Manea, Mihai
A icle
Bo leneck links, essen ial in e media ies, and compe ing
pa hs o di usion in ne wo ks
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Manea, Mihai (2021) : Bo leneck links, essen ial in e media ies, and compe ing
pa hs o di usion in ne wo ks, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y,
New Ha en, CT, Vol. 16, Iss. 3, pp. 1017-1053,
h ps://doi.o g/10.3982/TE4385
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/253536
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Theo e ical Economics 16 (2021), 1017–1053 1555-7561/20211017
Bo leneck links, essen ial in e media ies, and compe ing pa hs
o di usion in ne wo ks
Mihai Manea
Depa men o Economics, S ony B ook Uni e si y
We in es iga e how in o ma ion goods a e p iced and di used o e links in a ne -
wo k. A new equi alence ela ion be ween nodes cap u es he e ec s o ne wo k
a chi ec u e and loca ions o selle s on he di ision o p o i s, and cha ac e izes
he opology o compe ing (and po en ially o e lapping) di usion pa hs. Sell-
e s indi ec ly app op ia e p o i s o e in e media ion chains om buye s in hei
equi alence classes. Links wi hin he same class cons i u e bo lenecks o in o -
ma ion di usion and con e monopoly powe . Links ha b idge dis inc classes
a e edundan o di usion and gene a e compe i ion among selle s. In dense
ne wo ks, compe i ion limi s he scope o indi ec app op iabili y and in ellec ual
p ope y igh s os e inno a ion.
Keywo ds. Ne wo ks, di usion, indi ec app op iabili y, cap i e ma ke s, in e -
media ion, compe i ion, bo lenecks, edundan links, in o ma ion goods, copy-
ing, in ellec ual p ope y.
JEL classi ica ion. C78, D85, L14, O33.
1. In oduc ion
In o ma ion, knowledge and o he eplicable goods a e o en aded o e links in a ne -
wo k. Digi al goods (e.g., so wa e, music, and audiobooks) a e copied and sha ed be-
ween iends. Fa me s ep oduce high quali y plan and animal b eeds and sell hem
in local ma ke s (Bold in and Le ine 2008). App en ices a e willing o accep low wages
o e en pay ees o lea n a ade and hen se up hei own businesses in which hey,
in u n, ain app en ices in exchange o cheap labo (F aze 2006). Inside ips abou
co po a e e en s ha impac inancial ma ke s a e some imes ansmi ed o e ou o
mo e links in ne wo ks o med by amily and iends, and ips e s a e ewa ded o in-
side in o ma ion wi h gi s and jobs (Ahe n 2017). Chains o in e media ies analyze,
package, and dis ibu e inancial and indus y-speci ic in o ma ion ailo ed o business
solu ions, accoun ing pu poses, o epo ing (Sa a y 2011). This pape s udies how he
loca ions o he ini ial sou ces o a eplicable good in a ne wo k shape di usion pa hs
and de e mine he p o i s ha playe s a di e en posi ions in he ne wo k ob ain om
consuming and eselling he good.
Mihai Manea: [email p o ec ed]
I hank Dilip Ab eu, Kenne h Ahe n, Nageeb Ali, Alessand o Bona i, Aub ey Cla k, Ma c Cla e ia Mayol,
Glenn Ellison, And ea Galeo i, Ben Golub, Ma Jackson, E ik Madsen, Al Ro h, Miklos Sa a y, And ew
Sinclai , Sa o u Takahashi, Johan Walden, Andy Wei, Glen Weyl, and Alex Woli zky o commen s. This
esea ch has been unded by NSF G an 1260744.
©2021 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE4385
1018 Mihai Manea Theo e ical Economics 16 (2021)
We conside a ma ke wi h a ne wo k s uc u e in which some playe s a e endowed
wi h an iden ical in o ma ion good. In o ma ion is an indi isible consump ion good o
which playe s ha e uni demand and he e ogeneous alues. We assume ha he e a e
no consump ion ex e nali ies and ha he good does no dep ecia e. Ou model builds
on wo k by Polanski (2007), who analyzed a ma ke wi h a single selle and homoge-
neous alues. E e y playe who has he good can eplica e i a no cos and sell copies o
his neighbo s in he ne wo k. Playe s acqui e he good o he oppo uni y o consume
i and u he esell copies. We e e o he playe s who own he good a a ce ain ime
as selle s and o he o he s as buye s. A e e y da e, a buye –selle pai linked in he ne -
wo k is andomly selec ed o ba gain o e he p ice o he good. We p opose a Ma ko ian
solu ion concep unde which he e ms o ade o each ma ched buye and selle a e
de e mined by Nash ba gaining. The s a e o he ma ke a each da e, which de e mines
he disag eemen payo s in e e y ma ch, is gi en by he con igu a ion o selle s in he
ne wo k a ha da e.
While ou analysis applies b oadly o goods wi h he p ope ies ou lined abo e, i is
con enien o ame concep s and esul s in e ms o (indi isible) in o ma ion di using
h ough he ne wo k. The assump ion ha he same good is ansmi ed h ough he
ne wo k need no be aken li e ally. Some playe s may al e he good and esell cus-
omized e sions. Fo ins ance, Sa a y (2011) desc ibes he “ alue chain” in he in o -
ma ion indus y on a da a–in o ma ion–knowledge con inuum as in e media ies add
con ex , pa e ns, and causal links o aw da a, and sell di e en e sions o mul iple
ins i u ions.
Buye s se e as bo h consume s and in e media ies in he ma ke . The in e medi-
a ion ole can be bene icial o selle s when buye s p o ide access o pa s o he ne -
wo k ha selle s canno each di ec ly o when buye s enhance he good and make i
use ul o o he s. Each buye de i es a di ec u ili y om consuming he good and can
also ea n p o i s om selling copies o o he buye s. Selle s may indi ec ly ex ac p o -
i s om buye s ia in e media ion pa hs along which e e y playe demands a sha e o
he consump ion alue and he esale p o i s o he nex buye on he pa h. Liebowi z
(1985) coined he e m “indi ec app op iabili y” o he idea ha selle s can collec pa
o he p o i s gained by in e media ies who acqui e he o iginal good and esell copies.1
Ne e heless, as he s anda d a gumen o in ellec ual p ope y igh s sugges s, com-
pe i ion among selle s o he o iginal good and buye s who esell copies d i es p ices
down in seconda y ma ke s and elimina es oppo uni ies o indi ec app op ia ion.
Ou ne wo k o mula ion encapsula es he in e media ion ole o buye s who p o ide
indispensable ma ke access as well as compe i i e o ces ha es ic indi ec app o-
p iabili y.
The s a ing poin o ou analysis is he in ui ion ha a selle scan app op ia e p o -
i s di ec ly o indi ec ly om a buye bi and only i he ollowing condi ions hold:
(i) he e exis s a unique pa h be ween sand b; and (ii) any pa h om ano he selle
o bis in e media ed by s. When hese condi ions a e me , e e y playe along he pa h
1Liebowi z a gues ha indi ec app op iabili y explains why he in oduc ion o pho ocopie s in 1959
led publishe s o inc ease p ice disc imina ion o indi idual and lib a y jou nal subsc ip ions, bu has no
ha med publishe p o i s.
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1019
om s o bis he only po en ial sou ce o in o ma ion o he nex buye on he pa h. We
p o e ha uniqueness o he pa h be ween a pai o nodes de ines an equi alence ela-
ion o e nodes in any gi en ne wo k. This key equi alence ela ion acili a es a succinc
s a emen o condi ions (i) and (ii) abo e. Speci ically, selle sand buye bsa is y he wo
condi ions i and only i bbelongs o he equi alence class o sin an auxilia y ne wo k
ha mainly di e s om he o iginal one in linking all pai s o selle s.
The pa i ion o he se o nodes in o equi alence classes de i ed om he auxil-
ia y ne wo k e lec s he e ec s o compe i ion and he scope o indi ec app op iabili y
o e e y selle . This pa i ion deli e s a classi ica ion o links in e ms o compe i i e
and monopolis ic unc ions, which also de e mines he consequences o emo ing a link
o in o ma ion di usion and he dis ibu ion o p o i s in he ne wo k. Selle s ex ac
p o i s only om buye s who belong o hei blocks in he pa i ion. E e y block ha
does no include selle s con ains one buye — he deale —who in e media es all di u-
sion pa hs be ween selle s and o he nodes in he block. Links wi hin blocks cons i u e
bo lenecks o he di usion o in o ma ion. Remo ing such links disconnec s he ne -
wo k and s ops in o ma ion om eaching some buye s. Fo his eason, bo leneck
links con e monopoly powe o selle s and gene a e posi i e ex e nali ies o all play-
e s. When ade akes place ac oss a bo leneck link, he selle demands a ac ion o
he buye ’s consump ion alue and esale p o i s, and he pa i ion e ol es o e lec he
buye ’s akeo e o he subma ke o which he p o ides essen ial in e media ion. Links
be ween blocks a e edundan o di usion. Remo ing any such link does no a ec he
ul ima e sp ead o in o ma ion. Howe e , edundan links c ea e compe i ion and en-
able deale s o acqui e in o ma ion a ze o p ice. Hence, selle s ha e incen i es o se e
edundan links.
Beside i s economic ele ance, he ne wo k pa i ion we disco e p o ides g aph
heo e ic insigh s in o he s uc u e o compe ing di usion pa hs. The e is a mos one
selle in e e y block. In o ma ion in a iably en e s any block wi hou selle s h ough he
deale o he block. Deale s can ge in o ma ion om mul iple neighbo s and always e-
cei e i ia edundan links. Nodes along he unique pa h connec ing a pa icula buye
o he selle o deale in his block p o ide essen ial in e media ion o con eying in o -
ma ion o ha buye ; in o ma ion di uses wi hin blocks ia bo leneck links. In pa ic-
ula , e e y nondeale buye can only ob ain in o ma ion om a single neighbo o e a
bo leneck link. Mo eo e , all di usion pa hs ha each he same buye ia a gi en block
mus o e lap wi hin ha block.
Ou esul s indica e ha selle s’ p o i blocks a e small in ne wo ks ha a e su i-
cien ly well connec ed o clus e ed, as is he case o many la ge social and economic
ne wo ks (Jackson 2008,Easley and Kleinbe g 2010). In such ne wo ks, he possibili y o
ep oducing he good and i s compe i i e e ec s unde mine he indi ec app op iabil-
i y a gumen . I he c ea ion o he o iginal good equi es in es men s g ea e han he
low p o i s selle s can ea n in he ne wo k, hen g an ing in ellec ual p ope y igh s o
selle s may be socially op imal.2When eplica ion and esale a e no p ohibi ed, selle s
2Ne e heless, small ne wo ks in ol ing c iminal ac i i y such as he ne wo ks o inside ade s mapped
by Ahe n (2017) a e spa se, allowing selle s o indi ec ly app op ia e signi ican p o i s.
1020 Mihai Manea Theo e ical Economics 16 (2021)
can s ill a oid he ha m o compe i ion by enginee ing ce ain ea u es o he p o o ype
so as o es ic ade.3
The in e play be ween compe i ion and monopoly in his se ing is eminiscen o
he ma ke o ces eme ging in he in e media ion model o Manea (2018). In ha model,
a single non eplicable good is sequen ially esold be ween linked in e media ies in a
ne wo k un il i eaches a consume . A e e y poin in he esale p ocess, he p ice is
de e mined by compe i ion be ween buye s. Simila ly, p icing in he p esen model
is d i en by compe i ion be ween selle s, bu his analogy is supe icial, as he op ion
o selling mul iple uni s o neighbo s leads o dis inc s a egic conside a ions. E en
in ma ke s wi h no in e media ies whe e he selle is linked di ec ly o se e al buye s,
he selle may p e e o limi supply o enhance compe i ion among buye s and cha ge
highe p ices. This obse a ion o e s a ba gaining heo y pe spec i e on he p ice–
quan i y ade-o aced by a monopolis .4
In a e sion o he model wi h a single ini ial selle , Polanski (2007)p o ides ecu -
si e equa ions o he e olu ion o payo s as buye s acqui e in o ma ion. Those payo
equa ions cap u e ansi ions be ween “consecu i e” ma ke s a es and, as such, e lec
local ne wo k e ec s, bu do no elucida e how hese e ec s agg ega e o o e all p o -
i s. We ill his gap by p o iding explici payo o mulae ha e lec he global ne wo k
s uc u e. Ou ne wo k decomposi ion in o equi alence classes iden i ies he e ec i e
ma ke sha e o e e y selle . The no el g aph heo e ic concep s de eloped he e o e
a s a k delinea ion be ween de imen al compe i ion and bene icial in e media ion in
ne wo ks o in o ma ion goods, and deli e a pu ely opological cha ac e iza ion o
compe ing di usion pa hs.
In a con empo aneous wo king pape , Ali e al. (2020) s udy ma ke s o in o ma ion
goods in which e e y pai o playe s can ade. Thei se ing co esponds o a comple e
ne wo k.5Focusing on a comple e ne wo k a o ds a cha ac e iza ion o he bes and he
wo s equilib ia o he in o ma ion selle , and acili a es he design o a mechanism in
which he selle sells okens and delays he elease o in o ma ion un il all bu one buye
pu chase a oken. In he mechanism, buye s a e e ec i ely p epaying o in o ma ion,
and he selle ex ac s he p o i s a ainable when esale is p ohibi ed.
In o he ela ed wo k, No os and Waldman (1984), Besen and Ki by (1989), Bakos
e al. (1999), and Va ian (2000) in es iga e how p oduce p o i s and social wel a e a e a -
ec ed by copying and sha ing in o ma ion goods. Bold in and Le ine (2002) a gue ha
he c ea o o a good can ea n p o i s in a ma ke wi hou copy igh p o ec ion whe e
use s ep oduce he good a a cons an a e and en ou copies. Jo ano ic and Wang
3Roundup Ready seeds a e gene ically modi ied o be esis an o he he bicide Roundup bu a e a he
same ime designed o be s e ile, so ha a me s canno ep oduce and sha e hem. Digi al igh s manage-
men schemes con ol how digi al con en can be accessed and sha ed. Selle s can also elimina e edun-
dan links while au ho izing bo leneck links by es ic ing esale ma ke s ia sublicensing ag eemen s o
by selling enc yp ed e sions o he good o buye s who gene a e compe i ion on he p ima y ma ke , while
selling he p oduc ion echnology o bluep in o buye s who a e essen ial o se ing seconda y ma ke s.
4Ab eu and Manea (2021) o malize a gene al class o “exclusion commi men s” and cha ac e ize he
op imal commi men o he selle .
5Ali e al. conside ed he case o incomple e ne wo ks in an ea ly e sion o hei pape , and he exposi-
ion o some esul s he e bene i ed om a p e iew o hei i s d a .

Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1021
(2020) s udy di usion and p icing in an indus y whe e i ms can ei he inno a e o an-
domly mee and imi a e inno a o s, and he “idea” can be licensed o esold o imi a o s
and consume s. Mu o (1986)andTakeyama (1994) analyze he consequences o con-
sump ion ex e nali ies o he p icing o in o ma ion goods in a comple e ne wo k, and
Polanski (2019) p o ides a ea men o gene al ne wo ks wi h a ocus on link o ma-
ion and an applica ion o ci a ion g aphs. This pape con ibu es o he g owing li e a-
u e on in e media ion and ba gaining powe in ne wo ks (Condo elli and Galeo i 2016,
Manea 2016), which has ocused on he ade o non eplicable goods so a .
The es o his pape is o ganized as ollows. Sec ion 2de ines basic g aph he-
o y concep s needed o he analysis. Sec ion 3in oduces he in o ma ion selling
game and he solu ion concep . In Sec ion 4, we de elop he ne wo k decomposi-
ion in o equi alence classes and he cha ac e iza ion o payo s building on his de-
composi ion. In Sec ion 5, we discuss he independence o p ices om he his o y o
ades. Sec ion 6in oduces he no ions o bo leneck and edundan links, and elu-
cida es how hey shape di usion pa hs. In Sec ion 7, we de i e compa a i e s a ics
o link emo als, buye alues, and selle en y. Sec ion 8p o ides concluding e-
ma ks. The Appendix p esen s p oo s omi ed in he main ex , whe eas he Supple-
men al Ma e ial Appendix (a ailable in a supplemen a y ile on he jou nal websi e,
h p://econ heo y.o g/supp/4385/supplemen .pd ) explo es he implica ions o ou e-
sul s o la ge andom ne wo ks and con as s he p esen model wi h he in e media-
ion game o Manea (2018).
2. G aph heo y p elimina ies
This sec ion e iews s anda d g aph heo y no ions needed o he analysis: undi ec ed
ne wo ks, links, pa hs, dis ance, connec ed componen s, cycles, ees, and o es s.
Reade s amilia wi h hese concep s a e ad ised o p oceed o he nex sec ion.
Le Mbe a ini e se whose elemen s we call nodes.Ane wo k Hlinking he nodes
in Mis a subse o M×M {(i i)|i∈M}. The condi ion (i j) ∈His in e p e ed as he
exis ence o a link be ween nodes iand jin ne wo k H. Fo b e i y, we use he no a ion
ij o he link (i j).Thene wo kHis undi ec ed i ij ∈Hwhene e ji ∈H. All ne wo ks
in ou analysis a e assumed o be undi ec ed. I ij ∈H, we say ha iand ja e neighbo s
in H.Thesubne wo k Ho Hinduced by a subse o nodes M⊆Mis he ne wo k
linking he nodes in M o med by he se o links H∩(M×M).
Apa h connec ing nodes iand jin ne wo k His a sequence o dis inc nodes
(i0=i i1i¯
k=j)such ha ikik+1∈H o all k∈{01 ¯
k−1}.Thedis ance be-
ween nodes iand jin His he smalles leng h ¯
ko any pa h (i0=i i1i¯
k=j)con-
nec ing iand jin H(de ined o be in ini e i he e is no pa h be ween he wo nodes).
Aconnec ed componen o His he subne wo k o Hinduced by any maximal (wi h
espec o inclusion) se o nodes ha a e mu ually connec ed by pa hs in H.I is
known ha he se o connec ed componen s o an undi ec ed ne wo k pa i ions he
se s o nodes and links. A ne wo k is connec ed i i has a single connec ed compo-
nen . A cycle in His a sequence o nodes (i0=i i1i¯
k=i)such ha ikik+1∈H o
all k∈{01 ¯
k−1}wi h he p ope y ha he i s ¯
knodes a e dis inc . A connec ed
1022 Mihai Manea Theo e ical Economics 16 (2021)
ne wo k ha does no con ain any cycle is called a ee. A ne wo k wi hou cycles is a
o es (al e na i ely, a o es is a ne wo k whose connec ed componen s a e all ees).
3. The in o ma ion selling game
A ini e se o playe s Nis linked by an undi ec ed connec ed ne wo k G.Someo he
playe s— he ini ial selle s—a e endowed wi h an iden ical in o ma ion good. Le S⊂N
deno e he nonemp y se o ini ial selle s. We assume ha in o ma ion is a nondep eci-
a ing indi isible consump ion good o which e e y playe has uni demand. Selle s can
eplica e he good a ze o cos and sell i sequen ially o each o hei neighbo s in G.6
Upon acqui ing he good, playe i∈Nenjoys a consump ion alue i≥0and joins he
se o selle s.7The ma ke is open o an in ini e numbe o disc e e da es =01.
Playe s do no discoun u u e payo s.
The s a e o he ma ke a da e is desc ibed by he se o holde s o he in o ma ion
good S⊇Sa . Fo a gi en s a e S, we e e o he playe s in Sas selle s and o hose
in N Sas buye s. In s a e S, one andomly selec ed buye –selle pai linked in Gis
p esen ed wi h he oppo uni y o ade. Hence, he se o links ac oss which ade is
possible in s a e Sis gi en by L(S) ={bs ∈G|b∈N Ss ∈S}.Le Sdeno e he se o
selle con igu a ions ha may a ise om S ollowing a sequence o ades.8Fo e e y S∈
S {N}, a p obabili y dis ibu ion π(S) assumed o ha e ull suppo on L(S) speci ies
he p obabili y πbs(S) wi h which each link bs ∈L(S) is selec ed o ba gaining a any
da e when he selle con igu a ion is S.I band sag ee o ade in s a e Sa da e , hen
bpays he ag eed p ice o s, consumes he good, and becomes a selle in he new s a e
S∪ba +1.9The game ends when he ma ke eaches he s a e N, in which all playe s
ha e he good.
3.1 The solu ion
We p opose a coope a i e solu ion concep wi h a Ma ko s uc u e unde which he
expec ed payo o e e y playe and he p obabili y o ag eemen o he selec ed link
a each da e depend only on he se o selle s a da e .Le ui(S) deno e he expec ed
payo o playe i∈Nin s a e S∈S. When he link bs ∈L(S) is selec ed o ba gaining
a da e in s a e S, selle sand buye bnego ia e he p ice o he in o ma ion good as
ollows. In he e en o an ag eemen , he ma ke ansi ions o s a e S∪ba +1,and he
p ice in he ansac ion be ween sand bis de e mined acco ding o he Nash ba gaining
solu ion wi h weigh s (p1−p),whe ep∈(01)is an exogenous a iable common o
all selle –buye in e ac ions,10 unde he ollowing assump ions:
6The analysis ex ends o a model in which playe s ha e a common uni cos o p oducing copies o he
good.
7The e a e no consump ion ex e nali ies. Playe s iwi h i=0ac exclusi ely as in e media ies in he
ma ke . Selle s in Sa e assumed o ha e consumed he good be o e da e 0.
8Fo mally, S ep esen s he collec ion o se s S⊇Swi h he p ope y ha e e y node in Sis connec ed
o a node in Sby a pa h ha con ains only nodes in S.
9Fo no a ional con enience, we ou inely w i e X∪yand X y o he se s X∪{y}and X {y}, espec-
i ely.
10All esul s gene alize o a se ing in which o e e y s a e S∈Sand pai (sb) ∈S×(N S), selle sand
buye bdi ide he gains om ade acco ding o Nash ba gaining wi h weigh s (p(sb) 1−p(s b)).
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1023
•The o al su plus c ea ed by he ag eemen is b+ub(S ∪b)+us(S ∪b),which ep e-
sen s he sum o he consump ion alue o band he con inua ion payo s o band
sin he new s a e S∪b.
•The h ea poin s o band sa e gi en by hei co esponding disag eemen payo s,
ub(S) and us(S).
Hence, he easibili y o an ag eemen be ween band sin s a e Shinges on he gains
om ade
wbs(S) := b+ub(S ∪b) +us(S ∪b) −ub(S) −us(S) (1)
Speci ically, he p obabili y αbs(S) o an ag eemen be ween band sin s a e Smus sa -
is y he incen i e cons ain s
∀bs ∈L(S) :αbs(S)⎧
⎪
⎪
⎨
⎪
⎪
⎩
=1i wbs(S) > 0
∈[01]i wbs(S) =0
=0i wbs(S) < 0
(2)
Condi ional on sand bbeing ma ched o ba gain in s a e S, hei espec i e con in-
ua ion payo s a e gi en by us(S) +pαbs(S)wbs(S) and ub(S) +(1−p)αbs(S)wbs(S).In
he e en o an ag eemen be ween band sin s a e S, he con inua ion payo o playe
i∈N {bs}is gi en by ui(S ∪b), while in case o disag eemen , i emains ui(S).Hence,
he expec ed payo s o selle s s∈Sand buye s b∈N Sin s a e S∈S {N}sol e he
equa ions
∀s∈S:us(S) =
b:bs∈L(S)
πbs(S)us(S) +pαbs(S)wbs(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)us(S ∪b)+1−αbs(S)us(S)(3)
∀b∈N S:ub(S) =
s:bs∈L(S)
πbs(S)ub(S) +(1−p)αbs(S)wbs(S)
+
bs∈L(S):b=b
πbs(S)αbs(S)ub(S ∪b)+1−αbs(S)ub(S)(4)
I selle sand buye ba e ma ched o ba gain and each an ag eemen in s a e S, he
implici p ice bs(S) a which sand b ade sol es he equa ion us(S ∪b)+ bs(S) =us(S)+
pwbs(S).Hence, bs(S) =us(S) −us(S ∪b) +pwbs(S).
The equa ions abo e do no lead o any cons ain s on payo s o s a es in which
all ag eemen p obabili ies a e 0. To a oid his degene acy, we assume ha ade akes
place wi h posi i e p obabili y o a leas one link in e e y non- e minal s a e, i.e.,
∀S∈S {N}: ∃bs ∈L(S) s. . αbs(S) > 0(5)
1024 Mihai Manea Theo e ical Economics 16 (2021)
Na u ally, con inua ion payo s a he end o he game should be ze o:
ui(N) =0∀i∈N (6)
Fo selle con igu a ions Sin which ade akes place wi h posi i e p obabili y on a
single link bs (i.e., αbs(S) > 0and αbs(S) =0 o all bs∈L(S) {bs}), we need o im-
pose an addi ional condi ion on he ba gaining solu ion. In such si ua ions, he payo
equa ion o selle sin s a e Sboils down o
us(S) =us(S) +pπbs(S)αbs(S)wbs(S)
which is equi alen o wbs(S) = b+ub(S ∪b) +us(S ∪b) −ub(S) −us(S) =0(since
pπbs(S)αbs(S) > 0). The equa ion o ub(S) is equi alen o he same condi ion. The
payo equa ions o playe s i∈N {b s}do no p o ide any cons ain s on us(S) and
ub(S),as hey educe o
ui(S) =1−πbs(S)αbs(S)ui(S) +πbs(S)αbs(S)ui(S ∪b)
which is equi alen o ui(S) =ui(S ∪b). The inde e minacy o he ba gaining solu ion
o s a es Sin which αbs(S) > 0 o a single link bs ∈L(S) is a consequence o he as-
sump ion ha h ea poin s in he bila e al ba gaining game be ween band sa e gi en
by he solu ion i sel in s a e S.When(b s) is he only pai ha ades in con igu a ion
S, i is mo e na u al o assume ha bo h playe s’ h ea poin s a e 0since he ma ke is
pe manen ly shu down i band s ail o each an ag eemen in s a e S.Thus,we equi e
ha sand bspli he gains b+ub(S ∪b) +us(S ∪b) om a po en ial ag eemen acco d-
ing o he Nash ba gaining solu ion wi h weigh s (p 1−p) and disag eemen payo s o
0 o bo h playe s. Fo mally, we impose he condi ion
bs∈L(S)|αbs(S) > 0={bs}=⇒us(S) =p b+ub(S ∪b) +us(S ∪b)(7)
The o mula o us(S) in he condi ion abo e, along wi h he equa ion b+ub(S ∪b) +
us(S ∪b) −ub(S) −us(S) =0, implies ha ub(S) =(1−p)( b+ub(S ∪b) +us(S ∪b)).
We a e now p epa ed o de ine ou solu ion concep . The p o ile (u α) o expec ed
payo s u=(ui(S))i∈NS∈Sand ag eemen p obabili ies α=(αbs(S))bs∈L(S)S∈S {N}con-
s i u es a ba gaining solu ion i i sa is ies cons ain s (2)–(7) o e e y s a e S∈S(wi h
he a iables wbs(S) de i ed om u ia (1)). We say ha he payo s ua e consis en wi h
he ag eemen p obabili ies αi (u α) cons i u es a ba gaining solu ion.
3.2 Uniqueness o payo s o a gi en s uc u e o ag eemen s
A con ac ion a gumen shows ha he ag eemen p obabili ies αuniquely de e mine
he payo s uin e e y ba gaining solu ion.
P oposi ion 1. The e exis s a mos one p o ile o expec ed payo s ha is consis en wi h
a gi en p o ile o ag eemen p obabili ies.
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1031
in S.Fo s∈S∈S, de ine
s(S) =
i∈Cs(S) s
pδ(is) i
whe e δ(is) deno es he dis ance be ween nodes iand sin ne wo k G.15
Theo em 1. The p o ile (uα) cons i u es a ba gaining solu ion (unde he e inemen )
i and only i o e e y S∈S,
∀s∈S:us(S) = s(S)
∀b∈N S:ub(S) = b+ b(S ∪b) i b=dSCb(S)
(1−p) b+ b(S ∪b)i b= dSCb(S)
(9)
and αbs(S) ∈(01] o all bs ∈L(S).
The p oo shows ha he unique ba gaining solu ion payo s usa is y b+ub(S ∪
b) +us(S ∪b) =ub(S) +us(S) o all bs ∈L(S) and S∈S. Hence, playe s a e indi e en
be ween ading and no ading ac oss e e y link, and he payo p o ile uis consis en
wi h any p o ile o ag eemen p obabili ies α.16 The a iables s(S) and b(S ∪b) in o -
mulae (9) e lec he p o i s ha selle sand buye bindi ec ly app op ia e om hei
cap i e ma ke s Cs(S) sand Cb(S ∪b) b, espec i ely, in s a e S. Lemma 2implies he
ollowing es a emen o Theo em 1. Fo any selle con igu a ion S, selle sapp op ia es
a ac ionpδ(bs) o he consump ion alue bo each buye b o whom sis he essen ial
supplie in s a e S. Simila ly, ollowing any sequence o ades ha con eys he good
o buye b, he esale alue o buye bagg ega es a ac ion pδ(bb)o he consump ion
alue bo each buye b o whom bis an essen ial in e media y in s a e S.Thep ice
buye bpays o he good is ei he 0o a ac ion po his consump ion and esale alues
in s a e S ha co espond o whe he bis a deale o his equi alence class in G(S) o no .
P oposi ion 2and Theo em 1imply ha p ices decline along any ading pa h wi hin an
equi alence class and d op o 0when he good is sold o a new class. Mo eo e , i bis a
buye in s a e Sand (d(S Cb(S))b1bk=b) deno es he unique pa h in Gbe ween
he deale o Cb(S) and buye b, he consump ion alue o buye bis di ec ly o indi-
ec ly app op ia ed by he playe s in he in e media ion chain (d(S Cb(S))b1bk)
wi h co esponding sha es (pk(1−p)pk−1(1−p)p 1−p).
4.6 The case wi h a single selle
Polanski (2007) p o ides a ecu si e sys em o payo equa ions o a se ing simila o
he one s udied he e o ma ke s wi h a single ini ial selle . Fo he special case wi h a
15The dis ances δ(is) appea ing in he o mula o s(S) in ol e pai s (i s) wi h i∼G(S) sand, hence,
i∼Gs.In hiscase,δ(is) is simply he leng h o he unique pa h be ween iand sin G.
16This indi e ence in he ic ionless model is he main obs acle in ex ending he analysis o he case o
discoun ing. In a e sion o he model wi h discoun ing, i is di icul o p o e ha he indi e ence is always
b oken in a o o ade as pos ula ed by ou e inemen .

1032 Mihai Manea Theo e ical Economics 16 (2021)
single selle , S={s}, he equi alence ela ions ∼G({s})and ∼Gde eloped in ou ame-
wo k coincide (modulo he dummy playe ), so he o mula o selle p o i s om Theo-
em 1boils down o
us{s}=
i∼Gsi=s
pδ(is) i
The e o e, o de e mine he p o i o selle s, i is su icien o conside equi alence
classes unde ∼G. Examining equi alence classes in he auxilia y ne wo ks G(S) is nec-
essa y only o compu ing buye s’ payo s and acking he e olu ion o p o i s as o he
playe s acqui e he good. The cons uc ion o he auxilia y ne wo k G(S) has indeed
been mo i a ed by he in ui ion ha he co esponding equi alence ela ion ∼G(S) suc-
cinc ly desc ibes compe i ion among selle s in S.
Polanski’s ecu si e equa ions cap u e ansi ions be ween “consecu i e” ma ke
s a es by ela ing he payo ui(S) o payo s o he ype ui(S ∪b). He inds ha he e ms
o ade be ween a selle sand a buye bdepend on whe he bbelongs o a cycle ha in-
cludes a leas one selle . To ex end his esul o ou se ing wi h mul iple ini ial selle s,
we need o conside cycles in he ne wo k G(S) a he han G o he payo equa ions
ha co espond o s a e S.Fo S∈S,b∈N S, de ine
γb(S) =0i he e exis s a cycle in G(S) ha con ains band an elemen o S
1o he wise.
One can check ha o bs ∈L(S),weha eγb(S) =0i bis he deale o Cb(S) in s a e S
and γb(S) =1o he wise. This obse a ion, along wi h P oposi ion 2, leads o he ollow-
ing co olla y o Theo em 1, which gene alizes Polanski’s esul .
Co olla y 1. Fo any s∈S∈Sand b∈N Ssuch ha bs ∈G, he ba gaining solu ion
payo s sa is y
us(S) =us(S ∪b) +pγb(S) b+ub(S ∪b)
ub(S) =1−pγb(S) b+ub(S ∪b)
Fo s∈S∈Sand bb∈N Ssuch ha L(S) does no con ain any links o bo s,bu
con ains a link o b, we ha e us(S) =0and ub(S) =ub(S ∪b).
As Polanski poin s ou , he iden i ies om he co olla y p o ide a compu a ional
p ocedu e o e alua ing he ba gaining solu ion payo s based on ansi ions be ween
ma ke s a es. These ecu si e payo equa ions e lec local ne wo k e ec s. Ou closed-
o m payo o mulae elucida e how he global ne wo k s uc u e a ec s he di ision o
gains om ade, and he decomposi ion o he ne wo k in o equi alence classes de-
linea es oppo uni ies o indi ec app op iabili y and p o ides a classi ica ion o links
acco ding o hei monopolis ic o compe i i e oles. In Sec ion 6, we show ha his
classi ica ion ansla es in o a axonomy o links as ei he bo lenecks o edundan o
in o ma ion di usion and ne wo k connec i i y. We also demons a e ha he g aph
heo e ic concep s in oduced he e— he equi alence ela ion, he auxilia y ne wo k,
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1033
deale s, essen ial in e media ies, and bo leneck and edundan links—cha ac e ize he
o e lapping s uc u e o compe ing pa hs o di usion (Theo em 2) and p o ide ac able
ools o he de i a ion o compa a i e s a ics o buye alues (Co olla y 2), selle en y
(P oposi ion 4), and, mos ele an o applica ions, link emo als (Theo em 3). Ano he
esul ha elies on his oolki is ha he e inemen selec s he unique payo p o ile
unde which p ices a e his o y-independen (P oposi ion 3).17
5. Founda ion o he e inemen
Wi h he aim o p o iding a ounda ion o ou e inemen , we e e o using he e ms
“ba gaining solu ion” o any p o ile (u α) sa is ying condi ions (2)–(7) and “ e inemen
o he ba gaining solu ion” o p o iles ha addi ionally sa is y cons ain (8). Fix a ba -
gaining solu ion wi h payo p o ile u. Recall ha an ag eemen in s a e Sbe ween selle
sand buye ben ails he p ice bs(S) =us(S) −us(S ∪b) +pwbs(S). We say ha he p ices
gene a ed by ua e his o y-independen i o e e y bs ∈G,weha e bs(S) = bs(S) o any
pai o s a es SS∈Ssuch ha sis a selle and bis a buye in bo h con igu a ions Sand
S. The in e p e a ion o his o y independence o p ices is ha he ba gaining p ocess
o any buye –selle link does no equi e in o ma ion abou p io ades.
In Sec ion 3, we a gued ha p ices unde he ba gaining solu ion uled ou by he
e inemen in he ne wo k om Figu e 1a e no his o y-independen . The nex esul
gene alizes ha conclusion: in e e y ne wo k, p ices a e his o y-independen only o
he ba gaining solu ion payo s ha su i e he e inemen .18 Hence, ou e inemen
selec s he solu ions ha do no ely on he assump ion ha ma ched playe s obse e
he s a e o he ma ke (bu know he ini ial s a e S).
P oposi ion 3. The e inemen o heba gainingsolu iongene a es his o y-independen
p ices and selec s he unique payo p o ile o which p ices a e his o y-independen .
6. The ana omy o di usion pa hs
Conside a link ij ∈Gsuch ha no bo h iand ja e selle s in he ini ial s a e S.We
say ha ij is a edundan link i iand jbelong o dis inc equi alence classes in he ini-
ial ma ke , i.e., iG(S)j.Fo allS∈S,weha e ha G(S)⊆G(S),soiG(S)jimplies
iG(S) j.Thus,i ij is edundan , hen iand j emain in dis inc equi alence classes
as he ma ke e ol es. We say ha ij is a bo leneck link i i is no edundan , i.e.,
i∼G(S)j. Since equi alence classes induce ees in he ne wo k Gand each ade b eaks
up a mos one equi alence class in o wo dis inc classes, P oposi ion 2implies ha he
only pai o playe s linked in G ha can be sepa a ed in o di e en equi alence classes
ollowing a ade is he buye –selle pai ha is conduc ing he ade. Hence, i ij is a
bo leneck link, hen iand ja e membe s o he same equi alence class in s a e Sand
17Polanski also no es his o y independence o p ices in his model, bu does no a gue ha his e inemen
selec s he unique payo s wi h his p ope y.
18The e inemen has he addi ional p ope y o inducing selle -independen p ices, i.e., bs(S) = bs(S)
o any pai o s a es SS∈Ssuch ha s∈S,s∈S, and b/∈S∪S.
1034 Mihai Manea Theo e ical Economics 16 (2021)
con inue o sha e an equi alence class un il hey ade wi h each o he ; ha is, i∼G(S) j
o all S∈Ssuch ha {i j}S.
The e olu ion o equi alence classes as in o ma ion di uses (P oposi ion 2)canbe
es a ed in he language o edundan and bo leneck links as ollows. T ading o e a
edundan link does no change he s uc u e o equi alence classes, while ading o e
a bo leneck link b eaks up he equi alence class ha con ains he link in o wo classes
ha sepa a e he buye om he selle .
The pa i ion o he ne wo k in o equi alence classes and he ensuing concep s o
edundan and bo leneck links lead o a sys ema ic cha ac e iza ion o compe ing pa hs
o di usion in he ne wo k. By de ini ion, each deale buye can ecei e he good only
om neighbo s ou side his class. Since links ha span dis inc equi alence classes a e
edundan , deale buye s mus acqui e he good by means o edundan links. Mo eo e ,
Lemma 3implies ha deale buye s ha e a leas wo po en ial supplie s.
By con as , Lemma 3shows ha each nondeale buye has a single po en ial sup-
plie , which is he neighbo o he buye on he unique pa h connec ing he buye o he
deale o his equi alence class. This pa h is con ained wi hin he buye ’s equi alence
class and, hus, consis s o bo leneck links. In pa icula , he nondeale buye acqui es
he good om his only po en ial supplie o e a bo leneck link. The e o e, he e is an
implici low o ade o e bo leneck links: di usion wi hin each equi alence class is
desc ibed by a di ec ed ee oo ed a i s deale .
Conside now he collec ion o compe ing pa hs ha deli e he good o a gi en
buye . E e y pa h in his collec ion ha “c osses” a ce ain equi alence class has o en-
e he class ia i s deale . Logic simila o Lemma 3shows ha each such pa h mus
also exi he equi alence class h ough he same node. This implies ha all pa hs con-
eying he good o he chosen buye and in e sec ing a gi en equi alence class mus
c oss he class only once and o e lap wi hin he class. The nex esul summa izes hese
obse a ions.
Theo em 2. A buye is a deale in s a e Si and only i he has wo o mo e po en ial
supplie s in s a e S. The good always eaches deale buye s ia edundan links and non-
deale buye s ia bo leneck links. Fo any ma ke s a e S∈Sand buye b∈N S,all
pa hs in G ha connec any selle in S o buye band in e sec a gi en equi alence class
Ci(S) o ∼G(S) mus en e Ci(S) exac ly once and o e lap pe ec ly wi hin Ci(S).
7. Compa a i e s a ics
In his sec ion, we p esen compa a i e s a ics esul s o buye alues, selle en y and
link emo als, and discuss he op imal emo al o links o selle s.
7.1 Compa a i e s a ics o buye alues and selle en y
Since s(S) is inc easing in b o all s∈Sand b∈N S, and equi alence classes a e
de e mined en i ely by ne wo k opology, Theo em 1has he ollowing co olla y.
Co olla y 2. Fo any S∈Sand b∈N S, he payo s o all playe s in s a e Sa e (weakly)
inc easing in b.
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1035
Theo em 1also deli e s compa a i e s a ics wi h espec o he se o selle s. Sup-
pose ha new selle s en e he ma ke and he ini ial s a e expands om S o S. Since
G(S)is a subne wo k o G(S), e e y pai o nodes ela ed unde ∼G(S)is also ela ed
unde ∼G(S). I ollows ha Ci(S)⊆Ci(S) o all i∈N. In pa icula , he cap i e ma -
ke o e e y incumben selle in Ssh inks, and Theo em 1implies ha he p o i s o all
hese selle s dec ease ollowing he en y o he new selle s om S S. By Lemma 3,
deale buye s (ou side S) ha e a leas wo po en ial supplie s in s a e S,and hus
main ain deale s a us in s a e S. Howe e , he cap i e ma ke s o deale buye s may
sh ink, esul ing in lowe p o i s. Finally, he se o buye s o whom nondeale buy-
e s (ou side S) a e essen ial in e media ies sh inks as well— o e e y buye b,weha e
Cb(S∪b) ⊆Cb(S ∪b)—bu such buye s may become deale s in he new selle con igu-
a ion Sdue o compe i ion c ea ed by he addi ional selle s. Such buye s acqui e he
good a ze o p ice ollowing he en y o new selle s, which may ansla e in o highe
payo s. Whe he he en y o he new selle s bene i s nondeale buye s depends on he
ade-o be ween he lowe acquisi ion p ice and he smalle cap i e ma ke . The nex
esul summa izes hese indings.
P oposi ion 4. Conside wo ini ial ma ke s a es S⊂S. The payo s o e e y selle in
Sand e e y buye (ou side S) who is a deale in s a e Sa e weakly lowe in s a e S han
in S. The e ec o he expansion o he se o selle s om S o S o buye s who a e no
deale s in s a e Sis ambiguous.
7.2 Compa a i e s a ics o link emo als
We now in es iga e he e ec s o emo ing links om he ne wo k on in o ma ion di -
usion and in e media ion p o i s. Fix a connec ed ne wo k G, a selle con igu a ion
S∈S, and a link ij ∈G o which no bo h iand jbelong o S(links be ween selle s a e
i ele an in he game). Le Gdeno e he ne wo k ob ained by emo ing link ij om
G.19
Suppose i s ha ij is a bo leneck link. As a gued in Sec ion 6, he assump ion ha
{i j}Simplies ha i∼G(S) j. In pa icula , we ha e i∼Gjand, hence, dele ing he link
om Gdisconnec s he ne wo k in o wo connec ed componen s. We p o e ha he
selle s in Sbelong o he same connec ed componen o he esul ing ne wo k Gas he
deale d(SCi(S)) o he common equi alence class o iand jin G(S).Hence,playe s
in he o he connec ed componen o Gdo no ha e access o any selle and ob ain
no p o i s. The emo al o bo leneck link ij b eaks up he equi alence class o iand j
om G(S) in o wo subclasses and does no a ec he composi ion o o he equi alence
classes. Playe d(SCi(S)) emains he deale o his smalle equi alence class in G,bu
su e s a d op in p o i s. The loss o he link hu s bo h iand j: one o hem becomes
disconnec ed om selle s and ge s ze o payo , while he o he collec s in e media ion
p o i s om a smalle equi alence class. Since he o he equi alence classes o ∼G(S)
con ained in he connec ed componen o node d(SCi(S)) in Gand hei deale s a e
19While Gmay be disconnec ed, he esul s o p e ious sec ions apply o e e y connec ed componen
o G ha con ains selle s, and we use his s aigh o wa d ex ension he e.
1036 Mihai Manea Theo e ical Economics 16 (2021)
Figu e 4. Nondeale buye bis be e o i he se e s his link wi h b.Remo ing he edundan
link bb also bene i s deale buye bi b<(1−p) b .
una ec ed by he emo al o link ij , playe s in hose classes ob ain he same payo s
in Gand G. Fo an illus a ion, conside he pai o nodes b∼G({ss})slinked in he
ne wo k G om Figu e 3. The emo al o he link bs om Gdoes no a ec he payo s
o playe s in he equi alence classes o band sin G({ss}), bu disconnec s he buye s
in o he equi alence classes om he wo selle s.
I , ins ead, ij is a edundan link, hen we show ha i s emo al om Gdoes no
p e en any playe om acqui ing he good.20 The emo al o he edundan link ij leads
o a weak expansion in each playe ’s equi alence class in s a e S.Theo em1implies
ha e e y selle ’s p o i is weakly highe in G han in G. The e o e, edundan links
impose nega i e ex e nali ies on selle s. As he example om Sec ion 3demons a es,
a selle may bene i om se e ing one o his links. The se o playe s o whom each
buye se es as an essen ial in e media y also weakly expands. Lemma 2and Theo em 1
imply ha he payo s o buye s who a e no deale s in G(S) weakly inc ease a e link ij
is dele ed om G. The ne wo k om Figu e 4p o ides an example in which he p o i o
a nondeale buye s ic ly inc eases a e dele ing one o his edundan links. Indeed, i
bdele es his edundan link wi h b in ha ne wo k, hen his equi alence class expands
om {bs} o {b bbs}. Since bis no a deale ei he be o e o a e dele ing he link
bb, he link dele ion inc eases his payo om (1−p) b o (1−p)( b+p b+p2 b ).
Howe e , deale buye s may lose deale s a us when a edundan link is dele ed om
he ne wo k. Such buye s exploi compe i ion be ween selle s o ob ain he good o
ee in he o iginal ne wo k, bu ha e o pay a ac ion 1−po hei consump ion and
esale alues ollowing he dele ion o he edundan link, which may cause a decline
in hei o e all p o i s. Fo example, conside he link be ween nodes band s o which
bG({ss})sin he ne wo k G om Figu e 3. Remo ing he link bs om Gleads o he
me ge o he equi alence classes o nodes band s om G({ss}). A e he link emo al,
buye bis no longe a deale , and selle sis able o ge a sha e 1−po his consump ion
and esale alues. Hence, he emo al o link bs is bene icial o sand de imen al o b.
Remo ing edundan links can also ha e he opposi e e ec on deale buye payo s. Fo
ins ance, in he ne wo k om Figu e 4, bo h buye s band b a e deale s o single on
20Howe e , he ensuing ne wo k Gmay be disconnec ed. Fo ins ance, in he ne wo k Gwi h wo selle s
(sand s) linked o a single buye (playe b), we ha e bG({ss})s.Remo ing helinkbs om Gdisconnec s
he ne wo k, bu does no p e en b om acqui ing he good om s.

Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1037
equi alence classes. Remo ing he edundan link bb leads o a ne wo k wi h a single
equi alence class whe e buye bob ains payo (1−p)( b+p b ), which is g ea e han
his payo bin he o iginal ne wo k i (1−p) b >
b.
In he example om Figu e 4, emo ing he link bb unde mines he deale s a us o
buye b, bu expands his cap i e esale ma ke a he same ime. Howe e , we show ha
emo ing one o a deale ’s own links canno simul aneously p oduce bo h o hese ou -
comes o he deale . Tha is, i he emo al o one o a deale buye ’s own links dep i es
him o deale s a us, hen ha buye ’s cap i e ma ke canno expand; in his case, he
buye ’s payo unambiguously dec eases. I , ins ead, he link emo al does no a ec he
buye ’s deale ole, hen he buye con inues o secu e he good a ze o p ice a e los-
ing he link; as he buye ’s cap i e ma ke canno sh ink ollowing he link emo al, he
buye ’s payo weakly inc eases. Thus, o p edic he payo consequences o losing a link
o a deale buye , i is su icien o de e mine whe he he link loss changes ha buye ’s
deale s a us. A special ca ego y o edundan links is ele an o his phenomenon: a
link ij is pi o al o deale buye iin s a e Si ihas exac ly wo po en ial supplie s in s a e
S, one o which is j(by Theo em 2, links be ween deale s and hei po en ial supplie s
a e always edundan ). We p o e ha emo ing he link ij om he ne wo k esul s in
buye i’s loss o deale s a us i and only i ij is a pi o al link o deale iin s a e S.The e-
o e, i iis a deale buye in s a e S, heniis hu by he loss o he link ij i and only i ij
is a pi o al link o iin s a e S.
The ollowing esul , whose de ailed p oo is elega ed o he Supplemen al Ma e ial
gi en he ex ended ske ch abo e, summa izes he compa a i e s a ics.
Theo em 3. Conside a selle con igu a ion S∈Sin he connec ed ne wo k Gand a link
ij ∈Gwi h {i j}S.Le Gbe he ne wo k ob ained by dele ing he link ij om G.
(i) I ij is a bo leneck link, hen Gis a disconnec ed ne wo k o med by wo connec ed
componen s. In o ma ion does no each he playe s in he connec ed componen o
G ha does no con ain d(SCi(S)); hus, hese playe s’ payo s d op o 0when link
ij is emo ed. The payo s o playe s in Ci(S) om he same connec ed componen
as d(SCi(S)) in Gweakly dec ease a e emo ing link ij . The payo s o playe s i,j
and d(SCi(S)) s ic ly dec ease ollowing he link emo al i b>0 o all b∈N S.
The payo s o all o he playe s a e iden ical in Gand G.
(ii) I ij is a edundan link, hen in o ma ion di uses o all playe s in G. All selle s
and he buye s who a e no deale s in s a e S o ne wo k Gweakly bene i om he
emo al o link ij . In gene al, he e ec o emo ing he link on he payo s o buye s
who a e deale s o hei equi alence class in G(S) is ambiguous. Ne e heless, i
iis a deale in s a e Sand b>0 o all b∈N S, hen he loss o link ij s ic ly
educes i’s payo i and only i ij is a pi o al link o iin s a e S.
The esul abo e conside s he e ec s o emo ing a single edundan link om he
ne wo k. I , ins ead, we emo e all edundan links om Ga he same ime, which leads
o he o es F(G(S)), hen he p o i s o selle s do no change. Howe e , he simul ane-
ous emo al o edundan links blocks he sp ead o in o ma ion o buye s whose equi -
alence class unde ∼G(S)does no con ain selle s and educes hese buye s’ payo s o 0.
1038 Mihai Manea Theo e ical Economics 16 (2021)
The classi ica ion o links eme ging om Theo ems 2and 3leads o he ollowing
conclusions ega ding selle p o i s and in o ma ion ansmission. Bo leneck links con-
e monopoly powe o selle s. The dele ion o a bo leneck link disconnec s he ne -
wo k, blocks he sp ead o in o ma ion, and hu s selle s. Redundan links c ea e com-
pe i ion among selle s. The dele ion o a edundan link does no p e en he di usion
o in o ma ion and bene i s selle s.
7.3 Op imal link emo als
Conside now a si ua ion wi h a single selle s, who can p ohibi ade on a subse o links
by enginee ing ea u es o he good as explained in oo no e 3.Theo em1implies ha
selle swould op imally allow ade only o e he links o a ee T, which is a subne wo k
o G ha maximizes he exp ession

i∈N s
pδT(is) i
whe e δT(i s) ep esen s he dis ance be ween nodes iand sin ee T.No e ha any
es uc u ing o a ee whe eby a gi en buye bwho o iginally ecei es he good om a
node bse e s his link wi h band c ea es a new link wi h a node close o sis bene icial
o he selle . In pa icula , he s a ne wo k, in which he selle is linked o all buye s
and he e a e no links be ween buye s, maximizes selle p o i among all ne wo ks.
We can simila ly cha ac e ize he subne wo k o G ha maximizes he join p o i s
o a g oup o compe ing selle s S. In his case, he op imal subse o ading links is
desc ibed by a pa i ion (Ns)s∈So he se o nodes Nand a collec ion o associa ed ees
(Ts)s∈Ssuch ha s∈Nsand Tsconsis s o a subse o he links in Gbe ween pai s o
nodes in Ns o all s∈S. The pa i ion should maximize he exp ession

s∈S
i∈Ns s
pδTs(is) i
Selle s p e e o emo e all links om Gno belonging o he o es s∈STs. This means
ha selle s di ide he ma ke in o a se o non-o e lapping ees om which hey indi-
ec ly app op ia e p o i s and commi o no compe ing wi h one ano he o any buye .
8. Conclusion
We s udied a model in which playe s consume, eplica e, and esell copies o a good in
a ne wo k. In he model, buye s may in e media e ade and indi ec ly ans e p o i s
om a -away buye s o selle s as he good is sequen ially esold o e he links o he
ne wo k. Howe e , buye s who acqui e copies o he good may also c ea e compe i-
ion o selle s o he o iginal good, and his limi s oppo uni ies o indi ec p o i ap-
p op ia ion. Ou ne wo k o mula ion hus cap u es he an i hesis be ween wo cen al
concep s in he esea ch on copying and in ellec ual p ope y: indi ec app op iabili y
e sus compe i ion. We disco e ed a key equi alence ela ion ha desc ibes he oles o
essen ial supplie s and in e media ies o he di usion o he good. Selle s collec p o i s
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1039
om buye s o whom hey a e essen ial supplie s, while buye s make p o i s by con ey-
ing he good o o he buye s o whom hey p o ide essen ial in e media ion. Equi a-
lence classes o he ela ion delinea e he cap i e ma ke s o e e y selle and buye in
he ne wo k.
The p ice a buye pays o he good is ei he ze o o a ixed ac ion o his consump-
ion and esale alues co esponding o whe he he buye is able o exploi compe i ion
among mul iple neighbo s supplying he good o is subjec o a monopoly in which a
single neighbo p o ides access o he good. Links ha induce compe i ion among sell-
e s a e edundan o he di usion o he good h ough he ne wo k and gene a e neg-
a i e ex e nali ies o selle s, while links ha enable monopolies cons i u e bo lenecks
o di usion and p oduce posi i e ex e nali ies o all playe s. Redundan links b idge
dis inc equi alence classes, while bo leneck links a e enclosed in he same equi alence
class. The ne wo k pa i ion in o equi alence classes e eals ich s uc u al p ope ies
o compe ing pa hs o di usion. Ou analysis shows ha in ne wo ks ha a e well con-
nec ed o clus e ed, compe i ion obs uc s indi ec app op iabili y. In such si ua ions,
g an ing in ellec ual p ope y igh s os e s he c ea ion o in o ma ion goods.
To ob ain heo e ical esul s o gene al ne wo ks, we ha e made a numbe o simpli-
ying assump ions, among which we enume a e he ne wo k s uc u e and buye alues
a e exogenous and commonly known; playe s do no discoun payo s; he o iginal good
and i s copies a e pe ec subs i u es; he solu ion concep is coope a i e and a o s
ade; sales con ac s a e bila e al and canno speci y es ic ions on eplica ion and e-
sale. In u u e wo k, i would be use ul o ex end he analysis o ma ke s in which some
o hese modeling assump ions a e un ealis ic. Ne e heless, he g aph heo e ic by-
p oduc s o his esea ch—including he concep s o equi alence classes, essen ial sup-
plie s, and in e media ies, deale s, and bo leneck and edundan links—do no hinge
on he pa icula model speci ica ion and a e likely o play a ole in o he models o
di usion in ne wo ks.
Appendix:P oo s
P oo o P oposi ion 1. We p oceed by con adic ion. Suppose ha (u α) and
(uα) cons i u e wo ba gaining solu ions wi h dis inc payo s uand u,bu iden i-
cal ag eemen p obabili ies α.Le S∈Sbe a se o maximal ca dinali y o which he e
exis s i∈Nsuch ha ui(S) = u
i(S). By de ini ion, S= N,L(S) = ∅,and
ui(S ∪b) =u
i(S ∪b) ∀i∈Nb∈N S(s. . bs ∈L(S) o some s∈S) (10)
Then he payo equa ions o he solu ions (u α) and (uα)lead o
us(S) =
b:bs∈L(S)
πbs(S)1−pαbs(S)+
bs∈L(S):s=s
πbs(S)1−αbs(S)us(S)
+p
b:bs∈L(S)
πbs(S)αbs(S) b+ub(S ∪b)+us(S ∪b)−ub(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)us(S ∪b)(11)
1040 Mihai Manea Theo e ical Economics 16 (2021)
u
s(S) =
b:bs∈L(S)
πbs(S)1−pαbs(S)+
bs∈L(S):s=s
πbs(S)1−αbs(S)u
s(S)
+p
b:bs∈L(S)
πbs(S)αbs(S) b+ub(S ∪b)+us(S ∪b)−u
b(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)us(S ∪b) (12)
Le 1=maxs∈S|us(S)−u
s(S)|and 2=maxb∈N S|ub(S)−u
b(S)|.Wep o e ha 1=
2=0, which con adic s he assump ion ha ui(S) = u
i(S) o some i∈N.
Fix s∈Ssuch ha |us(S)−u
s(S)|=1.Le Xdeno e he p obabili y ha he ma ched
pai does no each ag eemen unde αin a pe iod wi h selle con igu a ion S,le Ys
deno e he p obabili y ha selle s eaches an ag eemen in such a pe iod, and le Zs
deno e he sum o e ms ha do no in ol e he a iables (ui(S))i∈Nin (11). Ma hema -
ically,
X=
bs∈L(S)
πbs(S)1−αbs(S)
Ys=
b:bs∈L(S)
πbs(S)αbs(S)
Zs=p
b:bs∈L(S)
πbs(S)αbs(S) b+ub(S ∪b)+us(S ∪b)
+
bs∈L(S):s=s
πbs(S)αbs(S)us(S ∪b)
We ha e
1−X−(1−p)Ys=
bs∈L(S):s=s
πbs(S)αbs(S) +p
b:bs∈L(S)
πbs(S)αbs(S) > 0
because p>0,π(S) places posi i e p obabili y on e e y link in L(S) = ∅, and condi ion
(5) equi es ha he p obabili y o ag eemen unde αis posi i e o a leas one link in
s a e S. Collec ing he a iables us(S) in (11)andu
s(S) in (12), we ob ain
us(S)1−X−(1−p)Ys=Zs−p
b:bs∈L(S)
πbs(S)αbs(S)ub(S)
u
s(S)1−X−(1−p)Ys=Zs−p
b:bs∈L(S)
πbs(S)αbs(S)u
b(S)
o , equi alen ly,
us(S) =Zs
1−X−(1−p)Ys
−p
b:bs∈L(S)
πbs(S)αbs(S)
1−X−(1−p)Ys
ub(S)
u
s(S) =Zs
1−X−(1−p)Ys
−p
b:bs∈L(S)
πbs(S)αbs(S)
1−X−(1−p)Ys
u
b(S)
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1047
Cb(S ∪b). Since band sa e selle s in he con igu a ion S∪b,weha e ha
us(S) = s(S) =
i∈Cs(S) s
pδ(is) i
us(S ∪b) = s(S ∪b) =
i∈Cs(S∪b) s
pδ(is) i
ub(S ∪b) = b(S ∪b) =
i∈Cb(S∪b) b
pδ(ib) i
By Lemma 2,bis an essen ial in e media y and sis he essen ial supplie in s a e S
o he buye s in Cb(S ∪b) b. Hence, o all i∈Cb(S ∪b) b, he link bs belongs o he
unique pa h connec ing s o iand δ(is) =δ(ib) +1. Since Cs(S) s=(Cs(S ∪b) s) ∪
b∪(Cb(S ∪b) b),δ(bs) =1,andδ(is) =δ(ib) +1 o i∈Cb(S ∪b) b, he o mula o
us(S) can be ew i en as
us(S) =
i∈Cs(S∪b) s
pδ(is) i+p b+
i∈Cb(S∪b) b
pδ(is) i
= s(S ∪b) +p b+
i∈Cb(S∪b) b
pδ(ib)+1 i
= s(S ∪b) +p b+ b(S ∪b)
As s∈Cb(S),weha ed(SCb(S)) =s, and hence
ub(S) =(1−p) b+ b(S ∪b)
The equali ies abo e imply ha
b+ub(S ∪b) +us(S ∪b) −ub(S) −us(S)
= b+ b(S ∪b) + s(S ∪b) −(1−p) b+ b(S ∪b)
− s(S ∪b) +p b+ b(S ∪b)=0
Fo a p oo o claim (d), suppose ha L(S) ={bs}. Then selle shas no neighbo le
o sell o when all playe s in S∪bha e he good. Hence, Cs(S ∪b) ={s}and us(S ∪b) =
s(S ∪b) =0. Since L(S) ={bs},weha eb∼G(S) s, which ia P oposi ion 2implies ha
Cb(S ∪b) =Cs(S) Cs(S ∪b) =Cs(S) s.Asδ(is) =1+δ(ib) o all i∈N S, i ollows
ha
us(S) = s(S) =
i∈Cs(S) s
pδ(is) i=
i∈Cb(S∪b)
pδ(is) i
=p b+
i∈Cb(S∪b) b
p1+δ(ib) i=p b+ub(S ∪b)
Then us(S ∪b) =0leads o us(S) =p( b+ub(S ∪b) +us(S ∪b)), as asse ed.

1048 Mihai Manea Theo e ical Economics 16 (2021)
Conside now a p o ile (u α) ha sa is ies he hypo heses o he heo em. To p o e
ha (uα) is a ba gaining solu ion, ix a s a e S∈S. Claim (c) implies ha wbs(S) =0
o all bs ∈L(S).Hence,(uα) sa is ies he incen i e cons ain s (2). Claims (a), (b), and
(c) imply ha he p o ile (uα) sol es he payo equa ions (3)and(4). I S= N, hen
he se L(S) is nonemp y because he ne wo k Gis assumed o be connec ed. Thus,
he ag eemen p o ile αmee s he equi emen (5) since i assigns posi i e p obabili y
o ag eemen o e e y link in L(S). By cons uc ion, he payo s usa is y condi ion
(6). Finally, o e i y ha (uα) has p ope y (7), suppose ha αbs(S) > 0 o a single
link bs ∈L(S).Asαspeci ies a posi i e p obabili y o ag eemen o any ading link in
e e y s a e, i mus be ha L(S) ={bs}. Claim (d) hen implies (7). We ha e shown ha
(uα) sa is ies condi ions (2)–(7) o e e y s a e S∈Sand, hus, cons i u es a ba gaining
solu ion. The p oo is comple ed as ou lined in he p eamble.
P oo o P oposi ion 3. We i s show ha he e inemen o he ba gaining solu ion
gene a es his o y-independen p ices. Le u∗be he payo s unde he e inemen wi h
associa ed gains om ade and p ices deno ed by w∗and ∗, espec i ely. S ep (c) in
he p oo o Theo em 1shows ha o all S∈Sand bs ∈L(S),weha ew∗
bs(S) =0,which
implies ha ∗
bs(S) =u∗
s(S)−u∗
s(S ∪b). To es ablish his o y independence o p ices unde
u∗, i is su icien o a gue ha ∗
bs(S) = ∗
bs(S ∪b) o any b∈N (S ∪b) such ha S∪b∈
S.Fixb,b,s,Swi h he p ope ies lis ed abo e. We ha e o check ha he payo s
selec ed by he e inemen sol e he equa ion u∗
s(S) −u∗
s(S ∪b) =u∗
s(S ∪b)−u∗
s(S ∪
{bb})o , equi alen ly, ha s(S)− s(S ∪b) = s(S∪b)− s(S∪{b b}). Gi en he o mula
o , he la e equa ion is equi alen o

i∈Cs(S) Cs(S∪b)
pδ(is) i=
i∈Cs(S∪b) Cs(S∪{bb})
pδ(is) i
The e o e, i is su icien o p o e ha
Cs(S) Cs(S ∪b) =CsS∪b CsS∪b b(16)
I bG(S) s, hen P oposi ion 2implies ha Cs(S) =Cs(S ∪b).Mo eo e ,bG(S∪b) s
and P oposi ion 2also leads o he conclusion ha Cs(S ∪b) =Cs(S ∪{bb}).Hence,(16)
holds in his case.
I bG(S) s, hen P oposi ion 2implies ha Cs(S) =Cs(S∪b),soCs(S) Cs(S∪b) =∅.
Mo eo e , bG(S∪b)sand P oposi ion 2also leads o Cs(S ∪b)=Cs(S ∪{b b}),which
means ha Cs(S ∪b) Cs(S ∪{b b})=∅.Hence,(16) holds in his case as well.
We a e le wi h he case b∼G(S) s∼G(S) b. Since S∪b∈S,i mus be ha bis linked
o a node in S. By Lemma 2, he ela ionship b∼G(S) simplies ha sis he essen ial
supplie o bin s a e Sand, hus, belongs o any pa h om a node in S o b, including
any link connec ing b o S. I ollows ha bs∈G. Since b∼G(S) s, P oposi ion 2implies
ha Cs(S) Cs(S ∪b) =Cb(S ∪b).No e ha bG(S∪b)bbecause band ba e connec ed
by he pa hs (bsb)and (b s 0b)in G(S ∪b). Applying P oposi ion 2again, we ha e
Cs(S) =Cs(S ∪b)∪Cb(S ∪b).Asb∈Cs(S) bu b/∈Cb(S ∪b), we in e ha b∈Cs(S ∪b)
and, hus, b∼G(S∪b)s. P oposi ion 2leads o Cs(S ∪b) Cs(S ∪{b b})=Cb(S ∪{b b}).
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1049
Then (16) ollows om he ac ha Cb(S ∪b) =Cb(S ∪{bb}), which is a consequence
o s ep (b) in he p oo o Theo em 1.
We nex p o e ha e e y ba gaining solu ion wi h his o y-independen p ices mus
gene a e he payo s selec ed by he e inemen . Fix a ba gaining solu ion (uα) un-
de which p ices a e his o y-independen . We need o show ha u(S) =u∗(S) o e e y
S∈S. The p oo o his claim p oceeds by induc ion on |N S|. Fo hebasecase
|N S|=0,weha e ha S=N, and he claim ollows i ially om assump ion (6).
Fo he induc i e s ep, ix S⊂Nand assume ha he induc ion hypo hesis holds o
e e y se in So g ea e ca dinali y han S.Inpa icula ,u(S ∪b) =u∗(S ∪b) o e e y
b∈N S ha is linked o a node in S. Since Gis connec ed and S⊂N, he e exis s a leas
one node b∈N Ssuch ha S∪b∈S. We conside wo cases, depending on whe he
he e exis s only one such node o mul iple ones.
Fi s , assume ha he e exis s only one b∈N Ssuch ha S∪b∈S. Then all links
in L(S) con ain node b. In his case, he payo equa ions along wi h condi ion (5)imply
ha ub(S) =ub(S ∪b) and u∗
b(S ∪b) =u∗
b(S) o all b∈N (S ∪b). Since ub(S ∪b) =
u∗
b(S∪b) by he induc ion hypo hesis, i ollows ha ub(S) =u∗
b(S) o all b∈N (S ∪b).
Fu he mo e, us(S) =u∗
s(S) =0 o all selle s sno linked o bin G.
The payo equa ion o buye bin s a e Sleads o
ub(S) =
s:bs∈L(S)
πbs(S)ub(S) +(1−p)αbs(S)wbs(S)
Since s:bs∈L(S) πbs(S) =1,andπbs(S) > 0and αbs(S)wbs(S) ≥0 o bs ∈L(S),i mus be
ha αbs(S)wbs(S) =0 o all ssuch ha bs ∈L(S).
The payo equa ion o any selle sin s a e Slinked o node bin ne wo k G educes
o
us(S) =πbs(S)us(S) +pαbs(S)wbs(S)
+
s=s:bs∈L(S)
πbs(S)αbs(S)us(S ∪b) +1−αbs(S)us(S)
=πbs(S)us(S) +
s=s:bs∈L(S)
πbs(S)1−αbs(S)us(S)
whe e we ook in o accoun ha αbs(S)wbs(S) =0and us(S ∪b) =0(sis no linked o any
buye in s a e S∪b). I ollows ha
us(S) 
s=s:bs∈L(S)
πbs(S)αbs(S) =0
which is possible only i ei he us(S) =0o αbs(S) =0 o all s= ssuch ha bs∈L(S).
Suppose i s ha
∃s∈Ss. . bs ∈L(S) and αbs(S) =0∀s= swi h bs∈L(S) (17)
Then cons ain (5) implies ha he e exis s exac ly one ssa is ying his condi ion and
αbs(S) > 0. Assump ion (7) leads o us(S) =p( b+ub(S ∪b) +us(S ∪b)) =p( b+ub(S ∪
b)) and ub(S) =(1−p)( b+ub(S ∪b) +us(S ∪b)) =(1−p)( b+ub(S ∪b)).
1050 Mihai Manea Theo e ical Economics 16 (2021)
I L(S) ={bs}, hen we also ha e ha u∗
s(S) =p( b+u∗
b(S∪b)+u∗
s(S∪b)) and u∗
b(S) =
(1−p)( b+u∗
b(S ∪b) +u∗
s(S ∪b)), which along wi h he induc ion hypo hesis implies
ha us(S) =u∗
s(S) and ub(S) =u∗
b(S).
We now conside he case |L(S)|≥2. In his case, he e exis s s∈S ssuch ha
bs∈L(S) and αbs(S) =0. As a gued abo e, αbs(S) > 0implies ha us(S) =0.Hence,
us(S) =us(S ∪b) =0. Since αbs(S) =0,weha ewbs(S) ≤0and, hus, b+ub(S ∪b) +
us(S ∪b) −ub(S) −us(S) = b+ub(S ∪b) −ub(S) ≤0. Howe e , αbs(S) > 0also means
ha wbs(S) ≥0, which leads o b+ub(S ∪b) −ub(S) −us(S) ≥0. I ollows ha b+
ub(S ∪b) −ub(S) =us(S) =0. We ha e es ablished ha us(S) =0 o all s∈Ssuch ha
bs∈L(S). Since he equa ion o selle payo s abo e also applies o any e inemen o
he ba gaining solu ion (u∗α∗)unde which α∗
bs(S) > 0 o all s∈Swi h bs∈L(S),we
conclude ha u∗
s(S) =0—in pa icula , us(S) =u∗
s(S)— o all s∈Ssuch ha bs∈L(S).
Suppose nex ha s a emen (17) is alse. Then i mus be ha us(S) =0 o all s∈S,
|L(S)|≥2,andbis a deale in s a e S, while each selle o ms a single on equi alence
class in G(S). I ollows ha us(S) =0=u∗
s(S) o all s∈S.The eexis ss∈Swi h bs ∈
L(S) such ha αbs(S) > 0, which implies ha wbs(S) =0.Fo suchans,weha eub(S) =
b+ub(S ∪b) +us(S ∪b) −us(S) = b+u∗
b(S ∪b) =u∗
b(S). The second equali y elies on
ub(S ∪b) =u∗
b(S ∪b) (induc ion hypo hesis) and us(S) =us(S ∪b) =0, while he hi d
ollows om he deale s a us o buye bin s a e S. We ha e shown ha he nega ion o
(17) implies ha u(S) =u∗(S), which comple es he p oo o he induc i e s ep o he
case in which S∪b∈S o a single b∈N S.
Finally, conside he case in which he e exis b= b∈N Swi h he p ope y ha
S∪band S∪b∈S. Fo such pai s (b b), he induc ion hypo hesis implies ha bs(S ∪
b)= ∗
bs(S ∪b)whene e bs ∈L(S). His o y independence o p ices unde uand u∗
equi es ha bs(S) = bs(S ∪b)and ∗
bs(S) = ∗
bs(S ∪b),and,hence, bs(S) = ∗
bs(S) o
bs ∈L(S).Weha eshown ha in hiscase, bs(S) = ∗
bs(S) o e e y link bs ∈L(S).
Fix s∈S. The payo equa ion o selle sin s a e Scanbe ew i enas
us(S) =
b:bs∈L(S)
πbs(S)1−αbs(S)us(S) +αbs(S)us(S ∪b) + bs(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)us(S ∪b) +1−αbs(S)us(S)
Since bs(S) = ∗
bs(S) and us(S ∪b) =u∗
s(S ∪b) in he equa ion abo e, we ha e
us(S) =
b:bs∈L(S)
πbs(S)1−αbs(S)us(S) +αbs(S)u∗
s(S ∪b) + ∗
bs(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)u∗
s(S ∪b) +1−αbs(S)us(S)
Theo e ical Economics 16 (2021) Links, in e media ies, and di usion in ne wo ks 1051
By Theo em 1, he payo s u∗a e consis en wi h any p o ile o ag eemen p obabili ies,
including α. The e o e, we also ha e ha
u∗
s(S) =
b:bs∈L(S)
πbs(S)1−αbs(S)u∗
s(S) +αbs(S)u∗
s(S ∪b) + ∗
bs(S)
+
bs∈L(S):s=s
πbs(S)αbs(S)u∗
s(S ∪b) +1−αbs(S)u∗
s(S)
Sub ac ing he wo equali ies abo e and ea anging e ms, we ob ain
us(S) −u∗
s(S)
bs∈L(S)
πbs(S)αbs(S) =0
Condi ion (5) implies ha he summa ion in he equa ion abo e is posi i e, so i mus
be ha us(S) =u∗
s(S).
We ha e a gued ha us(S) =u∗
s(S) o all s∈S. A simila logic p o es ha ub(S) =
u∗
b(S) o all b∈N Sand comple es he p oo o he induc i e s ep o he case unde
conside a ion.
P oo o Theo em 2. The i s wo s a emen s o he esul we e p o ed in Sec ion 6.
To p o e he hi d s a emen , conside a selle con igu a ion S∈Sand a buye b∈N S.
All pa hs in Gconnec ing any selle in S o buye b ha in e sec some equi alence class
Ci(S) mus en e Ci(S) ia i s deale d(SCi(S)) and, hus, can c oss Ci(S) only once. I
wo pa hs in his collec ion exi Ci(S) h ough nodes x= y∈Ci(S), hen we ob ain he
con adic ion ha xG(S) yby “pas ing” he subpa hs om x o band om b o y,and
elimina ing po en ial o e lap as in he p oo o Lemma 3. The e o e, e e y pa h ha
connec s a node in S o buye bin Gand in e sec s Ci(S) mus en e Ci(S) ia node
d(SCi(S)) and exi h ough he same node x. Hence, all such pa hs mus o e lap in
Ci(S) wi h he unique pa h be ween d(SCi(S)) and xin G.
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