Peski, Ma cin
A icle
Random u ili y coo dina ion games on 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: Peski, Ma cin (2025) : Random u ili y coo dina ion games on ne wo ks,
Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 20, Iss. 2, pp.
583-622,
h ps://doi.o g/10.3982/TE5653
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/320294
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc/4.0/
Theo e ical Economics 20 (2025), 583–622 1555-7561/20250583
Random u ili y coo dina ion games on ne wo ks
Ma cin P˛eski
Depa men o Economics, Uni e si y o To on o
We s udy s a ic bina y coo dina ion games wi h andom u ili y played on ne -
wo ks. In equilib ium, each agen chooses an ac ion only i a ac ion o he neigh-
bo s choosing he same ac ion is highe han an agen -speci ic i.i.d. h eshold.
A uzzy con en ion xis a p o ile whe e (almos ) all agen s choose he high ac ion
i hei h eshold is smalle han xand he low ac ion o he wise. The andom-
u ili y (RU) dominan ou come x∗is a maximize o an in eg al o he dis ibu ion
o h esholds. The de ini ion gene alizes Ha sanyi–Sel en’s isk dominance o co-
o dina ion games wi h andom u ili y. We show ha , on each su icien ly la ge
and ine ne wo k, he e is an equilib ium ha is a uzzy con en ion x∗.Onsome
ne wo ks, including a ci y ne wo k, all equilib ia a e uzzy con en ions x∗. Finally,
uzzy con en ions x∗a e he only beha io ha is obus o misspeci ica ion o he
ne wo k s uc u e.
Keywo ds. Random u ili y, coo dina ion games, ne wo ks.
JEL classi ica ion.C7.
1. In oduc ion
An indi idual’s beha io in social o economic si ua ions is o en posi i ely in luenced
by simila decisions made by hei iends, acquain ances, o neighbo s. An impo an
ecen example is he pos -Co id e a mask-wea ing: some people wea masks o p o ec
hemsel es o o he s, o he s do no wea hem because o incon enience o pe sonal
belie s, and many, including he au ho o his pape , a e posi i ely a ec ed by how
many people a ound hem wea masks. O he examples include he decision o main-
ain a nea on ya d, o obey speed limi s o ax laws, o engage in c iminal ac i i y,
o o adop a echnology wi h ne wo k ex e nali ies. A la ge li e a u e has es ablished
condi ions unde which a pa icula beha io becomes a con en ion: i is adop ed by
e e yone (see Young (1993), Ellison (1993), Mo is (2000), among many o he s). These
esul s ypically assume ha agen s ha e almos iden ical p e e ences, and show ha
a con agion-like p ocess, possibly ini ia ed by a small pe u ba ion o he p e e ences,
leads o uni o mi y.
Ma cin P˛eski: [email p o ec ed]
The pape has p e iously been ci cula ed unde he i le “Fuzzy con en ions.” I am e y g a e ul o S ephen
Mo is and Philip Nea y o de ailed commen s, Jacopo Pe ego o discussion, Denise C uz o he main
example, and an anonymous e e ee o insigh ul sugges ions. S. Mo is’s commen s g ea ly in luenced
he p oo o Theo em 1. I g a e ully acknowledge inancial suppo om he Insigh G an o he Social
Sciences and Humani ies Resea ch Council o Canada.
©2025 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/TE5653
584 Ma cin P˛eski Theo e ical Economics 20 (2025)
A he same ime, comple ely uni o m beha io is a ely obse ed in he eal wo ld.
E en in si ua ions which clea ly in ol e posi i e ex e nali ies, he e will o en be in e ac-
ions in which neighbo s make opposi e choices. An ob ious eason is ha indi iduals
a e di e en and hei as es and unique ci cums ances play jus as impo an ole in
de e mining hei decisions as he beha io o hei neighbo s. The goal o his pape is
o s udy coo dina ion games wi h he e ogeneous payo s wi h he ollowing ques ions
in mind. Is he e a use ul and cohe en way in which he e ogeneous-beha io equi-
lib ia can be unde s ood as con en ions? Can we explain how people coo dina e on a
con en ion? A e some con en ions mo e na u al han o he s?
Fo his pu pose, we s udy a andom u ili y bina y coo dina ion game played in a
ne wo k. Each ne wo k node con ains a single agen who in e ac s wi h he neighbo s.
We a e in e es ed in he asymp o ic o equilib ium beha io as he ne wo k becomes a -
bi a ily la ge, and impo an ly, as he g aph becomes su icien ly ine, ha is, he weigh
o he la ges neighbo in he neighbo hood o each agen becomes su icien ly small.
The la e ensu es ha no single indi idual has a disp opo iona e impac on ano he ,
and i is he i s key assump ion in ou model.
Each agen chooses a bina y (high o low) ac ion, and he ela i e gain om he ac-
ion is inc easing in he ac ion o neighbo s who make he same choice. Each agen has
an indi idual h eshold τi, wi h he in e p e a ion ha he high ac ion is he agen ’s bes
esponse i and only i mo e han ac ion τio he neighbo s do he same. Th esholds
a e dis ibu ed i.i.d., wi h dis ibu ion gi en by cd P(.). The independence assump ion
is he second key assump ion o ou model and i is app op ia e o some bu no all ap-
plica ions. An example o cd P(.)is d awn on Figu e 1; o each x,P(x)is he ac ion
o he popula ion wi h a h eshold equal o o smalle han x. Impo an ly, unlike in he
coo dina ion li e a u e men ioned abo e, he le el o p e e ence he e ogenei y cap u ed
by P(.)is nonze o and nondisappea ing.
A concep ual con ibu ion o his pape is a de ini ion o a con en ion app op ia e
o la ge andom u ili y coo dina ion games. De ine a uzzy con en ion xas an ac ion
p o ile whe e almos all agen s choose he high ac ion i τi<xand he low ac ion i
τi>x.I xis an a om o dis ibu ion P(.), he de ini ion allows o andomiza ion a
τi=x. Ou assump ions on ne wo ks imply ha , in a uzzy con en ion, almos all agen s
obse e app oxima ely P(x) ac ion o hei neighbo s choosing he high ac ion. This
de ini ion cap u es indi idual he e ogenei y o ac ions, wi h wo ypes o uni o mi y:
(a) almos all agen s choose hei ac ion as he same unc ion o hei h eshold and (b)
almos all agen s expe ience almos he same a e age beha io o hei neighbo s. Fo
a uzzy con en ion x o be an equilib ium, he choice in (a) mus be a bes esponse,
which implies ha i is an in e sec ion wi h 45◦line, x=P(x). Figu e 1illus a es wi h
mul iple candida e solu ions.
Nex , we de ine a pa icula ixed poin . Le andom u ili y-dominan ,o RU-
dominan ,ou comex∗be a solu ion o he maximiza ion p oblem
x∗∈a gmax
x
x
0y−P−1(y)dy.(1)
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 585
Figu e 1. Th eshold cd P.
The de ini ion implies ha P(x∗)=x∗. Geome ically, he maximized objec i e on he
igh -hand side is equal o he a ea abo e he 45◦line and below unc ion P(blue a ea
on Figu e 1) minus he a ea below he 45◦line and abo e P( ed a ea). The RU-dominan
ou come depends on he h eshold dis ibu ion, and gene ically, i is unique. Two ob-
se a ions abou special cases o ou model mo i a e his de ini ion u he . Fi s , (1)is
equi alen o a o mula om Mo is and Shin (2006), whe e i is de i ed as a po en ial
unc ion o he con inuum popula ion e sion o he model whe e agen s ea he en-
i e popula ion as hei neighbo s. Second, i he h eshold dis ibu ion is concen a ed
on a single ou come (i.e., all agen s’ p e e ences a e iden ical), hen he RU-dominan
ou come is equi alen o he s anda d isk-dominan ou come o a 2 ×2 coo dina ion
game (Ha sanyi and Sel en (1988)).
The esul s o he pape show ha uzzy con en ion x∗is he “ igh ” solu ion: In o -
mally, all ne wo ks ha e an equilib ium ha is a uzzy con en ion x∗, and on some ne -
wo ks, he e a e no o he equilib ia. Mo e p ecisely, i s , we show ha o each ne wo k
ha is su icien ly la ge and ine, wi h a p obabili y close o 1 (i.e., o almos all eal-
iza ions o h esholds), he e is an equilib ium ha is a uzzy con en ion x∗. The p oo
elies on a cha ac e iza ion o coo dina ion games as po en ial games. (Fo an a bi a y
ne wo k, a po en ial unc ion is necessa ily di e en han he one in (1).) Such games
a e in oduced in Monde e and Shapley (1996), whe e i is shown ha any p o ile ha
is a local maximize o he po en ial unc ion is an equilib ium o he unde lying game.
In he p oo , we show ha , ega dless o he s uc u e o he ne wo k, wi h a p obabili y
close o 1, he global maximize o he po en ial unc ion is a uzzy con en ion x∗.The
di icul pa o he p oo is o de i e a e sion o a uni o m law o la ge numbe s and
o show ha i gua an ees ha ac ion p o iles ha a e no uzzy con en ions x∗canno
maximize he po en ial.
586 Ma cin P˛eski Theo e ical Economics 20 (2025)
Second, we show he e exis ne wo ks, whe e wi h a la ge p obabili y, all equilib ia
a e uzzy con en ions x∗. An example o such a ne wo k is a ci y-like ne wo k, whe e
agen s li e on a 2-dimensional g id la ice and hey in e ac wi h agen s in a su icien ly
la ge neighbo hood. The idea o he p oo is o show ha , o each p o ile wi h an a -
e age beha io ha is no RU-dominan , con agion-like bes esponse dynamics would
b ing he beha io close o x∗. The p oo uses an idea om Blume (1995a)andLee and
Valen inyi (2000)(seealsoMo is (2000)) o show how a con agion wa e sp eads ac oss
la ice ne wo ks. The e a e wo no el di icul ies ela i e o ea lie li e a u e. Fi s , un-
like in he ea lie li e a u e, he agen p e e ences a e andom and he e ogeneous. In-
s ead o a bina y wa e (whe e he e is a sha p sepa a ion be ween isk-domina ed and
isk-dominan egions), he con agion wa e he e has mul iple alues as i desc ibes he
ac ion o agen s ha adop he high ac ion. Second, we mus compa e he likelihood
ha a a o able con igu a ion o payo shocks may ini ia e such a wa e, wi h he likeli-
hood ha such a wa e would no be s opped by an un a o able con igu a ion o payo
shocks. The p oblem wi h he la e is he eason why he 1-dimensional ne wo k o El-
lison (1993) is no a good example o he esul and a 2- (o mo e) dimensional la ice is
needed.
The wo esul s oge he sugges ha RU-dominan ou come x∗is he only p edic-
ion o agg ega e equilib ium beha io ha is ne wo k-independen . We o malize his
h ough a de ini ion ha is inspi ed by Kajii and Mo is (1997): Conside an analys who
p edic s agen s’ beha io bu she is no ce ain whe he he model co ec ly speci ies
he ne wo k in e ac ions, o whe he he agen s know he en i e ne wo k. We say ha
he beha io is obus o misspeci ica ions i , e en i she o he agen s a e w ong, he
p edic ion is close o some equilib ium o he ue model. Ou esul s imply ha uzzy
con en ion x∗is he only obus p edic ion.
1.1 Li e a u e e iew
This is he i s pape wi h p edic ions abou beha io in s a ic comple e-in o ma ion
andom-u ili y games on ne wo ks. The model and echniques used d aw om wo
s ands o he li e a u e: andom u ili y games on ne wo ks and models o lea ning (o
e olu ion) in games.
The i s andom-u ili y coo dina ion model was in oduced in G ano e e (1978).
G ano e e wo ks wi h a comple e (con inuum) ne wo k, whe e he agen s’ payo s de-
pend on he a e age beha io in he en i e popula ion. A la ge li e a u e gene alized
G ano e e ’s model o ne wo ks. Typically, each agen is a single node on a ne wo k and
adop s he new beha io (e.g., wea s a mask) only i he ac ion o he neighbo s doing
he same is la ge han he h eshold. Many pape s, like Wa s (2002)o López-Pin ado
(2008) (among many o he s) s udy G ano e e ’s model on an E d˝
os–Renyi s yle o a an-
dom g aph wi h he e ogeneous deg ee dis ibu ion. The limi a ion o such models is
ha hey do no cap u e many impo an aspec s o eal-wo ld ne wo ks, like clus e ing,
o o e lapping neighbo hoods, which a e known o play impo an ole in coo dina ion
o con agion phenomena.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 587
Jackson and Ya i (2007) (see also Galeo i, Goyal, Jackson, Vega-Redondo, and Ya i
(2010)) analyzes a Bayesian equilib ium, whe e he agen s choose hei ac ion wi h-
ou knowing he h esholds o hei neighbo s. This assump ion imp o es he model’s
ac abili y as he agen ’s beha io does no depend on he indi idual h esholds o he
neighbo s. A he same ime, his assump ion is no sa is ac o y i he equilib ium is o
be in e p e ed as a long- e m p ocess as each agen may change he beha io when she
obse es he ac ions o he neighbo s. This is he i s key di e ence om ou model,
whe e an equilib ium is a s eady-s a e beha io a e he h esholds a e ealized and
ac ions a e chosen. Because ou model is a s a ic, comple e in o ma ion equilib ium
o a gi en ealiza ion o h esholds, i is also much mo e di icul o analyze. Fu he ,
because he neighbo s in he Bayesian equilib ium o Jackson and Ya i (2007)a ese-
lec ed a andom, he neighbo hood s uc u e looks like a andom g aph. Like o he
andom-g aph-based models, he e a e ypically mul iple equilib ia. In con as , in his
pape , we a e se ious abou he opology o he ne wo k and explain an impo an ole
o o e lapping neighbo hoods ha canno be cap u ed in andom g aphs models.
The esul s o his pape a e closely ela ed o he li e a u e on e olu iona y lea n-
ing and con agion in ne wo ks. E olu iona y game heo y (Kando i, Maila h, and Rob
(1993), Young (1993), Blume (1993), New on (2021), and many o he s) s udies he long-
un beha io o pe u bed bes esponse p ocesses, whe e agen s commi mis akes wi h
a small p obabili y, and ins ead o choosing a bes esponse, ake some o he ac ion.
A majo con ibu ion o his li e a u e is a demons a ion o a con agion phe-
nomenon. Ellison (1993)(seealsoEllison (2000)) shows ha a bes esponse may sp ead
a isk-dominan ac ion om a small ini ial se o de ia o s o he es o a 1-dimensional
la ice ne wo k. Blume (1995b)andLee and Valen inyi (2000) ex end his obse a ion
o highe -dimensional la ices. Mo is (2000) desc ibes gene al p ope ies o ne wo ks
o which Ellison’s con agion wa e exis s. Mo is (2000) also shows ha isk-domina ed
ac ions canno sp ead h ough a bes esponse p ocess ega dless o he geome y o he
ne wo k.
A s and o he li e a u e s udies e olu iona y equilib ium selec ion in games wi h
he e ogeneous popula ions. Fo ins ance, F iedman (1991) desc ibes a gene al ame-
wo k wi h mul iple con inuum popula ions choosing ac ions and ecei ing payo s and
s udies e olu iona y s eady s a es o con inuous ime adjus men dynamics. Mo e
closely ela ed o his pape is Nea y (2012), which s udies a simila model o ou s bu
wi h wo payo shocks (mo e p ecisely, wo subpopula ions o de e minis ic size) and
agen s loca ed on a comple e g aph. The pape p esen s condi ions unde which he
e olu iona y dynamics o Kando i, Maila h, and Rob (1993) selec s a uzzy con en ion,
ha is, an equilib ium whe e membe s o di e en subpopula ions play di e en ac-
ions. Nea y and New on (2017) s udy gene al payo shocks and p esen s a su icien
condi ion unde which he logi dynamics o Blume (1993) selec a uzzy con en ion.
Ou cu en esul s (speci ically, Theo ems 1and 2) a e ela ed, bu wi h some key
di e ences. Fi s , he e, we a e in e es ed in s a ic equilib ia ins ead o a dynamic ad-
jus men p ocess. The e olu iona y li e a u e is subjec o he c i icism ha one may
need o wai o a e y long ime be o e eaching a s ochas ically s able ou come (El-
lison (1993)). Tha c i icism does no apply o ou s a ic model. Second, he p e ious
588 Ma cin P˛eski Theo e ical Economics 20 (2025)
pape s s udy games wi h homogeneous payo s and a beha io ha is subjec o small
and disappea ing pe u ba ions: small and disappea ing shocks in he case o Ellison
(1993)o Blume (1995b), and a ini e and small ac ion o socie y modi ying hei ac-
ions in Lee and Valen inyi (2000)o Mo is (2000). Ins ead, he payo shocks in ou
model a e signi ican , and as a esul , we a e se ious abou he e ogenei y. The non i ial
payo shocks make ou model mo e di icul o analyze, bu hey also ende i close o
eali y. Thi d, he e olu iona y li e a u e esul s show con e gence o Ha sanyi and Sel-
en (1988)’s isk-dominance. He e, due o payo he e ogenei y, we need a new solu ion
concep in he o m o he RU-dominance. We show ha he RU-dominance becomes
equi alen o he isk-dominance when payo s a e homogeneous. Finally, he ne wo k
opology plays an impo an ole in bo h e olu iona y models and in he cu en pape .
In e olu iona y models, he ne wo k a ec s he ime o he coo dina ion on he isk-
dominan ou come. Howe e , i does no a ec he inal ou come: one o he key esul s
o his li e a u e is ha isk-dominan coo dina ion is (uniquely) s ochas ically s able on
all ne wo ks (Peski (2010)). In ou case, simila ly o Lee and Valen inyi (2000)andMo is
(2000), he ne wo k opology a ec s he equilib ium ou come.
In a ecen con ibu ion, Leis e , Zenou, and Zhou (2022) s udy coo dina ion games
wi h a ixed ne wo k and a ixed (no andom) h eshold dis ibu ion. The pape wo ks
wi h a bi a y ne wo ks. To deal wi h a possible mul iplici y o equilib ia, hey use global
games as an equilib ium selec ion de ice. The au ho s de elop an algo i hm o compu e
he equilib ium adop ion. The ou come o he algo i hm depends on he de ails o pay-
o he e ogenei y and how hey in e ac wi h he opology o he ne wo k. In con as , in
ou pape , he assump ion ha h esholds a e andomly and independen ly d awn om
he same dis ibu ions allows us o sepa a e he e ec s o he payo dis ibu ions and
he opology o he ne wo k.
2. Nume ical example
Al hough ou esul s a e asymp o ic, he coo dina ion on RU-dominan ou come as well
as he ole o he ne wo ks can be demons a ed h ough simula ions in a nume ical
example.
We compa e he beha io unde wo h eshold dis ibu ions. In bo h cases, he high
ac ion is s ic ly dominan o 30% o he popula ion and he low ac ion is s ic ly domi-
nan o ano he 30%. Unde P1, he emaining 40% plays he high ac ion only i a leas
0.55 o hei neighbo s do he same. Unde P2, he emaining 40% plays he high ac ion
only i a leas 0.4 o hei neighbo s do he same. The dis ibu ions a e d awn in he op
ow o Figu e 2.
I , like in G ano e e (1978), he popula ion is a con inuum, and all agen s play
agains he en i e popula ion, he equilib ium a e age beha io can be ound as a ixed
poin o P(.), ha is, an in e sec ion o P(.)wi h he 45◦line. In bo h cases, he e a e wo
s able equilib ia: Awi h 0.3 and Bwi h 0.7 ac ions playing high. (In each case, he e
is also an uns able equilib ium in be ween.) Fo each dis ibu ion, only one o hese
ou comes is RU-dominan —Ain he case o dis ibu ion P1and Bin hecaseo P2.
Ins ead, conside a popula ion o agen s li ing on one o wo ne wo ks. Bo h ne -
wo ks ha e ∼60,000 agen s and each agen has, on a e age, ∼120 neighbo s.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 589
Figu e 2. Mon e-Ca lo simula ions o a e age equilib ium beha io in he lowes (blue, “ ”
ha ch a eas) and he highes equilib ia (yellow, “/” ha ch a eas). The dis ibu ions subs an ially
o e lap (b own colo ) in he las ow, co esponding o he ci y ne wo k.
•In a andom g aph (E d˝
os and Rényi (1959)), neighbo s a e andomly selec ed om
he popula ion.
•In a “ci y” ne wo k, people a e loca ed on a wo-dimensional g id. Each agen
neighbo hood is a squa e o agen s wi h a side equal o 11, cen e ed a he agen .
We use Mon e Ca lo simula ions o es ima e he p obabili y dis ibu ions o a e age
equilib ium beha io . In each simula ion, we d aw i.i.d. h esholds o all agen s. Fo
each ealiza ion o h esholds, we ind he highes and lowes equilib ia. Such equilib ia
a e well-de ined o bina y coo dina ion games. Fo example, o ind he highes equi-
lib ium, we s a wi h a p o ile whe e all agen s play he high ac ion, and hen un he
bes esponse p ocess un il none o he agen s wan s o change hei ac ion. Nex , o
each equilib ium, we compu e he a e age equilib ium beha io . By combining a e age
beha io s in wo equilib ia ac oss di e en h eshold ealiza ions, we ob ain he Mon e
Ca lo es ima es.
These dis ibu ions o each ne wo k, each h eshold dis ibu ion, and each equilib-
ium ype ( he lowes is ma ked wi h blue, “ ” ha ch a eas and he highes wi h yellow,
“/” ha ch a eas) a e plo ed in he wo bo om ows o Figu e 2. Because bo h dis ibu-
ions a e highly concen a ed a ound 0.3 (i.e., A) and 0.7 (i.e., B) alues, o cla i y, we
only show egions a ound hese wo alues.
590 Ma cin P˛eski Theo e ical Economics 20 (2025)
The e is a signi ican di e ence be ween andom and ci y ne wo ks. In he andom
g aph, he lowes and he highes equilib ia co espond o he lowes (A)andhighes (B)
equilib ia om he popula ion model o G ano e e (1978), ega dless o he h eshold
dis ibu ion. This is no unexpec ed as a andom g aph wi h a ela i ely la ge numbe o
agen s is a good app oxima ion o he con inuum model.
On he ci y ne wo k, he ange o equilib ium beha io s is much smalle and i de-
pends on a h eshold dis ibu ion. Unde P1, he lowes and he majo i y o ealiza ions
o he highes equilib ia a e concen a ed a ound A.Unde P2, he a e age beha io in
he highes and lowes equilib ia is essen ially equal o B. In o he wo ds, o a signi i-
can majo i y o h eshold ealiza ions, all equilib ia on he ci y ne wo k ha e agg ega e
beha io consis en wi h he RU-dominan p edic ion.
The goal o he es o he pape is o explain his pa e n.
3. Model
3.1 Model
We a e s udying agen s li ing in he nodes o a ne wo k. The ne wo k is de ined as an
undi ec ed weigh ed g aph wi h weigh s gij =gji ≥0 o i,j≤Ng,whe eNgis he size
o he ne wo k. The weigh s can be in e p e ed as a equency o in e ac ions be ween
wo agen s and we assume ha gii =0. Le gi=jgij >0 o each agen i.Eachagen i
has a h eshold τid awn i.i.d. om p obabili y dis ibu ion P. Each ne wo k g, and each
ealiza ion o h esholds τde ines a comple e in o ma ion s a ic game G(g,τ).
Each agen chooses a bina y ac ion ai∈{0, 1}and uses i in each in e ac ion. The
payo in in e ac ion wi h agen jis equal o ui(ai,aj,τi)=aiaj−aiτi,and he o alpay-
o o agen iin all (weigh ed) in e ac ions is equal o jgij ui(ai,aj,τi). Fo each ac ion
p o ile a,le βa=(βa
i)be a p o ile o a e age neighbo hood ac ions o agen s who play
ac ion 1, ha is, βa
i=1
gijgij aj. An ac ion p o ile is a Nash equilib ium i all agen s
bes espond, o al e na i ely, i each agen plays ac ion 1 ( esp., 0) i he a e age ac ion
in hei neighbo hood is s ic ly la ge ( esp., smalle ) han hei h eshold, ha is, o
each i,
1τi<β
a
i≤ai≤1τi≤βa
i.(2)
The model is s a egically equi alen o gene al andom-u ili y bina y-ac ion coo di-
na ion games on ne wo ks.1The no ion o equilib ium is a s anda d, s a ic equilib ium
o a comple e in o ma ion game. Al hough i is con enien o assume ha agen s know
he h esholds and he ne wo k s uc u e o he en i e socie y, his assump ion is nei-
he ealis ic no necessa y. Fo he in e p e a ion o he equilib ium, i is su icien ha
agen s obse e he ac ions o hei neighbo s. Because ou s is a coo dina ion game, we
1A gene al model is as ollows: Fo each agen iand j,i’s payo om in e ac ion wi h agen jis equal o
u(ai,aj,εi),whe eai,aj∈{0, 1}a e ac ions and εiis a andom shock o agen i’s u ili y d awn om some
dis ibu ion F. Assume ha , o each ε,
(ε):=u(1, 1, ε)+u(0, 0, ε)−u(1, 0, ε)−u(0, 1, ε)>0.
To ansla e his model o he h eshold model, o each x,le τi=1
(ε)(u(1, 0, εi)−u(0, 0, εi)).
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 597
(Because 1{.≤x∗}is no Lipschi z, he lemma is applied o a Lipschi z app oxima ion—
he de ails a e le o he Appendix.)
Second, ake an a bi a y equilib ium p o ile ha is no ε- uzzy con en ion o x∗.
Because o (2)and(3), we ge
ε≤1
Ngai−1τi≤x∗
≤1
Ng
i1βa
i≤x∗1τi∈βa
i,x∗+1βa
i≥x∗1τi∈βa
i,x∗.
By Lemma 1, wi h a la ge p obabili y, he ollowing bound holds:
1
Ng
iPβa
i−Px∗≥1
2ε. (11)
Thi d, we es ima e he po en ial o such a p o ile a. Applying Lemma 1once mo e,
we ob ain
V(a;τ)=1
2
i,j
gij aiaj−giaiτi
=1
2
i,j
gi1τi≤βa
i1τj≤βa
j−gi1τi≤βa
iτi
≈1
2
i,j
gij Pβa
iPβa
j−gi
βa
i
0
ydP(y).
Because 2P(βa
i)P(βa
j)≤P(βa
i)2+P(βa
j)2, he po en ial o ais no la ge han
≤1
2
i,j
gij Pβa
i2−gi
βa
i
0
ydP(y)=
i
giνβa
i.
By he ema k a he end o Sec ion 3.3,unlessβa
i=x∗, he abo e is s ic ly smalle han
he po en ial o a∗. Hence, oge he wi h he es ima e o po en ial o p o ile a∗, he
bound (11) implies ha an a bi a y equilib ium p o ile ha is no ε- uzzy con en ion
o x∗canno maximize po en ial.
Finally, ecall ha any po en ial maximize mus be an equilib ium. I ollows ha
he po en ial maximize mus be ε- uzzy con en ion o x∗.
5. RU-dominan selec ion
In he p e ious sec ion, we showed ha all su icien ly ine ne wo ks ha e equilib ia
ha a e uzzy con en ions x∗. He e, we show ha he e a e ne wo ks whe e, wi h a la ge
p obabili y, all equilib ia a e uzzy con en ions x∗:
598 Ma cin P˛eski Theo e ical Economics 20 (2025)
Fo each η>0, he p oo cons uc s a “ci y” ne wo k, whe e agen s li e on a wo-
dimensional g id and in e ac wi h o he agen s who li e a ound hem. The ne wo k is
pa ame e ized wi h Mand m.The ea eM2agen s li ing on squa e [0, M
m]2⊆R2a ac-
ional poin s (k
m,l
m) o k,l=1, ,M. Any wo agen s iand ja e connec ed, gij =1,
i he (Euclidean) dis ance be ween hem is no la ge han 1. To a oid sepa a ely deal-
ing wi h bo de cases, we assume ha all dis ance calcula ions a e done mod M
m,which
ans o ms he squa e [0, M
m]2in o a o us.
Theo em 2. Suppose ha x∗is he s ic ly RU-dominan ou come and ha ei he (a)
x∗∈(0, 1)and 0<P
(0)≤P(1)<1,(b)x∗=1and P(0)>0,o (c)x∗=0and P(1)<
1.Fo eachη>0,i mand M
ma e su icien ly la ge, hen wi h p obabili y 1−η, each
equilib ium on (M,m)ci y ne wo k is η- uzzy con en ion x∗.
The heo em says ha he e exis ne wo ks whe e all equilib ia a e uzzy con en ions
x∗, o ha all equilib ia ha e a o m iden i ied by Theo em 1.
We emphasize ha he heo em makes a s a emen abou s a ic,comple e in o ma-
ion game equilib ia. A he same ime, he p oo elies on a dynamic echnique o con-
agion wa es (Ellison (1993), Mo is (2000)). We show ha i an ac ion p o ile is, in some
sense, highe ( esp., lowe ) han uzzy con en ion x∗, hen bes esponse dynamics will
push he p o ile below ( esp., abo e) x∗. This shows ha he o iginal p o ile could no
ha e been an equilib ium. We desc ibe he in ui ion behind he p oo , including he
ela ion o he maximiza ion p oblem, below.
I P(0)>0( esp.,P(1)<1), hen wi h a posi i e p obabili y, he e a e agen s o
whom ac ion 1 ( esp., 0) is s ic ly dominan and i is played in any equilib ium. The
only assump ion o he heo em is ha he e is a posi i e p obabili y o such agen s. The
ole o such agen s is simila o he ole o ini ial in ec o s in Lee and Valen inyi (2000)
and Mo is (2000) o he ole o small p obabili y mis akes in e olu iona y models.
The ci y ne wo k is an example o a wo-dimensional la ice. The p oo could easily
ex end o K>2 dimensional la ices (bu , as we explain below, no o K=1). A e
we desc ibe he p oo , we poin o he p ope ies o mul idimensional la ices ha a e
impo an o he p oo . Ex ending he heo em o o he ne wo ks is beyond he goals o
his pape .
5.1 Con agion on line
Nex ,wedesc ibe hein ui ion o hep oo .Weassume ha x∗=0andP(1)<1.
We s a wi h he in ui ion behind he con agion a gumen . I is use ul ini ially o
wo k wi h a oy e sion o he line ne wo k om Ellison (1993) ( he gene al a gumen
does no wo k on a line and i equi es a leas wo-dimensional la ices). Suppose ha
agen s a e dis ibu ed uni o mly along a line a disc e e and equally spaced loca ions.
Each loca ion con ains a con inuum popula ion o mass 1. The popula ions in loca ions
iand ja e connec ed wi h each o he , wi h weigh s ha depend only on he dis ance
gij =gi−j=:gj−i. We assume he e a e no connec ions be ween agen s in he same lo-
ca ion, ha is, g0=0, and he weigh s a e no malized so ha gd=1. Finally, we as-
sume ha he e a e no connec ions be ween agen s a dis ance la ge han d:gi−j=0
o |i−j|>d.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 599
Figu e 4. Con agion wa e.
Take an ac ion p o ile a0such ha agen s in loca ions i∈[−2d,0
]play ac ion 0 and
all o he agen s play 1. In ou model (bu no i s con inuum oy e sion), assump ion
P(1)<1 implies ha he e is a posi i e p obabili y ha a con iguous g oup o agen s
ha e 0 as a s ic ly dominan ac ion. I he line ne wo k is long enough, he exis ence o
ag oupo 2dagen s who play 0 o su e can be gua an eed wi h a p obabili y a bi a ily
close o 1.
Going back o he oy line wi h a con inuum o agen s in each loca ion, conside a
e ision p ocess in which agen s in all loca ions apa om i≥0 swi ch o hei myopic
bes esponses. Complemen a i ies imply ha hey can swi ch a mos once, and i hey
do, hey swi ch om ac ion 1 o 0. Figu e 4illus a es he i s wo s ages o such a p o-
cess. In he i s s age, ac ions a e changed by agen s in loca ions i>0 o whom ac ion 0
is s ic ly dominan , as well as high- h eshold agen s in loca ions i∈[0, d] o whom 0 is
a bes esponse gi en a0. In he second s age, addi ional agen s in loca ions i≤2dmay
change ac ions, and so on. The p ocess will con inue un il a s able poin whe e no mo e
agen s i≥0 wan o swi ch o 0. Deno e he ac ion o agen s who play 1 in loca ion iin
s age nas an
iand he limi ac ion as limnan
i=ai. Due o he payo complemen a i ies,
p o iles an
i o each nand aimus be inc easing in i.
In his oy e sion, he con inuum law o la ge numbe s allows us o exp ess he
ac ion o agen s o whom 1 is a bes esponse gi en p o ile aas P(dgdai+d).Gi en
ha ais he limi o he bes esponse dynamics, we ha e, o each loca ion i≥−2d,
ai≤P
d
gdai+d.
Taking he in e se, we ob ain
P−1(ai)≤
d
gdai+d=
j
d≥j−i
gd(aj+1−aj),
whe e he equali y is due o a disc e e e sion o he in eg a ion-by-pa s o mula and
he ac ha ai≥0 o each i. A e mul iplying by ai+1−ai≥0, and summing up ac oss
all loca ions i,wege
i
P−1(ai)(ai+1−ai)≤
i,j
d≥j−i
gd(ai+1−ai)(aj+1−aj). (12)
600 Ma cin P˛eski Theo e ical Economics 20 (2025)
The le -hand side o he inequali y is app oxima ely equal o a
0P−1(y)dy when he
dis ance be ween loca ions is small and o la ge m. To compu e he igh -hand side,
no ice ha we can swi ch he oles o iand jin he summa ion wi hou a ec ing i s
alue. Toge he wi h he ac ha d≥j−igd+d≥i−jgd=gd=1, we ge
i,j
d≥j−i
gd(ai+1−ai)(aj+1−aj)
=1
2
i,j
d≥j−i
gd+
d≥i−j
gd(ai+1−ai)(aj+1−aj)
=1
2
i,j
(ai+1−ai)(aj+1−aj)=1
2a2
=1
2a2=
a
0
ydy.
Pu ing he wo sides oge he , inequali y (12) implies ha
a
0y−P−1(y)dy ≥0.
I a>0, his con adic s he ac ha x∗=0 is he unique maximize o he in eg al on
he igh -hand side o (4). Thus, in he limi o bes esponse e ision p ocess, i mus be
ha all loca ions play ai=0.
The con agion a gumen ex ends om a line o highe -dimensional la ices due o
an elegan a gumen om Blume (1995b)(seealsoLee and Valen inyi (2000)andMo is
(2000)). The idea is ha i he ini ial g oup is su icien ly la ge, we can app oxima e i
using a se wi h a smoo h (i.e., low cu a u e) bounda y. Then we can analyze he sp ead
o he con agion wa e beha io in he di ec ion ha is no mal o he bounda y. This ick
u ns he p oblem in o a one-dimensional one, and he abo e a gumen applies.
5.2 Obs acles
Al hough he con inuum assump ion is use ul in explaining he in ui ion, he a gumen
needs o be modi ied o ou model. Fo example, he assump ion igno es a posi i e
p obabili y o a con iguous g oup o “bad” agen s o whom 1 is he s ic ly dominan
ac ion. I su icien ly la ge, such a g oup o “bad” agen s will s op he bes esponse
e isions owa ds ac ion 0 and block he con agion wa e (see he le panel o Figu e 5).
“Bad” se s canno be elimina ed o a oided in he one-dimensional “line” ne wo k.
Howe e , “bad” se s a e in ui i ely less likely o block he con agion wa e on highe -
dimensional la ices (see he igh panel o Figu e 5). The eason is ha o block he
wa e, he “bad” se s would ha e o be a anged so as o su ound i . We show ha , on
a wo-dimensional la ice, i mand M
ma e su icien ly la ge, he likelihood o “bad” se s
su ounding he ini ial in ec o s is e y small.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 601
Figu e 5. Obs acles o he con agion wa e.
5.3 P oo summa y
Mo e gene ally, wi hou he con inuum assump ion, he a gumen behind con agion
wa es mus wo k wi h ini e laws o la ge numbe s. Below, we ske ch he main ideas o
howwedoi .Thede ailso hep oo canbe oundinAppendixB.
The la ice is di ided in o la ge and small cubes so ha he numbe o la ge cubes in
he la ice is e y la ge, each la ge cube con ains a e y la ge numbe o disjoin neigh-
bo hoods, each neighbo hood con ains a e y la ge numbe o small cubes, and each
small cube con ains a e y la ge numbe o agen s (see Figu e 6). These numbe s a e
chosen so ha he ollowing se ies o claims holds:
(1) The numbe o agen s in a small cube and he numbe o small cubes in a neigh-
bo hood a e su icien ly la ge, so ha he ac ion o sha ed agen s and he ac-
ion o sha ed small cubes in he neighbo hoods o any wo agen s iand jis well
app oxima ed by he a ea o he in e sec ion o wo 1- adius ci cles wi h cen e s
a iand j(Lemma 3).
(2) The size o each small cube is su icien ly la ge so ha , o each small cube, wi h a
p obabili y close o 1, he empi ical dis ibu ion o payo shocks wi hin he cube
is close o he ue dis ibu ion. We say ha a small cube is (γ-)bad i , o some
ac ion x, he a e age bes esponse ac ion o he agen s wi hin he cube is (γ-
)la ge han P(x). Agen s in bad cubes may il owa d highe bes esponses han
a s a is ical agen . Agen s in a small cube ha is no bad a e well app oxima ed
by he con inuum assump ion in he ollowing sense: he a e age bes esponse
in he small cube is no highe han P(β),whe eβis he a e age “belie ” (i.e., he
a e age neighbo hood ac ion) o membe s o he cube.
(3) A la ge cube is good i i con ains no bad small cubes. The a io o he size o a
small cube (i.e., he numbe o agen s wi hin each small cube) o he numbe o
small cubes in a la ge cube is su icien ly la ge, so ha he p obabili y p ha he
la ge cube is good is a bi a ily close o 1.
Ala gecubeisex ao dina y i i con ains only agen s o whom 0 is he s ic ly
dominan ac ion. Ex ao dina y cubes play he ole o ini ial in ec o s. The num-
be o la ge cubes is su icien ly la ge, so ha he p obabili y ha an ex ao dina y
la gecubeexis sisa bi a ilyclose o1.
602 Ma cin P˛eski Theo e ical Economics 20 (2025)
Figu e 6. La ice di ision.
(4) Two la ge cubes a e connec ed i hey sha e a wall. The numbe o la ge cubes is
su icien ly la ge, and he p obabili y p ha a la ge cube is no good is su icien ly
small, so ha he e exis s a gian componen o good la ge cubes—a se o good
la ge cubes ha con ains almos all la ge cubes on he la ice and such ha all o
i s elemen s a e connec ed wi h each o he by pa hs o good la ge cubes ha sha e
a wall. This a gumen is he con en o Lemma 7and i elies on de ini ions and
esul s om he pe cola ion heo y (Bollobás and Rio dan (2006)).
(a) Fi s , we show ha each connec ed se Scan be su ounded by a connec ed
“bounda y” ∂S ha isola es se S(and, possibly, some o he la ge cubes) om
he emaining la ge cubes. The o al numbe o la ge cubes isola ed away om
se Sis no la ge han |S|2. (On a wo-dimensional la ice, he wo s -case sce-
na io bound comes om elemen s o se Sa anged in a way ha su ounds
an in e io p opo ional in size o he squa e o i s pe ime e .)
(b) Fo a collec ion o connec ed se s S1,,SJ ha a e no connec ed wi h each
o he , he gian connec ed componen ha omi s all se s Sjcon ains all bu a
mos |Sj|2la ge cubes.
(c) Le S1,,SJbe he collec ion o all maximally connec ed collec ions o la ge
bad cubes. We es ima e he expec ed alue o |Sj|2as p opo ional o he
numbe o all la ge cubes mul iplied by he p obabili y p ha a single la ge
cube is bad (Lemma 5). An applica ion o he Ma ko inequali y shows ha ,
i pis su icien ly small, he gian connec ed componen ha con ains only
good cubes con ains a ac ion o all la ge cubes ha is a bi a ily close o 1.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 603
(5) Using he ideas om Blume (1995b), we show ha i he cu a u e o he wo-
dimensional con agion wa e is su icien ly small ela i e o he cu a u e o an
indi idual neighbo hood, he con agion wa e will sp ead, as long as i s pa h con-
ains only good small cubes (Lemma 9).
Pu ing i oge he , he con agion wa e is going o sp ead h ough a as majo i y o
he gian connec ed componen o good la ge cubes, and hus a as majo i y o he
la ice. Hence, wi h a la ge p obabili y, he a e age ac ion in he la ges equilib ium on
a su icien ly la ge wo-dimensional la ice is close o x∗.
5.4 Key p ope ies o he ci y ne wo k
We summa ize he abo e discussion by iden i ying ou p ope ies o (M,m)-ci y ne -
wo k ha play key oles in he p oo .
(1) La ge numbe o connec ions mallows us o app oxima e he empi ical dis ibu-
ion o h esholds in an agen ’s neighbo hood by he model dis ibu ion P.This
app oxima ion o ms a basis o he con inuum model discussed in Sec ion 5.1.
(2) La ge ne wo k: The popula ion mus be su icien ly la ge o ensu e ha , o each
ac ion, wi h a high p obabili y, he e is a su icien ly la ge numbe o agen s o
whom his ac ion is s ic ly dominan . Such agen s s a he con agion a gumen
and hey play a simila ole as ini ial in ec o s in Lee and Valen inyi (2000)o Mo -
is (2000). In he ci y ne wo k, we equi e ha M
mis su icien ly la ge.
(3) Slow neighbo hood g ow h: Fo he con agion a gumen o Sec ion 5.1 o hold,
he size o neighbo hoods mus g ow su icien ly slowly (see Mo is (2000) o he
de ini ion and p ope ies).
(4) Pe cola ion p ope y: The con agion canno be obs uc ed by he obs acle phe-
nomenon desc ibed in Sec ion 5.2. Using he language in oduced abo e, he
good se o cubes mus con ain a la ge connec ed componen o he g aph.
I is no immedia ely ob ious how o o malize he las p ope y in a simple way. (A
nonsimple way is o assume ha he hesis o Lemma 4 om he Appendix mus hold.)
We lea e his ask o u u e esea ch.
6. Equilib ium selec ion
In his sec ion, we poin ou wo equilib ium selec ion heo ies ha selec uzzy con en-
ion x∗as he unique solu ion o andom u ili y coo dina ion games on ne wo ks.
6.1 E olu iona y s abili y
The p oo o Theo em 1shows ha uzzy con en ion x∗is, wi h a la ge p obabili y, a
global maximize o a po en ial unc ion o he coo dina ion game. Recall ha global
maximize s o he po en ial unc ion a e selec ed in comple e in o ma ion s a ic coo di-
na ion games by wo di e en equilib ium selec ion heo ies: obus ness o incomple e
604 Ma cin P˛eski Theo e ical Economics 20 (2025)
in o ma ion (Ui (2001)) and s ochas ic s abili y unde logis ic dynamics (Blume (1993),
Blume (2018)).
6.2 Robus beha io
Nex , we explain ha uzzy con en ion x∗is he only beha io ha is obus o incom-
ple e in o ma ion abou he ne wo k. The idea is pa allel o he de ini ion o obus ness
o incomple e in o ma ion om Kajii and Mo is (1997). We ake a pe spec i e o a e-
sea che /analys who obse es a la ge popula ion o agen s and a emp s o p edic in-
di idual beha io αi(τi)∈[0, 1],whe eαi(τ)is he p obabili y o playing ac ion 1, as a
unc ion o indi idual h esholds τi. The esea che unde s ands ha he agen s play a
coo dina ion game wi h hei neighbo s on some la ge and ine ne wo k and she unde -
s ands he pa ame e s o he game, bu she does no necessa ily unde s and he de ails
o he ne wo k opology. She would like he p edic ion o be obus o a misspeci ica ion
o he ne wo k.
De ini ion 1. A h eshold beha io (αi(.))iis obus o he misspeci ica ion o he ne -
wo k i and only i , o each η, he eexis sd>0, such ha o each ne wo k g,i d(g)<d,
wi h p obabili y a leas 1 −η(o e he ealiza ion o h esholds τi), he e exis s an equi-
lib ium ai∈{0, 1}o he ne wo k game G(g,τ)such ha
1
Ng
i≤Ngai−αi(τi)≤η.
To in e p e he de ini ion, no ice ha h eshold beha io (αi(.))iis ne wo k-
independen : each agen ’s ac ion depends on hei own h eshold and no o whom
hey a e connec ed and wha hei neighbo s a e doing. I he beha io is obus o
misspeci ica ion, i p esc ibes a bes esponse beha io o a g ea majo i y o agen s,
wha e e is he ue ne wo k o in e ac ions, and whe he he agen s know he ne wo k
o no . In o he wo ds, he beha io is app oxima ely an equilib ium on he ue ne wo k
ega dless o whe he he esea che o he agen s know he ue ne wo k.
Recall ha he 0- uzzy con en ion o x∗is a ne wo k-independen p o ile whe e
agen s play 1 i and only i hei h eshold is smalle han x,a∗(τi)=1(τi≤x∗).
Theo em 3. Suppose ha x∗is he s ic ly RU-dominan ou come. Then a h eshold
beha io αis obus o misspeci ica ion o he ne wo k i and only i i is he 0- uzzy con-
en ion o x∗.
The abo e esul shows ha playing a∗is he only p o ile ha is obus o misspeci-
ica ion o he ne wo k.
P oo . The “i ” di ec ion ollows om Theo em 1. The “only i ” di ec ion ollows om
Theo em 2.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 605
7. Conclusions
This pape p esen ed a heo y o beha io in andom u ili y bina y coo dina ion games
on la ge ne wo ks. We showed ha on some ne wo ks, wi h a la ge p obabili y, la ge
coo dina ion games ha e essen ially a unique equilib ium. Because his equilib ium
exhibi s mic o-, bu no mac o-le el he e ogenei y o beha io , we e e o i as a uzzy
con en ion. The a e age beha io in such a con en ion co esponds o a na u al ex en-
sion o isk-dominance om de e minis ic o andom-u ili y coo dina ion games. We
also showed ha , wi h a la ge p obabili y, all su icien ly ine ne wo ks (i.e., ne wo ks
whe e each agen has su icien ly many neighbo s), coo dina ion on he special uzzy
con en ion o RU-dominan ou come is always an app oxima e equilib ium, ega dless
o he ne wo k s uc u e.
The pape lea es many impo an ques ions unanswe ed. Fi s , how do he esul s
ex end o small-deg ee ne wo ks? Second, in eal applica ions, bo h mac o- and mic o-
le el he e ogenei y a e obse ed. Likely, he la e is due o sys ema ic di e ences in
p e e ences (pe haps di e ences in he h eshold dis ibu ions) ac oss di e en pa s o
he ne wo k. Can he wo idiosync a ic and sys ema ic di e ences be combined in a
single model? Thi d, and ela ed, can eal-wo ld da a be used o es ima e pa ame e s
o he model, like he h eshold dis ibu ion unc ion P(.)? We lea e hese ques ions o
u u e esea ch.
Appendix A: P oo o Theo em 1
A.1 P oo o Lemma 1
De ine a dis ance on he space o (mixed) p o iles: Fo any a,b∈[0, 1]N,le
d(a,b)=
1
g2
ig2
i(ai−bi)2.
Recall ha B={βa:ais ac ion p o ile}is he space o neighbo hood ac ions. Fo
each δ>0, le N(δ,B)be he co e ing numbe o B, ha is, he smalles ca dinali y no
a lis o p o iles b1,,bn∈Bsuch ha , o each b∈B, he eisl≤nso ha d(b,bl)≤δ.
Lemma 2. The e exis s a uni e sal cons an c<∞such ha , o each δ>0,andeach
ne wo k g,
N(δ,B)≤exp1
δ2cw∗2d(g)N.
P oo . We will use Sudako ’s mino a ion inequali y (Theo em 7.4.1 om Ve shynin
(2018)), which p o ides an uppe bound on he co e ing numbe ia he expec a ion o
a ce ain Gaussian p ocess. Fo his, le Zi o each agen ibe an i.i.d. s anda d no mal
andom a iable. Fo each (possibly mixed) p o ile a∈A, de ine
Xa=1
i
g2
i
i
giaiZi.
606 Ma cin P˛eski Theo e ical Economics 20 (2025)
Fo any wo p o iles a,b∈A,
E(Xa−Xb)2=
1
g2
i
E
i
gi(ai−bi)Zi2
=
1
g2
i
i
gi(ai−bi)2=d(a,b).
Gi en he de ini ion and he abo e p ope y, Sudako ’s mino a ion inequali y implies
ha , o some uni e sal cons an c1>0 (i.e., a cons an ha is independen o pa ame-
e s and he cu en p oblem),
logN(δ,B)≤c1Esup
b∈B
Xb2
δ2.
We compu e
Esup
b∈B
Xb=Esup
a∈A
Xβa=E⎛
⎜
⎜
⎜
⎜
⎝
sup
a∈A
1
i
g2
i
i
giZi1
gigij aj⎞
⎟
⎟
⎟
⎟
⎠
=1
i
g2
i
Esup
a∈A
i
ai
j
gij Zj≤1
i
g2
i
E
i
j
gij Zj
≤2
π
1
i
g2
i
i
j
g2
ij ,
whe e he las inequali y is due o a bound on he expec a ion o he absolu e alue o he
no mal a iable gij Zj ia i s s anda d de ia ion σi=jg2
ij . Because jg2
ij ≤d(g)g2
i
and (igi)2≤N2w∗2g2
min ≤Nw∗2g2
i,weha e
logN(δ,B)≤2
πc1
1
δ2
1
i
g2
i
id(g)gi2
d(g)≤1
δ22
πc1w∗2d(g)N.
We p oceed wi h he p oo o Lemma 1. Fo he i s inequali y, suppose is K-
Lipschi z. Fix ε>0andδ>0so ha δ=1
12K√w∗ε. Find δ-co e b1,,bno B. Because
n≤N(δ,B), Lemma 2implies ha
P obsup
l≤n
i
gi τi,bl
i−
i
giE ., bl
i≥1
2εgi
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 613
(a) Wcon ains a leas a ac ion (1−γ)o cubes, |W|≥(1−γ)|Gb|,
(b) Wis connec ed as a subse o he cube ne wo k,
(c) i c∈Gbis γ-bad, hen db(c,c)>3R o each c∈W(in pa icula , each cube in W
is γ-good), and
(d) Wcon ains a cube c0such ha each cube csuch ha d(c,c0)≤Ris ex ao dina y.
We show ha la ge good se s o cubes exis wi h high p obabili y.
Lemma 4. Fo each γ,ρ>0,andR<∞, he eexis mγ,ρ,R>0, and o each m>m
γ,ρ,R,
he e exis Mγ,ρ,R(m)such ha , i m≥mγ,ρ,Rand M≥Mγ,ρ,R(m) hen, i Gis an (M,m)-
la ice, b=ρm,andGbis he associa ed cube ne wo k, hen
P he e exis s (γ,R)-good se W⊆Gb≥1−γ.
B.2.1 In e media e esul s We need wo in e media e esul s. The i s esul p o ides
a bound on he size o he la ges connec ed componen o he g aph ob ained om he
ne wo k o cubes a e emo ing a g oup o smalle and connec ed se s o cubes.
Lemma 5. Suppose ha {S1,,SJ}is a collec ion o connec ed subse s o Gbsuch ha
Si∪Sja e no 2-connec ed o any i= j. Then he e is a connec ed subse V⊆Gb !Sj
such ha |Gb V|≤j|Sj|2.
P oo . Fi s , obse e ha o each connec ed se Ssuch ha |S|2<|Gb| he e is a se
Sand a loop (i.e., a pa h wi h he same beginning and ending) cS
0,,cS
n=c0o cubes
cS
l/∈Ssuch ha
•S⊇Sand |S|≤|S|2,and
•loop cS
0,,cS
n igh ly su ounds se Sand sepa a es i om he es o he g aph:
|{c:d(c,S)=1}|⊆{cS
l}⊆|{c:d(c,S)≤2}|.
This obse a ion ollows om he Jo dan cu e heo em and om he ac ha each
connec ed se Ssuch ha |S|2<|Gb|can be con ained in a |S2|-elemen “squa e” o
cubes such ha he se ou side he squa e is connec ed.
Fo each se Si om he hypo hesis o he lemma, ind loop ciand se S
ias in he
obse a ion abo e. We will show ha se Gb !S
jis connec ed, which will conclude
he p oo o he lemma. Take any wo cubes c,c∈Gb !S
j, and an a bi a y pa h c=
c0,,cn=cbe ween hem. We will modi y his pa h so ha i a oids each se Si.Fo
each i, ei he he exis ing pa h a oids se S
i, o i in e sec s i . Find li
0=min{l:d(cl,Si)=
1}and li
1=max{l:d(cl,Si)=1}. Then eplace he in e al cli
0,,cli
1o he pa h wi h
he pa h om cli
0 o cli
0along pa h ci. The new pa h a oids se S
i. Because he modi ied
pa o he pa h s ays wi hin 2-dis ance o se S
i, he modi ica ion does no c ea e new
in e sec ions wi h o he se s S
j. A e possibly modi ying he pa h o any i,weob aina
pa h be ween cand c ha a oids each se S
i.Thus,se Gb !S
jis connec ed.
614 Ma cin P˛eski Theo e ical Economics 20 (2025)
The second esul p o ides an uppe bound on he numbe o di e en -connec ed
se s o cubes.
Lemma 6. The numbe o -connec ed se s in Gbo ca dinali y nisno la ge han22n(2 +
1)n|Gb|.
P oo . We i s ind an encoding o each -connec ed uple. Le m be he size
o he -neighbo hood o an elemen o Gb.Thenm ≤(2 +1)2.Conside uples
(s1,(l2,,ln),(k2,,kn)) such ha s1∈Gb,ki∈{1, .., m },andli≤iand li≤lj o
each 2 ≤i≤j.
We show ha each -connec ed se can be encoded as one o he abo e uples in
such a way ha any wo di e en -connec ed se s mus ha e a di e en encoding. Le
e:Gb→{1, ,|Gb|}be an enume a ion o se Gb. Fo each s∈Gb,le es:{s:d(s,s)=
1}→{1, ,4
}be he enume a ion o he immedia e neighbo hood o s ha has he
same anking in he neighbo hood as enume a ion e. Choose s1=a gmins∈Se(s).Sup-
pose ha s1,si−1a e chosen o 1 <i<n. Fo each x∈S {s1,,si−1},le l(x)=
mind(x,sl)=1land le i equal ∞i he se is emp y. Then l(x)<i o a leas onex.Le
k(x)=esl(x)(x). Choose
si=a g min
lexicog apically,x∈Sl(x),k(x),
so as o minimize lexicog aphically (l(x),k(x)) among all x∈S {s1,,si−1}.Le li=
l(si)and ki=k(si).
We de i e an uppe bound on he numbe o encoding uples. Say ha a sequence
li,,lnis (i,m)-sequence i i is inc easing, lj<j o each j,andli=i−m−1. Le
S(i,m)deno e he numbe o di e en (i,m)-sequences. I is easy o see ha
S(i,m)=
m+1
p=0
S(i+1, p),
whe e S(n,m)=1. We check by induc ion on i ha S(i,n)≤22(n−i)+m.
The numbe o choices o s1is no la ge han |Gb|.By heabo e, henumbe o
(2, 0)-sequences is no la ge han 22(n−2). The numbe o choices o k2,,knis no
la ge han (2 +1)n−1. I ollows ha he o al numbe o encodings, and hence he
numbe o connec ed se s is no la ge han 22n(2 +1)n|Gb|.
B.2.2 P oo o Lemma 4Lemma 4 ollows om he ollowing wo esul s. The i s e-
sul es ablishes he exis ence o a la ge connec ed componen ha is a om bad cubes.
Le Bγ={c∈Gb:cis γ-bad}be he ( andom) se o γ-bad cubes.
Lemma 7. Fo each γ>0and R<∞, he eexis sbγ,R>0such ha i b>b
γ,R, hen
P∃W0⊆Gb,s .W0is connec ed, W0≥(1−γ)Gb,dbW0,Bγ≥5R≥1−1
4γ.
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 615
P oo .Le pγ>0 be he p obabili y ha a cube is γ-bad. Due o he D o e zky–
Kie e –Wol owi z–Massa inequali y, he p obabili y ha a cube cis γ-bad is bounded
by
pγ≤Ce−2b2γ2
o some uni e sal cons an C.
Le S0
1,,S0
nbe he smalles di ision o he se o bad cubes Bγ=!S0
iin o se s
ha a e 11R-connec ed and such ha S0
i∪S0
ja e no 11R-connec ed o i=j.Le X=
|S0
i|2. We compu e he expec ed alue o X.Le mn=(22n(11R+1)n|Gb|)be an uppe
bound on he ca dinali y o all 11R-connec ed se s (ob ained om Lemma 6). Then
EX≤
n≥1
n2mnpn
γ≤|Gb|
n≥1
2n22n(6R+1)npn
γ
=|Gb|8(11R+1)pγ
1−8(11R+1)pγ
.
Le S1
i⊇S0
ibe he smalles connec ed se such ha se s S1
i∪S1
ja e no 11R-
connec ed o i= jandsuch ha |S1
i|≤11R|S0
i|. Such se s can be cons uc ed by con-
nec ing elemen s o S0
iby a pa h inside he in e sec ion o he 11R-neighbo hood o he
wo se s.
Le Sibe he 5R-neighbo hood o se S1
i. Clea ly, se s Sia e disjoin (and sepa a ed
by R). Because each 5R-neighbo hood o an elemen o a se S1
ihas no mo e han (11R+
1)2|S1
i|cubes, he ca dinali y o Siis a mos (11R+1)2|S1
i|≤(11R+1)3|S0
i|.
Le W0be he la ges connec ed componen o Gb ha does no con ain elemen s o
se s Si. By cons uc ion, each se Siis connec ed, bu se s Si∪Sja e no 2-connec ed. By
Lemma 5, he ca dinali y o W0is a leas |Gb|−4(11R+1)6X. By Ma ko ’s inequali y,
PW0≥(1−γ)Gb≤P4(11R+1)6X≤γGb
≤4(11R+1)6EX
γGb≤1
γ
32(11R+1)7pγ
1−8(11R+1)pγ
.
Assume ha bγ,R>0 is la ge enough so ha o each b>b
γ,R,1
γ
32(11R+1)7Ce−2b2γ2
1−8(11R+1)Ce−2b2γ2≤
1
4γ.
Say ha cube c∈GRis an ex ao dina y cen e i all cubes in U(c,R)a e ex ao di-
na y.
Lemma 8. The e exis s Kγ,R<0la ge enough so ha i M
b>K
γ,R, hen
P"∃W⊆Gb,s .W⊇W0,Wis connec ed, db(W,Bγ)≥3R
and Wcon ains an ex ao dina y cen e #≥1−γ,
whe e W0inside he p obabili y sa is ies he condi ions om Lemma 7.
616 Ma cin P˛eski Theo e ical Economics 20 (2025)
P oo . Recall ha K=M
bis he numbe o cubes. I Kis di isible by (2R+1),we
can ind a g id o cubes GR⊆Gbsuch ha any wo c,c∈G,d(c,c)=2Rand Gb=
!c∈GRU(c,R). Because he U(c,R)neighbo hoods a e disjoin , |Gb|=|GR|(2R+1)2,
whe e (2R+1)2is he size o each neighbo hood. Fo simplici y, he es o he a gu-
men s ely on he di isibili y assump ion. The a gumen is easily modi ied o he case
when he di isibili y does no hold (and band M
ba e su icien ly la ge).
Le W0be he ( andom) se om Lemma 7.Le W1=!cU(c,R+1)and W=
!cU(c,2R+1).Thend(W,Bγ)>2R. Because o each c∈U(c, ) he e is a pa h be-
ween cand c ha is inside se U(c, ),Wis connec ed.
We show ha |GR∩W1|≥(1−γ)|GR|. On he con a y, suppose ha |GR W1|>
γ|GR|.ThenA=!c∈GR WU(c,R)⊆Gb W0.Mo eo e ,|A|>γ|GR|(2R+1)2=γ|Gb|.
Howe e , his con adic s |Gb W0|≤γ|Gb|.
Le q>0 be he p obabili y ha a cube cis an ex ao dina y cen e . Then q≥
P(0)(2R+1)2b2.Le q∗be he p obabili y ha cube cis an ex ao dina y cen e , condi-
ional on c∈W1. Because being in c∈W1p o ides no o he in o ma ion abou he
dis ibu ion o as e shocks apa om cis no γ-bad and γ-bad cubes a e no ex ao di-
na y,i mus be ha q∗≥q. Simila ly, condi ional on c,c∈W1,i cand ca e sepa a ed
by 2R+1, he e en s ha he wo a e ex ao dina y cen e s a e independen . Hence, he
p obabili y ha none o he cubes in c∈GR∩W1is an ex ao dina y cen e is a mos
1−q∗|GR∩W1|≤1−P(0)(2R+1)2b2(1−γ)K2(2R+1)−2
≤e−(1−γ)Kγ,R(2R+1)−2P(0)(2R+1)2b2
.
I Kis su icien ly la ge, he abo e is smalle han 1
4γ.
To conclude he p oo o he lemma, we se mγ,ρ,R>1
ρbγ,Rand hen Mγ,ρ,R(m)≥
ρmKγ,R.
B.3 P oo o Theo em 2
Below, we will show he ollowing lemma.
Lemma 9. Fo each ε>0, he e exis s su icien ly small γ,ρ>0, and su icien ly la ge
R>0so ha i b=ρm,Wis a (γ,R)-good se in he ne wo k o cubes Gb,andais an
equilib ium p o ile, hen o each i∈c∈W,βa
i≤x∗+ε.
Toge he wi h Lemma 4, Lemma 9shows ha o each ε>0, i mand M
ma e su -
icien ly la ge, wi h p obabili y o a leas 1 −ε,i ais an equilib ium p o ile, hen
βa
i≤x∗+ε o all agen s ibu a ε- ac ion o he popula ion (i.e., all membe s o he
“good” se W).
A simila a gumen shows ha βa
i≥x∗−ε o elemen s o an analogously de ined
“good” se (wi h he app op ia e modi ica ion o wha good and ex ao dina y cubes
a e). Toge he , he wo a gumen s show ha , wi h p obabili y o a leas 1 −2ε, o each
agen in he good se , he agen ’s a e age neighbo hood beha io is wi hin εo x∗.All
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 617
such agen s, i hey ha e a h eshold ou side in e al [x∗−ε,x∗+ε], will choose he bes
esponse as in 0- uzzy con en ion x∗p o ile ax∗.
Finally, choose εsmall enough so ha P(x∗+ε)−P(x∗−ε)≤1
4η. Then, i he
ne wo k is su icien ly la ge, he p obabili y ha he ac ion o agen s wi h h eshold
τi∈[x∗−ε,x∗+ε]is la ge han 1
2ηis smalle han ε.
Take ε=1
4η. Then, wi h a p obabili y o a leas 1 −η,a mos η
2agen s ha e h esh-
olds in he ε−in e al, and a mos 2ε=η
2agen s obse e equilib ium neighbo hood
beha io ha is ou side he ε−in e al. All he o he agen s choose he same beha io
as in p o ile ax∗.
P oo . We di ide he p oo o he lemma in o wo s eps.
P epa a ion. Find ε0>0, such ha
σ∗=max
a≥x∗+ε
2
a
x∗+ε0P−1(y)−ydy > 0.
The exis ence o such ε0∈(0, ε
2)comes om he de ini ion o x∗as he unique maxi-
mize o a
x∗(y−P−1(y))dy.Le δρbe a ac ion o neighbo s o iwho a e no membe s
o a cube ha is ully con ained in he neighbo hood o i. I is easy o see ha δρ→0as
ρ→0.
Le abe an equilib ium p o ile. Fo each cube c, de ine
ac=1
|c|
j∈c
ajand βc=1
|c|
j∈c
βa
j.
Then |βc−βa
i|≤δρ,and
βc≤δρ+|c|
B(i,1
)
c⊆B(i,1)
ac. (16)
I cube cis γ-good, hen
ac=1
|c|
i∈c
1τi<β
a
i≤1
|c|
i∈c
1{τi<β
c+δρ}≤P(βc+δρ)+γ. (17)
F om now on, assume ha W⊆Gbis (γ,R)-good. I db(c,W)≤3R, hencubecis
γ-good.
De ine
C0=c:∀cdc,c≤R=⇒ ac≤x∗+ε0.
Fo each i∈C0, he a e age beha io in all he cubes ully con ained in he neighbo -
hood o iis ≤x∗+ε0, which, oge he wi h (16), implies ha
βa
i≤x∗+ε0(1−δρ)+δρ≤x∗+ε.
The las inequali y holds when ρis su icien ly small so ha δρ≤ε
2. Hence, o es ablish
ou claim, i is enough o show ha W⊆C0.
618 Ma cin P˛eski Theo e ical Economics 20 (2025)
No ice ha C0canno be emp y as i con ains a leas one ex ao dina y cube. Fo
each a>x
∗+ε
2, de ine
d(a)=min
c∈W:ac≥adb(c,C0)≥R,
whe e he alue is ∞i he se o e which he dis ance is minimized is emp y.
On he con a y o ou claim, suppose ha he e is a cube c∈W0such ha ac>a>
x∗+ε
2. Then he e exis s a>x
∗+ε
2such ha d(a)<∞. Find a∗≥x∗+ε0such ha
d(a∗)≤2Rand d(a∗+1
R)≥d(a∗)+1. Such a∗exis s: o he wise, i o each asuch ha
d(a)≤2R,d(a+1
R)≤d(a)+1, hen d(a+1)≤2R, which is impossible (as he e is no
cube wi h he ac ion a e age s ic ly la ge han 1).
Con agion wa e.No ice ha ac akes disc e e alues a∈A={0, 1
|c|,,1
},whe e|c|
is he size o a cube. Le ak=k
|c|be he enume a ion o se A∩{a:a≥x∗+ε
2}. Fo each
such cube c, and each i∈c,(16)implies
βc≤δρ+|c|
B(i,1
)
c⊆B(i,1)
ac
≤δρ+
a∈A
ac⊆B(i,1
):ac=a
B(i,1
)/c
≤δρ+x∗+ε0+
k
(ak+1−ak)c⊆B(i,1
):ac≥a
B(i,1
)/c
≤δρ+δR,ρ+x∗+ε0+
k
(ak+1−ak)1− d(ak)−db(c,C0),
whe e he hi d inequali y is a consequence o a disc e e e sion o he in eg a ion by
pa s (i.e., xi(yi−yi+1)=(xi+1−xi)yi+1), and he ou h one is due o Lemma 3,
whe e δR,ρ→0asRis su icien ly la ge and ρis su icien ly small. Le δ1
R,ρ=δρ+δR,ρ.
Addi ionally, o each al∈A,al≤a∗, ind a cube csuch ha db(c,C0)=dR(al)<2R
and ac≥al. Using he abo e inequali y and (17), we ob ain
P−1(al−γ)≤P−1(ac−γ)≤βc+δρ
≤δ1
R,ρ+x∗+ε0+
k
(ak+1−ak)1− d(ak)−d(al).
Le k∗=max{k:ak≤a∗}. Then he igh -hand side is no la ge han
≤δ1
R,ρ+x∗+ε0+
k≤k∗
(ak+1−ak)1− d(ak)−d(al)
+
k>k∗:ak≤a∗+ε
10
(ak+1−ak)1− d(ak)−d(al)
+
k:ak>a∗+ε
10
(ak+1−ak)1− d(ak)−d(al)
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 619
≤δ1
R,ρ+x∗+ε0+1
R+
k≤k∗
(ak+1−ak)1− d(ak)−d(al),
due o he second e m in he i s line being no la ge han ε
10 , and he hi d e m being
equal o 0 (as (d(ak)−d(al)) ≥ (1)=1).
Le =a∗−(x∗+ε0). Mul iplying by (al+1−al)and summing ac oss l≤k∗,we
ob ain
l≤K∗
P−1(al−γ)(al+1−al)
≤δ1
R,ρ+1
R+x∗+
l≤K∗
k≤K∗
(ak+1−ak)(al+1−al)1− dR(al)−dR(ak)
=δ1
R,ρ+1
R+x∗+1
2
l,k≤K∗
(ak+1−ak)(al+1−al)
=δ1
R,ρ+1
R+x∗+ε0+1
22
≤δ1
R,ρ+1
R+
a∗
x∗+ε0
ydy.
To ob ain he equali y, we use he ac ha is balanced.
Because P−1(.−γ)∈[0, 1]and al+1−al=1
|c|, he le -hand side o he abo e inequal-
i y is smalle han
a∗
x∗+ε0
P−1y−γ−1
|c|dy ≥
a∗−γ−1
|c|
x∗+ε0−γ−1
|c|
P−1(y)dy.
Assuming ha bis la ge enough so ha 1
|c|≤γ, he abo e is no smalle han
a∗
x∗+ε0(P−1(y)−y)dy −2γ. Pu ing i back in o he main inequali y, we ob ain
a∗
x∗+ε0P−1(y)−ydy ≤δ1
R,ρ+1
R+2γ.
I γ,ρ>0 a e su icien ly small and Rsu icien ly la ge, δ1
R,ρ+1
R+2γ<σ
∗.Thecon a-
dic ion shows ha W⊆C0, which concludes he p oo o he lemma.
Appendix C: P oo o Theo em 3
Fo each η>0, de ine Pη=P(x:|x−x∗|≤η)as he p obabili y ha he h eshold eal-
iza ion is wi hin ηo x∗.I Pdoes no ha e an a om a x∗, hen we can choose ηδsuch
620 Ma cin P˛eski Theo e ical Economics 20 (2025)
ha Pηδ≤1
30 δ. Assume w.l.o.g. ha ηδ≤δ.Le
Tδ=$τ:1
Nτi:τi−x∗≤ηδ≤1
3δ%.
The law o la ge numbe s implies ha o su icien ly high N,P ob(Tδ)≥1−δ.
Fix h eshold p o ile τ∈Tδ.Le I0={i:|τi−x∗|≤ηδ}. Suppose ha ais 1
3ηδ- uzzy
con en ion x∗.Le I(g)={i:|βa
i−x∗|>1
3ηδ}be he se o agen s ha is an equilib ium
in game G(g,τ).Le I=I0∩I(g).Then 1
N|I|≤2
3δ. Fo each i/∈I,ei he
•τi>x
∗+ηδand βa
i≤x∗+1
3ηδ, which implies ai=a∗
i=0, o
–τi<x
∗−ηδand βa
i≥x∗−1
3ηδ, which implies ai=a∗
i=1.
Hence, o any i/∈I,ai=a∗
i. This concludes he p oo o he heo em.
Re e ences
Blume, Law ence E. (1993), “The s a is ical mechanics o s a egic in e ac ion.” In Games
and Economic Beha io , olume 5, 387–424, Else ie . [0587,0591,0604]
Blume, Law ence E. (1995a), “The s a is ical-mechanics o bes - esponse s a egy e i-
sion.” Games and Economic Beha io , 11, 111–145. [0586]
Blume, Law ence E. (1995b), “The s a is ical mechanics o bes - esponse s a egy e i-
sion.” In Games and Economic Beha io , olume 11, 111–145, Else ie . [0587,0588,0600,
0603]
Blume, Law ence E. (2018), “Popula ion games.” In The Economy as an E ol ing Com-
plex Sys em II, 425–460, CRC P ess. [0604]
Bollobás, Béla and Oli e Rio dan (2006), Pe cola ion. Camb idge Uni e si y P ess.
[0602]
Ellison, Glenn (1993), “Lea ning, local in e ac ion, and coo dina ion.” Econome ica:
Jou nal o he Econome ic Socie y, 1047–1071. [0583,0586,0587,0588,0591,0598]
Ellison, Glenn (2000), “Basins o a ac ion, long- un s ochas ic s abili y, and he speed
o s ep-by-s ep e olu ion.” In The Re iew o Economic S udies, olume 67, 17–45, Wiley-
Blackwell. [0587]
E d˝
os, Paul and Al éd Rényi (1959), “On andom g aphs. I.” Publica iones Ma hema i-
cae, 6, 290–297. [0589]
F iedman, Daniel (1991), “E olu iona y games in economics.” Econome ica: Jou nal o
he Econome ic Socie y, 637–666. [0587]
Galeo i, And ea, Sanjee Goyal, Ma hew O. Jackson, Fe nando Vega-Redondo, and
Leea Ya i (2010), “Ne wo k games.” In The Re iew o Economic S udies, olume 77,
218–244, Wiley-Blackwell. [0587,0591]
G ano e e , Ma k (1978), “Th eshold models o collec i e beha io .” Ame ican Jou nal
o Sociology, 83, 1420–1443. [0586,0588,0590,0591,0593]
Theo e ical Economics 20 (2025) Random u ili y coo dina ion games 621
Ha sanyi, John C. and Reinha d Sel en (1988), “A gene al heo y o equilib ium selec ion
in games.” In MIT P ess Books, olume1.TheMITP ess.[0585,0588,0592]
Jackson, Ma hew O. (2010), Social and Economic Ne wo ks. P ince on Uni e si y P ess,
P ince on, NJ. [0594]
Jackson, Ma hew O. and Leea Ya i (2007), “Di usion o beha io and equilib ium
p ope ies in ne wo k games.” Ame ican Economic Re iew, 97, 92–98. [0586,0587,0591]
Kajii, A sushi and S ephen Mo is (1997), “The obus ness o equilib ia o incomple e
in o ma ion.” Econome ica, 65, 1283–1309. [0586,0604]
Kando i, Michihi o, Geo ge J. Maila h, and Ra ael Rob (1993), “Lea ning, mu a ion, and
long un equilib ia in games.” Econome ica: Jou nal o he Econome ic Socie y, 29–56.
[0587]
Lee, In Ho and Akos Valen inyi (2000), “Noisy con agion wi hou mu a ion.” In The Re-
iew o Economic S udies, olume 67, 47–56, Wiley-Blackwell. [0586,0587,0588,0591,
0598,0600,0603]
Leis e , C. Ma hew, Y es Zenou, and Junjie Zhou (2022), “Social connec edness and lo-
cal con agion.” In The Re iew o Economic S udies, olume 89, 372–410, Ox o d Uni e -
si y P ess. [0588]
López-Pin ado, Dunia (2008), “Di usion in complex social ne wo ks.” In Games and
Economic Beha io , olume 62, 573–590, Else ie . [0586]
Monde e , Do and Lloyd S. Shapley (1996), “Po en ial games.” Games and Economic
Beha io , 14, 124–143. [0585,0594]
Mo is, S ephen (2000), “Con agion.” In The Re iew o Economic S udies, olume 67,
57–78, Wiley-Blackwell. [0583,0586,0587,0588,0598,0600,0603]
Mo is, S ephen and Hyun Song Shin (2006), “He e ogenei y and uniqueness in in e -
ac ion.” In The Economy as an E ol ing Complex Sys em, III: Cu en Pe spec i es and
Fu u e Di ec ions, olume 3, 207, Ox o d Uni e si y P ess, London. [0585]
Mossel, Elchanan, Allan Sly, and Ome Tamuz (2015), “S a egic lea ning and he opol-
ogy o social ne wo ks.” Econome ica, 83, 1755–1794. [0594]
Nea y, Philip R. (2012), “Compe ing con en ions.” Games and Economic Beha io , 76,
301–328. [0587]
Nea y, Philip Ruane and Jona han New on (2017), “He e ogenei y in p e e ences and
beha io in h eshold models.” A ailable a SSRN 3035289. [0587]
New on, Jona han (2021), “Con en ions unde he e ogeneous beha iou al ules.” The
Re iew o Economic S udies, 88, 2094–2118. [0587]
Peski, Ma cin (2010), “Gene alized isk-dominance and asymme ic dynamics.” In Jou -
nal o Economic Theo y, olume 145, 216–248, Else ie . [0588,0591]
622 Ma cin P˛eski Theo e ical Economics 20 (2025)
Ui, Takashi (2001), “Robus equilib ia o po en ial games.” Econome ica, 69, 1373–1380.
[0604]
Ve shynin, Roman (2018), High-Dimensional P obabili y: An In oduc ion Wi h Appli-
ca ions in Da a Science. Camb idge Se ies in S a is ical and P obabilis ic Ma hema ics.
Camb idge Uni e si y P ess, Camb idge. [0605,0607]
Wa s, Duncan J. (2002), “A simple model o global cascades on andom ne wo ks.” In
P oceedings o he Na ional Academy o Sciences, olume 99, 5766–5771, Na ional Acad
Sciences. [0586]
Young, H. Pey on (1993), “The e olu ion o con en ions.” Econome ica: Jou nal o he
Econome ic Socie y, 57–84. [0583,0587]
Co-edi o Simon Boa d handled his manusc ip .
Manusc ip ecei ed 29 Ma ch, 2023; inal e sion accep ed 30 Sep embe , 2024; a ailable on-
line 1 Oc obe , 2024.
