Jou nal o Cleane P oduc ion 355 (2022) 131662
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Sus ainabili y in ou ism de e mined by an asymme ic game wi h mobili y
Manuel Chicaa,b,∗, Juan M. He nándezc, Ma jaž Pe cd,e, ,g
aAndalusian Resea ch Ins i u e DaSCI ‘‘Da a Science and Compu a ional In elligence", Uni e si y o G anada, 18071 G anada, Spain
bSchool o Elec ical Enginee ing and Compu ing, The Uni e si y o Newcas le, Callaghan, NSW 2308, Aus alia
cDepa men o Quan i a i e Me hods in Economics and Managemen , Uni e si y Ins i u e o Tou ism and Sus ainable Economic De elopmen (TIDES), Uni e si y
o Las Palmas de G an Cana ia, 35017 Las Palmas, Spain
dFacul y o Na u al Sciences and Ma hema ics, Uni e si y o Ma ibo , Ko oška ces a 160, 2000 Ma ibo , Slo enia
eDepa men o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y, Taichung, Taiwan
Alma Ma e Eu opaea, Slo enska ulica 17, 2000 Ma ibo , Slo enia
gComplexi y Science Hub Vienna, Jose s äd e s aße 39, 1080 Vienna, Aus ia
ARTICLE INFO
Handling Edi o : Zhi u Mi
Keywo ds:
Asymme ic game
Spa ial s uc u e
Mig a ion
E olu iona y game heo y
Sus ainable ou ism
O e - ou ism
ABSTRACT
Many coun ies wo ldwide ely on ou ism o hei economic well-being and de elopmen . Bu wi h issues
such as o e - ou ism and en i onmen al deg ada ion looming la ge, he e is a p essing need o de e mine
a way o wa d in a sus ainable and mu ually ewa ding manne . Wi h his mo i a ion, we he e p opose an
asymme ic e olu iona y game wi h mobili y whe e local s akeholde s and ou is s can ei he coope a e o
de ec in a spa ially s uc u ed se ing. Ou s udy e lec s ha sus ainable ou ism is p ima ily de e mined by
an op imal ade-o be ween economic bene i s o he s akeholde s and hei cos s ela ed o he applica ion
o sus ainabili y policies. In con as , he speci ic bene i s and cos s o he ou is s a e compa a i ely less
ele an . The eade can also obse e ha allowing o g ea e ou is mobili y dec eases coope a ion and
leads o as e pola iza ion among local s akeholde s. In ag eemen wi h obse a ions wo ldwide, we iden i y
dec easing popula ion densi ies in ou is a eas in e ms o bo h, s akeholde s and ou is s, o be a key aid o
g ea e coope a ion and o e all sus ainabili y o ou ism. These esul s a e oo ed in spa ial o ma ions and
complex alliances ha mani es spon aneously h ough he e olu iona y dynamics in a s uc u ed popula ion.
1. In oduc ion
Tou ism is a human ac i i y ha implies in e ac ion among wo
popula ions, ou is s and s akeholde s (i.e., local business, se ice
p o ide s, and esiden s). Con inuous and non- i uous ela ionships
be ween hese ac o s can gene a e o e - ou ism, which is a ype o
unsus ainable ou ism de elopmen cha ac e ized by a change o he
local li es yle and socio-cul u al iden i y, loss o ameni ies, and local
businesses. O e - ou ism is hen e ealed as a physical ‘‘ ou is i ica-
ion’’ o domina ion o ou ism business in he ci y cen e s (Cheung
and Li,2019;Koens e al.,2018;Mihalic,2020;Milano e al.,2019).
This is no a new phenomenon bu he e was a g owing conce n o
unsus ainable ou ism de elopmen in he las decade, mos ly a is-
ing in ci ies. Al hough he causes o o e - ou ism a e complex and
mul idimensional, he mos ele an ones a e he high inc ease o
accommoda ion uni s o ou ism since he i up ion o pla o ms
such as Ai bnb, excessi e a luence o isi o s, he de e io a ion o
en i onmen al condi ions, ood was e, and inapp op ia e beha io o
ou is and local s akeholde (Cheung and Li,2019;Koens e al.,2018;
Milano e al.,2019;Wang e al.,2021a;Se aphin e al.,2018).
∗Co esponding au ho a : Andalusian Resea ch Ins i u e DaSCI ‘‘Da a Science and Compu a ional In elligence", Uni e si y o G anada, 18071 G anada, Spain.
E-mail add esses: [email p o ec ed] (M. Chica), [email p o ec ed] (J.M. He nández), [email p o ec ed] (M. Pe c).
Se e al measu es o a oid o e - ou ism and a o sus ainabili y
ha e been implemen ed in ecen yea s. Some o hem a e o ien ed
o es ic he ac i i y o local s akeholde s. Fo example, he ci y o
Ba celona has applied limi s o accommoda ion g ow h by imi a ing
simila solu ions implemen ed in coas al des ina ion wo decades be-
o e (Dodds,2007;Wea e ,2012). Local go e nmen om Ams e dam
has op ed by es ic ing ou ism- ela ed comme cial ac i i ies in he
ci y cen e (Goodwin,2019). O he policies ha e been ocused on
egula ing some speci ic ou is ic ac i i ies such as banning isi s o
ce ain a eas (e.g., he Maya Beach in Thailand BBC (2018)), imposing
highe ees o axes o isi ce ain ci ies (e.g., Venice Be linghie i
(2021)), se ing s ic egula ions o ou is s’ beha io (e.g., Rome and
Ba celona Goodwin (2019)), and a oiding he ad e ising o some s a
a ac ions (e.g., he Taj Mahal monumen in India Khalil,2017).
Unde s anding he uling o ces o o e - ou ism and he dynamics
o he di e en ou ism playe s a e necessa y o selec he mos ap-
p op ia e policy o handle his phenomenon and mo e owa ds a mo e
sus ainable ou ism de elopmen . This is he main goal o his wo k,
h ps://doi.o g/10.1016/j.jclep o.2022.131662
Recei ed 10 Decembe 2021; Recei ed in e ised o m 10 Ma ch 2022; Accep ed 2 Ap il 2022
Jou nal o Cleane P oduc ion 355 (2022) 131662
2
M. Chica e al.
whe e we p opose and analyze an e olu iona y game model o ep e-
sen he phenomenon o o e - ou ism. We ocus on unde s anding he
dynamics and social in e ac ions d i ing o unsus ainable o sus ainable
ou ism, he ole o he agen s’ beha io , and how o e - ou ism can
appea in a social ne wo k om he dynamic in e ela ionship among
ou is s and local ou ism businesses (we call hem s akeholde s, om
now on).
Ou p oposed model is an e olu iona y asymme ic game which
in oduces wo ime-in a ian oles o he popula ions’ playe s: ou is s
and s akeholde s. Al hough e olu iona y game heo y has been adi-
ionally used o explain s able coope a i e in e ac ions among biolog-
ical species (Ho baue and Sigmund,2003), he ange o applica ion
has widened in he las decade. New e olu iona y models ha e been
ecen ly buil o analyze he condi ions o coope a ion among agen s
in ol ed in di e se con empo a y socioeconomic p oblems, such as
ac ions agains clima e change (Pacheco e al.,2014) and eopening
policies a e COVID-19 pandemic (Chica e al.,2021).
Mo e speci ically, bo h playing oles o he p oposed model can
choose o ei he coope a e (being sus ainable) o de ec (being unsus-
ainable). This sus ainabili y can be unde s ood om an en i onmen al
poin o iew (e.g., dec easing amoun o was e, adop ing low ca bon
emission policies, o ha ing a p ope use o wa e ), o om a social
poin o iew (e.g., by suppo ing local cul u e and business, educing
ensions wi h esiden s, o o e ing au hen ic local and eco- iendly
p oduc s and expe iences by he s akeholde s). The combined ac ions
o he playe s in he spa ial ecosys em p oduce a payo o each playe
when i in e ac s wi h playe s o an opposing ole. The e o e, he model
is an asymme ic game wi h wo species ( ou is and s akeholde s)
and wo s a egies ( hese models a e called bi-ma ix games Ho baue
and Sigmund,2003). In gene al, asymme ic games include in e ac ion
among any indi idual belonging o he same ole o no . Bu in ou case,
in e ac ions a e only possible o indi iduals belonging o di e en
species (i.e., opposing oles).
The playe s o he game occupy nodes in a s uc u ed ne wo k o
in e ac ions and accumula es i s payo s om pai -wise in e ac ions
wi h neighbo s ha ing opposing ole. The asymp o ic s able s a egies
in asymme ic games ha e been long analyzed in in ini e popula ions
using eplica o equa ion (C essman and Tao,2014;Haue e al.,2019;
McA oy and Haue ,2015) and o ini e popula ions using Ma ko
p ocesses (Oh suki,2010). Howe e , when including s uc u ed pop-
ula ions, he dynamic e olu ion o asymme ic models is inc eased in
complexi y and esul s om well-mixed popula ions do no apply in
his con ex (Szabó and Fá h,2007). Some gene al esul s o his kind
o asymme ic games ha e been ound (McA oy and Haue ,2015),
bu assuming es ic i e condi ions o e he s a egy upda e a e e y
ime s ep. In o he cases, as in ou model, simula ions a e a use ul
ool o s udy he e olu iona y dynamics o such game (Guo e al.,
2020;Szolnoki and Pe c,2018;Wang e al.,2021c), and his is why
we employ agen -based simula ions o compu ing he ou comes o he
model (Macal and No h,2005;Adami e al.,2016).
Addi ionally, he model inco po a es a mig a ion p ocess. Bu , o be
close o eali y, his mig a ion p ocess is only pe o med by ou is s,
one o he wo oles in he asymme ic game, which a e he dynamic
playe s while s akeholde s ha e a mo e s a ic na u e. Tou is s can
hen mig a e o emp y nodes a e e y ime s ep when a neighbo ing
posi ion o he ne wo k is acan . This mig a ion p ocess p oduces new
in e ac ions wi h ou is s akeholde s. The e ec o mig a ion in spa ial
e olu ion o e olu iona y games has also been analyzed, a o ing inal
coope a ion (Wang e al.,2021c;Zhang e al.,2021), bu has no been
applied o a single ole in an asymme ic game.
Ou o emos aim is o explo e he condi ions o acili a ing a
coope a i e beha io o he popula ion o a oid o e - ou ism and
help ou ism sus ainabili y. We also p esen he e he i s e olu ion-
a y model o explain how o e - ou ism can unin en ionally appea in
a ou is a ea om he collec i e in e ac ions be ween ou is s and
s akeholde s. Thus, he expe imen a ion and he esul s o his wo k
will consis o : (a) explo ing he ou come o he asymme ic model and
main dynamics o he game and i s pa ame e s, (b) inding he bes
condi ions o a oiding he de ec i e beha io o bo h oles o playe s,
(c) showing he impac o in oducing mig a ion o ou is s and (d),
obse ing he spa ial o ma ion o he s uc u ed popula ion a he end
o he simula ion.
2. Theo e ical amewo k and ela ed wo k
The e m sus ainable ou ism began o be used by academic e-
sea che s a he beginning o he 90s, as an applica ion o he case o
ou ism o he gene al concep o sus ainable de elopmen , popula -
ized by he B und land epo (WCED,1987). The concep ualiza ion
and p ac ice o sus ainable ou ism has been long deba ed h ough-
ou hese las h ee decades, bu he de ini ion gi en by he Wo ld
Tou ism O ganiza ion (UNWTO) has achie ed an ex ended consen-
sus (Mihalic,2020). Acco ding o UNWTO, sus ainable ou ism is a
‘‘ ou ism ha akes ull accoun o i s cu en and u u e economic,
social and en i onmen al impac s, add essing he needs o isi o s, he
indus y, he en i onmen and hos communi ies’’ (UNWTO,2022).
Then, sus ainable ou ism e e s o hese h ee p inciples, he economic,
en i onmen al and socio-cul u al, and a sui able balance among hem
o gua an ee long- e m sus ainabili y (UNWTO,2004).
In his pape we deal wi h o e - ou ism, a ela i ely ecen e m
aimed by i s di usion in social media and social con es . O e - ou ism
can be desc ibed as ‘‘des ina ion whe e hos s o gues s, local o isi o s,
eel ha he e a e oo many isi o s and he quali y o li e in he a ea o
he quali y o he expe ience has de e io a ed unaccep ably’’ (Goodwin,
2017). The e o e, o e - ou ism is a o m o unsus ainable ou ism ha
ocuses on he socio-psychological and socio-poli ical dimensions o
sus ainabili y (Mihalic,2020). I s an i hesis is esponsible ou ism, a
kind o ou ism based on he espec o he local cul u e (Goodwin,
2017). Responsible ou ism is he e o e a sus ainable ou ism in ac ion,
o in o he wo ds, a ou ism whe e isi o s and s akeholde s beha e in
a sus ainable manne (Mihalic,2016).
The ole o ou is beha io and encoun e s wi h s akeholde s as
a cause o o e - ou ism has been also s essed in o he explo a o y
s udies (Koens e al.,2018). O he con ibu ions ela e o e - ou ism o
esiden s’ nega i e a i udes on ou ism, main ained o e ime al hough
he isi o s change beha io (Cheung and Li,2019). Then, o e - ou ism
and esponsible ou ism lean on he ole o ac o s’ beha io o di ec
o p omo e sus ainabili y, in line wi h he ecen in e es in he s udies
o sus ainable ou ism (B amwell e al.,2017).
The model p esen ed in his pape connec s wi h his beha io al
aspec o sus ainabili y by p oposing an e olu iona y game model o
show how o e - ou ism can na u ally appea om he in e ela ionship
among ou is s and local s akeholde s. Fig. 1 shows he aming o
ou wo k wi h espec o he c i ical a iable o s udy. E olu iona y
game models ha e been p e iously used o ep esen he adop ion
o sus ainable p ac ices in ou ism by mean o g een incen i es (He
e al.,2018;An oci e al.,2013) and also o he p omo ion o eco-
ou ism ac i i ies and i s implica ions in a cleane p oduc ion (Wang
e al.,2021b). In hese con ibu ions, he au ho s s udy he dynamics
o ou is s akeholde s, esiden s, and go e nmen s in a mul iplaye
coo dina ion e olu iona y game. Unlike o p e ious s udies, ou model
ocuses on he ela ionships and syne gies be ween ou is s and local
s akeholde s, wi hou assuming any incen i e o ex e nal in luence o
o ien he sys em o sus ainabili y. By doing so, we analyze he inne
o ces in he ou is -s akeholde dyad ha can lead o sus ainable o
unsus ainable ou ism. The de ails o he e olu iona y model a e gi en
in he nex Sec ion 3.
3. Model
The e olu iona y game is asymme ic since consis s o a ini e se o
𝑍agen s ha ing wo di e en and ime in a ian oles: being a ou is
Jou nal o Cleane P oduc ion 355 (2022) 131662
3
M. Chica e al.
Fig. 1. Theo e ical amewo k and ou line o ou p oposed e olu iona y model o sus ainable ou ism.
(𝑇) o a s akeholde (𝑆). The model sa is ies ha 𝑍=𝑍𝑇+𝑍𝑆wi h 𝑍𝑇
being he numbe o ou is s and 𝑍𝑆 he numbe o s akeholde s. All he
playe s a e dis ibu ed on he nodes o a social ne wo k. Speci ically,
we conside a egula la ice, gi en i s simplici y wi h espec o o he
in e ac ion ne wo ks such as scale- ee ne wo ks bu s ill ensu ing he
undamen al p ope y o a limi ed playe s in e ac ion ange. Thanks o
his limi a ion in he in e ac ion ange p o ided by he squa e la ice,
we p o ide su icien condi ions o obse e all possible esul s ha a e
due o pa e n o ma ion in he s udied sys em. The egula la ice has
size 𝐿×𝐿, being 𝑍≤𝐿×𝐿so he la ice can ha e acan cells. We call
𝜌=𝑍
𝐿×𝐿 o he ime in a ian popula ion densi y o he la ice and
𝜌𝑇= 1 − 𝑍𝑆
𝑍 o he ou ism p essu e.
Independen ly om i s ole, an agen 𝑖 om bo h popula ions can
adop wo s a egies: coope a ion 𝐶o de ec ion 𝐷. Tou is coope a o s
a e no ed by 𝑇 𝐶 while ou is de ec o s a e no ed by 𝑇 𝐷. We use he
same no a ions o s akeholde s, being 𝑆𝐶 and 𝑆𝐷 he s akeholde
coope a o s and s akeholde de ec o s, espec i ely. In ou model, co-
ope a ion means o be sus ainable while de ec ion is unde s ood as
a ou is op ion ha does no conside sus ainabili y in he ou ism
ansac ion.
3.1. Payo s ma ix
We de ine he e he pai -wise in e ac ion be ween wo playe s in
he game ha ing opposing oles. In he model, a coope a i e beha io
owa ds sus ainabili y can be seen as ob aining an added alue o he
ou ism expe ience and some e enue o s akeholde s while bo h o
hem pay a cos in a sho - e m ho izon. Table 1 shows he payo
ma ix o he asymme ic game and hei in e ac ions o hese wo
ypes o oles and bo h coope a ing and de ec ing s a egies.
In he de ined payo s ma ix, e e y coope a ing ou ism s ake-
holde 𝑆𝐶 o e s a sus ainable p oduc o se ice o a ou is . A ou is
pays hen an addi ional cos 𝜖∈ [1,2] o his sus ainable ansac ion
bu ob ains an added alue 𝜈(we se 𝜈= 1 o simplici y in he es o
he pape ). This addi ional cos is also e e ed as he willingness o pay
alue o a consume when ha ing sus ainable p oduc s o se ices o
be chosen in hei decision p ocess (Zhao e al.,2018). On he o he
hand, he e enue ob ained by he s akeholde om he sus ainable
ansac ion is equal o he addi ional cos 𝜖paid by he ou is .
I a ou is playe de ec s (𝑇 𝐷) and is no sus ainable when he
s akeholde o e s a sus ainable asse , he coope a o s akeholde 𝑆𝐶
educes he /his p o i by an addi ional cos 𝛾∈ [0,1]. Tou is s, bo h
Table 1
Payo s ma ix o each playe in he asymme ic game o sus ainable ou ism.
Tou ism s akeholde (𝑆)
𝐶 𝐷
Tou is (𝑇)
𝐶 𝜈 −𝜖,𝜖−1 − 𝜏, 1
𝐷 𝜈 −𝜖,𝜖−𝛾−1 , 1
No es:𝜖is he cos o he ou is when choosing sus ainable ou ism ( he e enue o
he s akeholde ); 𝜈is he added sus ainabili y alue o he ou is ; 𝛾is he addi ional
cos o s akeholde o a ending a ou is de ec o ; and 𝜏is he ou is discom o when
he p o ided o e is no ul illing hei expec a ions.
coope a o s 𝑇 𝐶 and de ec o s 𝑇 𝐷, ob ain he added alue om he
sus ainable se ice 𝜈minus i s addi ional cos 𝜖 om he ansac ion
wi h coope a ing s akeholde s. When a s akeholde adop s a de ec ing
s a egy 𝑆𝐷, i s payo is a no malized alue o 1independen ly om
he s a egy ollowed by ou is s. Coope a ing ou is s 𝑇 𝐶 also pay a
cos when s akeholde s do no p o ide a sus ainable asse , modeled by
a discom o pa ame e 𝜏∈ [0,1]. In his way, ou is s ob ain a lowe
payo o −1 − 𝜏i hey adop a sus ainable o coope a ion s a egy 𝐶
and hey do no ob ain he expec ed enue.
3.2. Payo s accumula ion and playe s’ s a egy upda e
Playe s o bo h oles in e ac wi h hei di ec neighbo s in a pai -
wise in e ac ion in he la ice. A ocal playe 𝑖only plays wi h hose
ha ing opposing oles and accumula es i s payo in 𝛱𝑖 om all i s
in e ac ions i a leas he e is one possible in e ac ion; being ou he
maximum numbe o possible in e ac ions i all he neighbo ing cells
a e occupied by agen s o opposing oles.
The dynamics o he game includes an imi a ion p ocess o he
neighbo ing agen s in he spa ial la ice and a mu a ion o andomly
change he s a egies o he playe s a e e y ime s ep 𝑡. A playe
𝑖changes i s s a egy a andom wi h a mu a ion p obabili y 𝜇and
imi a es he s a egy o a local neighbo wi h p obabili y 1 − 𝜇. As
playe s will keep hei ole o he whole simula ion, hey can only
imi a e o he playe s in he la ice ha ing he same ole in he game.
A e playing he pai wise in e ac ions in he la ice and accumula ing
i s payo s in 𝛱𝑖, a playe 𝑖has he oppo uni y o upda ing i s s a egy
acco ding o his payo alue in p e ious ime s ep 𝑡− 1 and he ones
om hei neighbo s.
Jou nal o Cleane P oduc ion 355 (2022) 131662
4
M. Chica e al.
Fig. 2. We obse e om he simula ion esul s how he sys em easily achie es a s a iona y s a e o wo di e en densi ies 𝜌∈ {0.3,0.8}, wi h and wi hou mig a ion. The es o
pa ame e s a e 𝜖= 2.0,𝛾= 1.0,𝜏= 0.5wi h equal ini ial equency o s a egies.
In his imi a ion p ocess, a playe 𝑖imi a es a playe 𝑗ha ing he
same ole as 𝑖wi h a p obabili y 𝑝𝑖→𝑗 ha inc eases wi h hei payo
di e ence (𝛱𝑗−𝛱𝑖, being 𝛱𝑖and 𝛱𝑗 he payo s o 𝑖and 𝑗in 𝑡− 1,
espec i ely), as done in (T aulsen e al.,2006):
𝑝𝑖→𝑗=1
1 + 𝑒−(𝛱𝑗−𝛱𝑖)∕𝜅,(1)
whe e he ampli ude o noise 𝜅equals o 0.1 as done in Wang e al.
(2021c) and Zhang e al. (2021). I is impo an o no ice ha a playe
𝑗canno be imi a ed i i did no in e ac wi h any o he playe and did
no accumula e any payo in p e ious ime s ep 𝑡− 1. This can happen
i , o ins ance, a playe is isola ed in he la ice o all he neighbo ing
playe s ha e he same ole.
3.3. The mig a ion p ocess
The model also conside s a mig a ion p ocess o playe s ha ing
a ou is ole. Tou is playe s mo e h ough he la ice a each ime
s ep 𝑡 o a acan neighbo ing cell independen ly om hei s a egy.
In case he e a e mo e han one neighbo ing acan posi ions, he
agen picks one acan posi ion a andom o he mig a ion mo emen .
S akeholde playe s canno mo e and keep he same posi ion o he
spa ial cell du ing he simula ion. I popula ion densi y 𝜌= 1, he
spa ial la ice has no acan posi ions and he e o e, no mig a ion is
allowed.
3.4. Agen -based compu e simula ions
We use Mon e-Ca lo (MC) agen -based simula ions (Macal and
No h,2005;Adami e al.,2016), pe o med in compu ing clus e s
and eso ing o pa allel compu ing a chi ec u es. E olu ion p oceeds
in disc e e s eps in ol ing he payo s accumula ion, upda e ules, and
mig a ion p ocesses in line wi h he dynamics desc ibed abo e. The
mu a ion (o explo a ion) p obabili y (𝜇) equals 1
𝑍in all expe imen s.
Fo he agen -based simula ions, he size o he popula ion is 𝑍=
4.9×103in a squa ed egula la ice o 70 ×70 wi h pe iodic bounda y
condi ions and Von Neumann neighbo hood. Each node o he la ice is
ei he acan o occupied by one playe . The acan posi ions a e gi en
by densi y 𝜌. We un he model o 30 independen MC ealiza ions
and a maximum numbe o 103synch onous ime s eps, whe e all he
ealiza ions each a s a iona y s able s a e and de ia ion om he MC
ealiza ions is low. Finally, all he simula ion esul s we e ob ained by
a e aging he las 25% o he simula ion ime s eps in he independen
MC ealiza ions.
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M. Chica e al.
Fig. 3. Final equency o 𝑇 𝐶 and 𝑆𝐶 o a sensi i i y analysis on 𝛾and 𝜖pa ame e s (𝜏= 0.5). We see how mig a ion educes coope a ion o ou is s and gene a es wo clea
di e en egions wi h global coope a ing and de ec ing s a egies (pola iza ion o he popula ion). Highe densi ies also help o achie e his pola iza ion.
4. Resul s
We i s analyze he gene al dynamics o he asymme ic model.
La e , we e alua e he implica ions o he main pa ame e s o bo h sub-
popula ions and how popula ion densi y and mig a ion a ec he ou -
come o he model. Finally, we p esen he spa ial o ma ions ob ained
in he la ice.
4.1. Tempo al e olu ion o he model and s a iona y s a e
As s a ed in Sec ion 3.4, he model needs o be analyzed h ough
simula ions since gene al esul s o asymme ic games do no apply
in he con ex o a game in s uc u ed ne wo k allowing mig a ion.
Ne e heless, we can i s analyze he asymp o ic end o he model
in a simple case, well-mixed and in ini e popula ion, by using he
eplica o equa ion. Calcula ion de ails can be ound in Appendix.
Unde hese condi ions, he asymp o ic s able s a egy depends on he
ela i e alues be ween he s akeholde s’ bene i ma gin when being
sus ainable and a ending a coope a o ou is (𝜖− 1) and he cos o
a ending a de ec o ou is (𝛾). The inal s a egy is pu e de ec ion
in bo h popula ions when 𝛾 > 𝜖 − 1 and pu e coope a ion only o
s akeholde s when 𝛾≤𝜖− 1. The speci ic asymp o ic s able s a egy
o ou is s in he la e case anges be ween pu e coope a ion and
pu e de ec ion, depending on he ini ial coope a ion le els in bo h
popula ions.
The la e obse a ion helps o o esee he ou comes o he gene al
model. Fi s , we s udy whe he he simula ions con e ge o s able
s a egies o di e en alues o he pa ame e s. Fig. 2 p esen s he
empo al e olu ion o he s a egies adop ed by ou is and s akeholde
playe s unde di e en popula ion densi ies 𝜌and mig a ion se ings.
The esul s con i m ha he s a egy s abilizes a e 400 ime s eps
o mos o he cases. The highes a iabili y in he inal equency
o s a egies a e when ha ing low popula ion densi ies (𝜌= 0.3)
wi h mig a ion. Ne e heless, hese igu es con i m he s abili y o he
coope a ion le els in he long e m o di e en condi ions o he game.
Addi ionally, no signi ican changes we e obse ed when modi ying he
ini ial equency o s a egies o bo h oles o playe s. The e o e, we
will se an equal ini ial equency o s a egies o ha e he same numbe
o playe s wi h 𝑇 𝐶,𝑇 𝐷,𝑆𝐶, and 𝑆𝐷 in he es o he expe imen s.
4.2. Mig a ion and popula ion densi ies e ec s
In his sec ion we un di e en sensi i i y analysis on he main
pa ame e s o he model o unde s and he dynamics o he game when
ha ing mig a ion unde di e en densi ies o he popula ion. Fi s ,
Fig. 3 shows a sensi i i y analysis on ou is s’ cos (𝜖) and s akeholde s’
cos (𝛾) when ha ing di e en popula ion densi ies (𝜌), by also compa -
ing he dynamics wi h and wi hou mig a ion. Values o he hea maps
show he inal equency o ou is s coope a o s 𝑓𝑇 𝐶 and s akeholde s
coope a o s 𝑓𝑆𝐶 .
F om Fig. 3 we can obse e how he esul s o high densi y alues
(𝜌= 0.8) a e simila o he analy ical ou come o he well-mixed and
in ini e popula ion shown in Appendix. When 𝛾 > 𝜖 − 1, pu e de ec ion
is he expec ed s able s a egy o bo h ypes o playe s 𝑇 𝐶 and 𝑆𝐶.
When 𝛾≤𝜖− 1, s akeholde s show high le els o coope a ion in he
popula ion. In he la e case, he s able s a egy o ou is s depends
on he speci ic ini ial condi ions.
When mig a ion is in oduced, we a e a o ing mo e in e ac ions
among ou is and s akeholde s by consequen ly elaxing he e ec
o he ixed la ice. This gene a es wo clea egions wi h global co-
ope a ing and de ec ing s a egies o s akeholde s, abo e and below
he line 𝛾=𝜖− 1. Some combina ions o pa ame e s gene a e a
ull coope a ion in he sub-popula ion o s akeholde s (𝑓𝑆𝐶 is 1). In
con as , mig a ion is clea ly dec easing coope a ion o ou is s o
di e en densi y alues. The e o e, when mig a ion is no induced, he
bounda y be ween ha ing ei he coope a ing o de ec ing popula ions
is smoo he o s akeholde s while inal equency o 𝑇 𝐶 is almos
independen om he pa ame e s o he model and highe han when
ha ing mig a ion.
In o de o u he s udy he e ec s o he pa ame e s in he model,
Fig. 4 shows he inal equency o coope a o s 𝑓𝑇 𝐶 and 𝑓𝑆𝐶 o a
sensi i i y analysis on s akeholde s’ cos s (𝛾) and ou is discom o (𝜏).

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M. Chica e al.
Fig. 4. Final equency o 𝑇 𝐶 and 𝑆𝐶 o a sensi i i y analysis on 𝛾and 𝜏pa ame e s (𝜖= 1.2). We show ha low densi ies inc ease coope a ion in ou is s as well as educes
pola iza ion in he esul s. The inclusion o mig a ion a o s he opposi e: highe pola iza ion in he esul s and a lowe ou is s’ coope a ion le el.
We es di e en densi y alues 𝜌as done be o e, wi h and wi hou
inducing mig a ion o ou is s. The esul s con i m ha he e ec o
he ou is discom o is less signi ican han he o he pa ame e s,
as shown in he analy ical s udy o he well-mixed popula ion in
Appendix.
Mo e impo an ly, we obse e he same insigh s wi h espec o he
e ec s o densi ies and mig a ion in he inal equency o coope a o s
o bo h sub-popula ions. We can summa ize he main obse a ions
below:
•When allowing mig a ion, esul s a e mo e pola ized han when
mig a ion is no allowed and highe le els o coope a ion a e
achie ed o s akeholde s. Howe e , coope a ion in he ou is s
sub-popula ion (𝑓𝑇 𝐶 ) is dec eased wi h espec o no mig a ion.
This phenomenon was also obse ed in Fig. 3.
•Dec easing he popula ion densi y educes esul s’ pola iza ion as
well as inc eases homogenei y in he esul s. S akeholde s a e
he playe s ha ing he highes sensi i i y wi h espec o he
pa ame e s’ alues and he inal coope a ion. Fo ins ance, when
𝜌= 0.8and mig a ion is induced, he e is a s eep and apid
ansi ion om 𝑓𝑆𝐶 = 1 o 𝑓𝑆𝐶 = 0.
•Tou is s ob ain highe le el o coope a ion when popula ion den-
si ies a e low. The inal s able s a egy depends on he speci ic
ini ial condi ions. The e o e, c owded popula ions inc ease he
chances o ha ing de ec ing ou is s, apa om apidly mo ing
om de ec ing o coope a ing s a es in he sub-popula ion o
s akeholde s.
4.3. Spa ial o ma ion
Fig. 5 shows he la ices a ime s eps 𝑡= 0 and 𝑡= 900 o he
popula ion o playe s wi hou mig a ion and ull densi y. In addi ion,
Fig. 6 shows he la ices o he same ime s eps 𝑡= 0 and 𝑡= 900
when ha ing mig a ion and 𝜌= {0.3,0.8}. Tou is s a e ep esen ed by
ed cells (ligh ed o 𝑇 𝐶 and in ense ed o 𝑇 𝐷) while s akeholde s
a e blue cells (ligh blue o 𝑆𝐶 and in ense blue o 𝑆𝐷). When he
densi y o he popula ion is no ull, whi e cells a e acan cells.
Fo all he cases we can see how, when showing he s a iona y
s a e a 𝑡= 900, he e a e clea spa ial o ma ions o coope a o s and
de ec o s o opposing oles. Coope a o s in he wo sub-popula ions
a e joined in isola ed o weakly in e connec ed clus e s. The same
spa ial disposi ion occu s o de ec o s in he wo popula ions. When
including mig a ion, he coope a o s clus e s a e smalle and mo e
isola ed, whe eas he de ec i e s a egy p edomina es.
5. Discussion
We p esen in his sec ion he heo e ical implica ions o ou s udy
in Sec ion 5.1, he p ac ical insigh s ob ained om he model in Sec-
ion 5.2, and he limi a ions o ou wo k in Sec ion 5.3.
5.1. Theo e ical implica ions
The e olu iona y game model p esen ed in his pape explains how
he playe s’ beha io in he ou ism sys em ( ou is and s akeholde )
de e mines o e - ou ism in a ou is a ea. P e ious analyses ha e de-
sc ibed mul iple causes o o e - ou ism, such as he i up ion o Ai bnb,
low-cos ips and unsus ainable beha io (Koens e al.,2018;Goodwin,
2017). Howe e , he mechanism om which he des ina ion slides in o
o e - ou ism has no been add essed in he scien i ic li e a u e ye . In
his ega d, his pape a emp s o ep esen his mechanism by using
a dynamic model whe e ou is and s akeholde in e ac in a spa ial
s uc u e.
P e ious applica ions o e olu iona y games ha analyze he adop-
ion o sus ainable p ac ices in ou ism assume he exis ence o an
exogenous incen i e o a o sus ainable p ac ices. Fo example, sub-
sidies and penal ies o s akeholde (He e al.,2018;Wang e al.,
2021b) o en i onmen al bonus o ou is (An oci e al.,2013). This
kind o incen i es a e absen in ou model. Ins ead, he model de-
sc ibes how sus ainable o unsus ainable ou ism can a ise om he
economic and psychological condi ions o he playe s, wi hou any
ex e nal in e en ion.
Then, he model and i s esul s p esen an inno a i e ocus o
he ques ion o sus ainabili y, showing he ac o s ha help o hinde
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M. Chica e al.
Fig. 5. La ices showing spa ial o ma ions wi h playe s o simila s a egies om 𝑡= 0 (le plo ) o 𝑡= 900 ( igh plo ) when ha ing ull densi y 𝜌= 1 and wi hou mig a ion.
Colo s mean: ed (𝑇 𝐷); ligh ed (𝑇 𝐶); Blue (𝑆𝐷), ligh blue (𝑆𝐶). The es o pa ame e s a e 𝜇=,𝜖= 1.2,𝛾= 0.3,𝜏= 0.5.
Fig. 6. La ices showing spa ial o ma ions wi h playe s o simila s a egies om 𝑡= 0 (le column) o 𝑡= 900 ( igh column) o 𝜌= {0.3,0.8} when allowing mig a ion. Colo s
mean: ed (𝑇 𝐷); ligh ed (𝑇 𝐶); Blue (𝑆𝐷), ligh blue (𝑆𝐶), and whi e cells a e acan cells. The es o pa ame e s a e 𝜇=,𝜖= 1.2,𝛾= 0.3,𝜏= 0.5.
Jou nal o Cleane P oduc ion 355 (2022) 131662
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M. Chica e al.
agen s ( ou is and s akeholde s) o adop sus ainable beha io . The
indings p o ide use ul in o ma ion o esea che s and p ac i ione s
abou he inne o ces ha lead o (un)sus ainable ou ism. Speci -
ically, wo ele an indings can be ex ac ed om he simula ions
o he model. Fi s , we show ha sus ainable ou ism p ima ily de-
pends on he ela i e impo ance be ween e enue ma gin and cos s o
s akeholde when o e ing a sus ainable p oduc , whe eas he ou is
discom o wi h unsus ainable se ice does no play a majo ole. And
second, we gene ally obse e ha educing he densi y and mig a ion o
he popula ion o ou is s gene a es homogeneous ou come o di e se
alues o he model’s pa ame e s such as ou is s and s akeholde s cos s
o ou is discom o . In o he wo ds, long- e m o e - ou ism is mo e
likely i he popula ion densi y and ou is mobili y in he des ina ion
is high.
Those p e ious indings de e mine a s a ing poin o u he ana-
lyze he ac o s in luencing o e - ou ism in ou is ic a eas by including
new eal-based condi ions o he model, such as ano he social s uc-
u e and go e nmen al in e en ions. Mo eo e , al hough he model
was applied o ou ism, i s de ini ion and esul s can be applied o
o he simila phenomenon whe e in e ac ions be ween cus ome s and
s akeholde s gene a e a sus ainabili y p oblem.
5.2. P ac ical implica ions
The indings in his pape can be used o check he sui abili y o he
policy measu es cu en ly implemen ed in ce ain des ina ions o man-
age o e - ou ism, such as limi ing accommoda ion and se ice g ow h,
banning ou is isi s, imposing highe ees and axes o ou is s, and
egula ing ou is ’s beha io . Among hem, hose policies o ien ed o
limi ou ism g ow h and p e en high a luence o ou is , conse-
quen ly educing popula ion densi y in he des ina ion, a e alida ed
by ou indings. In addi ion, measu es o discou age o a oid spa ial
ou is mobili y will a o long- e m sus ainabili y as well. Howe e ,
egula ions o ien ed o inc ease cos s o ou is s, such as ees o axes,
a e no jus i ied by ou esul s.
Addi ionally, ou indings can be employed o p opose no el ecom-
menda ions o help des ina ion manage s and decision make s when
implemen ing sus ainable policies. The ou come o he model shows
ha he elemen s go e ning he economic bene i and cos o ou ism
s akeholde s a e he key ac o s explaining o e - ou ism, independen ly
on o ces explaining ou is ’s decision. The e o e, manage s ha e o i s
poin ac ions o inc ease s akeholde e enues ob ained wi h sus ain-
able ou is p ac ices, such as hose espec ing local cul u e, p ese ing
en i onmen and dis u bing he leas o he local li es yle.
5.3. Limi a ions
This wo k p esen s some limi a ions. The phenomenon o o e -
ou ism is complex and some aspec s a e no conside ed in his e o-
lu iona y model. One o hese aspec s is ha he model only ocuses on
ou is s and s akeholde s’ beha io , gi en aside o he e ec s d i ing
o o e - ou ism, such as he en i onmen al condi ions. Mo eo e , we
do no include he ea u es o he p oduc s o se ices o e ed by
s akeholde s no he adap a ion o hei po olio o asse s o ou is s.
6. Concluding ema ks and u u e wo k
The s udy we p esen he e models how o e - ou ism can a ise om
he dynamic in e ac ions among ou is s and local s akeholde s. Playe s
o he e olu iona y game om bo h oles decide a e e y ime s ep
whe he ha ing a sus ainable o unsus ainable beha io and ob ain a
pay-o om hese pai -wise in e ac ions. Then, he model includes a
dynamic mechanism in he agen s’ beha io o achie e a long- e m
si ua ion wi h o wi hou o e - ou ism. The model allows analyzing he
in luence o some ac o s on achie ing long- e m sus ainable ou ism.
Ou esul s show ha , in a eas wi h high ou is densi y, he ela i e
alues be ween he bene i ma gin o coope a i e s akeholde s when
in e ac ing wi h a sus ainable ou is and he cos o a ending a ou is
de ec o de e mines he inal ou come. Thus, when he bene i ma gin
is g ea e han he cos , a sus ainable si ua ion in he long e m is
expec ed. The opposi e occu s i he bene i ma gin is lowe han he
cos . O he ac o s, such as he ou is discom o and he added alues
o ou is , play a lowe ole in sus ainabili y.
The e ec o he spa ial mig a ion o mo emen s o he ou is s as
well as popula ion ha e also been analyzed. The gene al inding is ha
mig a ion and high densi y educes he gene al le els o coope a ion
o ou is s while pola ize he esul s o he s akeholde s popula ion.
In ac , coope a ion is bo h maximum and minimum o s akeholde s
when ha ing high densi y and ou is s’ mig a ion. Finally, we also
obse ed spa ial o ma ions in he la ice whe e we ind islands o bo h
oles (i.e., ou is s and s akeholde s) ha ing he same s a egy. These
esul s a e also ound when he densi y is dec eased and bo h wi h and
wi hou mig a ion.
Fu u e esea ch can be ex ended in di e en ways. Fi s , ewa ding
policies (Chen e al.,2015) can be applied o one o bo h o he playe
oles o he game o aid decision make s in hei ask o decide how
hey con igu e hei incen i iza ion policies. Second, mo e ad anced
mig a ion p ocesses, close o he ou ism eali y, can be inco po a ed
in o he model. Finally, ano he in e es ing u u e pa h will be s udying
a mo e gene al e olu iona y amewo k able o analyze and unde s and
he e ec s o mig a ion p ocesses in asymme ic e olu iona y games.
CRediT au ho ship con ibu ion s a emen
Manuel Chica: So wa e, Me hodology, Valida ion, In es iga ion,
W i ing – o iginal d a , Funding acquisi ion. Juan M. He nández:
Concep ualiza ion, In es iga ion, Fo mal analysis, W i ing – o iginal
d a . Ma jaž Pe c: Me hodology, Fo mal analysis, W i ing – e iew
edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
M.C. was suppo ed by he Spanish Minis y o Science, Andalusian
Go e nmen , Uni e si y o G anada, and ERDF unde g an s SIMARK
(P18-TP-4475), RYC-2016-19800, and PPJIA2020-09 (TURCOMPLEX).
J.H. was suppo ed by he Uni e si y o Las Palmas de G an Cana ia,
unde g an COVID-19 04. M.P. was suppo ed by he Slo enian Re-
sea ch Agency (G an Nos. P1-0403 and J1-2457). Funding o open
access cha ge: Uni e sidad de G anada/CBUA.
Appendix
He e we analyze he asymp o ic s able poin s o he bi-ma ix
game wi h pay-o shown in Table 1, assuming a well-mixed in ini e
popula ion.
A simple algeb aic manipula ion o he pay-o ma ix leads o
conclude ha double-de ec ion s a egy is a s ic Nash equilib ium
(NE) i and only i 𝛾 > 𝜖 −1 and he e o e i is he asymp o ically s able
s a egy (C essman and Tao,2014).
To analyze he case 𝛾≤𝜖− 1, we make use o he eplica o
equa ion (Ho baue and Sigmund,2003). Fi s , by adding con enien
cons an s o he columns, we ans o m he pay-o ma ix in Table 1
in o wo squa e pay-o ma ices wi h diagonal e ms equal ze o, hey
a e
Jou nal o Cleane P oduc ion 355 (2022) 131662
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M. Chica e al.
𝑇=(0 −𝜏
0 0 ), 𝑆 =(0𝜖−1−𝛾
−(𝜖− 1) 0 ).
Ma ix 𝑇and 𝑆ha e he same eplica o equa ion han he pay-o
ma ices o ou is s and s akeholde s, espec i ely. We no e 𝑥and 𝑦
he p opo ion o coope a o s in he ou is and s akeholde popula ion,
espec i ely. I is clea ha 0≤{𝑥, 𝑦}≤1. Then, he dynamic e olu ion
o hese a iables is go e ned by he ollowing di e en ial equa ions:
𝑥 = −𝜏𝑥(1 − 𝑥)(1 − 𝑦),
𝑦 =𝛾𝑦(1 − 𝑦)(𝜖−1−𝛾
𝛾+𝑥).(2)
Gi en ha 𝜖−1−𝛾≥0, he equilib ium o es poin s o his sys em
a e (𝑥1, 𝑦1) = (0,0),(𝑥2, 𝑦2) = (1,0) and he segmen o med by 𝑦= 1
and any alue 𝑥∈ [0,1]. Analyzing he Jacobian ma ix o he sys em
(2), poin s (𝑥1, 𝑦1) = (0,0) and (𝑥2, 𝑦2) = (1,0) a e uns able. Howe e ,
any poin in he segmen 𝑦= 1 and 0≤𝑥≤1is s able. Then, hese
in ini e poin s a e po en ial asymp o ic s able s a egies, which a e
cha ac e ized by pu e coope a ion o s akeholde s and any coope a ion
le el o ou is s. The speci ic asymp o ic s a egy o ou is s depends
on he ini ial coope a ion le els in he wo popula ions.
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