L , Yali; Yang, Jian; Sun, Xiaoning; Wu, Hua ei
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E olu iona y game analysis o s akeholde p i acy
managemen in he AIGC model
Ope a ions Resea ch Pe spec i es
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E olu iona y game analysis o s akeholde p i acy managemen in he AIGC
model
Yali L , Jian Yang ∗, Xiaoning Sun, Hua ei Wu
School o In o ma ion, Shanxi Uni e si y o Finance and Economics, Taiyuan, 030006, Shanxi, China
ARTICLE INFO
Keywo ds:
AIGC
Da a p i acy
E olu iona y game
Replica o dynamic equa ions
ABSTRACT
The echnological de elopmen powe ed by A i icial In elligence Gene a ed Con en (AIGC) models, exem-
pli ied by Gene a i e P e- ained T ans o me 4 (GPT-4) and Bidi ec ional Encode Rep esen a ions om
T ans o me s (BERT), has comple ely ans o med machine language p ocessing and os e ed subs an ial ech-
nological ad ancemen s. Howe e , hei ex ensi e deploymen has ampli ied conce ns ega ding da a p i acy
isks, which a e a ibu ed no only o echnological ulne abili ies bu also o he in ica e con lic s o in e es
among model p o ide s, applica ion se ice p o ide s, and p i acy egula o s. To ackle his challenge, his
esea ch de elops a ipa i e e olu iona y game model ha examines he s a egic in e ac ions and dynamic
ela ionships among la ge language model p o ide s, applica ion se ice p o ide s, and p i acy egula o y
agencies. By employing eplica o dynamic equa ions and Jacobian ma ices, he esea ch in es iga es he
s abili y o s a egic equilib ia and simula es op imal adjus men pa hs ac oss di e se policy scena ios. D awing
on he esea ch indings, his pape o e s p ac ical ecommenda ions o s eng hen da a p i acy p o ec ion in
la ge language models, deli e ing a solid heo e ical ounda ion o policymake s and indus y p ac i ione s.
1. In oduc ion
La ge language models as a signi ican b eak h ough in a i icial
in elligence (AI) echnology, a e eshaping se ice modes ac oss mul-
iple domains. A i icial In elligence Gene a ed Con en (AIGC) mod-
els ep esen ed by Gene a i e P e- ained T ans o me 4 (GPT-4) and
Bidi ec ional Encode Rep esen a ions om T ans o me s (BERT) [1]
demons a e excep ional capabili ies in seman ic unde s anding [2] and
con en gene a ion [3], pa icula ly in highly specialized ields such as
medical diagnos ic assis ance [4] and inancial analysis, signi ican ly
enhancing se ice e iciency and decision suppo capabili ies h ough
accu a e comp ehension o p o essional ex s and con ex ual ela ion-
ships [5]. These models no only handle ou ine language asks bu also
deeply unde s and domain-speci ic equi emen s, b inging inno a i e
solu ions o a ious indus ies.
Wi h he ex ensi e applica ion o la ge language models in sen-
si i e ields such as he apy [6], educa ion [7], and heal hca e [8–
10], hei da a secu i y isks ha e become inc easingly p ominen .
Resea ch indica es hese isks mani es in h ee aspec s: i s , po en ial
leakage o use s’ pe sonal in o ma ion due o model memo y mecha-
nisms [11]; second, malicious a acks agains models, including in e -
ence econs uc ion [12] and exploi a ion o pe sonalized con igu a ion
ulne abili ies [13]; and hi d, da a leakage isks a he echnical in-
e ace le el [14]. These mul i-dimensional secu i y challenges equi e
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (J. Yang).
no only echnical p o ec ion measu es bu also he es ablishmen o
comp ehensi e egula o y amewo ks and indus y s anda ds.
P i acy p o ec ion o la ge language models is a complex sys-
ems enginee ing challenge in ol ing collabo a ion among mul iple
s akeholde s [15]. This sys em encompasses a ious en i ies including
da a p o ide s, echnology de elope s, se ice use s, and egula o y
agencies, wi h complex in e ac ions and ade-o s among hem [16].
Each pa icipan ’s decisions and beha io s a ec he o e all e ec i e-
ness o p i acy p o ec ion: om da a p o ide s’ p i acy awa eness
o he secu i y design o echnical solu ions o he o mula ion and
implemen a ion o egula o y policies, all equi e coo dina ion and
op imiza ion wi hin a uni ied amewo k. This s udy ocuses on hese
complex sys emic cha ac e is ics, a emp ing o cons uc an analy ical
model ha e lec s he in e ac ion mechanisms among all pa ies.
This s udy adop s e olu iona y game heo y (EGT) as i s heo e -
ical amewo k, le e aging i s unique ad an ages in analyzing mul i-
agen dynamic decision p ocesses [17–19]. By cons uc ing a ipa i e
e olu iona y game model, his s udy dynamically acks he s a egy
e olu ion p ocesses among la ge language model p o ide s, applica ion
se ice p o ide s, and egula o y agencies, while iden i ying he Nash
equilib ium—a s able s a e whe e no pa y can imp o e i s payo by
unila e ally changing i s s a egy [20]. The model examines se e al key
h ps://doi.o g/10.1016/j.o p.2025.100327
Recei ed 13 Sep embe 2024; Recei ed in e ised o m 13 Janua y 2025; Accep ed 5 Feb ua y 2025
Ope a ions Resea ch Pe spec i es 14 (2025) 100327
A ailable online 13 Feb ua y 2025
2214-7160/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
Y. L e al.
a iables, including he economic bene i s o all pa ies, iola ion cos s,
egula o y e iciency, and he social epu a ion impac s. This analy ical
amewo k no only unco e s he op imal s a egies o di e en pa ies
unde a ying condi ions bu also p edic s he long- e m e olu iona y
ends o he sys em, he eby p o iding obus heo e ical suppo
o de eloping p i acy p o ec ion policies ha balance e iciency and
secu i y.
This s udy es ablishes a no el ipa i e e olu iona y game model
o analyze he in e ac ion mechanisms among la ge language model
p o ide s, applica ion se ice p o ide s, and p i acy egula o y agen-
cies, demons a ing signi ican alue in bo h heo e ical con ibu ions
and p ac ical implica ions:
(a) F om a heo e ical pe spec i e, he model ho oughly exam-
ines key ac o s such as p o ide p o i g ow h a es, penal y
amoun s, social epu a ion o egula o y agencies, and egula-
o y cos s, o e ing a no el analy ical pe spec i e o la ge-scale
a i icial in elligence (AI) p i acy go e nance.
(b) F om a p ac ical s andpoin , he esea ch indings p o ide ac-
ionable policy ecommenda ions o balancing compliance e-
qui emen s and s akeholde incen i es.
Howe e , he s udy is limi ed by i s simpli ied assump ions in he
e olu iona y game amewo k and eliance on simula ion da a, which
may es ic he applicabili y o i s indings o eal-wo ld scena ios.
Fu he empi ical esea ch is equi ed o enhance i s gene alizabili y
and p ac ical impac .
2. Rela ed wo k
2.1. La ge language models
La ge language models, epi omized by Cha GPT, ha e showcased
hei as onishing eme gen capabili ies, u he p opelling echnolog-
ical b eak h oughs in he di ec ion o Gene al A i icial In elligence
(AGI). In his e olu iona y shi wi hin he AI pa adigm, he academic,
indus ial, and esea ch communi ies a e ac i ely explo ing and s udy-
ing he po en ial o la ge language models, emba king on a se ies o
expe imen s and applied esea ch.
Vaswani, A e al. [21] in oduced he sel -a en ion mechanism
in hei p oposed T ans o me a chi ec u e, a signi ican inno a ion
ha enhanced he capabili ies o NLP and laid he ounda ion o he
aining o la ge language models. Ouyang [22] buil upon he GPT-3
a chi ec u e, inco po a ing ins uc ion based lea ning and ein o ce-
men lea ning om human eedback o guide he model’s aining
u ilizing ine uning and policy alignmen and success ully de eloped
Ins uc GPT and Cha GPT. Zeng e al. [23] ha e ained he PanGu-
alpha la ge-scale au o eg essi e language model, which shows ema k-
able capabili ies in a ious scena ios based on massi e high-quali y
Chinese indus y da a wi hin he MindSpo e amewo k.
In he esea ch and de elopmen o la ge language models, s udies
on da a p i acy p o ec ion and co esponding AI go e nance a e o
pa amoun impo ance. Xu e al. [24] e iewed key p i acy-p ese ing
machine lea ning (PPML) echniques, such as di e en ial p i acy,
homomo phic enc yp ion, and secu e mul i-pa y compu a ion. They
iden i ied signi ican challenges, including high compu a ional cos s,
scalabili y limi a ions, and ade-o s be ween p i acy and u ili y, while
p oposing s a egies o in eg a e hese echniques in o la ge-scale AI
sys ems. Li e al. [25] conduc ed a comp ehensi e analysis o p i acy
a acks and de ense s a egies o la ge language models, ca ego izing
he ypes o a acks based on he capabili ies o po en ial a acke s
and e ealing c i ical ulne abili ies wi hin LLMs. P e ious s udies
ha e demons a ed ha a acke s can ex ac o econs uc p ecise
aining samples om LLMs, po en ially leading o he leakage o
pe sonal iden i y in o ma ion. To mi iga e his isk, Rehnia e al. [26]
p oposed a no el amewo k known as EW-Tune. This amewo k
employs ad anced g adien pe u ba ion echniques o sa egua d a
limi ed numbe o samples while in oducing minimal noise. Kandpal
N e al. [27] disco e ed ha educing duplica e da a in he aining
se s o la ge language models signi ican ly dec eases he likelihood
o p i acy b eaches when handling sensi i e in o ma ion, he eby
enhancing he o e all secu i y o he models ega ding da a p i acy.
Recen ad ancemen s in p i acy-p ese ing compu a ion echnologies
ha e u he expanded his ield. Fo ins ance, Xu e al. [28] p oposed
a ede a ed lea ning amewo k ha inco po a es p i acy-p ese ing
da a p icing, ensu ing sensi i e da a emains p o ec ed while enabling
equi able alua ion ac oss s akeholde s. This app oach unde sco es he
po en ial o in eg a ing ad anced p i acy-p ese ing mechanisms in o
la ge language model ecosys ems o balance da a u ili y and p i acy
in mul i-s akeholde scena ios. Rajagopal M e al. [29] in oduced
a concep ual amewo k o AI go e nance in public adminis a ion,
combining egula o y heo y and e hical p inciples. The amewo k
emphasizes anspa ency, accoun abili y, and s akeholde collabo a ion
while add essing challenges such as bias, p i acy conce ns, and he
complexi y o AI sys ems.
2.2. Game heo y
Game heo y has become a widely used ool [30,31]. Schola s in
academia ha e analyzed he game beha io o a ious s akeholde s
in di e en scena ios, p o iding solid heo e ical suppo o logical
decision-making p ocesses wi hin hese ields. Zhang e al. [32] p o-
posed a game- heo e ic amewo k o p i acy-p ese ing ede a ed
lea ning, e e ed o as he Fede a ed Lea ning P i acy Game. This
amewo k conside s he s a egic in e ac ions be ween de ende s and
a acke s, accoun ing o compu a ional cos s, model u ili y, and p i-
acy leakage isks. By add essing incomple e in o ma ion scena ios, he
s udy p o ides a s uc u ed app oach o balancing p i acy p o ec ion
and pe o mance in ede a ed lea ning en i onmen s. Shah H e al. [33]
e iewed he applica ions o game heo y models in p i acy p o ec-
ion, cybe secu i y, in usion de ec ion, and esou ce op imiza ion. Xu
e al. [34] ans o med he p i acy issues a ising om da a collec ion,
anonymiza ion, and elease in o a game p oblem. Wi hin his ame-
wo k, hey explo ed he in e ac i e beha io s amongs da a p o ide s,
collec o s, and use s, u ilizing a game model based on k-anonymi y o
p opose a gene al me hod o inding Nash equilib ia.
‘‘F ee- iding’’ is a common beha io s udied in game heo y, o en
obse ed among s akeholde s in supply chains whe e some pa ici-
pan s bene i om sha ed esou ces o coope a i e e o s wi hou
con ibu ing p opo ionally o he associa ed cos s. Fo example, Ju
e al. [35] highligh ed ha in he adop ion o blockchain echnology
wi hin shipping supply chains, ce ain s akeholde s may s a egically
a oid in es ing in he echnology while s ill eaping i s bene i s. Sag-
duyu Y E e al. [36] in oduces a game- heo e ic amewo k o analyze
ee- iding beha io in ede a ed lea ning (FL) o e wi eless ne wo ks.
The s udy highligh s how sel ish clien s, seeking o a oid compu a ional
and communica ion cos s, engage in ee- iding by no pa icipa ing in
model upda es while s ill bene i ing om he global model. This be-
ha io ad e sely impac s he accu acy o he global model and educes
o e all sys em u ili y. By o mula ing a non-coope a i e game, he e-
sea ch de i es Nash equilib ium s a egies o ee- iding p obabili ies
and quan i ies he ade-o s be ween pa icipa ion cos s and global
accu acy. The esul s emphasize he need o incen i e mechanisms
o mi iga e ee- iding and enhance FL’s esilience while p ese ing
p i acy h ough decen alized da a sha ing.
In addi ion o e olu iona y game heo y (EGT), se e al ounda ional
app oaches ha e been explo ed in he con ex o p i acy p o ec ion
wi hin mul i-s akeholde AI go e nance. S a ic game heo y is e ec i e
o analyzing single-sho in e ac ions wi h ully a ional pa icipan s;
howe e , i lacks he capaci y o model he long- e m e olu ion o
s a egies, ende ing i unsui able o cap u ing he i e a i e adjus -
men s obse ed in dynamic mul i-pa y scena ios like p i acy in es -
men s and egula o y ac ions [37,38]. Agen -based modeling o e s
Ope a ions Resea ch Pe spec i es 14 (2025) 100327
2
Y. L e al.
de ailed simula ions o mic o-le el in e ac ions and accoun s o s ake-
holde he e ogenei y, making i aluable o s udying decen alized
sys ems and eme gen beha io s; ye , i lacks he heo e ical gene -
alizabili y and analy ical p ecision o equilib ium-based me hods like
EGT, which a e mo e adep a de i ing global s a egic insigh s. Sys em
dynamics excels a analyzing mac o-le el ends and modeling eed-
back loops and long- e m sys em beha io s bu s uggles o ep esen
he nuanced and s a egic in e ac ions among indi idual s akeholde s
essen ial in p i acy p o ec ion and in es men decisions [39,40]. In
con as , EGT add esses hese limi a ions by modeling bounded a io-
nali y and he dynamic e olu ion o s a egies o e ime, e ec i ely
cap u ing he in e play o compe i ion and coope a ion among s ake-
holde s unde condi ions o unce ain y, he eby making i well-sui ed
o analyzing p i acy go e nance in AI ecosys ems.
Building upon he s eng hs o EGT, his s udy explo es i s applica-
ion o he domain o use p i acy p o ec ion in he con ex o la ge
language models, whe e dynamic and i e a i e in e ac ions among
s akeholde s a e pa icula ly p ominen . To achie e his, a ipa i e
e olu iona y game model is de eloped, o e ing an in-dep h analysis
o p i acy p o ec ion s a egies and hei e olu ion among key pa ic-
ipan s. Unlike adi ional game heo y, which assumes ully a ional
beha io , his s udy adop s he mo e ealis ic amewo k o bounded
a ionali y, enabling a nuanced unde s anding o s a egic decision-
making unde unce ain y. Th ough nume ical simula ions, he s udy
u he in es iga es how es ed in e es s d i e s a egic adjus men s
du ing ongoing in e ac ions, p o iding aluable insigh s in o he chal-
lenges and oppo uni ies o p i acy go e nance in la ge language model
ecosys ems.
3. Basic assump ions and model cons uc ion
3.1. Model assump ions
To ensu e he ealism and alidi y o he cons uc ed model, his
s udy g ounds i s assump ions in empi ical e idence and indus y p ac-
ices obse ed in he AI supply chain. Speci ically, he oles o la ge
language model p o ide s (Pa icipan 1), applica ion se ice p o ide s
(Pa icipan 2), and p i acy egula o y au ho i ies (Pa icipan 3) a e
consis en wi h he s akeholde in e ac ions desc ibed in exis ing li -
e a u e. Fo ins ance, [41] highligh s he complex dynamics among
s akeholde s in he la ge language model supply chain, including he
need o coo dina ed in es men in p i acy and secu i y measu es.
Simila ly, [42] emphasizes he bounded a ionali y o s akeholde s and
he impo ance o balancing cos s, bene i s, and egula o y p essu es. In
his con ex , he s a egic choices o hese pa icipan s a e modeled as
e ol ing o e ime and s abilizing a op imal s a egies, as summa ized
in Table 1.
Assump ion 1. The e a e h ee key game pa icipan s in he da a
p i acy game and in es men decision-making. Fi s ly, la ge language
model p o ide s (S) possess he echnology amewo ks o deep lea n-
ing and machine lea ning, la ge language model in e aces, as well as
ela ed s o age and compu a ional capabili ies. They choose o in es
in da a p i acy p o ec ion wi h a p obabili y o 𝑥, and choose no o
wi h a p obabili y o 1 −𝑥. Secondly, applica ion se ice p o ide s (H)
u ilize he basic echnology o e ed by la ge language model p o ide s
o p o ide sec o -speci ic so wa e solu ions, suppo , and main enance.
They choose o in es in da a p i acy p o ec ion wi h a p obabili y
o 𝑦, and no o wi h a p obabili y o 1 −𝑦. The hi d en i y is
p i acy egula o y au ho i ies (G). Thei esponsibili ies include issu-
ing p i acy p o ec ion guidelines and conduc ing echnical audi s and
ce i ica ions. They also ha e he powe o in es iga e and penalize
non-complian beha io s, wi h he assump ion ha he p obabili y o
hem en o cing s ic egula ion is 𝑧, and lax egula ion is 1 −𝑧.𝑥,𝑦,
and 𝑧a e all de ined wi hin he in e al [0,1].
Assump ion 2. F om he pe spec i e o la ge language model p o ide s,
when nei he hey no he applica ion se ice p o ide s in es in
p i acy p o ec ion, hei p o i is 𝑃𝑆. Howe e , unde he supe ision
o egula o y au ho i ies, la ge language model p o ide s will ace a
ine o 𝐹. E en i la ge language model p o ide s choose o in es in
p i acy p o ec ion (a a cos o 𝐶𝑆), hey canno ensu e ha he p i acy
p o ec ion measu es a e 100% e ec i e, and he e is a isk o ailu e.
This means hey may s ill ace ines o 𝐹due o p i acy b eaches.
Ne e heless, in es ing in p i acy p o ec ion can gene ally be expec ed
o inc ease hei p o i o (1 +𝛼0)𝑃𝑆, whe e 𝛼0is he p o i g ow h
a e when la ge language model p o ide s in es in p i acy p o ec ion
alone. Addi ionally, unde s ic egula ion by egula o y au ho i ies,
la ge language model p o ide s who in es in p i acy p o ec ion may
ecei e an addi ional ewa d subsidy o 𝐹. When la ge language model
p o ide s do no in es while applica ion se ice p o ide s do, hey
can ee- ide and ob ain an addi ional bene i o 𝜉𝑆. Howe e , due o
no in es ing in p i acy p o ec ion hemsel es, he e is a highe isk o
p i acy b eaches, which may lead o ines o 𝐹.
Assump ion 3. F om he pe spec i e o applica ion se ice p o ide s,
when nei he hey no he la ge language model p o ide s in es in
p i acy p o ec ion, hei p o i is 𝑃𝐻. Howe e , unde s ic egula ion,
hey will ace a ine o 𝐹. E en i applica ion se ice p o ide s choose
o in es in p i acy p o ec ion (a a cos o 𝐶𝐻), he p i acy p o ec ion
measu es may no be comple ely e ec i e, and he e is a isk o ailu e,
leading o he possibili y ha hey may s ill ace ines o 𝐹due o
p i acy b eaches. Ne e heless, in es ing in p i acy p o ec ion can
gene ally be expec ed o inc ease hei p o i o (1 +𝛽0)𝑃𝐻, whe e 𝛽0
is he p o i g ow h a e when applica ion se ice p o ide s in es in
p i acy p o ec ion alone. Fu he mo e, unde s ic egula ion, applica-
ion se ice p o ide s who in es in p i acy p o ec ion may ecei e an
addi ional ewa d subsidy o 𝐹. When applica ion se ice p o ide s do
no in es while la ge language model p o ide s do, hey can ee- ide
and ob ain an addi ional bene i o 𝜉𝐻. Howe e , due o no in es ing
in p i acy p o ec ion hemsel es, he e is a highe isk o p i acy
b eaches, which may lead o ines o 𝐹.
Assump ion 4. The mo i a ions o egula o y au ho i ies o en o ce
s ic egula ions a ise om iscal, epu a ional, and e hical conside a-
ions. S ic egula ion incu s a cos o 𝐶𝐺. Non-in es ing p o ide s a e
penalized wi h a ine 𝐹, while in es ing p o ide s ecei e an equi alen
subsidy o 𝐹. S ic egula ion enhances he social epu a ion o egu-
la o y au ho i ies, wi h an inc ease o 𝑅0. In con as , lax egula ion
esul s in no epu a ion gain, and ine ec i e egula ion leads o a
epu a ion loss o 𝐿. When all p o ide s choose o in es , egula o y
au ho i ies achie e he maximum epu a ion gain, 𝑅1, whe e 𝑅1> 𝑅0.
In scena ios whe e egula o y au ho i ies ac as go e nmen en i ies,
hei decisions a e no solely d i en by iscal o epu a ional ac o s.
E hical esponsibili ies, such as p o ec ing public in e es s and ensu ing
p i acy s anda ds, may also in luence hei choices. While he cu en
model simpli ies his complexi y by ocusing p ima ily on epu a ional
conside a ions, u u e ex ensions could inco po a e addi ional ac o s
(e.g., e hical esponsibili ies o public expec a ions) in o he decision-
making amewo k o be e e lec he mul i ace ed mo i a ions o
go e nmen egula o s.
Assump ion 5. In indus ial p ac ice, due o cons ain s o unding,
echnical challenges, and di icul ies in he e alua ion p ocess, egula-
o y au ho i ies a e unable o comp ehensi ely moni o s a egies like
p i acy in es men o ee- iding. The e o e, he ac ual ines may be
lowe han he epu a ion gains due o egula ion, i.e., 2𝐹 < 𝑅0.
Assump ion 6. When bo h la ge language model p o ide s and appli-
ca ion se ice p o ide s choose o in es in p i acy p o ec ion, bo h
pa ies achie e a win–win si ua ion. A his ime, he p o i g ow h a e
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Table 1
Main symbols used in he pape .
Symbol Desc ip ion Uni
𝑥The p obabili y ha la ge language model p o ide s (S) in es in da a p i acy p o ec ion, wi h 𝑥∈ [0,1] –
𝑦The p obabili y ha applica ion se ice p o ide s (H) in es in da a p i acy p o ec ion, wi h 𝑦∈ [0,1] –
𝑧The p obabili y ha p i acy egula o y au ho i ies (G) en o ce s ic egula ion, wi h 𝑧∈ [0,1] –
𝑃𝑆The p o i o la ge language model p o ide s when no in es ing in p i acy p o ec ion Mone a y Uni (e.g., $)
𝑃𝐻The p o i o applica ion se ice p o ide s when no in es ing in p i acy p o ec ion Mone a y Uni (e.g., $)
𝐶𝑆The in es men cos o p i acy p o ec ion o la ge language model p o ide s Mone a y Uni (e.g., $)
𝐶𝐻The in es men cos o p i acy p o ec ion o applica ion se ice p o ide s Mone a y Uni (e.g., $)
𝐶𝐺The iscal expendi u e p oduced by egula o y au ho i ies o en o cing s ic egula ion Mone a y Uni (e.g., $)
𝐹The ine imposed by egula o y au ho i ies o non-compliance Mone a y Uni (e.g., $)
𝛼0The p o i g ow h a e o la ge language model p o ide s when in es ing alone in p i acy p o ec ion F ac ion (e.g., 0.1)
𝛼1The p o i g ow h a e when bo h la ge language model p o ide s and applica ion se ice p o ide s
in es , 𝛼1> 𝛼0>0
F ac ion (e.g., 0.1)
𝛽0The p o i g ow h a e o applica ion se ice p o ide s when in es ing alone in p i acy p o ec ion F ac ion (e.g., 0.1)
𝛽1The p o i g ow h a e when bo h applica ion se ice p o ide s and la ge language model p o ide s
in es , 𝛽1> 𝛽0>0
F ac ion (e.g., 0.1)
𝜉𝑆The addi ional bene i ha la ge language model p o ide s ob ain om ee- iding on he in es men s o
applica ion se ice p o ide s
Mone a y Uni (e.g., $)
𝜉𝐻The addi ional bene i ha applica ion se ice p o ide s ob ain om ee- iding on he in es men s o
la ge language model p o ide s
Mone a y Uni (e.g., $)
𝑅0The social epu a ion gained by egula o y au ho i ies om en o cing s ic egula ion –
𝑅1The mo e signi ican social epu a ion gained when bo h p o ide s in es , 𝑅1> 𝑅0–
𝐿The epu a ion loss aced by egula o y au ho i ies due o lax egula ion –
Fig. 1. T ipa i e E olu iona y Game Decision T ee.
o la ge language model p o ide s inc eases o 𝛼1,i.e., 𝛼1> 𝛼0>0;
he p o i g ow h a e o applica ion se ice p o ide s inc eases o 𝛽1,
i.e., 𝛽1> 𝛽0>0. Al hough p i acy p o ec ion measu es may s ill ail, he
collabo a i e in es men o bo h pa ies can educe he isk o ailu e,
enhance he o e all e ec i eness o p i acy p o ec ion, and he eby
educe he likelihood o ines and epu a ional losses.
To del e deepe in o s a egic in e ac ions in la ge language model
da a p i acy p o ec ion, we ha e cons uc ed a decision ee o e o-
lu iona y game in ol ing h ee pa icipan s. As shown in Fig. 1, his
decision ee illus a es he po en ial s a egies and e olu iona y pa hs
o la ge language model p o ide s, applica ion se ice p o ide s, and
p i acy egula o y au ho i ies in he da a p i acy game. Each game
pa icipan has wo choices a e e y decision node; hese decisions un-
old h oughou he decision ee, e en ually o ming eigh end-s a es
ep esen ing he e olu iona y esul s o he ipa i e bodies unde
di e en s a egy combina ions. This model highligh s he unce ain y
o s a egic choices and he complexi y o s a egy e olu ion, p o iding
a heo e ical amewo k o unde s anding and p edic ing he beha io
pa e ns o each pa icipan in da a p i acy p o ec ion.
3.2. Game pa icipan ela ions
Fig. 2depic s a ipa i e e olu iona y game model o da a p i acy
p o ec ion, in ol ing la ge language model p o ide s, applica ion se -
ice p o ide s, and p i acy egula o y au ho i ies as p incipal playe s.
Fig. 2. T ipa i e E olu iona y Game F amewo k.
In his amewo k, use s and hacke s play pi o al oles. Use s, as da a
gene a o s, supply in o ma ion o bo h se ice p o ide ca ego ies.
Con e sely, hacke s aim o b each hese sys ems o pil e c i ical use
da a, di ec ly jeopa dizing sys em da a p i acy. La ge language model
p o ide s possess key echnologies and compu a ional esou ces, which
applica ion se ice p o ide s le e age o o e ailo ed solu ions. Thei
in es men decisions in p i acy p o ec ion a e mu ually dependen : i
one in es s and he o he does no , he non-in es o may indi ec ly ben-
e i om he in es o ’s commi men . Howe e , mu ual in es men in
p i acy p o ec ion enables bo h o achie e a symbio ic gain, enhancing
hei p o i abili y.
P i acy egula o y au ho i ies play a c i ical ole in his game by
in luencing he in es men beha io o se ice p o ide s h ough he
o mula ion and en o cemen o p i acy p o ec ion policies. The s ic -
ness o hei egula o y s a egy is di ec ly linked o he ne bene i s and
in es men mo i a ions o se ice p o ide s: oo s ic egula ion may
lead o ines o non-complian p o ide s, while ewa d mechanisms
may encou age p o ide s o comply wi h egula ions. The e ec i eness
o he egula o y ins i u ions no only a ec s he economic bene i s o
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Table 2
The p o i and loss ma ix o he h ee game pa icipan s.
LLM p o ide App p o ide P i acy au ho i y
S ic egula ion(z) Lax egula ion(1-z)
In es (x)
In es (y)
(1 +𝛼1)𝑃𝑆−𝐶𝑆(1 +𝛼1)𝑃𝑆−𝐶𝑆
(1 +𝛽1)𝑃𝐻−𝐶𝐻(1 +𝛽1)𝑃𝐻−𝐶𝐻
𝑅1−𝐶𝐺𝑅1
No In es (1-y)
(1 +𝛼0)𝑃𝑆−𝐶𝑆+𝐹(1 +𝛼0)𝑃𝑆−𝐶𝑆
𝜉𝐻−𝐹 𝜉𝐻
𝑅0−𝐶𝐺−𝐿
No In es (1-x)
In es (y)
𝜉𝑆−𝐹 𝜉𝑆
(1 +𝛽0)𝑃𝐻−𝐶𝐻+𝐹(1 +𝛽0)𝑃𝐻−𝐶𝐻
𝑅0−𝐶𝐺−𝐿
No In es (1-y)
𝑃𝑆−𝐹 𝑃𝑆
𝑃𝐻−𝐹 𝑃𝐻
2𝐹−𝐶𝐺−𝐿
he se ice p o ide s bu is also ela ed o hei own social c edibili y
and au ho i y. E ec i e egula ion canno only imp o e hei epu a-
ion among he public bu can also enhance social wel a e; con e sely,
i may lead o damage o hei epu a ion.
3.3. Model es ablishmen
A e syn hesizing he assump ions and analyses p oposed in Sec-
ions 3.1 and 3.2, we ha e cons uc ed a de ailed payo ma ix o
quan i a i ely desc ibe he in e ac ions and expec ed payo s o he
game en i ies — la ge language model p o ide s, applica ion se ice
p o ide s, and p i acy egula o y au ho i ies — unde di e en s a egy
combina ions. De ailed in o ma ion is ou lined in Table 2.
3.3.1. Replica o dynamics equa ion and phase diag am o la ge language
model p o ide s
Based on Table 2, i is known ha la ge language model p o ide s
ace wo s a egic choices: o in es o no in es in p i acy p o ec ion.
When hey choose he o me , he expec ed payo is 𝐸𝑆1; o he
la e , i is 𝐸𝑆2. We de ine he speci ic calcula ion o mulas o 𝐸𝑆1
and 𝐸𝑆2as ollows:
I choosing o in es in p i acy p o ec ion, he expec ed payo 𝐸𝑆1
is calcula ed ia he o mula:
𝐸𝑆1=𝑦𝑧[(1 +𝛼1)𝑃𝑆−𝐶𝑆]
+𝑦(1 −𝑧)[(1 +𝛼1)𝑃𝑆−𝐶𝑆]
+ (1 −𝑦)𝑧[(1 +𝛼0)𝑃𝑆−𝐶𝑆+𝐹]
+ (1 −𝑦)(1 −𝑧)[(1 +𝛼0)𝑃𝑆−𝐶𝑆]
=𝑃𝑆−𝐶𝑆+𝑃𝑆𝛼0+𝐹 𝑧−𝑃𝑆𝛼0𝑦+𝑃𝑆𝛼1𝑦−𝐹 𝑦 𝑧
(1)
I choosing no o in es in p i acy p o ec ion, he expec ed payo
𝐸𝑆2is calcula ed ia he o mula:
𝐸𝑆2=𝑦𝑧[𝜉𝑆−𝐹] +𝑦(1 −𝑧)𝜉𝑆
+ (1 −𝑦)𝑧[𝑃𝑆−𝐹]
+ (1 −𝑦)(1 −𝑧)𝑃𝑆
=𝑃𝑆−𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦
(2)
Subsequen ly, he a e age expec ed payo 𝐸𝑆 o la ge model
p o ide s can be ep esen ed by he o mula:
𝐸𝑆=𝑥𝐸𝑆1+ (1 −𝑥)𝐸𝑆2(3)
To del e in o he pa hways and equilib ium poin s o s a egy
e olu ion o he ipa i e game pa icipan s, we sol e he eplica o
dynamics equa ion o la ge model p o ide s:
𝐹(𝑥) =d𝑥
d𝑡=𝑥(𝐸𝑆1−𝐸𝑆)
=𝑥(𝑥− 1) (𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦
+𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧)
(4)
Designa ing 𝑦0=𝑃𝑆𝛼0−𝐶𝑆+2𝐹 𝑧
𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧and calcula ing he pa ial
de i a i e o he eplica o dynamics equa ion 𝐹(𝑥)wi h espec o
a iable 𝑥, we ob ain:
𝑑 𝐹(𝑥)
𝑑 𝑥=(2𝑥− 1)(𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦
+𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧)
=(2𝑥− 1)[(𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧)𝑦
−(−𝐶𝑆+𝑃𝑆𝛼0+ 2𝐹 𝑧)]
(5)
I 𝑦=𝑦0, we can ob ain 𝐹(𝑥) = 0, whe e ega dless o he alue o
𝑥, he s a egic choice o la ge language model p o ide s is in a s able
s a e.
I 𝑦 < 𝑦0, we can de i e ha 𝑑 𝐹(𝑥)
𝑑 𝑥||||𝑥=0
>0and 𝑑 𝐹(𝑥)
𝑑 𝑥||||𝑥=1
<0, a
which poin 𝑥= 1is an equilib ium poin . When he p obabili y o
applica ion se ice p o ide s choosing o ‘‘in es in p i acy p o ec ion’’
is lowe han a ce ain h eshold, la ge language model p o ide s will
choose he ‘‘in es in p i acy p o ec ion’’ s a egy.
I 𝑦 > 𝑦0, we can deduce ha 𝑑 𝐹(𝑥)
𝑑 𝑥||||𝑥=0
<0and 𝑑 𝐹(𝑥)
𝑑 𝑥||||𝑥=1
>0, a
which poin 𝑥= 0is an equilib ium poin . When he p obabili y o
applica ion se ice p o ide s choosing o ‘‘in es in p i acy p o ec ion’’
exceeds a ce ain h eshold, la ge language model p o ide s will op o
he ‘‘no in es in p i acy p o ec ion’’ s a egy.
Acco ding o he abo e analysis, he la ge language model
p o ide s’ eplica ion dynamic phase diag am can be ob ained, as
shown in Fig. 3.
3.3.2. Applica ion se ice p o ide ’s eplica o dynamics equa ion and
phase diag am
Fo applica ion se ice p o ide s, hei decision-making s a egies
can be di ided in o ‘‘in es ing in p i acy p o ec ion’’ and ‘‘no in-
es ing in p i acy p o ec ion’’. When choosing o ‘‘in es in p i acy
p o ec ion’’, he expec ed payo is de ined as 𝐸𝐻1; when choosing ‘‘no
in es ing in p i acy p o ec ion’’, he expec ed payo is de ined as 𝐸𝐻2.
The speci ic o mulas a e:
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Fig. 3. Replica ion dynamic phase diag am o la ge language model p o ide s: (a)
𝑦=𝑦0; (b) 𝑦 < 𝑦0; (c) 𝑦 > 𝑦0.
I choosing o in es in p i acy p o ec ion, he expec ed payo 𝐸𝐻1
is calcula ed as:
𝐸𝐻1=𝑥𝑧[(1 +𝛽1)𝑃𝐻−𝐶𝐻] +𝑥(1 −𝑧)[(1 +𝛽1)𝑃𝐻−𝐶𝐻]
+ (1 −𝑥)𝑧[(1 +𝛽0)𝑃𝐻−𝐶𝐻+𝐹]
+ (1 −𝑥)(1 −𝑧)[(1 +𝛽0)𝑃𝐻−𝐶𝐻]
=𝑃𝐻−𝐶𝐻+𝑃𝐻𝛽0+𝐹 𝑧
−𝑃𝐻𝛽0𝑥+𝑃𝐻𝛽1𝑥−𝐹 𝑥𝑧
(6)
I no choosing o in es in p i acy p o ec ion, he expec ed payo
𝐸𝐻2is calcula ed as:
𝐸𝐻2=𝑥𝑧[𝜉𝐻−𝐹] +𝑥(1 −𝑧)𝜉𝐻
+ (1 −𝑥)𝑧(𝑃𝐻−𝐹)
+ (1 −𝑥)(1 −𝑧)𝑃𝐻
=𝑃𝐻−𝐹 𝑧−𝑃𝐻𝑥+𝑥 𝜉𝐻
(7)
The a e age expec ed payo 𝐸𝐻 o applica ion se ice p o ide s
can be exp essed by he ollowing o mula:
𝐸𝐻=𝑦𝐸𝐻1+ (1 −𝑦)𝐸𝐻2(8)
The eplica o dynamics equa ion o applica ion se ice p o ide s
is:
𝐹(𝑦) =d𝑦
d𝑡
=𝑦(𝐸𝐻1−𝐸𝐻)
=𝑦(𝑦− 1) (𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻
+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧)
(9)
Se ing 𝑧0 o make he g ow h a e neu al:
𝑧0=𝐶𝐻−𝑃𝐻𝛽0−𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥
2𝐹−𝐹 𝑥, calcula ing he pa ial de i a i e
o he eplica o dynamics equa ion 𝐹(𝑦)wi h espec o he a iable 𝑦,
Fig. 4. Replica ion dynamic phase diag am o applica ion se ice p o ide s: (a) 𝑧=𝑧0;
(b) 𝑧 < 𝑧0; (c) 𝑧 > 𝑧0.
we ge :
𝑑 𝐹(𝑦)
𝑑 𝑦=(2𝑦− 1)(𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻
+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧)
=(2𝑦− 1)[(𝐹 𝑥− 2𝐹)𝑧+𝐶𝐻−𝑃𝐻𝛽0
−𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥]
(10)
I 𝑧=𝑧0, we ha e 𝐹(𝑦) = 0, so no ma e he alue o 𝑦, he s a egic
choice o he applica ion se ice p o ide is in a s able s a e.
I 𝑧 < 𝑧0,𝑑 𝐹(𝑦)
𝑑 𝑦||||𝑦=0
<0and 𝑑 𝐹(𝑦)
𝑑 𝑦||||𝑦=1
>0, hus a 𝑦= 0 he e
is an equilib ium poin . When he p obabili y o he egula o y body
choosing a ‘‘s ic egula ion’’ s a egy is below a speci ic h eshold, he
applica ion se ice p o ide will choose he ‘‘no in es ing in p i acy
p o ec ion’’ s a egy.
I 𝑧 > 𝑧0,𝑑 𝐹(𝑦)
𝑑 𝑦||||𝑦=0
>0and 𝑑 𝐹(𝑦)
𝑑 𝑦||||𝑦=1
<0, hus a 𝑦= 1 he e
is an equilib ium poin . When he p obabili y o he egula o y body
choosing a ‘‘s ic egula ion’’ s a egy exceeds a ce ain h eshold,
he applica ion se ice p o ide will choose he ‘‘in es ing in p i acy
p o ec ion’’ s a egy.
Acco ding o he abo e analysis, he applica ion se ice p o ide s’
eplica ion dynamic phase diag am can be ob ained, as shown in Fig. 4.
3.3.3. Regula o y au ho i y’s eplica o dynamics equa ion and phase dia-
g am
Fo p i acy egula o y au ho i ies, when implemen ing he s a egy
o ‘‘s ic egula ion’’, he expec ed payo is de ined as 𝐸𝐺1; while im-
plemen ing ‘‘lax egula ion’’, he expec ed payo is 𝐸𝐺2. The o mulas
a e:
𝐸𝐺1=𝑥𝑦 [𝑅1−𝐶𝐺]
+𝑥(1 −𝑦)[𝑅0−𝐶𝐺]
+ (1 −𝑥)𝑦[𝑅0−𝐶𝐺]
+ (1 −𝑥)(1 −𝑦)[2𝐹−𝐶𝐺]
(11)
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𝐸𝐺2=𝑥𝑦𝑅1
+𝑥(1 −𝑦)[−𝐿]
+ (1 −𝑥)𝑦[−𝐿]
+ (1 −𝑥)(1 −𝑦)[−𝐿]
(12)
To ully e alua e he impac o hese s a egies, we calcula e he
a e age expec ed bene i 𝐸𝐺 o he egula o y au ho i ies using he
ollowing o mula:
𝐸𝐺=𝑧𝐸𝐺1+ (1 −𝑧)𝐸𝐺2(13)
The eplica o dynamics equa ion o he egula o y au ho i y is:
𝐹(𝑧) =𝑑 𝑧
𝑑 𝑡=𝑧(𝐸𝐺1−𝐸𝐺)(14)
Se ing 𝑥0 o make he g ow h a e neu al: 𝑥0=2𝐹−𝐶𝐺+𝐿−2𝐹 𝑦+𝑅0𝑦
2𝐹−𝑅0−2𝐹 𝑦+𝐿𝑦+2𝑅0𝑦,
calcula ing he pa ial de i a i e o he eplica o dynamics equa ion
𝐹(𝑧)wi h espec o he a iable 𝑧, we ob ain:
𝑑 𝐹(𝑧)
𝑑 𝑧=(2𝑧− 1)(𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦
−𝑅0𝑥−𝑅0𝑦− 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦)
=(2𝑧− 1)[(2𝐹−𝑅0− 2𝐹 𝑦+𝐿𝑦 + 2𝑅0𝑦)𝑥
−(−𝐶𝐺+ 2𝐹+𝐿− 2𝐹 𝑦+𝑅0𝑦)]
(15)
I 𝑥=𝑥0,𝐹(𝑧) = 0, and no ma e he alue o 𝑧, he s a egy choice
o he egula o y au ho i y is in a s able s a e.
I 𝑥 < 𝑥0,𝑑 𝐹(𝑧)
𝑑 𝑧||||𝑧=0
>0and 𝑑 𝐹(𝑧)
𝑑 𝑧||||𝑧=1
<0, hus a 𝑧= 1 he e is
an equilib ium poin . When he p obabili y o he big language model
p o ide choosing o ‘‘in es in p i acy p o ec ion’’ is below a ce ain
h eshold, he egula o y au ho i y will choose he ‘‘s ic egula ion’’
s a egy.
I 𝑥 > 𝑥0,𝑑 𝐹(𝑧)
𝑑 𝑧||||𝑧=0
<0and 𝑑 𝐹(𝑧)
𝑑 𝑧||||𝑧=1
>0, hus a 𝑧= 0 he e is
an equilib ium poin . When he p obabili y o he big language model
p o ide choosing o ‘‘in es in p i acy p o ec ion’’ exceeds a ce ain
h eshold, he egula o y au ho i y will choose he ‘‘lax egula ion’’
s a egy.
Acco ding o he abo e analysis, he p i acy egula o y au ho i ies’
eplica ion dynamic phase diag am can be ob ained, as shown in Fig. 5.
4. S abili y analysis o he model’s equilib ium poin s
4.1. Jacobian ma ix
Th ough Eqs. (4),(9) and (14), we de i e he s a e equa ions o he
ipa i e game in ol ing la ge language model p o ide s, applica ion
se ice p o ide s, and p i acy egula o y au ho i ies in he con ex o
da a p i acy p o ec ion:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
𝐹(𝑥) =𝑥(𝑥− 1) (𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦
+𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧 )
𝐹(𝑦) =𝑦(𝑦− 1) (𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥
+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧 )
𝐹(𝑧) =𝑧(𝑧− 1) (𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦
−𝑅0𝑥−𝑅0𝑦− 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦 )
(16)
In sys em dynamics, an analysis o he eigen alues o he Jacobian
ma ix is a key s ep in assessing local s abili y. The de e mina ion o
E olu iona ily S able S a egy (ESS) o en elies on a local s abili y
analysis o he Jacobian ma ix nea he equilib ium poin . Speci ically,
an equilib ium poin ’s ESS is conside ed s able only i all eigen alues
o i s Jacobian ma ix a e nega i e; o he wise, he equilib ium poin
is conside ed uns able. On his heo e ical basis, we i s de i e he
Jacobian ma ix based on Fo mula (16) and u he use Lyapuno ’s
me hod o conduc a comp ehensi e assessmen o he s abili y o each
equilib ium poin in he game sys em, p o iding a s ic ma hema ical
Fig. 5. Replica ion dynamic phase diag am o egula o y au ho i y: (a) 𝑥=𝑥0; (b)
𝑥 < 𝑥0; (c) 𝑥 > 𝑥0.
ounda ion o he sys em’s s abili y analysis. The Jacobian ma ix is
shown in Fo mula (17):
𝐽=⎡⎢⎢⎢⎣
𝑗11 𝑗12 𝑗13
𝑗21 𝑗22 𝑗23
𝑗31 𝑗32 𝑗33
⎤⎥⎥⎥⎦
(17)
Among hem,
𝑗11 = (2𝑥− 1)(𝐶𝑆−𝑃𝑆𝛼0− 2𝐹 𝑧−𝑃𝑆𝑦+𝜉𝑆𝑦+𝑃𝑆𝛼0𝑦
−𝑃𝑆𝛼1𝑦+𝐹 𝑦𝑧)
𝑗12 =𝑥(𝑥− 1)(𝜉𝑆−𝑃𝑆+𝑃𝑆𝛼0−𝑃𝑆𝛼1+𝐹 𝑧)
𝑗13 =𝐹 𝑥(𝑥− 1)(𝑦− 2)
𝑗21 =𝑦(𝑦− 1)(𝜉𝐻−𝑃𝐻+𝑃𝐻𝛽0−𝑃𝐻𝛽1+𝐹 𝑧)
𝑗22 = (2𝑦− 1)(𝐶𝐻−𝑃𝐻𝛽0− 2𝐹 𝑧−𝑃𝐻𝑥+𝑥𝜉𝐻+𝑃𝐻𝛽0𝑥
−𝑃𝐻𝛽1𝑥+𝐹 𝑥𝑧)
𝑗23 =𝐹 𝑦(𝑥− 2)(𝑦− 1)
𝑗31 =𝑧(𝑧− 1)(2𝐹−𝑅0− 2𝐹 𝑦+𝐿𝑦 + 2𝑅0𝑦)
𝑗32 =𝑧(𝑧− 1)(2𝐹−𝑅0− 2𝐹 𝑥+𝐿𝑥 + 2𝑅0𝑥)
𝑗33 = (2𝑧− 1)(𝐶𝐺− 2𝐹−𝐿+ 2𝐹 𝑥+ 2𝐹 𝑦−𝑅0𝑥−𝑅0𝑦
− 2𝐹 𝑥𝑦 +𝐿𝑥𝑦 + 2𝑅0𝑥𝑦)
4.2. S abili y analysis
4.2.1. Eigen alues a equilib ium poin s
By inse ing he 8 local equilib ium poin s in o he Jacobian ma ix
and ollowing he assump ions p o ided, we ob ain he espec i e
eigen alues o each equilib ium poin . The esul s a e as shown in he
Table 3.
The able clea ly demons a es a signi ican ela ionship be ween
pa ame e s such as egula o y cos s, egula o y bene i s, p o ide s’
p o i g ow h a es, and ines/subsidies, and he ESS o he h ee
Ope a ions Resea ch Pe spec i es 14 (2025) 100327
7
Y. L e al.
Table 3
Sys em equilib ium poin s and eigen alues.
𝜆1𝜆2𝜆3
𝐸𝑝1(0,0,0) 𝑃𝑆𝛼0−𝐶𝑆𝑃𝐻𝛽0−𝐶𝐻2𝐹−𝐶𝐺+𝐿
𝐸𝑝2(0,0,1) 2𝐹−𝐶𝑆+𝑃𝑆𝛼02𝐹−𝐶𝐻+𝑃𝐻𝛽0𝐶𝐺− 2𝐹−𝐿
𝐸𝑝3(0,1,0) 𝐶𝐻−𝑃𝐻𝛽0𝐿−𝐶𝐺+𝑅0𝑃𝑆−𝐶𝑆−𝜉𝑆+𝑃𝑆𝛼1
𝐸𝑝4(0,1,1) 𝐶𝐺−𝐿−𝑅0𝐶𝐻− 2𝐹−𝑃𝐻𝛽0𝐹−𝐶𝑆+𝑃𝑆−𝜉𝑆+𝑃𝑆𝛼1
𝐸𝑝5(1,0,0) 𝐶𝑆−𝑃𝑆𝛼0𝐿−𝐶𝐺+𝑅0−𝜉𝐻+𝑃𝐻−𝐶𝐻+𝑃𝐻𝛽1
𝐸𝑝6(1,0,1) 𝐶𝐺−𝐿−𝑅0𝐶𝑆− 2𝐹−𝑃𝑆𝛼0−𝜉𝐻+𝐹−𝐶𝐻+𝑃𝐻+𝑃𝐻𝛽1
𝐸𝑝7(1,1,0) −𝐶𝐺𝐶𝑆−𝑃𝑆+𝜉𝑆−𝑃𝑆𝛼1𝜉𝐻+𝐶𝐻−𝑃𝐻−𝑃𝐻𝛽1
𝐸𝑝8(1,1,1) 𝐶𝐺𝐶𝑆−𝐹−𝑃𝑆+𝜉𝑆−𝑃𝑆𝛼1𝜉𝐻+𝐶𝐻−𝐹−𝑃𝐻−𝑃𝐻𝛽1
Table 4
Pa ame e alues o di e en p oposi ions.
𝑃𝑆𝑃𝐻𝐶𝑆𝐶𝐻𝐶𝐺𝐹 𝛼0𝛼1𝛽0𝛽1𝜉𝑆𝜉𝐻𝑅0𝑅1𝐿
P oposi ion 130 25 10 8 44 0 0.24 1.09 0.05 0.31 70 60 5 15 20
P oposi ion 248 37 15 10 40 0 0.19 0.25 0.51 0.59 65 55 2 12 10
P oposi ion 348 37 15 10 40 0 0.41 0.49 0.19 0.62 65 55 10 110 10
P oposi ion 448 37 15 10 40 0 0.92 0.94 0.79 0.86 65 55 2 12 10
P oposi ion 548 37 15 10 33 10 0.07 0.10 0.27 0.62 65 55 30 35 10
P oposi ion 648 37 20 15 27 6 0.45 0.57 0.07 0.17 65 55 45 55 10
P oposi ion 748 37 20 15 16 6 0.00 0.30 0.01 0.10 65 55 17 22 10
majo game pa icipan s. Fo simpli ica ion o he analysis p ocess,
we assume ha he e is only one egula o y au ho i y esponsible o
o e seeing all la ge language model p o ide s and applica ion se ice
p o ide s. Based on his assump ion, we se pa ame e alues unde
di e en p oposi ions and analyze h ee c i ical pa ame e anges. The
pa ame e alues a e shown in Table 4.
4.2.2. When 𝐶𝐺> 𝑅0+𝐿
When 𝐶𝐺> 𝑅0+𝐿, he egula o y au ho i y will op o a lax
egula ion policy ega dless o whe he he p o ide s in es due o he
high cos o egula ion.
P oposi ion 1. When he condi ions 0< 𝛼0< 𝐶𝑆∕𝑃𝑆,𝛼0< 𝛼1<
𝜉𝑆+𝐶𝑆−𝑃𝑆
𝑃𝑆
,0< 𝛽0< 𝐶𝐻∕𝑃𝐻, and 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻
𝑃𝐻
a e me , as
shown in Fig. 6, he sys em ends o ake he s a egy combina ion o non-
in es men and lax egula ion (0, 0, 0), which cons i u es an ESS. E alua ing
he cos –bene i o he in es men e u n 𝐸𝑆(1,0,0) = (1 +𝛼0)𝑃𝑆−𝐶𝑆
o la ge language model p o ide s, we ind 𝐸𝑆(1,0,0) o be less han he
non-in es men e u n 𝐸𝑆(0,0,0) = (1 +𝐶𝑆∕𝑃𝑆)𝑃𝑆−𝐶𝑆=𝑃𝑆. A simila
cos –bene i assessmen o applica ion se ice p o ide s shows ha in es ing
in da a p i acy p o ec ion 𝐸𝐻(0,1,0) = (1 +𝛽0)𝑃𝐻−𝐶𝐻does no exceed
he s aigh o wa d e u n 𝐸𝐻(0,0,0) = (1 +𝐶𝐻∕𝑃𝐻)𝑃𝐻−𝐶𝐻=𝑃𝐻. In
addi ion, he go e nmen aces a si ua ion whe e egula o y cos s 𝐶𝐺exceed
he sum o basic ines and losses 𝑅0+𝐿, which in i sel is g ea e han wice
he ines and losses 2𝐹+𝐿. In his con ex , he go e nmen is mo e inclined
o op o lax egula ion. In summa y, due o limi ed p o i ma gins, bo h
la ge language model p o ide s and applica ion se ice p o ide s will choose
no o in es in da a p i acy p o ec ion, while he go e nmen op s o lax
egula ion. The e o e, he sys em will end o e ol e in o a s a e whe e all
pa ies choose no o in es and no o s ic ly egula e, solidi ying (0,0,0)
as he sys em’s ESS unde his condi ion.
P oposi ion 2. When he condi ions 0< 𝛼0< 𝐶𝑆∕𝑃𝑆,𝛼0< 𝛼1<
𝜉𝑆+𝐶𝑆−𝑃𝑆
𝑃𝑆
, and 𝐶𝐻∕𝑃𝐻< 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻
𝑃𝐻
a e me , as shown in
Fig. 7, he sys em eaches he sys em’s equilib ium poin (0,1,0) a e 50
e olu ions.
I 𝐶𝐺> 𝑅0+𝐿, a cos –bene i analysis o he egula o y au ho i y
yields:
𝐸𝐺(0,1,1) =𝑅0−𝐶𝐺< 𝐸𝐺(0,1,0) (18)
Fig. 6. Diag am o he e olu ion pa h unde P oposi ion 1.
In his case, he egula o y au ho i y will en o ce lax egula ion. Due
o 𝐶𝐻∕𝑃𝐻< 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻
𝑃𝐻
, a cos –bene i assessmen o he
in es men e u n o applica ion se ice p o ide s is conduc ed:
𝐸𝐻(0,1,0) = (1 +𝛽0)𝑃𝐻−𝐶𝐻
>(1 +𝐶𝐻
𝑃𝐻)𝑃𝐻−𝐶𝐻
=𝑃𝐻=𝐸𝐻(0,0,1)
(19)
This indica es ha he applica ion se ice p o ide is inclined o
unde ake da a p i acy p o ec ion. A cos –bene i assessmen o he
in es men e u n o la ge language model p o ide s shows:
𝐸𝑆(1,1,0) = (1 +𝛼1)𝑃𝑆−𝐶𝑆
<(1 +𝜉𝑆+𝐶𝑆−𝑃𝑆
𝑃𝑆)𝑃𝑆−𝐶𝑆
=𝜉𝑆=𝐸𝑆(0,1,0)
(20)
The e o e, he la ge language model p o ide is no inclined o
in es unde hese condi ions.
P oposi ion 3. When he condi ions 𝐶𝑆∕𝑃𝑆< 𝛼0< 𝛼1<𝜉𝑆+𝐶𝑆−𝑃𝑆
𝑃𝑆
,
0< 𝛽0< 𝐶𝐻∕𝑃𝐻, and 𝛽0< 𝛽1<𝜉𝐻+𝐶𝐻−𝑃𝐻
𝑃𝐻
a e me , as shown in Fig. 8,
he sys em eaches he sys em’s equilib ium poin (1,0,0) a e 50 e olu ions.
Simila ly, i is ound ha 𝐸𝐺(1,0,1) =𝑅0−𝐶𝐺< 𝐸𝐺(1,0,0), hence
he egula o y au ho i y is inclined o choose lax egula ion. Based on
he gi en pa ame e ange, a cos –bene i assessmen o he in es men
e u ns o bo h la ge language model p o ide s and applica ion se ice
Ope a ions Resea ch Pe spec i es 14 (2025) 100327
8
