Reinwald, Pa ick; Lei ne , S ephan; Wall, F iede ike
A icle
Pe o mance-based pay and limi ed in o ma ion access:
An agen -based model o he hidden ac ion p oblem
Jou nal o Economics and S a is ics
P o ided in Coope a ion wi h:
De G uy e B ill
Sugges ed Ci a ion: Reinwald, Pa ick; Lei ne , S ephan; Wall, F iede ike (2024) : Pe o mance-based
pay and limi ed in o ma ion access: An agen -based model o he hidden ac ion p oblem, Jou nal
o Economics and S a is ics, ISSN 2366-049X, De G uy e Oldenbou g, Be lin, Vol. 244, Iss. 4, pp.
381-423,
h ps://doi.o g/10.1515/jbns -2023-0101
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/333289
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Pa ick Reinwald, S ephan Lei ne * and F iede ike Wall
Pe o mance-Based Pay and Limi ed
In o ma ion Access. An Agen -Based Model
o he Hidden Ac ion P oblem
h ps://doi.o g/10.1515/jbns -2023-0101
Recei ed No embe 30, 2023; accep ed July 26, 2024
Abs ac : Models in ol ing human decision-make s o en include idealized
assump ions, such as a ionali y, pe ec o esigh , and access o ele an in o ma-
ion. These assump ions usually assu e he models’in e nal alidi y bu , a he same
ime, migh limi he models’powe o explain empi ical phenomena. This pape
add esses he well-known model o he hidden ac ion p oblem, which p oposes an
op imal pe o mance-based sha ing ule o si ua ions in which a p incipal assigns
a ask o an agen and he ask ou come is sha ed be ween he wo pa ies. The
p incipal canno obse e he ac ion aken by he agen o ca y ou his ask. We
in oduce an agen -based e sion o his p oblem in which we elax some o he
idealized assump ions. In he p oposed model, he p incipal and he agen only ha e
limi ed in o ma ion access and a e endowed wi h he abili y o gain, s o e and
e ie e in o ma ion om hei (fini e) memo y. We ollow an e olu iona y
app oach and analyze how he p incipal’s and he agen ’s decisions affec hei
espec i e u ili ies, he sha ing ule, and ask pe o mance o e ime. The esul s
sugges ha he op imal (o a close- o-op imal) sha ing ule does no necessa ily
eme ge in all cases. The esul s indica e ha he p incipal’s u ili y is ela i ely obus
o a ia ions in memo y. On he con a y, he agen ’s u ili y is significan ly affec ed
by limi a ions in he p incipal’s memo y, whe eas he agen ’s memo y appea s o
only ha e a mino effec .
Keywo ds: obus ness; eplica ion; limi ed a ionali y; agen -based modeling;
simula ion
JEL Classifica ion: M20; C63; D00
*Co esponding au ho : S ephan Lei ne , Uni e si y o Klagen u , Uni e si ä ss aße 65–67, 9020
Klagen u , Aus ia, E-mail: [email p o ec ed]. h ps://o cid.o g/0000-0001-6790-4651
Pa ick Reinwald and F iede ike Wall, Uni e si y o Klagen u , Uni e si ä ss aße 65–67, 9020
Klagen u , Aus ia. h ps://o cid.o g/0000-0002-2907-7939 (P. Reinwald). h ps://o cid.o g/0000-0001-
8001-8558 (F. Wall)
Jou nal o Economics and S a is ics 2024; 244(4): 381–423
Open Access. © 2024 he au ho (s), published by De G uy e . This wo k is licensed unde he
C ea i e Commons A ibu ion 4.0 In e na ional License.
1 In oduc ion
In he his o y o esea ch on beha io al con ol, he concep o a ional expec a ions
eme ged as he dominan pa adigm (Mu h 1961), which wen hand in hand wi h he
ex ensi e de elopmen o sophis ica ed ma hema ical me hods and models wi h
some imes idealized assump ions abou indi iduals. Up o oday, such app oaches
ha e played an essen ial ole in many fields, such as managemen and economics.
Mos o hese models assume agen s who a e a ional in hei beha io , employ
sophis ica ed op imiza ion me hods, usually possess all (o a leas mos ) pieces o
in o ma ion o come up wi h op imal solu ions immedia ely, and mos ly make no
e o s in doing so (Fa me and Foley 2009; Thale 2000). Howe e , i has al eady
been ecognized ha echnically co ec and alid models o en lack he powe o
explain empi ical phenomena (F anco e al. 2020) and calls o include beha io al
insigh s om o he disciplines –such as cogni i e psychology –ha e eme ged
(Hämäläinen, Luoma, and Saa inen 2013; Lei ne and Beh ens 2015a; Roys on 2013;
Wall, Chen, and Lei ne 2024; Wall and Lei ne 2021). F anco and Hämäläinen (2016)
poin ou ha inc eased a en ion o beha io al aspec s becomes p ominen when-
e e scien ificfields each ma u i y and a gue ha , amongs o he s, his is he case in
economics, accoun ing, and s a egic managemen .
We place ou esea ch in he s eam o esea ch ha analyzes he obus ness o
economic models o elaxa ions o he included and o en es ic i e assump ions. In
pa icula , in ou esea ch we ocus on he assump ions o in o ma ion a ailabili y
(and limi a ions he eo ) in he well-known model o he hidden ac ion p oblem
in oduced by Holms öm (1979). This model is based on a p incipal who assigns a
ask o an agen . The agen ac s on behal o he p incipal by makingan effo o ca y
ou he ask assigned o him, while he p incipal’s ole is o p o ide capi al and
incen i es. The modeled si ua ion is u he cha ac e ized by in o ma ion asym-
me y in he o m o hidden ac ions. The p incipal can only obse e he ask ou come
bu no he ac ion aken by he agen , which is why he p incipal can only employ a
pe o mance-based compensa ion mechanism. The model p o ides his mechanism
by c ea ing a ule o op imally sha e he ask ou come be ween he p incipal and he
agen (Caillaud and He malin 2000; Eisenha d 1988; Lambe 2001). The assump ions
conside ed in his model include wha Ax ell (2007) e e s o as he economic swee
spo : Ra ionali y, homogenei y, and equilib ium solu ions. We a e pa icula ly
in e es ed in he condi ions o a ionali y ha also include issues ela ed o in o -
ma ion a ailabili y.
Economic models o en implici ly ollow he idea ha decision-make s possess
he necessa y capabili ies o access all ele an pieces o in o ma ion. Keeping in
mind ha in o ma ion can ake diffe en o ms, such as complex ex s, mul imedia
382 P. Reinwald e al.
ma e ials, o in o ma ion ha is ga he ed by obse ing he en i onmen , hese
equi ed capabili ies a e uly mani old and ich (McC eadie and Rice 1999). This
ende s he assump ions implici ly included in economic models non i ial. In a
mic oeconomic con ex , Lei ne , Rausch, and Beh ens (2017) and Lei ne , B auneis,
and Rausch (2015) analyze he effec s o limi ed in o ma ion in capi al budge ing by
modeling o e confiden decision-make s, and p incipals and agen s wi h limi ed
o esigh , espec i ely. In he con ex o he hidden ac ion p oblem, Lei ne and Wall
(2021, 2022) limi he p incipal’s and he agen ’s espec i e capabili ies o sea ch o
he op imal sha ing ule. In he ein o his ela ed esea ch, we ans e he hidden
ac ion model in o an agen -based model (Gue e o and Ax ell 2011; Lei ne and
Beh ens 2015a). By doing so, we p opose an agen -based model o he hidden ac ion
p oblem wi h p incipals and agen s who suffe om limi a ions in in o ma ion
access. We ocus on he ollowing ques ions: How do limi a ions in he p incipal’s and
he agen ’s in o ma ion a ailabili y in he con ex o he hidden ac ion p oblem affec
(i) he ule o sha e he ask ou come be ween he p incipal and he agen , (ii) he
agen ’seffo , and (iii) he p incipal’s and he agen ’s espec i e u ili ies.
To demons a e ha ou agen -based model effec i ely in eg a es he hidden
ac ion model, we fi s es ablish ha he solu ions de i ed om ou model align wi h
hose sugges ed by Holms öm (1979) unde condi ions o unlimi ed in o ma ion
o bo h he p incipal and he agen . When in o ma ion is es ic ed, ou findings
sugges ha he alloca ion o ask ou comes be ween he p incipal and he agen
depends solely on he p incipal’s in o ma ion. Specifically, g ea e in o ma ional
access o he p incipal leads o an inc eased sha e o ou comes o he agen .
Mo eo e , he p incipal’s in o ma ion p edominan ly influences he agen ’seffo ;
consis en wi h many economic models, he agen ’s beha io is d i en by he
incen i es p o ided. We u he obse e ha he agen ’s in o ma ion does no
impac he effo exe ed.
Rega ding u ili ies, ou analysis indica es ha he p incipal’s u ili y emains
la gely unaffec ed by his in o ma ion s a e, as he can adjus he incen i e mecha-
nisms o consis en ly maximize u ili y. In con as , he agen ’s u ili y is significan ly
influenced by en i onmen al ola ili y and he p incipal’s le el o in o ma ion. This
sugges s po en ial scena ios whe e a isk-neu al p incipal migh wi hhold a isk
p emium om a isk-a e se agen . Fu he mo e, since he p incipal consis en ly
achie es maximum u ili y, he e is minimal incen i e o enhance he in o ma ion
s a e. This leads o scena ios whe e he agen ’s dependence on he p incipal is
disp opo iona ely high.
The emainde o his pape is s uc u ed as ollows: We discuss he ele an
backg ound in Sec ion 2. Sec ion 3 discusses ou app oach o limi he p incipal’s and
he agen ’s in o ma ion a ailabili y, o malizes he p oposed agen -based model, and
An ABM o he Hidden Ac ion Model 383
in oduces he simula ion se up. The esul s a e p esen ed in Sec ion 4, and Sec ion 5
discusses he esul s. Finally, Sec ion 6 concludes he pape .
2 Rela ed Wo k
2.1 Holms öm’s Hidden Ac ion Model
The hidden ac ion model in oduced in Holms öm (1979) desc ibes he ela ionship
be ween one p incipal who assigns a ask o one agen . In pa icula , he p incipal
designs a con ac ha includes he ask o be ca ied ou and a ule o sha e he
ou come, and offe s his con ac o he agen . I he agen accep s he con ac , he
makes an effo (o en also e e ed o as ac ion) o ca y ou he ask assigned o him.
Toge he wi h an exogenous ac o , his ac ion gene a es he ask ou come. A he
same ime, ac ing leads o disu ili y o he agen . The sha ing ule included in
he con ac defines –be o e he ac ion is aken –how he ou come is o be sha ed
be ween he p incipal and he agen . The model p esen ed by Holms öm (1979) is a
non- epea ed model, indica ing ha i encapsula es a single execu ion o he speci-
fied sequence ou lined abo e. As a esul , i does no accoun o empo al dynamics,
including he dis ibu ion o effo o e se e al pe iods, con ac e-nego ia ions, and
he implica ions o p esen ac ions on u u e payoffs (as, o example, done in Ma
1991). The hidden ac ion model in oduced in Holms öm (1979) is capable o
desc ibing ela ions be ween one p incipal and one agen in a wide ange o
a eas, including he ela ionship be ween employe and employee, buye and sup-
plie , and clien and con ac o (Caillaud and He malin 2000; Eisenha d 1989;
Lei ne and Wall 2021; Reinwald, Lei ne , and Wall 2020).
The sequence o e en s wi hin he hidden ac ion model is included in Figu e 1,
whe eby he main ea u es can be summa ized as ollows: In τ= 0, he p incipal
designs he con ac and offe s i o he agen who makes his decision abou whe he
o no o accep i in τ=1.Inτ= 2, he agen selec s an effo le el a∈A⊆R om a se
o effo le els A ha a e easible o ca y ou he ask. The agen ’s selec ed effo
le el ais hidden o he p incipal, i.e. he p incipal canno obse e i because he cos s
o obse ing i a e p ohibi i ely high o his in o ma ion is no accessible o he .
Consequen ly, he e is an in o ma ion asymme y ega ding he effo le el in a o
o he agen . The exogenous ac o θ∼N(μ,θ) akes effec in τ= 3, and i is a andom
a iable ha desc ibes he s a e o na u e, which includes, o example, he beha io
o supplie s o cus ome s. The ou come xma e ializes in τ= 4; i is a unc ion o he
agen ’seffo le el aand an exogenous ac o θand ollows he p oduc ion unc ion
x=x(a,θ). Bo h he p incipal and he agen can obse e he ou come. The e is
in o ma ion asymme y ega ding he exogenous ac o : Unlike he p incipal, he
384 P. Reinwald e al.
agen can obse e he exogenous ac o o deduce i om he ou come. I he p in-
cipal knows o he exogenous ac o , she can deduce he agen ’seffo le el om
he ou come, which would ende he hidden ac ion p oblem i ial since all in o -
ma ion asymme y would be esol ed, and hep incipal couldpay he agen basedon
his effo . As a consequence o he in o ma ion asymme y, he p incipal can only
base he sha ing ule on he ou come.
The p incipal is isk-neu al. He u ili y comes om he gene a ed ou come x
minus he agen ’s compensa ion s(x) so ha
UP(x,s(x)) = x−s(x).(1)
The agen is isk-a e se and cha ac e ized by he u ili y unc ion
UA(s(x),a)=V(s(x)) − G(a),(2)
whe e V(s(x)) s ands o he u ili y om his sha e o he ou come and G(a) indica es
he disu ili y o he effo he makes o ca y ou he ask, wi h V′> 0 and x
a
≥0.
1
Gi en hese cha ac e is ics o he hidden ac ion model, he p incipal’s op imi-
za ion p oblem o gene a e he op imal sha ing ule is o malized as ollows:
max
s(⋅),aEU
Px,s(x)
()() (3a)
s. E U
A(s(x),a)
()
≥U
(3b)
a∈a g max
a′∈A
EU
A(s(x),a′)
() (3c)
Equa ions (3b) and (3c) a e cons ain s ha ha e o be conside ed by he p incipal. In
pa icula , Eq. (3b) ep esen s he pa icipa ion cons ain ha ensu es ha he agen
ge s a minimum u ili y U
and he e o e accep s he con ac . This minimum u ili y is
also called ese a ion u ili y and ep esen s he agen ’s bes ou side op ion. Equa-
ion (3c) is he incen i e compa ibili y cons ain and aligns he agen ’s objec i e ( o
Figu e 1: Sequence o e en s wi hin Holms öm’s hidden ac ion model.
1The subsc ip adeno es he pa ial de i a i e conce ning a.
An ABM o he Hidden Ac ion Model 385
maximize his u ili y) wi h he p incipal’s objec i e. This cons ain affec s he agen ’s
choice o effo le el a. The no a ion ‘a g max’ ep esen s he se o all a gumen s ha
maximizes he objec i e unc ion ha ollows. The solu ion o he p oblem in Eqs.
(3a)–(3c) is p o ided in Appendix A.
2.2 Rela ed Wo k on Ex ensions o Repea ed Models o he
Hidden Ac ion P oblem
The hidden ac ion model in oduced in Holms öm (1979) has been ex ended in
se e al di ec ions, whe eby we a e pa icula ly in e es ed in wo ks ha ex end he
model owa ds a mul i-pe iod o epea ed model, and his sec ion aims o gi e an
illus a i e o e iew o hese ex ensions. Ini ial s eps owa ds a model ha spans
mul iple pe iods a e in oduced by Rubins ein and Yaa i (1983) and Radne (1985),
ocusing on p oblems epea ed indefini ely wi h no discoun ing om bo h he agen
and he p incipal. They explo e how he agen ’s ewa ds adap based on hei
his o ical pe o mance, demons a ing ha a con ac can be o mula ed o elimi-
na e inefficiencies associa ed wi h mo al haza d. Con e sely, Lambe (1983)
p oposes a model wi h a fini e ho izon, inco po a ing dynamic p oduc ion unc ions
and discoun ing o he p incipal’s and agen ’s u ili ies. This model highligh s he
significance o long- e m con ac s o add essing mo al haza d, by offe ing he
agen a long- e m commi men and u ilizing hei pe o mance his o y o mi iga e
unce ain y in hei ac ions. Roge son (1985) also akes in o accoun discoun ing om
bo h he p incipal’s and he agen ’s pe spec i es, demons a ing ha he his o y o
pe o mance is c ucial in c a ing an op imal con ac . Building upon his ounda-
ion, Spea and S i as a a (1987) u he de elop he concep by also in oducing a
epea ed mo al haza d model ha inco po a es discoun ing. U ilizing his model,
hey explo e how con ac s can easibly inco po a e dependency on his o ical
ac ions and examine he e olu ion o hese con ac s o e ime.
B oadening he analysis, Holms öm and Milg om (1987) p opose a con inuous
ime model and in es iga e ins ances in which agen s a e compensa ed a heend o a
fini e du a ion, wi h Holms öm and Milg om (1991) acknowledging ha hei find-
ings accu a ely eflec sho - e m dynamics. Holms öm and Milg om (1987)
demons a e ha con ac s adop a linea o ma wi h espec o o al ou pu unde
ce ain condi ions, such as agen s wi h exhibi ing exponen ial u ili y and he effo
en ailing mone a y cos s. This finding has been u he explo ed by esea che s
like Schä le and Sung (1993), who de ise a mo e gene al ma hema ical amewo k,
and Hellwig and Schmid (2002), who examine he p e equisi es o disc e e- ime
models o align wi h he findings o Holms öm and Milg om (1987). Williams (2015)
also ex ends he model in oduced in Holms öm and Milg om (1987). In pa icula ,
386 P. Reinwald e al.
he is conce ned wi h si ua ions in which he agen has hidden sa ing, which is why
his consump ion and weal h canno be moni o ed. Sanniko (2008) con ibu es o
his line o esea ch by showing ha op imal long- e m con ac s exhibi complex
non-linea wage and effo pa e ns. They de ail he ci cums ances unde which an
agen can e i e wi h an op imal con ac , whe e he decision o con inue o no a
each s ep is influenced by p ospec i e wages and effo le els.
Edmans and Gabaix (2011) a e conce ned wi h fixed con ac s in he hidden
ac ion con ex . They e e ence he wo k o G ossman and Ha (1992), who demon-
s a ed he complexi ies a ising om fixed con ac s in disc e e- ime scena ios. In
esponse, Edmans and Gabaix (2011) in oduce a model ea u ing con ac s ha a e
easie o manage by e ising a c ucial p emise: he agen has he abili y o pe cei e
en i onmen al noise p io o exe ing effo . They con end ha wi hou his modi-
fica ion, agen s would beha e in a manne aimed a ulfilling expec ed incen i es.
Resea ch has also ocused on he issue o an agen ’s decision o s ay indefini ely wi h
he p incipal. While many s udies assume such indefini e commi men , he e a e
models ha accoun o limi ed commi men . A significan amoun o his wo k is
aimed a c ea ing con ac s ha , in any si ua ion, gi e agen s no eason o exi ,
ensu ing he con ac ’sinfini e alidi y (e.g. Koche lako a 1996; Phelan 1995; Thomas
and Wo all 1988). Ano he app oach is ollowed by Wang and Yang (2019); hey
add ess his in e ac ion be ween he p incipal and he agen , in oducing a model
ha ea u es s ochas ic al e na i es, dynamic con ac s, and a mechanism o
endogenous sel -en o cemen .
The e a e a numbe o p e ious wo ks ha ocus on unce ain y in in o ma ion
abou pe o mance me ics. Fo ins ance, Chaigneau, Edmans, and Go lieb (2014)
examine in o ma ion’s ole and i s cons ain s conce ning pe o mance me ics
wi hin hidden ac ion scena ios. They d aw on he in o ma i eness p inciple
(Holms öm 1979; Sha ell 1979), which posi s ha p incipals should seek ou highly
p ecise pe o mance indica o s. Emphasizing his p inciple, Chaigneau, Edmans,
and Go lieb (2014) highligh he necessi y o weighing he in o ma ional benefi s
agains he associa ed cos s, a ask complica ed by he indi ec ela ionship be ween
addi ional in o ma ion and con ac ing p oblem pa ame e s when de i ing op imal
con ac s p o es challenging. U ilizing he mo al haza d amewo k, hey in oduce
a model o sc u inize he benefi s p incipals gain om enhanced in o ma ion
accu acy. MacLeod (2003) akes up he a gumen p o ided in P ende gas (1993) who
a gues ha jobs in which objec i e pe o mance measu es a e a ailable a e e y
a e, and a he bonuses a e o en based on subjec i e pe o mance e alua ions. In
his ein, MacLeod (2003) explo es (one-pe iod) models whe e pe o mance mea-
su es a e no always ully accessible o may e en be en i ely una ailable. In hese
models, agen e alua ions a e based mo e on subjec i e assessmen s by he p in-
cipal. Simila ly, Fuchs (2007) a gues ha ce ain labo ma ke phenomena, like wage
An ABM o he Hidden Ac ion Model 387
comp ession and pe iodic e iews, can se e as effec i e s a egies o mi iga e mo al
haza d on he agen ’s pa , especially when pe o mance e alua ions a e subjec i e
and held p i a ely.
Some o p e ious wo ks ocused on unce ain y ega ding he agen ’s cha ac-
e is ics. Fo example, Cohen, Deligkas, and Ko en (2022) examine scena ios
in ol ing hidden ac ions, whe e he p incipal is unawa e o no only he agen ’s
effo bu also hei u ili y unc ion and ac ion space. They p esen a model whe ein
he p incipal i e a i ely disce ns he cha ac e is ics o an op imal con ac by
ex ending offe s o iden ical agen s and moni o ing he esul s. Addi ionally, Cohen,
Deligkas, and Ko en (2022) in oduce a lea ning algo i hm ailo ed o his pu pose.
P a and Jo ano ic (2014) ocus in o hidden ac ion scena ios wi h long- e m con-
ac s, especially when an agen ’s capabili ies o pe o m asks emain unknown.
They assume he pe sis ence o an agen ’s capabili ies and base hei a gumen a ion
and he accumula ion o in o ma ion o e ime o acili a e incen i e p o ision. In a
simila ein, Lai, Liu, and Li (2021) add ess scena ios wi h unknown agen capabil-
i ies, a guing ha as he con ac ual pe iod ex ends owa ds infini y, unce ain y
diminishes because he agen ’s capabili ies become ully e ealed. Con e sely, He
e al. (2017) and DeMa zo and Sanniko (2016) ackle s ochas ic unce ain y h ough
lea ning, p oposing models ha equa e s ochas ic u u e p ofi s om ou pu wi h
unce ain y abou agen abili ies. Mekonnen (2017) builds upon hese wo ks,
add essing he limi ed knowledge ega ding he p oduc ion unc ion, which
in oduces unce ain y abou ou come dis ibu ions. In his amewo k, bo h he
p incipal and he agen o m indi idual belie s abou his dis ibu ion, upda ing hei
belie s based on pas ou comes while he agen can influence he p incipal’s pe -
cep ions. To add ess hese in o ma ion limi a ions, Mekonnen (2017) inco po a es an
addi ional s a e a iable in compu ing op imal con ac s.
In addi ion, he e is a line o esea ch ha employs simula ion-based esea ch
app oaches o assess he obus ness o he hidden ac ion model o i s included
assump ions. Lei ne and Wall (2021) modi y he hidden ac ion model’s assump ions,
in oducing in o ma ion unce ain y ega ding he en i onmen al a iable and he
ac ion space. In his adap a ion, he p incipal and he agen a e equipped wi h
dis inc in o ma ion sys ems o da a e ie al. This s udy obse es he pa ame e s
o incen i e mechanisms ha eme ge a he o ganiza ional le el. Following
his, Lei ne and Wall (2022) conduc a de ailed examina ion o he mic o-le el
beha io al dynamics in scena ios cha ac e ized by limi ed in o ma ion abou he
ac ion space and en i onmen . Fu he ing his esea ch, Reinwald, Lei ne , and Wall
(2022) ocus solely on he pa ies’knowledge o he en i onmen al a iable and,
ex ending beyond p e ious esea ch, conside he memo y capaci ies o bo h he
p incipal and he agen .
388 P. Reinwald e al.
whe e ηdeno es he A ow-P a measu e o isk-a e sion (A ow 1973). The
no a ion used in he agen -based model is summa ized in Table 2.
3.2.2 Simul aneous and Sequen ial Lea ning Model
The p incipal and he agen dispose o an indi idual memo y m
P
and m
A
, espec-
i ely. The highe m
P
and m
A
, he mo e es ima ions and obse a ions o he exoge-
nous ac o he p incipal and he agen can s o e in hei memo y. Due o diffe en
in o ma ion s a es, he agen ’s ac ual effo a
migh de ia e om he inci ed effo a
.
Howe e , he p incipal has no in o ma ion abou he ac ual effo a
and, he e o e,
Table :No a ion o he agen -based model.
Desc ip ion No a ion
Endogenous a iables:
P incipal’s u ili y U
P
Agen ’s u ili y U
A
Times eps
Ou come in x
=a
+θ
Agen ’seffo in a
Inci ed effo in a
Exogenous a iable in θ
P incipal’s expec ed ou come o effo le el a′in x
P ða′Þ
Agen ’s expec ed ou come o effo le el a′in x
A ða′Þ
Agen ’s sha e o he ou come in s(x
)=x
⋅ρ
P emium pa ame e included in he con ac in ρ
P emium pa ame e o effo le el a′in ρ
(a′)
P incipal’s se o all easible ac ions in A
P
P incipal’s candida es o inci ed effo in A
P
Agen ’s se o all easible ac ions in A
A
Es ima ions o he exogenous ac o s a ailable o he p incipal in Θ
P
Obse a ions o he exogenous ac o s a ailable o he agen in Θ
A
P incipal’slea ned expec a ion o he exogenous ac o in θ
P
Agen ’slea ned expec a ion o he exogenous ac o in θ
A
Exogenous a iables:
Maximum imes eps T
Agen ’s A ow-P a measu e o isk-a e sion η
P incipal’s memo y m
P
Agen ’s memo y m
A
An ABM o he Hidden Ac ion Model 395
bases he es ima ion o he exogenous ac o in on he inci ed effo . She compu es
he es ima ion acco ding o
θ
=x −a
.(8)
The agen , in con as , knows he ac ual effo a
he made and, he e o e, can
compu e he ac ual exogenous ac o in acco ding o
θ =x −a .(9)
The p incipal and he agen s o e hei es ima ions and obse a ions in hei
memo ies. Once hei capaci ies, m
P
o he p incipal and m
A
o he agen , a e
eached, he oldes in o ma ion is eplaced by he la es es ima ion o obse a ion.
Once he simula ion mo es on o he nex imes ep, he p incipal and he agen
upda e he lea ned expec ed alue o he exogenous ac o by a e aging all p i a ely
s o ed es ima ions/obse a ions. To do so, hey e ie e he in o ma ion om hei
memo ies. Le us deno e he es ima ions o he exogenous ac o a ailable o he
p incipal in by
ΘP =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
[θ
1,…,θ
−1]i mP=∞,
[θ
−mP,…,θ
−1]i mP<∞and ≥mP,
[θ
1,…,θ
−1]i mP<∞and <mP.
(10)
Fo he agen , we deno e he obse a ions o he exogenous ac o a ailable in by
ΘA =⎧
⎪
⎨
⎪
⎩
[θ1,…,θ −1]i mA=∞
[θ −mA,…,θ −1]i mA<∞and ≥mA
[θ1,…,θ −1]i mA<∞and <mA
(11)
The p incipal and he agen compu e hei lea ned expec ed alue o he exogenous
ac o in as he mean ∅(⋅) o he in o ma ion a ailable o hem: Fo he p incipal, he
lea ned expec ed alue o he exogenous ac o is θ
P =∅(ΘP ), while o he agen i
is compu ed acco ding o θ
A =∅(ΘA ). No e ha he lea ned expec ed exogenous
ac o can also be in e p e ed as he p incipal’s and he agen ’s in o ma ion abou
(and pe cep ion o ) he en i onmen since i ep esen s how hey ega d he en i-
onmen in ha imes ep.
3.2.3 The P incipal’s and Agen ’s Decisions
The p incipal’s and he agen ’s decisions e ol e a ound he selec ion o ac ions and
he compu a ion o he co esponding p emium pa ame e s. We deno e he se o
easible ac ions om he pe spec i e o he p incipal and he agen by A
P
and A
A
,
espec i ely. The pa icipa ion cons ain defines he lowe bounda y o hese
396 P. Reinwald e al.
spaces, and he uppe limi is gi en by he incen i e compa ibili y cons ain
(Holms öm 1979; Lei ne and Wall 2021). Recall ha he compu a ion o he wo
cons ain s includes he expec a ion abou he exogenous ac o (see Eqs. (3b) and
(3c)). Thus, i he p incipal and he agen ha e he same (diffe en ) expec a ions abou
he en i onmen , A
P
and A
A
pe ec ly coincide (a e diffe en ).
3.2.3.1 The P incipal’s Decision
In e e y imes ep, he p incipal can adap he p emium pa ame e in he con ac . To
do so, she andomly disco e s wo al e na i e effo le els in he ac ion space A
P
,a
1
and a
2, which oge he wi h he inci ed effo o he p e ious pe iod a
−1se e as
candida es o he effo she wan s he agen o make in pe iod . Le us deno e he se
o candida es o he inci ed effo in by A
P =[a
1,a
2,a
−1]. Then, he p incipal
unde akes a s ep-by-s ep local sea ch, adhe ing o a hill climbing s a egy. This
implies ha , based on he in o ma ion a ailable o he a any gi en momen , she op s
o he mos p omising choice (Co men e al. 2022). In line wi h he p oduc ion
unc ion in Eq. (5), he p incipal compu es he expec ed ou come o all al e na i es
a′∈A
P acco ding o
x
P (a′)=a′+θ
P .(12)
The p incipal also compu es he p emium pa ame e s o all candida e effo le els
a′∈A
P acco ding o
ρ (a′)=a g maxρ∈[0,1]UP(x
P (a′),s(x
P (a′))),(13)
and finally selec s he candida e which maximizes he u ili y as he inci ed effo o
he pe iod acco ding o
a
=a g maxa′∈A
P UP(x
P (a′),s(x
P (a′))) (14)
Toge he wi h he ask ha is o be ca ied ou (which is he same h oughou all ime
s eps), he co esponding p emium pa ame e ρ ≔ρ (a
)is he main elemen o he
con ac ha is offe ed o he agen .
3.2.3.2 The Agen ’s Decision
In e e y imes ep, he agen makes wo decisions. Fi s , he decides whe he o accep
o ejec he con ac offe ed by he p incipal. In pa icula , i he u ili y o he offe ed
con ac exceeds he ese a ion u ili y he agen accep s he con ac . To make his
decision, he agen compu es he effo ha maximizes his u ili y gi en he offe ed
con ac acco ding o
a*
=a g maxa′∈AA UA(s(x
A (a′)),a′),(15)
An ABM o he Hidden Ac ion Model 397
whe e x
A (a′)=a′+θ
A . I he u ili y o his effo le el exceeds o is equal o he
agen ’s ese a ion u ili y, UA(s(x ),a*
)≥U
, he agen accep s he con ac and
makes his effo in . In consequence, a ≔a*
.
3
3.3 Pa ame e Se ings and Obse a ions
3.3.1 Pa ame e Se ings
3.3.1.1 Pa ame e s o Scena ios wi h Unlimi ed In o ma ion Access
The solu ion p oposed in Holms öm (1979) aids as he benchma k solu ion in ou
s udy. To show ha he p oposed model is capable o eplica ing he benchma k
solu ion, we fi s un simula ions on scena ios wi h unlimi ed in o ma ion access
o he p incipal and he agen , i.e. we se he a iables m
A
and m
P
equal o ∞.In
addi ion, we ake in o accoun he u bulence o he en i onmen : Recall he exog-
enous a iable ollows a no mal dis ibu ion, which allows us o con ol he u bu-
lence ia he s anda d de ia ion. In pa icula , we se he mean o he no mal
dis ibu ion o ze o and define i s s anda d de ia ion ela i e o he op imal
ou come x* o Holms öm’s hidden ac ion model (see and Appendix A), so ha
σ= {0.05x*, 0.25x*, 0.45x*}.
All o he pa ame e s a e kep cons an du ing he simula ion uns. Fo scena ios
wi h unlimi ed in o ma ion access, ou analysis ocuses memo y on he fi s T= 200
imes eps in e e y simula ion ound. E e y scena io is epea ed R= 700 imes.
4
We
se he A ow-P a measu e o isk a e sion a 0.5 o ep esen agen s wi h mode a e
isk a e sion, aligning wi h he findings o G aham, Ha ey, and Pu i (2013) and
B enne (2015). These s udies indica e ha CEOs exhibi lowe isk a e sion
compa ed o he b oade popula ion and ha he beha io o execu i es is consis en
wi h a mode a e le el o isk a e sion.
3.3.1.2 Pa ame e s o Scena ios wi h Limi ed In o ma ion Access
To cap u e he effec s o limi ed in o ma ion access in hidden ac ion se ups,
we analyze scena ios wi h h ee diffe en alues o he p incipal’s memo y
3Please no e ha , wi hou loss o gene ali y, we no malize he ese a ion u ili y o ze o du ing he
simula ion expe imen s, and make su e ha he agen accep s he con ac in all cases.
4We ollow he app oach p oposed in Lee e al. (2015) and selec he numbe o simula ion ounds
based on he coefficien o a ia ion. Fo ou se ings, his measu e s abilizes a ϵ≤0.01 a a ound 700
epe i ions.
398 P. Reinwald e al.
(m
P
= {1, 3, 5}) and he agen ’s memo y (m
A
= {1, 3, 5}), whe eby his pa ame e can be
in e p e ed so ha he poo e he memo y, he lowe he in o ma ion a ailabili y.
5
The analysis o scena ios wi h limi ed memo y ocuses on he fi s T=20
imes eps in e e y simula ion ound; we do so because mos o he dynamics can be
obse ed wi hin his pe iod. All o he pa ame e s a e kep cons an du ing he
simula ion uns. An o e iew o he pa ame e s conside ed in his simula ion s udy
is p o ided in Table 3.
3.3.2 Obse a ions
In ou simula ion expe imen s, we obse e ou measu es: (i) he p emium
pa ame e de e mined by he p incipal, (ii) he le el o effo exe ed by he agen in
comple ing he gi en ask, (iii) he u ili y expe ienced by he agen h oughou he
expe imen s, and (i ) he u ili y o he p incipal.
As in oduced in Eq. (6), he p emium pa ame e defines he agen ’s sha e o he
ask ou come. We deno e he p emium pa ame e ha is effec i e in pe iod and
simula ion un by ρ
and he p emium pa ame e effec i e in he op imal solu ion
by ρ* (see Appendix A). We compu e he a e age no malized p emium pa ame e in
e e y imes ep acco ding o
Table :Simula ion pa ame e s o scena ios wi h unlimi ed and limi ed memo y.
Pa ame e No a ion Limi ed memo y Unlimi ed memo y
Subjec o a ia ion:
P incipal’s memo y m
P
,,∞
Agen ’s memo y m
A
,,∞
Exogenous ac o : s anda d de ia ion σ.x*, .x*, .x*.x*, .x*, .x*
Cons an s:
Exogenous ac o : mean μ
Agen ’s A ow-P a measu e η..
Obse a ion pe iod T
Simula ion ounds R
5We limi he pa ame e space because o he compu a ional complexi y o he simula ion model.
The compu a ion ime Tcomp o a simula ion model wi h ppa ame e s and lle els pe pa ame e is lp,
scaling polynomially wi h he numbe o le els lpe pa ame e . Fo example, wi h p= 3 and l= 3, he
pa ame e space is 33= 27, and he ini ial compu a ion ime is Tcomp
0=27 ⋅Tsim, wi h Tsim ≈640 min o
T= 20 ime s eps pe simula ion un. Adding nle els pe pa ame e changes he pa ame e space o
(l+n)pand he compu a ion ime o Tcomp
n=(l+n)p⋅Tsim, esul ing in a compu a ion ime a io o
Tcomp
n/Tcomp
0=(l+n)/
(l)p. Thus, o n= 1 he compu a ion ime app oxima ely doubles (≈2.37), wi h
n= 2 i quad uples (≈4.63), and wi h n= 3 i oc uples.
An ABM o he Hidden Ac ion Model 399
ρ
=1
R∑
R
=1
ρ
ρ*.(16)
To cap u e he effo he agen makes o ca y ou his ask, we epo he a e age
no malized ac ual effo le el in e e y imes ep . We compu e his me ic as ollows:
a
=1
R∑
=R
=1
a
a*,(17)
whe e a
indica es he effo made by he agen in imes ep and simula ion un ,
and a* s ands o he op imal le el o effo sugges ed in Holms öm (1979) (see
Appendix A). Please no e ha he a e age no malized effo is also a p oxy o he ask
ou come a he mac o le el, as he ou come is a unc ion o he agen ’seffo and he
exogenous a iable (see Eq. (5)).
Las ly, we documen he u ili y expe ienced by bo h he p incipal and he agen
h oughou he simula ion uns. Fo he p incipal, we deno e he u ili y expe ienced
in ime s ep and simula ion un by U
P
(see Eq. (4)) and no malize i by he u ili y
ha can be achie ed wi h he solu ion o Holms öm’s hidden ac ion model, which
we deno e by U*
P( o i s compu a ion see Appendix A). Then, he p incipal’s a e age
no malized u ili y a ime is compu ed acco ding o
U
P =1
R∑
R
=1
UP
U*
P
.(18)
Simila ly, we deno e he agen ’s u ili y (see Eq. (7)) in imes ep and simula ion un
and he u ili y ollowing Holms öm’s model by U
A
and U*
A, espec i ely. We
compu e he agen ’s a e age no malized u ili y a ime by
U
A =1
R∑
R
=1
UA
U*
A
.(19)
4 Resul s
The esul s a e o ganized in o ou sec ions. In Sec ion 4.1, we show ha he solu ion
eme ging om he agen -based model con e ges o he op imal solu ion sugges ed by
Holms öm (1979). Then, Sec ion 4.2 analyzes he scena ios wi h limi ed in o ma ion
access. Specifically, Sec ion 4.2.1 analyzes he effec s o limi ed in o ma ion on he
p emium pa ame e , Sec ion 4.2.2 p o ides an analysis o he effo made by he
agen , and Sec ion 4.2.3 ocuses on he dynamics wi hin he agen -based model and
pa icula ly emphasizes he p incipal’s and he agen ’s espec i e u ili ies ha esul
om he choices ela ed o he p emium pa ame e and he effo .
400 P. Reinwald e al.
4.1 Scena ios wi h Unlimi ed In o ma ion Access: Replica ing
he Benchma k Solu ion
Ou fi s simula ions aim a demons a ing ha he obse a ions om ou
agen -based model align wi h he op imal solu ion p oposed by Holms öm (1979)
(gi en he u ili y unc ions specified in Sec ion 3.2.1). In hese simula ions, bo h he
p incipal and he agen a e endowed wi h unlimi ed memo y conce ning he en i-
onmen al a iable. I can be expec ed ha wi h a sufficien ly long obse a ion
pe iod, he p incipal’s and he agen ’s es ima ions o he en i onmen al a iable will
con e ge o a alue close o i s expec ed alue, and ollowing his, we expec he
eme gence o solu ions ha a e close o he ones p oposed in Holms öm (1979) om
he agen -based model.
To obse e he model’s ac ual beha io , we un simula ions o an obse a ion
pe iod o 200 imes eps and ack all obse a ions as de ailed in Sec ion 3.3.2. Fo a
summa y o all pa ame e s, see Table 3. To accoun o he equi emen o long
(p ac ically infini e) obse a ion pe iods, we ha e fi ed powe models o he o m x
( )=a⋅ b+c o ex apola e he obse ed ime se ies o e longe pe iods. The
pa ame e s o hese models a e lis ed in Table 4. Gene ally, we can see om he
esul s ha he exponen bis nega i e o all models. When inc eases, he e m b
con e ges o ze o because o nega i e alues o b; in consequence, he e m a⋅ balso
app oaches ze o, and he esul o he unc ion x( ) will con e ge o he alue o he
cons an c. The alues o c(conside ing he confidence in e als wi h α= 0.05) a e
sufficien ly close o 1. The e o e (and as we no malize ou obse a ions), weconclude
ha , o a sufficien ly ex ended obse a ion pe iod, ou model p edic ions align
closely wi h he heo e ical solu ions p oposed by Holms öm (1979), gi en he u ili y
unc ions specified in Sec ion 3.2.1.
To assess he con e gence a e o x( ) owa d he cons an c,wedefine he
coefficien -exponen influence a io (CEIR), R=|a/b|. This a io is calcula ed o each
pai (a,b) and p esen ed in Table 4. The CEIR acili a es a compa a i e analysis; a
lowe CEIR indica es as e con e gence o x( ) oc, whe eas a highe CEIR sugges s
slowe con e gence.
6
The esul s e eal ha en i onmen al u bulence has a
negligible impac on he con e gence a es o bo h he agen ’seffo and he
6The con e gence a e o x( ) owa ds cis dic a ed by i s de i a i e’s magni ude, exp essed as |x′
( )|=|a|⋅|b|⋅ b−1. Please no e ha he e m a⋅ bin he powe unc ion x( )=a⋅ b+cis domina ed by b,
due o he powe law dynamics; bde e mines he a e o change and aonly scales his effec .
Consequen ly, a highe |b| esul s in a mo e esponsi e model o a ia ions in . High CEIRs, whe e |a|
exceeds |b|, sugges ha he scaling effec o ap edomina es, leading o a ela i ely slowe con e -
gence o x( ) owa ds c. Con e sely, low CEIRs imply ha b’sinfluence is g ea e ela i e o a,
acili a ing a quicke con e gence o x( ) oc.
An ABM o he Hidden Ac ion Model 401
Table :Es ima ed pa ame e s o fi ed powe models o scena ios wi h unlimi ed in o ma ion access.
Obse a ion Es ima ed powe model pa ame e s CEIR Goodness
aCI bCI cCI |a/b|RRMSE
Rela i ely s able en i onmen :
P emium pa ame e −. ±. −. ±. . ±. . . .
Effo le el −. ±. −. ±. . ±. . . .
P incipal’s u ili y −. ±. −. ±. . ±. . . .
Agen ’s u ili y −. ±. −. ±. . ±. . . .
Mid- u bulen en i onmen :
P emium pa ame e −. ±. −. ±. . ±. . . .
Effo le el −. ±. −. ±. . ±. . . .
P incipal’s u ili y −. ±. −. ±. . ±. . . .
Agen ’s u ili y −. ±. −. ±. . ±. . . .
Tu bulen en i onmen :
P emium pa ame e −. ±. −. ±. . ±. . . .
Effo le el −. ±. −. ±. . ±. . . .
P incipal’s u ili y −. ±. −. ±. . ±. . . .
Agen ’s u ili y −. ±. −. ±. . ±. . . .
Es ima ed powe models ha e he o m o x( )=a⋅ b+c. CI columns epo he confidence in e als o α=.. CEIR column epo s he coefficien -exponen influence a io, |a/b|.
Goodness columns epo he coefficien o de e mina ion (R) and he oo mean squa ed e o (RMSE) as wo measu es o he models’fi .
402 P. Reinwald e al.
p incipal’s u ili y. In con as , he con e gence o he p emium pa ame e and he
agen ’s u ili y o cis no ably slowe in u bulen condi ions. Mo eo e , he analysis o
he CEIRs indica es ha he agen ’seffo and he p incipal’s u ili y consis en ly
con e ge mo e apidly o c han he p emium pa ame e and he agen ’s u ili y.
4.2 Scena ios wi h Limi ed In o ma ion Access
Fo e e y scena io, we moni o R ime se ies o he measu emen s desc ibed in
Sec ion 3.3.2, each wi h a du a ion o Tpe iods. This includes obse ing 700 ime
se ies o leng h 20 o scena ios wi h limi ed memo y, co e ing all ou me ics.
To e i y i a se ies leng h o 20 is adequa e, we define wo ime windows wi hin
he se ies and compa e hei cen al endencies wi h a Mann-Whi ney U es (Mann
and Whi ney 1947), and hei a iances wi h a Le ene es (Le ene 1960). Ou ocus is
on he se ies’las qua e , iden i ying segmen s om pe iods 15 o 17 and 18 o 20,
u ilizing samples om 700∗3 = 2, 100 obse a ions. We conside a ime se ies leng h
o 20 adequa e i we can accep he null hypo heses o he Mann-Whi ney U
es (indica ing equal cen al endencies) and he Le ene es (sugges ing equal
a iances) a significance le els o ei he p≤0.01 o 0.05. The co esponding esul s
a e included in Tables 5 h ough 8.
To e alua e i al e ing he p incipal’s o he agen ’s memo y impac s he
obse ed ime se ies, we conduc a Mann-Whi ney U es compa ing wo ela ed ime
se ies (Lee e al. 2015). I we canno accep he null hypo hesis o equal cen al
endencies a p≥0.05 o 0.01, we in e ha he memo y capaci y influences he
cen al endency. Fu he mo e, o de e mine he magni ude o his impac , we
calcula e he ank bise ial co ela ion as an effec size indica o . This is done using
he app oach ou lined in Wend (1972) who compu es he ank bise ial co ela ion
as ollows:
b
=1−(2 ⋅U)/(n
1
⋅n
2
), whe e Uis he Mann-Whi ney U es s a is ic, and
n
1
and n
2
a e he sizes o he wo compa ed samples. The esul s a e de ailed in
Tables 9 and 10.
4.2.1 P emium Pa ame e
This sec ion examines he impac o in o ma ion cons ain s on he p emium
pa ame e . We documen ou findings by fi s p esen ing he mean o he a e age
no malized p emium pa ame e s h oughou he en i e obse a ion pe iod and, a
he end o his pe iod, alongside hei espec i e confidence in e als in Table 5.
Fu he , he analysis o he ime se ies beha io is also included in Table 5. All ime
se ies o he p emium pa ame e demons a e s abili y in bo h hei cen al
endencies and a iances.
An ABM o he Hidden Ac ion Model 403
The esul s indica e ha , in gene al, he p emium pa ame e dec eases wi h
highe en i onmen al u bulence. Please no e ha he u bulence pe cei ed by he
p incipal and he agen is also affec ed by hei memo ies. Recall he p incipal’s and
Table :Expec ed p emium pa ame e s and s a iona i y o ime se ies.
Memo y Expec ed p emium pa ame e S a iona i y
P incipal Agen Pe iods : CI Pe iod CI Cen al
endency
Va iance
Rela i ely s able en i onmen :
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
Mid- u bulen en i onmen :
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗
. ±. . ±. ∗∗∗
Tu bulen en i onmen :
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
. ±. . ±. ∗∗ ∗∗
CI columns epo he confidence in e als o α=.. S a iona i y columns epo esul s o a Mann-Whi ney U es
(cen al endency) and a Le ene es ( a iance). ∗null hypo hesis can be accep ed wi h p≤..∗∗ null hypo hesis can
be accep ed wi h p≤..
404 P. Reinwald e al.
ises, highligh ing ha he mos subs an ial posi i e effec s a e seen in mo e s able
condi ions.
5 Discussion
The esul s p esen ed in Sec ion 4 allow o pu he effec s o limi a ions in he
p incipal’s and agen ’s espec i e in o ma ion and en i onmen al u bulence in he
ollowing amewo k (see Figu e 3): Recall, he p incipal and he agen es ima e and
obse e he exogenous a iable in e e y imes ep, espec i ely. Then, hey s o e
hei es ima ions/obse a ions in hei memo ies. Limi a ions in hei espec i e
in o ma ion ake effec in he o m o cons ain s in hei memo ies. The mo e
in o med he p incipal and he agen a e, he mo e in o ma ion s o ed in hei
memo ies a e conside ed when compu ing he lea ned expec a ion o he exogenous
a iable. Thus, hei espec i e in o ma ion mode a es he p incipal’s and agen ’s
pe cep ions o he en i onmen . The p incipal’s pe cep ion affec s he choice o he
p emium pa ame e . One migh expec ha he agen ’s pe cep ion o he en i on-
men impac s his effo choice, bu we could no obse e significan influences o
his ela ionship. Toge he wi h he ac ual en i onmen , he p incipal’s and he
agen ’s decisions abou he p emium pa ame e and he effo , espec i ely, affec
hei u ili ies. The analysis in his pape ocuses mainly on he effec s o u bulence in
he en i onmen and limi a ions in he p incipal’s and agen ’s espec i e in o ma-
ion in his amewo k (which is indica ed by he g ay boxes in Figu e 3).
Figu e 3: F amewo k o limi ed in o ma ion in hidden ac ion si ua ions.
An ABM o he Hidden Ac ion Model 411
5.1 Resul s Rela ed o he P emium Pa ame e
The main findings p esen ed in Sec ion 4.2.1 a e ha (i) he alue o he p emium
pa ame e dec eases wi h en i onmen al u bulence and (ii) inc eases wi h he
p incipal’s in o ma ion. This means ha he agen ’ssha eo heou comeis
ela i ely small (la ge) i he en i onmen is a he u bulen (s able). Howe e ,
i he u bulenceishigh, heagen canno easilycon ol heou come,which
pu s his compensa ion a isk (Miceli and Heneman 2000). Acco dingly, he
finding ha he (pe cei ed) u bulence dec eases he p emium pa ame e is
(in pa ) con a y o he isk p emium hypo hesis.Thishypo hesiss a es ha he
p incipal will ha e o inc ease he isk-a e se agen ’s o al compensa ion in
u bulen en i onmen s o p o ec him om isk (Bu ns and S alke 2001;
Eisenha d 1988; Umana h, Ray, and Campbell 1993). Coun e -in ui i ely, he
esul s p esen ed in Sec ion 4.2.1 indica e he con a y. E en hough he isk-
bea ing is ini ially wi hin he p incipal’s ole (Fama and Jensen 1983), she shi s a
pa o he isk o he agen .
This finding no only con adic s he isk p emium hypo hesis bu is also
in con as o p e ious esea ch on ask p og ammabili y and he p edic ions o
classical o ganiza ion heo y. S oh e al. (1996) and Sung and Choi (2012) link
en i onmen al u bulence wi h ask p og ammabili y by a guing ha mo e (less)
u bulence can be in e p e ed in e ms o less (mo e) p og ammable asks since
ask si ua ions and p oblems change mo e equen ly in u bulen en i onmen s.
Fu he mo e, i is o en sugges ed ha ask p og ammabili y is nega i ely co e-
la ed wi h he magni ude o a iable compensa ion componen s (Gomez-Mejia
and Balkin 1992). This ansla es in o he expec a ion ha mo e en i onmen al
u bulence (and less ask p og ammabili y) should be linked o highe p opo ions
o a iable compensa ion and ice e sa (Ge ha and Milko ich 1990; S oh e al.
1996). This expec a ion is also in line wi h classical o ganiza ion heo y (Thompson
1967), which p edic s ha o ganiza ions ha ope a e in u bulen en i onmen s
ely mo e hea ily on a iable compensa ion o ensu e he app op ia e beha io s
o hei agen s. Ou findings do no suppo his expec a ion bu can be linked o he
a gumen a ion p o ided in Smi h (1984). The e a e no jus d awbacks bu also
ad an ages o u bulen en i onmen s (Eisenha d 1989), and an analogy o agen s
who a e hi ed in isky en i onmen s and ‘un ai lo e ies’can be es ablished: The
a e age pay is no necessa ily high, bu he e is a chance o ecei e a high
compensa ion, e en hough pe haps his chance is ela i ely sligh (see also Miceli
and Heneman 2000).
412 P. Reinwald e al.
5.2 Resul s Rela ed o he Agen ’sEffo
The esul s p esen ed in Sec ion 4.2.2 indica e ha he agen ’seffo ollows he
pa e ns obse ed o he p emium pa ame e s. The effo dec eases wi h en i-
onmen al u bulence so ha he agen makes mo e (less) effo in ela i ely s a-
ble ( u bulen ) en i onmen s. I a mo e in o med p incipal se s he p emium
pa ame e , he agen makes significan ly mo e effo . This ela ion is a undamen al
assump ion in economic con ex s and is suppo ed by expe imen al esea ch
(Dickinson 1999; Takahashi, Shen, and Ogawa 2016) and field esea ch (Banke , Lee,
and Po e 1996; Lazea 2000).
Since he agen ’s a e age no malized effo is also a p oxy o how well he
o ganiza ion pe o ms, hese obse a ions can be ela ed o en i onmen al u -
bulence and fi m pe o mance esea ch. En i onmen al unce ain y and dyna-
mism (Ald ich 2008; Chen, Reilly, and Lynn 2005; Milliken 1987) inc ease he
difficul y o o ganiza ional decision-making and significan ly affec o ganiza ional
pe o mance (Reinwald, Lei ne , and Wall 2022). This is in line wi h he esul s
p esen ed in Yu, Wang, and B ou he s (2016), who find ha (pe cei ed) en i on-
men al unce ain y affec s he iden ifica ion o compe i o s so ha fi ms iden i y
mo e compe i o s i he en i onmen is ce ain. This di ec ly ansla es in o a mo e
compe i i e ad an age and, as a consequence, a be e (wo se) pe o mance in
ce ain (unce ain) en i onmen s (Du a and King 1980; Po e 1997). Fo he hid-
den ac ion con ex , i is also shown in Reinwald, Lei ne , and Wall (2020) ha
inc eases in (pe cei ed) en i onmen al u bulence lead o a d op in fi m pe o -
mance. This finding is also suppo ed by he con ingency managemen accoun ing
li e a u e, which is, amongs o he s, conce ned wi h he fi be ween o ganiza ions
and hei en i onmen (O ley 1999, 2016). I he en i onmen is u bulen and/o he
p incipal is no e y well in o med abou he en i onmen , she canno design
he incen i e scheme so ha i fi s he ac ual en i onmen (Chenhall and Mo is
1986; Ezzamel 1990; Ghosh and Olsen 2009; Hoque 2005). Since he agen esponds o
he incen i es se by he p incipal, subop imal incen i e pa ame e s di ec ly
ansla e in o ad e se effec s on pe o mance.
5.3 Resul s Rela ed o he P incipal’s and he Agen ’s U ili ies
Abo e, i was es ablished ha he p incipal’s choice o he p emium pa ame e migh
indica e ha she ans e s some o he isks o he agen by educing he p emium
pa ame e as en i onmen al u bulence inc eases. Now, as we ake he esul s
ela ed o he agen ’s u ili y p esen ed in Sec ion 4.2.3 in o accoun , his conjec u e
becomes e en mo e e iden . One would expec ha he agen compensa es o he
An ABM o he Hidden Ac ion Model 413
addi ional isk and he missing isk p emium by making less effo , which –in u n –
e en ually inc eases his u ili y. Howe e , he esul s indica e he opposi e: The
agen ’s u ili y dec eases wi h inc eases in en i onmen al u bulence. I a mo e
in o med p incipal se s he p emium pa ame e , he agen ’s u ili y appea s o
inc ease. Fo he p incipal, he esul s p esen ed in Sec ion 4.2.3 indica e ha he
ola ili y o his u ili y is ela i ely high. Howe e , he p incipal’s u ili y appea s o be
less sensi i e o limi a ions in he own and he agen ’s in o ma ion. As a consequence
we can conclude ha he isk-neu al p incipal wi hholds a isk p emium om he
isk-a e se agen , which, in u n, assu es he obus ness o he p incipal’s u ili y o
en i onmen al u bulence.
The esul s indica e ha he agen is o e -dependen on he p incipal, which
esul s in a dilemma o he agen : Fi s , i he agen we e able o inc ease his
memo y, doing so would no allow him o escape he si ua ion, since his memo y has
no significan effec on his u ili y. Second, he p incipal appea s o se he p emium
pa ame e o punish he agen o he isky en i onmen . Howe e , his would s ill
be he bes op ion o he agen since o he wise, he would ha e ejec ed he con ac
and ollowed he ou side op ion. Shi king (i.e. pu ing in less effo ) would no be an
op ion ei he , since i would dec ease he agen ’s u ili y (Nilakan and Rao 1994).
Thi d, e en i he agen we e mo e in o med, he would ha e no incen i e o disclose
his p i a e in o ma ion abou he en i onmen . I he p incipal was in o med
abou he ac ual exogenous a iable (ins ead o es ima ing i ), she could deduce he
agen ’seffo om he ou come (Caillaud and He malin 2000). As soon as he p in-
cipal ealizes ha he agen discloses his in o ma ion, she could swi ch om
pe o mance-based pay o effo -based compensa ion. As a consequence, he p in-
cipal could u he inc ease he u ili y a he cos o he agen . P e ious esea ch
me ely add esses he issue o o e ly dependen agen s. I ends o ocus on o e -
dependence on he p incipal’s side: Huang, Raimo, and Hum ey (2016), o example,
s a e ha he p incipal’s powe o exe con ol dec eases wi h he dependence o
he agen . Willcocks and Choi (1995) also ocus on he agen ’s pe spec i e and a gue
ha in endo –clien ela ionships, he clien migh be o e ly dependen on he
endo (see also Hancox and Hackney 2000). Ou esul s indica e ha limi ed
in o ma ion –o bo h he p incipal and he agen –appea s o empowe he p incipal
o (unin en ionally) siphon-offu ili y om he agen by capi alizing on he con ol
o e he compensa ion (Da id, Kochha , and Le i as 1998).
Recall ha one undamen al ounda iono he p incipal-agen heo y is ha bo h
he p incipal and he agen a e d i en by sel -in e es (Huang, Raimo, and Hum ey
2016). Now ha we know ha he p incipal expe iences (almos ) he same u ili y in
all cases, she has no incen i e o s op ans e ing isk o he agen o o ga he
in o ma ion abou he en i onmen on he own (i.e. o inc ease he in o ma ion).
The p incipal, hus –pe haps unin en ionally –beha es in a way ha migh be
414 P. Reinwald e al.
in e p e ed as oppo unism, i.e. sel -in e es -seeking wi h guile (Williamson 1975),
whe eby limi a ions in he p incipal’s in o ma ion appea o ein o ce beha io al
pa e ns ha appea as guile.
5.4 Me hodological Con ibu ions
F om a me hodological poin o iew, we ha e in oduced an agen -based model o
he hidden ac ion p oblem (Holms öm 1979) by employing he esea ch app oach
pu o wa d by Gue e o and Ax ell (2011) and Lei ne and Beh ens (2015a), and we
show ha he solu ion o he agen -based model con e ges o he solu ion p oposed
by he o iginal model. Ou app oach is diffe en om he classical p incipal-agen
heo y a a concep ual le el. In pa icula , agen -based modeling and simula ion
allow sys ema ically analyzing he obus ness o he solu ions de i ed om closed-
o m models o de ia ions om he assump ions included in hese models (Gue e o
and Ax ell 2011; Lei ne 2024; Wall and Lei ne 2021). The p incipal-agen heo y
o en includes some idealized assump ions. Some esea che s a e conce ned ha
(o e -)simplified beha io al assump ions migh come a he cos o he alidi y and
explana o y powe o he findings (Eden 1989; Minge s 2011; Minge s and Rosenhead
2004). Usually, i is assumed ha indi idual beha io is d i en by op imiza ion and
a ionali y, he modeled indi iduals’choices a e ep esen a i e o he en i e pop-
ula ion, and equilib ium solu ions can be achie ed. Ou app oach, howe e , allows
o he explici conside a ion o eme gence, limi a ions in in o ma ion, and
he e ogenei y (Chang and Ha ing on J 2006; Chen 2017; Mealy, Fa me , and
Tey elboym 2019). These ea u es o ou app oach allow o o e come some o he
limi a ions o he o mal app oaches in analy ical esea ch: While solu ions de i ed
om analy ical models a e ela ed o na ow beha io al assump ions, he model
p esen ed he e allows o ake ich en i onmen al con ex s and elaxed–and pe haps
e en mo e ealis ic –beha io al assump ions in o accoun (Wall 2024; Wall and
Lei ne 2021). Howe e , he use o agen -based modeling and simula ion in mic o-
economic con ex s is a he sca ce. The e o e, he app oach p esen ed he e can be
ega ded as a s ep owa ds a mo e open app oach in mic oeconomic esea ch ha
allows o elaxing (some o ) he well-es ablished assump ions.
6 Conclusions
In his pape , we p oposed an agen -based model o he hidden ac ion p oblem. In
pa icula , we ans e ed he closed- o m model in oduced in Holms öm (1979)
in o an agen -based model (Gue e o and Ax ell 2011; Lei ne and Beh ens 2015a).
An ABM o he Hidden Ac ion Model 415
Doing so allowed us o elax some o he idealized assump ions in Holms öm’s
model ela ed o he p incipal’s and he agen ’s espec i e in o ma ion. We ocus on
he memo y o in o ma ion abou he en i onmen . Ou esul s indica e ha he
p incipal’s in o ma ion is he key o good pe o mance, whe eas he agen ’s in o -
ma ion does no significan ly affec pe o mance. Su p isingly, he p incipal appea s
o beha e e y selfishly by siphoning offu ili y om he agen o main ain nea -
op imal pe sonal u ili y by exe ing he con ol o e he agen ’s compensa ion. We
model (mo e ealis ic) human beha io by employing ope a ional esea ch me hods
and by conside ing (and b inging oge he ) findings om he disciplines o cogni i e
psychology, economics, managemen , and ope a ional esea ch.
O cou se, ou esea ch is no wi hou limi a ions. Se e al u he incen i e
mechanisms –also non o mal ones –migh be employed in he modeled si ua ion.
G an ing he p incipal deg ees o eedom in he choices ela ed o he con ol
mechanism migh be a ui ul a enue o u u e esea ch. We limi he p incipal’s
and he agen ’s espec i e in o ma ion conce ning memo y only. Fu he esea ch
migh ocus on ex ending he limi a ions in in o ma ion by also aking in o accoun ,
o example, calcula ion e o s, limi a ions in o he ypes o in o ma ion, and biases
in in o ma ion p ocessing. Addi ionally, he agen makes decisions based on he
lea ned mean alue o he en i onmen al a iable. Fu u e esea ch could enhance
he agen ’s u ili y unc ion by inco po a ing he a iance o he en i onmen al
a iable, he eby mo e accu a ely eflec ing he agen ’s isk a e sion. Cu en ly, we
model scena ios whe e he e is a one- o-one delega ion ela ionship be ween a
single p incipal and a single agen . Fu u e s udies could b oaden his amewo k o
include delega ion ela ionships in ol ing mul iple agen s. Al hough he ocus o
his pape is on a heo e ical analysis o how he hidden ac ion model wi hs ands
limi a ions in in o ma ion access, subsequen esea ch migh include labo a o y
expe imen s o empi ically alida e he model.
Resea ch unding: This wo k was suppo ed by unds o he Ös e eichische
Na ionalbank [Aus ian Cen al Bank Anni e sa y Fund, p ojec numbe : 17930].
Da a and code a ailabili y: Simula ion da a and code a e a ailable ia he ollowing
link: h ps://gi hub.com/s o s ephan/JBNST24.
Appendix A: Solu ion o Holms öm’s Hidden
Ac ion P oblem
The e a e wo diffe en app oaches ha can be used o sol e he p og am o malized
in Eqs. (3a)–(3c). Impo an o us is he app oach o Mi lees (1976), who supp esses
416 P. Reinwald e al.
θand iews xas a andom a iable wi h dis ibu ion F(x,a). In his app oach, i is
assumed ha ∀a∈A∃x∈IR:F
a
(x,a) < 0 so ha a change in ahas non i ial effec on
he dis ibu ion o x. Fo a gi en dis ibu ion o θ,F(x,a) is he dis ibu ion induced
on x=x(a,θ) (Holms öm 1979).
In he ollowing p og am, (x,a) is he densi y unc ion o Fwi h
a
and
aa
well
defined o all (x,a) and Eq. (3c) is eplaced wi h a fi s -o de cons ain . Fu he -
mo e, s(x) is es ic ed o lie in he in e al [c,d+x] o gua an ee an exis ing solu ion
o Eqs. (3a)–(3c) o he class o unc ions in Eq. (20), whe e Vb′
bis he o al a ia ion o
sin he in e al [b,b′] (Holms öm 1979; Kolmogo o and Fomin 1970).
SK=s(x)∈[c,d+x]|Vb′
b(s)≤K⋅(b′−b)
{}
,(20)
max
s(x)∈[c,d+x],a∫G(x−s(x)) (x,a)dx(21a)
subjec o ∫[U(s(x)−V(a)] (x,a)dx≥H
,(21b)
∫U(s(x)) a(x,a)dx=V′(a).(21c)
We deno e he mul iplie s o Eqs. (21b) and (21c) by λand μ, espec i ely. A e a
poin wise Lag angian op imisa ion, he op imal sha ing ule yields he ollowing
cha ac e iza ion:
G′(x−s(x))
U′(s(x)) =λ+μ⋅ a(x,a)
(x,a),(22)
o almos e e y x o which Eq. (22) has a solu ion s(x)∈[c,d+x]. Also, μis gi en as
solu ion o he adjoin equa ion and is de e mined by Eq. (21c) (Holms öm 1979).
∫G(x−s(x)) a(x,a)dx+μ∫U(s(x)) aa(x,a)dx−V″(a)
{}
=0 (23)
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