Maly, Leona d Ome ; A inada , Tal
A icle
Sma alloca ion o a de elope 's spending on p oduc
quali y and non-sala y employee bene i s in a supply chain
o apps
Ope a ions Resea ch Pe spec i es
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spending on p oduc quali y and non-sala y employee bene i s in a supply chain o apps,
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Sma alloca ion o a de elope ’s spending on p oduc quali y and
non-sala y employee bene i s in a supply chain o apps
Leona d Ome Maly
*
, Tal A inada
Depa men o Managemen , Ba -Ilan Uni e si y, Rama Gan 5290002, Is ael
ARTICLE INFO
Keywo ds:
Supply chain managemen
Re enue sha ing
Quali y
Non-sala y bene i s
ABSTRACT
Quali ied and capable employees a e c ucial o he success o high- ech companies. Wi h an e e -sh inking pool
o alen , employe s a e o ced o de ise c ea i e ec ui men and e en ion me hods, which inc easingly ake he
o m o hea y spending on non-sala y bene i s. The p esen s udy con ibu es o he exis ing supply-chain
li e a u e h ough examining he ole played by such bene i s in a wo-agen sys em consis ing o a pla o m
and an app de elope . In pa icula , we examine he e ec o non-sala y bene i s on he ou going quali y c ea ed
by he employees o he app de elope . The pa ies ollow a S ackelbe g sequen ial game led by he pla o m o
accu a ely e lec he in e ac ion in he ma ke , allowing us o each equilib ium using backwa d induc ion. Ou
esul s indica e ha when app de elope s a e mo e isk a e se o ace g ea e unce ain y, hey spend a g ea e
amoun on non-sala y bene i s and compa a i ely less on app quali y. This inding highligh s he impo ance o
in es ing in wo ke s, pa icula ly in unce ain imes. We u he ex end he applicabili y and obus ness o ou
indings by in oducing mul iple de elope s o ou wo-agen sys em. The ex ension p o es ha he pla o m
cha ges a uni e sal commission a e, i espec i e o he numbe o de elope s –a inding ha is consis en wi h
cu en p ac ice. Gi en he non-linea e ec o key model pa ame e s on he p o i s o he supply-chain membe s
in bo h he single and he mul iple-de elope se ups, we also u ilize nume ical analyses and a i e a elling
manage ial implica ions o all pa ies.
1. In oduc ion
While base sala ies con inue o play a undamen al ole in a ac ing
capable employees o wo kplaces [15], ecen ly, non-sala y bene i s and
“pe ks”seem o ha e aken cen e s age. High- ech and
inno a ion-based companies egula ly publicize hei o e ings,
including leisu e and elaxa ion acili ies, quali y ca e ing and onsi e
wellness ac i i ies [7]. Fo example, Is aeli high- ech i ms o ganize
ex a agan pa ies and all-inclusi e aca ions o he Ca ibbean. Some
ech wo ke s epo edly app oach ec ui e s in an icipa ion o a se
s anda d o such bene i s when joining a company, sugges ing hei
capaci y o acili a e ec ui men e o s [17]. This s a egy co esponds
o he e e -g owing sho age o ech p o essionals [8], encou aging
high- ech i ms o di e si y hei spending on non-sala y bene i s and se
hemsel es apa om he es .
Fi ms ha de elop mobile and compu e apps (he ea e “apps”)
seem o ha e adop ed his app oach as high- ech en u es. Tinde , o
example, o e s i s employees legal assis ance, paid aca ions, and
aining men o ship, among o he bene i s.
1
Al hough e enue om
digi al apps was p ojec ed o each $430B in 2022 wi hin he mobile
sec o alone (wi h an a e age annual g ow h o app oxima ely 10 %),
2
as men ioned, he pool o alen wi hin he ech indus y emains
limi ed. The e o e, i could be wo hwhile o e ing non-sala y bene i s in
o de o a ac quali y employees, al hough cau ion needs o be exe -
cised when using such a s a egy gi en he unce ain na u e o he e ec
o such pe ks on employee pe o mance [17].
Due o he special ea u es o apps as i ual p oduc s [10], an app
de elopmen i m p ima ily incu s cos s ela ed o he quali y o he
p oduc . Mo e speci ically, spending on elemen s such as isual design,
unc ionali y, eliabili y, and secu i y usually aises he quali y o he
app, which is essen ial o i s compe i i e posi ioning [1]. Since human
de elope s can exe cise a subs an ial in luence on app quali y ( o
empi ical e idence o he phenomenon o so wa e de elope s, see
[16]), in addi ion o he a o emen ioned quali y-inducing elemen s, his
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (L.O. Maly).
1
h ps://www.glassdoo .com/Bene i s/Tinde -US-Bene i s-EI_IE916118.0,6_IL.7,9_IN1.h m.
2
h ps://www.s a is a.com/ou look/dmo/app/wo ldwide.
Con en s lis s a ailable a ScienceDi ec
Ope a ions Resea ch Pe spec i es
jou nal homepage: www.else ie .com/loca e/o p
h ps://doi.o g/10.1016/j.o p.2024.100320
Recei ed 23 July 2024; Recei ed in e ised o m 3 Decembe 2024; Accep ed 3 Decembe 2024
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
2
s udy conside s he e ec o non-sala y bene i s on app-quali y
achie emen .
De elope s o mobile apps commonly each hei end cus ome s by
o e ing he app ia a dis ibu ion pla o m (he ea e , “pla o m”). The
bigges pla o ms a e Apple’s App S o e and Google Play ( oge he ac-
coun ing o 95 % o he ma ke ou side China
3
). Wi h millions o apps
on each o hese pla o ms, bo h o e a uni e sal con ac o app de-
elope s, which s ipula es ha he pla o m will keep a de ined pe -
cen age o he e enue gene a ed by he app (15–30 %). The pla o m
acili a es he dis ibu ion e o s and billing p ocess, while he de-
elope s main ain owne ship o hei apps.
4
The e o e, he in e ac ion
be ween he wo pa ies esembles a consignmen con ac wi h e enue
sha ing [1].
This s udy models he business in e ac ion as a wo-agen sys em
consis ing o a pla o m (“i ”) and a de elope (“he/she”). Using a game-
heo e ic app oach, we cons uc he ollowing S ackelbe g game led by
he pla o m: Fi s , he pla o m se s he con ac e m, i.e., he pe -
cen age o he p oduc ’s e enue ha i cha ges as commission. Second,
he de elope makes wo decisions simul aneously –his/he in ended
le el o app quali y and he amoun spen on non-sala y bene i s o his/
he employees. The game’s na u e e lec s he in e ac ion in he app
ma ke , as de elope s ollow he pla o m’s exis ing e ms when
launching hei app on i (as in [2,4]). As sugges ed by p e ious s udies
[1,2,10], we only conside he cos o quali y incu ed by he de elope ;
he pla o m is assumed o ha e negligible ma ginal cos s. This
assump ion aligns wi h he cha ac e is ics o apps as digi al p oduc s
since hei incep ion o e a decade ago [9], allowing pla o ms o
imelessly-dis ibu e millions o apps
5
wi h ample capaci y o ul ill
demand. Since we assume ha non-sala y bene i s ha e an e ec on app
quali y –bu one ha is unce ain in na u e –we conside he de-
elope ’s a i ude owa ds isk in his/he objec i e unc ion.
We ocus on non-sala y bene i s in he o m o ini ia i es (such as
onsi e ac i i ies, pa ies, and ips) a he han common pe -employee
bene i s (such as insu ance o e i emen bene i s). Ou easoning is
wo old. Fi s , common pe -employee bene i s end o be ag eed upon
when hi ing each indi idual wo ke nowadays, and gi en he compe i-
i e ma ke o ech wo ke s, hey end o be ma ched ac oss he in-
dus y (simila ly o he base sala y).
6
The e o e, pe -employee bene i s
a e assumed o be ixed –unlike bene i s issued h ough ini ia i es,
which a e ex e nal o con ac s and can be designed wi h g ea e lexi-
bili y. Fu he mo e, he ac ha ini ia i e-based bene i s a e ex e nal o
con ac s means ha hey a e mo e likely o be able o a ec employee
pe o mance, pa icula ly since employees o en ecei e he bene i ee
o ax.
7
Ou second eason o ocusing on ini ia i e-based bene i s is
ha his s a egy has become inc easingly popula in ecen yea s,
pa icula ly in he high- ech sec o , such ha i is wo hwhile analyzing
he e ec o his app oach independen ly o o he o ms o compensa-
ion. No e, ha ou model ocuses on any ini ia i e-based non-sala y
bene i ha in ol es spending by he employe . These include indi ec
mone a y bene i s (e.g., ee conce s by leading a is s o luc a i e ex-
cu sions, see [17]) as well as in angible elemen s (such as p o iding
wo k lexibili y o new pa en s, o inducing a posi i e wo king a mo-
sphe e, e.g., he slides ins alled in many Google o ices, see [33]).
Gi en he ecen ise in popula i y o ini ia i e-based bene i s in he
o m o subs an ial spending [17], combined wi h he heo e ical
amewo k de eloped in his s udy, ou pape p o ides a unique and
impo an ou look on he po en ial alue o such bene i s bo h o
employee ec ui men and o he achie emen o quali y. To he bes o
ou knowledge, his s udy is he i s o in oduce non-sala y bene i s
in o a model o he in e ac ion be ween pa ies in a supply chain.
Using he mean- isk amewo k o model he de elope ’s isk a i-
ude (p o ed o be consis en wi h second-deg ee s ochas ic dominance
by [38]), we aim o answe he ollowing esea ch ques ions:
•How does he de elope ’s a i ude owa ds isk a ec he pa ies’
decisions, as well as hei expec ed p o i s, a equilib ium?
•How is he de elope ’s budge di ided, a equilib ium, be ween
in es ing in p oduc quali y and unding non-sala y bene i s o em-
ployees, and which ac o s a ec his alloca ion?
The es o he s udy is s uc u ed as ollows. Sec ion 2 e iews he
ele an li e a u e, he eby highligh ing he o iginal con ibu ion o his
s udy. In Sec ion 3, we cons uc he analy ical model o he wo-agen
sys em, while in Sec ion 4, we p esen i s esul s a equilib ium. Sec-
ion 5 ex ends he p ima y model o include mul iple compe ing de-
elope s, hus s eng hening he obus ness and applicabili y o ou
analysis. Las ly, Sec ion 6 expands upon he conclusions and manage ial
implica ions de i ed om ou esul s, and o e s po en ial di ec ions o
u u e s udy.
2. Li e a u e e iew
The p esen s udy ela es o h ee main a eas o esea ch: (a) non-
sala y compensa ion and i s e ec on employee pe o mance; (b) sup-
ply chains o i ual p oduc s; and (c) he mean- isk c i e ion. The
ollowing sec ions e iew he exis ing li e a u e unde each o hese
domains. Key ele an s udies a e mapped and classi ied by hei cha -
ac e is ics in he Au ho -con ibu ion able below Table 1.
2.1. Non-sala y bene i s and hei e ec on employee pe o mance
A mul i ude o empi ical s udies ha e ound a posi i e co ela ion
be ween wage and p oduc i i y, as i ms minimize hei labo cos s
while main aining e iciency [28]. Le ine [32] u he elabo a ed on his
hypo hesis, and p o ed empi ically ha he inc eased p oduc i i y
ollowing a sala y inc ease mo e han o se s he inc ease in labo cos s.
Mos ecen ly, Ma ie e al. [35] es ed he e ec o se e al ac o s on
wo ke p oduc i i y in a ious Russian i ms, including om he
high- ech sec o . In e es ingly, hey ound ha sala y was he mos
impo an ac o in luencing a wo ke ’s willingness o con ibu e. Ye ,
Fishe e al. [19] showed ha a salespe son’s pe o mance eaches
sa u a ion once hey a e paid beyond a speci ic h eshold.
In con as , li e a u e on non-sala y bene i s is a he limi ed.
Schmid -Sø ensen [40] in oduced non-sala y bene i s in o he basic
e iciency-wage model, on he basis ha hey we e becoming a mo e
p ominen sha e o labo cos s. His heo e ical analysis highligh ed he
complex e ec o such bene i s on p oduc i i y, which may explain why
la e s udies do no p oduce consis en indings. On he one hand, Gil-
ch is e al. [23] disco e ed, using an empi ical app oach, ha unex-
pec ed and uncondi ional bene i s gi en o wo ke s boos p oduc i i y in
a simila manne o hi ing addi ional wo ke s. An empi ical s udy o
Ame ican uni e si y s a showed ha college ui ion wai e s o e ed o
he wo ke s’dependen s inc eased bo h e en ion and p oduc i i y [41].
On he o he hand, Sung and Choi [42] ound ha when employe s und
ex e nal educa ion o wo ke s (as opposed o in e nal aining), he
o ganiza ion’s inno a ion pe o mance can de e io a e.
Ou s udy, o he bes o ou knowledge, is he i s o conside
ini ia i e-based bene i s in he high- ech sec o . Unde he assump ion
ha such bene i s ha e an unce ain ye no able e ec on p oduc i i y as
3
As Google Play is banned in China, see h ps://www.businesso apps.com/
da a/app-s o es/.
4
h ps://suppo .google.com/googleplay/and oid-de elope /answe /
112622?hl=en;h ps://de elope .apple.com/p og ams/wha s-included/.
5
By he las qua e o 2022, o e 3.5 million and 1.6 million applica ions
we e a ailable on Google Play and App S o e, espec i ely (h ps://www.s a is
a.com/s a is ics/276623/numbe -o -apps-a ailable-in-leading-app-s o es/).
6
h ps://www.be e up.com/blog/ ypes-o -employee-bene i s.
7
See, o example, he e ms in he USA: h ps://digi .business/ inancial-li e
acy/o ice-ch is mas-pa ies- inge-bene i s- ax; o he UK: h ps://www.go .
uk/ ax-company-bene i s/ ax ee-company-bene i s.
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
3
exp essed h ough app quali y, we use he mean- isk amewo k o
analyze he decisions o he de elope , who is an employe ac ing unde
unce ain y. Unlike he empi ical app oach aken by almos all p e ious
pape s on non-sala y bene i s (excep o [40]), ou model p o ides an
analy ical in es iga ion o he opic.
2.2. Supply chains o i ual p oduc s
The ollou o ad anced echnologies in he 21s cen u y esul ed in a
su ge o new ca ego ies o in angible goods, commonly classi ied as
i ual p oduc s [9]. Unlike angible p oduc s, which in ol e in en o y,
deli e y, dis ibu ion and o he cos ly p ocedu es, he supply o i ual
p oduc s is ca ied ou ins an aneously a negligible uni cos [3,4].
These unique ea u es ha e piqued he cu iosi y o nume ous e-
sea che s in he ields o ope a ions and supply-chain managemen .
Mos esea che s conside ed a wo-echelon chain, consis ing o a
manu ac u e (o en an app/so wa e de elope ) and a dis ibu e (e.g.,
a pla o m o e aile ), which ope a es unde a S ackelbe g game led by
ei he he dis ibu e [1–4,11,26] o he manu ac u e [10]. The in e -
ac ion has gene ally been analyzed unde ei he a ixed- ee con ac
( ele an o supply chains o mobile games; see [25,26]) o a
commission- a e con ac ( ele an o mobile apps; see [3]). Only a ew
s udies in oduced addi ional membe s in o he chain, such as an
in es o [1], ano he pla o m [12] o mul iple compe ing de elope s
[4].
A inada e al. [3] ecognized he e ec o quali y on demand in
such chains –along wi h he possibili y o in luencing his ela ionship
ia quali y in es men . Mos subsequen s udies conside ed he i ual
p oduc ’s p ice, and ei he he in es men in quali y o he desi ed le el
o quali y, as he sole in luence s o demand –whe e hese decisions a e
commonly assumed o be se by he manu ac u e . To p e en he sce-
na io whe e he in es men in quali y a equilib ium is in ini e, he cos
o quali y is usually conside ed o ake on a quad a ic o m (assuming
diminishing e u ns; see [1]). O he a iables ha ha e been conside ed
include he sales e o [25] and ma ke ing in es men [26]. Che nonog
[11] u he gene alized hese decisions in o a ec o o ac i i ies o be
pe o med, wi h he goal o in luencing ei he he e enue o he cos s.
The in ol emen o unce ain y in he majo i y o ope a ional de-
cisions [14] has led esea che s o inco po a e i in o hei heo e ical
analyses o supply chains o i ual p oduc s. Many esea che s ha e
in es iga ed unce ain demand by inco po a ing a andom a iable in o
he de e minis ic demand unc ion, using ei he addi ion o mul ipli-
ca ion (e.g., [2–4]). Using he mean- a iance (MV) c i e ion o ep esen
he a i udes o supply-chain membe s owa ds isk, mos au ho s ha e
epo ed ha he e aile ’s isk a i ude is i ele an o equilib ium de-
cisions due o he e aile ’s p o i s uc u e (e.g., [3]). Only Che nonog
[11] has in es iga ed unce ain y wi h espec o he cos o c ea ing
quali y. In pa icula , she assumed ha , unde in o ma ion asymme y,
he e aile es ima es he manu ac u e ’s cos unc ion. In o ma ion
asymme y has also been conside ed in o he ecen s udies o supply
chains o i ual p oduc s [2,26].
Ou s udy makes a unique con ibu ion o he exis ing li e a u e, as
we in oduce a new decision a iable – he non-sala y bene i s o e ed by
he de elope o his/he employees, which does no di ec ly a ec he
demand o he app. Speci ically, we cap u e he unce ain na u e o he
e ec o non-sala y bene i s on he cos o quali y (using a andom
a iable). In addi ion, ou model inco po a es mul iple de elope s
wi hin he supply chain, a scena io ha has a ely been in es iga ed in
p e ious publica ions (aside om [4]). No e ha ou decision o igno e
bo h he cos s and he isk a i ude o he pla o m is in line wi h he
indings o he a o emen ioned s udies.
8
2.3. The mean- isk c i e ion
Decision-making unde unce ain condi ions ine i ably e lec s he
decision-make ’s a i ude owa ds isk. I only he expec ed alue o he
decision’s ou come is conside ed, hen his e lec s isk-neu al
beha io , he eby igno ing he isk-a e se o isk-seeking pa e ns ha
equen ly cha ac e ize decision-make s. Ha ing made his obse a ion,
he Nobel lau ea e Ma kowi z c ea ed he MV c i e ion o inancial
decisions, which conside s he sp ead o he andom a iable in he o m
o i s a iance [13]. Walls and Dye [44] la e showed empi ically ha
he MV c i e ion success ully p edic s op imal decisions ac oss
indus ies.
In o de o ocus he analysis on nega i e ou comes (which a e
cen al o isk a i udes), esea che s o en eplace a iance wi h semi-
a iance cha ac e is ics, such as he s anda d de ia ion (SD) o abso-
lu e semi-de ia ion o he andom a iable. The use o such modi ied MV
c i e ia has been u he jus i ied by he inding ha hey a e analy i-
cally consis en wi h he ules o second-deg ee s ochas ic dominance
9
–
unlike he o iginal MV c i e ion [38]. Subs i u ing a iance wi h a
di e en cha ac e is ic o he andom a iable’s dis ibu ion (usually he
SD, as in he inancial calcula ion o VaR) is commonly e e ed o as he
mean- isk (MR) c i e ion [13].
O e ecen decades, s udies in ope a ions and supply-chain man-
agemen ha e inco po a ed he MV c i e ion in o hei analyses o de-
cisions unde unce ain y [13]. As men ioned p e iously, chains o
i ual p oduc s o en conside unce ain demand, and he as majo i y
o s udies ha e used he o iginal MV c i e ion (e.g., [2,3]). The in o-
duc ion o an in es o as a hi d playe in he supply chain led A inada
and Bunke [1] o disco e ha he mo e isk-a e se he de elope , he
Table 1
Au ho -con ibu ion able (N/A deno es no applicable).
Au ho s S ackelbe g
game
Nash
game
Ho izon al
compe i ion
Wholesale p ice
con ac
Re enue-
sha ing con ac
Linea cos
unc ion
Quad adic cos
unc ion
Unce ain cos
unc ion
Risk
in es iga ion
[3] V V V VMV
[4] V V V V V MV
[10] V VV UF
[26] V V VN/A
[1] V VVMV
[25]VV VMV
[11] V V V N/A
[2] V VVMV
Cu en
pape
V V V VV V MR
Risk in es iga ion legend: UF –U ili y Func ion; MV –Mean Va iance; MR –Mean Risk.
8
The pla o m is assumed o incu negligible dis ibu ion cos s [4]. Addi-
ionally, nume ous s udies wi h simila o mula ions ha e concluded ha he
pla o m’s a i ude owa ds isk has no e ec on decisions a equilib ium [1,3,4,
10].
9
Og yczak &Ruszczy´
nski [38] limi ed his conclusion o cases whe e he
" ade-o " coe icien λis smalle han 1, gi en wo possible unce ain al e -
na i es o he decision-make . Since ou analysis conside s only one, his
limi a ion does no apply o ou use o he c i e ion.
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
4
mo e likely he/she is o seek ex e nal unding.
Al hough s udies o supply chains o angible goods ha e e ol ed o
inco po a e MR analyses [14], esea ch on supply chains o i ual
p oduc s ends o adhe e o he o iginal MV c i e ion. Ou s udy is
o iginal in i s use o he MR c i e ion, while elying on i s p o en
analy ical c edibili y. The MR c i e ion has he po en ial o ease
analy ical p ocessing [31] and yield esul s ha a e mo e in elligible
han he basic MV c i e ion.
As men ioned, unce ain cos s in supply chains o i ual p oduc s
ha e only been in es iga ed in one p e ious s udy [11] and ha e ne e
been examined using he MR c i e ion. To de e mine whe he unce -
ain y has a posi i e o nega i e e ec on a company’s spending pa e ns
(in e ms o cos s), we e e o se e al empi ical s udies conduc ed on he
ma e . Mos o hese de ec ed a nega i e ela ionship, i.e., g ea e un-
ce ain y leads o cu bed spending –pa icula ly o high unce ain y
le els [6]. Mo eo e , g ea e isk-a e sion le els we e ound o exac-
e ba e his nega i e ela ionship [36]. The e o e, in he cu en s udy,
whe e he MR c i e ion is used o model he de elope ’s a ge unc ion,
he p oduc o unce ain y ( ep esen ed by he SD) and isk-a e sion
le els is sub ac ed om he expec ed alue. Thus, we employ he
same nega i e ela ionship as ha adop ed in p e ious MV applica ions
(i.e., E−λV), bu no e ha in hese applica ions, i was used o in oduce
unce ain y in o o he elemen s o he a ge unc ion (e.g., unce ain
base demand in [1]).
3. Model o mula ion
Conside a wo-agen sys em o an app consis ing o he app de el-
ope and a pla o m. Use s can ei he acqui e he app o ee o a a cos ,
whe e in he o me case, he app can gene a e e enue h ough one o
he ollowing me hods: subsc ip ion ees (e.g., a subsc ip ion o he New
Yo k Times); a paid, p emium e sion o he app; o in-app ads o pu -
chases. No e ha all no a ions used in his s udy appea in Table A in he
Appendix, and key assump ions a e summa ized in he ollowing
subsec ion.
The demand o he app is a ec ed by wo elemen s –i s quali y, q,
and he a e age e enue pe use (ARPU), p. Speci ically, he demand is
gi en by
D(q) = a
q
√−
α
p(1)
whe e ais he ma ke scale pa ame e , and
α
ep esen s he sensi i i y o
demand wi h ega d o p. In line wi h exis ing li e a u e, we use he
squa e oo unc ion o con ey he diminishing e u ns o consume s
om he app’s quali y le el [1]. The ARPU o he app ( ep esen ing he
e enue gene a ed by he app pe use om all i s po en ial sou ces) is
assumed o be exogenous, e lec ing he in ense compe i ion among
apps [39]. Logically, i has a nega i e linea e ec on demand (since he
ARPU is equi alen o he uni -p ice a iable used in [3,9], and [10]). By
using he ARPU o ep esen p, we accu a ely e lec eali y in he sense
ha apps end o make use o di e se e enue s eams, in line wi h he
cu en indus y s anda d. In con as , he a o emen ioned s udies (lis-
ed in he p e ious se o pa en heses) a i icially cons ained hei
models such ha only paid apps we e conside ed. Simila o hose
s udies, we in oduce he pa ame e
α
, which o paid apps simply s ands
o he p ice sensi i i y o demand. Fo ee apps, howe e , we assume
ha a highe ARPU would imply ha he use ’s wel a e is educed in a
simila ashion o when paying a highe selling p ice ( h ough mo e ads,
highe subsc ip ion ees, e c.), and he ex en o his e ec is depic ed
h ough
α
. No e ha he demand o he app is de e minis ic h oughou
ou analysis, despi e he a i y o such a scena io in eal li e. This
assump ion howe e allows us o ocus on unce ain y ied o cos
( a ely discussed in closely ela ed li e a u e, e.g., [11]), while main-
aining sa is ac o y esemblance o he beha io o an unce ain demand
unc ion.
The de elope , in o de o de elop he app and posi ion i compe i-
i ely on he ma ke , in es s in quali y incu ing he ollowing cos :
C(q, ) =
ε
ρ
q2(2)
whe e
ε
is a no mally dis ibu ed andom elemen wi h mean 1, is he
amoun ha he de elope spends on non-sala y bene i s, and
ρ
is a scale
pa ame e ep esen s he de elope ’s economic e iciency in ansla ing
he in es ed sum in o quali y. The cos o c ea ing quali y clea ly de-
pends on he desi ed app-quali y le el, q, and he assumed quad a ic
ela ionship e lec s he expec ed diminishing e u ns o quali y in-
es men s [18,34]. The e iciency in achie ing he desi ed quali y le el
has been assumed o be cons an by p e ious esea che s (e.g., [1,25]),
igno ing he possibili y ha i s alue migh change acco ding o he
decisions o he de elope . The e o e, ou model b eaks down he e i-
ciency in c ea ing quali y in o wo elemen s. Wi h espec o
ρ
, he
highe i s alue, he mo e cos ly i would be o achie e a desi ed quali y
le el. The ac ion
ε
/ co esponds o he p esumed e ec o non-sala y
bene i s on he c ea ion o app quali y by he de elope ’s employees. We
assume ha ( he amoun ha he de elope spends on non-sala y
bene i s) has a hype bolic e ec
10
wi h ega ds o he e iciency in
achie ing quali y, exp essed h ough he ecip ocal p esence o in Eq.
(2). Namely, he ma ginal wo ke e iciency gained om inc easing
when i is al eady high, is smalle han when is ini ially o a low alue
( ollowing he indings o Fishe e al., 2006
11
). Conside ing he
p e iously-discussed unce ain e ec o , we in oduce a no mally
dis ibu ed andom a iable o e he icini y o one, exp essed by
ε
∼
N(1,
σ
2). We u he assume ha
σ
≤0.33, such ha he p obabili y o
ε
being nega i e is su icien ly small o be dis ega ded. In so doing, we
e lec he assump ion ha bene i s o e ed o employees esul in some
posi i e ma ginal u ili y o he a e age wo ke .
Using a S ackelbe g non-coope a i e sequen ial game, we model he
in e ac ion be ween he wo membe s o he chain as p esen ed in Fig. 1.
The wo la ges pla o ms in he wo ld o apps –Apple’s App S o e and
Google Play –o e a non-nego iable con ac o app de elope s, which
Fig. 1. The sequence o e en s.
10
The hype bolic unc ion, exp essed h ough he appea ance o he a iable
in he ecip ocal, has been chosen o i s ollowing wo cha ac e is ics o
ep esen he ela ion be ween ( he amoun ha he de elope spends on non-
sala y bene i s) and he e iciency coe icien o c ea ing quali y: (a) I de-
c eases wi h espec o he a iable, i.e., g ea e spending on non-sala y bene i s
inc eases e iciency. (b) I s inc emen s a e smalle as he alues o he a iable
inc ease, simila o law o diminishing e u ns, i.e., when e iciency is low –
spending on non-sala y bene i s would imp o e i mo e signi ican ly han when
e iciency is al eady high, as a mo e subs an ial spending is equi ed o inc ease
i in he same manne . Fu he mo e, he unc ion allows o each analy ical
solu ions o ou heo e ical se up, leading o use ul conclusions and implica-
ions. Addi ionally, he hype bolic unc ion has been used p e iously in Ope -
a ions Managemen li e a u e o demons a e a simila e ec [24].
11
Thei esea ch is ele an only o a limi ed ex en , since hey did no ocus
on non-sala y bene i s. Howe e , hei inding o employee sa u a ion om
compensa ion i s he logic behind he law o diminishing ma ginal
p oduc i i y.
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
5
allows hese pla o ms o in e ac e ec i ely wi h hund eds o housands
o app de elope s. A he co e o he con ac lies he e enue-sha ing
ag eemen , which secu es he pla o m 15–30 %
4
o he e enue
gene a ed om he app. The e o e, he i s s age o he game is he
pla o m’s announcemen o i s desi ed ac ion
η
(0<
η
<1).No e ha
he only op ions open o he de elope a e o accep o ejec he con-
ac en i ely; hence, we only add ess he scena io in which he de el-
ope is willing o en e in o he ag eemen (simila o [3,4,25]). A he
second s age, he de elope de e mines bo h qand in p epa a ion o
his/he app being launched ia he pla o m. Las ly, once he app has
been launched, he demand is ealized and he e enue is spli be ween
he wo membe s o he supply chain. Deduc ion o he de elope ’s cos s
om his/he sha e o he e enue (as he only pa y o incu cos s in he
analysis) esul s in he ollowing p o i s o he pla o m and he
de elope , espec i ely:
π
p(
η
) =
η
pD(q)(3)
π
d(q, ) = (1−
η
)pD(q)− −C(q, ).(4)
No e ha appea s in he de elope ’s p o i unc ion as an inde-
penden cos (in addi ion o i s e ec on he a o emen ioned e iciency in
c ea ing quali y). The e o e, does no include common pe -employee
non-sala y bene i s (e.g., heal h insu ance, pension plans), which a e
ha dly adjus ed (i a all) by high- ech employe s due o hei commi -
men o ma ching he ma ke ’s hi ing s anda ds. Na u ally, sala y-
ela ed bene i s (such as bonus paymen s) adhe e o simila ma ke
p essu es [20], and consequen ly canno be iewed in he same manne
as , i.e., as an independen cos and decision a iable.
Fu he mo e, we assume ha issuing ini ia i e-based bene i s (e.g.,
cos ly and leng hy aca ions ab oad, [17]) does no aim o imp o e
ou going quali y di ec ly,
12
albei hei ega ded possible e ec on i s
cos . Sala y a es and sala y- ela ed bene i s, on he o he hand, a e
deployed p edominan ly as ins umen s o le e aging employee pe -
o mance
13
(i.e., o in luencing he quali y o he ou pu ). The e o e,
any spending ha is p ima ily in ended o imp o e quali y (including
inc eased sala ies, imp o ed equipmen , and aining) appea s unde
he cos o quali y C(q, )(and no unde ). To summa ize, ou unique
o mula ion cap u es all signi ican cos s associa ed wi h unning a
con empo a y app-de elopmen i m.
3.1. Key assump ions
•The pla o m incu s negligible ma ginal cos s.
•Demand is p opo ional o he squa e- oo o he quali y le el o he
app.
•The app’s ARPU is exogenously de e mined by ma ke o ces.
•The cos o quali y c ea ion is p opo ional o he squa e o he app’s
quali y le el.
•The de elope ’s spending on non-sala y bene i s has a hype bolic
e ec on he e iciency in c ea ing quali y.
•Ini ia i e-based bene i s do no aim o imp o e app quali y di ec ly.
4. Equilib ium esul s
Each playe ’s objec i e is o maximize i s own expec ed u ili y by
ine- uning i s espec i e decision a iables. We adop he MR c i e ion
as a su oga e u ili y unc ion o he de elope , who aces cos unce -
ain y, as i cap u es he p e e ence o a high expec ed p o i and a low
SD o he p o i . Since he pa ies’decisions a e made in wo s ages (see
Fig. 1), we use backwa d induc ion o each equilib ium. This well-used
me hod aligns wi h he na u e o he sequen ial game lead by he pla -
o m (as used by he as majo i y o esea che s in he ield, e.g., [2–4,
26]), namely, he pla o m se s he con ac e m by ex apola ing he
de elope ’s decisions (i.e., bes esponse) –in con as o a Nash Equi-
lib ium, in which he pa ies make hei decisions simul aneously (see
Sec ion 5).
4.1. Second s age o decision making: he de elope se s and q
As illus a ed in Fig. 1, he de elope makes his/he decisions only
a e he pla o m has announced he con ac e m, i.e., he de el-
ope is he second pa y o mo e in he sequen ial game. The de el-
ope se s ( he mone a y alue o non-sala y bene i s o his/he
employees) and q( he app’s desi ed quali y le el) simul aneously,
aiming o maximize he u ili y o his/he p o i , as cap u ed by he
mean- isk c i e ion. The c i e ion exp esses he desi e o he de el-
ope o ob ain a high expec ed p o i while, a he same ime,
a oiding unce ain y, which is cap u ed by sub ac ing a p opo ion
o he p o i ’s s anda d de ia ion om i s mean alue. As in p e ious
s udies using he MV c i e ion (e.g., [1]), λ(whe e commonly, |λ|≪1;
see, e.g., [3]) ep esen s he de elope ’s le el o isk a e sion (when
posi i e) o isk-seeking beha io (when nega i e), while
σ
s ands o
he SD o he andom elemen .
Fo simplici y o p esen a ion, we subs i u e he exp ession (1+λ
σ
)
wi h
ψ
, which is o posi i e alue h oughou ou en i e analysis (see
Appendix). Then, by inse ing
ψ
in o Eq. (4), alongside he demand
unc ion in Eq. (1) and he cos o quali y in Eq. (2), we a i e (as
de ailed in he Appendix) a he ollowing o mula ion o he de elope ’s
op imiza ion p oblem wi h ega d o he mean- isk o his/he p o i :
max
q, {MRd(q, )=(1−
η
)p(a
q
√−
α
p)− −1
ρψ
q2}.(5)
Since MRd(q, )is a conca e unc ion wi h a single local (which is
also a global) maximum o (q, ), we a i e a he ollowing p oposi ion:
P oposi ion 1.The de elope ’s bes - esponse is gi en by
q(
η
) = 1
ρψ
(ap(1−
η
)
4)2
, (
η
) = 1
ρψ
√(ap(1−
η
)
4)2
.(6)
P oo . See Appendix.
The decision a iables a equilib ium a e p opo ional wi h coe i-
cien
ρψ
√. I can be seen ha , al eady a his s age, he exp essions
depend di ec ly on all he pa ame e s o he analysis wi h he excep ion
o he p ice (equi alen ) sensi i i y o demand (
α
). The wo decision
a iables a e di ec ly p opo ional o a2,p2and (1−
η
)2(i.e., he squa e
o he de elope ’s e enue sha e). While q(
η
)is p opo ional o he in-
e se o
ρ
and
ψ
, (
η
)is p opo ional o he squa e- oo o hei in e se. A
la ge ma ke (a) o a g ea e ac ion o he e enue o he de elope
(1−
η
)logically leads o g ea e in es men s in bo h quali y and
employee bene i s, as well as g ea e e iciency in c ea ing quali y
(exp essed by a lowe alue o
ρ
)o g ea e ce ain y in he e iciency o
c ea ing quali y (a lowe alue o
σ
).
Since
ψ
>0 o bo h nega i e and posi i e alues o λ(i.e., o isk-
seeking and isk-a e se beha io s, espec i ely), ou analysis is obus
o bo h o hese scena ios. Acco ding o Eq. (6), he less isk-a e se he
de elope (which e e s o dec easing |λ|when λ>0) o he mo e isk-
seeking he de elope is (inc easing |λ|when λ<0), he g ea e his/he
in es men s in bo h quali y and employee bene i s.
No e ha he bes esponse q(
η
)is di ec ly p opo ional o p2. In ou
analysis, howe e , he de elope is me ely a ‘p ice- ake ’and has no
con ol o e he p ice. Tha is, he de elope adjus s his/he sou ces o
e enue so as o ma ch he ma ke ’s ARPU ( h ough he app’s selling
p ice, he in-app ad in ensi y, and he p ice o in-app pu chases). This
cha ac e is ic i mly e lec s he eali y obse ed in he as majo i y o
12
P ima ily, ini ia i e-based non-sala y bene i s a e epo edly issued wi h
he goal o ec ui ing " op quali y new employees while e aining exis ing ones"
[17].
13
h ps://www.w wco.com/en-NL/Insigh s/2021/12/compensa ion- ends-
spo ligh - ech-and-media.
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
6
app ma ke s, which seem o closely esemble pe ec compe i ion.
14
Ne e heless, we conclude ha a highe (lowe ) ma ke ARPU would
push he de elope o aise ( educe) his/he app’s quali y in o de o
ma ch he cus ome s’expec a ion o a highe (lowe ) u ili y om using
he app (a phenomenon p o ed empi ically o paid apps by [45]
15
).
4.2. Fi s s age o decision making: he pla o m se s
η
Based on he de elope ’s bes esponse (see Eq. (6)), he pla o m
ini ia es he game by se ing i s desi ed commission a e (0 <
η
<1),
while aiming o maximize i s p o i . This simula es he pla o m’s de-
cision making in eali y, se ing he con ac e ms o i s millions o app
de elope s, aiming o assess how hey would eac p io o making i s
inal decision. This es ima ion would ha e been conduc ed o se he
cu en e ms o bo h exis ing and new apps, s anding a 15–30 % on
Google Play and App S o e. Highligh ing he impo ance o he pla -
o m’s decision, amending exis ing e ms a ely akes place nowadays
due o i s signi ican e ec on he ma ke , al hough ecen ly bo h Apple
and Google decided o lowe he commission a e on hei espec i e app
s o es o he majo i y o de elope s om 30 % down o 15 % [21]. Thei
decision is a di ec esul o p edic ions on he de elope ’s bes - esponse
o new con ac e ms, p ima ily h ough his/he quali y in es men s
[1].
Unlike he de elope , he p o i o he pla o m is essen ially de e -
minis ic since all i s ac o s p esen ed in Eq. (3) a e independen o he
andom elemen
ε
. This comes as a esul o he de ini ion o he demand
unc ion as de e minis ic h oughou ou analysis, ocusing on unce -
ain y ela ed o he de elope ’s e iciency in c ea ing quali y (see Sec-
ion 3). Ne e heless, i s p o i is indi ec ly a ec ed by his o m o
unce ain y due o he sequen ial na u e o he game –u ilizing he
de elope ’s a o emen ioned esponse o make i s decisions, which
indeed is ied o he andom elemen ( h ough he in eg a ed pa ame e
ψ
, as indica ed in Eq. (6)). The e o e, i is possible o exp ess he pla -
o m’s p o i by simply inse ing he demand unc ion (1) in o Eq. (3),
while inco po a ing he de elope ’s bes esponse o q(
η
), gi en in Eq.
(6). Applying udimen a y algeb aic educ ions o he esul ing
exp ession leads he pla o m o sol e he ollowing maximiza ion
p oblem:
max
η
{
π
p(
η
)=p2
η
(a2(1−
η
)
4
ρψ
√−
α
)}.(7)
The pla o m’s p o i is a conca e unc ion o
η
. The e o e, a equi-
lib ium, i se s
η
o he alue o he single maximum o
π
p(
η
), as p e-
sen ed in he ollowing p oposi ion:
P oposi ion 2.A equilib ium,
η
∗=0.5−2
α
a2
ρψ
√.(8)
P oo . See Appendix.
4.3. Equilib ium esul s and discussion
Co olla y 1.A equilib ium, he pla o m s ipula es a commission a e ha
is <50 % o he de elope ’s e enue.
P oo . S aigh o wa d om P oposi ion 2.
In e es ingly, he commission a e a equilib ium illus a es a well-
es ablished p ac ice in he wo ld o apps. By Co olla y 1, he pla o m
would ne e se a commission a e in excess o 50 % o he de elop ’s
e enue, which is consis en wi h he eali y ha his a e ypically
anges om 15 o 30 %
4
. This inding u he es i ies o he applicabili y
o ou analysis.
Co olla y 2.A equilib ium, he pla o m’s eques ed commission a e
inc eases when
i. The ma ke scale pa ame e inc eases;
ii. The demand is less sensi i e o p ice (o equi alen );
iii. The de elope is mo e economically e icien in c ea ing quali y;
i . Unce ain y wi h ega d o he e ec o non-sala y bene i s on quali y
c ea ion is lowe ;
. The de elope is ei he less isk-a e se o mo e isk-seeking.
P oo . S aigh o wa d om P oposi ion 2.
The conclusions o Co olla y 2 seem o ollow common sense, and
esemble some o he indings o p e ious pape s (e.g., [1]). In essence,
he pla o m pe mi s i sel o cha ge a highe commission a e when he
de elope ’s ci cums ances a e be e , i.e., se ing mo e cus ome s,
se ing cus ome s who a e less sensi i e o p ice (o , in e changeably,
ARPU), enjoying g ea e ce ain y, o p oducing quali y mo e e i-
cien ly. Simila ly, he pla o m’s commission a e inc eases when he
de elope is ei he less isk-a e se (which would mean ha |λ|declines
when λ>0) o mo e isk-seeking (|λ|g ows when λ<0).This appa en
pu sui o a ai a e co esponds o he ac ha he de elope canno
en e in o nego ia ions wi h ega d o he pla o m’s eques ed com-
mission a e –bo h in ou analysis and in eali y5 (as well as in p e ious
li e a u e; see, e.g., [3,4]).
By inse ing he alue o
η
∗ om P oposi ion 2 in o he a o emen-
ioned exp essions ( ollowed by algeb aic manipula ions), we a i e a
he equilib ium alues o he de elope ’s decisions, he de elope ’s cos
o c ea ing quali y, and he pa ies’p o i s:
Co olla y 3.A equilib ium:
i. The de elope ’s decisions a e gi en by q∗=1
ρψ
(p(a2+4
α
ρψ
√)
8a)2
, ∗=
1
ρψ
√(p(a2+4
α
ρψ
√)
8a)2
;
ii. The de elope ’s expec ed cos o c ea ing quali y is gi en by E[C(q∗,
∗)] = 1
ψ
ρψ
√(p(a2+4
α
ρψ
√)
8a)2
;
iii. The pla o m’s p o i and he de elope ’s expec ed p o i a e gi en by
π
p(
η
∗) = 1
ρψ
√(p(4
α
ρψ
√−a2)
4a)2
,
E[
π
d(q∗, ∗)] = p2(a2+4
α
ρψ
√)(a2(3
ψ
−1)− 4
α
(5
ψ
+1)
ρψ
√)
64a2
ψ
ρψ
√;
i . The expec ed alue o he p o i o he channel is gi en by E[
π
ch(
η
∗,q∗,
∗)] = p2(a4(7
ψ
−1)− 8
α
ρψ
√(a2(5
ψ
+1)+2
α
(
ψ
+1)
ρψ
√))
64a2
ψ
ρψ
√.
P oo . S aigh o wa d om P oposi ion 2, along wi h Eqs. (5),(6)
and (7).
P oposi ion 3.A equilib ium, ∗/E[C(q∗, ∗)] =
ψ
; Thus, he ollowing
s a emen s apply o he unce ain y
σ
and he de elope ’s le el o isk sensi-
i i y λ:
i. These a e he only pa ame e s ha a ec he de elope ’s alloca ion o his/
he spending be ween p oduc quali y and non-sala y bene i s;
14
App ma ke s mee mos o he condi ions o pe ec compe i ion (see [39]):
millions o independen app de elope s and consume s; mos apps ha e a
mul i ude o nea ly iden ical compe i o s; de elope s and consume s ha e
abundan in o ma ion abou he apps (mos ly a ailable ia he pla o m); e y
ew ba ie s o en e /lea e he ma ke (h ps://www.in es opedia.com/ e
ms/p/pe ec compe i ion.asp, and [30]).
15
Zolkepli e al. [45] p o ed ha "Use s a e […] willing o pay o apps ha
ha e a highe a ing"; hese app s a - a ings ha e been epea edly used o
indica e use -pe cei ed quali y (see [37]).
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
7
ii. An inc ease (dec ease) in he alue o each pa ame e would esul in he
de elope alloca ing a la ge (smalle ) sha e o his/he in es men o non-
sala y bene i s.
P oo . See Appendix.
P oposi ion 3 e eals a no ewo hy cha ac e is ic ega ding he de-
elope ’s alloca ion decision, which lies a he co e o his s udy.
Al hough all equilib ium decisions and p o i s o bo h playe s (appea -
ing in Co olla y 3) depend on all o he model pa ame e s (p esen ed in
Table A in he Appendix), he de elope ’s alloca ion o esou ces (be-
ween non-sala y bene i s and he c ea ion o quali y) depends solely on
wo pa ame e s. The i s is he le el o unce ain y, pa icula ly wi h
ega d o he e ec o in es ing in non-sala y bene i s on he e iciency
o quali y c ea ion (as inco po a ed in Eq. (2)). In e es ingly, he lowe
he le el o ce ain y ega ding his e ec , he highe he de elope ’s
in es men in non-sala y bene i s (ins ead o quali y c ea ion). A plau-
sible analy ical explana ion o he de elope ’s beha io is he desi e o
inc ease his/he de e minis ic spending ( ) while educing he unce ain
cos o quali y c ea ion (C(q, )). The second pa ame e , which is he
de elope ’s le el o isk sensi i i y, seems o ollow a simila pa e n.
The mo e isk-a e se (o less isk- aking) he de elope , he lowe he
budge he/she alloca es o quali y c ea ion, p e e ing ins ead o in es
in non-sala y bene i s. Thus, he conse a i e de elope p e e s o in es
in his/he wo ke s, which is he sa e al e na i e o enhancing e i-
ciency in he long un. Ou indings a e consis en wi h e idence p o-
ided in a ious publica ions, highligh ing ha in es men in human
capi al (pa icula ly non-sala y bene i s) is a supe io ool o boos ing
p oduc i i y o capi al in es men s [22], pa icula ly in unce ain imes
[5].
Gi en he cen al in luence o bo h pa ame e s encompassed wi hin
ψ
( ≡ 1+λ
σ
)on equilib ium decisions, we he eby cons ain i s alue in
o de o ensu e ha he pla o m’s commission a e is a posi i e ac ion
(0 <
η
∗<1). Since Co olla y 1 gua an ees an uppe limi o he com-
mission a e o 0.5, i is only necessa y o limi he pa ame e s o i s
exp ession o ensu e ha
η
∗>0. Pe o ming udimen a y algeb aic
manipula ions on his inequali y, we s a e he ollowing condi ion:
ψ
<1
ρ
(a2
4
α
)2
.(9)
Co olla y 4.While he pla o m’s p o i a equilib ium is always pos-
i i e, he de elope ’s expec ed p o i is posi i e only when a >2
α
ψσ
√
√⋅
5
ψ
+1
3
ψ
−1
√and he channel’s expec ed p o i is posi i e only when a
>2
α
ψσ
√
√⋅
5
ψ
+1+4
ψ
(1+2
ψ
)
√
7
ψ
−1
√.
P oo . S aigh o wa d om he exp essions in Co olla y 3 and gi en
he condi ion in Eq. (9).
In line wi h he decision o dis ega d any cos s associa ed wi h he
pla o m (see subSec ion 2.2), by Co olla y 4, he pla o m will always
be posi i ely ewa ded as a esul o in e ac ing wi h he de elope . On
he con a y, he de elope ’s expenses on quali y c ea ion and non-
sala y bene i s lead him/he o lose when he abo e condi ion is no
me . Combining he p o i s o he wo, he en i e channel could also
expe ience losses, albei o a less igh condi ion (because he pla o m
always has a posi i e p o i ). Ul ima ely, he de elope will lose when
his/he ma ke appea s o be oo small (i.e., when ais lowe han he
minimum s ipula ed in Co olla y 4), al hough bo h he pla o m and he
channel may s ill be p o i able. The e o e, u u e s udies could conside
a scena io in which he pla o m u he lowe s i s commission a e in
o de o ensu e ha he de elope can be p o i able (e.g., by o e ing a
side paymen ), hus making i wo hwhile o he la e o ope a e unde
he non-nego iable con ac o he pla o m.
4.4. Nume ic explo a ion
Assuming ha he de elope ’s ope a ions a e p o i able (i.e., he
condi ion om Co olla y 4 is me ), we a e s ill unable o explo e
analy ically he e ec o
ψ
( ≡ 1+λ
σ
)on he playe s’p o i s (gi en he
Fig. 2. The e ec o
ψ
on he pla o m’s, he de elope ’s and he channel’s
expec ed p o i s. The se ing used is a=100, p=1,
α
=1 and
ρ
=0.05.
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
8
non-linea e ec o
ψ
on he p o i exp essions p esen ed in Co olla y 3).
The e o e, we pe o m a nume ical analysis o each exp ession ha
sa is ies he condi ion in Eq. (9) and mee s he addi ional condi ions in
Co olla y 4. As demons a ed in Fig. 2, we use a=100,p=1,
α
=1,and
ρ
=0.05 as he pa ame e alues ( esembling he nume ical analysis o
[1]) and a y he in eg a ed pa ame e
ψ
be ween i s ex eme alues
(0.67 <
ψ
<1.33; see Appendix). No e ha ou analysis also holds o
mo e ex eme pa ame e alues.
16
The pla o m’s p o i dec eases (albei nonlinea ly) wi h an inc ease
in
ψ
, meaning ha he pla o m achie es lowe p o i s as he de elope
becomes mo e isk-a e se (i.e., as |λ|g ows when λ>0 o as |λ|declines
when λ<0). This inding eplica es he esul s o mul iple p e ious
s udies o app supply chains (e.g., [4]). Howe e , he ela ion be ween
he pla o m’s p o i and he le el o unce ain y (pa icula ly he le el
o unce ain y wi h ega d o he e ec o non-sala y bene i s on he
e iciency o quali y c ea ion) is discon inuous. Speci ically, when he
unce ain y inc eases (i.e., highe
σ
)and he de elope is mo e
isk-a e se ( isk-seeking), he pla o m’s p o i will dec ease (inc ease).
Thus, he pla o m is ha med by non-sala y ambigui y when he de el-
ope is isk-a e se, bu bene i s om i when he de elope is
isk-seeking. This non-in ui i e esul implies ha he pla o m would
wish o inc ease he unce ain y aced by he isk-seeking de elope wi h
ega d o he e ec i eness o non-sala y bene i s. This could be achie ed
by he pla o m issuing such bene i s o i s own wo ke s, and hen
en e ing in o compe i ion o employees wi h he de elope . We he e-
o e ecommend ha u u e s udies should conside a pla o m ha also
ac s as an employe o app de elope s (simila ly o Apple and Google).
The impo ance o s udying his ex ension is u he jus i ied by
conside ing he social planne ’s pe spec i e, as he expec ed p o i o he
en i e channel is domina ed by he end o he pla o m’s p o i (i.e., he
channel’s expec ed p o i dec eases in
ψ
simila ly o he pla o m’s
p o i ).
Unlike he pla o m’s and he channel’s p o i s, he de elope ’s ex-
pec ed p o i beha es di e en ly depending on whe he he de elope is
isk-a e se (λ>0) o isk-seeking (λ<0). Peaking a ound isk-
neu ali y (λ=0 implying
ψ
=1, see Fig. 2(b)), he expec ed p o i
dec eases as he de elope aces g ea e unce ain y (highe
σ
) o as he
de elope adop s a mo e ex eme isk a i ude (highe |λ|). Thus, he
de elope could conside adop ing me hods o educe one (o bo h) o
hese pa ame e s, such as business in elligence o de e mine he non-
sala y bene i s o e ed by his/he compe i o s o objec i e decision-
suppo sys ems (DSS)
17
o allow he de elope o adop a mo e
neu al a i ude owa ds isk.
To s eng hen he alidi y o ou nume ic explo a ion abo e, we ha e
epea ed he analysis o six addi ional se ings o pa ame e alues. In
pa icula , we modi ied he alues o p,
α
and
ρ
by 50 % abo e and below
he o iginal alues used in his sec ion (i.e., p={0.5, 1.5},
α
={0.5, 1.5}
and
ρ
={0.025, 0.075}), while keeping he o he pa ame e alues ixed
(in o de o isola e he e ec o each pa ame e on he esul s). The e ec
o
ψ
on he pla o m’s, he de elope ’s and he channel’s expec ed
p o i s unde his ex ended analysis appea in Figu es A1-A6 in he
Online Appendix. Compa ing he esul s in Figu es A1-A6 wi h hose o
Fig. 2 shows ha ou main indings a e obus o such modi ica ions o
he pa ame e alues.
5. Model ex ension: mul iple de elope s
5.1. In oduc ion and ma ke con ex ualiza ion
Ini ia i e-based non-sala y bene i s a e g an ed p ima ily in o de o
a ac new wo ke s and e ain exis ing ones
12
, as a esponse o global
labo sho ages –pa icula ly in he high- ech sec o . Employe s a e
compelled o compe e o he exis ing pool o capable wo ke s, o e ing
uncon en ional bene i s in an a emp o win o e majo alen s.
The e o e, i is impe a i e o conside employe compe i ion when dis-
cussing non-sala y bene i s. Mo eo e , esea che s o supply chains o
i ual p oduc s ha e a ely discussed compe i ion be ween de elope s
(one o he ew s udies ha has conside ed such compe i ion is [4]) and,
o he bes o ou knowledge, ha e ne e conside ed compe i ion be-
ween de elope s wi h ega d o a ac ing employees.
We ex end he o iginal wo-agen sys em o a sys em o Napp de-
elope s who compe e o employees h ough he use o ini ia i e-
based non-sala y bene i s. No e ha h oughou he ollowing anal-
ysis, we use he same no a ions as hose p esen ed in Table A (see
Appendix), while in oducing he index i o deno e he pa icula app
de elope (ou o a o al o Nde elope s) o whom he a iable o
pa ame e applies. An addi ional adap a ion in ol es he de ini ion o
α
, which, om his poin on, deno es he a e age p ice-equi alen
sensi i i y o demand o all he o he (N−1) compe ing apps.
Las ly, we in oduce pa ame e β(0<β<1) o ep esen he in ensi y
o he compe i ion be ween app de elope s o a ailable employees; a
highe alue o βindica es ha he employees wo king o a pa icula
de elope a e mo e s ongly in luenced by he ini ia i e-based non-
sala y bene i s o e ed by compe ing de elope s (and ice e sa o a
lowe alue o β).
To allow o an elabo a e analy ical in es iga ion, we conside app
de elope s wi h simila alues o all he pa ame e s p esen ed in
Table A (see Appendix). Besides he uni e sali y achie ed hanks o
he analy ical na u e o ou explo a ion (a oiding he eso o case-
speci ic nume ical analyses, e.g., o speci ic company sizes and
ma ke s), ou concen a ion on de elope s wi h simila pa ame e
alues has been adop ed by simila s udies p e iously [4] as well as
depic s eal examples om he wo ld o apps (see Table 2). Acco d-
ingly, ou analysis ocuses on app de elope s who se e he same
ca ego y o ma ke (i.e., wi h simila alues o p, a, and
α
) and ha e
compa able in e nal o ganiza ion (simila
ρ
and λ). The le el o un-
ce ain y ( ep esen ed h ough
σ
), pa icula ly ega ding he e ec o
in es ing in non-sala y bene i s on he e iciency o quali y c ea ion,
is assumed o be iden ical o all de elope s gi en he lack o
conclusi e e idence on he ma e ac oss he en i e indus y, as well
as o gi es ise o explici esul s o he op imal solu ion. Since we
wish o isola e he e ec o employe compe i ion, in his ex ension,
we assume ha he apps do no incu a pu chase p ice (paid apps
accoun ed o <6 % o a ailable apps on Apple’s App S o e and
Google Play in 2023),
18
while he ARPU ep esen s he pe -use e -
enue om all non-p ice equi alen s (e.g., he a e age e enue pe
use om iewing in-app ads).
Ou se up closely esembles nume ous no able examples om he
wo ld o compe ing apps, and se e al o hem appea on Table 2. Spe-
ci ically, he i als on each ca ego y appea o sha e analogous cha -
ac e is ics ha align wi h ou assump ion o iden ical pa ame e s.
Fi s ly, all apps a e downloaded ee-o -cha ge and o e in-app pu -
chases o compa able sums (wi hin each ca ego y), suppo ing ou
assump ion o a sha ed p. When conside ed alongside wi h simila
a e age e enues pe use ,
19
i is possible o suppose ha he demand
aced by each app de elope beha es simila ly, i.e., wi h co esponding
ma ke scale (a) and p ice sensi i i y (
α
). Las ly, he compe ing apps
sha e a simila economic e iciency in c ea ing quali y (
ρ
), as e lec ed
16
We pe o med nume ical analyses using pa ame e alues much highe
han hose s a ed abo e, as well as alues close o hose equi ed o mee he
condi ion in equa ion (9) and he addi ional condi ions in Co olla y 4.
17
h ps://www.in es opedia.com/ e ms/d/decision-suppo -sys em.asp.
18
App oxima ely 95% o mobile apps (bo h in Google Play and in Apple’s App
S o e) a e ee o download: h ps://www.s a is a.com/s a is ics/263797/nu
mbe -o -applica ions- o -mobile-phones/
19
Commonly used o cha ac e ize he app’s ac i e use base: h ps://www.
apps lye .com/glossa y/a pu/
L.O. Maly and T. A inada
Ope a ions Resea ch Pe spec i es 14 (2025) 100320
15
maximizing he pla o m’s p o i , d
d
ηπ
p(
η
) = − Np2
2(a2(
η
−0.5)
ρψ
√+2
α
)=0, leads o
η
∗=0.5−2
α
a2
ρψ
√.
P oo o P oposi ion 6
i. Based on he esul s p esen ed in Co olla y 6(i) and 6(ii), he a io ∗
i
E[Ci(q∗
i, ∗
i)] =
ψ
1−β=1+λ
σ
1−β. The e o e, he balance be ween hese wo elemen s
en i ely depends on he deg ee o unce ain y – ep esen ed by he SD o he andom a iable (
σ
), he isk sensi i i y o he de elope (λ), and he
in ensi y o compe i ion among de elope s o employees (β).
ii. While he abo e a io is linea in he p oduc o
σ
and λ, i is a ising-hype bolic unc ion wi h espec o β(0<β<1).
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.
Re e ences
[1] A inada T, Bunke AE. The e ec o isk a e sion and inancing sou ce on a
supply chain o in-app p oduc s. In T ans Ope Res 2021;29(4):2145–71.
[2] A inada T, Che nonog T, Meilijson IE, Pe lman Y. A consignmen con ac wi h
e enue-sha ing be ween an app de elope and a dis ibu ion pla o m. In J P od
Econ 2022;243:108322.
[3] A inada T, Che nonog T, Pe lman Y. The e ec o isk sensi i i y on a supply
chain o mobile applica ions unde a consignmen con ac wi h e enue sha ing
and quali y in es men . In J P od Econ 2015;168:31–40.
[4] A inada T, Che nonog T, Pe lman Y. Consignmen con ac o mobile apps
be ween a single e aile and compe i i e de elope s wi h di e en isk a i udes.
Eu J Ope Res 2015;246(3):949–57.
[5] Banholze M, D eischmeie R, LaBe ge L, Liba ikian A. Business building: he pa h
o esilience in unce ain imes. McKinsey &Company 2022. h ps://www.mckinse
y.com/capabili ies/ isk-and- esilience/ou -insigh s/business-building- he-pa h-
o- esilience-in-unce ain- imes.
[6] Bo H, Lensin R. Is he in es men -unce ain y ela ionship nonlinea ? An empi ical
analysis o he Ne he lands. Economica 2005;72(286):307–31.
[7] B ad o d L. 13 ech companies ha o e cool wo k pe ks. Fo bes 2016.
h ps://www. o bes.com/si es/lau enceb ad o d/2016/07/27/13- ech-compa
nies- ha -o e -insanely-cool-pe ks/?sh=155ea02779d1.
[8] Ca lin D, Madga ka A, Mao D, Reich A. O e coming he ea ac o in hi ing ech
alen . McKinsey &Company 2022. h ps://www.mckinsey.com/capabili ies/peop
le-and-o ganiza ional-pe o mance/ou -insigh s/o e coming- he- ea - ac o -in-hi
ing- ech- alen .
[9] Che nonog T, A inada T. P o i c i e ia in ol ing isk in p ice se ing o i ual
p oduc s. Eu J Ope Res 2014;236(1):351–60.
[10] Che nonog T, A inada T, Ben-Z i T. How o se p ice and sales e o in a supply
chain o i ual p oduc s unde bi-c i e ia and isk conside a ion. In J P od Econ
2019;209:156–63.
[11] Che nonog T. S a egic in o ma ion sha ing in online e ailing unde a
consignmen con ac wi h e enue sha ing. Ann Ope Res 2021;300(2):621–41.
[12] Che nonog T, Le y P. Co-c ea ion o mobile app quali y in a wo-pla o m supply
chain when pla o ms a e asymme ic. Eu J Ope Res 2023;308(1):183–200.
[13] Chiu CH, Choi TM. Supply chain isk analysis wi h mean- a iance models: a
echnical e iew. Ann Ope Res 2016;240(2):489–507.
[14] Choi T, Chiu C. Risk analysis in s ochas ic supply chains: a mean- isk app oach
(In e na ional se ies in ope a ions esea ch &managemen science. 2012 h ed.
Sp inge ; 2012. p. 178.
[15] Dal B´
o E, Finan F, Rossi MA. S eng hening s a e capabili ies: he ole o inancial
incen i es in he call o public se ice. Q J Econ 2013;128(3):1169–218.
[16] Da a S. How does de elope in e ac ion ela e o so wa e quali y? an examina ion
o p oduc de elopmen da a. Emp So wa e Eng 2017;23(3):1153–87.
[17] Do i R. Ha e ech companies gone oo a wi h exo bi an pe ks o a ac
employees? CTECH; 2022. h ps://www.calcalis ech.com/c ech/a icles/0,7340,
L-3929178,00.h ml.
[18] El Oua dighi F, Kim B. Supply quali y managemen wi h wholesale p ice and
e enue-sha ing con ac s unde ho izon al compe i ion. Eu J Ope Res 2010;206
(2):329–40.
[19] Fishe M, K ishnan J, Ne essine S. Re ail s o e execu ion: an empi ical s udy. SSRN
Elec onic Jou nal 2006.
[20] F ancis T, Mickle T. Tech gian s u n o a classic ec ui men ool: cash. The Wall
S ee J 2022. h ps://www.wsj.com/a icles/ ech-gian s- u n- o-a-classic- ec ui
men - ool-cash-11645957980.
[21] Ga enbe g C. Google will educe Play S o e cu o 15 pe cen o a de elope ’s i s
$1M in annual e enue. The Ve ge 2021. h ps://www. he e ge.com/2021/3/16/
22333777/google-play-s o e- ee- educ ion-de elope s-1-million-dolla s.
[22] Ga on E. The case o in es ing mo e in people. Ha Bus Re 2018. h ps://hb .
o g/2017/09/ he-case- o -in es ing-mo e-in-people.
[23] Gilch is DS, Luca M, Malho a D. When 3 +1>4: gi s uc u e and ecip oci y in
he ield. Manage Sci 2016;62(9):2639–50.
[24] Hanuko G, A inada T, Che nonog T, Yechiali U. Pe o mance imp o emen o a
se ice sys em ia s ocking pe ishable p elimina y se ices. Eu J Ope Res 2018;
274(3):1000–11. h ps://doi.o g/10.1016/j.ejo .2018.10.027.
[25] Hu B, Meng C. The e ec o isk ole ance in mobile game supply chain p icing and
e o decisions. J Ope Res Soc 2021;72(10):2301–16.
[26] Hu B, Liu L, Meng C. In o ma ion sha ing in an online game supply chain. IEEE
T ans Eng Manage 2020. h ps://doi.o g/10.1109/TEM.2020.3040767.
[27] Kalia A. Newly single? A beginne ’s guide o he bes da ing apps. The Gua dian
2020. h ps://www. hegua dian.com/li eands yle/2019/dec/02/newly-single-a-
beginne s-guide- o- he-bes -da ing-apps.
[28] Ka z LF. E iciency wage heo ies: a pa ial e alua ion. NBER 1986. Re ie ed
Sep embe 27, 2022 om, h p://www.nbe .o g/books-and-chap e s/nbe -mac oe
conomics-annual-1986- olume-1/e iciency-wage- heo ies-pa ial-e alua ion.
[29] Küble RV, Pauwels K, Yildi im G, Fand ich T. App popula i y: whe e in he wo ld
a e consume s mos sensi i e o p ice and use a ings? J Ma k 2018;82(5):20–44.
[30] Kung M, Mon oe KB, Cox JL. P icing on he in e ne . J P oduc B and Manag 2002;
11(5):274–88.
[31] Lau HS. The newsboy p oblem unde al e na i e op imiza ion objec i es. J Ope
Res Soc 1980;31(6):525–35.
[32] Le ine DI. Can wage inc eases pay o hemsel es? Tes s wi h a p oduc i e
unc ion. Econ J 1992;102(414):1102–15.
[33] Lynley M. RANKED: he bes slides and wo s slides in Google’s o ices. Business
Inside 2012. h ps://www.businessinside .com/googles-o ice-slides-2012-5.
[34] Pe lman Y. Es ablishing a dual ood supply chain o o ganic p oduc s in he
p esence o show ooming –A game heo e ic analysis. J Clean P od 2021:128816.
[35] Ma ie O, Nagie a K, Pushka e A, Da idson N, Sohag K. E ec s o R&D spending
on p oduc i i y o he Russian i ms: does echnological in ensi y ma e ? Empi
Econ 2021;62(5):2619–43.
[36] Nakamu a T. Risk-a e sion and he unce ain y–in es men ela ionship: a no e.
J Econ Beha O ganiza 1999;38(3):357–63.
[37] Noei E, Sye MD, Zou Y, Hassan AE, Kei anloo I. A s udy o he ela ion o mobile
de ice a ibu es wi h he use -pe cei ed quali y o And oid apps. Emp So wa e
Eng 2017;22(6):3088–116.
[38] Og yczak W, Ruszczy´
nski A. F om s ochas ic dominance o mean- isk models:
semide ia ions as isk measu es. Eu J Ope Res 1999;116(1):33–50.
[39] Pe sas T, Papadogiannakis A, Polych onakis M, Ma ka os EP, Ka agiannis T. Rise o
he plane o he apps. In: P oceedings o he 2013 Con e ence on In e ne
Measu emen Con e ence; 2013.
[40] Schmid -Sø ensen JB. Non-wage labou cos s and p oduc i i y shi s in an
e iciency-wage-hou s model. Reche ch ´
Econ de Lou ain /Lou ain Econ Re 1991;
57(2):205–11.
[41] Spence M, Ge ek D, Chambe s V, Bowden R. Labo supply and p oduc i i y
esponses o non-sala y bene i s. Pe sonnel Re iew 2016;45(5):1047–68.
[42] Sung SY, Choi JN. Do o ganiza ions spend wisely on employees? E ec s o aining
and de elopmen in es men s on lea ning and inno a ion in o ganiza ions. J O gan
Beha 2013;35(3):393–412.
[43] Unal P, Temizel TT, E en PE. Wha ins alled mobile applica ions ell abou hei
owne s and how hey a ec use s’download beha io . Telema In 2017;34(7):
1153–65.
[44] Walls MR, Dye JS. Risk p opensi y and i m pe o mance: a s udy o he pe oleum
explo a ion indus y. Manage Sci 1996;42(7):1004–21.
[45] Zolkepli IA, Mukhia SNS, Tan C. Mobile consume beha iou on apps usage: he
e ec s o pe cei ed alues, a ing, and cos . J Ma ke Communica 2020;27(6):
571–93.
L.O. Maly and T. A inada
