Network effects on information acquisition by DeGroot updaters

Risco, Miguel
A icle — Published Ve sion
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo
upda e s
Economic Theo y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Risco, Miguel (2024) : Ne wo k e ec s on in o ma ion acquisi ion by DeG oo
upda e s, Economic Theo y, ISSN 1432-0479, Sp inge , Be lin, Heidelbe g, Vol. 79, Iss. 1, pp.
201-234,
h ps://doi.o g/10.1007/s00199-024-01568-7
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Economic Theo y (2025) 79:201–234
h ps://doi.o g/10.1007/s00199-024-01568-7
RESEARCH ARTICLE
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo
upda e s
Miguel Risco1
Recei ed: 27 May 2023 / Accep ed: 9 Ma ch 2024 / Published online: 4 Ap il 2024
© The Au ho (s) 2024
Abs ac
In oday’s wo ld, social ne wo ks ha e a signi ican impac on in o ma ion p ocesses,
shaping indi iduals’ belie s and in luencing hei decisions. This pape p oposes a
model o unde s and how boundedly a ional (DeG oo ) indi iduals beha e when
seeking in o ma ion o make decisions in si ua ions whe e bo h social communica ion
and p i a e lea ning ake place. The model assumes ha in o ma ion is a local public
good, and indi iduals mus decide how much e o o in es in cos ly in o ma ion
sou ces o imp o e hei knowledge o he s a e o he wo ld. Depending on he ne -
wo k s uc u e and agen s’ posi ions, some indi iduals will in es in p i a e lea ning,
while o he s will ee- ide on he social supply o in o ma ion. The model shows ha
mul iple equilib ia can a ise, and uniqueness is con olled by he lowes eigen alue
o a ma ix de e mined by he ne wo k. The lowes eigen alue oughly cap u es how
wo-sided a ne wo k is. Two-sided ne wo ks ea u e mul iple equilib ia. Unde a u ili-
a ian pe spec i e, agen s would be mo e in o med han hey a e in equilib ium. Social
wel a e would be imp o ed i in luen ial agen s inc eased hei in o ma ion acquisi ion
le els.
Keywo ds In o ma ion acquisi ion ·Lea ning ·Public goods ·Ne wo k e ec s ·
In o ma ion di usion ·Bounded a ionali y
JEL Classi ica ion D61 ·D83 ·D85 ·H41
Suppo by he Ge man Resea ch Founda ion (DFG) h ough CRC TR 224 (P ojec B05) and unding
om he Eu opean Resea ch Council (ERC) unde he Eu opean Union’s Ho izon 2020 esea ch and
inno a ion p og am (G an ag eemen 949465) a e g a e ully acknowledged. I hank S en Rady, F ancesc
Dilmé, Alexande F ug, Alexande Win e , Simon Block, Jus us P eusse , Axel Niemeye , an anonymous
e e ee and pa icipan s a he semina s a Uni e si y o Bonn and Uni e si a Pompeu Fab a o hei
help ul commen s and discussions.
BMiguel Risco
[email p o ec ed]
1Bonn G adua e School o Economics, Uni e si y o Bonn, Bonn, Ge many
123
202 M. Risco
1 In oduc ion
In o ma ion is key o making decisions. Nowadays, social ne wo ks ha e a signi ican
impac on in o ma ion p ocesses. We discuss a ious issues wi h amily, iends, and
colleagues, a ec ing hei opinions and shaping ou own. Random con e sa ions abou
an upcoming elec ion, he job ma ke , o s ock ma ke pe o mances can in luence ou
belie s. B eak h ough news sp eads apidly, and indi iduals a e cons an ly upda ing
hei opinions. Apa om his social supply o in o ma ion, indi iduals can lea n
p i a ely, such as by sea ching on he in e ne o consul ing a book. The e o e, i is
essen ial o unde s and how indi iduals beha e when hey seek o acqui e in o ma ion
o make decisions in si ua ions whe e bo h social communica ion and p i a e lea ning
ake place. To wha ex en do people exe e o hemsel es, and o wha ex en do
hey ely on o he s?
In his pape , we p opose a model o in o ma ion acquisi ion in ne wo ks in which
indi iduals a e boundedly a ional, beha ing as mechanical upda e s when i comes
o lea ning. Wi h his in hand, hey decide how much o in es in a cos ly in o ma ion
sou ce o imp o e hei knowledge o he s a e o he wo ld. Mechanical upda ing
he e consis s o agen s me ely aking weigh ed a e ages o he signals ecei ed— he
so-called DeG oo upda ing ule om DeG oo (1974).
Despi e conside ing boundedly- a ional agen s, we s ill apply he concep o Nash
equilib ium a he s age whe e hey de e mine hei in o ma ion acquisi ion. This is
done in he spi i o an e olu iona y concep o Nash equilib ium. An e olu iona y
model consis s o a la ge popula ion o boundedly- a ional playe s playing some game
epea edly o e ime (Maila h 1998). E olu iona y heo y shows ha such playe s
e en ually lea n o play Nash equilib ium,1e en in he absence o pe ec a ionali y.
The c ucial assump ion is ha mo e success ul beha io s become mo e p e alen due
o a combina ion o imi a ion and he ailu e o unsuccess ul beha io s.2In ou model,
boundedly- a ional agen s ha upda e mechanically ace he p oblem o p o ision o a
local public good. The ask o ga he ing in o ma ion o subsequen decision-making
ecu s nume ous imes h oughou an indi idual’s li e. In he spi i o e olu iona y
heo y, we hink o boundedly- a ional agen s who, despi e hei cogni i e cons ain s,
ha e lea ned o each Nash equilib ium ou comes h ough hei choices.
To p o ide an in ui ion o he o mal model, conside an agen who wishes o
become mo e in o med abou a pa icula issue. We assume ha she has some p io
knowledge, and ha in o ma i e con e sa ions ake place in he neighbo hood— o
example, a he o ice. The e, each colleague exe s a di e en and ixed in luence ha
shapes he agen ’s inal opinion. An icipa ing his, she decides how much e o o
spend on p i a e lea ning. Gi en ha lea ning ools a e simila among neighbo s, we
assume a posi i e co ela ion when i comes o p i a e lea ning signals. Hence, each
1In pa icula , he cen al no ion in e olu iona y game heo y is ha o e olu iona y s able s a egy and
heo y shows ha any symme ic s ic Nash equilib ium is indeed an e olu iona y s able s a egy. See
Maila h (1998) o Samuelson (2002) o an o e iew.
2This is also discussed by Aumann (1997), asse ing ha o dina y people, in hei daily ac i i ies, do
no consciously adhe e o a ionali y bu e ol e “ ules o humb” h ough e olu ion. I hese ules p o e
e ec i e, hey p oli e a e and mul iply, e en ually eaching he equilib ium ha s ic a ionali y would
ha e p edic ed.
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 203
agen aces a p oblem o in o ma ion acquisi ion in which in o ma ion is a local public
good. Indi iduals ha e o decide how much o in es in p i a e lea ning, knowing ha
ee social lea ning will ake place la e . Depending on he subs i u abili y be ween
in o ma ion acqui ed pe sonally and in o ma ion acqui ed by o he s, bu also on he
neighbo s’ choices, agen s will aise o lowe hei lea ning e o s. F ee- ide beha io s
will a ise.3
This pape p o ides h ee main con ibu ions. Fi s , we analyze and cha ac e ize he
equilib ia a ising in he model. Depending on he ne wo k s uc u e and hei posi ions,
agen s will con ibu e wi h some lea ning o comple ely ee- ide. In p inciple, he e
a e mul iple equilib ia, and all o hem can be calcula ed. A su icien condi ion o
equilib ium uniqueness is ou second con ibu ion. I his condi ion does no hold, he
equilib ia compu a ions un in exponen ial ime. Equilib ium uniqueness is con olled
by he lowes eigen alue o a ma ix gi en by he ne wo k. This eigen alue essen ially
cap u es how wo-sided he co esponding ne wo k is, ha is, whe he agen s can be
di ided in o wo se s wi h many links be ween hem bu jus a ew wi hin. In a game o
s a egic subs i u abili ies, when an agen inc eases he e o , he neighbo s dec ease
hei s in esponse, so ha he neighbo s’ neighbo s ha e o inc ease, and so on. When
he ne wo k is wo-sided, hese di ec e ec s accumula e and lead o se e al dis inc
equilib ia. Howe e , i he lowes eigen alue is su icien ly la ge, he ne wo k will no
be wo-sided enough o he ac ions o ebound, and he equilib ium will be unique.
Finally, we p o ide a wel a e analysis. F om a social (u ili a ian) pe spec i e, e e y
agen would be mo e in o med han she is in equilib ium. To sa is y his demand, a
leas he in luen ial agen s ( hose agen s om which he o he s ge he majo i y o
in o ma ion) ha e o inc ease hei con ibu ion. I he ne wo k is oo unbalanced,
his becomes a bu den and he wel a e o he in luen ials dec eases. In gene al, he
u ili a ian op imum does no Pa e o-domina e he equilib ium ou come.
The choice o he upda ing ule, i.e., how indi iduals p ocess and inco po a e he
in o ma ion ecei ed, is a ele an decision when ying o model social lea ning.
One has o decide whe he o employ he ully a ional Bayesian ocus o he nai e,
boundedly- a ional app oach, mainly ep esen ed by he al eady men ioned DeG oo
ule. Quo ing Acemoglu and Ozdagla (2011), “which ype o app oach is app op ia e
is likely o depend on he speci ic ques ion being in es iga ed”. We a gue he e ha he
DeG oo upda ing ule i s bes wi hin ou con ex .
Bayesian upda ing equi es an un ealis ic cogni i e demand o lea ning in la ge
ne wo ks. Howe e , he DeG oo ule p o ides a con enien al e na i e, gi en i s sim-
plici y and lack o es ic i e equi emen s. In a simul aneous se ing, Bayesian agen s
who ecei e Gaussian p i a e signals beha e like DeG oo upda e s when subjec o
pe suasion bias, as shown by DeMa zo e al. (2003). In ac , i he game is one-sho
(as i is in his pape ), pe suasion bias is no e en necessa y o such a esul o hold.
S ill, hei model de ia es sligh ly om s anda d a ional assump ions, as neighbo s’
signal p ecision is unknown. The ela ionship be ween Bayesian and DeG oo ules,
especially o one-sho games, suppo s ou model and is u he analyzed in he
“Appendix”. None heless, a e he i s pe iod, a pu e Bayesian (no su e ing om
pe suasion bias) would adjus o he in o ma ion bu ied in he ne wo k, while DeG-
3Fo an axioma ic cha ac e iza ion o cos ly in o ma ion acquisi ion p ocesses, see Du aj and Lin (2022).
123
204 M. Risco
oo ian agen s would no .4The li e a u e on ne wo ks has widely used he DeG oo
ule in di e en se ings. Golub and Jackson (2010) show ha unde some mild con-
di ions on connec edness and in luence, DeG oo agen s con e ge o he belie ha
would esul om he ull agg ega ion o e e yone’s signal. Bo h Golub and Jack-
son (2012), de o ed o s udy homophily, and Acemoglu e al. (2010), which analyzes
he ension be ween he sp ead o misin o ma ion and in o ma ion agg ega ion, also
e lec how con enien DeG oo upda ing is o la ge ne wo ks analysis. Howe e , he
majo d awback o he ule is ha he choice o weigh s migh seem a bi a y, pa icu-
la ly when communica ion las s longe han one pe iod. Fu he mo e, he assump ion
ha e e yone is in o med a he ou se may be oo demanding. Bane jee e al. (2021)
adap ed he ule o spa se signals o add ess his issue.
This ha ing been said, he empi ical e idence hea ily suppo s DeG oo upda -
ing. Va ious pape s con on i agains Bayesian lea ning in an expe imen al se ing,
concluding ha i app oxima es be e people’s in o ma ion agg ega ion ules (see
Co azzini e al. 2012; Muelle -F ank and Ne i 2013; G imm and Mengel 2020; Chan-
d asekha e al. 2020). Al hough he e is no de ini i e app oach, many ecen pape s
end o use a boundedly a ional model o bo h sequen ial and simul aneous se -
ings. Fo example, Dasa a ha and He (2020) assume ha agen s neglec edundancies
o in o ma ion and hen agg ega e heu is ically, and Muelle -F ank and Ne i (2021)
conside a la ge class o boundedly a ional o quasi-Bayesian ules, espec i ely.
Al hough modelling lea ning h ough a mechanical upda ing ule is o e ly sim-
plis ic, i allows us o isola e he ne wo k e ec s, which is he p ima y conce n o
his pape . Fu he mo e, we a gue ha assuming exogenous and ixed weigh s e lec s
human beha io . The in luence ha ou neighbo s exe on a conc e e opic is almos
p ede e mined. A wide ange o ac o s such as pas in e ac ions, us wo hiness, and
expec ed le el o knowledge de ines an in luence le el be o e communica ion occu s.
Simila ly, i seems sensible ha agen s can endogenously se he in luence o hei
own p i a e lea ning on hei iews: he mo e ime de o ed o esea ching, he mo e
eliable he agen pe cei es i o be. Thus, he expendi u e o cos ly a en ion will
educe playe -speci ic noise.
Galeo i and Goyal (2010) is a key pape in he li e a u e on in o ma ion acquisi-
ion in ne wo ks. In his pape , ne wo k-placed agen s s a egically selec hei links
o access he in o ma ion held by hei neighbo s. E e y equilib ium displays he so-
called “law o he ew”: he majo i y o indi iduals end o ge mos o hei in o ma ion
om a iny subse o he g oup, he in luen ials. Ou model shows ha his esul holds
ue o ne wo ks whe e a subse o agen s, he popula s, has a signi ican ly highe
weigh han he es , such as he co e-pe iphe y ne wo k. In such ne wo ks, popu-
la agen s become in luen ial and acqui e mos o he in o ma ion, while he o he s
ee- ide. This inding con as s wi h Bane jee e al. (2021), whe e he spa se-signals
s uc u e indica es ha being popula alone is insu icien o being in luen ial. How-
e e , wo assump ions in Galeo i and Goyal (2010) di e om ou model: links
a e endogenous, allowing an agen o each any o he indi idual in a po en ially la ge
ne wo k, and homogeneous, meaning pe ec subs i u abili y. Ne wo k e ec s on in o -
ma ion acquisi ion ha e also been analyzed om a Bayesian pe spec i e. In Mya and
4SeeMola ie al.(2018) o an axioma ic ounda ion o he DeG oo ule unde impe ec ecall.
123

Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 205
Wallace (2019), a ional agen s acqui e in o ma ion abou he s a e o he wo ld om
sou ces ha p o ide noisy signals. Paying cos ly a en ion educes noise, and signals
a e possibly co ela ed ac oss playe s, simila o ou model. Howe e , incen i es di e
as agen s no only wan o ma ch he s a e o he wo ld bu also ca e abou coo dina ion
asymme ically. Fu he mo e, he e is no communica ion s age. The playe ’s cen al-
i y (in he sense o Bonacich) and co ela ions de e mine in o ma ion acquisi ion, bu
cen ali y en ails less expendi u e, in con as o Galeo i and Goyal (2010) and ou
pape . Finally, Den i (2017) in oduces he concep o en opy educ ion o s udy how
playe s endogenously acqui e cos ly in o ma ion o dec ease hei unce ain y abou
undamen als. Ne wo k e ec s induce ex e nali ies in in o ma ion acquisi ion and a e
a sou ce o mul iple equilib ia.
Rega ding equilib ium analysis, ou wo k closely ollows ha o B amoullé e al.
(2014). Following p e ious a emp s in he li e a u e o cha ac e ize condi ions o
equilib ium in linea games o s a egic complemen s (c . Balles e e al. 2006) and
s a egic subs i u es (especially in public goods, c . B amoullé e al. 2007), he au ho s
showed ha equilib ium uniqueness and s abili y depend on he lowes eigen alue o
he ne wo k ma ix.5This is dependen on he wo-sidedness o he ne wo k. Al hough
ou pape di e s in se ing and mo i a ion, he bes esponse unc ion de i ed om ou
model is simila o ha o B amoullé e al. (2014). Consequen ly, he esul ega ding
he lowes eigen alue cha ac e izing equilib ium uniqueness is also simila . Howe e ,
hei model assumes ha agen s’ con ibu ions a e ecip ocal and weigh ed equally,
which di e s om ou assump ions. This has wo consequences. Fi s , he po en ial
heo y in oduced in Monde e and Shapley (1996), on which B amoullé e al. base
hei esul s, canno be applied he e; second, a wide ange o ne wo ks can be analyzed.
Ne e heless, i we es ic ou se ing o symme ic, homogeneous ne wo ks, an almos
equi alen condi ion a ises. Finally, ou pape is also ela ed o B amoullé e al. (2007)
model o pu e public goods in exogenous ne wo ks, whe e again all con ibu ions a e
weigh ed equally and he e is pe ec subs i u abili y. In ha model, he au ho s ind ha
mul iple equilib ia ypically exis , and he e is always one in which some indi iduals
con ibu e while o he s ee- ide. This equilib ium is ypically unique.
The es o he pape is o ganized as ollows. Sec ion2desc ibes and analyzes ou
model, Sec .3s udies he equilib ia, p o ides a uniqueness condi ion and p esen s
some examples, and Sec .4analyzes he model om a social planne pe spec i e.
Sec ion5b ie ly in oduces a dynamic e sion o he model, and Sec .6concludes.
2Model
We conside a ini e se o nagen s in e ac ing ia a social ne wo k ep esen ed by an
n×nma ix G=(gij), which is p ede e mined and s ochas ic: he en ies in each ow
a e non-nega i e and sum o one. In e ac ions need no be symme ic o wo-sided, so
in gene al gij = gji and gij >0 does no imply gji >0.
5Using he lowes eigen alue o he ne wo k ma ix o de e mine he uniqueness o an ou come is a
echnique o en employed in he social ne wo ks li e a u e. See, o example, Melo (2022).
123
206 M. Risco
Each agen holds a p i a e signal siabou a common unde lying s a e o he wo ld
μ∈R, d awn independen ly om a no mal dis ibu ion wi h mean μand a iance
σ2>0. The e a e wo lea ning esou ces a ailable o imp o e his signal, p esen ed
in he o de in which hey become accessible o he agen : ac i e p i a e lea ning om
a mo e in o ma i e bu cos ly sou ce, and social lea ning om neighbo s. The i s
esou ce in ol es d awing a signal Ii om a no mal dis ibu ion wi h mean μand
a iance ˜σ2<σ
2, while he second esou ce in ol es agg ega ing he signals o he
agen ’s di ec neighbo s in he ne wo k.
Bo h ypes o lea ning ake he o m o DeG oo upda ing o signals, ollowing
DeG oo (1974). Agen s ake a weigh ed a e age o hei signals, i.e., hey agg ega e
se e al indica o s in jus one. In he case o p i a e lea ning, agen idecides he weigh s
in he con ex combina ion be ween siand Ii. The cos ly signal Ii ecei es weigh
xi∈[0,1]a linea cos xicwi h c>0. Cos ly signals a e posi i ely co ela ed ac oss
agen s, Co (Ii,Ij)=α>0 o all i,j. The o iginal p i a e signals a e independen
ac oss agen s and independen o all cos ly signals. Rega ding social lea ning, agen
i akes he weigh ed a e age o he neighbo s’ signals and he own. Weigh s a e
exogenously6gi en by he ne wo k ma ix, and hey ep esen in luence o us : agen
ilis ens o agen jp ecisely a in ensi y gij.
The mechanical upda ing p ocess desc ibed can be iewed as ac i e lea ning wi h
a en ion cos s o boundedly a ional agen s. In addi ion o no mal signals, i can also
be in e p e ed om a Bayesian pe spec i e, as demons a ed in DeMa zo e al. (2003).
Agen s assign subjec i e p ecisions πij o each o he , a emp ing o es ima e he ue
p ecision o hei signals. I he signals a e independen and no mal, Bayesian upda ing
is equi alen o DeG oo upda ing, wi h weigh s gi en by πij
n
j=1πij o social lea ning.7
A simila a gumen applies o he ac i e lea ning p ocess; see he “Appendix” o a
mo i a ion o he p esen amewo k in e ms o quasi-Bayesian upda ing as de ined
in DeMa zo e al. (2003).
In he ollowing, we use he e m “belie s” o e e o he mos ecen ly upda ed
signal an agen holds abou μ. A p ecise desc ip ion o he lea ning p ocess is as
ollows: The agen ecei es si∼N(μ, σ 2)and decides how much o spend on lea ning
Ii∼N(μ, ˜σ2). Once xiis selec ed, he belie becomes pi=(1−xi)si+xiIia cos
xic. Finally, he social communica ion s age yields belie s
n

j=1
gij pj=
n

j=1
gij (1−xj)sj+xjIj.
No e ha i iand ja e no neighbo s, gij =0, so summing o e i’s neighbo s is
equi alen o summing o e all nindi iduals. A his poin , only one communica ion
s age is assumed, bu conside ing s ages would imply he subs i u ion o Gby G ,
as shown in Sec . 5.
6Ra ional lea ne s migh adjus he weigh s based on neighbo s’ in o ma ion le els, as discussed in Galeo i
and Goyal (2010). Howe e , in his case, we wan o ocus on si ua ions whe e weigh s a e p e-de e mined
o a naï e lea ne due o pas in e ac ions, in luence, o epu a ion, and canno be modi ied.
7I gij =0, hen πij =0.
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 207
Agen iaims o ob ain he mos p ecise belie abou μa minimum cos , as de ia ions
a e penalized h ough a quad a ic loss unc ion. This is speci ied in he payo unc ion
−⎛
⎝μ−
n

j=1
gij pj⎞
⎠
2
−xic=−⎛
⎝μ−
n

j=1
gij((1−xj)sj+xjIj)⎞
⎠
2
−xic.
Al hough agen iis a nai e, mechanical lea ne , we assume, based on e olu iona y
heo y, ha she is capable o eaching Nash equilib ium ou comes. Speci ically, decid-
ing how much o con ibu e o a public good is a ypical example o a p ocess in which
boundedly- a ional agen s e ol e owa d Nash equilib ium ou comes in he long un
(Maila h 1998). Hence, we allow agen i o o m expec a ions and bes espond o
o he s’ choices, as i she we e a ional a his s age. She chooses he amoun xi ha
maximizes he expec ed u ili y:
max
xi∈[0,1]⎧
⎪
⎨
⎪
⎩
E⎡
⎢
⎣−⎛
⎝μ−
n

j=1
gij((1−xj)sj+xjIj)⎞
⎠
2⎤
⎥
⎦−xic⎫
⎪
⎬
⎪
⎭
.(1)
3 Equilib ium
The equilib ium concep used in his model is Nash equilib ium, whe e each agen i
chooses he in o ma ion le el xiby bes esponding o o he s’ equilib ium choices. I is
impo an o no e ha E[si]=E[Ii]=μ o all i. Addi ionally, e e y pai o signals is
independen excep o Iiand Ij. As a esul , En
j=1gij(xjIj+(1−xj)sj)=μ,
while Va (xjIj+(1−xj)sj)=x2
j˜σ2+(1−xj)2σ2and Co (xjIj+(1−xj)sj,xkIk+
(1−xk)sk)=αxjxk. These equali ies, along wi h he payo equa ion, imply ha only
second momen s ma e . In ac ,
E⎡
⎢
⎣−⎛
⎝μ−
n

j=1
gij(xjIj+(1−xj)sj)⎞
⎠
2⎤
⎥
⎦=−Va ⎡
⎣
n

j=1
gij(xjIj+(1−xj)sj)⎤
⎦.
Using ha o any sequence o andom a iables {˜
Xj}n
j=1i holds ha Va (n
j=1˜
Xj)=
n
j=1Va (˜
Xj)+2n
j=1j−1
k=1Co (˜
Xj,˜
Xk), he maximiza ion p oblem o agen i
can be simpli ied as ollows:
max
xi∈[0,1]⎧
⎨
⎩−˜σ2
n

j=1
g2
ijx2
j−σ2
n

j=1
g2
ij(1−xj)2−2α
n

j=1
j−1

k=1
gijgikxjxk−cxi⎫
⎬
⎭
.
(2)
123
208 M. Risco
The objec i e is s ongly conca e in he choice a iable. The i s o de condi ion o
an in e io solu ion yields
xi=2σ2−c/g2
ii
2(˜σ2+σ2)−α
gii(˜σ2+σ2)
j=i
gijxj.
No e ha his exp ession is bounded abo e by 1 bu could be nega i e. As xi∈[0,1]
by assump ion, he op imal choice o ac i e lea ning o agen igi en o he s’ choices
x−iis
x∗
i=max ⎧
⎨
⎩
0,2σ2−c/g2
ii
2(˜σ2+σ2)−α
gii(˜σ2+σ2)
j=i
gijxj⎫
⎬
⎭
.
This bes esponse unc ion is simila o he one ob ained when sol ing a maximiza ion
p oblem in a local public goods se ing. Games o nega i e ex e nali ies o Cou no
compe i ion also yield simila o ms. The only di e ence is ha he e, δis di ided by
gii, a pa ame e ha a ies ac oss agen s. In he o he cases, he subs i u abili y ac o
is he same o all agen s.
No e ha only he weigh ed ou -deg ee ma e s o in o ma ion acquisi ion, bu
no he weigh ed in-deg ee.8In o he wo ds, agen s ca e abou who hey a e lis ening
o ( he gijs), bu no who lis ens o hem ( he gjis). Fu he mo e, i gii =0, hen
x∗
i=0 i ially, as ac i e lea ning is a was e o esou ces o someone who does no
assign posi i e weigh o he sel . The e o e, wi hou loss o gene ali y we can assume
gii >0. By se ing
¯xi=2σ2−c/g2
ii
2(˜σ2+σ2),
and
δ=α
(˜σ2+σ2),
we ob ain
x∗
i=max ⎧
⎨
⎩
0,¯xi−δ
gii 
j=i
gijxj⎫
⎬
⎭
.
He e, in o ma ion e e s o indi iduals’ cos ly lea n signals and is a local public
good. Each agen bene i s om o he s’ p i a e lea ning ia ne wo k communica ion.
The quo ien δ
gii scales he bene i and indica es he subs i u abili y be ween an agen ’s
and he neighbo s’ in o ma ion. Agen iseeks o each a leas he in o ma ion a ge
8The ou -deg ee o agen iis he o al weigh o links di ec ed away om he . The in-deg ee is he o al
weigh o links di ec ed o he .
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 215
Fig. 8 C iminal ne wo k
Fig. 9 In o ma ion acquisi ion o he c iminal ne wo k
As soon as δinc eases, B-class agen s ake ad an age o subs i u abili y and ee- ide
on A and he C-class agen s. Agen A has only B-class neighbo s, so al hough she
ex ac s in o ma ion om hem, she has o make up he di e ence. In con as , C-class
agen s ha e some C-class neighbo s, so he ee- iding beha io o B-class agen s does
no a ec hem as se e ely. Figu e9shows ha x∗
Ais highe han x∗
C o all δ>0.
3.2 Equilib ium cha ac e iza ion
Le us di ide he agen s in o wo g oups: ac i e (A) agen s, who a e ac i e lea ne s
(x∗
i>0), and passi e (P) agen s. An equilib ium in which all agen s belong o A
is known as a dis ibu ed equilib ium, as e o is dis ibu ed among all agen s. In
con as , a specialized equilib ium is such ha only a ew indi iduals ( he specialis s)
lea n, while he o he s ee- ide.
This pa mainly ollows B amoullé e al. (2014). Wi hou loss o gene ali y, we
can eo de he agen s such ha he i s a e ac i e and he las n− a e passi e. As
xj=0 o all j∈P, o any indi idual i,weha ej=igijxj=j∈A {i}gijxj.
123

216 M. Risco
Thus, o i∈1,..., , an equilib ium equi es ha :
x∗
i=¯xi−δ
gii 
j∈A {i}
gijx∗
j>0.
Fo i∈{ +1,...,n}, an equilib ium equi es ha :
¯xi−δ
gii 
j∈A {i}
gijx∗
j≤0.
Le ¯
xA=(¯x1,..., ¯x )and ¯
xP=(¯x +1,..., ¯xn). The diagonal o a ma ix Ais
deno ed by dA.Le GAbe he × mino co esponding o he ac i e agen s o he
ne wo k, while GPis he (n− −1)×(n− −1)mino o Gco esponding o he
passi e agen s. The (n− −1)× mino GP,Ao Gis gi en by (gij)whe e i∈P
and j∈A. Rea anging he abo e exp essions, we ob ain he ollowing esul :
P oposi ion 3.2 The p o ile o in o ma ion le els x=(x∗
1,...,x∗
n)=(xA,0)wi h
xA=(x∗
1,...,x∗
)∈(0,1] cons i u es an equilib ium i and only i
dGA¯
xA=(1−δ)dGA+δGAxA,
dGP¯
xP≤δGP,AxA.(3)
No e ha gi en nagen s, he e a e 2npo en ial pa i ions. Ob aining all possible
equilib ia equi es sol ing he sys em (3) o each pa i ion. This can be done in wo
s eps:
(i) Fi s , sol e o xAin dGA¯
xA=(1−δ)dGA+δGAxA. The solu ion is unique i
and only i de [(1−δ)dGA+δGA] = 0.
(ii) Then, check whe he all componen s o xAa e s ic ly posi i e and dGP¯
xP≤
δGP,AxA.
I he diagonal elemen s o Ga e iden ical, i.e., gii =gjj o all i,j, he numbe o
equilib ia is weakly lowe han 2nand can be compu ed in exponen ial ime. In his case,
he condi ion de [(1−δ)dGA+δGA]=0 simpli ies o de −(δ−1)gii
δId +GA=0,
which holds i and only i GAhas an eigen alue λ=(δ−1)gii
δ. Consequen ly, o almos
all δ, he equa ion has a unique solu ion. While B amoullé e al. (2014) assume no
only dG=dId bu also ma ix symme y, we ha e shown ha hese assump ions a e
no necessa y o ob ain an explici exp ession o equilib ia.
3.3 Equilib ium uniqueness
In gene al, he e migh be mul iple equilib ia in his model. We p esen wo examples.
The i s example is a h ee-agen ne wo k ha is incomple e and can also be isu-
alized as a s a . The weigh s o he connec ions be ween he agen s a e ep esen ed by
he ma ix G. Figu e10 shows he g aph co esponding o his ne wo k, wi h hicke
a ows indica ing la ge weigh s.
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 217
Fig. 10 Incomple e h ee-agen ne wo k
Fig. 11 Fou -agen s eye
Since gAA =gBB =gCC,i ollows ha ¯xA=¯xB=¯xC=¯x. Assuming δ=1
2,
he bes eply unc ions become:
xA=max{0,¯x−xB+xC
2},
xB=max{0,¯x−xA},
xC=max{0,¯x−xA}.
The e a e wo dis ibu ed equilib ia: (x∗
A,x∗
B,x∗
C)=(¯x
2,¯x
2,¯x
2)and (x∗
A,x∗
B,x∗
C)=
(¯x
3,2¯x
3,2¯x
3). Addi ionally, he e exis specialized equilib ia whe e ei he agen A o
bo h agen s B and C pu chase ¯xwhile he o he s ee- ide. Ano he simila example
holds o δ=1
kand a s a wi h kagen s, demons a ing ha he mul iplici y o
equilib ia does no depend on he ex eme assump ion ha δ=1
2(which is ex eme
in he sense ha i implies ˜σ2=0).
The second example is a ou -agen s eye, shown in Fig. 11. Weigh s a e gi en by
he ma ix G.
Once again, we assume δ=1
2. Since ¯xi=¯x o all i, he bes eplies a e as ollows:
xA=max 0,¯x−xB+xC+xD
2,
xB=max 0,¯x−3(xA+xD)
4,
xC=max 0,¯x−3(xA+xD)
4,
123
218 M. Risco
xD=max 0,¯x−xA+xB+xC
2.
The e a e wo specialized equilib ia: 2
3¯x,0,0,2
3¯xand (0,¯x,¯x,0). The ough idea
behind mul iplici y is ha agen s can be di ided in o wo dis inc g oups so ha ac i e
lea ning con ibu ions a y be ween hem. When one g oup lea ns mo e, he o he
dec eases i s e o , and ice e sa. We will discuss his in de ail in Sec . 3.3.2.
Nex , we seek a s uc u al condi ion on he ne wo k ha gua an ees uniqueness. I
u ns ou ha , gi en δ, he posi i e de ini eness o a ma ix ha we deno e Qensu es
equilib ium uniqueness. This ma ix Qcan be de e mined om Gin a one- o-one
co espondence once δis ixed.
Recall ha agen i’s expec ed payo s a e gi en by he ollowing equa ion:
ui(x1,...,xn)=E⎡
⎢
⎣−⎛
⎝μ−
n

j=1
gij((1−xj)sj+xjIj)⎞
⎠
2⎤
⎥
⎦−xic.
P oposi ion 3.3 The p o ile o ac i e lea ning choices x∗=(x∗
1,...,x∗
n)is an equi-
lib ium o he game i and only i
(θ−
ˆ
Qx∗)T(x∗−x)≥0(4)
o any x∈[0,1]n, wi h he ma ix
ˆ
Q=(σ2+˜σ2)⎛
⎜
⎜
⎜
⎝
2g2
11 2δg11g12 ···2δg11g1n
2δg22g21 2g2
22 ···2δg22g2n
.
.
..
.
.....
.
.
2δgn1gnn 2δgn2gnn ··· 2g2
nn
⎞
⎟
⎟
⎟
⎠
and he ec o
θ=(2σ2g2
11 −c,...,2σ2g2
nn −c).
P oo Fi s , he ollowing equi alence is es ablished: he p o ile x∗is an equilib ium
i and only i
∂
∂xiui(x∗
i,x∗
−i)(x
i−x∗
i)≤0
o all iand x
i∈[0,1].
Fixing a p o ile x∗∈[0,1]nand an agen i, le us de ine
g( ):= ui(x
i+ (x∗
i−x
i), x∗
−i)
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 219
o 0 ≤ ≤1 and x
i∈[0,1]. The de i a i e wi h espec o is gi en by g( )=
∂
∂xi(ui(xi,x∗
−i))|xi=x
i+(x∗
i−x
i)(x∗
i−x
i).I x∗is an equilib ium, g( )has a maximum
a =1 and g(1)≥0. Hence,
∂
∂xiui(x∗
i,x∗
−i)(x
i−x∗
i)≤0.
Now, le us show he con e se. Conca i y o g ollows om he conca i y o ui,11 and
hen g( )≤g(y)+g(y)( −y) o any ,y∈[0,1]. Choosing =0 and y=1, we
see ha g(0)≤g(1)−g(1). Mo eo e , g(1)≥0 by assump ion, so ha −g(1)≤0
and g(0)≤g(1). This inequali y implies ha
ui(x∗
i,x∗
−i)≥ui(x
i,x∗
−i)
o all x
i∈[0,1], and x∗is an equilib ium.
Summing up wi h espec o all agen yields
n

i=1 ∂
∂xiui(x∗
i,x∗
−i)(x
i−x∗
i)!≤0.
Deno ing by "∂
∂xiui(x∗
i,x∗
−i)#i he ec o gi en by s acking up all ∂ui
∂xi, he p e ious
inequali y can be ew i en as
∂
∂xi[ui(x∗
i,x∗
−i)]!T
i
(x−x∗)≤0.
The p o ile o ac i e lea ning choices x∗is an equilib ium i and only i his inequali y
holds o any x∈[0,1]n.12 I jus emains o explici ely de i e he ec o componen s,
which a e gi en by
∂
∂xi[ui(x∗
i,x∗
−i)]=2g2
iiσ2−2g2
iix∗
i(σ2+˜σ2)−2αgii 
j=i
gijx∗
j−c.
11 The unc ion uiis clea ly wice di e en iable wi h espec o xiand ∂2ui
∂x2
i=−2(−gii(Ii−si))2<0.
12 I ˆ
xis no an equilib ium, hen he e is some agen jsuch ha
∂
∂xjuj(ˆxj,ˆx−j)(xj−ˆxj)>0
o some xj∈[0,1]. Hence, de ining he p o ile ˜
xas ˜xj=xjand ˜xi=ˆxi o i= j,

i ∂
∂xi
(ui(ˆxi,ˆx−i)!(˜xi−ˆxi)=∂
∂xjuj(ˆxj,ˆx−j)(xj−ˆxj)>0.
123
220 M. Risco
Finally, i is a me e e i ica ion o check ha de ining
ˆ
Qand θas abo e, x∗is an
equilib ium i and only i
(θ−
ˆ
Qx∗)T(x∗−x)≥0.

I
ˆ
Qis posi i e de ini e, he e is jus one ec o o in o ma ion le els x∗ ha sa is ies
(4). This is he su icien condi ion o equilib ium uniqueness.
P oposi ion 3.4 I he ma ix
ˆ
Qis posi i e de ini e, he equilib ium is unique.
P oo Suppose x∗
1and x∗
2a e wo di e en equilib ia. Then, (θ−
ˆ
Qx∗
1)T(x∗
1−x∗
2)≥0
and (θ−
ˆ
Qx∗
2)T(x∗
2−x∗
1)≥0. Summing up bo h inequali ies yields (θ−
ˆ
Qx∗
1)T(x∗
1−
x∗
2)+(θ−
ˆ
Qx∗
2)T(x∗
2−x∗
1)≥0, which holds i and only i
(x∗
2−x∗
1)Tˆ
Q(x∗
2−x∗
1)≤0.
Bu
ˆ
Qis posi i e de ini e, i.e. xTˆ
Qx>0 o all x= 0. Consequen ly, x∗
1=x∗
2and
he equilib ium is unique. 
Di iding
ˆ
Qby σ2+˜σ2does no change i s de ini eness and simpli ies he
exp ession— ecall ha σ2+˜σ2>0.13 Thus, Qis gi en by
Q=1
σ2+˜σ2
ˆ
Q=⎛
⎜
⎜
⎜
⎝
2g2
11 2δg11g12 ···2δg11g1n
2δg22g21 2g2
22 ···2δg22g2n
.
.
..
.
.....
.
.
2δgn1gnn 2δgn2gnn ··· 2g2
nn
⎞
⎟
⎟
⎟
⎠
.(5)
The ollowing esul shows ha Qis comple ely de e mined by G, once δis ixed.
Consequen ly, equilib ium uniqueness o his model depends solely on he in luence
ne wo k G.
P oposi ion 3.5 Gi en δ, he e is a one- o-one co espondence be ween Qand G.
P oo Gi en δ, he ma ix Qis de ined elemen -wise om Gas in (5). Assume δis ixed
and deno e his ans o ma ion by φδ. Le us show ha i is possible o eco e G om
Q. Deno ing by qij he elemen s in Q, le us de ine (elemen -wise) he ans o ma ion
τδby τδ(qii)=√qii o all iand τδ(qij)=qij
δ√qii o all i= j. I is i ial o check
ha τδ(φδ(G)) =Gand φδ(τδ(Q)) =Q.
In gene al, he ma ix Qis an n×nma ix ha need no be symme ic. No e ha
xTQx =1
2xT(Q+QT)x, and Qis posi i e de ini e i and only i A:= 1
2(Q+QT)
is posi i e de ini e. Since Ais symme ic, we can use he cha ac e iza ion o posi i e
de ini eness in e ms o eigen alues: a symme ic ma ix is posi i e de ini e i and only
i all o i s eigen alues a e posi i e. Le λ1(A)deno e he lowes eigen alue o A.
13 I would be possible o di ide by 2(σ 2+˜σ2)ins ead, bu keeping he ac o 2 simpli ies he exp ession
o he ma ix Ala e .
123

Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 221
Co olla y 3.6 I λ1(A)>0, hen he equilib ium is unique.
The explici exp ession o Ais gi en by:
A=⎛
⎜
⎜
⎜
⎝
2g2
11 δ(g12g11 +g21g22)···δ(g1ng11 +gn1gnn)
δ(g21g22 +g12g11)2g2
22 ···δ(g2ng22 +gn2gnn)
.
.
..
.
.....
.
.
δ(gn1gnn +g1ng11)δ(gn2gnn +g2ng22)··· 2g2
nn
⎞
⎟
⎟
⎟
⎠
.
No e ha his su icien condi ion is independen o he cos co ac i e lea ning bu
depends on he in luences be ween agen s and he subs i u abili y o in o ma ion acqui-
si ion.
The scope o his condi ion is he ocus o ou subsequen analysis. We will make
mo e es ic i e assump ions on he model o explo e pa icula cases o in e es , which
will e en ually lead o a esul simila o ha o B amoullé e al. (2014). La e , we will
apply he su icien condi ion o he examples in Sec .3.1. We i s p o e an auxilia y
lemma.
Lemma 3.7 Le Abe a symme ic ma ix and β,δ > 0. The ma ix βId+δAis posi i e
de ini e i and only i λ1(A)≥−β
δ.
P oo The ma ix βId +δAis posi i e de ini e i and only i all he solu ions λ
o de [λId −(β Id +δA)]=0 a e s ic ly posi i e. The equa ion is equi alen o
de [λ−β
δId−A]=0. No e ha he eigen alues o Aa e he solu ions o he equa ion
de [ Id−A]=0. Consequen ly, as =λ−β
δ, he condi ion λ>0 can be ansla ed in o
all eigen alues o A e i ying >−β
δ. This is p ecisely he condi ion λ1(A)>−β
δ.

Nex , we conside wo pa icula cases ha a e wo h explo ing. Assuming ha all
agen s pay he same a en ion o hemsel es, i.e., gii =gjj o all i,j, we can deno e
he diagonal e ms o Gby β:= gii >0. We de ine ¯
Aas
¯
A=⎛
⎜
⎜
⎜
⎜
⎝
0g12+g21
2... g1n+gn1
2
g21+g12
2
...... g2n+gn2
2
.
.
....
....
.
.
gn1+g1n
2
gn2+g2n
2... 0
⎞
⎟
⎟
⎟
⎟
⎠
.
Using Lemma 3.7, we see ha λ1(A)>0 i and only i λ1(¯
A)>−β
δ. No e ha ¯
A
is simply ¯
A=1
2(G+GT)−βId. He e, ¯
A e lec s he a e age low o in o ma ion
be ween a pai o ne wo ks, o he undi ec ed ne wo k associa ed wi h G.
Now, assume ha he ne wo k displays ecip ocal ela ions, i.e., gij =gji,in
addi ion o same sel -impo ance ac oss agen s. This means ha he in luence o agen
ion agen jis he same as ha o agen jon agen i, and so he ma ix Gis symme ic
and can be seen as undi ec ed. Again, λ1(A)>0 i and only i λ1(¯
A)>−β
δ,bu ¯
A
is now simply G−βId. The ma ix ¯
A=G−βId can be seen as a gene aliza ion
123
222 M. Risco
o he ma ix Gin B amoullé e al. (2014), whe e ¯aij ∈[0,1]ins ead o gij ∈{0,1}.
The su icien condi ion is equi alen o hei s. Howe e , o de i e such a esul hey
use he po en ial heo y de eloped by Monde e and Shapley (1996), which equi es
symme y— his is why we canno apply i o he gene al model.
P oposi ion 3.8 The su icien condi ion o he uniqueness o equilib ium can be spe-
cialized o wo pa icula cases:
•I sel -impo ance is equal ac oss agen s (gii =gjj =β o all i,j), he condi ion
becomes λ1(¯
A)>−β
δwi h ¯
A=1
2(G+GT)−βId.
•I on op o ha he in luences a e ecip ocal (gij =gji o all i,j), he condi ion
becomes λ1(¯
A)>−β
δwi h ¯
A=G−βId.
This p oposi ion summa izes he esul s ob ained so a , which show ha he con-
di ion o he uniqueness o equilib ium can be specialized o wo pa icula cases:
when all agen s ha e he same le el o sel -impo ance and when he ne wo k exhibi s
ecip ocal ela ions be ween agen s. In bo h cases, he condi ion in ol es he eigen-
alue o a ma ix ¯
A, which can be calcula ed based on he p ope ies o he ne wo k.
The p ecise de ini ion o ¯
Ais gi en o each case.
3.3.1 Examples: uniqueness
The ne wo ks analyzed in Sec .3.1 a e e iewed again o apply he equilib ium unique-
ness condi ion.
Fi s , we e isi he class o k- egula g aphs wi h nagen s ha sha e hei a en ion
homogeneously. P oposi ion 3.8 applies, and he lowes eigen alue o ¯
Ais λ1(¯
A)=
−1
k. The equilib ium is unique i δ<1, which always holds. As an example o his
class o ne wo ks, he ma ix ¯
Aassocia ed wi h he comple e g aph is gi en by
¯
A=⎛
⎜
⎜
⎜
⎜
⎝
01
n... 1
n
1
n0... .
.
.
.
.
. ... ...1
n
1
n
1
n... 0
⎞
⎟
⎟
⎟
⎟
⎠
.
Nex , we conside he class o s a s. Due o he asymme y o Qand he di e en
e ms in he diagonal (sel -impo ance is no equal ac oss agen s), only Co olla y 3.6
applies. The equilib ium is unique i λ1(A)>0, which depends on bo h δand ε.
Ma ix Ais gi en he e by:
A=⎛
⎜
⎜
⎜
⎜
⎝
2
n2δ((1−ε)ε +1
n2)...δ((1−ε)ε +1
n2)
δ((1−ε)ε +1
n2)2(1−ε)2... 0
.
.
. ... ....
.
.
δ((1−ε)ε +1
n2)0... 2(1−ε)2
⎞
⎟
⎟
⎟
⎟
⎠
.
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 223
Fig. 12 The lowes eigen alue o he s a
Figu e12 shows he alues o which a unique equilib ium is ensu ed—e e y pai
(δ, ε) such ha he blue su ace is abo e he o ange plane.
A pa icula ne wo k s uc u e belonging o he class o co e-pe iphe y ne wo ks
was se in Fig.6. He e,
A=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
2
9δ2
81 δ2
81 δ9
100 00
δ2
81
2
9δ2
81 0δ9
100 0
δ2
81 δ2
81
2
900δ9
100
δ9
100 002
100 00
0δ9
100 002
100 0
00δ9
100 002
100
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
and we apply Co olla y 3.6. I u ns ou ha λ1(A)>0 o all δ∈[0,1
2],so he
equilib ium is always unique.14
The c iminal ne wo k om Balles e e al. (2006) was ep esen ed in Fig.8.P o-
ceeding as be o e, we calcula e he lowes eigen alue o A.15 We ind ha λ1(A)>0
o all δ<0.45011, which gua an ees a unique equilib ium o such alues.
Finally, we conside he incomple e ne wo k depic ed in Fig.10. Recall ha δ=1
2
and all diagonal e ms a e equal: β=1
3. To apply P oposi ion 3.8, we compu e
14 The explici exp ession o he lowes eigen alue o Ais λ1(A)=
981−100δ−√670761−163800δ+541441δ2
8100 . We see ha λ1(A)is a dec easing unc ion o δin [0,1/2].
As i is s ic ly posi i e a δ=1/2, λ1(A)>0 o all δ∈0,1
2.
15 Le y1(δ),y2(δ) and y3(δ) be he h ee oo s o −32400 −97200δ+259200δ3+(3096 +6192δ−
7200δ2)y+(−97 −97δ)y2+y3.Le y1(δ) be he smalles oo in δ∈0,1
2. Then, λ1(A)=1
450 y1(δ)
and λ1(A)>0⇔δ<0.45011.
123
224 M. Risco
¯
A=⎛
⎝
01
2
1
2
1
200
1
200
⎞
⎠.
The uniqueness condi ion λ1(¯
A)>−β
δis no sa is ied because λ1(¯
A)=−1
√2<
−2
3. This was expec ed, as we had al eady ob ained wo di e en equilib ia o his
pa icula ne wo k.
3.3.2 The lowes eigen alue
The p esen subsec ion explo es he meaning o he uniqueness condi ion and p o ides
an in ui ion. A ne wo k is bipa i e i agen s can be di ided in o wo se s, say Rand
S, such ha i i∈R,iis no connec ed o any j∈Rexcep o he sel . The ne wo k
is comple ely bipa i e i e e y i∈Ris connec ed o all j∈S. Bipa i e ne wo ks
ep esen disjoin o independen communi ies. An a ilia ion ne wo k is a classic
example. Ano he bipa i e ne wo k migh be ound when ep esen ing supe iso -
candida e communica ion. A comple e bipa i e ne wo k ep esen s one ex eme o
wo-sidedness. The o he ex eme is he comple e egula g aph. In his subsec ion,
we alk abou wo-sidedness as an in ui i e measu e o how close a ne wo k is o he
comple e bipa i e g aph.
Fi s , le us b ie ly cha ac e ize λ1(A).16 By de ini ion, λ1(A)=min{λ∈R:
∃∈Rnsa is ying λ =A}. Assuming = 0, λ =Aimplies Tλ =TA,
which leads o λT=TA, and inally o λ||||2=TA. So, i |||| = 1, hen
λ=TA. Hence, λ1(A)=min $λ∈R:λ=TAand |||| = 1%.
Following B amoullé e al. (2014), we can use an eigen ec o associa ed o λ1(A)
o sepa a e he agen s in o wo g oups. I i≥0, agen ibelongs o se R. O he wise,
she belongs o se S. This leads o he decomposi ion
λ1(A)=εTAε=
>0
&'( )

i,j∈R
ijqij +
>0
&'( )

i,j∈S
ijqij +
<0
&'( )
2
i∈R,j∈S
ijqij .
The g ea e he lowes eigen alue, he mo e weigh he ne wo k pu s wi hin se s and
he less i pu s be ween se s. Hence, he size o λ1(A)is ela ed o he wo-sidedness
o he g aph A. The close he ne wo k is o he comple e bipa i e g aph, he lowe
λ1(A). This is because ans e ing weigh om links wi hin Ro S o links be ween
bo h se s dec eases λ1(A). C ea ing new links be ween se s o emo ing links wi hin
Ro Sbelong o ha kind o weigh ans e . Thus, making he g aph mo e wo-sided
dec eases he lowes eigen alue.
Le us show how he di ision o agen s in o he wo g oups is induced by agen s’
lis ening s uc u es. We ha e λ1(A)= A=i,jqijijwi h |||| = 1. Wi hou
loss o gene ali y, le us assume ha λ1(A)>0 (i no , a simila easoning holds).
Then, agen ibelongs o Ri and only i λ1(A)i>0. Since λ1(A)i=Ai,wesee
16 Remembe ha when Qis symme ic, hen A=Q.
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 231
Fig. 14 Acquisi ion o he c iminal ne wo k in he long- un
In he long un, only in-deg ee ma e s, bu no ne wo k posi ion. Hence, agen A no
longe has a dis inc ole and he e a e jus wo classes o agen s: B-class and C-class
( o which agen A belongs now). B-class agen s ha e in-deg ee 6, and πj=6
59 , and
C-class agen s ha e in-deg ee i e, so πj=5
59 . Bes eply unc ions a e gi en by:
x∗
B=max 0,¯xB−δ59
6(3xB+7xC),
x∗
C=max 0,¯xC−δ59
5(4xB+6xC).
Simila o he one-pe iod communica ion game, B-class agen s acqui e less in o -
ma ion. In pa icula , o he con igu a ion o pa ame e s used in Sec . 3.1, we can
obse e in Fig.14 ha B-class agen s comple ely ee- ide on C-class agen s. This
happens because B-class agen s conside acqui ing p i a e in o ma ion oo cos ly,
elying ins ead on he in o ma ion ob ained om he se en C-class agen s.
Finally, he examples om Sec .3.3 a e i ial in he long un. The incomple e h ee-
agen ne wo k con e ges o a ma ix cha ac e ized by he limi ec o π=1
2,1
4,1
4,
while he ou -agen s eye con e ges o a ma ix cha ac e ized by he limi ec o
π=3
10 ,1
5,1
5,3
10 . Bo h cases lead o unique equilib ium con igu a ions.
6 Conclusion
This pape has analyzed he beha io o DeG oo upda e s in a ne wo ked en i on-
men and s udied he impac o subs i u abili y and ne wo k s uc u e on in o ma ion
acquisi ion and wel a e. We ha e shown ha he subs i u abili y o agen s’ ac i e
lea ning e o s induces ee- iding beha io and can lead o mul iple equilib ia. We
ha e also p o ided a su icien condi ion o equilib ium uniqueness in e ms o he
lowes eigen alue o he ma ix A, which is de e mined by Gand he pa ame e o
123

232 M. Risco
subs i u abili y δ. When his eigen alue is posi i e, he equib ium is unique. E en i
he e a e mul iple equilib ia, we ha e p oposed a p ocedu e o calcula ing hem.
In e ms o wel a e, we ha e ound ha he in o ma ion a ge is lowe in equilib ia
han unde he u ili a ian pa adigm. This is signi ican since he a ge is p ecisely
he le el o in o ma ion an agen will ha e a he end o he game. We ha e shown
ha i is socially desi able o inc ease he in o ma ion le el o e e y agen . While
inc easing agen s’ ac i e lea ning may seem like a solu ion, we show ha in he
one-sho game i is no . No only he anking in a ge s does no imply a anking in
acquisi ion le els, bu he u ili a ian op imum does no Pa e o domina e he equilib ium
alloca ion. Ne e heless, o e he long un, neighbo hood ic ions a e elimina ed and
he u ili a ian alloca ion always exceeds he equilib ium alloca ion.
An in e es ing a enue o u he esea ch would be he implemen a ion p oblem
o a planne ying o incen i ize DeG oo upda e s o mo e om equilib ium le els
o ac i e lea ning o he u ili a ian op imum. Public in o ma ion policies, such as
subsidizing ex e nal in o ma ion sou ces, ewa ding lea ning con ibu ions, o c ea ing
new links o os e communica ion, could also be explo ed.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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Appendix: Quasi-Bayesian ounda ion
Rega ding agen s’ cogni i e sophis ica ion, his pape ollows he boundedly a ional
app oach, which assumes ha agen s ha e limi ed cogni i e esou ces and do no
possess p ecise knowledge o hei en i onmen . None heless, i is use ul o connec
he assump ions o his pape o he s anda d Bayesian amewo k. In his “Appendix”,
we p o ide a pu e heo e ical mo i a ion o DeG oo upda ing in ne wo ks, ollowing
DeMa zo e al. (2003). DeG oo upda ing can be iewed as a Bayesian upda ing
p ocess o agen s ha ecei e no mally dis ibu ed signals bu do no know he ue
a iances o hei neighbo s’ signals.
Conside nagen s who wan o es ima e some unknown pa ame e μ∈R. Agen i
ecei es an independen signal x0
i∼N(μ, σ 2
i), and she assigns some p ecision πij =
1
Va i(x0
j) o agen j’s signal, which may o may no be he ue p ecision. No e ha his
assump ion does no align wi h he s anda d Bayesian app oach, which assumes ha
agen s ha e p ecise knowledge o he signal s uc u e. Agen s communica e acco ding
123
Ne wo k e ec s on in o ma ion acquisi ion by DeG oo … 233
o a social ne wo k ˜
G, which is a di ec ed g aph ha indica es whe he agen ilis ens o
agen j;˜gij =1 i agen ilis ens o agen j, and ˜gij =0 o he wise. Each agen knows
he own in o ma ion, so ˜gii =1. T u h ul epo ing is assumed. Gi en no mali y and
he assigned p ecisions, a su icien s a is ic o he signals is hei weigh ed a e age,
wi h weigh s gi en by he p ecisions. DeMa zo e al. (2003) deno e such a s a is ic by
x1
i, and e e o i as agen i’s belie a e communica ion:
x1
i=
n

j=1
˜gijπij
n
j=1˜gijπij
x0
j.
The su iciency o he s a is ic x1
icomes om he applica ion o he Fishe -Neyman
ac o iza ion heo em. De ining gij := n
j=1˜gijπij
n
j=1˜gijπij , we ob ain he s ochas ic
ma ix G=(gij). A DeG oo ian popula ion communica ing acco ding o Gholds
he same belie s as he quasi-Bayesian popula ion om DeMa zo e al. (2003).20 This
insigh p o ides addi ional mo i a ion o he model desc ibed in his pape .
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and ins i u ional a ilia ions.
123