Random informative advertising with vertically differentiated products

Lahmandi-Ayed, Rim; Laussel, Didie
A icle
Random in o ma i e ad e ising wi h e ically
di e en ia ed p oduc s
Games
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Lahmandi-Ayed, Rim; Laussel, Didie (2024) : Random in o ma i e ad e ising
wi h e ically di e en ia ed p oduc s, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 2, pp. 1-29,
h ps://doi.o g/10.3390/g15020010
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Ci a ion: Lahmandi-Ayed, R.; Laussel,
D. Random In o ma i e Ad e ising
wi h Ve ically Di e en ia ed
P oduc s. Games 2024,15, 10.
h ps://doi.o g/10.3390/g15020010
Academic Edi o s: Ul ich Be ge and
S e ano Colombo
Recei ed: 12 Feb ua y 2024
Re ised: 14 Ma ch 2024
Accep ed: 16 Ma ch 2024
Published: 22 Ma ch 2024
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games
A icle
Random In o ma i e Ad e ising wi h Ve ically
Di e en ia ed P oduc s
Rim Lahmandi-Ayed 1,* and Didie Laussel 2
1Cen e o Un amed Thinking (CUT), Rennes School o Business, 35065 Rennes, F ance
2CNRS & EHESS, Aix-Ma seille School o Economics, Aix-Ma seille Uni e si y, 13009 Ma seilles, F ance;
didie [email p o ec ed]
*Co espondence: [email p o ec ed] o [email p o ec ed]
Abs ac : We s udy a simple model in which wo e ically di e en ia ed i ms compe e in p ices and
mass ad e ising on an ini ially unin o med ma ke . Consume s di e in hei p e e ence o quali y.
The e is an uppe bound on p ices since consume s canno spend mo e on he good han a ixed
amoun (say, hei income). Depending on his income and on he a io be ween he ad e ising cos
and quali y di e en ial ( ela i e ad e ising cos ), ei he he e is no equilib ium in pu e s a egies
o he e exis s one o he ollowing h ee ypes: (1) an in e io equilib ium, whe e bo h i ms ha e
posi i e na u al ma ke s and cha ge p ices lowe han he consume ’s income; (2) a cons ained
in e io equilib ium, whe e bo h i ms ha e posi i e na u al ma ke s, and he high-quali y i m
cha ges he consume ’s income o (3) a co ne equilib ium, whe e he low-quali y i m has no na u al
ma ke selling only o unin o med cus ome s. We show ha no co ne equilib ium exis s in which
he high-quali y i m would ha e a null na u al ma ke . A an equilib ium (whene e he e exis s
one), he high-quali y i m always ad e ises mo e, cha ges a highe p ice and makes a highe p o i
han he low-quali y one. As he ela i e ad e ising cos goes o in ini y, p ices become equal and he
ad e ising in ensi ies con e ge o ze o as well as he p o i s. Finally, he ad e ising in ensi ies a e, a
leas globally, inc easing wi h he quali y di e en ial. Finally, in all cases, as he ad e ising pa ame e
cos inc eases unboundedly, bo h p ices con e ge inc easingly owa ds he consume ’s income.
Keywo ds: andom ad e ising; ad e ising cos ; e ical di e en ia ion
JEL Classi ica ion: D83; L13; M37
1. In oduc ion
The global ad e ising and communica ion ma ke oday weighs mo e han 1370 bil-
lion dolla s (i.e., app oxima ely 1.5 pe cen o global GDP, in 2019) and con inues g owing
as e han he wo ld GDP. Ad e ising appea s o be a key ac o in he compe i ion be-
ween i ms. In F ance, o ins ance, comme cial communica ion “expendi u e”, s ic ly
speaking (excluding human esou ces in pa icula ), weighs 31 billion, which is nea ly he
equi alen o p i a e in es men in R&D (32 billion).
The ques ion o di e en ia ion and quali y is a na u al pa o he deba e. Ad e ising
has been s udied mainly in he case o ho izon ally di e en ia ed ma ke s. Only a ew pa-
pe s deal wi h he case o ad e ising in e ically di e en ia ed ma ke s, lea ing impo an
ques ions pending. We aim a illing he gap by s udying andom (o mass) ad e ising in
a e ical di e en ia ion model wi h a p ice compe i ion. We aim a de e mining whe he
he ad e isemen inc eases wi h he quali y sold and whe he an inc ease in he ad e is-
ing cos has a di e en ial o a simila impac on he i ms’ p ices, ad e ising in ensi ies
and p o i s.
Ad e ising is gene ally conside ed ei he as pe suasi e o in o ma i e. In he i s
case, i does no p o ide any ac ual in o ma ion on he p oduc bu ies o appeal o
consume s’ desi es and d i es o ha e hem buy he good. In he second one, i p o ides
Games 2024,15, 10. h ps://doi.o g/10.3390/g15020010 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 10 2 o 29
in o ma ion on he exis ence o he p oduc , he p ice, he cha ac e is ics o he good and so
on. Ad e ising is also o en conside ed as a quali y signal i highe quali y p oduc s a e
mo e ad e ised han lowe quali y ones, in which case i is indi ec ly in o ma i e.
In his pape , we deal di ec ly wi h in o ma i e ad e ising in he amewo k o an
oligopolis ic compe i ion. We analyze a simple e ical di e en ia ion duopoly model
wi h a low-quali y i m and a high-quali y one, whe e consume s di e in hei p e e ence
o quali y. The consume s a e ini ially unin o med o he exis ence o he i ms. When
ecei ing an ad om one i m, hey lea n i s exis ence, i s p ice and i s p oduc quali y.
1
We ocus on andom (o mass) ad e ising, i.e., he dissemina ion o p omo ional con en
wi hou any a ge ing o consume s. Each i m chooses i s p ice and i s ad e ising in ensi y
(which, he e, amoun s o choosing a a e o he consume s popula ion o be in o med
uni o mly). We assume ha consume s canno spend on he good mo e han hei (iden ical)
income (o , possibly, a p ede e mined sha e o i ), which pu s an-uppe bound on p ices.2
We conside a e ical di e en ia ion model, i.e., whe e consume s a e unanimous
on he anking o a ian s sold a he same p ice (as conside ed in many pape s, such as
Gabszewicz and Thisse, 1980 [
1
]; Shaked and Su on, 1982 [
2
], among o he s).
3
Consume s
di e , ne e heless, wi h ega ds o he in insic cha ac e is ic ha we call an “in ensi y o
p e e ence o quali y”, which measu es how s ongly a consume is sensi i e o quali y,
and hus how much a p io i s/he is willing o pay o acqui e a be e quali y. Mo eo e , we
suppose consume s o be limi ed by a budge cons ain , o , equi alen ly, ha p ices a e
uppe -bounded by an exogenous limi . In doing so, we a e supposing ha he he e ogenei y
in he in ensi y o a p e e ence o quali y is no equi alen o he he e ogenei y in income.
The abundan li e a u e in ma ke ing and psychology may ind his hypo hesis by mainly
using mo i a ion heo y (Ree e, 2017) [
4
]. Any pu chase occu s, as any beha io , o sa is y
physical (hunge ; hi s ) o psychological needs ( ecogni ion; es eem; belonging). When
he need is ac i a ed, he consume expe iences a s a e o ension, d i ing he consume
o y o sa is y he need. The s eng h o he ension de e mines he in ensi y wi h which
he indi idual is going o seek o he sa is ac ion o his/he need. Suppose he quali y
e e s o en i onmen al a ibu es, i.e., i measu es he e o made by he i m o espec
he en i onmen in he en i e p ocess. Consume s di e ing in e ms o indi idual and
amilial his o ies, physical and in ellec ual capabili ies, cul u al backg ounds, eac ion and
sensi i i y o ma ke ing
4
would necessa ily di e in insically wi h ega ds o he e o s
hey a e willing o make o be iendly o he en i onmen , possibly independen ly om
hei budge cons ain o income. The in ensi y o he p e e ence o quali y (
θ
in he
model), ep esen ing he mo i es we ha e jus desc ibed, is di e en in na u e om he
income (
y
), which ep esen s wha ex insically limi s he expenses. Conside ing bo h in he
same model gi es ise o in e es ing esul s ha we would no ha e been able o obse e,
had we conside ed only one o hem.
Fi s , we cha ac e ize he i ms’ choices a equilib ium. We show ha , depending on
he consume ’s income and he a io be ween he ad e ising cos and quali y di e en ial
( ela i e ad e ising cos ), ei he he e is no equilib ium in pu e s a egies o , one o
h ee possible ypes o equilib ium holds. (1) In e io Equilib ium (IE), whe e bo h i ms
ha e posi i e “na u al ma ke s”
5
and cha ge p ices lowe han he consume ’s income;
(2) cons ained In e io Equilib ium (CIE), whe e bo h i ms ha e posi i e na u al ma ke s
and he high-quali y i m cha ges he consume ’s income; (3) co ne equilib ium (COR),
whe e he low-quali y i m has no na u al ma ke . The in ui ion is ha i he e is no uppe -
bound o i he uppe -bound on p ices is oo high, a leas one among he wo i ms may
bene i om de ia ing om he candida e equilib ium o a highe p ice and se ing he
cus ome s who a e unin o med o he exis ence o i s i al. The necessi y o he exis ence
o an in e io equilib ium o an uppe -limi on p ices is a i s con ibu ion o he exis ing
li e a u e. Once his uppe limi is in oduced, i plays an explici ole in he exis ence
and he na u e o he equilib ium, i.e., in pa icula , i en ails he possible exis ence o a
co ne equilib ium and o a cons ained in e io equilib ium. This is a second con ibu ion
o his pape . All o hese ea u es ha e indeed been o e looked in he li e a u e (see, o
Games 2024,15, 10 3 o 29
ins ance, G ossman and Shapi o, 1984 [
6
]; Ti ole, 1988 [
7
]) which co espond o a speci ic
con ibu ion o his pape .
Secondly, we p o ide se e al compa a i e s a ics esul s a equilib ium, s udying
how he ou come a equilib ium a ies wi h he ela i e ad e ising cos , o some gi en
he consume ’s income. Depending on his income, when his ela i e ad e ising cos
a ies, we may go h ough wo o h ee egimes, and e en go h ough a hole (wi h no
equilib ium). In e es ingly, con a y o wha has been supposed by he exis ing li e a u e
ha conside ed only he in e io equilib ium o e looking he p oblem o exis ence and he
possible exis ence o equilib ia o di e en ypes, he equilib ium may ne e be an in e io
one ( o su icien ly low consume s’ income). Fo su icien ly high consume s’ income, he
equilib ium is an in e io equilib ium o low enough alues o he ela i e ad e ising
cos , bu he ype o equilib ium necessa ily changes as he ela i e ad e ising cos goes
beyond some h eshold; and o s ill highe consume s’ income, we may come up agains
an exis ence p oblem. Mo eo e , he highe he consume ’s income, he la ge he segmen
o he ela i e ad e ising cos o which he e is no equilib ium. Hence, looking o all
he possible ypes o equilib ium and in es iga ing he exis ence p oblem p ope ly, a e no
supe luous ma hema ical exe cises.
Ve y in ui i ely, p ices a e inc easing (in a b oad sense) wi h he ela i e ad e ising
cos . Beyond some c i ical h eshold o his ela i e cos , bo h p ices become equal o he
consume ’s income.
Conce ning he ad e ising in ensi ies, bo h a e dec easing (in a b oad sense) in he
cases o in e media e and high consume s’ income, as i can be p edic ed in ui i ely. Bu ,
in he case o low consume s’ income, he ad e ising in ensi y o he low-quali y i m is
inc easing on a ange o in e media e le els o he ela i e cos .
Rega ding he p o i s, bo h go h ough h ee phases (no synch onized): hey a e
dec easing a Phase 1, inc easing a Phase 2 and hen dec easing a Phase 3, con e ging
each o ze o as he a io goes o in ini y.
As o he p o i s a io, equal o he high-quali y i m’s p o i o e he low-quali y
i m’s one, i is always la ge han 1, meaning ha i always pays o be he high-quali y
i m. The a ia ion o his a io is, howe e , no simple. I goes h ough an inc easing phase
o a ange o ela i e ad e ising cos s close o ze o, and i is dec easing o su icien ly
high le els o his ela i e cos , con e ging o 1, as he ela i e cos ad e ising goes o
in ini y, meaning ha i pays less and less o be he high-quali y i m, as he ela i e cos
inc eases unboundedly.
Li e a u e Re iew
The e is impo an li e a u e in his ield o igina ing wi h he wo ks o Bu e s (1977) [
8
],
in he case o homogeneous goods (see, also, he mo e ecen con ibu ion o Roy, 2000 [
9
]).
G ossman and Shapi o (1984) [
6
] and Ti ole (1988) [
7
] launched he basis o he case o
ho izon ally di e en ia ed ma ke s, dis inguishing be ween mass ( andom) ad e ising
(whe e he e is no co ela ion be ween ad e ising in ensi ies and consume s’ ypes) and
a ge ed ad e ising (when he i ms ad e ise hei mo e in e es ing consume s mo e
equen ly). Mo e ecen pape s conside he ques ion o ad e ising wi hin he ho izon al
di e en ia ion amewo k, such as Celik (2007) [
10
], Ben Elhadj-Ben B ahim e al. (2011) [
11
]
and Es eban and He nandez (2014) [
12
]. Simbanega i (2009) [
13
] also conside s ad e ising
wi hin a ho izon ally di e en ia ed ma ke , bu deals wi h he di e en ques ion o coop-
e a ion be ween he i ms on ad e ising o p ices. The ho izon al di e en ia ion pape s
gene ally deal wi h ex an e symme ic i ms, con a y o e ical di e en ia ion ones.
A s and o li e a u e conside s ques ions ela i e o ad e ising a ge ing, suppos-
ing an exogenous segmen a ion o consume s: Es e es and Resende (2016, 2019) [
14
,
15
];
Iye e al. (2005) [16]
; Es eban and He nandez (2016) [
17
]; Zhang and He (2019) [
18
]; Zhang,
Cao and Yue (2018) [
19
]; Galeo i and Mo aga-Gonzalez (2003) [
20
]. E en i he models in
he ci ed pape s may e lec some di e en ia ion, o he ex en ha consume s do no eac in
Games 2024,15, 10 4 o 29
he same way o a p ice di e en ial be ween i ms, i is de ini ely no e ical di e en ia ion,
as he e is no unanimi y on he anking o he p oduc s by well-in o med consume s.
Conce ning asymme y, all hese pape s conside symme ic i ms excep Zhang
and He (2019) [
18
], who conside he exogenous cos asymme y be ween i ms, while
in ou model, his asymme y is inhe en o he e ical di e en ia ion; he cos asym-
me y eme ges endogenously h ough he asymme y in he choice o i ms in e ms o
ad e ising in ensi ies.
Colombo and Lambe ini (2003) [
21
] is one o he ew pape s we iden i ied ha deals
wi h ad e ising in a e ically di e en ia ed ma ke . Bu , ou wo k is di e en in se e al
espec s. Fi s , hey conside pe suasi e ad e ising, while ad e ising is in o ma i e in
ou pape . Second, hey deal wi h he endogenous in e play be ween ad e ising and
p oduc quali y, while we conside exogenous quali ies. Thi d, hey conside a e ical
di e en ia ion model di e en om he one we conside . T emblay and Ma in-Filho
(2001) [
22
] and T emblay and Polasky (2002) [
23
]conside ad e ising wi hin a e ically
di e en ia ed ma ke bu wi h pe suasi e ad e ising. Ellio (2004) [
24
], Es eban and
He nandez (2007) [
25
] Es eban and He nandez (2018) [
26
] conside a e ical di e en ia ion
model bu conside mass ad e ising as he dis ibu ion o ads o he en i e ma ke wi h no
choice o ad e ising in ensi ies.
Loosely ela ed o ou pape , Shen and Villas-Boas (2018) [
27
] deal wi h beha io al-
based ad e ising, bu , he alua ion o consume s o he p oduc in he second pe iod, in
he case o monopoly in a wo-pe iod model, may be co ela ed wi h he alua ion in he
i s pe iod. Johnson (2013) [
28
] deals wi h a ge ing wi hin a model, whe e a con inuum o
i ms choose he ad e ising amoun , while consume s ha e he possibili y o block ads,
wi h no compe i ion among i ms.
This pape is o ganized as ollows. Sec ion 2desc ibes he model. Sec ion 3p o ides
he esul s. Sec ion 4p o ides some compa a i e s a ics. Sec ion 5concludes.
All p oo s a e gi en in Appendix B.
2. The Model
Two i ms p oduce wo e ically di e en ia ed p oduc s wi h exogenous quali ies.
Fi ms 1 and 2 a e, espec i ely, he high- and he low-quali y i m. The quali y di e en ial
is deno ed by
∆=q1−q2
. We assume ha ma ginal cos s a e ze o and ha Fi ms 1 and 2
compe e in uni o m p ices, espec i ely,
p1
and
p2
. Fi ms in es in ad e ising o in o m
consume s abou hei exis ence, p oduc s’ cha ac e is ics and p ices.
No p oduc ion cos is supposed, o simplici ies’ sake. I would be na u al o suppose
asymme ic cos s wi h a highe cos o he high-quali y i m. We we e compelled o such a
simpli ica ion, which al eady esul ed in ough calcula ions. Mo eo e , cos asymme y
a ises endogenously. Indeed, we will p o e ha , a equilib ium, he highe quali y i m
spends mo e on ad e ising han he lowe quali y one. This amoun s somehow o an
endogenous ixed cos ( ela i e o quan i ies) ha is p o ed o be highe o a highe quali y.
The e is a uni mass o consume s. Consume s a e ini ially o ally unawa e o he
exis ence o he i ms and may become in o med only h ough ad e isemen s.
6
Consume s
in o med only o he exis ence o Fi m
i
buy one uni o
i
’s good, p o ided ha i s p ice is
no g ea e han hei income
Y
. Consume s in o med o he exis ence o bo h i ms buy
he p oduc which be e i s hei needs.
7
A ype
θ
-cus ome de i es a g oss u ili y
U+θqi
om consuming one uni o he quali y
i
-good in a gi en pe iod, hence he indi ec u ili y
U+θqi−pi
. Cha ac e is ic
θ
is uni o mly dis ibu ed o e
[0, 1]
, wi h a densi y no malized
o 1.
The p ices a e hus assumed o belong o
[
0,
Y]
.
U
is assumed o sa is y
U>Y
, so ha
all consume s who a e awa e o he exis ence o a leas one i m buy he good.
We de ine ˆ
θas he ma ginal consume , i.e., he consume who, when in o med o he
exis ence o bo h i ms, is indi e en be ween pu chasing a Fi m 1 o a Fi m 2. Tha is:
ˆ
θ=p1−p2
∆. (1)

Games 2024,15, 10 5 o 29
When ully in o med, consume s wi h ypes g ea e han
ˆ
θ
buy om Fi m 1, while
consume s o ypes smalle han
ˆ
θ
buy om Fi m 2. F om now on, we call
ˆ
θ, 1
Fi m
1’s “na u al ma ke ” and
0, ˆ
θ
Fi m 2’s “na u al ma ke ”. When
p1≥∆+p2
, Fi m 2’s
na u al ma ke is he whole ma ke , and Fi m 1 has a null na u al ma ke sha e. When
p2<p1<∆+p2, bo h i ms ha e a s ic ly posi i e na u al ma ke sha e. When p1≤p2,
Fi m 1’s na u al ma ke is he whole ma ke , and Fi m 2 has a null na u al ma ke .
We conside he e he case o mass o andom ad e ising, in which he ad e ising
in ensi y
Ψi
o Fi m
i
, i.e., he p opo ion o consume s who a e in o med abou p oduc
i
,
is uni o m o e all consume s’ ypes. This means ha a ac ion
Ψ1Ψ2
o consume s a e
in o med o he exis ence o bo h i ms ( hus, hey may ac ually compa e be ween bo h
and choose he one ha ensu es he bes u ili y, as in he s anda d li e a u e), a ac ion
(
1
−Ψ1)(
1
−Ψ2)
a e in o med o he exis ence o none o hem ( hus, hey buy no hing), a
ac ion
Ψ1(
1
−Ψ2)
a e only in o med o he exis ence o Fi m 1 (and buy om i ) and a
ac ion
Ψ2(
1
−Ψ1)
a e only in o med o he exis ence o Fi m 1 (and buy om i ). The
cos o eaching a ac ion o ype θ-consume s is simply
C(gi(θ)) = a
2Ψ2
i. (2)
Fo con enience, we de ine he ela i e p ices
i=pi
∆
, he ela i e income
y=Y
∆
and
he ela i e cos α=a
∆.
The game: he i ms choose simul aneously
8
p ices
pi
in
[
0,
Y]
(o , equi alen ly, he
ela i e p ices iin [0, y]) and ad e ising in ensi ies Ψi∈[0, 1].
3. The Equilib ium Ou comes
A he equilib ium, whene e he e exis s any, h ee cases a e possible. (1) The wo
i ms ha e posi i e na u al ma ke s and cha ge p ices lowe han he consume ’s e enue
(in e io equilib ium); (2) he wo i ms ha e posi i e na u al ma ke s wi h he high-quali y
i m cha ging a p ice equal o he consume ’s e enue (cons ained in e io equilib ium);
(3) only one o he i ms has a posi i e na u al ma ke (co ne equilib ium). When no
equilib ium candida e among he h ee desc ibed is an equilib ium, he game admi s no
pu e-s a egy equilib ium.
F om he de ini ions, he p o i s o he wo i ms a e, espec i ely:
Π1=










p1Ψ1(1−Ψ2)−a
2Ψ2
1i p1≥∆+p2,
p1Ψ1((1−ˆ
θ) + ˆ
θ(1−Ψ2)) −a
2Ψ2
1i p2<p1<∆+p2,
p1Ψ1−a
2Ψ2
1i p1≤p2.
, (3)
Π2=










p2Ψ2−a
2Ψ2
2i p1≥∆+p2,
p2Ψ2(ˆ
θ+ (1−ˆ
θ)(1−Ψ1)) −a
2Ψ2
2i p2<p1<∆+p2,
p2Ψ2(1−Ψ1)−a
2Ψ2
2i p1≤p2.
(4)
The p o i s a e hus de ined abo e in he h ee possible p ice con igu a ions: (i) When
p1≥∆+p2
, Fi m 1 has no na u al ma ke and sells only o consume s unawa e o he
exis ence o i s i al bu in o med o i s own exis ence, while Fi m 2 can sell o all cus ome s
in o med o i s exis ence; (ii)
p2<p1<∆+p2
, bo h i ms ha e a posi i e na u al ma ke ,
and i sells bo h o consume s in hei na u al ma ke and o consume s unawa e o he
exis ence o hei i al, p o ided hey a e in o med o hei exis ence; (iii) i p1≤p2, Fi m
2 has no na u al ma ke and sells only o consume s unawa e o he exis ence o i s i al
bu in o med o i s own exis ence while Fi m 1 can sell o all cus ome s in o med o i s
exis ence. We de ine he ela i e p o i s o be πi=Πi
∆.
Games 2024,15, 10 6 o 29
A pu e-s a egy Nash equilib ium o his game is a quad uple
 ∗
1,Ψ∗
1, ∗
2,Ψ∗
2
, such
ha
( ∗
i
,
Ψ∗
i)∈[
0,
y]×[
0, 1
]
is he bes eply o
( ∗
j
,
Ψ∗
j)
o each
i
,
j=
1, 2, and
i=j
.
P oposi ion 1cha ac e izes he equilib ium whene e i exis s in he space
(α
,
y)
, and
Figu e 1pic u es, in his space, he a eas co esponding o he di e en ypes o equilib ia
and o he nonexis ence o a pu e-s a egy equilib ium.
Figu e 1. Equilib ium in he (α,y)-space
P oposi ion 1 (Equilib ium).Whene e an equilib ium exis s, i is unique. Depending on he
posi ion o
(α
,
y)
ela i e o he zones depic ed in Figu e 1and de ined analy ically in Appendix A,
he e a e ou main cases in e ms o he exis ence and ype o equilib ium.
1.
Zone IE (in e io equilib ium): bo h i ms ha e posi i e na u al ma ke s and cha ge p ices
lowe han he e enue o he consume s. This zone is di ided in o h ee sub-zones, depending
on whe he o no he i ms each all consume s.
IE(i)
Bo h i ms each all consume s (
Ψ∗
1=Ψ∗
2=
1), and ela i e p ices a e
∗
1=
2
/
3,
∗
2=1/3.
IE(ii)
The high-quali y i m eaches all consume s, while he low-quali y i m eaches only a
ac ion o hem: Ψ∗
1=1,Ψ∗
2=1
9α1/3; he equilib ium ela i e p ices a e:
∗
1=2α
31/3, ∗
2=α
31/3.
IE(iii)
Bo h i ms each only a ac ion o consume s.9
2.
Zone CIE (cons ained in e io equilib ium): Bo h i ms ha e posi i e na u al ma ke s, bu
he high-quali y i m cha ges he e enue o consume s. This zone is also di ided in o h ee
sub-zones, depending on whe he o no i ms each all consume s.
CIE(i)
Bo h i ms each all consume s:
Ψ∗
1=Ψ∗
2=
1, and cha ge he ela i e p ices
∗
1=y
,
∗
2=y/2.
CIE(ii)
The high-quali y i m eaches all cus ome s (
Ψ∗
1=
1), he low-quali y i m only a
ac ion Ψ∗
2=y2/4αo hem and ela i e p ices a e gi en by: ∗
1=y, ∗
2=y/2;
CIE(iii) Bo h i ms each only a ac ion o cus ome s; he high-quali y i m cha ges he ela i e
p ice ∗
1=y and he low-quali y i m cha ges a lowe p ice.10
3.
Zone COR (CORne equilib ium): he low-quali y i m has a ze o na u al ma ke , wi h he
ollowing ela i e p ices and ad e ising in ensi ies:
∗
1= ∗
2=y;Ψ∗
1=y
α,Ψ∗
2=y
α(1−y
α).
4. Zone X: The e is no equilib ium in pu e s a egies.
Games 2024,15, 10 7 o 29
F om Figu e 1, i appea s clea ha a pu e s a egy equilib ium exis s wha e e
he alue o y when he ela i e ad e ising cos is small enough (smalle han 8/9).
The ques ion why is qui e clea . When he ad e ising cos is small and/o he quali y
di e en ial is high, Fi m 1 in o ms all cus ome s, which lea es no possibili y o Fi m
2 o se e unin o med consume s a a high p ice. Ano he ea u e is ha he ange o
alues o
y
o which an in e io equilib ium
11
exis s sh inks when he ela i e ad e ising
cos inc eases. This is because, as his ela i e cos inc eases, less and less consume s a e
in o med; hus, a de ia ion owa d se ing a a high p ice only becomes mo e and mo e
p o i able, because consume s a e unawa e o he exis ence o one’s i al, and his p e en s
he equilib ium candida e om being an equilib ium.
To p o e P oposi ion 1, we deal consecu i ely wi h each possible case.
Fo he i s case (in e io equilib ium), we w i e he i s -o de condi ions o he
associa ed Lag angian, supposing ha each i m has a posi i e na u al ma ke . A e
elimina ing he i ial solu ion wi h null p ices and ad e ising a es, we examine he ou
possible cases: (i) Bo h i ms each all cus ome s (
Ψ1=Ψ2=
1); (ii) Fi m 1 eaches all
cus ome s, bu Fi m 2 only eaches a ac ion o hem (
Ψ1=
1,
Ψ2<
1); (iii) bo h i ms
each only a ac ion o hei cus ome s (
Ψ1=
1,
Ψ2<
1); (i ) Fi m 1 eaches only a ac ion
o i s cus ome s, and Fi m 2 eaches i s whole na u al ma ke (Ψ1<1, Ψ2=1).
Fo sub-cases (i), (ii) and (iii), we calcula e he equilib ium candida es and de e mine
necessa y and su icien condi ions o each candida e o co espond o a maximum o he
se o p ices, such ha bo h i ms ha e posi i e na u al ma ke s. As o case (i ), i u ns
ou ha i can ne e co espond o an equilib ium.
The easoning abo e elimina es he de ia ions such ha each i m has a posi i e
na u al ma ke , bu no de ia ions such ha one o hem has no na u al ma ke . Look a
he p o i o Fi m
i
when i has no na u al ma ke sha e and i s compe i o does no each
he en i e ma ke (
Ψj<
1). We see easily ha his p o i may inc ease unboundedly wi h
he p ice, and hence may become highe han he p o i a he equilib ium candida e, hus
cons i u ing a p o i able de ia ion. The e o e, i he uppe -bound on p ices is oo high,
he equilib ium candida e canno be an equilib ium. In o he wo ds, o ensu e ha he
iden i ied candida e is an equilib ium, he p ice mus no be allowed o be oo high, so ha
he bes possible de ia ion is no p o i able. In each sub-case o case 1) o P oposi ion 1,
we w i e condi ions o
y
and
α
, such ha , on he one hand, he p o i a he bes possible
de ia ion is lowe han he p o i a he equilib ium candida e; and on he o he hand, he
p ice candida es a e less han Y.
Rega ding case 2 (cons ained in e io equilib ium), we p oceed exac ly in he same
way as o case 1, excep ha we ake he cons ain on p ices in o accoun in he Lag angian.
As o case 3 (co ne equilib ium), we iden i y he co ne equilib ium in each con-
side ed si ua ion (ei he Fi m 1 o Fi m 2 has a null na u al ma ke ). Then, we conside
possible de ia ions.
The e is an asymme y be ween i ms ega ding he exis ence o co ne equilib ia.
While a co ne equilib ium wi h a null na u al ma ke o he low-quali y i m may exis ,
he e is ne e an equilib ium wi h a null na u al ma ke o he high-quali y i m. Indeed,
he low-quali y i m is he one which has less incen i es o ha e cus ome s who would
buy he p oduc when hey know i s “ ue alue”. Thus, i may be in e es ed in elying
comple ely on unin o med cus ome s.
4. Compa a i e S a ics
We a e now going o p o ide some compa a i e s a ics a he equilib ium whene e
i exis s. The e a e quali a i ely h ee cases depending on he posi ion o he consume ’s
income ela i e o he wo c i ical alues (2/3 and app oxima ely 2.0477), as depic ed in
Figu e 1. We e e o he h ee cases as low, in e media e and high consume income,
which is sel -explana o y. We a e going o s udy, in he h ee cases, consecu i ely, he
p ices/ad e ising in ensi ies, p o i s and p o i s’ a io ( he high-quali y i m’s p o i o e
he low-quali y i m’s p o i ), as a unc ion o
α
, he ela i e ad e ising cos . This amoun s
Games 2024,15, 10 8 o 29
o mo ing along a ho izon al line in Figu e 1. In doing so, we may go h ough mul iple
egions cha ac e ized by di e en ypes o an equilib ium. Fo ins ance, o
y<
2
/
3,
inc easing
α
s a ing om ze o, we go h ough CIE (i), hen CIE (ii), hen CIE (iii) and
inally COR and emain he e.
Co olla y 1p o ides he compa a i e s a ics o p ices and ad e ising in ensi ies.
Figu es 2–4depic he ela i e equilib ium p ices as a unc ion o he ela i e ad e ising
cos , espec i ely, in he low, in e media e and high consume s’ income. Figu es 5–7
depic he equilib ium ad e ising in ensi ies as a unc ion o he ela i e ad e ising cos ,
espec i ely, in he h ee cases and in he same o de .
Co olla y 1 (P ices and ad e ising in ensi ies).A he equilib ium (whene e i exis s), he high-
quali y i m cha ges a highe p ice and ad e ises mo e (in a b oad sense) han he low-quali y one.
The p oo co esponds o he ep esen a ion o he ela i e p ices
∗
i
and ad e ising
in ensi ies
Ψ∗
i
gi en in P oposi ion 1in each case (low, in e media e and high consume s’
income), as a unc ion o α.
Figu e 2. Compa a i e s a ics o p ices: he case o low consume s’ income.
Figu e 3. Compa a i e s a ics o p ices: he case o in e media e consume s’ income.
Games 2024,15, 10 15 o 29
(iii) Le
Ψ∗
1(α)
be he equilib ium alue o
Ψ1
as a unc ion o
α
, and w i e he equilib ium
p o i o Fi m 2 as ∆π∗
2(α),
IE(iii) = 

(α,y)∈R+×R+∗, such ha α>8/9 and y≤q2απ∗
2(α)
1−Ψ∗
1(α)

.
Zone (CIE)
(i)
CIE(i) = (α,y)∈R+×R+∗, such ha 2√α≤y≤2/3.
(ii)
CIE(ii) = {(α
,
y)∈R+×R+∗
,
such ha y−y4/
8
α−α≥
0,
y≤
2
√α
and
y≤2(α/3)1/3}.
(iii)
CIE(iii) = {(α
,
y)∈R+×R+∗
,
such ha y−y4/
8
α−α≤
0,
y≥1
2(−
1
+√1+4α)
and y≤ IE
1(α)},
whe e
IE
1(α)co esponds o he p ice equilib ium o case IE (iii).
Zone (COR)
COR =((α,y)∈R+×R+∗, such ha y≤−1+√1+4α
2).
P oposi ion A1. De ails o P oposi ion 1Zone IE (iii). A he equilib ium, ad e ising in ensi ies
a e such ha
Ψ∗
1= ∗
1(α− ∗
1 ∗
2+ ∗2
2)
α2− ∗
1( ∗
1− ∗
2) ∗
2(1− ∗
1+ ∗
2),Ψ∗
2= ∗
2(α+ ∗2
1− ∗
1(1+ ∗
2))
α2− ∗
1( ∗
1− ∗
2) ∗
2(1− ∗
1+ ∗
2),
wi h equilib ium ela i e p ices o
∗
1= IE
1(α) = 1
21+q1+4 ∗2
2> ∗
2, ∗
2=6−1/3
2√A+s−A+12α
√A,
whe e A =21/39α2−√3√256α6+27α41/3 −2α2(2/3)1/3
(9α2−√3√256α6+27α4)1/3 .
Zone CIE (iii). A he equilib ium, he low-quali y i m’s ela i e p ice
∗
2
is he eal solu ion
o he hi d-o de polynomial, e c.
P3( 2) = α((−1+y)y+α)−2yα 2+y2 2
2−y 3
2=0. (A1)
Tha is: ∗
2=21/3C
3(D+√4C3+D2)1/3y−D+√4C3+D2
3 21/3y+y
3,
whe e C =−y4+6y2αand D =−2y6+27y3α−9y4α−27y2α2.
Appendix B. P oo s
P oo o P oposi ion 1. Elimina ion o he i ial solu ion pi=Ψi=0.
Mo e p ecisely, i
p1=p2=
0, hen necessa ily
Ψ1=Ψ2=
0. Indeed, wi h a null
p ice, a i m has no e enues and should no in es in ad e ising.
Howe e ,
pi=Ψi=
0 does no co espond o an equilib ium. Indeed, o
p2=ψ2=
0,
he p o i o Fi m 1 is gi en by: π1=p1ψ1−a
2ψ2
1, which is no maximal a p1=ψ1=0.
Fi s -O de Condi ions wi h a posi i e na u al ma ke o each i m (0 <ˆ
θ<1).

Games 2024,15, 10 16 o 29
Gi en he exp essions o he p o i s p o ided in Equa ions (3) and (4), he p o i
maximiza ion by Fi ms 1 and 2 unde he cons ain s
Ψi≤
1,
i=
1, 2, wi h he associa ed
non-nega i e Lag angian mul iplie s µi,i=1, 2, yields he necessa y condi ions:
∂L1
∂p1
=Ψ11+Ψ2(p2−2p1)
∆=0, (A2)
∂L2
∂p2
=Ψ21+Ψ1(p1−2p2−∆)
∆=0 (A3)
∂L1
∂Ψ1
=p1−aΨ1+p1(−p1+p2)Ψ2
∆−µ1=0, (A4)
∂L2
∂Ψ2
=p2−aΨ2−p2(∆−p1+p2)Ψ1
∆−µ2=0 (A5)
A he equilib ium, Ψi>0, o i=1, 2.
Indeed, i one o he
Ψi=
0, hen necessa ily, by Equa ions (A2) and (A3), he second
Ψj=
0. Hence, by Equa ions (A4) and (A5), p ices a e
pi=µi
. Bu ,
Ψi=
0
<
1, hence
µi=
0, hen
pi=
0. Bu , we ha e jus p o ed ha
pi=Ψi=
0 does no co espond o
an equilib ium.
Second-O de Condi ions: Any solu ion o Equa ions (A2)–(A5), such ha
Ψi>
0,
i=1, 2, co esponds, o each i m, o a p o i maximum.
Indeed, as
Ψ1>
0, Equa ion (A2) implies 1
+Ψ2(p2−2p1)
∆=
0, which co esponds
p ecisely o ∂2L1
∂p1∂Ψ1, which is hus equal o ze o. We p o e, simila ly, ha ∂2L2
∂p2∂Ψ2=0.
The Hessian ma ix o Fi m 1 is gi en by:
H1=

−2Ψ1Ψ21
∆(1+Ψ2(p2−2p1)
∆) = 0
(1+Ψ2(p2−2p1)
∆) = 0−a



which is a de ini e nega i e.
As o Fi m 2,
H2=

−2Ψ1Ψ21
∆(1+Ψ1(p1−2p2−∆)
∆) = 0
(1+Ψ1(p1−2p2−∆)
∆) = 0−a



which is also a de ini e nega i e.
We now deal wi h ou possible cases, depending on whe he he cons ain s a e
binding o no . One o hem u ns ou o ne e be possible.
Case (i): Bo h i ms each all cus ome s.
Bo h cons ain s a e binding, so ha
Ψi=
1,
i=
1, 2. The e is a unique solu ion o
Equa ions (A2)–(A5), which is
p1=
2
∆/
3,
p2=∆/
3,
µ1=4
9∆−a
,
µ2=1
9∆−a
. The
condi ion
a∈[
0,
∆/
9
]
is necessa y and su icien o ensu e ha bo h mul iplie s a e indeed
non-nega i e. Finally, as bo h
Ψi>
0, he solu ion co esponds o a maximum o each i m.
Case (ii): Fi m 1 eaches all cus ome s, Fi m 2 only a ac ion o hem.
We mus hen ha e
Ψ1=
1 and
µ2=
0. Sol ing Equa ions (A2)–(A5), we ob-
ain a unique solu ion, which is
p1=
2
a∆2
31/3
,
p2=a∆2
31/3
,
Ψ2=∆
9a1/3
and
µ1=−a+4a1/3∆2/3
3×31/3
. No ice ha
µ1
is non-nega i e i
a∈[0, 8∆/9]
, while
Ψ2<
1 i
a>∆/9.
As bo h Ψi>0, hen he solu ion co esponds o a maximum o each i m.
Consequen ly, case (ii) co esponds o an equilib ium i a∈(∆/9, 8∆/9].
Case (iii): Bo h i ms each only a ac ion o cus ome s (in e io equilib ium).
He e,
µ1=µ2=
0. Le us i s sol e o ad e ising a es as unc ions o p ices.
We ob ain:
Games 2024,15, 10 17 o 29
Ψ1=∆p1(a∆−p1p2+p2
2)
a2∆2−p1(p1−p2)p2(∆−p1+p2), (A6)
Ψ2=∆p2(a∆+p2
1−p1(∆+p2))
a2∆2−p1(p1−p2)p2(∆−p1+p2). (A7)
We decompose he easoning in o h ee s eps in o de o acili a e he eading.
S ep 1: We p o e ha he e is no equilib ium, such ha
pi=
0 and/o
Ψi=
0,
i=
1, 2.
(a) Conside i s
pi=
0. F om (A6), i ollows ha
Ψi=
0. Now, om Equa ions (A4)
and (A5), ∂Lj
∂Ψj=pj−aΨj=0, so ha Ψj=pj/a.
Now, om (A2), o j=i,∂Lj
∂pj=pj/a=0, which implies pj=0 and hen Ψj=0.
(b) Conside hen Ψi=0. F om (A2), we ob ain Ψj=0 and hen pi=pj=0.
As shown abo e, howe e , we canno ha e pi=Ψi=0, i=1, 2, a he equilib ium.
S ep 2: A necessa y and su icien condi ion o he solu ion.
Subs i u ing he exp essions ob ained in Equa ions (A6) and (A7), in o Equa ions (A2)
and (A3), and accoun ing o he ac ha
Ψi>
0 a he equilib ium, one ob ains he wo
equilib ium condi ions which he equilib ium p ices mus sa is y:
a2∆2+p2−2a∆p1−p3
1+a∆p2+p2
1(∆+p2)
(a2∆2−p1(p1−p2)p2(∆−p1+p2)) =0, (A8)
and a2∆2+p2
1(a∆+p2
2)−p1(a∆2+2a∆p2+p3
2)
(a2∆2−p1(p1−p2)p2(∆−p1+p2)) =0. (A9)
Sub ac ing (A8) om (A9), one ob ains ha he ollowing condi ion mus hold a
he equilib ium
−((∆−p1)p1+p2
2)(a∆+p1p2) = 0
We can hen conclude ha he equilib ium p ices mus sa is y
p1=1
2∆+q∆2+4p2
2, (A10)
whe e p1is s ic ly g ea e han p2and s ic ly smalle han14 ∆+p2.
Subs i u ing his alue o
p1
in (A9), we ob ain ha he equilib ium alue o
p2
mus sa is y:
a2∆2+p2a∆p2−a∆∆+q∆2+4p2
2+1
4(∆+p2)(∆+q∆2+4p2
2)2−1
8(∆+q∆2+4p2
2)3
a2∆2−1
2p2∆+q∆2+4p2
2∆+p2+1
2∆+q∆2+4p2
2−p2+1
2∆+q∆2+4p2
2 =0. (A11)
Le α=a/∆and 2=p2/∆. The equilib ium condi ion (A11) can be ew i en as:
F(α, 2) = 2α2+2 4
2−2α 2(1+q1+4 2
2)− 3
2(1+q1+4 2
2) + 2
2(1+2α+q1+4 2
2)
2α2+ 2(−2(1+q1+4 2
2) + 2(1−4 2+q1+4 2
2))
=0 (A12)
S ep 3: Exis ence and uni y o he solu ion o Equa ion (A12).
Using he abo e change o a iables and 1=p1/∆, we can w i e
Ψ1= 1(α− 1 2+ 2
2)
α2+ 1( 1− 2) 2(−1+ 1− 2)
Le us hen use he equilib ium ela ionship 1=1
21+q1+4 2
2 o ob ain
Games 2024,15, 10 18 o 29
Ψ1(α, 2) =
1
21+q1+4 2
2α−1
21+q1+4 2
2 2+ 2
2
α2+1
21+q1+4 2
21
21+q1+4 2
2− 2 2−1+ 1
21+q1+4 2
2− 2, (A13)
Equa ion (A12) has wo posi i e eal solu ions which a e depic ed in Figu e A1 in he
(α, 2)
-space using he Con ou Plo unc ion o Ma hema ica. In he same igu e, using he
RegionPlo unc ion, we ha e depic ed in blue he a ea in his space whe e
Ψ1(α
,
2)≤
1. I
u ns ou ha only he smalles solu ion o (A12) (co esponding o he exp ession o
∗
2
o
case IE (iii) o P oposi ion 1) is an equilib ium. The g ea es one belongs o he whi e a ea,
whe e Ψ1(α, 2)>1.
Figu e A1. Rep esen a ion in he (α, 2)-space o Equa ion (A12).
Deno ing by
2(α)
, he equilib ium alue om Equa ion (A12), we ob ain ha
Ψ1
(8/9, 2(8/9)) = 1 and Ψ1(α, 2(α)) <1 o all α>8/9.
We de ine
Ψ2(α
,
2(α))
simila ly o
ψ1(α
,
2(α))
. We use Equa ion (A7), gi ing
Ψ2
as a
unc ion o p ices, hen Equa ion (A10) o elimina e
p1
. We use he same change in a iables
o exp ess
Ψ2
as a unc ion o
α
and
2
; inally, we use
2(α)
, he alue o
2
, sa is ying
Equa ion (A12).
Plo ing
Ψ1(α
,
2
(α))
and
Ψ2(α
,
2(α))
on he same Figu e A2, o all
α>
0, we ob ain
ha (i)
Ψ1(α
,
2(α)) <
1, i and only i
α>
8
/
9, implying ha he solu ion we jus desc ibed
is alid, i and only i , α>8/9; and (ii) Ψ2<Ψ1<1.
Finally, he ob ained
Ψi
a e bo h posi i e; hus, he ob ained solu ion co esponds o a
maximum o each i m.
Case (i ): We p o e ha he e is no equilib ium whe e Fi m 2 eaches all consume s,
while Fi m 1 eaches only pa o hem.
Suppose his is he case; we should hen ha e
Ψ2=
1 and
µ1=
0. Using he i s -o de
condi ions wi h ega ds o p ices (A2), we would hen ob ain:
p1=∆1+Ψ1
3Ψ1
,
p2=∆2−Ψ1
3Ψ1
.
Subs i u ing o p1and p2 he abo e alues in o Equa ion (A4), we ob ain (α=a
∆):
Games 2024,15, 10 19 o 29
1+2Ψ1+Ψ2
1−9αΨ3
1=0. (A14)
On he o he hand, om Equa ion (A5), we ob ain
µ2=∆
9−4−9α+4
Ψ1
+Ψ1. (A15)
F om (A14) one ob ains
α=1+2Ψ1+Ψ2
1
9Ψ3
1
, whe e he alue o
Ψ1
is he equilib ium
candida e alue. Subs i u ing his alue o αin (A15), we should ha e
µ2=−4−1+2Ψ1+Ψ2
1
Ψ3
1
+4
Ψ1
+Ψ1. (A16)
As pic u ed in Figu e A3 ( ep esen ing he exp ession gi en in Equa ion (A16)), he
RHS is always nega i e o all alues o
Ψ1∈[0, 1]
. Since he mul iplie has o be posi i e,
his does no co espond o an equilib ium.
Figu e A2. Rep esen a ion o he exp essions o Ψ1(α, 2(α)) and Ψ2(α, 2(α)) as unc ions o α.
Figu e A3. Rep esen a ion o he RHS o Equa ion (A16).
De ia ions: We deal wi h each sub-case o case 1 o p o e ha no i m admi s a
p o i able de ia ion.
Games 2024,15, 10 20 o 29
Case (i): He e,
Ψ∗
i=
1, o
i=
1, 2. As long as he p ice candida es a e less han
Y
,
no p o i able de ia ions exis o i ms. In ac , each i m’s p o i is always null ou side o
i s na u al ma ke , as all consume s a e in o med o he compe i o ’s p oduc , lea ing no
oom o make a p o i on unin o med ones. Hence, o he equilib ium candida e o be an
equilib ium, i su ices o ha e Y≥p∗
1.
Case (ii): He e, he e is no possible p o i able de ia ion by Fi m 2, since
Ψ1=
1, i
p∗
2≤Y, o he same eason explained in case (i).
As o Fi m 1, as long as
p∗
1≤Y
, only de ia ions owa d p ices
pD
1>p∗
2+∆
may
po en ially be p o i able. Bu , i Y<p∗
2+∆, such de ia ions do no exis a all.
Mo eo e , we know ha
p∗
1<p∗
2+∆
as
ˆ
θ<
1. Hence, he in e al
[p∗
1
,
p∗
2+∆]
has a
posi i e measu e and o all
Y∈[p∗
1
,
p∗
2+∆]
, he e is no de ia ion o Fi m 1, showing ha
a null na u al ma ke is possible.15
Case (iii). He e, we ha e o conside de ia ions by Fi m 1 and Fi m 2.
Le us begin wi h Fi m 1. We ha e o conside de ia ions in p ices
pD
1≥p∗
2+∆
,
esul ing in a null na u al ma ke o Fi m 1.
We conduc he same easoning as in case (ii). No ing ha
p∗
1<p∗
2+∆
, o all
Y∈[p∗
1,p∗
2+∆], such de ia ions a e no possible.
Le us u n o Fi m 2. The p ices
pD
2
, such ha Fi m 2 has no na u al ma ke , sa is y
pD
2≥p∗
1and p o ide he i m he p o i :
p2Ψ2(1−Ψ∗
1)−aΨ2
2/2,
which is maximal a p2=Y.
The op imal alue in e ms o Ψ2is equal o ΨD
2=Y(1−Ψ∗
1)
a, which yields he p o i :
πD
2(Y) = Y2(1−Ψ∗
1(α))2
2a.
I has o be smalle han he candida e equilib ium p o i , which can be w i en as
∆π∗
2(α). This is equi alen o
y≤q2απ∗
2(α)
1−Ψ∗
1(α).
P oo o P oposi ion 1, Case 2 (CIE).
Gi en he exp essions o he p o i s p o ided in
Equa ions (3) and (4), he p o i maximiza ion by Fi ms 1 and 2 unde he cons ain s
Ψi≤
1,
i=
1, 2, and
pi≤Y
wi h, espec i ely, he associa ed non-nega i e Lag angian
mul iplie s µi,i=1, 2, and λi,i=1, 2, yields he necessa y condi ions:
∂L1
∂p1
=Ψ11+Ψ2(p2−2p1)
∆−λ1=0, (A17)
∂L2
∂p2
=Ψ21+Ψ1(p1−2p2−∆)
∆−λ2=0 (A18)
∂L1
∂Ψ1
=p1−aΨ1+p1(−p1+p2)Ψ2
∆−µ1=0, (A19)
∂L2
∂Ψ2
=p2−aΨ2−p2(∆−p1+p2)Ψ1
∆−µ2=0 (A20)
We a e looking o an equilib ium such ha
p1=Y
, and each i m has a posi i e
na u al ma ke . This necessa ily implies p2<Y, hence λ2=0.
Now, we conside each possible case, one by one. We i s w i e he i s -o de
condi ions allowing us o iden i y he candida e, hen we check he second-o de condi ions.

Games 2024,15, 10 21 o 29
(i) Fo
Ψ1=Ψ2=
1, om Equa ion (A18), one ob ains
p2=Y/
2. This co esponds o
an in e io equilib ium only i ˆ
θ=Y/2∆<1⇔y<2.
The F.O.C. wi h espec o
p1
(Equa ion (A17)), oge he wi h he condi ion
λ1≥
0 and
he exp essions o p1and p2, imply y≤2/3.
On he o he hand, om condi ion A19 and he necessa y condi ion µ1≥0, we mus
ha e Y−a−1
∆Y2
2≥0⇔α≤y−y2
2.
F om condi ion A20 and he necessa y condi ion
µ2≥
0, we mus ha e
α≤y2/
4
⇔
y≥2√α.
No ice hen ha
y2/
4
<y−y2
2
whene e
y<
8
/
3. Thus, when
y≤
2
/
3, we ha e
y2/4 <y−y2
2.
To sum up, only he condi ions 2√α≤y≤2/3 a e necessa y.
The second-o de condi ions.
Fo Fi m 1, he wo cons ain s a e binding, while he e a e also wo a iables. Then,
we ha e no hing o check o he second-o de condi ions.
Fi m 2’s bo de ed Hessian, gi en ha he cons ain on Ψ2is he only one ac i e, is:
BH2=





0 0 −1
0−2/∆0
−1 0 −a






We ha e o conside he sign o he las p incipal mino , as he e a e wo a iables and
one binding cons ain . The las p incipal mino ( he hi d) is equal o 2/∆, which has he
same sign as (−1)2. Thus, he second-o de condi ions a e sa is ied o Fi m 2.
(ii) Case
Ψ1=
1 and
Ψ2<
1. Conduc ing he same easoning as in case (i), we ob ain
p2=Y/2 and he necessa y condi ion y<2 o ha e an in e io equilib ium.
As
Ψ2<
1, hen
µ2=
0 and Equa ion (A20) implies
Ψ2=y2/
4
α
, which is smalle han
1 i y≤2√α.
Now, om Equa ion (A17) and he condi ion λ1≥0, we ob ain y≤2(α/3)1/3.
Using Equa ion (A19) and he condi ion µ1≥0, we ob ain y−(y4/8α)−α≥0.
No ice, inally, ha he la e condi ion implies16 y<2.
To sum up, only he condi ions
y−y4/
8
α−α≥
0,
y≤
2
√α
and
y≤
2
(α/
3
)1/3
a e necessa y.
The second-o de condi ions.
Fo Fi m 1, as in case (i), he e is no hing o check, since he e a e wo a iables and
wo binding cons ain s.
As o Fi m 2, gi en ha no cons ain is binding, we ha e o conside he Hessian.
H2=−2/∆0
0−a.
I is ob iously a de ini e nega i e; hus, he second-o de condi ions a e sa is ied as
well o Fi m 2.
(iii) Case
Ψ1<
1 and
Ψ2<
1. We ha e
µ1=µ2=
0. Using Equa ions (A19) and (A20)
simul aneously, we exp ess
Ψ1
and
Ψ2
, each as a unc ion o he wo p ices, and hus ob ain
Equa ions (A6) and (A7) again. Then, we subs i u e hese exp essions in o Equa ion (A18),
which yields Equa ion (A9) again, i.e.:
a2∆2+p2
1(a∆+p2
2)−p1(a∆2+2a∆p2+p3
2) = 0.
In his equa ion, subs i u e
∆y
o
p1
,
α∆
o
a
and
2∆
o
p2
, hen we ob ain
condi ion (A1).
We a e going o show ha (1) his equa ion has one and only one accep able eal
posi i e solu ion17 unde he condi ions indica ed in P oposi ion 1case CIE (iii); (2) o he -
Games 2024,15, 10 22 o 29
wise (when hese condi ions a e no sa is ied), ei he no solu ion exis s o he solu ion is
no accep able.
The de i a i e o P3wi h ega ds o 2is gi en by:
P′
3( 2) = −3y 2
2+2y2 2−2yα.
The disc iminan o his second-o de polynomial is gi en by
y2(y2−
6
α)
, which is o
he same sign as (y2−6α). The analysis depends on his sign.
(1) Suppose, i s , ha
y2≤
6
α
, and he disc iminan o
P′
3
is always nega i e; hus,
P3
is always dec easing.
The limi o
P3
as
2
goes o in ini y is
−∞
. Condi ion (A1) has one and only one eal
non-nega i e oo , i and only i P3( 2=0)≥0.
We ha e P3( 2=0) = α(α+y(y−1)).
(a) Fo
(α
,
y)
, simul aneously sa is ying
y(y−
1
) + α≥
0 and
y2≤
6
α
,
∗
2
, he unique
eal non-nega i e oo o
P3
co esponds o an in e io equilib ium only i
y> ∗
2>y−
1
(so ha 0 <ˆ
θ<1).
Since
P3
is dec easing, in his case, and
P3( ∗
2) =
0, hen Equa ion
y> ∗
2
is equi -
alen o
P3( 2=y)<
0, which w i es as
−y2−y+α<
0, and hus is equi alen o
y>1
2(−1+√1+4α).
In he same way, we use he dec ease in
P3
,
∗
2>y−
1, i and only i
P3( 2=y−
1
)>
0,
which is equi alen o y3+α2+y(1+α)−y2(2+α)>0.
We g aphically p o e ha he wo condi ions
y>1
2(−
1
+√1+4α)
and
y−y4/
8
α−
α≤0 imply he condi ion y3+α2+y(1+α)−y2(2+α)>0.
The in e io equilib ium iden i ied in case IE (iii) o P oposi ion 1co esponds o
he solu ion o he p esen sys em ha is composed o Equa ions (A17)–(A20), wi h
λ1=λ2=µ1=µ2=0.
The h ee Equa ions (A18)–(A20) a e sa is ied he e in he same way as in case IE (iii).
Fo
p∗
1=Y
o co espond o he choice o Fi m 1 a he equilib ium, necessa ily
y≤ IE
1(α)
.
Equa ion (A18) can we easily be ew i en ( eplacing p1wi h y) as
1−Ψ1+Ψ1(y−2 2) = 0
The e o e, Ψ∗
1<1 is equi alen o ∗
2>y/2.
Bu , we ha e supposed
y2<
6
α
, which implies
P′
3( 2)<
0 o all
2≥
0. Thus,
P3(y/2)>P3( ∗
2) = 0. We ha e
P3(y/2) = α2−αy+y4/8.
This way, we ha e shown Ψ∗
1<1 o be equi alen o y−y4/8α−α≤0.
Finally, we conside he alue o
Ψ2
, as gi en by Equa ion (A7) and subs i u e
Y
o
p1
and
p∗
2
o
p2
. We ob ain ha an inequali y
Ψ2=g(α
,
y)≤
1 mus hold a he equilib ium.
18
G aphical analysis
19
shows ha
y3+α2+y(
1
+α)−y2(
2
+α)≥
0 and
g
(
α
,
y)≤
1
hold when y−y4/8α−α≤0, y≥1
2(−1+√1+4α)and y≤ in
1.
Second-o de condi ions:
Fi m 1’s bo de ed Hessian, gi en ha only he cons ain on p1is ac i e:
BH1=






0−1 0
−1−2Ψ1Ψ2
∆1+Ψ2(p2−2p1)
∆
0 1 +Ψ2(p2−2p1)
∆−a







Games 2024,15, 10 23 o 29
We ha e o conside he sign o he las p incipal mino : he de e minan o he ma ix,
which equals a>0, so ha he SOC a e sa is ied o Fi m 1.
Fi m 2’s Hessian (since no cons ain is binding) gi en by
H2=−2Ψ1Ψ2
∆0
0−a
is a de ini e nega i e; hus, he SOC a e also sa is ied o Fi m 2.
(b) Fo
(α
,
y)
sa is ying
y2≤
6
α
bu
y(y−
1
) + α<
0, he polynomial
P3
is dec easing
wi h
P3(
0
)<
0, which implies
P3( 2)<
0 o all
2≥
0. This means ha he p o i o Fi m
2 is dec easing wi h
p2
; hus, he equilib ium candida e in his case is
∗
2=
0,
Ψ∗
2=
0,
Ψ∗
1=y
α
. The inequa ion
Ψ∗
1<
1 is equi alen o
y<α
. Bu ,
y<α
implies, on he one hand,
y(y−
1
) + α>
0, and on he o he hand
y−(y4/
8
α)−α<
0, o which he equilib ium
co esponds o case (iii) o he p oposi ion wi h a posi i e
p2
. Thus, no new case appea s
wi h a null p ice p2
(2) Suppose now ha
y2>
6
α
,
P′
3
is a second-deg ee polynomial ha admi s wo oo s:
′
2= (1/3)(y−qy2−6α)
and
′′
2= (1/3)(y+qy2−6α)
wi h
′
2< ′′
2<y
.
P′
3
is posi i e be ween he wo oo s and nega i e ou side, which means
ha P3is dec easing be o e ′
2, inc easing be ween ′
2and ′′
2, and hen begins o dec ease.
We ha e ha
P3( 2=y) = α(−y−y2+α)
, which is nega i e when
y2>
6
α
. Polyno-
mial P3admi s po en ially h ee oo s, depending on he sign o P3(0),P3( ′
2)and P3( ′′
2).
When hey exis , hese oo s (
Ri)
sa is y necessa ily
R1< ′
2
,
′
2<R2< ′′
2
and
′′
2<R3<y.
The oo R1, whene e i exis s, is ne e ele an because R1< ′
2<y/2.
The oo
R2
exis s, i and only i ,
P3( ′
2)<
0,
P3( ′′
2)>
0. Fo his oo o be accep able,
i mus sa is y
R2>y/
2, and he mul iplie
λ
calcula ed a
R2
mus sa is y
λ1≥
. Recall ha
Ψ1( 2) = y(α−y 2+ 2
2)
α2−y(y− 2) 2(1−y+ 2),
Ψ2( 2) = 2(α+y2−y(1+ 2))
α2−y(y− 2) 2(1−y+ 2)
and
λ1( 2) = Ψ1( 2)(1+Ψ2( 2)( 2−2y)).
The ep esen a ion o he se o
(α
,
y)
, such ha we ha e, simul aneously
y2>
6
α
,
P3( ′′
2)>0, P3( ′
2)<0, R2>y/2 and λ1≥0, leads o an emp y se .
Finally, ega ding R3, i exis s i and only i P3( ′′
2)≥0.
Again, o his oo o be accep able,
(α
,
y)
has o sa is y simul aneously
y2>
6
α
,
P3( ′′
2)>0, R3>y/2 and λ1≥0. And, his se is p o ed, g aphically, o be emp y.
De ia ions. Finally, we ha e o check ha , o each i m, no p o i able de ia ion exis s
among he p ices, such ha i s na u al ma ke is null20.
Fo Fi m 1, such p ices mus sa is y
p1≥∆+p∗
2
, o equi alen ly
1≥ ∗
2+
1. Such
p ices do no exis as ∗
2+1>y.
As o Fi m 2, he p ices such ha i has no na u al ma ke mus sa is y
p2≥p∗
1=Y
,
hus
p2=Y
, o
2=y
. Bu , his p ice canno be a p o i able de ia ion o Fi m 2 as
∗
2
Games 2024,15, 10 24 o 29
sa is ies he i s -o de condi ions o e he segmen o p ices
y−
1
< 2<y
, and he p o i
is conca e in 2o e his segmen ; hus, i is dec easing in he neighbo hood o y.
P oo o P oposi ion 1, Case 3 (co ne equilib ium).
We deal successi ely wi h each pos-
sible case: i s , when Fi m 2 has no na u al ma ke , and second, when Fi m 1 has no
na u al ma ke . Fo each case, we iden i y he equilib ium candida e, hen check whe he
p o i able de ia ions exis .
(1) I a co ne equilib ium exis s such ha Fi m 2 has no na u al ma ke , his means
ha ˆ
θ≤0, hus p1≤p2.
We i s p o e ha necessa ily p∗
1>0. Indeed, Fi m 1’s p o i w i es:
π1=p1Ψ1−(a/2)Ψ2
1.
I e e p1=0 hen, necessa ily, Ψ1=0, which leads o p o i π1=0.
Fi m 2’s p o i w i es:
π2=p2Ψ2(1−Ψ1)−(a/2)Ψ2
2=p2Ψ2−(a/2)Ψ2
2
which would be maximal o
p2=Y
. Then, Fi m 1 has in e es in de ia ing o a posi i e
p ice
p1<p2
and a su icien ly small
Ψ1
ha ensu e a posi i e
π1
. Thus, necessa ily,
p∗
1>0.
Fi m 2’s p o i w i es:
π2=p2Ψ2(1−Ψ1)−(a/2)Ψ2
2.
Fi s , no e ha necessa ily
Ψ∗
1<
1. Indeed, o he wise he bes p o i in his si ua ion
would be
π2=
0, ob ained a
Ψ2=
0, whe eas Fi m 2 may ob ain a posi i e p o i i i
de ia es o a p ice p2<p1(which is possible since p∗
1>0) and a su icien ly small Ψ2.
Hence, π2is inc easing wi h p2. Thus, p∗
2=Yand Ψ2=Y(1−Ψ1)
a.
Fi m 1’s p o i
π1=p1Ψ1−(a/2)Ψ2
1.
is inc easing in p1, and hen i eaches i s maximum a p∗
1=p∗
2=Y.
The op imal alue o
Ψ1
is gi en by
Ψ1=min(
1,
Y
a)
. Since
Ψ1<
1, hen, necessa ily,
Y<a
, o equi alen ly
y<α
, which is hus a necessa y condi ion. Hence, we ha e
Ψ∗
1=Y
a
,
which implies Ψ∗
2=Y
a(1−Y
a).
De ia ions: Fo his equilib ium candida e o be an equilib ium, we ha e o check
whe he he i ms ha e in e es in de ia ing.
Fo Fi m 1, when p2=Y, i s p o i has only one exp ession, gi en by
π1=p1Ψ1−(a/2)Ψ2
1,∀p1≤Y.
The bes op ion o Fi m 1 in absolu e e ms is he one p o ided by he equilib ium
candida e. This implies ha Fi m 1 has no in e es in de ia ing.
As o Fi m 2, i Y>∆,
π2=


p2Ψ2−(a/2)Ψ2
2i p2≤Y−∆,
p2Ψ2(ˆ
θ+ (1−ˆ
θ)(1−Ψ∗
1)) −(a/2)Ψ2
2i Y−∆<p2≤Y
I
Y≤∆
, only he second line o he abo e p o i applies. We ha e o conside
wo ypes o de ia ions:
Y−∆<p2<Y
and
p2≤Y−∆
, when
Y>∆
and only
Y−∆<p2<Y o Y≤∆.
Le us begin wi h de ia ions Y−∆<p2<Y.
The exp ession o
π2
o
Y−∆≤p2≤Y
is a con inuous and conca e unc ion
21
in
(p2
,
Ψ2)
, which necessa ily eaches i s maximum. When an in e io solu ion o he