Ke kho , Anna; Müns e , Johannes
A icle
S a egic complemen a i ies in a model o comme cial
media bias
Games
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Ke kho , Anna; Müns e , Johannes (2025) : S a egic complemen a i ies in a
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Published: 23 Ap il 2025
Ci a ion: Ke kho , A., & Müns e , J.
(2025). S a egic Complemen a i ies in
a Model o Comme cial Media Bias.
Games,16(3), 21. h ps://doi.o g/
10.3390/g16030021
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A icle
S a egic Complemen a i ies in a Model o Comme cial
Media Bias
Anna Ke kho 1and Johannes Müns e 2,*
1I o Ins i u e o Economic Resea ch, and CESi o, Uni e si y o Munich, 80539 Munich, Ge many;
[email p o ec ed]
2Depa men o Economics, Uni e si y o Cologne, 50923 Cologne, Ge many
*Co espondence: [email p o ec ed]; Tel.: +49-221-470-4411
Abs ac : Media con en is an impo an p i a ely supplied public good. While i has
been shown ha con ibu ions o a public good c owd ou o he con ibu ions in many
cases, he issue has no been ho oughly s udied o media ma ke s ye . We show ha
in a s anda d model o comme cial media bias, quali ies o media con en a e s a egic
complemen s, whe eby in es men s in o quali y can c owd in u he in es men s and
engage compe i o s in a ace o he op. The e o e, inancially s ong public se ice media
can mi iga e comme cial media bias: he con en o comme cial media can be mo e in line
wi h he p e e ences o he audience and less ad e ise - iendly in a dual (mixed public
and comme cial) media sys em han in a pu ely comme cial media ma ke .
Keywo ds: comme cial media bias; public se ice media; ad e ising; wo-sided ma ke s;
supe modula games; s a egic complemen s; public goods
JEL Classi ica ion: C70; H41; L13; L51; L82
1. In oduc ion
Media con en belongs o he mos impo an cases o p i a ely supplied public goods.
I s consump ion is non- i al, and in many cases like ee TV o eely a ailable In e ne
con en no exclusion is aking place. Media con en di e s ma kedly om o he public
goods, hough, because media ou le s ypically ely on ad e ising e enues ins ead o
cha ging hei consume s a pecunia y p ice. The economic analysis o he p i a e supply
o media con en as a public good mus ake his mul i-sided na u e o media ma ke s
in o accoun (Ande son & Coa e,2005). Recen li e a u e has made majo p og ess in his
esea ch a ea (see Ande son & Jullien,2015;Jullien e al.,2021).
One impo an esul om he heo y o p i a e public good supply is ha , unde
ai ly gene al condi ions, p i a e con ibu ions o a public good a e s a egic subs i u es, i.e.,
highe p i a e con ibu ions o he public good a e c owding ou o he p i a e con ibu ions
( o o e iews, see Chap e 6 o (Ba ina & Iho i,2005), and Finding F9 in (Buchholz &
Sandle ,2021)). Su p isingly, his issue has no been ho oughly examined o media
ma ke s, e en hough i is highly ele an o he wel a e analysis o media policy. E.g.,
in discussions abou he p ope ole and scope o public se ice media (PSM), one c ucial
ques ion is whe he aising he quali y o a egula ed PSM will inc ease o dec ease he
quali y o i s comme cial—i.e., p o i -maximizing—compe i o s.
The e a e wo con lic ing iews. On he one hand, PSM could c owd ou p i a e in es -
men and inno a ion in media ma ke s. E.g., he exis ence o PSM may lead o less en y o
Games 2025,16, 21 h ps://doi.o g/10.3390/g16030021
Games 2025,16, 21 2 o 46
comme cial media; see (Be y & Wald ogel,1999) o empi ical e idence. Simila ly, (A m-
s ong & Weeds,2007a) show ha in a duopoly whe e a PSM and a comme cial b oadcas e
compe e, aising he quali y o PSM pa ially c owds ou he comme cial b oadcas e and
lowe s i s quali y. This easoning is echoed by egula ion au ho i ies like O com (O com,
2004) in he UK and he Scien i ic Ad iso y Boa d a he Fede al Minis y o Finance in
Ge many (Wissenscha liche Bei a beim Bundesminis e ium de Finanzen,2014).
Howe e , PSM migh also os e a “compe i ion o quali y”, whe eby public and
comme cial media compe e o audiences. This easoning goes back o (Coase,1947),
ponde ing ha PSM migh induce a “na u al i al y o u nish he mos a ac i e p og ams”
(p. 197). Indeed, ecen empi ical e idence sugges s ha in coun ies whe e PSM in es in o
high-quali y media con en , he quali y o comme cial media ends o be high, oo (Simon,
2013). Simila ly, (Sehl e al.,2020) ind ha , con olling o GDP, pe capi a e enues o
PSM and comme cial b oadcas e s a e posi i ely co ela ed ac oss EU coun ies. These
co ela ions a e in line wi h a c owding in e ec o PSM, i.e., he p esence o s ong PSM
coincides wi h lou ishing comme cial media.1
In his pape , we show ha in a model o comme cial media bias, p og am quali ies in
e ms o unbiased epo ing a e s a egic complemen s a he han s a egic subs i u es.
2
Unbiased epo ing he e e e s o a p og am ha ully and u h ully epo s ac s as
opposed o wi hholding in o ma ion. E.g., ad e ise s migh p e e he media o hide
un a o able ac s abou hei p oduc s; p ime examples include he obacco and ca bon-
emi ing indus ies. Viewe s p e e high p og am quali y, while ad e ise s p e e he
opposi e. The s a egic complemen a i y s ems om he media’s undamen al ade-o in
hese models: Raising p og am quali y inc eases he alue o he p og am o he audience
bu dec eases he willingness o pay o he ad e ise s o each consume s.
3
The la e
e ec becomes less impo an when a media company has a smalle audience; hence, i s
incen i es o aise p og am quali y a e highe . Thus, in a media ma ke wi h bo h PSM
and comme cial media, aising he PSMs’ p og am quali y educes he comme cial media’s
audiences and he eby also hei implici cos o inc easing hei own p og am quali y. As
a esul , he PSM c owd in p og am quali y and engage he comme cial media in a ace
o he op.
Ou main model ocuses on PSM and comme cial media whose p og ams a e eely
a ailable o he consume s. Howe e , ou esul s gene alize o a model ea u ing bo h
eely a ailable and pay media, when p og am quali y in ol es e ealing in o ma ion ha
he media al eady possess. We also discuss condi ions unde which ou indings gene alize
o mul idimensional s a egy spaces, spillo e e ec s o p og am quali y on ad e ising
e enue o o he media ou le s, endogenous en y and exi , and biases o PSM.4
Ou pape ela es o ou s ands o li e a u e. Fi s , we con ibu e o he li e a u e
on comme cial media bias. Se e al empi ical pape s documen he e ec o ad e ising
on media co e age in e ms o mu ual und ecommenda ions (Reu e & Zi zewi z,2006),
p oduc men ions (Gamb o & Puglisi,2015), co e age o go e nmen scandals (Tella &
F anceschelli,2011) and clima e change (Bea ie,2020). We p esen a ai ly s anda d model
o comme cial media bias. Ou model is in many ways simila o he models s udied by
(Ellman & Ge mano,2009), (Ge mano & Meie ,2013), and (Ke kho & Müns e ,2015), as i
cap u es bias h ough a p og am ha ca e s o he p e e ences o ad e ise s a he han
consume s. Ou pape is especially close o (Ellman & Ge mano,2009) and (Ge mano &
Meie ,2013) who show ha compe i ion in media ma ke s mi iga es comme cial bias, and
o (Ke kho & Müns e ,2015) who ind ha compe i ion be ween media ou le s inc eases
he likelihood ha a cap on ad e ising quan i ies is wel a e enhancing. Rela edly, (Blasco
e al.,2016) ind ha i he media can aise hei audience sha e h ough educing hei
bias, hen compe i ion in he ma ke may also inc ease he expec ed p og am quali y.
5
Games 2025,16, 21 3 o 46
These p edic ions a e in line wi h he empi ical esul s o (Bea ie e al.,2021) who ind
ha newspape s p o ide less co e age o ca ecalls by hei ad e ise s, bu compe i ion
o eade s mi iga es his bias. Simila ly, (Focke e al.,2016) show ha comme cial media
bias is likely mi iga ed by epu a ional conce ns on behal o he media, e.g., i hey ace a
demanding audience.
In con as o he exis ing li e a u e, he p esen s udy conside s compe i ion be ween
comme cial media and PSM, whe e PSM a e no p o i -maximizing, po en ially egula ed,
and do no depend on ad e ising e enue o und hei ope a ions. This allows us o in o m
policy deba es ega ding he p ope ole and scope o PSM in media ma ke s. Mo eo e , in
con as o p e ious wo k, he p esen pape explici ly models compe i ion be ween media
ou le s as a supe modula game, enabling us o d aw ai ly gene al conclusions ega ding
he impac o aising PSMs’ budge on he p og am quali y p o ided by comme cial media.
Second, we ad ance he b oad esea ch on he p i a e supply o public goods (Ba ina
& Iho i,2005;Be gs om e al.,1986). The p o ision o public goods ia ad e ising is
s udied by (Ande son & Coa e,2005;Luski & We s ein,1994). These pape s do no s udy
media bias, howe e .
Thi d, ou pape ela es o he li e a u e on supe modula games, i.e., games in which
he bes esponse o any playe is inc easing in he ac ions o i s compe i o s (F ankel e
al.,2003;Milg om & Robe s,1990;Topkis,1979;Van Zand & Vi es,2007;Vi es,1985,
1990,2005a,2005b). To he bes o ou knowledge, we a e he i s o apply he heo y o
supe modula games o a model o comme cial media bias. This app oach allows us o
ob ain ai ly gene al esul s in a model wi h many asymme ic media ou le s. Speci ically,
we show ha in ou model o comme cial media bias, p og am quali ies a e s a egic
complemen s a he han s a egic subs i u es.
Fou h, as a consequence o s a egic complemen a i ies, public in es men s in o p o-
g am quali y induce comme cial media o p o ide high quali y, oo. Hence, ou esul s
suppo media policies ha ad oca e inancially s ong PSM. In his way, we also con ibu e
o he economics li e a u e on PSM (see (A ms ong & Weeds,2007b;S ömbe g,2015;
Weeds,2020) o su eys). To he bes o ou knowledge, he issue how PSM a ec he
p og am o comme cial media has no been s udied ye in he li e a u e on comme cial
media bias. O he aspec s o his deba e ha e, o cou se, been analyzed; in addi ion o
he empi ical li e a u e e e enced abo e, se e al heo e ical s udies on he ma ke im-
pac o PSM exis . (A ms ong & Weeds,2007a) s udy in es men s in a e ical quali y
dimension. (Richa dson,2006) in es iga es how a publicly-p o ided adio s a ion o e ing
local p og ams a ec s he p o ision o local p og ams by comme cial s a ions. (Ga cia
Pi es,2016) compa es p og am di e si y in comme cial e sus mixed public and p i a e
duopolies. Ou pape complemen s his line o esea ch by s udying comme cial media
bias. E.g., nei he (A ms ong & Weeds,2007a) no (Richa dson,2006) conside ad e ise s
who alue p og am quali ies in e ms o (un-)biased epo ing. In (A ms ong & Weeds,
2007a), iewe s a e ad a e se and highe ad e ising quan i ies educe iewe s’ u ili y.
PSM maximize iewe wel a e, whe eby iewe s a e be e o han in a pu ely comme cial
ma ke . Howe e , in con as o ou pape , his is because PSM pa ially c owd ou com-
me cial media, whe eby subsc ip ion p ices and ad e ising quan i ies dec ease. Simila ly,
(Richa dson,2006) shows ha in a Ho elling model wi h ad a e se iewe s, PSM educe
p o i s o comme cial media, bu inc ease iewe wel a e. Thus, in bo h models, iewe s a e
be e o because audience- iendly PSM displace comme cial media. Ou pape conside s
a di e en mechanism: inancially s ong PSM enhance iewe s’ u ili y because hey c owd
in p og am quali ies by comme cial media.
The emainde o his pape is s uc u ed as ollows. Sec ion 2in oduces ou he-
o e ical amewo k. In Sec ion 3, we demons a e ha p og am quali ies in e ms o
Games 2025,16, 21 4 o 46
unbiased epo ing a e s a egic complemen s, which is ou main inding, and desc ibe
he implica ions o c owding in e ec s o PSM. Sec ion 4conside s he case whe e some
comme cial media a e pay media. Sec ion 5discusses se e al ex ensions o ou model.
Sec ion 6concludes.
2. Model
This sec ion in oduces a ai ly s anda d model o comme cial media bias (Ellman &
Ge mano,2009;Ge mano & Meie ,2013;Ke kho & Müns e ,2015), see (Blasco e al.,2012)
o a su ey). Conside a model wi h
n
comme cial media deno ed by 1,...,
n
and
m
PSM
deno ed
n+
1,
. . .
,
n+m
. The se o comme cial media is deno ed by
C={1, . . . , n}
, he se
o PSM is
P={n+1, . . . , n+m}
. Each media ou le
i∈C∪P
chooses a p og am quali y
i∈Vi⊂R+
. (An ex ension o mul idimensional s a egy spaces is conside ed in Sec ion 5).
P og am quali y
i
is abou unbiased epo ing, i.e., abou ully and u h ully epo ing
ac s, as opposed o wi hholding in o ma ion.
6
The audience p e e s high p og am quali y,
while ad e ise s p e e he opposi e. We assume ha he s a egy se s
Vi
a e compac and
con ain i=0.
A consume ’s u ili y om consuming ou le
i
is
ui= i( i)
, whe e
i
is con inuous,
s ic ly inc easing, and sa is ies
i(0)=
0. Unless o he wise no ed, we simply assume
i( i)= i
. In Sec ions 2and 3, no hing is los in se ing
ui= i
; he dis inc ion be ween
u ili y
ui
and p og am quali y
i
becomes impo an when conside ing pay media o mul i-
dimensional s a egies. Fo a comme cial ou le
i∈C
, le
uC
−i=(u1, . . . , ui−1,ui+1, . . . , un)
deno e he ec o o he u ili ies o
i
’s comme cial compe i o s,
uP=(un+1, . . . , un+m)
he
ec o o u ili ies o he PSM, and u−i=uC
−i,uP.7
The size o he audience o a media ou le is deno ed by
si
. We impose he ollowing
assump ions.8
Assump ion 1. Fo all
i∈C
,
si
is posi i e, con inuous, weakly inc easing in
ui
, and weakly
dec easing in uj o all j ∈P∪C {i}.
Assump ion 1 is easonable i consume s ca e abou quali y, and he media ou le s a e
subs i u es o he consume s.
Assump ion 2. Fo all i ∈C,sihas weakly inc easing di e ences in (ui,u−i).
I
si
is wice con inuously di e en iable, and he s a egy spaces a e in e als, Assump-
ion 2 means ha ∂2
∂uj∂uisi≥0
o all
j=i
.
9
No e ha Assump ion 2 only assumes ha he di e ences a e weakly inc easing.
In pa icula , i is ul illed in he case o cons an di e ences, whe e he abo e inequali y
holds wi h equali y.
As we discuss in de ail in Appendix A, Assump ions 1 and 2 a e sa is ied by many—
bu no all—models o audience demand ha a e equen ly used in media economics. Fo
example, any model whe e
si
is linea in
ui
and
uj
o all
j=i
, and does no include any
in e ac ion e ms, sa is ies Assump ion 2 because
si
has cons an di e ences in
(ui
,
u−i)
.
This class o models comp ises he Ho elling duopoly model o ho izon ally di e en ia ed
goods en iched by a e ical quali y di e en ia ion, and gene aliza ions o he Ho elling
model o mo e han wo ou le s such as he Spokes model and he Salop ci cle model.
Simila ly,
si
has cons an di e ences in
(ui
,
u−i)
in ep esen a i e consume models wi h
quad a ic u ili y unc ions. See Appendix A o he unc ional o ms o
si
in hese models,
and o e e ences o publica ions in media economics using hese speci ica ions.
Games 2025,16, 21 5 o 46
No e, howe e , ha ou assump ions a e a mo e gene al han simply assuming linea
demand unc ions. Fo example, when he audience demand unc ions include linea and
quad a ic e ms,
si(ui,u−i)=ai+biui−∑
j=i
bijuj+∑
k
cikukui,
Assump ion 2 holds as long as cik ≥0 o all i∈Cand all k=i.
On he o he hand, he logi model iola es Assump ion 2 whene e he e a e
n≥
2
comme cial ou le s. We deal wi h possible iola ions o Assump ion 2 in wo dis inc
ways. Fi s , in Appendix Fwe gi e a su icien condi ion o ou main esul s o hold when
Assump ion 2 is iola ed, and he eby show he obus ness o ou esul s o su icien ly
small iola ions o Assump ion 2. Second, we explo e ou model wi h a weake e sion o
Assump ion 2:
Assump ion (2 log) Fo all
i∈C
,
ln(si)
has weakly inc easing di e ences in
(ui,u−i)
.
Gi en Assump ion 1, Assump ion 2 implies Assump ion (2 log), bu no ice e sa.
Fo example, he logi model iola es Assump ion 2, bu sa is ies assump ion (2 log) (see
Appendix F). Mo eo e , he nes ed logi model—which is o en used in empi ical s udies o
audience demand in media economics (see Be y & Wald ogel,2015 o a su ey)—sa is ies
Assump ion (2 log) as well (see Appendix F).
10
Fo some o ou esul s, Assump ion (2 log)
is su icien . Unless o he wise no ed, below we will assume ha Assump ion 2 holds; we
will explici ly s a e when we weaken i o Assump ion (2 log).
Deno e he ad e ising e enue o ou le
i
, pe membe o he audience, by
Ri
. A
c ucial assump ion in models o ad e ise bias is ha , o a gi en audience, ad e enue
depends nega i ely on p og am quali y:
Assump ion 3. Fo all
i∈C
,
Ri
is posi i e, con inuous, weakly dec easing in
i
,and independen
o j o all j =i.
By Assump ion 3,
Ri
is independen om he p og am quali y o o he ou le s as in
(Ellman & Ge mano,2009) and (Ke kho & Müns e ,2015). (Ge mano & Meie ,2013) model
spillo e e ec s o p og am quali y on he ad e ising e enue o o he ou le s; we will
discuss spillo e e ec s in Sec ion 5.11
Each media ou le ihas a cos ci( i) ha may depend on i s p og am quali y.12
Assump ion 4. Fo all media ou le s
i∈C∪P
,
ci
is con inuous, weakly inc easing in
i
,and ze o
a i=0.
We dis inguish be ween wo cases. Fi s , p og am quali y could be abou ully and
u h ully epo ing ac s ha he media al eady ha e. In his case, he only cos o quali y is
lowe ad e ising e enue, bu he e is no addi ional di ec cos o ob aining he in o ma ion
in he i s case. Fo mally, in he cu en model i means ha
ci( i)=
0 o all
i∈Vi
.
We e e o his case as “wi hholding ac s”. Second, p og am quali y could also be abou
in es iga i e jou nalism, abou es ablishing new ac s and in o ma ion. Then i seems
plausible ha
ci( i)
is s ic ly inc easing in
i
. Fo example, he media migh ha e o hi e
mo e jou nalis s o inc ease p og am quali y (see (Hamil on,2016) o a de ailed desc ip ion
o he economics o in es iga i e jou nalism). We e e o his case as “in es iga ing ac s”.
A guably, he case o wi hholding ac s is highly ele an o he s udy o comme cial
media bias; indeed se e al pape s in he li e a u e ocus on his case (Blasco e al.,2016;
Ellman & Ge mano,2009;Ge mano & Meie ,2013;Ke kho & Müns e ,2015). Fo ins ance,
ad e ising has been known o in luence edi o s on c ucial opics such as he heal h isks o
smoking (e.g., (Bagdikian,2008) Chap e 12) and clima e change (Bea ie,2020;Boyko &
Boyko ,2004). The scien i ic ac s abou hese opics we e long well es ablished and easily
Games 2025,16, 21 6 o 46
accessible, bu media co e age and public pe cep ion signi ican ly lagged in ime behind
he scien i ic consensus. Mo eo e , as shown in (Bea ie,2024), comme cial media bias in he
one o co e age abou clima e change, measu ed based on compa isons o en i onmen al
and skep ical ex s, can ha e impo an beha io al consequences, and me ely changing he
one o co e age does no impac i s cos .
On he o he hand, p essu e om ad e ise s may also de e media om in es iga ing
ac s. Ou main esul s on eely a ailable media do no depend on whe he we s udy
wi hholding o in es iga ing ac s. Fo pay media, we show ha he dis inc ion ma e s.
Comme cial media in ou main model a e unded by ad e ising, and hei p og am
is eely a ailable o consume s; pay media will be conside ed in Sec ion 4. The p o i o a
comme cial media ou le i=1, ..., nis (subs i u ing i=uiand −i=u−iin o si)
πi( i, −i)=si( i, −i)Ri( i)−ci( i).
Comme cial ou le
i
maximizes
πi( i, −i)
by choosing
i∈Vi
. No e ha we dis ega d
ixed cos s which could be sa ed by going ou o business; we de e a discussion o exi
and en y o Sec ion 5.
The PSM in ou model a e no - o -p o i and inanced independen o ad e ising.
Thei p og am is eely a ailable o all consume s.
13
The budge o PSM
i∈P
is
bi
. We
assume he PSM spend hei budge o maximize consume u ili y by choosing
i∈Vi
subjec o
ci( i)≤bi
.
14
The easible se s
Vi⊂R+
a e compac , con ain
i=
0, and may
depend on he budge
bi
. We assume ha a la ge budge enla ges he easible se : i
bi<b′
i
,
hen
Vi(bi)⊆Vib′
i
. The model allows o ine iciencies o PSM, since di e en media
ou le s can ha e di e en cos unc ions and di e en easible se s o p og am quali y. We
discuss po en ial biases o PSM in Sec ion 5.
Some (bu no all) o ou conside a ions below impose he addi ional assump ion ha
a su icien ly high p og am quali y is necessa y o a PSM o a ac an audience. To exp ess
his o mally, o
i∈P
le
P
−i=( n+1, ..., i−1, i+1,..., n+m)
deno e he ec o o p og am
quali ies o he o he PSM.
Assump ion 5. A PSM ou le wi h ze o p og am quali y
( i=0)
a ac s no audience:
si0, C, P
−i=
0 o all
i∈P
and
C, P
−i
, and demand o he o he media ou le s is as
i ou le i did no exis .
Assump ion 5 seems easonable when he audience has a su icien ly a ac i e ou -
side op ion no o consume any media. No e ha unde Assump ion 5, a PSM wi h an
insu icien budge canno p oduce a p og am ha a ac s any audience; hen he game
educes o a game be ween he emaining media ou le s only. We will explici ly indica e
whe e we use Assump ion 5.
3. Main Resul s
Conside he PSM i s .
Lemma 1. A PSM i chooses
i=¯
i(bi):=max
i∈Vi(bi){ i|ci( i)≤bi}.
Mo eo e ,
¯
i(bi)
is weakly inc easing in
bi
,and independen o he s a egies o he o he media
ou le s.
Games 2025,16, 21 7 o 46
P oo . Ou le i∈Psol es
max
i∈Vi
is. . ci( i)≤bi.
An inc ease o
bi
elaxes he PSM’s budge cons ain and enla ges he easible se
Vi
, hence
¯
i
is weakly inc easing in
bi
. Mo eo e
¯
i
is unique and independen o he s a egies chosen
by he o he media ou le s.
We now u n o he comme cial media. Lemma 1 allows us o iew he game be ween
he comme cial media as pa ame e ized by he budge s o he PSM
b:=(bn+1, ..., bn+m)
.
Le
¯
P(b)=(¯
i(bi))m
i=n+1
deno e he ec o o p og am quali ies chosen by he PSMs. Fo
i∈C, le
˜
πi i, C
−i,b:=πi i, C
−i,¯
P(b)
and le
Γb=C,(˜
πi)n
i=1,Πn
i=1Vi
deno e he esul ing game be ween he comme cial media
ou le s: he se o playe s is C, payo unc ions a e ˜
πi, and s a egy spaces a e Vi.
To s a e ou main esul , we need he concep o a pa ame e ized supe modula
game.
15
Conside a amily o games wi h se o playe s
N
, s a egy spaces
Xi
, and payo
unc ions
ui
pa ame e ized by
in some pa ially o de ed se o pa ame e s alues
T
. The
game
(N,(ui,(Xi)i∈N)i∈N,T)
is a pa ame e ized supe modula game i , o each
i∈N:
(i)
Xi⊆Rmi
is a la ice and is compac , (ii)
ui(xi,x−i, )
is con inuous in
xi
o ixed
x−i
and
,
(iii)
ui(xi,x−i, )
is supe modula in
xi
and has weakly inc easing di e ences in
(xi;x−i, )
.
P oposi ion 1. Γbis a pa ame e ized supe modula game.
P oo .
The s a egy spaces
Vi⊂R+
a e compac by assump ion, hence compac la ices,
and he objec i e unc ions ˜
πia e con inuous in i o ixed −iand b.
Nex , we show ha
πi
has weakly inc easing di e ences in
( i, −i)
. Fo simplici y
o he exposi ion, we will assume he e ha he unc ions
Ri
,
si
and
ci
a e di e en iable;
Appendix Bgi es he p oo wi hou assuming di e en iabili y. F om
πi( i, −i)=si( i, −i)Ri( i)−ci( i)
we ob ain ∂πi
∂ i=∂si( i, −i)
∂ iRi( i)+si( i, −i)R′
i( i)−c′
i( i)
and ∂2πi
∂ j∂ i=∂2si( i, −i)
∂ j∂ iRi( i)+∂si( i, −i)
∂ jR′
i( i)≥0, j=i(1)
whe e he inequali y ollows because o
∂2si( i, −i)
∂ j∂ i≥
0 by Assump ion 2,
∂si( i, −i)
∂ j≤
0 by
Assump ion 1, and R′
i( i)≤0 by Assump ion 3.
We ha e shown ha
πi
has weakly inc easing di e ences in
( i, −i)
. The e o e,
˜
πi
has
weakly inc easing di e ences in
i, C
−i
. Mo eo e ,
πi
has weakly inc easing di e ences
in
i, P
. I emains o show ha
˜
πi
has weakly inc easing di e ences in
( i,b)
. By Lemma
1,
¯
k(bk)
is inc easing in
bk
, while
¯
k′
does no depend on
bk
o
k′=k
. Since
πi
has weakly
inc easing di e ences in
i, P
, i ollows ha
˜
πi
has weakly inc easing di e ences in
( i,b).
P oposi ion 1 shows ha he p og am quali ies a e s a egic complemen s. The eco-
nomics o he esul is s aigh o wa d. The undamen al ade-o o a comme cial ou le
in a model o comme cial media bias is as ollows: p o iding a p og am in line wi h he
p e e ences o he audience a ac s a bigge audience, bu leads o lowe ad e ising e -
enue pe consume . I he p og am quali ies o compe ing media inc ease, he audience o
Games 2025,16, 21 8 o 46
a gi en ou le is smalle , hence also he implici cos o inc easing i s own p og am quali y.
The logic is closely ela ed o he inding in (Ge mano & Meie ,2013) ha wi hholding ac s
ypically inc eases wi h he concen a ion o owne ship on he media ma ke , which has
ound empi ical suppo in (Bea ie e al.,2021).
No e ha iola ions o Assump ion 2 do no necessa ily o e u n P oposi ion 1. In
Appendix F, we gi e a su icien condi ion o P oposi ion 1 o hold i Assump ion 2
is iola ed. Mo eo e , a di e en su icien condi ion is a ailable in he case o hiding
in o ma ion:
Co olla y 1. Suppose ha Assump ions 1, (2 log), and 3 hold. Conside he case o hiding
in o ma ion, i.e.,
ci( i)=
0 o all
i∈Vi
. Suppose ha
si
and
Ri
a e s ic ly posi i e o all
( i, −i)∈Vi×V−i. Then Γbis a pa ame e ized supe modula game.
P oo .
We only gi e he p oo assuming di e en iabili y o simplici y. Comme cial ou le
i
maximizes
πi( i
,
−i) = si( i
,
−i)Ri( i)
. By assump ion,
si( i
,
−i)Ri( i)>
0. Thus he
p o i maximizing quali y o i m ialso has o maximize
ln πi=ln si+ln Ri.
Since
ln si
has weakly inc easing di e ences by Assump ion (2 log), and
Ri
is independen
o
j
o
j=i
, i ollows ha
ln πi
has weakly inc easing di e ences in
( i, −i)
. The es o
he p oo is as in he p oo o P oposi ion 1.
The logi model, and he nes ed logi model, a e wo equen ly used audience demand
unc ions in media economics. As men ioned abo e, hey sa is y Assump ion (2 log).
Co olla y 1 hus implies ha in he case o hiding in o ma ion ou main esul s apply o
he (nes ed) logi model.
Le e aging he heo y o supe modula games (see (Vi es,2005a), (Vi es,2005b) o
(Sa e ,2023) o exposi ions) allows us o gene a e ai ly gene al esul s in ou model
ea u ing many asymme ic media ou le s. Deno e a s a egy p o ile in game
Γb
by
C=
( 1, ..., n)
. The ollowing co olla y collec s s anda d esul s o pa ame ized supe modula
games ha a e use ul in ou con ex :
Co olla y 2. I
Γb
is a pa ame e ized supe modula game, hen
Γb
has, o any
b
,a lowes
equilib ium
C,low
and a highes equilib ium
C,high
,such ha any equilib ium
C
sa is ies
C,low ≤ C≤ C,high
.Mo eo e , he equilib ia
C,low
and
C,high
a e weakly mono one inc easing
in b.
In pa icula , an equilib ium exis s; he s anda d p oo o equilib ium exis ence in
supe modula games uses he Ta ski ixed poin heo em. Tu ning o he compa a i e
s a ics, no e Co olla y 2 does no imply ha all equilib ia a e mono one inc easing in
b
:
only he highes and lowes equilib ia a e gua an eed o be weakly mono one inc easing
in
b
(see (Sa e ,2023) o a gene al elabo a ion o his poin o supe modula games).
16
When he equilib ium is unique, a s onge mono one compa a i e s a ic esul is a ailable,
which we highligh in he ollowing P oposi ion 2.
P oposi ion 2. Suppose
Γb
is a pa ame e ized supe modula game and has a unique equilib ium.
Then he equilib ium p og am quali y o each comme cial media ou le
i∈C
is weakly inc easing in
he budge bjo any PSM j ∈P.
P oposi ion 2 s a es ha PSM do no c owd ou , and may e en c owd in p og am
quali y, in line wi h he idea ha PSM engage comme cial media in a ace o he op. To
Games 2025,16, 21 15 o 46
6. Conclusions
In his pape , we show ha in a model o comme cial media bias, p og am quali ies
in e ms o unbiased epo ing a e s a egic complemen s a he han s a egic subs i u es.
The s a egic complemen a i y s ems om he media’s undamen al ade-o in hese
models: Inc easing p og am quali y inc eases he alue o he p og am o he audience
bu dec eases he willingness o pay o he ad e ise s o each consume s. The la e e ec
becomes less impo an when a media company has a smalle audience; hence, i s incen i es
o inc ease p og am quali ies a e highe . Thus, in a media ma ke wi h bo h PSM and
comme cial media, PSMs’ wi h high p og am quali ies gi e comme cial media incen i es
o p o ide high quali y hemsel es, oo. As a esul , he PSM c owd in quali y and engage
he comme cial media in a ace o he op. This is in line wi h ecen empi ical e idence on
public and p i a e in es men s in o p og am quali y (Sehl e al.,2020;Simon,2013).
Ou esul s hold unde ai ly gene al condi ions. One impo an assump ion is ha
audience demand unc ions ha e weakly inc easing di e ences, a condi ion me by a ious
s anda d demand unc ions. These include linea demand unc ions, quad a ic demand
unc ions wi h posi i e coe icien s o he in e ac ion e ms, as well as models such as
Ho elling, Salop, and Spokes. We also gi e su icien condi ions o ou esul s o hold o
audience demand unc ions ha do no ha e weakly inc easing di e ences, such as he logi
model. While ou main analysis conside s media ou le s who o e hei p og ams o ee,
ou esul s also ex end o pay media when comme cial media bias is abou wi hholding
ac s ha he media al eady ha e. Simila ly, ou esul s hold o mul idimensional s a egy
spaces in he case o wi hholding ac s; o he case o in es iga ing ac s, we p o ide
condi ions unde which ou main esul s gene alize. A guably, ad e ising e enue o
all ou le s migh be nega i ely a ec ed when some ou le s epo abou de iciencies o
a p oduc , hence he e may be spillo e e ec s. Howe e , we show ha ou esul s a e
obus when hese spillo e e ec s a e small compa ed o he di ec e ec o an ou le ’s
own p og am quali y on i s ad e ising e enue. I en y o exi o comme cial media was
possible, comme cial en an s would ha e o p o ide su icien ly high quali y in o de o
o e u n ou main esul s. Finally, we poin ou ha ou esul s can allow o some biases in
PSM as long as a highe budge o a PSM will ansla e in o less se e e biases o
his ou le .
We ha e also show, howe e , ha p og am quali ies a e no always s a egic comple-
men s. We gi e examples ha show ha , o pay media (o mul idimensional s a egy
spaces) in he case o cos ly in es iga i e jou nalism, highe quali y PSMs can inc ease o
dec ease he quali y o comme cial media ou le s, depending on de ails o he model. Mo e-
o e , s ong PSM may p e en en y o comme cial ou le s in he i s place. Fu he mo e,
highe budge s o PSMs could, in case o poli ically cap u ed PSMs, inc ease hei biases,
and by s a egic complemen a i ies he biases o comme cial media would inc ease as a
consequence.
The pape con ibu es o ecu en media policy deba es abou he p ope ole and
scope o PSM. While se e al egula ion au ho i ies ea ha aising he p og am quali y
o PSM could c owd ou p i a e in es men s in o p og am quali y, ou esul s suppo
policies ha ad oca e s ong and inancially well-equipped PSM. Reduc ions in he unding
o PSM migh esul in a wo se media landscape al oge he .
Ou insigh s migh be especially impo an o mode n media ma ke s like social
media, whe e sys ema ic quali y con ols a e missing and quali y s anda ds a e o en
claimed o be low (Zhu a skaya e al.,2020). While PSM ha e ypically played a mino ole
he e, ou esul s encou age PSM o de elop a s onge p esence and p o ide high-quali y
con en on social media, oo. This easoning is in line wi h ecen schola ly ad ances
calling on PSM o become “Public Se ice In e ne pla o ms” wi h he objec i e o p o ide
Games 2025,16, 21 16 o 46
oppo uni ies o public deba e, pa icipa ion, and he ad ancemen o social cohesion
(Un e be ge & Fuchs,2021).
An in e es ing a enue o u u e esea ch would be o es ou p edic ions empi ically.
In pa icula , we hypo hesize ha an inc ease in PSMs’ budge would ce e is pa ibus ansla e
bo h in o highe p og am quali ies o he PSM hemsel es as well as in o highe p og am
quali ies o he PSMs’ comme cial compe i o s. Simila ly, educ ions in PSM budge s
would lead o lowe p og am quali ies o bo h PSM and comme cial media. Howe e , he
iden i ica ion o a causal ela ionship be ween PSM budge s and p og am quali ies would
equi e some exogenous a ia ion in PSM budge s.
Au ho Con ibu ions: Concep ualiza ion, A.K. and J.M.; o mal analysis, J.M.; w i ing—o iginal
d a p epa a ion, A.K. and J.M.; w i ing— e iew and edi ing, A.K. and J.M. All au ho s ha e ead
and ag eed o he published e sion o he manusc ip .
Funding: This p ojec is unded by he Ba a ian S a e Minis y o Science and he A s in he ame-
wo k o he bid G adua e Cen e o Pos docs. Funded by he Deu sche Fo schungsgemeinscha
(DFG, Ge man Resea ch Founda ion) unde Ge many’s Excellence S a egy—EXC2126/1-390838866.
Da a A ailabili y S a emen : The o iginal con ibu ions p esen ed in he s udy a e included in he
a icle, u he inqui ies can be di ec ed o he co esponding au ho .
Acknowledgmen s: We hank La a Mai o excellen esea ch assis ance.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A. Assump ions 1 and 2 A e Sa is ied by Many F equen ly
Used Models o Audience Size in Media Economics
In he media economics li e a u e, he e a e ou commonly used ways o model
audience size when media quali y ma e s (see he su eys by (Ande son & Gabszewicz,
2006;Ande son & Jullien,2015). These models a e also widely used in o he ields, see
(Huang e al.,2013) o a su ey. This appendix discusses which o hese models sa is y
Assump ions 1 and 2, and gi es e e ences o publica ions in media economics ha employ
hese models.
Fi s , Ho elling model o ho izon ally di e en ia ed goods, en iched by a e ical
quali y di e en ia ion. In hese models, he e a e wo compe ing ou le s. When all ma ke
sha es a e posi i e,
siui,uj=1
2+ui−uj
2τ,
whe e
τ>
0 is a pa ame e o he deg ee o p oduc di e en ia ion. This speci ica ion is
used (wi h
ui= i−pi
whe e
pi
is he p ice) o example, in (A ms ong & Weeds,2007a,
2007b;Bisceglia,2023;D’Annunzio,2017;Gonzalez-Maes e & Ma ínez-Sánchez,2015;
Li e al.,2023;Li & Zhang,2016;Liu e al.,2004;S ennek,2014). Gene aliza ions o mo e
han wo compe ing ou le s include he Spokes model (Chen & Rio dan,2007) used by
(Ge mano & Meie ,2013), and he equen ly used Salop ci cle model (used by (Ke kho
& Müns e ,2015) o s udy comme cial media bias
24
). Assuming all ma ke sha es o be
s ic ly posi i e, hese models sa is y Assump ions 1 and 2; in pa icula ,
si
has cons an
di e ences in (ui,u−i).
Second, ep esen a i e consume models wi h a quad a ic u ili y unc ion a e used
in media economics by se e al pape s, including (Dewen e e al.,2011;Godes e al.,
2009;Kind e al.,2007,2009,2016;Mo a & Polo,2003). The model can allow o i m
speci ic demand in e cep s (see (Choné & Linneme ,2020)) and in his way be used o s udy
Games 2025,16, 21 17 o 46
compe i ion in quali ies (as in (Banke e al.,1998) and (Mo a & Polo,2003)). The esul ing
audience demand is
si(ui,u−i)=ai+biui−∑
j=i
bijuj,
whe e
ai
,
bi>
0 and
bij ≥
0. In his class o models, Assump ions 1 and 2 hold; in pa icula ,
sihas cons an di e ences in (ui,u−i).
A hi d ype o model o audience size is he andom u ili y model o disc e e choice.
The u ili y o choosing ou le
i
is composed o
ui+εi
, he
εi
a e assumed o be i.i.d.
dis ibu ed, and he consume chooses he ou le o e ing he highes u ili y. Assuming he
εi o be Gumbel (o ex eme alue ype I) dis ibu ed, his esul s in he logi model
si=exp(µui)
∑n+m
j=0expµuj,
whe e
j=
0 co esponds o he ou side op ion, and
µ>
0 is a pa ame e ela ed o he
a iance o he
εi
. The logi model iola es Assump ion 2 whene e he e a e
n≥
2
comme cial ou le s, bu i sa is ies Assump ion (log 2). In addi ion o he logi model, he
nes ed logi model—which is o en used in empi ical s udies o audience demand in media
economics (see (Be y & Wald ogel,2015) o a su ey)—sa is ies Assump ion (2 log) as
well (see Appendix F). The log-sepa able model ((Be ns ein & Fede g uen,2004), see also
(Huang e al.,2013)) also sa is ies Assump ion (2 log) when all p ices a e ze o.
Fou h, models o e ical quali y di e en ia ion in he adi ion o (Mussa & Rosen,
1978); see o example (Roge ,2017) o an applica ion o media economics. These models
ha e he p ope y ha , i all goods a e a ailable o ee, hen all consume s choose he good
wi h he highes quali y. The e o e, he esul ing audience unc ions o eely a ailable
media a e discon inuous in quali y a he highes quali y o he compe i o s, iola ing ou
assump ions. (Wi h posi i e p ices, he discon inui ies a e smoo hed ou because consume s
di e in hei willingness o pay o quali y.) While hese a e aluable models o pay
media, we do no conside hem in his pape , whe e ou main ocus is on ee media.
Appendix B. Weakly Inc easing Di e ences Wi hou Di e en iabili y
In he p oo o P oposi ion 1, we assumed ha he unc ions
si
,
Ri
, and
ci
a e di e -
en iable in o de o p o e ha
πi
has weakly inc easing di e ences in
( i, −i)
. In his
appendix we gi e he p oo wi hou assuming di e en iabili y. Conside one ou le
j=i
,
hold all o he
k(k=i,j)
cons an and supp ess hem in he o mulas o a oid no a ional
clu e . Then
πi i, j=si i, jRi( i)−ci( i).
Suppose ha h
i> l
iand h
j> l
j. Then
πi i, h
j−πi i, l
j=si i, h
j−si i, l
jRi( i)
and
πi h
i, h
j−πi h
i, l
j−πi l
i, h
j−πi l
i, l
j
=si h
i, h
j−si h
i, l
jRi h
i−si l
i, h
j−si l
i, l
jRi l
i
=si h
i, h
j−si h
i, l
j−si l
i, h
j−si l
i, l
jRi h
i
+si l
i, h
j−si l
i, l
jRi h
i−Ri l
i
Games 2025,16, 21 18 o 46
By Assump ion 2, sihas weakly inc easing di e ences in i, j, i.e.,
si h
i, h
j−si l
i, h
j≥si h
i, l
j−si l
i, l
j
o equi alen ly
si h
i, h
j−si h
i, l
j≥si l
i, h
j−si l
i, l
j.
Since Ri h
i≥0, i ollows ha
si h
i, h
j−si h
i, l
j−si l
i, h
j−si l
i, l
jRi h
i≥0.
Mo eo e , by (2) si l
i, h
j≤si l
i, l
jand by (3), Ri h
i≤Ri l
i, hus
si l
i, h
j−si l
i, l
jRi h
i−Ri l
i≥0.
I ollows ha
πi h
i, h
j−πi h
i, l
j≥πi l
i, h
j−πi l
i, l
j
o equi alen ly
πi h
i, h
j−πi l
i, h
j≥πi h
i, l
j−πi l
i, l
j
i.e., πihas weakly inc easing di e ences in h
i, h
j.
Appendix C. Pay Media and In es iga i e Repo ing: An Example
In his appendix we conside an example o a pay media ou le in he case o in es iga -
ing ac s. Conside a Ho elling duopoly. Ou le 1 is a pay media ou le . We in es iga e he
compa a i e s a ics o he p o i maximizing choices o ou le 1 wi h espec o
u2
; o his
exe cise i does no ma e whe he ou le 2 is ano he comme cial (pay o eely a ailable)
media ou le o a PSM.
Example A1. Suppose ha
V1=[0, ¯
]
wi h 0
<¯
<
1
/β
,
c1( 1)=k 2
1/
2whe e
k>
0is a
pa ame e , and
R1( 1)=
1
−β 1
wi h 0
<β<
1. The o al audience has a ixed size no malized
o 1. The ma ke sha e o ou le 1is gi en by he Ho elling speci ica ion
s1( 1,p1,u2)=
0, i 1
2+ 1−p1−u2
2τ≤0,
1
2+ 1−p1−u2
2τ,i 0<1
2+ 1−p1−u2
2τ<1,
1, o he wise,
whe e τ>0is a pa ame e o he deg ee o p oduc di e en ia ion. The p o i o ou le 1is
π1( 1,p1,u2)=s1( 1,p1,u2)(R1( 1)+p1)−k 2
1
2.
Assume ha
4kτ>(1−β)2(A1)
in o de ha π1is s ic ly conca e in ( 1,p1)in he ele an ange.
No e ha in his example
k
has o be su icien ly high o he second o de condi ion
o hold, hence he case o wi hholding ac s is no a limi case o his example. In o de o
ha e a uni ied ea men o he cases o in es iga ing ac s (whe e
k>
0) and wi hholding
Games 2025,16, 21 19 o 46
ac s (whe e
k=
0), we will also commen on he case whe e inequali y A1 is e e sed in
Appendix C.2.
Be o e we p oceed, we poin ou ha Example A1 is s a egically equi alen o a model
whe e
u1
is s ic ly conca e in quali y, and quali y has linea cos s. To see his, hink o
ou le 1 choosing
w1:=k 1/
2. Then
u1= 1−p1=√2w1/k−p1
is s ic ly conca e in
w1
,
and
w1
has cons an ma ginal cos s o 1. A model wi h linea cos s and conca e consume
u ili y could esul , o example, i quali y ep esen s he numbe o ealiza ions om a
noisy signal acqui ed by media i ms, o he ime spend on in es iga ion be o e epo ing.
The conca i y o consume u ili y hen ep esen s dec easing ma ginal u ili y o new signal
ealiza ions o mo e ime spend on in es iga ing he same issue.
Appendix C.1. Analysis o Example A1
Rema k A1. In Example A1, suppose ha he p o i maximiza ion p oblem o 1has an in e io
solu ion whe e 0
< 1<¯
,
p1>
0,
R1>
0and 0
<s1<
1.
25
Then
1
and
p1
a e s ic ly
dec easing in
u2
.Mo eo e ,
u1= 1−p1
is s ic ly inc easing in
u2
i 2
kτ>(1−β)2
, and
s ic ly dec easing i 2kτ<(1−β)2.
P oo . In he ele an ange,
π1( 1,p1,u2)=1
2+ 1−p1−u2
2τ(1−β 1+p1)−k 2
1
2.
The pa ial de i a i es a e
∂π1
∂p1=−1
2τ(1−β 1+p1)+1
2+ 1−p1−u2
2τ,
∂π1
∂ 1=1
2τ(1−β 1+p1)−β1
2+ 1−p1−u2
2τ−k 1.
Mo eo e ,
∂2π1
∂p2
1
=−1
τ<0,
∂2π1
∂ 2
1
=−β
τ−k<0,
∂2π1
∂p1∂ 1=1+β
2τ.
Hence he de e minan o he Hessian is
1
τβ
τ+k−1+β
2τ2
>0
i 4
kτ>(1−β)2
. This shows
π1
is s ic ly conca e in he ele an ange i inequali y (A1)
holds.
The i s o de condi ions o an in e io solu ion a e
1
2τ(1−β 1+p1)=1
2+ 1−p1−u2
2τ,
1
2τ(1−β 1+p1)=β1
2+ 1−p1−u2
2τ+k 1.
Games 2025,16, 21 20 o 46
Sol ing he i s o de condi ions gi es
∗
1(u2)=(τ+1−u2)(1−β)
4kτ−(1−β)2,
p∗
1(u2)=(β(1−β)+2kτ)(τ−u2)+1−β−2kτ
4kτ−(1−β)2.
Di e en ia e
∂ ∗
1(u2)
∂u2=−1−β
4kτ−(1−β)2<0,
∂p∗
1(u2)
∂u2=−β(1−β)+2kτ
4kτ−(1−β)2<0.
Mo eo e , om u∗
1(u2)= ∗
1(u2)−p∗
1(u2),
∂u∗
1(u2)
∂u2=2kτ−(1−β)2
4kτ−(1−β)2.
The e o e,
u∗
1(u2)
is s ic ly inc easing in
u2
i 2
kτ>(1−β)2
, and
u∗
1(u2)
is s ic ly
dec easing in u2i 2kτ<(1−β)2.
I emains o check unde which pa ame e cons ella ions an in e io solu ion exis s.
No e ha ∗
1(u2)>0 i
τ+1>u2, (A2)
and p∗
1(u2)>0 i
τ+1−β−2kτ
(β(1−β)+2kτ)>u2. (A3)
No e ha 1−β−2kτ
(β(1−β)+2kτ)<1
by inequali y (A1). Thus inequali y (A2) is implied by inequali y (A3).
We u n o ad e ising e enue nex . No e ha
R1( ∗
1(u2)) =1−β(τ+1−u2)(1−β)
4kτ−(1−β)2
is s ic ly posi i e i
u2>τ+1−4kτ−(1−β)2
β(1−β). (A4)
Inequali ies (A3) and (A4) hold simul aneously i
τ+1−β−2kτ
(β(1−β)+2kτ)>u2>τ+1−4kτ−(1−β)2
β(1−β). (A5)
By inequali y (A1), he igh hand side is s ic ly smalle han he le hand side;
he e o e (A5) is sa is ied in a non-emp y open se o alues o u2.
The equi emen 1<¯
is sa is ied whene e ¯
is su icien ly high.
Finally, we need o make su e ha 0 <s1u∗
1(u2),u2<1. This is he case i
0<1
2+ ∗
1(u2)−p∗
1(u2)−u2
2τ<1,
Games 2025,16, 21 21 o 46
o equi alen ly
−τ< ∗
1(u2)−p∗
1(u2)−u2<τ.
We ha e
∗
1(u2)−p∗
1(u2)−u2
=(τ+1−u2)(1−β)
4kτ−(1−β)2−(β(1−β)+2kτ)(τ−u2)+1−β−2kτ
4kτ−(1−β)2−u2
=τ2k−2kτ+(1−β)2−2ku2
4kτ−(1−β)2.
Thus 0 <s1u∗
1(u2),u2<1 i
−1<2k−2kτ+(1−β)2−2ku2
4kτ−(1−β)2<1,
o equi alen ly
−4kτ−(1−β)2<2k−2kτ+(1−β)2−2ku2<4kτ−(1−β)2. (A6)
The exp ession in he middle is a s ic ly dec easing unc ion o u2.
Since u2<τ+1 by (A2),
2k−2kτ+(1−β)2−2ku2>2k−2kτ+(1−β)2−2k(τ+1)
=−4kτ−(1−β)2,
hus he i s inequali y in (A6) holds.
Simila ly, by (A4),
2k−2kτ+(1−β)2−2ku2
<2k−2kτ+(1−β)2−2k τ+1−4kτ−(1−β)2
β(1−β)!
=1
β(1−β)β2−β+2k−β2+2β+4kτ−1.
The e o e, a su icien condi ion o he second inequali y in (A6) is ha
4kτ−(1−β)2−1
β(1−β)β2−β+2k−β2+2β+4kτ−1
=2(β(1−β)−k)4kτ−(1−β)2
β(1−β)>0,
which is ue i
β(1−β)>k. (A7)
We ha e es ablished ha he p oblem has an in e io solu ion unde he condi ions
(A7), (A1), and (A5), which we epea he e o con enience:
β(1−β)>k,
4kτ>(1−β)2,
τ+1−β−2kτ
(β(1−β)+2kτ)>u2>τ+1−4kτ−(1−β)2
β(1−β).
Games 2025,16, 21 22 o 46
To see hey can be sa is ied simul aneously, i s choose
β
and
k
such ha he i s line
holds. Then choose
τ
such ha he second line holds; no e ha depending on how you
choose τ, ei he 2kτ>(1−β)2o 2kτ<(1−β)2. Finally, choose u2 o he las line.
A nume ical example ha sa is ies all he cons ain s may be eassu ing. Le
β=
0.5,
τ=
1.25, and
u2=
1.5. Fo
k=
0.11, 2
kτ=
2
∗
0.11
∗
1.25
=
0.275
>(1−β)2=
0.25 and
u∗
1(u2)
is s ic ly inc easing in
u2
. Fo
k=
0.09, 2
kτ=
2
∗
0.09
∗
1.25
=
0.225
<
0.25
<
4kτ=0.45, and u∗
1(u2)is s ic ly dec easing.
Wi hin ou pa ame e es ic ions,
u∗
1(u2)
is s ic ly inc easing in
u2
i
k
is la ge. An
economic in ui ion is ha he ma ginal cos s o
1
a e apidly inc easing i
k
is la ge, and
hence hen he alling p ice domina es he dec ease in p og am quali y. To gi e mo e de ails,
ecall ha
∗
1(u2)
and
p∗
1(u2)
a e s ic ly dec easing in
u2
. I
k
is la ge, he e ec o
u2
on
∗
1(u2)becomes less impo an (smalle in absolu e alue):
∂
∂k
∂ ∗
1(u2)
∂u2=4τ(1−β)
4kτ−(1−β)22>0.
On he o he hand, he e ec o u2on p∗
1(u2)also becomes less impo an :
∂
∂k
∂p∗
1(u2)
∂u2=∂
∂k −β(1−β)+2kτ
4kτ−(1−β)2!
=2τ1−β2
4kτ−(1−β)22>0
Bu no e ha 4τ(1−β)−2τ1−β2=2τ(1−β)2>0, hus
∂
∂k
∂ ∗
1(u2)
∂u2
>∂
∂k
∂p∗
1(u2)
∂u2.
Tha is, i
k
inc eases, he change o
∗
1(u2)
in
u2
is anishing quicke han he change
o
p∗
1(u2)
in
u2
. Fo la ge enough
k
,
u∗
1(u2)
inc eases in
u2
because he alling p ice o e -
compensa es o he alling quali y.
Appendix C.2. The Case Whe e Inequali y (A1) Is Re e sed
This subsec ion conside s he case whe e 4
kτ<(1−β)2
. This is o pa icula in e es
in o de o ha e a uni ied ea men wi h he case o hiding in o ma ion whe e
k=
0. In
his case, he p o i maximiza ion p oblem o 1 canno ha e an in e io solu ion whe e
0
< 1<¯
,
p1>
0,
R1>
0 and 0
<s1<
1, since he necessa y second o de condi ion o
a maximum would be iola ed.
We con inue o s udy pay media, i.e., we will ocus on cons ella ions whe e
p1>
0 in
he solu ion o he p o i maximiza ion p oblem; as we will show, a su icien condi ion o
his o be he case is ha he compe i o ’s quali y is no oo high.
Rema k A2. Conside Example A1 bu suppose ha 4
kτ<(1−β)2
,and he solu ion o he p o i
maximiza ion p oblem o i m 1in ol es
p1>
0,
R1>
0and 0
<s1<
1.
26
I
u2<τ−
1, hen
1=¯
and p1is s ic ly dec easing in u2.
P oo . In he ele an ange,
π1( 1,p1)=1
2+ 1−p1−u2
2τ(1−β 1+p1)−k 2
1
2.
Games 2025,16, 21 23 o 46
Since in he p o i maximum p1>0, he ollowing i s o de condi ion has o hold:
∂π1( 1,p1)
∂p1=−1
2τ(1−β 1+p1)+1
2+ 1−p1−u2
2τ=0.
Sol ing o p1gi es
p1=p∗
1( 1):=1
2(τ−u2+(1+β) 1−1).
No e ha he assump ion ha u2<τ−1 implies ha p∗
1( 1)>0 o all 1∈[0, ¯
].
De ine
π1( 1):=π1( 1,p∗
1( 1))
= 1
2+ 1−1
2(τ−u2+(1+β) 1−1)−u2
2τ!1−β 1+1
2(τ−u2+(1+β) 1−1)
−k 2
1
2.
Di e en ia ing π1( 1)shows ha
π′
1( 1)=1
4τ(1−β)(τ−u2+ 1−β 1+1)−k 1.
E alua ed a 1=0, his is
π′
1(0)=1
4τ(1−β)(τ−u2+1)>0.
Mo eo e ,
π′′
1( 1)=1
4τ(1−β)2−k>0.
I ollows ha in he p o i maximum, 1=¯
, and
p1=1
2(τ−u2+(1+β)¯
−1),
which is s ic ly dec easing in u2.
I emains o check he e a e pa ame e cons ella ions whe e
R1>
0 and 0
<s1<
1 in
his solu ion. No e
R1=
1
−β¯
>
0 since
¯
<
1
/β
. Mo eo e , he ma ke sha e o ou le 1
is
1
2+¯
−1
2(τ−u2+(1+β)¯
−1)−u2
2τ!
=1
4τ(¯
(1−β)+τ−u2+1)>0
since by assump ion
u2<τ−
1. Finally, he ma ke sha e o ou le 1 is smalle han
100% i
¯
(1−β)−u2+1<3τ,
which is ue i τis su icien ly high.
To ela e Rema k A2 o Lemma 2, no e ha he e
R′
1( 1)=−β>−
1 o all
1
, he e o e
choosing he highes possible quali y is op imal.
Games 2025,16, 21 24 o 46
Appendix C.3. Summa y
To summa ize, we ound ha in all cases conside ed in his Appendix, he p ice
p1
is s ic ly dec easing in
u2
. Mo eo e , he quali y
1
is s ic ly dec easing in
u2
unless
k
is small in which case
1
is cons an in
u2
. Finally, he u ili y o consuming ou le 1,
u1= 1−p1
, is s ic ly inc easing in
u2
i
k
is small (see Rema k A2) o
k
is la ge, bu
u1
is
s ic ly dec easing in
u2
i
k
is in an in e media e ange (see Example A1). These p edic ions
could, in p inciple, be es ed empi ically.
Appendix D. Mul idimensional S a egy Spaces
This appendix p o ides a o mal analysis o mul idimensional s a egy spaces. As
in he main ex , suppose ha ou le
i
epo s abou
ki
opics, and le
i,k
deno e p og am
quali y o opic
k
. We assume ha
i,k∈h0, high
i,ki
wi h
high
i,k>
0. Ou le
i
chooses a ec o
i∈Vi=×ki
k=1h0, high
i,ki
. Consume u ili y om consuming ou le
i
is
ui= i( i)
, whe e
i
is con inuous, s ic ly inc easing wi h espec o each a gumen
i,k
, and sa is ies
i(0)=
0.
Ad e ising e enue pe consume is
Ri( i)
, whe e
Ri:Vi→R+
is posi i e, con inuous,
weakly dec easing in each a gumen
i,k
, and independen o
−i
. The p o i o
i∈C
is
πi=si(ui,u−i)Ri( i)−ci( i)
. Tu ning o he PSM, suppose ha
i∈P
chooses
i∈Vi(bi)
o maximize
ui= i( i)
subjec o
ci( i)≤bi
. As abo e, a highe budge may enla ge he
easible se Vi.
Appendix D.1. Wi hholding Fac s
In he case o wi hholding ac s, he e a e no di ec cos s o aising p og am quali y.
Thus, ci( i)=0 o all i∈C. We show ha ou main esul s gene alize.
In ou analysis, we make use o he ac ha i he quali ies
i
o a comme cial ou le
i
maximize i s p o i and gene a e u ili y
ui
o a consume , hen
i
mus maximize ad e is-
ing e enue pe consume
Ri
subjec o he cons ain ha he u ili y o he consume is a
leas
ui
. (I no , hen he e exis s a easible
ˆ
i
ha gene a es weakly highe u ili y, hence
weakly highe demand, and a he same ime gene a es s ic ly highe ad e ising e enue
Ri
, con adic ing he op imali y o he
i
.) We can he e o e decompose he p oblem o com-
me cial media ou le
i
in o wo s eps: The i s s ep asks which
i
maximizes ad e ising
e enue subjec o he cons ain ha he u ili y o he consume is a leas equal o some
gi en ui. The second s age hen op imizes o e ui.27
S ep 1
In he i s s ep, he ec o o p og am quali ies is chosen o maximize ad e ising
e enue pe consume , subjec o he cons ain ha he u ili y o he consume is a leas
equal o some gi en
ui
. The maximal alue o ad e ising e enue unde his cons ain is
R∗
i(ui)=max
i∈Vi{Ri( i)| i( i)≥ui}.
We show ha
R∗
i
has all he ea u es assumed abou
Ri
in ou main model
(see Assump ion 3)
:
R∗
i
is posi i e, con inuous, weakly dec easing in
ui
and independen
o u−i.
Fi s , R∗
iis posi i e since Riis posi i e by assump ion.
Second, we use he Maximum Theo em o show ha
R∗
i
is con inuous.
Ri
is con inuous
by assump ion. Le
Ui:={ui∈R+|∃ i∈Vi: i( i)≥ui}.
Since 0
∈Vi
and
i(0)=
0, 0
∈Ui
. Mo eo e , since
Vi
is compac and
i
is con inuous, by he
Weie s ass Theo em a maximum achie able u ili y exis s, hus Ui=0,max i∈Vi i( i).
Games 2025,16, 21 31 o 46
Appendix E.2. La ge Spillo e E ec s: An Example
In his appendix, we show by example ha
πi
may ha e cons an di e ences in
( i, −i)
i spillo e e ec s a e la ge. We conside a Ho elling duopoly. Ou le 1 is a comme cial
ou le . We in es iga e he compa a i e s a ics o he p o i maximizing choices o ou le
1 wi h espec o
2
; o his exe cise i does no ma e whe he ou le 2 is ano he eely
a ailable comme cial media ou le o a PSM.
Example A3. Suppose ha V1=R+,
R1( 1, 2)=max{1−(α 1+β 2), 0}
whe e
α>
0and
β>
0a e exogenous pa ame e s,
s1
is gi en by a Ho elling speci ica ion. Mo eo e ,
suppose ha he p o i maximiza ion p oblem o 1has an in e io solu ion whe e
1>
0,
R1>
0,
and 0<s1<1.31
No e ha R1in Example A3 sa is ies Assump ion A1.
Rema k A4. Conside Example A3. I
α>β
, hen
π1
has s ic ly inc easing di e ences in
( 1, 2)
in he ele an ange. I α=β, hen π1has cons an di e ences in ( 1, 2)in he ele an ange.
P oo . In he ele an ange,
s1( 1, 2)=1
2+ 1− 2
2τ,
hence ∂2s1( 1, 2)
∂ 2∂ 1=0
and ∂s1( 1, 2)
∂ 1=−∂s1( 1, 2)
∂ 2=1
2τ.
The p o i o comme cial ou le 1 is
π1( 1, 2)=s1( 1, 2)(1−(α 1+β 2)) −c1( 1).
Hence ∂π1
∂ 1=1
2τ(1−(α 1+β 2)) −αs1( 1, 2)−∂c1( 1)
∂ 1
and ∂2π1
∂ 1∂ 2=α−β
2τ.
The e o e, i
α>β
,
π1
has s ic ly inc easing di e ences in
( 1, 2)
. On he o he hand,
i α=β, hen π1has cons an di e ences in ( 1, 2).
An implica ion o Rema k A4 is ha , i he cos unc ion
c1
is s ic ly con ex and wice
di e en iable, he p o i maximizing p og am quali y
∗
1( 2)
is s ic ly inc easing in
2
i
α>β, and ∗
1( 2)is cons an in 2i α=β.32
To conclude his appendix, we assume a quad a ic cos unc ion o illus a e ha all he
assump ions in Example A3 a e consis en wi h each o he . Suppose ha
c1( 1)=k 2
1/
2,
k>0. Then he bes eply unc ion is
∗
1( 2)=
1
2τ−α
2+α−β
2τ 2
α
τ+k.
Games 2025,16, 21 32 o 46
Example A3 assumed an in e io solu ion wi h
1>
0,
R1>
0 and
s1∈(0,1)
. To see
hese assump ions a e consis en wi h each o he , conside he symme ic case whe e bo h
i ms a e comme cial media and ha e he same cos and ad e ising e enue unc ions. In
a symme ic equilib ium,
1= 2=1−ατ
α+β+2kτ>0
i ατ <1. Mo eo e , o i=1, 2,
Ri( i)=1−(α+β)(1−ατ)
α+β+2kτ=τα2+αβ +2k
α+β+2kτ>0.
Finally, by symme y s1( 1, 2)=s2( 1, 2)=1/2.
Appendix F. Audience Func ions wi h Dec easing Di e ences, he Logi
and Nes ed Logi Models
Appendix F.1. Audience Func ions wi h Dec easing Di e ences: A Su icien Condi ion o
P oposi ion 1
Viola ions o Assump ion 2 do no o e u n ou esul s when he elas ici y o ad e is-
ing e enue wi h espec o p og am quali y is su icien ly high. To see his, no e ha he
c ucial inequali y (1) in he p oo o P oposi ion 1 holds i
R′
i( i)
Ri( i)≥
∂2si( i, −i)
∂ j∂ i
∂si( i, −i)
∂ j
.
Unde Assump ion 2, he igh hand side is nega i e hence he abo e inequali y is
always sa is ied; when Assump ion 2 is iola ed ad e ising e enue mus eac su icien ly
s ong o p og am quali y o he inequali y o hold.
Appendix F.2. The Logi Model
To illus a e, we conside a gene aliza ion o he logi model. Suppose he e is mass o
consume s no malized o one, and he ma ke sha e o ou le iis
si( i, −i)= i( i)
∑n+m
j=0 j j,
whe e he unc ions
i( i)
a e s ic ly posi i e and s ic ly inc easing, and
0
is he u ili y o
he ou side op ion. We allow (bu do no equi e) he unc ions
i
o di e ac oss media
ou le s. The logi model is a special case whe e
i( i)=exp(µ i)
o some exogenous
pa ame e µ>0.
No e ha his audience unc ion sa is ies Assump ion 1, bu in gene al iola es As-
sump ion 2. In pa icula , i he e a e wo o mo e comme cial media ou le s,
si
canno
ha e inc easing di e ences o all
i∈C
, as we show below. We also p o e, howe e , ha a
su icien condi ion o Γb o be a supe modula game is ha
R′
i( i)
Ri( i)≥ ′
i( i)
i( i)
o all
i∈C
. In he logi model, his su icien condi ion educes o
R′
i( i)/Ri( i)≥µ
o
all i∈C.
Games 2025,16, 21 33 o 46
This illus a ion shows ha , while Assump ion 2 is es ic i e, dec easing di e ences
in he demand unc ions do no necessa ily o e u n ou esul s when ad e ising e enue
eac s s ongly on p og am quali y.
Appendix F.3. The Logi Model Viola es Assump ion 2
As in he las subsec ion, suppose ha
si( i, −i)= i( i)
∑n+m
j=0 j j,
whe e he unc ions
i( i)
a e s ic ly posi i e and s ic ly inc easing, and
0
is he u ili y o
he ou side op ion. Fo k=i,
∂si
∂ k=− i( i) ′
k( k)
∑n+m
j=1 j j2,
∂2si
∂ i∂ k= ′
k( k) ′
i( i) i( i)−∑j=i j j
∑n+m
j=1 j j3.
This implies ha , i he e a e wo o mo e comme cial ou le s,
si
canno ha e weakly
inc easing di e ences o all i∈C, so Assump ion 2 is iola ed.
Appendix F.4. A Su icien Condi ion o P oposi ion 1 in he Logi Model
No e ha
∂2πi
∂ i∂ k= ′
k( k) ′
i( i) i( i)−∑j=i j j
∑n+m
j=1 j j3Ri( i)− i( i) ′
k( k)
∑n+m
j=1 j j2R′
i( i)
= ′
k( k)
∑n+m
j=1 j j2
′
i( i) i( i)−∑j=i j j
∑n+m
j=1 j jRi( i)− i( i)R′
i( i)
> ′
k( k)
∑n+m
j=1 j j2
′
i( i)−∑j=i j j
∑n+m
j=1 j jRi( i)− i( i)R′
i( i)
> ′
k( k)
∑n+m
j=1 j j2− ′
i( i)Ri( i)− i( i)R′
i( i)
so a su icien condi ion o πi o ha e weakly inc easing di e ences in ( i, −i)is ha
− i( i)R′
i( i)≥ ′
i( i)Ri( i)
o equi alen ly
R′
i( i)
Ri( i)≥ ′
i( i)
i( i).
In he logi model,
i( i)=exp(µ i)
and hence
′
i( i)/ ( i)=µ
. The e o e, ou main
esul s hold in he logi model whene e R′
i( i)/Ri( i)≥µ o all i∈C.
Games 2025,16, 21 34 o 46
Appendix F.5. The Logi Model Sa is ies Assump ion (2 log)
He e we es ablish ha he logi model sa is ies Assump ion (2 log). Suppose he e is
mass o consume s no malized o one, and he ma ke sha e o ou le iis
si( i, −i)= i( i)
∑n+m
j=0 j j,
whe e he unc ions
i( i)
a e s ic ly posi i e and s ic ly inc easing, and
0
is he u ili y o
he ou side op ion. We allow (bu do no equi e) he unc ions
i
o di e ac oss media
ou le s. The logi model is a special case whe e
i( i)=exp(µ i)
o some exogenous
pa ame e µ>0.
Then
ln si=ln( i( i)) −ln n+m
∑
j=0
j j!
∂ln si
∂ i= ′
i( i)
i( i)− ′
i( i)
∑n+m
j=0 j j
o j=i, his is s ic ly inc easing in j, hus
∂
∂ j
∂ln si
∂ i
>0.
The e o e, ln sihas s ic ly inc easing di e ences in (si,s−i).
Appendix F.6. The Nes ed Logi Model Sa is ies Assump ion (2 log)
This subsec ion conside s he nes ed logi model, which is o en used in empi ical
s udies o media demand (see (Be y & Wald ogel,2015) o a su ey). Ou exposi ion o
he nes ed logi model ollows (Be y,1994). Fi ms a e pa i ioned in o g oups. Suppose
i m
i
belongs o g oup
g
. The ma ke sha e o a i m
i
is gi en by
si=si|gsg
, whe e
si|g
is
he sha e o i m iwi hin i s g oup g, and sgis he ma ke sha e o g oup g. Mo eo e ,
si|g=expui
1−σ
∑k∈gexpuk
1−σ
and
sg=∑k∈gexpuk
1−σ1−σ
∑g′∑k∈g′expuk
1−σ1−σ,
whe e σis a pa ame e wi h 0 ≤σ<1.
We now show ha he nes ed logi model sa is ies Assump ion (2 log).
Rema k A5. In he nes ed logi model,
∂2
∂uj∂uiln si>0.
Games 2025,16, 21 35 o 46
P oo .
ln si=ui
1−σ−ln ∑
k∈g
expuk
1−σ!
+(1−σ)ln ∑
k∈g
expuk
1−σ!
−ln
∑
g′
∑
k∈g′
expuk
1−σ
1−σ
=ui
1−σ−σln ∑
k∈g
expuk
1−σ!
−ln
∑
g′
∑
k∈g′
expuk
1−σ
1−σ
.
Di e en ia e wi h espec o ui, keeping in mind i m ibelongs o g oup g
∂ln si
∂ui=1
1−σ−σ
1−σ
expui
1−σ
∑k∈gexpuk
1−σ
−∑k∈gexpuk
1−σ−σexpui
1−σ
∑g′∑k∈g′expuk
1−σ1−σ.
The sign o he c osspa ial
∂2ln si
∂uj∂ui
can be de e mined by conside ing how he e ms in
his sum depend on uj.
Suppose
j
belongs o a di e en g oup han
i
. Then he i s wo e ms a e independen
o
uj
, and he nume a o o he hi d e m is independen o
uj
as well. The sign o
c osspa ial is equal o he sign o
∂
∂uj
−1
∑g′∑k∈g′expuk
1−σ1−σ
>0.
Suppose
i
and
j
belong o he same g oup
g
. Conside he e ms one by one. The
i s e m 1
/(1−σ)
does no depend on
uj
. The second e m is nondec easing in
uj
. Con-
side he hi d e m. Since
σ≥
0,
∑k∈gexpuk
1−σ−σ
is noninc easing in
uj
. Mo eo e ,
∑g′∑k∈g′expuj
1−σ1−σ
is s ic ly inc easing in
uj
. This implies he hi d e m is s ic ly
inc easing in uj.
We conclude ha ∂2ln si
∂uj∂ui>0.
Appendix G. En y: A Ho elling Example
In his Appendix, we illus a e he ou conside a ions on en y in a Ho elling model.
Example A4. Assume ha he e a e a mos wo media ou le s ac i e in he ma ke . Condi ional
on en y and assuming ha 0<si<1and i∈h0, 1
βi, he p o i o a comme cial ou le i is
πi=1
2+ i− j
2τ(1−β i)−F.
Games 2025,16, 21 36 o 46
The ixed cos s
F>
0can be sa ed by s aying ou o he ma ke . Le 3
βτ >
1
>βτ
and
F<βτ/2.
We compa e a mixed comme cial and public media ma ke (dual media ma ke )
wi h a pu ely comme cial media ma ke . In he dual ma ke , he e is one PSM wi h an
exogenously gi en quali y
P
, and one comme cial ou le decides whe he o en e . In a
pu ely comme cial ma ke , he e is no PSM, and up o wo comme cial ou le s may en e .
We model en y by a s anda d wo s age game, whe e in he i s s age en y decisions
a e aken, and in he second s age quali ies a e chosen. The solu ion concep is subgame
pe ec equilib ium.
The assump ions on pa ame e s
β
,
τ
, and
F
allow us o ocus on he mos in e es ing
cases. Speci ically, he assump ion 1
>βτ
ules ou si ua ions whe e he p o i maximizing
quali y equals ze o. The assump ion 3
βτ >
1 ules ou si ua ions whe e he bes eply o a
comme cial ou le o a PSM wi h low quali y is such ha he comme cial ou le has 100%
ma ke sha e. The assump ion ha
F<βτ/
2 ensu es ha in a pu ely comme cial media
ma ke wo media ou le s en e .
In he dual ma ke , he comme cial ou le will en e i he PSM’s quali y is no oo
high. De ine
ˆ
:=1+βτ −p8βτF
β.
We will show below ha
ˆ
is he ele an cu o o he PSM’s quali y, below which he
comme cial ou le en e s.
Rema k A6. Conside Example A4. In a pu ely comme cial ma ke , bo h comme cial media ou le s
en e . (i) In a dual ma ke whe e he PSM’s quali y is
P<(1−βτ)/β
, he comme cial ou le
en e s. The quali y o he PSM and he quali y o he comme cial ou le a e bo h s ic ly lowe han
he equilib ium quali ies in a pu ely comme cial duopoly. (ii) In a dual ma ke whe e he PSM’s
quali y sa is ies
(1−βτ)/β< P<ˆ
, he p i a e ou le en e s. The quali ies o he PSM and o
he comme cial ou le a e bo h s ic ly highe han he equilib ium quali ies in a pu ely comme cial
duopoly. (iii) In a dual ma ke whe e he PSM’s quali y is
P>ˆ
, he p i a e ou le will no en e .
P oo .
We sol e he game by backwa d induc ion. Suppose he e a e wo i ms in he
ma ke . Fi m
i
is a comme cial ou le . Fi m
j
may be a comme cial ou le o a PSM. The
i s o de condi ion o ou le iis
∂
∂ i1
2+ i− j
2τ(1−β i)=1
2τ1−βτ −2β i+β j=0.
The eac ion unc ion o iis
∗
i j=1−βτ +β j
2β>0
whe e he inequali y ollows om he assump ion 1 >βτ.33
No e ha ∗
i jis s ic ly inc easing in j.
Conside he pu ely comme cial ma ke . The e a e wo symme ic po en ial en an s.
I bo h o hem en e , in equilib ium o he esul ing subgame
∗
i j= j
o
i
,
j=
1,2,
hus
1= 2=1−βτ
β,
and p o i s a e
πi=1
21−β1−βτ
β−F=βτ
2−F.
Games 2025,16, 21 37 o 46
Since by assump ion F<βτ/2, in equilib ium bo h comme cial i ms en e .
Now conside he dual ma ke . No e ha
Ri( ∗
i( P)) =1−β1+β P−βτ
2β
=1+βτ −β P
2
is s ic ly posi i e i
P<(1+βτ)/β
. In case ha
P≥(1+βτ)/β
,
i
canno gene a e a
s ic ly posi i e e enue, and hence will no en e .34
Fo he es o he p oo , conside he case whe e
P<(1+βτ)/β
, unless o he wise
no ed. The ma ke sha e o i m iis
si( ∗
i( P), P)=1
2+
1+β P−βτ
2β− P
2τ
=1+βτ −β P
4βτ .
No e his is posi i e when
P<(1+βτ)/β
; mo eo e
si ∗
i( P), P
is smalle han
one ( o all
P≥
0
)
i 1
+βτ <
4
βτ
o 1
<
3
βτ
which we assume o be he case. The p o i
o i m iis
πi( ∗
i( P), P)=(1+βτ −β P)2
8βτ −F.
The comme cial ou le will en e i
(1+βτ −β P)2
8βτ >F,
o equi alen ly
P<ˆ
:=1+βτ −p8βτF
β.
No e ha he assump ion F<βτ/2 implies
ˆ
=1+βτ −p8βτF
β>
1+βτ −q8βτ βτ
2
β=1−βτ
β.
We a e now in a posi ion o comple e he p oo .
Case (i):
P<(1−βτ)/β
. Then
P<ˆ
, hus he he comme cial ou le en e s.
Mo eo e ,
P
is by assump ion s ic ly lowe han he equilib ium quali y in a pu ely
comme cial duopoly
(1−βτ)/β
. Because o he s a egic complemen a i ies,
i
is also
s ic ly lowe han he equilib ium quali ies in a pu ely comme cial duopoly.
Case (ii):
(1−βτ)/β< P<ˆ
. Since
P<ˆ
, he p i a e ou le en e s. Mo eo e ,
P
is by assump ion s ic ly highe han he equilib ium quali y in a pu ely comme cial
duopoly
(1−βτ)/β
. Because o he s a egic complemen a i ies,
i
is also s ic ly highe
han he equilib ium quali ies in a pu ely comme cial duopoly.
Case (iii): I
ˆ
< P<(1+βτ)/β
, he comme cial ou le will no en e because he
e enue i can gene a e does no co e he ixed cos s
F
. Mo eo e , as a gued abo e, i
P≥(1+βτ)/β
he comme cial ou le canno gene a e any s ic ly posi i e e enue and
hence will no en e .
Appendix H. Income E ec s
Consume s in ou model ha e o pay axes o licence ees o co e he budge s o he
PSM. These paymen s a e independen o indi idual media consump ion. They could,
Games 2025,16, 21 38 o 46
howe e , a ec demand ia income e ec s. Such income e ec s can s eng hen ou main
esul s, howe e , when he media a e no mal goods, i.e., demand inc eases in income.
Suppose ha o
i∈C
,
si( i, −i,b)
is weakly dec easing in
b
( he highe
b
, he lowe
consume s’ emaining income; i media a e no mal goods, demand is lowe ). Mo eo e ,
suppose ha
si
has weakly inc easing di e ences in
( i,b)
, i.e., demand eac s mo e on
quali y di e ences when income is lowe . The s a egic complemen a i ies be ween he
comme cial media a e no a ec ed by he income e ec s. Fo
i∈C
and
j∈P
, conside he
c oss-pa ial
∂2˜
πi
∂bj∂ i= ∂2si( i, −i,b)
∂ j∂ iRi( i)+∂si( i, −i,b)
∂ jR′
i( i)!¯
′jbj
+∂2si( i, −i,b)
∂bj∂ iRi( i)+∂si( i, −i,b)
∂bjR′
i( i).
The i s line desc ibes he e ec s s udied in ou main model abo e: an inc ease o
bj
inc eases
¯
j
and his has he e ec s s udied abo e ( he e ms in he b acke a e he same
as in inequali y (1) in he p oo o P oposi ion 1). The second line s ems om he income
e ec . No e ha
∂2si( i, −i,b)
∂bj∂ i≥
0 because
si
has weakly inc easing di e ences in
( i,b)
, and
∂si( i, −i,b)
∂bj≤
0 because good
i
is no mal; hence he second line is posi i e. This shows ha
income e ec s s eng hen he s a egic complemen a i ies ha d i e ou esul s.
On he o he hand, PSM migh lead comme cial media o exi he ma ke . Income
e ec s can s eng hen his ype o c owding ou : he PSM do no only o e compe ing
p oduc s, bu also lowe demand o comme cial media ia income e ec s.
Appendix I. Equilib ium Uniqueness
In his Appendix, we use he con ac ion app oach o gi e su icien condi ions o
equilib ium uniqueness in he main model wi h a linea audience unc ion
si
, and in he
case o wi hholding in o ma ion wi h a logi audience unc ion. Conside he case o a
linea audience unc ion i s :
Rema k A7. Conside he case o a linea audience unc ion
si( i, −i)=ai+bi i−∑
j=i
j∈C∪P
cij j.
Assume ha
ai≥∑
j=i
cij max
j∈Vj
j,
bi>0,
cij >0 o all j =i.
Mo eo e , assume ha
Ri
is smoo h wi h
R′
i( i)<
0 o all
i∈Vi
.A su icien condi ion o
a unique equilib ium is ha
2bi≥∑
j=i
cij,
R′′
i( i)≤0, and c′′
i( i)≥0 o all i∈Vi
wi h a leas one o hese inequali ies s ic .
Games 2025,16, 21 39 o 46
The assump ion
ai≥∑j=icij max j∈Vj j
ensu es ha
si≥
0 o all
( i, −i)∈Vi×V−i
;
bi>
0 ensu es ha
si
is s ic ly inc easing in
i
;
cij >
0 ensu es
si
is s ic ly dec easing in
j
o j=i.
P oo . Wi h he linea audience unc ion,
∂2si
∂ 2
i
=∂2si
∂ j∂ i=0,
so sihas cons an di e ences. I ollows ha ∂2πi
∂ i∂ j>0.
No e ha ∂πi
∂ i=bRi+siR′
i−c′
i
and ∂2πi
∂ 2
i
=2bR′
i+siR′′
i−c′′
i<0
whe e he s ic inequali y ollows because b>0>R′
iby assump ion.
The su icien condi ion o a con ac ion is
∂2πi
∂ 2
i
+∑
j=i
∂2πi
∂ i∂ j
<0.
He e,
∂2πi
∂ 2
i
+∑
j=i
∂2πi
∂ i∂ j
= 2bi−∑
j=i
cij!R′
i+siR′′
i−c′′
i
The e o e, a su icien o uniqueness is ha
2bi≥∑
j=i
cij,
R′′
i( i)≤0, and c′′
i( i)≥0 o all i∈Vi
wi h a leas one o hese inequali ies s ic .
In he Ho elling o Spokes model in he ele an ange whe e all ma ke sha es a e
in e io , bi=∑j=icij, so 2bi>∑j=icij holds au oma ically.
Rema k A8. Conside he logi model
si( i, −i)=exp(µ i)
∑n+m
k=0exp(µ k)
in he case o wi hholding in o ma ion whe e
ci( i)=
0 o all
i
.Mo eo e , assume ha
Ri
is
smoo h,
Ri( i)>
0 o all
i∈Vi
,and
Ri
is s ic ly log conca e in
i
.Then he game
Γb
is a
pa ame e ized supe modula game. Mo eo e , he equilib ium is unique.
Games 2025,16, 21 40 o 46
P oo .
Co olla y 1 implies ha
Γb
is a pa ame e ized supe modula game. No e ha
πi>
0
o all ( i, −i). The e o e, we can hink o he comme cial media ou le s as maximizing
ln πi=ln si( i, −i)+ln(Ri( i)).
Conside
ln si=µ i−ln ∑
j
expµ j!.
Di e en ia e o ob ain
∂ln si
∂ i=µ−µexp(µ i)
∑kexp(µ k).
Fo j=i, he e m µexp(µ i)
∑kexp(µ k)is s ic ly dec easing in j, hus
∂
∂ j
∂ln πi
∂ i=∂
∂ j
∂ln si
∂ i
>0.
The e o e, ln πihas s ic ly inc easing di e ences in ( i, −i).
Mo eo e ,
ln πi
is s ic ly conca e in
i
. Since
ln(Ri( i))
is s ic ly conca e in
i
by
assump ion, his esul ollows om
∂2ln si
∂ 2
i
=−∑jexpµ jexp(µ i)µ2−exp(µ i)exp(µ i)µ2
∑jexpµ j2
=−∑j=iexpµ jexp(µ i)µ2
∑jexpµ j2<0.
To p o e uniqueness, we show ha ln πisa is ies he con ac ion condi ion
∂2ln πi
∂ 2
i
+∑
j=i
j=0
∂2ln πi
∂ j∂ i
<0.
He e his is equi alen o
∂2ln si
∂ 2
i
+∂2ln Ri
∂ 2
i
+∑
j=i
∂2ln si
∂ j∂ i
<0.
By s ic log conca i y o Ri, i is enough o show ha
∂2ln si
∂ 2
i
+∑
j=i
∂2ln si
∂ j∂ i≤0.
F om
si=exp(µ i)
∑kexp(µ k)
i ollows ha
ln si=µ i−ln ∑
j
expµ j!
