Greenwashing risks in environmental quality competition: Detection and deterrence

Mukhe jee, A ka; Ghosh, Subhadip
A icle
G eenwashing isks in en i onmen al quali y compe i ion:
De ec ion and de e ence
Games
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Sugges ed Ci a ion: Mukhe jee, A ka; Ghosh, Subhadip (2025) : G eenwashing isks in en i onmen al
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Published: 11 Ma ch 2025
Ci a ion: Mukhe jee, A., & Ghosh, S.
(2025). G eenwashing Risks in
En i onmen al Quali y Compe i ion:
De ec ion and De e ence. Games,
16(2), 14. h ps://doi.o g/10.3390/
g16020014
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A icle
G eenwashing Risks in En i onmen al Quali y Compe i ion:
De ec ion and De e ence
A ka Mukhe jee and Subhadip Ghosh *
Depa men o Decision Sciences, School o Business, MacEwan Uni e si y, Edmon on, AB T5J 4S2, Canada;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : The ising p e alence o g eenwashing by i ms has eme ged as a majo con-
ce n o egula o y au ho i ies o e he pas decade. This pape examines he impac
o egula ion on i ms’ s a egic decisions ega ding g eenwashing and en i onmen al
quali y in an oligopolis ic ma ke . We model wo i ms ha compe e on en i onmen al
quali y and g eenwashing le els, ope a ing unde he o e sigh o a egula o y au ho i y.
The au ho i y’s policy ins umen s include a de ec ion mechanism and ines imposed on
i ms engaging in g eenwashing. Using a di e en ial game- heo e ical amewo k, we
examine he e ec i eness o egula o y in e en ions like de ec ion and penal ies in e-
ducing g eenwashing beha io and enhancing en i onmen al quali y. Addi ionally, we
discuss he pos -de ec ion ajec o ies o bo h i ms, p o iding insigh s in o he e ec s on
consume pe cep ions and ma ke compe i ion. We ind ha while egula ion can educe
g eenwashing as expec ed, i may also educe i ms’ en i onmen al quali y e o s. Indeed,
when penal ies a e su icien ly high, he ma ginal e u ns on in es men in g eenwashing
exceed hose om ac ual g een quali y imp o emen s.
Keywo ds: di e en ial game; g eenwashing; de ec ion; en i onmen al quali y; egula ion
1. In oduc ion
In ecen yea s, i ms ha e been pushed o make hei p oduc s mo e en i onmen ally
iendly due o ising consume in e es in sus ainable p oduc s and egula o y p essu es
o educe ca bon emissions. Howe e , in he ace o achie e a high le el o “g eenness”,
i ms ha e o en eso ed o exagge a ing he ue en i onmen al quali y o p oduc s
(Boncinelli e al.,2023;Szabo & Webs e ,2021). This is usually e e ed o as g eenwashing.
Fo mally, g eenwashing is “ he ac o misleading consume s ega ding he en i onmen al
p ac ices o a company o he en i onmen al pe o mance and posi i e communica ion
abou en i onmen al pe o mance” (Te achoice,2010).
Al hough g eenwashing is no a no el phenomenon, i s p e alence has inc eased
signi ican ly in ecen yea s, impac ing many indus ies, om ene gy o consume goods.
A ecen UK Compe i ion and Ma ke s Au ho i y e iew examined o e 500 websi es.
I e ealed ha o e 40 pe cen o employed g een ad e ising s a egies e iewed we e
misleading and po en ially iola ed consume p o ec ion laws (CMA,2021). Simila ly, an in-
es iga ion by he Eu opean Commission ound ha 42 pe cen o he g een claims assessed
we e ei he exagge a ed, alse, o decep i e, po en ially cons i u ing un ai comme cial
p ac ices (EC,2021).
The p e alence o g eenwashing has consequen ly led o a ising body o academic
li e a u e add essing i , mos o hem o e he las decade. A ecen comp ehensi e
Games 2025,16, 14 h ps://doi.o g/10.3390/g16020014
Games 2025,16, 14 2 o 17
sys ema ic e iew o he g eenwashing li e a u e obse ed ha nea ly 70 pe cen o he
310 a icles published by 2021 we e concen a ed in he p eceding i e yea s (2017–2021)
(San os e al.,2024). This su ge in academic a en ion e lec s he g owing ecogni ion o
g eenwashing as an impo an issue.
A subs an ial body o empi ical esea ch, along wi h a ela i ely lowe bu g owing
numbe o heo e ical s udies, has explo ed a ious dimensions o g eenwashing (Tang,
2018). These s udies span a ange o opics, om he d i e s and consequences o g een-
washing o i s impac on consume beha io , e hics, and egula o y esponses. Ou pape
con ibu es o he body o heo e ical esea ch on g eenwashing.
Se e al pape s ha explo e g eenwashing using ma hema ical modeling conside
s a egic in e ac ion be ween i ms, o which we will ci e only a ew. A de ailed li e a u e
su ey is p o ided by San os e al. (2024). One o he ea ly wo ks on g eenwashing
was by Shlei e (2004), who ound ha high compe i i e p essu e can lead o une hical
beha io like g eenwashing, which can lowe cos s associa ed wi h g een inno a ion o
quali y imp o emen s. Mo e ecen ly, Wu e al. (2020) examined he in e ac ion be ween
a po en ial g eenwashe and a genuinely g een i m in di ec p ice compe i ion. They
ound ha in o ma ion anspa ency plays a c i ical ole in ma ke s whe e g eenwashing
occu s, in luencing how consume s pe cei e en i onmen al claims and make pu chasing
decisions. Subsequen ly, J. Zhang and Yang (2022) o e ed addi ional insigh s, showing ha
g eenwashing i ms may educe p ices while imp o ing p oduc quali y in he p esence
o genuinely g een compe i o s, he eby a ac ing g ea e ma ke demand. Awas hy e al.
(2022) no ed ha g eenwashing by one i m nega i ely a ec s i s compe i o s. Howe e ,
when such nega i e “spillo e ” is minimal, he compe ing i m may s ill in es in posi i e
en i onmen al e o s, he eby inc easing i s g een quali y and p ices. Using a wo-pe iod
S ackelbe g game o a dual-channel manu ac u e – e aile se ing, Zong e al. (2022) ound
ha g eenwashing can be p o i able o manu ac u e s in a wo-pe iod game, hough i may
no bene i he o e all supply chain. They a gued ha g eenwashing, o “mis epo ing”,
has a minimal impac on p ices ac oss he supply chain.
Howe e , mos o hese pape s, and se e al o he s— o example, hose by Ce in
e al. (2023); Ruiz-Blanco e al. (2021); and Q. Zhang e al. (2020)—examine he e ec s
o g eenwashing pe o med by compe ing i ms in a ma ke wi hou he p esence o a
egula o y au ho i y. Howe e , mos de eloped coun ies and many eme ging economies
ha e implemen ed egula ions o add ess g eenwashing, wi h egula o y bodies imposing
penal ies when i ms a e caugh engaging in i . Fo ins ance, he U.S., he EU, he UK,
Aus alia, Canada, and China ha e all implemen ed legal amewo ks ha explici ly
p ohibi g eenwashing, subjec ing i ms ha engage in decep i e en i onmen al claims o
egula o y penal ies and sanc ions. The p esence o egula o y au ho i ies and he h ea o
de ec ion and subsequen penal ies al e he incen i es o he i ms o g eenwash.
One o he mos no able cases o g eenwashing de ec ion and en o cemen in ol ed
Volkswagen (VW), which aced widesp ead legal epe cussions a e he Fede al T ade
Commission (FTC) e ealed ha he company had manipula ed emissions es ing so wa e
in i s diesel ehicles. VW alsely p omo ed hese ehicles as low-emission and en i onmen-
ally iendly, leading o subs an ial ines and legal ac ions ac oss mul iple ju isdic ions
(Shepa dson,2016). In a mo e ecen case om 2021, Kohl’s and Walma we e ined
USD 5.5 million by he U.S. FTC o misleadingly ma ke ing hei bamboo-based ex iles
as eco- iendly despi e he p oduc s being manu ac u ed wi h en i onmen ally ha m ul
chemicals— iola ing he FTC’s G een Guides (FTC,2021). A simila inciden occu ed
in Canada, whe e Keu ig Canada was ined CAD 3 million by he Compe i ion Bu eau
o making alse en i onmen al claims abou i s K-Cup pods, misleading consume s by
Games 2025,16, 14 3 o 17
ad e ising hem as ecyclable in municipali ies whe e ecycling acili ies did no accep
he pods (Compe i ion-Bu eau-Canada,2022).
Gi en he impo ance o he e ec o egula ion on g eenwashing, we y o conside
how he p esence o such an agency a ec s he incen i e o i ms o g eenwash. The e ec
o some o m o egula ion on g eenwashing has been discussed by Huang e al. (2020); Lee
e al. (2018); Sun and Zhang (2019); and Wang e al. (2022). F om a supply chain pe spec i e,
Lee e al. (2018) demons a ed ha egula ing g eenwashing in compe i i e supply chains
can some imes unde mine i ms’ CSR (Co po a e Social Responsibili y) e o s due o high
cos s. In e es ingly, hei indings sugges ha pe mi ing some le el o g eenwashing may
p o ide incen i es o ce ain i ms o in es mo e in genuine en i onmen al quali y, as i
allows hem o balance cos p essu es while s ill compe ing on g een a ibu es. Howe e ,
ou ocus is on he e ec o egula ions on he p o i s o he i ms, he wel a e and pe cep ion
o inal consume s, and on he le el o g eenwashing and en i onmen al quali y. Ano he
no ewo hy s udy in his ield was by Sun and Zhang (2019), who employed e olu iona y
game heo y o analyze he e ec i eness o go e nmen egula ions in con olling g een-
washing ac oss i ms wi h di e en le els o ma ke dominance. Thei indings sugges
ha while go e nmen -imposed penal ies a e e ec i e in de e ing g eenwashing among
dominan en e p ises, ax subsidies alone may no be su icien o p e en g eenwashing,
pa icula ly among smalle o in e io en e p ises. They assumed g eenwashing is always
de ec ed by egula o s, which is o en un ealis ic in p ac ice. In con as , ou pape assumes
he p obabilis ic de ec ion o g eenwashing by egula o y bodies, wi h iden ical i ms,
making no a bi a y dis inc ion be ween hose engaging in g eenwashing and genuinely
g een i ms. Ou amewo k is also di e en in ha we use a di e en ial game heo e ic
amewo k. La e , Huang e al. (2020) used a S ackelbe g game wi h an incumben and an
en an i m o in es iga e how compe i i e p icing s a egies a e a ec ed by g eenwashing,
consume beha io , and an i-g eenwashing en o cemen . They ound ha mo e lax en-
o cemen o g eenwashing can some imes pa adoxically lead o highe social wel a e and
ha g eenwashing can bene i bo h incumben g een i ms and o e all cus ome su plus
when he ma ke ’s g eenness gap is small. In con as , we conside a di e en ial game
amewo k whe e he i ms mo e simul aneously, and he i ms a e symme ic, wi h no
a bi a y dis inc ion be ween g een and g eenwashing i ms.
Ano he ecen s udy ha examined he e ec o egula ion on g eenwashing was
conduc ed by Wang e al. (2022). They employed a h ee-s age Bayesian game o examine
how alse-claims ban (FCB) egula ions in luence i ms’ g eenwashing beha io when hey
possess imp ecise in o ma ion abou hei p oduc s’ en i onmen al impac . Thei indings
sugges ha while s ic FCB egula ions a e e ec i e in cu bing in en ional g eenwashing,
hey may also lead o unin o ma i e non-g eenwashing i he p ecision o g eenness
in o ma ion acqui ed by i ms is lowe han a ce ain h eshold le el. Thei model also
assumes a bina y s a e, ei he g een o non-g een. In con as , ou s udy assumes ha he
deg ee o g eenwashing a ies dynamically based on a ange o model pa ame e s.
Mo e ecen ly, he e ec o egula ion on g eenwashing was s udied by Mukhe jee
and Ghosh (2024) in a di e en ial game con ex . Howe e , hey conside ed a duopoly
ma ke wi h one g eenwashe and one g een i m. They also assumed a cons an penal y
o g eenwashing. In con as , we conside a scena io whe e no p ede ined dis inc ion
exis s be ween i ms, allowing bo h o po en ially engage in g eenwashing. Mo eo e ,
we assume ha he penal y is a p opo ion o he i m’s e enue, which is how penal ies
a e imposed on a ious coun ies, like Canada and he EU. Fo example, unde Canada’s
new Bill C-59, co po a ions ound guil y o g eenwashing may ace penal ies o up o
CAD 10 million o a i s o ense, CAD 15 million o subsequen o enses, h ee imes
he alue o he bene i de i ed om he decep i e conduc , o i ha amoun canno be
Games 2025,16, 14 4 o 17
easonably de e mined, 3 pe cen o he co po a ion’s annual wo ldwide g oss e enues
(Compe i ion-Bu eau-Canada,2024).
Be o e del ing in o ou model, we would like o poin ou ha , in a b oade sense,
ou esea ch p oblem examines how o con ol pollu ion unde he isk o a change in
egime. Al hough his ield has a ich and ex ensi e body o esea ch, we will discuss only
a ew s udies o he sake o b e i y. One o he ea lie wo ks in his ield was by Tsu and
Zemel (1998), who, using a no el h
λ
me hod, examined how he h ea o en i onmen al
ca as ophes, anging om e e sible o i e e sible e en s, a ec s decisions abou pollu-
ion con ol. They ound ha e e sible e en s consis en ly lead o g ea e conse a ion
(lowe pollu ion) as planne s mi iga e isks. In con as , i e e sible e en s will lead o
lowe pollu ion only unde ce ain speci ic condi ions. Nkuiya (2015) also con ibu ed o
he li e a u e by examining how he isk o inc eased en i onmen al damage in luences
he s a egic emissions decisions o coun ies. Using a non-coope a i e dynamic game
amewo k, his s udy demons a ed ha in e na ional coope a ion in pollu ion con ol
yields subs an ial bene i s. Fu he mo e, i highligh ed he impo ance o policymake s
accoun ing o he possibili y o sudden egime shi s when o mula ing and implemen ing
en i onmen al policies.
In a mo e ecen a icle, G omo e al. (2024) examined op imal pollu ion con ol
p oblems in he con ex o a ying en i onmen al condi ions. They p o ided an excellen
e iew o he exis ing li e a u e on his s eam o esea ch. In addi ion, hey ound ha
op imal solu ions con e ge o a hyb id limi cycle, which balances p o i maximiza ion
wi h en i onmen al p ese a ion. Thei indings sugges ha adjus ing he discoun a e
can be a egula o y ool o encou age sus ainable p ac ices among i ms.
In his pape , we conside wo simila i ms compe ing in a di e en ial game ame-
wo k wi h espec o en i onmen al quali y and he g eenwashing le el. The i ms ope a e
in a ma ke go e ned by a egula o y body wi h an en o cemen mechanism. I he agency
de ec s g eenwashing by a i m, i mus pay a penal y in p opo ion o i s e enue ea ned
in he i s pe iod. Unde such a si ua ion, we examine he e ec o g eenwashing on each
i m’s s a egies, ou comes, and consume pe cep ion. Ou objec i e was o in es iga e he
ollowing esea ch ques ions:
1.
Wha a e he equilib ium en i onmen al quali y and g eenwashing e o s o he i ms?
2.
How do he en i onmen al quali y and g eenwashing le els o each i m
i
a y wi h
he penal y mul iplie ρiand likelihood o de ec ion χ?
3. How do he i ms’ decisions a y wi h consume us ?
4.
Is e u n on in es men on en i onmen al quali y highe han ha on g eenwashing?
In add essing ou esea ch ques ions, a di e en ial game is he mos app op ia e
because i cap u es he dynamic e olu ion o pe cei ed quali y de e mined by each i m’s
ongoing decisions abou en i onmen al quali y and g eenwashing. This amewo k models
he s a egic compe i ion o e ime, whe e each i m’s ac ions impac no only i s own
demand unc ion bu also he compe i o ’s. Piece-wise cons an s a egies e lec he eali y
o i ms adjus ing decisions pe iodically a he han con inuously.
Ou indings sugges ha a high isk o exposu e o a high penal y may educe
g eenwashing and s op i al oge he , bu de e io a e he en i onmen al quali y o he
p oduc s as well. Howe e , a low penal y and ma ke wi h conscious consume s can esul
in highe en i onmen al quali y. Lack o consume us may also educe he en i onmen al
quali y pos g eenwashing.
Ou main con ibu ions a e he ollowing:
•
As opposed o s a ic games, we model a di e en ial game ha cap u es he dynamic
na u e o pe cei ed quali y and i s e olu ion o e ime, being in luenced by he i ms’
en i onmen al quali y and g eenwashing.

Games 2025,16, 14 5 o 17
•
While he p e ious li e a u e has ocused on egula ions on g eenwashing, we p opose
and sol e a model whe e i ms compe e while an icipa ing he isk o exposu e and
inco po a e his isk in hei decisions.
•
Con a y o in ui ion and compo ing wi h Lee e al. (2018) we show ha oo s ingen
egula ions on g eenwashing in an e ol ing ma ke may be de imen al o he o e all
en i onmen al quali y o he i ms’ p oduc s.
•
We also show ha he e u n on in es men s o g eenwashing may be highe han
ha o en i onmen al quali y unde ce ain condi ions.
In he nex sec ion, we de ine he model. The ea e , in Sec ion 3, we discuss he
equilib ium s a egies and hei implica ions. Sec ion 4is dedica ed o nume ical analysis
and some impo an obse a ions. We conclude he pape in Sec ion 5.
2. Model
2.1. Demand Func ions and S a e Va iables
We conside a duopoly ma ke whe e wo i ms compe e on he en i onmen al quali y
o hei p oduc s. In addi ion, he i ms g eenwash o enhance he pe cei ed quali y o
hei p oduc s o he consume s, he eby gaining a compe i i e ad an age. We model he
abo e si ua ion using a con inuous- ime di e en ial game model. The i ms decide on he
en i onmen al quali y
qi( )
o he p oduc s and g eenwashing le el
wi( )
, whe e
i∈ {
1, 2
}
is he index o he i m. The en i onmen al quali y o a p oduc can be jus i ied as a con ol
a iable in a di e en ial game model because i ms ha e di ec in luence o e i h ough
decisions such as in es ing in cleane echnologies o sou cing sus ainable ma e ials. Fo
example, a i m can inc ease en i onmen al quali y by adop ing eco- iendly packaging
o by educing emissions in he p oduc ion p ocess, ac ions ha di ec ly impac he i m’s
cos dynamics and compe i i e posi ioning o e ime. Mo e speci ically,
qij( )
deno es he
e o s he i ms make o imp o e en i onmen al quali y ( o example pollu ion con ol),
and
wij( )
deno es he g eenwashing e o s ( o example, misleading ad e ising). A
simila use and in e p e a ion o hese decisions can be ound in he wo ks by Dockne and
Van Long (1993) and Xue and Wang (2024).
Ou demand unc ion is simila o ha o Mukhe jee and Chauhan (2021). Speci ically,
he demand unc ion o i m iis exp essed as ollows:
Dij( ) = αi+βij(Gij( )−G(3−i)j( )) (1)
whe e
αi
is he ma ke po en ial,
j
is he index o he ime pe iod, and
βij
deno es he
sensi i i ies o he p oduc di e en ia ion in e ms o en i onmen al a ibu e. The decision
a iables o he wo i ms in he wo pe iods a e en i onmen al quali y
qij( )
and he
g eenwashing e o s
wij( )
,
i∈ {
1, 2
}
. The s a e a iables o ou model a e he pe cei ed
en i onmen al quali ies o he i m gi en by
Gij( )
,
i
,
j∈ {
1, 2
}
. The s a e a iables a e
posi i ely a ec ed by he en i onmen al quali y decisions
qij( )
and he g eenwashing
decisions
wij( )
. In essence, he s a e a iable e lec s consume s’ pe cep ion o a p oduc ’s
o e all en i onmen al quali y. This pe cep ion is shaped by bo h genuine e o s o imp o e
en i onmen al quali y and decep i e p ac ices (g eenwashing) by he i ms, which c ea e a
misleading imp ession o en i onmen al quali y. The e olu ion o he s a e a iables a e
gi en by he ollowing di e en ial equa ions:
dG1j
d =l1jw1j( ) + k1jq1j( )−δ1jG1j( ),j∈ {1, 2},G11(0) = G10
dG2j
d =l2jw2j( ) + k2jq2j( )−δ2jG2j( ),j∈ {1, 2},G21(0) = G20. (2)
Games 2025,16, 14 6 o 17
In he abo e equa ions,
kij
and
lij
a e he ma ginal e ec s o en i onmen al quali y
and g eenwashing on he pe cei ed en i onmen al quali y e olu ion o i m
i
in egime
j
.
δij is he decay in he pe cei ed g een quali y o i m iin egime j.
De ec ion o g eenwashing: The i ms an icipa e ha he e is a isk o being exposed,
and i de ec ed a a andom ime
τ
, he egula o y body will impose a penal y. We assume
ha he ime o de ec ion is a andom a iable ollowing a exponen ial dis ibu ion. Thus,
we ha e an associa ed haza d a e χ, which is de ined by
lim
d →0
P n ≤τ< +d |τ≥ o
d =χ. (3)
I he i ms do no g eenwash, hen
χ=
0 and
w1j( ) =
0, and he e o e, he e is only
one egime. This de ec ion ime spli s he decision ho izon in o wo egimes—
[
0,
τ]∪(τ
,
∞)
,
whe e
[
0,
τ]
is he pe iod when he g eenwashing is no de ec ed, and
(τ
,
∞)
is he pe iod
when i is de ec ed and he penal y is imposed. I g eenwashing is de ec ed, he i ms do
no g eenwash in he second egime.
2.2. The P o i Maximizing P oblems o he Fi ms
In no ing ha
j
is he index o he ime pe iod, he ins an aneous p o i s o i m
i
in egime
j
a e gi en by
πij( ) = Dij( )mij −Cij( )
, whe e
mij
is he uni p o i ma gin
(di e ence be ween p ice and p oduc ion cos ), and
Cij( )
deno es he o he cos s o i m
i
, like he cos s o g eenwashing, cos s o en i onmen al quali y, o he penal y cos i he
i m is caugh o g eenwashing. The e m
mij
is cons an o e e y egime
j
. I can happen
ha mi1=mi2.
Penal y
(Pi)
:The penal y o g eenwashing can be in se e al o ms. Acco ding o Fo bes
(McGowan,2024), such a penal y can be a pe cen age o g oss e enue, a ixed cons an
(like USD 10 million), o a mul iple o he e enue made om he sales o he i ems wi h
misleading ad e ising. While mos o he li e a u e conside s a ixed penal y, in his model,
we conside ha he penal y is a p opo ion o he i s pe iod’s e enue o i m
i
, and
his is deno ed by
ρi∈[
0, 1
]
. The e o e, he penal y
Pi=ρimi1Rτ
0Di1( )
, whe e
mi1
is he
p o i ma gin o i m
i
in egime 1. We highligh ha i he i ms a e exposed, in he second
egime,
τ
will be known o he i ms and he alues o
Di1( )
a e also known. The e o e,
he penal y
Pi=ρimi1Rτ
0Di1( )
is jus a numbe and is ea ed me ely as a cons an e m
o he solu ion o he second pe iod’s game.
The ins an aneous p o i unc ions o i m iin egime j(πij( )) is
πij( ) = Dij( )mij −µ
2q2
ij( )−ω
2wij( )2,wi2=0. (4)
We conside an in ini e ho izon egime swi ching di e en ial game whe e he e a e
wo egimes. The i s egime is when he i ms g eenwash, and egime 2 s a s when
he i ms a e exposed a andom ime
τ
. We a e in e es ed in he long- e m decisions and
p o i s o he i ms. Such games a e widely used in he applica ions o managemen science
and economics (Dockne e al.,2000). One s anda d way o sol ing such a game is o use
backwa d induc ion as we do he e. We s a wi h he second egime’s p oblem. The second
egime’s long- e m p o i s a e gi en by
Ji2(G12,G22) = Z∞
τe− πi2( )d −Pi, gi en Gi2(τ) = ˆ
Gi(5)
Games 2025,16, 14 7 o 17
and he p o i maximiza ion p oblem o he i ms a e gi en by
Vi2(G12,G22) = Max
qi2Z∞
τe− πi2( )d −Pi, gi en Gi2(τ) = ˆ
Gi(6)
whe e
(6)
is subjec o he s a e equa ions gi en by
(2)
. He e,
πij( )
is he ins an aneous p o i
o i m
i
in egime
j
,
Jij( )
is he long- e m p o i o i m
i
in egime
j
, and
Vij(Gij
,
G(3−i)j)
is
he alue unc ion o i m iin egime j.
De i a ion o i s pe iod’s p o i : A he beginning o he i s pe iod, he possible ime
τ
o he de ec ion o g eenwashing is andom, and he i ms do no know ex an e he da e
τ∈[
0,
∞)
. Consequen ly he ne p esen alue o p o i
J1
is also a andom a iable, and
he expec ed alue o p o i a he beginning o he planning ho izon ( = 0) is gi en by
Ji1(G11,G21) = EZτ
0e− πi1( )d +e− τ(Ji2(G12,G22)−Pi)(7)
whe e he expec a ion
E[
.
]
is o e he s ochas ic p ocess o he de ec ion o g eenwashing.
Ji1(G11,G21) = EZ
0e− sπ1(s)ds +e− Ji2−ρimi1Z
0Di1( )d 
=Z∞
0Z
0e− sπi1(s)ds +e− Ji2−ρimi1Z
0Di1( )d χe−χ d
=Z∞
0χe−χZ
0e− sπi1(s)dsd +χZ∞
0e−( +χ) Ji2d −ρ1mi1Z∞
0χe−χ Z
0Di1( )d
≜I1+χI2−ρimi1I3. (8)
To e alua e he h ee in eg als, we in eg a e by pa s. S a ing wi h
I1
and le ing
U=R
0e− sπi1(s)ds and V=−e−χ , we ob ain dU =e− πi1( )d ,dV =χe−χ , and
I1=Z∞
0Z
0e− πi1(s)dsχe−χ d ,
= (−Ue−χ )|∞
0−Z∞
0(e−χ )(e− πi1( ))d ,
=0+Z∞
0e−( +χ) πi1( )d .
The abo e esul s ollows om
(−Ue−χ )|∞
0=
0 since
lim →∞e−χ =
0 and
U(0) = R0
0e− sπi1(s)ds =0. Simila ly, we ha e
I3=Z∞
0e−( +χ) Di1( )d .
The e o e, (8) becomes
Ji1=I1+χI2−ρimi1I3
=Z∞
0e−( +χ) πi1( )d +χZ∞
0e−( +χ) Ji2d −χρimi1Z∞
0e−( +χ) Di1( )d
=Z∞
0e−( +χ) (πi1( ) + χ(Ji2−ρimi1Di1( )))d ,
Games 2025,16, 14 8 o 17
Consequen ly, he o e all op imiza ion p oblem is as ollows:
Vi1(G11,G21) = max
qi1,wi1Z∞
0e−( +χ) Di1( )mi1( )−µ
2q2
i1( )−ω( )
2w2
i1(9)
+χVi2(G12,G22)−ρimi1Di1( ))d
Subjec o
˙
Gi1( ) = ki1qi1( ) + li1wi1( )−δi1Gi1( ), and Gi1(0) = G0
No e ha he i s pe iod’s s a e a iable con ains g eenwashing
wi1( )
along wi h
en i onmen al quali y qi1( ).
3. Discussion o Equilib ium S a egies
We use he Hamil on–Jacobian–Bellman (HJB) equa ions o sol e he abo e p oblems.
HJB Equa ions
(11)
and
(12)
a e linea quad a ic and such linea quad a ic games ha e
been ex ensi ely s udied in he li e a u e unde di e en con ex s. I can be shown ha he
alue unc ions o such games a e linea quad a ic (Dockne e al.,2000). We posi ha he
alue unc ions will be o he o m
Vij(Gij,Gij) = XijG1j+YijG2j+Zij (10)
whe e
Xij
,
Yij
, and
Zij
a e he cons an coe icien s. The second pe iod’s HJB equa ions a e
V12(G12,G22) = max
q12 hD12( )m12( )−µ
2q2
12( ) + ∂V12
∂G12
˙
G12( ) + ∂V12
∂G22
˙
G22( )
−ρ1m12 Zτ
0D11( )d ]
V22(G12,G22) = max
q22 hD22( )m22( )−µ
2q2
22( ) + ∂V22
∂G12
˙
G12( ) + ∂V22
∂G22
˙
G22( )
−ρ2m21 Zτ
0D21( )d ](11)
F om Equa ion (9), he i s pe iod’s HJB equa ions a e gi en by
( +χ)V11(G11,G21) = max
q11,w11"D11( )m11( )−µ
2q2
11( )−ω
2w2
11( ) +
∂V11
∂G11
˙
G11( ) + ∂V11
∂G21
˙
G21( ) + χV12(G12,G22)−ρ1m11D11( )
( +χ)V21(G11,G21) = max
q21,w21"D21( )m21( )−µ
2q2
21( )−ω
2w2
21( ) +
∂V22
∂G11
˙
G11( ) + ∂V22
∂G21
˙
G21( ) + χV22(G12,G22)−ρ2m21D21( )#(12)
F om he abo e equa ions, i is easy o see ha he s uc u e o he game is linea quad a ic
and he solu ion o he game will yield piece-wise cons an s a egies in he wo egimes. The
illus a ion and impo ance o such games can be ound in Dockne e al. (2000).
3.1. Equilib ium Analysis
We assume ha i he e is a second egime, a e he i ms a e possibly exposed, he
i ms do no g eenwash anymo e. The e o e, he i ms’ only decisions in he second egime
a e he en i onmen al quali ies.
Games 2025,16, 14 15 o 17
Da a A ailabili y S a emen : The o iginal con ibu ions p esen ed in he s udy a e included in he
a icle, u he inqui ies can be di ec ed o he co esponding au ho .
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Appendix A. P oo o P oposi ion 1
We wan o p o e ha he equilib ium quali y decisions o he wo i ms in he egime
pe iod a e
q12 =k12m12β12
µ( +δ12)
q22 =k22m22β22
µ( +δ22)(A1)
and he alue unc ions a e gi en by
V12 =X12G12 +Y12G22 +Z12
and
V22 =X22G12 +
Y22G22 +Z22, whe e
X12 =m12β12
+δ12
;Y12 =−m12β12
+δ22
;
P oo . We assume ha he alue unc ions a e o he o m
V12(G12,G22) = X12G12 +Y12G22 +Z12
V22(G12,G22) = X22G12 +Y22G22 +Z22 (A2)
F om (11), he second pe iod’s HJB equa ions a e
V12(G12,G22) = max
q12 hD12( )m12( )−µ
2q2
12( ) + ∂V12
∂G12
˙
G12( ) + ∂V12
∂G22
˙
G22( )−P1]
V22(G12,G22) = max
q22 hD22( )m22( )−µ
2q2
22( ) + ∂V22
∂G12
˙
G12( ) + ∂V22
∂G22
˙
G22( )−P2](A3)
By no ing ha
Pi
is a numbe , subs i u ing he demand unc ions, he s a e a iables,
and he de i a i es in Equa ion (A3), and using he i s -o de condi ions o he decision
a iables qi2, we ob ain
−µq12( ) + X12k12 =0=⇒q12( ) = k12(X12)
µ
−µq22( ) + Y22k22 =0=⇒q22( ) = k22(Y22)
µ(A4)
Replacing he alue unc ions in (A3) wi h hei linea o ms in (A2), we ob ain
(Xi2Gi2+Yi2G(3−i)2+Zi2) = max
qi2hDi2( )mi2( )−µ
2q2
i2( ) + ∂Vi2
∂G12
˙
G12( ) + ∂Vi2
∂G22
˙
G22( )] (A5)
Subs i u ing he demand unc ions, he s a e a iables, and he de i a i es in
Equa ion (A5), we ob ain he ollowing wo equa ions.
Rema k A1. Fo b e i y, we a e d opping he ime index ( ) in he p oo s.

Games 2025,16, 14 16 o 17
(X12G12 +Y12G22 +Z12) = α1+β12(G12 −G22)m12 −(k12X12
µ)2+
(X12)(k12(k12X12
µ)−δ12G12) + (Y12(k22
k22X22
µ−δ22G22))
(X22G12 +Y22G22 +Z22) = α2+β22(G22 −G21)m22 −(k22X22
µ)2+
(X22)(k22(k12X12
µ)−δ12G12) + (Y22(k22
k22X22
µ−δ22G22))
Equa ing he coe icien s o s a e a iables
G12
and
G22
and he cons an e ms om
bo h sides o he abo e equa ions, we ob ain he ollowing se o six equa ions:
X12 =m12β12 −X12δ12 (A6)
Y12 =−m12β12 −Y12δ22 (A7)
Z12 =1
2−2P1+2m12α1+m12β12k2
12m12β12
µ( +δ12)2−2k2
22m22β22
µ( +δ22)2) (A8)
X22 =m12β12 −X12δ12 (A9)
Y22 =−m12β12 −Y12δ22 (A10)
Z22 =1
2−2P2+2m22α2+m22β22k2
22m22β22
µ( +δ22)2−2k2
12m12β12
µ( +δ12)2) (A11)
F om (A6), we ob ain X12 =m12β12
( +δ12). F om (A7), we ob ain Y12 =−m12β12
( +δ22).
F om (A9), we ob ain X22 =−m22β22
( +δ12). F om (A10), we ob ain Y22 =m22β22
( +δ22).
Plugging hese alues o coe icien s in Equa ion (A4), we ob ain
q12( ) = k12
µ
m12β12
( +δ12)
.
Simila ly, q22( ) = k22
µ
m22β22
( +δ22). This comple es he p oo o P oposi ion 1.
Rema k A2. The p oo o P oposi ion 2 is simila o ha o P oposi ion 1.
No e
1P oo s o he p oposi ions a e p o ided in Appendix A.
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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
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