Lu, Yi-Shan; Tsai, Chien-Shu; Lee, Jen Yao; Lee, Chung-Yang
A icle
Collusi e s abili y wi h ela i e pe o mance and ne wo k
ex e nali ies
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Lu, Yi-Shan; Tsai, Chien-Shu; Lee, Jen Yao; Lee, Chung-Yang (2024) : Collusi e
s abili y wi h ela i e pe o mance and ne wo k ex e nali ies, Games, ISSN 2073-4336, MDPI, Basel,
Vol. 15, Iss. 3, pp. 1-7,
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Ci a ion: Lu, Y.-S.; Tsai, C.-S.; Lee,
J.-Y.; Lee, C.-Y. Collusi e S abili y wi h
Rela i e Pe o mance and Ne wo k
Ex e nali ies. Games 2024,15, 21.
h ps://doi.o g/10.3390/g15030021
Academic Edi o s: Ul ich Be ge ,
Kons an inos Se es and
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Recei ed: 5 May 2024
Re ised: 10 June 2024
Accep ed: 17 June 2024
Published: 20 June 2024
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games
A icle
Collusi e S abili y wi h Rela i e Pe o mance and
Ne wo k Ex e nali ies
Yi-Shan Lu 1, Chien-Shu Tsai 2, Jen-Yao Lee 3and Chung-Yang Lee 3,*
1Depa men o Logis ics Managemen , Guangdong Uni e si y o Science and Technology,
Dongguan 523083, China; [email p o ec ed]
2Ins i u e o Ma ine A ai s and Business Managemen , Na ional Kaohsiung Uni e si y o Science and
Technology, Kaohsiung 811213, Taiwan; [email p o ec ed]
3Depa men o In e na ional Business, Na ional Kaohsiung Uni e si y o Science and Technology,
Kaohsiung 807618, Taiwan; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : In his pape , we aim o in es iga e he collusi e s abili y in he p esence o ne wo k
ex e nali ies among i ms wi h ela i e pe o mance in he i m’s objec i e unc ions. We demons a e
ha collusi e s abili y is inc easing (dec easing) in he deg ee o ela i e pe o mance, p oduc
subs i u abili y and ne wo k e ec when he ne wo k e ec is su icien ly la ge (small). A com-
pe i ion agency migh need o p o ide di e en guidance o an i-compe i i e egula ion in he
ne wo k indus y.
Keywo ds: ela i e pe o mance; ne wo k ex e nali ies; s abili y o collusion
JEL Classi ica ion: D43; O34; L13; L41
1. In oduc ion
In ecen yea s, oligopolis ic ma ke s uc u es ha e become p e alen in many indus-
ies, such as he elecommunica ions and mobile phone ma ke s. Concu en ly, ela i e
pe o mance and ne wo k e ec s play an impo an ole in indus ial compe i ion. The
decision-making o he boa d o di ec o s no only ca es abou he pe o mance o he i m
i sel bu pays a en ion o he pe o mance o compe i o s as well; meanwhile, commodi y
p oduc ion be ween i ms also a ec s consume s’ willingness o pay p ices due o ne wo k
e ec s, a ec ing i ms’ compe i i e and coope a i e beha io s.
Cla i ying he condi ions unde which i ms can easily collude will assis an i-monopoly
agencies in moni o ing an i-compe i i e beha io s mo e e ec i ely. F iedman [
1
] was he
i s o show ha coope a ion could be achie ed in an in ini ely epea ed game by using
igge s a egies ha swi ch o e e o he s age-game Nash equilib ium ollowing any
de ia ion, while Denecke e [
2
] p o ided indings abou he abili y o collude in epea ed
Cou no and Be and duopolies; collusion be ween i ms becomes unsus ainable as i ms
p oduce close subs i u es o he commodi y.1
On he collusion p oblem in he adi ional li e a u e, mos se ings o he i ms’
objec i e unc ions a e conside ing hei p o i s. In he business ope a ion model wi h
manage ial delega ion, se e al schemes o manage ial compensa ion ha e been conside ed
by Vicke s [
6
], Fe sh man and Judd [
7
], Mille and Pazgal [
8
] and Jansen e al. [
9
,
10
], among
o he s. In an oligopolis ic p oduc ma ke , sha eholde s s a egically use in o ma ion on
i al i ms’ pe o mances when designing managemen -incen i e con ac s ( o example,
Joh [
11
]; Chowdhu y and Gü le [
12
]; Bloom ield e al. [
13
]). Bloom ield e al. [
13
] exam-
ined whe he he po en ial o cos ly sabo age ac ed as a de e en o i ms’ use o ela i e
pe o mance e alua ion (RPE) in CEO pay plans and exploi ed illegal ca el membe ship
as a sou ce o a ia ion in he po en ial o cos ly sabo age, while documen ing ha i ms
Games 2024,15, 21. h ps://doi.o g/10.3390/g15030021 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 21 2 o 7
a e mo e likely o use RPE i hey a e cu en ly ca el membe s. Ma sumu a and Ma -
sushima [
14
] showed ha when he compe i ion becomes mo e in ensi e among i ms
wi h ela i e pe o mance in he i m’s objec i e unc ions in an oligopoly ma ke wi h
homogenous goods, i des abilizes he collusion. Yang and Zeng [
15
] examined he collu-
si e e ec o c oss-holding wi h asymme y in cos unc ions ac oss i ms in an in ini ely
epea ed Cou no duopoly game and ound ha c oss-holding may ei he acili a e o
hinde collusion. Sun and Wang [
16
] analyzed ups eam i ms’ collusi e sus ainabili y
when downs eam i ms adop he ela i e-pe o mance delega ion in an in ini ely epea ed
Cou no o Be and game and demons a ed ha ela i e pe o mance delega ion makes
manage s’ ac ions mo e agg essi e and ha leads o mo e di icul y in sus aining ups eam
collusion compa ed wi h sales- e enue delega ion ega dless o he compe i ion modes.
Ano he s eam o wo k in he li e a u e has s udied he s abili y o collusion in
e ms o ne wo k ex e nali ies. In a di e en ia ed oligopoly wi h ne wo k ex e nali ies,
Song and Wang [
17
] demons a ed ha he close he subs i u es o p oduc s we e, he
mo e s able he collusion unde ela i ely s ong ne wo k ex e nali ies.
2
Choi and Lee [
19
]
u he showed ha i ne wo k ex e nali ies a e weak (s ong), collusion in quan i ies
(p ices) is mo e s able han in p ices (quan i ies).
3
Lee e al. [
21
] showed ha la ge ne wo k
ex e nali ies lead o less collusi e incen i e o an ine icien i m, while o an e icien i m,
his depends on he e iciency gap. An inc ease in ne wo k ex e nali ies will des abilize he
downs eam collusion when he cos asymme y is la ge and he ne wo k ex e nali ies a e
ela i ely weak.
In his pape , we aim o in es iga e he collusi e s abili y in he p esence o ne wo k
ex e nali ies among i ms wi h ela i e pe o mance in he i m’s objec i e unc ions. The e
a e wo depa u es o ou amewo k om ha o Ma sumu a and Ma sushima [
14
]: one
is ne wo k ex e nali y, and he o he is p oduc di e en ia ion. Ne wo k ex e nali ies
a ec he compe i i eness o he ma ke , playing an impo an ole in he s abili y o
collusion. We show ha he collusi e s abili y is inc easing (dec easing) in he deg ee o
ela i e pe o mance, p oduc subs i u abili y and ne wo k e ec when he ne wo k e ec
is su icien ly la ge (small). Ou inding is di e en om Ma sumu a and Ma sushima [
14
]
in ha he compa a i e s eng h o he posi i e and nega i e e ec s o compe i i eness on
c i ical discoun ac o s is condi ional on he pa ame e s’ alues; o no e, since an inc ease
in he ne wo k e ec will weaken he compe i ion among i ms and he e ec i eness o
punishmen , i will des abilize he collusion.
The es o his pape is a anged as ollows. The basic model is p o ided in Sec ion 2.
Sec ion 3 epo s he main esul s o his pape . Sec ion 4p o ides he concluding ema ks.
2. Basic Model
Suppose ha he e a e wo i ms p oducing di e en ia ed goods wi h cons an p o-
duc ion ma ginal cos ,
c
. Following Hoe nig [
22
], Bha acha jee and Pal [
23
] and Song and
Wang [17], he u ili y unc ion o he ep esen a i e consume is gi en by he ollowing:4
U(x1,x2;y1,y2)=α(x1+x2)
1−γ−(x12+2γx1x2+x22)
2(1−γ2)
+n(y1+γy2)x1+(y2+γy1)x2
1−γ2−y12+2βy1y2+y22
2(1−γ2)+m
whe e
xi
deno es he quan i y o i m
i
’s p oduc ion;
yi
deno es he consume ’s expec a ion
ega ding i m
i
’s o al sales and
α>
0.
m
is he nume ai e ha deno es he consump ion o
all o he goods, while pa ame e s
γ
and
n
indica e he deg ee o p oduc di e en ia ion and
s eng h o ne wo k e ec s, espec i ely.
5
A g ea e
γ
co esponds o a lesse di e en ia ion
o p oduc . We assume
γ∈(0, 1)
and
n∈(0, 2)
.
n<
2 ensu es ha he ou pu is always
posi i e. I is easy o check, as ollows:
∂
yi∂U
∂xi=n
1−γ2;∂
yj∂U
∂xi=γn
1−γ2(1)
Games 2024,15, 21 3 o 7
Hence, a highe magni ude o
n
indica es a s onge posi i e ne wo k e ec gene a ed
om no only one’s own p oduc ion bu also he i al’s p oduc ion.
The in e se demand unc ion o goods iis as ollows:
pi(xi,xj;yi,yj) = α
1−γ−xi+γxj
(1−γ2)+nyi+γyj
1−γ2,i,j=1, 2, i=j(2)
whe e
pi
deno es he p ice o good
i
. A s onge ne wo k e ec can be seen om
Equa ion (2),
as i clea ly shows a highe willingness o pay (p ice) when he expec a ions o he ne wo k
sizes inc ease.
The payo o i m
i(i=1, 2)
is conside ed by ela i e pe o mance and is gi en by
he ollowing:
Ui=πi−λπj,i=j(3)
whe e πiis he p o i o i m i,
πi=pixi−cxi,i=j(4)
Assuming
α
1−γ>c>
0, he consume s’ ese a ion p ice is la ge han he ma ginal
cos and
λ∈(−1, 1)
, indica ing he weigh o ela i e pe o mance o i m
i
’s managemen .
Rela i e pe o mance schemes o manage ial compensa ion ha e been conside ed by Mille
and Pazgal [
8
] and Jansen e al. [
9
,
10
], among o he s. In Jansen e al. [
10
], he weigh on he
i al’s p o i anges be ween ze o and one, whe eas in Mille and Pazgal [8], i may ange
be ween minus one and one. Ma sumu a and Ma sushima [14] in e p e λas a pa ame e
indica ing he se e i y o compe i ion, and i may ange be ween minus one and one.
λ=
0
indica es he s anda d Cou no case,
λ=
1 he pe ec ly compe i i e case and
λ=−
1 he
monopoly case.
Fo p o i accoun ing unde c oss-sha eholding, Zhang and Zhang [
29
] and Yang
and Zeng [
15
] applied he di ec inancial in e es s app oach p oposed by Reynold and
Snapp [
30
]. Zhang and Zhang [
29
] s udied i al y be ween s a egic alliances and de e -
mined ha one na u al in e p e a ion o his o mula ion was c oss-sha eholding in equi y
alliances, whe e he weigh on he i al’s p o i anges be ween ze o and one. Yang and
Zeng [
15
] es ic ed he weigh on he i al’s p o i anges be ween ze o and hal due o
he assump ion ha he sha eholding i m only ecei es inancial in e es s in he in es ed
i m. In ou amewo k, a nega i e
λ
ep esen s he p o i o he in es ed i m aking in o
accoun he sha eholding i m; a posi i e
λ
ep esen s he owne pu ing mo e weigh on
he ela i e pe o mance on his managemen .
By le ing
δ∈(0, 1)
deno e he discoun ac o be ween pe iods in an in ini ely
epea ed game, which implies ha he alue oday o a dolla has o be ecei ed one pe iod
la e , we examine he e ec s o
γ
,
n
,
λ
on he s abili y o he collusion. The g im- igge
s a egy o F iedman [1] is used o analyze he punishmen ph ase.
In each pe iod, i ms and consume s make choices and ake ac ions in a h ee-
s age game:
S age 1: Gi en he las pe iod’s pe o mance, each i m
i
decides whe he o collude
o no .
S age 2: I collusion is ag eed on, i ms choose p oduc ion join ly o maximize agg e-
ga e p o i s. I no , each i m chooses i s p oduc ion independen ly.
S age 3: Consume s make expec a ions o i ms’ p oduc ion a ionally and decide on
he consump ion o bo h goods.
Following Ka z and Shapi o [
31
], ou equilib ium concep is ha o ul illed expec a ion
equilib ium, whe e each i m chooses i s ou pu le el unde he assump ion ha he
consume ’s expec a ions abou he sizes o he ne wo ks a e gi en and he ac ual ou pu
le el o he o he i ms is ixed. In he equilib ium, we ha e he ollowing:
x1=y1,x2=y2(5)
Games 2024,15, 21 4 o 7
Fi s ly, i he i ms choose o collude in he ou pu decision, he join payo (
UJ
) is
as ollows:
UJ=πi−λπj+πj−λπi=(1−λ)πi+πj(6)
The join payo is maximized when
x1C=x2C=α−c(1−γ)
2−n(7)
The collusion payo o each i m is
UC
1=(1−λ)(α−c(1−γ))2
(1−γ)(2−n)2(8)
whe e he supe sc ip ‘C’ deno es he ou come unde collusion.
Secondly, gi en he coope a i e ou pu o i m 2, i m 1 chooses o de ia e he coope -
a i e ou pu and maximizes i s payo U1. The i s -o de condi ion is
α
1−γ−c−2x1+γ(1−λ)x2−n(y1+γy2)
1−γ2=0 (9)
Subs i u ing a ionali y condi ion
x1=y1
and
x2=y2
in o he i s -o de condi ion,
we ob ain he ollowing eac ion unc ion
x1=(1+γ)[α−c(1−γ)] +γ(1−n−λ)x2
2−n(10)
Subs i u ing coope a i e p oduc ion o i m 2,
x2C=α−c(1−γ)
2−n
, we ob ain
he ollowing:
xD
1=(α−c(1−γ))(2−n+γ(1+λ))
(2−n)2(11)
whe e he supe sc ip ‘
D
’ deno es de ia ion om collusion. The esul ing de ia ion pay-
o is
UD
1=(α−c(1−γ))2γ2(1−λ)2+(4−2n)(1+γ)(1+λ)+n2−2n(1+γλ)2+1−γ2λ
(2−n)4(1−γ2)(12)
Thi dly, i i m 1 chooses o de ia e om he coope a i e ou pu , he g im- igge
s a egy is employed, and he i ms compe e in he Cou no ashion a e he de ian pe iod
o game. By sol ing he eac ion unc ion (10) and using a ional expec a ion condi ion,
xE
1=xE
2=(α−c(1−γ))(1+γ)
2−n(1+γ)+γ(1−λ)(13)
whe e ‘
E
’ deno es he Cou no –Nash equilib ium. The non-coope a i e p o i and payo
o he i ms a e as ollows:
πE
1=πE
2=(α−c(1−γ))2(1+γ)(1−γλ)
(1−γ)(2+γ(1−λ)−n(1+γ))2(14)
UE
1=UE
2=(1+λ)(α−c(1−γ))2(1+γ)(1−γλ)
(1−γ)(2+γ(1−λ)−n(1+γ))2(15)
3. Sus ainabili y o Collusion
Assume ha once de ec ion occu s be ween i ms, he opponen will adop a g im-
igge s a egy. Gi en he de ia ion payo
UD
1
, he Cou no –Nash compe i ion payo
Games 2024,15, 21 5 o 7
UE
1
, and he collusion payo
UC
1
, he aci collusion is sus ainable i and only i he p esen
alue o collusion is no less han he one o de ia ion, ha is
UC
1
(1−δ)≥UD
1+δUE
1
(1−δ)(16)
subs i u ing
UC
1
,
UD
1
and
UE
1
in o inequali y (16), and le ing
δ∗
sa is ying Equa ion (17) wi h
equali y. As F iedman [
1
] poin ed ou , his c i ical discoun ac o o he ca el main ains
collusion be ween he wo i ms. The aci collusion is sus ainable i and only i
δ≥δ∗
. We
ob ain he ollowing:
δ∗=UD
1−UC
1
UD
1−UE
1
=((2−n)(1+γ)−γ(1+λ))2(γ(1+λ)+(2n−n2)(1−γλ))
(2−n)2(−λ(2n−n2)(1−γ2)−γ(2−2n+λ−n2λ−λ2)−γ2(2−2n)(1−λ2))+γ3(1−n−λ)2(1−(1−n)2λ)
(17)
He e, i can be seen ha when
δ>δ∗
, he discoun a e o he i m is ela i ely la ge,
indica ing ha he cu en u ili y is p e e ed o he u u e u ili y and he i m is p one o
be ayal. Collusion s abili y in e ms o c i ical discoun ac o
δ∗
wi h espec o
γ(n,λ
) is
summa ized in he ollowing P oposi ion 1.
P oposi ion 1. The sign o
∂δ∗
∂λ
is condi ional on he alues o
γ
,
n and λ
, and
δ∗
is dec easing in
λwhen n is su icien ly la ge (n >n=2+γ−γλ
1+γ).
P oo . F om (17), we ha e ∂δ*
∂λ <0, ∂δ*
∂n<0, ∂δ*
∂γ <0, i n>n=2+γ−γλ
1+γ.□
The in luence o he weigh o ela i e pe o mance on he discoun ac o depends
on he e ec o he ne wo k ex e nali ies. Ou inding is di e en om Ma sumu a and
Ma sushima’s [
14
] wo k. Ma sumu a and Ma sushima [
14
] demons a ed ha
δ*
is inc eas-
ing in
λ
, and so, an inc ease in
λ
causes a g ea e ins abili y in collusi e beha io . As is
analyzed in he p e ious li e a u e, an inc ease in he ela i e pe o mance pa ame e
λ
,
on he one hand, leads o a less coope a i e ela ionship be ween i ms and inc eases hei
compe i ion while imp o ing he e ec i eness o punishmen (in his case, he o iginal
Cou no –Nash compe i ion), which will s abilize he collusion. On he o he hand, i leads
o mo e gains om de ia ion by cu ing down i s i al’s p o i , which will des abilize he
collusion. In Ma sumu a and Ma sushima [
14
], he o me des abilizing e ec is always
domina ed by he la e , so he s abili y o collusion imp o es wi h
λ
. Bu , in oducing
γ
and
n
in o ou model, he compa a i e s eng h o he posi i e and nega i e e ec s o
λ
on
δ∗a e condi ional on he s eng h o ne wo k e ec s.
In ou amewo k, when
n
is ela i ely la ge (
n>n
), his indica es a high deg ee
o ne wo k ex e nali y, which implies a less compe i i e ma ke en i onmen ; hence, he
s abilizing e ec o
λ
on
δ*
is s eng hened. Due o ha
∂δ*
∂λ <
0, he collusion is sus ainable
and mo e s able; howe e , i
n
is ela i ely small (
n<n)
, he collusion becomes less s able
wi h a ising deg ee o ela i e pe o mance, which is consis en wi h Ma sumu a and
Ma sushima’s [14] indings.
Song and Wang [
17
] poin ed ou ha he impac o p oduc di e en ia ion on collusion
s abili y is non-mono onic and depends c ucially on he deg ee o ne wo k ex e nali ies. In
ou amewo k, when
n
is ela i ely la ge (
n>n>
1), indica ing a low deg ee o p oduc
di e en ia ion (an inc ease in
γ
), hen his implies a mo e compe i i e ma ke en i onmen ,
so he s abilizing e ec o
γ
on
δ*
is s eng hened. The e o e,
∂δ*
∂γ <
0, he collusion is
sus ainable and mo e s able. Ou analy ical esul s con i m he obus ness o Song and
Wang [17].
Games 2024,15, 21 6 o 7
4. Concluding Rema ks
In his pape , we show ha collusi e s abili y is inc easing in he deg ees o ela i e
pe o mance, p oduc subs i u abili y and ne wo k e ec when he indus y-wide ne wo k
e ec is su icien ly la ge. The policy implica ion o ou indings is ha a compe i ion agency
migh need o p o ide di e ing guidance s a egies o an i-compe i i e o p o-compe i i e
egula ions in he ne wo k indus y.
Au ho Con ibu ions: Concep ualiza ion, Y.-S.L., C.-S.T., J.-Y.L. and C.-Y.L.; o mal analysis, Y.-S.L.,
J.-Y.L. and C.-Y.L.; me hodology, C.-S.T., J.-Y.L. and C.-Y.L.; so wa e, Y.-S.L. and C.-Y.L.; w i ing—
o iginal d a , Y.-S.L., J.-Y.L. and C.-Y.L.; w i ing— e iew and edi ing, Y.-S.L., C.-S.T., J.-Y.L. and
C.-Y.L. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: We hank he Fund a Guangdong Uni e si y o Science and Technology (G an No.
(GKXY)LX-DXFY-C-2024003) o hei unding suppo .
Da a A ailabili y S a emen : No new da a we e c ea ed o analyzed in his s udy. Da a sha ing is
no applicable o his a icle.
Acknowledgmen s: The au ho s would like o hank L.F.S. Wang and he anonymous e iewe s o
hei aluable commen s and cons uc i e sugges ions.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
No es
1Fo in ini ely epea ed games, see Ab eu [3], A amendia and Wen [4,5], among o he s.
2
Pal and Sc imi o e [
18
] demons a ed ha he ela ionship be ween collusion sus ainabili y and ma ke concen a ion in a
homogenous oligopoly wi h ne wo k ex e nali ies depended on he s eng h o ne wo k ex e nali ies.
3
Basak and Pe akis [
20
] ecen ly showed om he iewpoin o social desi abili y ha as long as he cos o en y is high, en y
is socially insu icien i ne wo k goods a e comple ely incompa ible, he goods a e su icien ly di e en ia ed, and he le el o
ne wo k ex e nali ies is low.
4The same u ili y unc ion is also used by many wo ks; o example, see Naska and Pal [24], Pal [25,26], Toshimi su [27,28], and
Choi and Lee [19].
5Basak and Pe akis [20] p o ided a simple exp ession o he in e se demand unc ion.
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