coun o he e iew. All o he pe iods coun , including he ic i ious one i his is no
he pe iod in which Swas p esc ibed. The ule χapplied o he Tpe iods ha coun
de e mines whe he o no he agen passes he es . I he agen passes he es , hen
he is paid he bonus τ. I he agen ails he es , hen he p incipal pays no bonus,
bu he ela ionship is e mina ed by ei he playe wi h a p obabili y such ha he low
cos o he p incipal equals τ. No ma e whe he he agen passes o ails, he is paid
an adjus men . The adjus men compensa es he agen o playing Sin a la e a he
han an ea lie pe iod o o no playing Sa all ( ha is, o choosing he ic i ious pe iod
T+1), so ha he agen is indi e en ega ding he pe iod in which he plays S.Mo e
p ecisely, he adjus men is (1−δ S−1)c/δT−1i S≤Tand c/δT−1i S=T+1.
Finally, i will be con enien o assume ha he alue o he agen is w.Thisiswi h-
ou loss o gene ali y, because he base wage and he bonus can be adjus ed o gua an ee
his condi ion.
14Tha is, (1−δ)τ=δαV P,whe eVPdeno es he alue o he p incipal o he T-pe iod game played
epea edly o e he in ini e ho izon.
1044 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
5.1.3 Tes ing p incipal’s epo s Finally, he p incipal’s epo is es ed. Mo e speci i-
cally, an εis added o he agen ’s bonus, no ma e wha he ou come o he es . Tha
is, he bonus becomes τ+εi he agen passes he es and εi he agen ails. This εis
nex sub ac ed i he p incipal’s epo passes he ollowing es .
(i) A pe iod o he han Sis d awn in a ai lo e y o e he emaining Tpe iods
(including he ic i ious pe iod T+1i S= T+1).
(ii) A i y– i y lo e y chooses ei he (a) So (b) he pe iod chosen in (1) (in which i
is assumed ha he agen epo ed E).
(iii) I he signal epo ed by he p incipal and he agen ’s ac ion in he selec ed pe iod
coincide, ha is, hey a e {G,E}o {B,S}, hen he p incipal passes. O he wise,
ha is, i hey a e {G,S}o {B,E}, hen he p incipal ails.
The eade may wonde why he p incipal is es ed in his speci ic manne . This
speci ic es gua an ees ha a he beginning o a block he p incipal assigns a i y– i y
chance o bo h ac ions in he pe iod ha will be aken o he es . The e o e, he signals
ob ained by he p incipal a e mo e likely o coincide wi h he ac ion aken o he es
han he opposi e signals.
The eade may also wonde wha he ole o he ic i ious pe iod is. I we did no
include he ic i ious pe iod and he agen was p esc ibed o choose one o he Tblock
pe iods o shi king, hen he p incipal wi h all signals B(o wi h all signals G)would
s ill assign a i y– i y chance o bo h ac ions in he pe iod aken o he es . The e o e,
he p incipal would no ha e s ic incen i es o epo he signals u h ully.
Wi h hose de ini ions, we can p esen he esul o his app oach o cons uc ing a
obus and e icien equilib ium.
Lemma 1. Fix δand T, and suppose ha (σ∗,ρ∗)=(τ,χ)is an agen -s ic T-pe iod
e iew s a egy equilib ium o he game wi hou cheap alk. Then, o high enough δ,
he e is an augmen a ion o (σ∗,ρ∗) ha is a pu i iable equilib ium o he game wi h
cheap alk.
Thep oo isgi enin heAppendix.15
P oposi ion 4. Le >0. Then he e exis s Tsuch ha o δ<1bu close enough o
1, he e is a pu i iable equilib ium in T-pe iod block s a egies wi h cheap alk whe e he
agen exe s e o in a leas T−1pe iods o he block, and he p incipal’s expec ed payo
is a leas ¯
y−¯
w−c−.
P oo . The p oposi ion ollows by applying Lemma 1 o he s a egies in Fuchs (2007)
plus he obse a ion ha when playe s become pa ien and Tis high enough, a single
pe iod o playing Sin e e y block c ea es only a negligible e iciency loss, as well as
ha ing a negligible e ec on he p incipal’s payo .
15We will show ha he equilib ia a e pu i iable e en when he agen ’s cos o e o is pe u bed. How-
e e , we will assume ha he cos shocks a e i.i.d. We only conjec u e, bu ha e no p o ed o mally, ha
he equilib ia would be pu i ied e en when he cos shocks we e au oco ela ed.
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1045
5.2 Shi king in 0 o Tpe iods
In ou second app oach, he agen andomizes be ween exe ing e o in all Tpe iods
and shi king in all Tpe iods. Tha is, he agen mus be indi e en be ween he wo
sequences—“always E“and “always S”—a he beginning o he block, and also de e ed
om de ia ing o o he ac ion sequences. As δapp oaches 1, he e is a sequence o such
equilib ia whose payo s asymp o e o e icien payo s. In addi ion, hese equilib ia will
ha e some, p esumably a ac i e, ealis ic ea u es: (a) he pe -pe iod bonus paid by
he p incipal will be close o he pe -pe iod cos o he agen ’s e o ; (b) ge ing a bonus
will be highly unlikely when he agen ne e wo ks, while i will be almos ce ain when
he agen always wo ks; (c) he base wage will be close o he ou side op ion o he agen .
Mo e p ecisely, we ha e he ollowing esul .
P oposi ion 5. Le >0. (i) Then he e exis s Tsuch ha o high enough δ he e is
a pu i iable equilib ium o ∞wi h cheap alk in T-pe iod block s a egies, whe e he
agen exe s e o in all Tpe iods o in none, and he p incipal’s expec ed payo is a
leas ¯
y−¯
w−c−.
(ii) In addi ion, he equilib ium has he ollowing ea u es: (a) c−<τ/T<c+;(b)
π(T)>1−and π(0)<,whe eπ(k)s ands o he p obabili y o he agen ob aining
he bonus τwhen she has wo ked in exac ly kpe iods du ing a block; (c) ¯
w+>w>
¯
w−.
The es o his sec ion is o ganized as ollows: We will i s desc ibe ou cons uc ion
o equilib ia and s a e some o hei ea u es. Since he cons uc ion is simila o he
cons uc ion in Ma sushima (2004), and especially ha in Ely, Hö ne , and Olszewski
(2005), we pos pone o mally p o ing (i) o he Appendix. The ea u es we s a e will
allow o p o ing (ii). We will nex compa e ou equilib ia o hose om Fuchs (2007)
and show ha in his equilib ia, (a) hepe -pe iodbonusgoes o∞as he leng h o he
block, T, goes o in ini y, (b) he agen is e y likely o be paid he bonus e en when he
shi ks in e e y pe iod, and (c) he base wage goes o −∞ as Tgoes o ∞.
To desc ibe ou cons uc ion, we will need wo lemmas, which display some p op-
e ies o binomial dis ibu ions. They a e known and easy o p o e, so hei p oo s will
be omi ed. (A e sion o hese lemmas was i s no ed in Ma sushima (2004).) Deno e
by F(n,T,k) he p obabili y o he e en ha he p incipal ecei es mo e han nsignals
Gcon ingen on he agen aking ac ion Ein exac ly kpe iods o a block o Tpe iods.
Deno e also by (n,T,k) he p obabili y o he e en ha he p incipal ecei es exac ly
nsignals G.
Lemma 2. Fo any ε>0, he e exis s a Tsuch ha o e e y T≥T, he eexis sann=n(T)
such ha F(n,T,T)>1−ε,F(n,T,0
)<ε,andT (n,T−1, T−1)>1/(p−q).
The lemma says ha he e exis s a cu o numbe o good signals wi h h ee p op-
e ies. An agen who always wo ks will each he cu o wi h p obabili y close o 1, an
agen who always shi ks will each i wi h p obabili y close o 0, and he p obabili y o
exac ly hi ing he cu o in pe iod T−1a e T−1 pe iods o wo king is la ge ela i e
o 1/T.
1046 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
Lemma 3. Fo e e y T>1and n=0, 1, ,T−1, (n,T−1, k)as a unc ion o k=
0, ,T−1is single-peaked, ha is, i (n,T−1, k)≥ (n,T−1, k+1), hen (n,T−
1, k+1)> (n,T−1, k+2) o k=0, ,T−3.
The key idea o ou cons uc ion is o se he cu o numbe o good ou comes o
paying a bonus a an in e media e alue, be ween ac ion qand ac ion po he Tpe-
iods. In addi ion, he cu o is se qui e close o pso as o gua an ee ha he agen who
is indi e en be ween always Eand always Sp e e s always wo king o wo king only
in T−1 pe iods. The choice o he cu o is sub le, because we also wan i no o be
oo close o ac ion po he Tpe iods o assu e ha ge ing a bonus is almos ce ain
when he agen always wo ks. Howe e , such a choice is possible by Lemma 2.(Ge -
ing a bonus will be highly unlikely when he agen ne e wo ks, because he cu o is
bounded away om ac ion qo he Tpe iods.) Lemma 3in u n gua an ees ha he
ma ginal bene i o wo king in k+1 e suskpe iods is i s nega i e and hen posi-
i e, and oge he wi h he indi e ence be ween “all S”and“allE,” i gua an ees ha
he agen p e e s he wo cons an sequences o ac ions o any o he sequence wi hin a
block.
As in Sec ion 4, he equilib ium is pu i iable because he playe s ha e s ic incen-
i es, excep he agen ’s choice o e o , which is no pe u bed in ou model. Howe e ,
he agen ’s choice o e o would be pu i ied e en i his cos was pe u bed and au oco -
ela ed by p esc ibing him S(in all pe iods o a block) i , a he beginning o he block,
his cos exceeds some high cu o c+ξx∗,asinSec ion4. In he unpe u bed game, once
he agen makes a choice be ween wo king and shi king a he beginning o a block, he
has s ic incen i es o con inue wi h he same ac ion o he es o he block. Conse-
quen ly, i ξ, he scaling o he shocks, is close enough o 0, he agen con inues o ha e
s ic incen i es o con inue wi h he ac ion chosen a he beginning o he block also in
he pe u bed game.
We will now elabo a e on ea u es (a)–(c). No e i s ha π(k) om P oposi ion 5is
equal o F(n,T,k). Thus, (b) ollows di ec ly om Lemma 2. Since he agen is indi -
e en be ween always Eand always S,i mus be ha (1+···+δT−1)c=δT−1τ[π(T)−
π(0)].Thus,τ/T is close o c o la ge enough alues o δby (b).16 So (a) ollows om
(b). (Ac ually, hese wo ea u es a e equi alen , ha is, (b) also ollows om (a).) Finally,
since he agen is indi e en be ween he con ac and he ou side op ion, i.e., he base
wage wmus be close o w, (c) ollows om (a).
Quali a i ely, ou equilib ia di e om he T-block s a egies in which he ela ion-
ship is no e mina ed by ei he playe as long as he ou come was Gin a leas one
pe iod o a block (as in Fuchs, 2007). These equilib ia ea u e (a)–(c). Indeed, o a -
ain e iciency (in he limi as δ ends o 1), he blocks mus become a bi a ily la ge
(T→∞); o he wise, playe s would b eak a p o i able ela ionship wi h a p obabili y
bounded away om 0 in e e y Tpe iod, whe e Twould be bounded away om ∞.
Wi h T ending o ∞, he agen misses he cu o (and loses a bonus) wi h p obabili y
close o 0, no ma e wha ac ions she akes in he block o Tpe iods; mo e speci i-
cally, (1−p)Tand (1−q)Tdeno e he p obabili y o losing he bonus when wo king
16No e ha Tin P oposi ion 5is chosen o all la ge enough alues o δ.

Theo e ical Economics 19 (2024) Robus ela ional con ac s 1047
in all pe iods and when shi king in all pe iods, espec i ely. So, o gi e he agen incen-
i es o wo k in all pe iods, o e shi king in all pe iods, he bonus τmus weakly exceed
(1+···+δT−1)c/δT−1[(1−q)T−(1−p)T]. Such a bonus becomes in ini ely la ge, e en
in pe -pe iod e ms, as T→∞(because (1−q)T−(1−p)T→0). So, i he p incipal
wan s o ex ac all (o mos ) su plus om he ela ionship, he base wage mus be se
nega i e; ac ually i mus end o −∞ when T→∞.
The equilib ium cons uc ion o ou i s app oach, used in P oposi ion 4,buildson
he s a egies in Fuchs (2007). I u ilizes sel - epo s by he agen , bu i sha es he nega-
i e base wage and la ge bonus o he ea lie wo k. We no e, howe e , ha i is possible
o cons uc a T-pe iod block-s a egy equilib ium o he baseline game in which he
agen shi ks in a mos one pe iod, he base wage is posi i e, and he size o he bonus
τis close o Tc. The co esponding e iew speci ies, like he one om his sec ion, ha
he agen passes he es as long as he p incipal obse es a leas ngood signals, whe e
n>1 is non i ial: always wo king ensu es ha he agen passes he es wi h high p ob-
abili y, bu always shi king means ha he agen is e y likely o ail he es . We can hen
build on hese s a egies (ins ead o he s a egies in Fuchs (2007)) o cons uc equilib ia
wi h p ope ies simila o hose o he equilib ia cons uc ed in his sec ion, in which he
agen shi ks in a mos one pe iod ins ead o always shi king o always exe ing e o .
Conside inally a a ian o ou model, whe e he p incipal can sign a binding con-
ac whe eby she commi s o paying ei he he agen o a hi d pa y such as a cha i y,
bu whe e pe o mance e alua ion emains subjec i e as in MacLeod (2003). P oposi-
ions 4and 5would hen apply, bu wi hou any equi emen on δ. Since he p incipal is
commi ed con ac ually o make paymen s ei he o he agen o o cha i y, she canno
enege on making a leas one o he paymen s. Ou cons uc ion p o ides incen i es o
he p incipal o u h ully choose be ween hese op ions. Fu he mo e, he ela ionship
could ha e a ini e ho izon T, and he p incipal could ob ain a pe -pe iod payo wi hin
o y−w−cas long as Twas su icien ly la ge.
6. Concluding commen s
Requi ing obus ness o small payo shocks, such as a e p esen in many economic ap-
plica ions, has majo consequences o ela ional con ac ing be ween a p incipal and
an agen . No equilib ium whe e he agen always exe s e o is obus o payo shocks.
We ha e shown, howe e , ha ex ending he basic in e ac ion by allowing cheap alk can
make e o sus ainable. Cheap alk, when coupled wi h he agen ’s andom e o , al-
lows p incipal and agen o c oss-check each o he , p o iding s ic incen i es o u h-
elling. Agen ’s andomiza ion would be unnecessa y i he agen could also obse e a
signal o his pe o mance ha is co ela ed wi h he p incipal’s subjec i e e alua ion. In
his case, cheap alk would su ice as a way o c oss-check hei obse a ions. Also, i
he agen ’s e o had a pe sis en componen , hen he p incipal would be less keen o
con inue he ela ionships a e obse ing low ou pu han a e high ou pu , so ha one
migh be able o dispense wi h cheap alk al oge he . Ou con ibu ion is o show ha
one can ob ain obus equilib ia e en wi hou modi ica ions o he undamen als o he
model.
1048 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
We cons uc equilib ia wi h ealis ic p ope ies: he base wage is posi i e, he size
o he bonus paid o he agen o good pe o mance so as o incen i ize e o is p o-
po iona e o he cos o e o , and he h eshold o ea ning he bonus is se so ha he
agen can likely each i by exe ing e o bu no o he wise. The op imal con ac s p e-
iously s udied in he li e a u e, in con as , ea u e bonuses ha a e much la ge han
he ac ual e o cos and h esholds ha a e nea ly ce ain o be eached e en wi hou
e o . Ou cons uc ion also highligh s he ole o sel - e iew o wo ke s’ pe o mance.
The cons uc ion elies on simul aneous messages, which sugges s a mo i a ion o as-
signing he ask o e alua ing an employee’s pe o mance o a manage a he han o
he i m owne who pays he bonuses. Tha way, he employee and owne do no know
he o he ’s epo when making hei own.
Appendix
A.1 P oo s o nega i e esul s
A.1.1 P oo o P oposi ion 1Conside he pe u bed game. Then, o e e y public his-
o y ω −1on he equilib ium pa h, he ac ion sequence (ˆ
a1,,ˆ
a )is co ec ly an ic-
ipa ed by he p incipal. Conside he p incipal making he decision in pe iod .The
p incipal’s obse a ion o ou pu y does no a ec he con inua ion alue; i only a -
ec ed he low payo ha she ob ained a da e , bu no he belie s abou he agen ’s
con inua ion s a egy. Le pe iod +kbe he i s pe iod in which he ealiza ion o y
a ec s ei he he bonus paymen o he p incipal’s e mina ion decision.
Suppose ha he p incipal makes wo dis inc bonus paymen s τ<τ
a e p i a e
his o ies ha only di e wi h ega d o y . Incen i e compa ibili y o he p incipal im-
plies ha he agen ’s con inua ion s a egy a e τmus di e om ha a e τ,wi hτ
inducing ei he lowe e o o a g ea e e mina ion p obabili y. Thus, o he alues
o z +kabo e some cu o , he p incipal will p e e he agen ’s con inua ion s a egy a -
e τ, and o he alues o z +kbelow ha cu o , he p incipal will p e e he agen ’s
con inua ion s a egy a e τ. Thus, he se o alues o z +ksuch ha he p incipal con-
di ions he bonus paymen on y is negligible. I he ealiza ion o y does no a ec he
bonus paymen in pe iod +k, hen o almos all alues o z +k, he p incipal mus
s ic ly p e e ei he he e mina ion decision p esc ibed a e y =Go he e mina ion
decision p esc ibed a e y =B.
A.1.2 P oo o P oposi ion 2Conside he pe u bed game and a pe iod a he end
o he T-pe iod block. Suppose ha he p incipal paid a bonus le el τ(which could
possibly be 0) and ha playe s obse ed a ealiza ion o he public andomiza ion de-
ice θ. Condi ional on he public e en (τ,θ), he agen ’s qui ing s a egy in pe iod
is a unc ion ρ2
(τ,θ):{E,S}T→{C,Q}, i.e., i speci ies con inue o qui depending on
his e o choices in he block. The p incipal’s i ing s a egy in pe iod is a unc ion
σ2
(τ,θ):{G,B}T×Z→{R,F}, i.e., i speci ies i ing o e aining he agen depending
on he ou pu s in he block and on he alue o he p incipal’s payo shock. Obse e
ha ei he playe ’s choice on whe he o con inue o e mina e he ela ionship only
ma e s when he o he playe chooses o con inue, since he ela ionship e mina es
Theo e ical Economics 19 (2024) Robus ela ional con ac s 1049
i one playe chooses o e mina e i . Suppose ha he equilib ium speci ies ha a e
(τ,θ), he ela ionship con inues wi h posi i e p obabili y, so ha he e exis s some e -
o sequence and a nonnegligible se o payo shocks o which he agen chooses C.
Since ou pu signals ha e ull suppo , o any ou pu sequence in {G,B}T, he p incipal
assigns posi i e p obabili y o he agen con inuing. Thus, o almos any ealiza ion o
he p incipal’s payo shock z , i i is op imal o he p incipal o con inue ( esp. e -
mina e) he ela ionship a some sequence in {G,B}T, i is op imal o con inue ( esp.
e mina e) a e e y o he sequence in {G,B}T. In o he wo ds, a any (τ,θ)whe e he
ela ionship con inues wi h posi i e p obabili y, he p incipal’s i ing decision does no
depend on he ou pu sequence in he block.
A simila a gumen es ablishes ha a such (τ,θ)(whe e he ela ionship con in-
ues wi h posi i e p obabili y), he agen ’s qui ing decision does no depend on he se-
quence o e o choices wi hin he block. This es ablishes ha o any (τ,θ), hep ob-
abili y ha he ela ionship con inues does no depend ei he on e o choices wi hin
he block o on he ou pu sequence.
Now conside he p incipal’s choice be ween wo dis inc bonus alues τand τ.I
τ>τ
, hen he p obabili y o e mina ing he ela ionship mus be g ea e a e τ.The
p eceding a gumen has es ablished ha he p incipal’s con inua ion alue om choos-
ing τ(o τ) does no depend on he obse ed ou pu sequence. Thus, i i is s ic ly op-
imal o choose τa some ou pu sequence and some ealiza ion z o he payo shock,
hen she will choose τa e e y o he ou pu sequence o ha same alue o z .Tha
is, he bonus paymen does no depend on he ou pu sequence and, consequen ly, i is
subop imal o he agen o exe any e o .
This a gumen es ablishes ha o any z , he se o block-s a egy equilib ia whe e
he agen chooses e o wi h posi i e p obabili y is emp y. Thus, any block-s a egy
equilib ium o he unpe u bed game ha ea u es posi i e e o is no pu i iable.
A.2 P oo o Lemma 1
We show i s ha he speci ied s a egies a e an equilib ium, and hen ha he equilib-
ium is pu i iable. The e a e ou se s o equilib ium condi ions ha mus be e i ied.
The agen has s ic incen i es o shi k in one and only one pe iod The agen is happy o
shi k in one pe iod, because he pe iod chosen by him does no coun in his e iew and
he sa es on he cos o p o iding e o (in pe iods ≤T), o he is paid an adjus men (in
pe iods >1), o bo h (in pe iods 1 < ≤T). Shi king in mo e han one pe iod will a ec
he e iew. By de ini ion o a T-pe iod e iew s a egy, he bonus τand es χa e such
ha he agen p e e s o exe e o in e e y pe iod excep he one chosen o shi king.
The agen is indi e en ac oss all T+1pe iods ega ding hechoiceo pe iod o shi king
This ollows because all pe iods (excep he one selec ed o shi king) equally ma e o
he agen ’s e iew, and he adjus men s (1−δ −1)c/δT−1 o ∈{1, ,T}and c/δT−1
o =T+1 make he sa ing on cos s equal o c(in e ms o pe iod 1) ac oss all T+1
pe iods. In addi ion, he es ing p ocedu e has he p ope y ha he chance o losing
εis independen o he choice o he pe iod o shi king, gi en he hones epo o he
p incipal.
1050 Bhaska , Olszewski, and Wiseman Theo e ical Economics 19 (2024)
The agen has s ic incen i es o hones ly epo he pe iod in which he has shi ked I
he agen mis epo s he pe iod in which he has shi ked, he eplaces in he e iew a
pe iod in which he played Ewi h a pe iod in which he played S.Soamis epo has he
same e ec on he e iew as aking ac ion Sins ead o aking ac ion E.Incon as o
aking Sins ead o aking E, he agen does no sa e on he cos o e o by a mis epo ,
bu he educes he p obabili y o losing ε. Howe e , his las bene i is assumed o be
e y close o 0.
The p incipal has s ic incen i es o hones ly epo he signals Tes ing o he p incipal’s
epo s is designed such ha om he ex an e pe spec i e each pe iod (including he
ic i ious one) is chosen o compa ing he agen ’s ac ion and he p incipal’s signal wi h
p obabili y 1/(T+1); in addi ion, he p obabili y ha ac ion E(o ha ac ion S)was
aken in he chosen pe iod is 1/2. The p incipal’s objec i e is o epo Go B ha is
consis en wi h he agen ’s ac ion (i.e., Gi he ac ion was E,andBi he ac ion was S).
By ecei ing he signal, she upda es he belie in a o o Ewhen he signal is Gand in
a o o Si he signal is B. Thus, any mis epo educes he p obabili y o a aining he
p incipal’s objec i e.
O cou se, he p incipal co ec ly an icipa es he agen ’s s a egy o playing Sin ex-
ac ly one pe iod. So he compu a ion o p obabili ies is con ingen on he e en ha
he agen akes ac ion Sin exac ly one pe iod (possibly he ic i ious one, ha is, in
none) and con ingen on all he Tsignals. This a ec s he p obabili ies assigned by
he p incipal o ac ions Eand Sin a gi en pe iod, gi en he sequence o signals ha
she ob ained, bu i does no change which o he wo p obabili ies is highe . Fo ex-
ample, suppose ha he p incipal ob ained signals Gin all Tpe iods, and conside s a
pe iod ∈{1, ,T}. I he agen had andomized 50–50 be ween ac ions Eand Sin pe-
iod , hen he p incipal would ha e a chance o p o he epo being consis en wi h
he agen ’s ac ion i she epo ed G, and she would ha e only a chance o q<pi she
epo ed B.
This is o cou se no he compa ison ha he p incipal is conduc ing. Ins ead, she
compu es he p obabili y o (i) being he pe iod in which he agen shi ked, and (ii)
being a pe iod in which he agen wo ked and being selec ed by he es ing p ocedu e,
and compa es hese wo p obabili ies. Acco ding o he es ing p ocedu e, she sa es ε
wi h p obabili y 1/2 con ingen on each o hese wo e en s i he epo in he e en is
consis en wi h he agen ’s ac ion. The p obabili y o he o me e en (i) is q/(Tq+p),
becausei oneo heac ualTpe iods is selec ed o shi king, he p obabili y o all sig-
nals being Gis qpT−1, i he ic i ious pe iod is selec ed, he p obabili y o all signals
being Gis pT, and each o he T+1 pe iods is selec ed wi h p obabili y 1/(T+1).The
p obabili y o he la e e en (ii) in u n is [(T−1)q+p]/T(Tq+p), because he p ob-
abili y o selec ing one o he pe iods s∈{1, ,T},s= , o shi king is q/(Tq+p),and
hen selec ing by he es ing p ocedu e is 1/T; and he p obabili y o selec ing he ic-
i ious pe iod o shi king is p/(Tq+p), and hen selec ing by he es ing p ocedu e is
1/T. Since [(T−1)q+p]/T(Tq+p)>q/
(Tq+p), he p incipal sa es εwi h a highe
p obabili y when she epo s signal Gin pe iod .
The p oo o o he sequences o he p incipal’s signals is analogous.