American Economic Association
The Law of the Few
Author(s): Andrea Galeotti and Sanjeev Goyal
Source: The American Economic Review, Vol. 100, No. 4 (SEPTEMBER 2010), pp. 1468-1492
Published by: American Economic Association
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American Economie Review 100 [September 2010) 1468-1492
100.4.1468
http'J/www.aeaweb. org/articles.php?doi=10.1257/aer.
The Law of theFew
By Andrea
Galeotti
and
Sanjeev
Goyal*
Empirical work shows that a large majority of individuals get most of their
informationfrom a very small subset of the group, viz., the influencers;more
over, there exist only minor differences between the observable characteris
tics of the influencers and the others. We refer to these empirical findings as
the Law of the Few. This paper develops a model where players personally
acquire information and form connections with others to access their informa
tion. Every {robust) equilibrium of thismodel exhibits the law of thefew. (JEL
D83, D85, Z13)
Individuals often have tomake a choice between alternatives whose advantages are imper
fectly known. In order tomake an informed decision, they personally acquire information and
gather information through social contacts. Personal acquisition of information is costly; simi
larly, creating and maintaining personal contacts takes time and resources. Rational individu
als therefore compare the relative costs of these different sources of information. This paper
explores the implications of such individual choices for social communication and the aggregate
information available in a society.
Our point of departure is a series of empirical studies about information acquisition and com
munication in social groups. The classical early work of Paul F. Lazarsfeld, Bernard Berelson,
and Hazel Gaudet (1948) and Elihu Katz and Lazersfeld (1955) investigated the impact of per
sonal contacts and mass media on voting and consumer choice with regard to product brands,
films, and fashion changes. They found thatpersonal contacts play a dominant role in dissemi
nating informationwhich in turn shapes individuals' decisions. In particular, they identified 20
percent of their sample of 4,000 individuals as the primary source of information for the rest.
Similarly, Lawrence F. Feick and Linda L. Price (1987) found that 25 percent of their sample
of 1,400 individuals acquired a great deal of information about food, household goods, nonpre
scription drugs, and beauty products and that theywere widely accessed by the rest.1
Research on virtual social communities reveals a similar pattern of communication. Jun
Zhang, Mark S. Ackerman, and Lada Adamic (2007) study the Java Forum, an online commu
nity of users who ask and respond to queries concerning Java. They identify 14,000 users and
*Galeotti:
Park, Colchester C04
3SQ, UK
(e-mail:
Department of Economics, University of Essex, Wivenhoe
Goyal: Faculty of Economics and Christ's College, University of Cambridge, Sidgewick Avenue,
agaleo@essex.ac.uk);
We thank the editor and three anonymous referees for
Cambridge CB3 9DD, UK (e-mail: sg472@econ.cam.ac.uk).
useful suggestions. We are grateful to Bhaskar Dutta, Francis Bloch, Yann Bramoulle, Myeonghwan
Cho, Matthew
Jackson, Willemien Kets, Marco van der Leij, Brian Rogers and a number of seminar audiences for suggestions which
have significantly improved the paper. We thank the ESRC for financial support via Grant RES-000-22-1901.
1
Recent work in political science arrives at similar conclusions;
see, e.g., Paul Beck et al. (2002) and Robert
Huckfeldt, Paul E. Johnson, and John Sprague (2004). For recent work on information acquisition about products, see
Gary L. Geissler and Steve W. Edison (2005). For evidence on patterns and the importance of informal communication
in firms, see Robert Cross and Andrew Parker (2004).
1468
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VOL.
100 NO.
4
GALEOTTI
AND GOYAL:
1469
Core-periphery architecture with 2 hubs
Core-periphery architecture with 3 hubs
Figure
THE LAW OF THE FEW
1.Core-Periphery
Networks
find that 55 percent of these users only ask questions, 12 percent both ask and answer queries,
and about 13 percent only provide answers.2
The Law of the Few subsumes these empirical findings: in social groups, a majority of indi
viduals get most of their information from a very small subset of the group, viz., the influ
encers. Moreover, research suggests that there are minor differences between the observable
economic and demographic characteristics of the influencers and the others.We are thus led to
ask: can the law of the few be understood as a consequence of strategic interaction among identi
cal individuals?
We propose a game inwhich individuals choose to personally acquire informationand to form
connections with others to access the information these contacts acquire. Our main finding is
thatevery (strict)equilibrium of thegame exhibits the law of the few (propositions 1-3). The net
work has a core-periphery architecture', the players in the core acquire information personally,
while the peripheral players acquire no information personally but form links and get all their
information from the core players. Figure 1 illustrates equilibrium configurations.
We informally outline the ideas underlying this result. In our model, returns from information
are increasing and concave, while the costs of personally acquiring information are linear. This
we
implies thaton his own an individual would acquire a certain amount of information,which
denote by 1.The second element in our model is the substitutabilityof information acquired by
different individuals. This implies that ifA acquires informationon his own and receives infor
mation from player B, then in the aggregate he must have access to 1 unit of information (else
he could strictly increase his payoff by modifying personal information acquisition). The third
2
Eytan Adar and Bernardo A. Huberman
to-peer network Gnutella.
(2000)
report similar findings with regard to provision of files in the peer
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1470
THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
and key element is that links are costly and rationally chosen by individuals. The implication
of this is that ifA finds itoptimal tomaintain a link with B, then so must every other player.
Hence, the group of individuals who acquire informationmust be completely linked, and the
aggregate information acquired in the society must equal exactly 1.Moreover, since linking is
costly,A will linkwith B only ifB acquires a certain minimum amount of information. Since
total information acquired is 1, it follows that there is an upper bound to the number of people
who will acquire information,and so the proportion of information acquirers in a large group is
very small. Finally, we observe that since the aggregate information acquired in the group is 1,
everyone who does not personally acquire informationmust be linked to all those who acquire
information,yielding the core-periphery network.
The resultmentioned above is derived in a settingwhere individuals are ex ante identical. A
recurring theme in the empirical work is that influencers have similar demographic character
istics as the others. But thiswork also finds that they have distinctive attitudes which include
higher attention to general market informationand enjoyment in acquiring information (see, e.g.,
Feick and Price 1987). This motivates a study of the consequences of small heterogeneity in indi
vidual characteristics. Our main finding is that a slight cost advantage in acquiring information
(or a greater preference for information) leads to a unique equilibrium inwhich the low cost (or
high informationneed) individual player is the single hub: he acquires all the information,while
everyone else simply connects with him (Proposition 3). Small heterogeneities thus help select
individuals who play dramatically different roles in social organization.
In actual practice, we receive information from friends and colleagues which theyhave them
selves received from others.We then extend the basic model to allow for indirect information
transmission. Our main insighthere is that indirect information transmission gives rise to a new
type of influencer: the connector (Proposition 5). A connector is someone whose primary role
is thatof an "informediary": he does not acquire (much) informationhimself but connects indi
viduals who personally acquire most of the information.
We now relate our paper to the theory of network formation as well as the theory of games
played on fixed networks.3 Our model builds on the approach to link formation introduced in
Goyal (1993) and Venkatesh Bala and Goyal (2000) and themodel of local public goods devel
the
oped inYann Bramoull? and Rachel Kranton (2007). We show that two economic ideas?(i)
and
information
between
information
others,
acquired personally
substitutability
acquired by
and (ii) thepossibility of forming costly linkswith others who acquire information?explain the
specialization in information acquisition and social communication reflected in the law of the
few.Moreover, in line with the findings of empirical work, we show that a small difference?
with regard to the costs of acquiring information, the need for information, or with regard to
personal sociability?is sufficient to perfectly select influencers in a social group.We observe
that existing models cannot explain the law of the few: the pure link formationmodel cannot
account for the specialization in information acquisition, while the public goods model cannot
account for the specific patterns of social communication.
Our results also resolve important theoretical questions in these two strands of the literature.
First, in the public goods model with exogenous networks, Bramoulle and Kranton (2007) show
thatmultiple equilibria typically exist, and these equilibria exhibit very different individual and
aggregate informationacquisition. In contrast, introducing link formation yields clear cut predic
tions on individual as well as aggregate information acquisition.
Second, in a model of network formationwith pure local information sharing, the equilibrium
network is either the complete or the empty network, depending on whether the cost of linking is
3
For surveys of this literature, see Goyal
(2007) and Matthew
O. Jackson (2008).
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VOL.
100 NO. 4
GALEOTTIAND GOYAL: THE LAWOF THE FEW
1471
lower or higher than the value of individual information.By contrast,we find the unique equilib
rium is a core-periphery network with the hubs acquiring all the information.Thus endogenous
information acquisition provides an alternative theoretical foundation for core-periphery (and
star) networks. Furthermore, our model yields an interesting cost of link effect: an increase in
this cost leads to a fall in the number of players who acquire information, an increase in the
amount of information acquired by each player who acquires information, and a decrease in
the total number of links. By contrast, changes in costs of linking have no effect on personal
information acquisition in a model of pure link formation, since everyone has exogenously speci
fied information (see, e.g., Bala and Goyal 2000; Galeotti 2006; Galeotti, Goyal, and Jurjen
Kamphorst 2006; and Daniel Hojman and Adam Siezdl 2008).4
Nonrival information acquisition is an instance of the private provision of public goods. For
global public goods, Theodore Bergstrom, Lawrence Blume, and Hal Varian (1986) showed that
the contributorswill be those with higher endowments. The key difference in our model is that
access to public good is a matter of individual choice; it is costly and takes the form of bilateral
connections. The findings with regard to the existence of information hubs and connectors and
the core-periphery network structure are, to the best of our knowledge, novel in this literature.
The rest of the paper is organized as follows. Section I develops the basic model with local
information flow, and Section II analyzes thismodel. Section III considers a model with indi
rect information transmission. Section IV discusses two important aspects of our model, linear
costs of acquiring information and forming links and the link formation protocol, respectively.
Section V concludes.
I. Model
n > 3 be the set of players and let i and j be typicalmembers of this
Let N=
{1,2,..., n) with
set. Each player /chooses a level of personal information acquisition xt G X and a set of links
with others to access their information,which is represented as a (row) vector gf= (giUgu_h
where gtj G {0,1}, for each j G N\{i}. We will suppose thatX = [0, -foe) and that
8a+\>
?^/n)?
?
We say thatplayer ihas a linkwith player j ifgtj= 1.A link between player
e
g. d
{0, l}n_1.5
/and j allows both players to access the information personally acquired by the other player.6
x Sn as the set
The set of strategies of player i is denoted by St ? X x Gt. Define S ? Sx x
of strategies of all players. A strategy profile s = (x,g) G S specifies the personal information
=
=
acquired by each player, x
(gt,g2,g?).
(xlfx2, ...,*?), and the network of relations g
The network of relations g is a directed graph; letG be the set of all possible directed graphs
on n vertices. Define A^(g) = {j G N :gtj= 1} as the set of players with whom i has formed a
=
=
link. Let ///(g)
c7(g), where
N^g)
|
|.The closure of g is an undirected network denoted by g
=
a
In
involves
in
N.
the
closure
of
directed
network
for
each
/
and
words,
j
gij
max{gijy g?}
=
=
an
one.
Define
TV
undirected
G
of
directed
g by
edge
replacing every
N^g)
1} as the
{j
igy
set of players directly connected to /.The undirected link between two players reflects bilateral
information exchange between them.
4
There
is a small body of work which combines network formation and play in games (see, e.g., Goyal and Fernando
and Yves
2005; Jackson and Alison Watts 2002; Bramoull? et al. 2004; and Antoni Calv?-Armengol
Vega-Redondo
Zenou 2004). The game studied and the questions addressed in the present paper are quite different from this literature,
and so a detailed discussion of the relation with these papers is omitted.
5
We have completely solved themodel with discrete information choice variable. We find that themain results on
equilibrium information acquisition and networks are robust to a change in action sets. The details of these derivations
are available in aWeb Appendix for the paper.
6
The Web Appendix presents and analyzes a model inwhich link formation and information flow are both one way.
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SEPTEMBER 2010
THE AMERICANECONOMIC REVIEW
1472
The payoffs to player i under strategyprofile s =
(i) nf.(s)= fL + ?
(x, g) are:
xj)- cXi-m(g)k,
where c > 0 reflects the cost of informationand k > 0 is the cost of linkingwith one other person.
assume pure local externalities: player /accesses only the informationpersonally acquired
his
immediate neighbors. Section III studies a model with indirect information transmission.
by
We will assume thatf(y) is twice continuously differentiable, increasing, and strictly
concave in y. To focus on interesting cases we will assume that/(0) = 0, f'(Q) > c and
= m < c. Under these
assumptions there exists a number y > 0 such that y
f'(y)
lim^^
= c.
=
cy, i.e., y solves/'(y)
argmaxyex/(y)
We now discuss thekey elements of our model.
First, consider the returns from information.We may thinkof the action x as draws from a dis
tribution,e.g., the price distribution for a product. If the differentdraws are independent across
individuals and players are interested in the lowest price, then the value of an additional draw,
which is the change in the average value of the lowest order statistic, is positive but declining
in the number of draws. Another possible interpretation is in terms of individuals choosing an
We
action whose payoffs are unknown. Every individual has access to a costly sample of observa
tions?which may reflectpersonal experience with a product or a technology.A linkwith another
player then allows access to his personal experience. Under reasonable conditions, the returns
from accessing more samples of information?own and others?are increasing but concave.7
Second, consider theprotocol of link formation.We assume a person can form a binary link (it
takes value one or zero) and that this link is formed once a cost is incurred.One possible inter
pretation of unilateral link formation and two-way exchange of information is that one player
pays fora telephone call and the call involves an exchange of information.Alternatively, we may
interpretthe cost incurred in the formation of a link as a "gift" or a social favor that the player
forming the linkmakes to the person receiving the link. In this case, k becomes a transfer,and
the payoffs to player i in a strategyprofile s = (x, g) are given by:
-
+?
*7/(g)*
gjik
The last term involving transfers is independent of the strategy of /. It then follows that for all
s_fe S_h ands,, s/G S^ft^s,-, s_,) > ft^s/, s_f), ifand only ifUfa, s_,) > II^s/, s_f).Therefore
our methods of analysis and our equilibrium findingswith payoffs (1) carry over to the alterna
tivemodel, where link formation costs are transfersfrom one individual to another.
Third, we assume thata player has no interest inmisleading others and that there are no incen
tives for refusing to share information.This assumption is reasonable in a number of contexts
such as informal social communication between individual consumers.
7
For a detailed
Hojman
and Szeidl
discussion
of these examples
on information sharing, see Bramoull?
(2008).
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and Kranton
(2007)
and
VOL.
GALEOTT1ANDGOYAL: THE LAWOF THE FEW
100 NO. 4
1473
Finally, we assume that there is no bargaining and no prices for information.We are aware
that this is a strong assumption. As a firststep toward incorporating prices, in Section IV, we
study the case where players can ask for transfers in exchange for sharing information.8
A Nash equilibrium is a strategy profile s* = (x*,g*) such that:
(2)
nf(s?, sV)
>
s*_f),Vsf G Si9 V? G N.
Ufa,
An equilibrium is said to be strict if the inequalities in the above definition are strict for every
player.
We define social welfare to be the sum of individual payoffs. For any profile s social welfare
is given by:
w(s)
(3)
=
En,-(s).
A profile s* is socially efficient ifW(s*) > W(s), Vs G S.
We say that there is a path in g between /and j ifeither gy = 1 or there exist players jh ...,jm
? ... =
?
=
distinct from each other and i and j such that
1}. A network g is
gjmj
gjij2
{gij]
a
we
connected if there exists path between every pair of players;
say that a network g ismini
mally connected if it is connected and there exists only one path between every pair of players.
In a core-periphery network there are two groups of players, A^g) and N2(g), with the feature
=
=
thatAfy(g)
Nodes whichhave
N2(g) forall i GNx(g)9andNj(g) N\{j} forall j G /V2(g).
n?
1 links are referred to as central nodes or as hubs, while the complementary set of nodes
are referred to as peripheral nodes or as spokes. A core-periphery network with a single hub is
referred to as a periphery-sponsored star.Figure 1 illustrates core-periphery networks. There are
n = 8 players; in each architecture the black nodes are the hubs (the setN2(g))9 thewhite nodes
are the spokes (the set^(g))
and an edge starting at /with the arrowhead pointing at j indicates
that i sponsors a link to/ We say thatg is a regular network of degree v ifeach player has v con
nections
in g. A
complete
network
is a regular
II.
network
with
v?
n
?
1.
Analysis
The main result of this section is thatevery (strict)equilibrium exhibits the law of the few: a
small subset of individuals personally acquire information,while the restof the population of indi
viduals form connections with this small set of informationacquirers. This differentiation in turn
generates an elegant architecture of social communication: players who personally acquire infor
mation constitute hubs, while the restof theplayers are spokes in a core-periphery social network.
We start by noting that in an equilibrium every player must access at least y information
(where y is the optimal information acquired by an isolated player). Moreover, the perfect
substitutability of own and neighbor's information and the linearity in the costs of acquiring
information imply that if a player personally acquires information then the sum of the informa
tion he acquires and the information acquired by his neighbors must equal y.We next observe
that if some player acquires y9 and if k < cy, then it is optimal for all other players to acquire
no information personally and to form a link with this player. Lemma
1 summarizes
8
For other recent studies of models
Cabrales
and Piero Gottardi
(2007)
these
inwhich players can charge prices or bargain for their information, see Antonio
and Myeonghwan Cho (2007).
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1474
THE AMERICAN
ECONOMIC
REVIEW
SEPTEMBER
2010
observations. For a strategy profile s = (x, g), let us define /(s) = {/ G N\xl > 0} as the set of
=
be the information that i
players who acquire informationpersonally, and let v,-(g)
Y,jeN(g) Xj
accesses from his neighbors.
1: In any equilibrium s* = (x*,g*), x* + v*(g) > y,for all i G N, and ifx* >
x* + y*(g) = y.Moreover, ifk< ey and x* = y thenx* = 0, for all j ^ i.
LEMMA
0 then
The first statement in the lemma is similar to a result obtained in Bramoulle and Kranton
(2007) in the context of local public good provision in a fixed network; we note that the second
statement arises out of the endogenous linking. The proof is given in theAppendix.
This lemma tells us a great deal about the information accessed by individuals but relatively
little about the distribution of personal information acquisition, the aggregate information
acquired in a social group, and the structureof social communication. Our next result addresses
these
concerns.
PROPOSITION
1:Suppose payoffs are given by (1). Ifk > ey then there exists a unique equilib
rium inwhich every player acquires information y and no one forms any links. Suppose k < ey
and let s* = (x*,g*) be an equilibrium.
?
y tnen g* Is a core-periphery network, hubs acquire
(0 V ? ieNx?
and
ally,
spokes acquire no informationpersonally.
(") IfY,ieNx*>
informationperson
y then:
?
=
=
a) Every player i G /(s*) chooses x*
k/c and has A G {1, ...,n
(y/(A + 1))
2}
linkswithin /(s*),while everyplayer j ? I(s*) forms A + 1 linkswith players in /(s*).
=
b) High information level players choose x*
k/c, low information level players have
links
with
level
rj
high information
players, theyare not neighbors of each other and
choose informationx* = y ? rj (k/c),where (yc/k) ? 1< rj < yc/k.
If the cost of a link exceeds the cost of acquiring the threshold information, k > ey, then no
one forms any links. Otherwise every equilibrium is characterized by linking activity, i.e., the
network is nonempty. Figure 2 illustrates equilibrium outcomes for n ? 8, y = 1 and k < c.
There are two types of equilibria: one, where aggregate information is equal to y (Figure 2A)
and two,where itexceeds y (Figure 2B). When aggregate information equals y, equilibrium net
works are connected, theyhave the core-periphery structure,and players in the core are the only
ones who acquire any informationpersonally. Moreover, as the relative cost of linking
k/c grows
the number of hubs decreases, each hub player acquires more information,and the total number
of links decreases. When k/c G (y/2, y) there is only one hub, and the social communication
structure takes the form of a periphery sponsored star.
When
aggregate information exceeds y, equilibrium networks may not be connected, but
players acquire at most two levels of information (as in Figure 2B). Moreover, since aggregate
information acquired exceeds y we know, fromLemma 1, that theremust exist players who are
accessed by some but not by other players. This means that the cost of linking is exactly equal to
the cost of information acquired by such a player. But thenplayers are indifferentbetween form
ing a linkwith such a player and acquiring informationpersonally. In otherwords, the strategies
of the players are not a strict best response to the strategies of others. The following result brings
out the general implications of this observation.
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VOL.
GALE0TT1AND GOYAL: THE LAWOF THE FEW
100 NO. 4
Panel
A. Nash
equilibrium
inwhich
aggregate
information acquisition
Panel
B. Nash
equilibrium
inwhich
aggregate
information acquisition
Figure
2. Example
of Nash
Equilibrium,
n=
8, y
?
1475
is 1
exceeds
1
1
2: Suppose payoffs are given by (1) and k < cy. In every strict equilibrium
PROPOSITION
=
s* = (x*,g*): (/)
y> (") tne network has a core-periphery architecture, hubs acquire
J^ieNx*
and
informationpersonally,
spokes acquire no informationpersonally, and (Hi)for given c and
? oo.
?>
n
as
ratio
the
0
k,
I(s*) \/n
\
PROOF:
firstprove that in every strictNash equilibrium s = (x, g) the aggregate information is
?
=
to
y, i.e., YlieNxi
S- Let s
equal
(x, g) be a Nash equilibrium in which aggregate infor
mation exceeds y. First, suppose s satisfies part ii(a) of Proposition 1.We know that a positive
informationplayer i chooses xt = kjc < y and formsA links such thatxt[A + 1]= y, and that
A + 1< /(s).Moreover cxt= k. Then it is immediate that this player is indifferentbetween a
link and acquiring additional information k/c. This means that equilibria in part ii(a) are not
strict.Second, suppose s satisfies part ii(b) of Proposition 1.Again a player with positive infor
mation acquisition is indifferentbetween forming a link and acquiring extra information him
? k.
Hence, equilibria in part ii(b) of the proposition are not strict.Taken together
self, since ex
We
with Proposition 1 this implies that in every strict equilibrium aggregate information acquisi
tion is equal to y. The core-periphery architecture of equilibrium networks follows directly from
Proposition 1.
We now consider the proportion |/(s) \/nin every strict equilibrium s. Recall that in a strict
=
9 anc* that xt+ j/(g) > y, for all i G N. This means that every player
equilibrium Y,ieNxi
who acquires information personally is accessed by every player in equilibrium. This implies
that there is at most one player /G I(s) with no incoming links, i.e., g? = 0, for all j G N. For
all other players /G /(s), itmust be the case there is at least one player j G I(s) such thatgjt = 1;
but this implies thatxt > k/c. So the number of accessed players who acquire informationper
?
1, is bounded above by (yc)/k. It follows that I(s)/n <
sonally and have incoming links, I(s)
can
which
be
made
+
arbitrarily small by raising n. This completes the proof.
l)/n,
([yc/k]
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1476
SEPTEMBER 2010
THE AMERICANECONOMIC REVIEW
If a positive information player strictlyprefers to retain a linkwith another player j, itmust
be the case that the costs of linkingwith j are strictly lower than the cost of the information that
player j acquires. But then all other players find itoptimal to access fs information, and so all
positive information players are linked among themselves. Lemma 1 tells us thatany player who
acquires information personally must access exactly y information.Hence the desired conclu
sion: aggregate information acquisition must equal y. Part (ii) of Proposition 2 then follows from
Proposition 1.
The last part of the result derives bounds on the number of hub players. If player /links with
j then the cost of the linkmust be less than the cost of personally acquiring the information
accessed fromj, i.e., cxj > k, and so player j must be acquiring at least k/c information. Since
aggregate information is y, itfollows that there is an upper bound on the number of players who
acquire information, /(s*), and this number is independent of n. It then follows that the ratio
can be made arbitrarily small by suitably raising n. Proposition 2 shows that these prop
I/(s*) \/n
erties arise in every equilibrium of our game.9 Thus, the law of the few obtains as a consequence
of strategic interaction among ex ante identical and rational individuals.
Proposition 2 tells us that the number of players personally acquiring information is small
relative to the number of players in large societies. But for fixed n, it does not determine the
number of hubs, nor does it tell us who the hub players are. A recurring theme in the empirical
literature is that even though hubs seem to have similar demographic characteristics as the oth
ers, they have distinctive attitudes that include higher attention to general market information
and greater enjoyment in collecting information.A natural way tomodel this difference is to
suppose that some players have slightly lower costs of acquiring information.We consider a
? e>
situationwhere ct= e for all / =^ 1,while cx = c
0, where e > 0 is a small number. Let
?
?
e
as
as
>
>
0,
y, and yx?> y as e ?> 0. We focus on
yx
y y SiTgmaXy
cxy.
long
Clearly,
f(y)
strictNash equilibria.
3: Suppose payoffs are given by (1), ct= cfor all i^ 1 and
PROPOSITION
e > 0. If k <f(yx) ?f{y) + cy then in a strict equilibrium s* = (x*,g*):
(ii) the network is a periphery-sponsored star and player 1 is the hub, and
?
?
?
?
=
and spokes choose x* ? 0, OR x* = [(n
l)y yx]/[n 2] and x*
[yx
=
c?
e, where
?
$\>
(/)YjieNx?
?
either
xx
yx
(Hi)
?
all
for
2],
y]/[n
cx
Ml.
The proof is given in theAppendix. Proposition 3 shows that a very small difference in the
cost of acquiring information is sufficient to separate the player who will acquire information
and act as a hub from those who will acquire little or no information personally and will only
form connections.10
We now discuss the ideas underlying this result. First, observe that for the low cost player
the optimal information level is greater than the optimal information level for other players,
i.e.,yx> y. From the arguments developed inProposition 2 we know thataggregate information
acquired by all players other than player 1will be at most y. This implies that in equilibrium,
?
yx, the best reply of every other
player 1must acquire information personally, xx > 0. If xx
no
a
to
to
and
form
link
with
is
information
player 1. In case xx < yxwe know,
acquire
player
fromLemma 1, thatxx + yx(g) = yx and so there is a player i^ 1with xt > 0 and xt+ y,(g) = y.
9
Indeed, part (iii) of Proposition 2 can be strengthened to read: for every strict equilibrium, lim^^^ I(s*)/na ?> 0,
for all a > 0. We thank a referee for this observation.
10
We have focused on slight differences in costs of acquiring information; analogous arguments show that if one
player derives greater marginal benefits from acquiring information, as compared to others, then he will constitute the
hub of the social network and acquire more
information than the others.
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VOL.
100 NO. 4
GALEOTTIAND GOYAL: THE LAWOF THE FEW
1477
If some player wants to link with / then so must everyone else. But then player i accesses all
information yx\ since yx > y, this contradicts Lemma 1.Thus no player must have a linkwith
player i^ 1 in equilibrium. Hence, /must form a link with player 1, and, from the optimality
of linking, somust every other player. Finally, since every player is choosing positive effort, the
?
=
equilibrium values of xx and xt can be derived from the two equations xx+ (n
l)jcf yx and
=
x\+ xi
y-n
A. Efficient Outcomes
Given their salience it is important to understand thewelfare properties of specialization in
information acquisition and social communication. Proposition 1 tells us that in a Nash equilib
rium for every player /G N, xt -h y,(g) = y. Thus, in every equilibrium, aggregate gross returns
are nf(y). lfk< cy, given the linearity in costs of information and linking, the efficientequilib
riumminimizes the total costs of information and links. This immediately implies that the effi
cient equilibrium is a periphery-sponsored star network inwhich the hub acquires informationy
and every spoke chooses 0.
However, individual information acquisition is a local public good, and this implies that so
long as equilibrium entails any links, therewill be underprovision of information acquisition
relative to the social optimum. To see this, note that in the star the hub player chooses y, and at
this point/'(y) = c. But marginal social returns are given by nf'(y), which are larger than c, for
n > 2. The following proposition characterizes efficientoutcomes.
PROPOSITION
4: Suppose payoffs are given by (I). For every c, there exists a k > cy such that
then
the
ifk<k
socially optimal outcome is a star network inwhich the hub chooses y (where
?
while
all other players choose 0. If k> k, then in the socially optimal outcome
nf'(y)
c)>
chooses
every player
y and no one forms links.
PROOF:
=
Suppose s
(x, g) corresponds to an efficientprofile.We firstshow that ifg is not empty, then
a
star.
Let g be a nonempty network and suppose thatC is a component in g. Let C
g is
| |> 3
be the number of players in C. Suppose that y is the total information acquired in component
?
C. Then it follows that the total payoff of all players in component C is at most C
| \f(y)
?
?
cy
l)k. Consider a star network with \C \players inwhich the hub player alone chooses
(\C\
y. It then follows that this configuration attains themaximum possible aggregate payoff given
efforty.Moreover, note that aggregate payoff in any profile s in which two or more players
acquire positive information is strictly less than this, since itwill entail the same total costs of
information acquisition and a strictlyhigher cost of linking or a strictly lower payoff to at least
one of the players. So the star network with the hub acquiring all the information personally is
the optimal profile for each component.
Next consider two ormore components in an efficientprofile s. It is easy to see that in a com
= c. If the
components
ponent of size m, efficiency dictates that information y satisfymf'(y)
are of unequal size then information acquisition effortswill be unequal and a simple switching
of spoke players across components raises social welfare. So in any efficientprofile with two or
11
In a recent paper on network formation with exogenous
information levels, Hojman and Szeidl (2008) obtain a
result on how small differences between players can help select the identity of the hub players. Their result is derived
in a dynamic model and relies on stochastic stability arguments. By contrast, our result shows that with endogenous
information acquisition, a slight amount of individual heterogeneity is sufficient for the selection of hubs in a one-shot
static model.
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1478
THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
more components, the components must be of equal size. Let m be the size, and let the efforty
= c.
satisfymf'(y)
Suppose now that the network contains two components Cx and C2 of size m.
Consider thenetwork inwhich the spoke players in component 2 are all switched to component 1.
This yields a network g' with components C[ and C'2 with the former containing 2m ? 1 players
while the lattercontains 1 player. Then thepayoff remains unchanged. However, the information
level y is no longer optimal in either of the components. So, for instance, informationcan be low
ered in component 2 and the aggregate payoff thereby strictly increased, under the assumptions
on/(-). A similar argument also applies to networks with three ormore components, and so we
have proved thatno profilewith two ormore components can be efficient.Thus, ifg is not empty
?
theng is a star,and the informationof the central player is y = arg
cy.The social
maxyEX nf(y)
?
?
=
welfare associated to such profile is: SW
cy
nf(y)
\)k.
(n
Finally, note that if s is socially efficient and g is not a star, then g must be empty and
?
every player will choose information y. The social welfare is then SW = n[f(y)
cy]. The
expression for k is obtained by equating the social welfare in these two configurations, i.e.,
?
?
?
?
?
(n
l)k
n[f(y)
f(y)] + c[(n
l)y
y] + cy. To see thatk > cy, note that ifk < cy, then
~
~
?
?
?
n\f(y)
f(y)\+ ci(n tyy y] + c9 < (n l)cy,whichholds ifand only ifnf(y) cy<
~
=
=
cy. Given that y
cy, y
argmaxyeX nf(y)
argmaxy X/(y)
cy, and that/( ) is
nf(y)
strictlyconcave, the above inequality cannot hold. This concludes the proof of Proposition 4.
The key point to note is that given any profile of information acquisition and linking, there
is a corresponding star network inwhich the hub does all the information acquisition, which is
strictlybetter.This is a consequence of the linear costs of information acquisition and the posi
tive costs of linking. So, if the optimal social organization is a nonempty network, then itmust
be a starwhere the hub acquires all the information.The value of k is obtained by equating the
social welfare attained in the empty network and such a star network. The following example
illustrates the relation between equilibrium and socially efficientoutcomes.
? 1. In
=
=
=
1,while y = 2n
Example 1: Suppose c
1/2 and f(y)
ln(l + y). In this case y
Figure 3 we plot k as a function of the number of players. For a given n there are three regions.
For low costs of linking, k < 1/2, the efficient equilibrium is a star where the hub acquires
1 and the spokes choose 0. As compared to socially optimal outcomes, in equilibrium there
is underinvestment in information. For moderate costs of linking, k G (1/2, k), in equilibrium
we have underinvestment and underconnectivity relative to socially optimal outcomes
(noting
that k > 1/2). In the remaining region, equilibrium outcomes coincide with socially optimal
outcomes.
III. Indirect Flow of Information
In the basic model, a person can either acquire informationpersonally or get it from another
person who has directly acquired it himself. In this context, Propositions 2 and 3 show that
equilibrium leads to core-periphery social communication structureswhere players in the core
acquire all information. In actual practice, we often receive information from friends and col
leagues which they have themselves received from other friends. The aim of this section is to
examine the implications of this form of indirect information transmission.We show that infor
mation spillovers give rise to a new type of hub player/social influencer: the connector. A con
nector does not acquire informationpersonally but acts as an
intermediary between other people
who acquire information.
Given two players i and j in g, the geodesic distance, d(i, j; g), is defined as the length of the
shortestpath between i and j in g. If no such path exists, the distance is set equal to infinity.Let
Nli(g)
=
=
who areatdistance/from/ing.
{j e N: d(i,j; g)
/},thatisNj(g) is thesetofplayers
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VOL.
GALEOTTIAND GOYAL: THE LAWOF THE FEW
100 NO. 4
1479
0.5
Underinvestment
I->
3
n
Figure
3. Equilibrium
and Efficient
Outcomes
the level of spillovers by a vector a = {ax, a2, ...,an_{}, where ax > a2,>
an_x
?
and at G [0,1] for all /G {1,n
1}. It is assumed that ifj G N/(g), then the value of agentfs
information to i is given by atXj. Observe that the case ax = 1 and a2 = 0 corresponds to the
pure local spillovers model analyzed in Section II. To bring out the role of indirect information
transmission in the simplest formwe start by considering the polar case of no decay in flow
We measure
across
links,
i.e., an_x
?
l.12
The payoffs to player i under strategyprofile s =
(4) nl.(s)=/U + i; E
\
(x, g) can be written as:
xA-cxi-^k.
'=1;ew/(g) /
?
?
=
In network g, define
i.e., the informationwhich /accesses exclu
g
yt(g
y,(g)
^(g)
next
via
result
derives
the
Our
j.
sively
properties of equilibrium with indirect information
transmission.
5: Suppose payoffs are given by (4). If k> cy, there exists a unique equilib
rium: every player personally acquires information y, and no one forms links. Ifk< cy, then
=
s* = (x*,g*) is an equilibrium ifand only if: (i)
y, (ii) g* isminimally connected, and
^ieNx*
= 1, i, G N.
<
all
k
j
{Hi)
g*j
cyij{g*) for
PROPOSITION
12
See Hojman and Szeidl (2008) for an elegant model of interpersonal communication
value of information with respect to distance in a social network.
which
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leads to declining
1480
THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
PROOF:
The proof fork > cy of Proposition 5 is straightforward,and it is omitted. Hereafter, we focus
.
on k < cy We
firstprove that if s satisfies properties (i)-(Hi) in the proposition then s is a Nash
=
a player /;since
Take
y> g isminimally connected, and there is no decay,
equilibrium.
Yjj?Nxj
thenxt+ y^g) = y, so player /does not want to change his own information level, and also he
does notwant to form an additional link.The payoffs to /at equilibrium s are/(y) ? cxt? rj^k.
If ^/(g) ? 0>tnen player iplays a best reply. Suppose ^(g) > 0, thengtj= 1 for some j. Note that
player / is indifferentbetween keeping the linkwith j and switching the link fromj to a player
that i accessed via j. Also, property (Hi) says thatk <
and thereforeplayer /does not gain
cy^g),
a
s
the
link
with
is
Nash
j. Hence,
by deleting
equilibrium.
We now prove the converse. Let s = (x, g) be an equilibrium. Frictionless information flow
implies thatevery component of g must be minimal. Also, frictionless information flow together
with Lemma 1 imply that in every component the aggregate information is y. Next, suppose g
is not connected. Let Cx be a component of g. If xt = y for some /G Q, then all /'sneighbors
choose information 0 and sponsor a link to /,so Vs payoffs are f(y) ? cy and k < cy. But then
player /strictlygains ifhe chooses 0 and forms a linkwith a player j ? C2. Thus, inQ there are
at least two players choosing positive level of personal information acquisition; since Q ismini
mal itmust be the case that there is a link gty ? 1 for some /',
/ G Q, such thatplayer /'accesses
via the link with / strictly less information than y, say z < y. It is then clear that if player /'
deletes the linkwith/ and forms a new linkwith a player inC2, he will incur the same costs, but
he will access strictlyhigher information.Therefore player /'can strictly improve his payoffs, a
contradiction. Thus, g is connected. Finally, it is readily seen that ifgtj= 1 and s is equilibrium,
thenk <
This concludes the proof.
cy^g).
Frictionless informationflow implies thatequilibrium networks areminimal.13 From Lemma 1
we know that in equilibrium every individual must access at least y information. If the costs of
linking k are smaller than the costs of acquiring the threshold level of informationy, then stan
dard considerations imply that the network is connected. Finally note that the costs of a link that
player i formswith player j must be lower than the value of information thatplayer i accesses
exclusively via the linkwith j, i.e., k < cy^?g*). This implies thateither player j acquires enough
information on his own, or that player j is a connector and accesses others who have enough
information.
We explore the distribution of information acquisition and the architecture of social commu
nication via an examination of differentclasses of equilibria.
the hubs personally acquire information.Figure 4A illus
Hubs Acquire Information.?Here
trates these equilibria. At a superficial level these equilibria are similar to the core-periphery
equilibria of thebasic model. However, there is a key difference: in thepresent context, the infor
mation hubs acquire as well as aggregate information.This can be seen in the equilibrium on the
right inFigure 4A: each hub personally acquires informationbut also passes on informationfrom
the other hub. This transmission in turnexplains theminimality of the equilibrium network.
hub is an active connector in the sense that he acquires no
Hub as Active Connector.?The
information himself but forms links with all players who are acquiring information.All play
ers who acquire no information in turn linkwith the hub. Figure 4B provides examples of such
equilibria. An equilibrium with |/(s*) |peripheral players acquiring information exists whenever
the costs of linking are smaller than the benefits of accessing a single information acquirer, i.e.,
13
This builds on a result of Bala and Goyal (2000), which establishes minimality
of pure network formation and frictionless information flow.
of equilibrium
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networks in a model
VOL.
4
100 NO.
Panel
A.
GALEOTTI
AND GOYAL:
THE LAW OF THE FEW
1481
Information hubs
Information
hubs
Panel
B. Hub as active
connector
Information
mmm
collectors
,..Active
connector
' Information
collectors
Passive
connector
Figure
4. Hubs, Connectors,
and Others,
n=
8
k < cy/\ /(s*) I.As k increases the number of informationacquirers falls, and each positive infor
mation agent must acquire more information.This is in line with one of the results in the basic
model: the number of players acquiring informationdeclines with the cost of linking.
Hub as Passive Connector.?All
players link with a single player who himself acquires no
information but serves as a gateway to those who do. Figure 4C illustrates this type of equi
librium. As costs of linking increase, initially, the number of players acquiring information
increases, and each of them acquires less information.That is, an increase in costs of linking
necessitates a greater number of active information sources. If every peripheral player acquires
information, an increased cost of linking implies thateach periphery player acquires less infor
mation, which is possible only if the hub acquires some information. In other words, an equilib
riumwhere the hub acts as a passive connector is no longer sustainable for large costs of linking.
We conclude this section with three remarks.First, the equilibria illustrated inFigure 4 exhibit
patterns which are consistent with the empirical work on social communication. In their study
of personal influence, Katz and Lazersfeld (1955) emphasize that social influencers typically
have more social ties and also acquire more information (via radio, newspapers, and television).
We interpret this as a situation in which influencers acquire information. In other instances,
a small amount of?information personally, but their numerous
hubs acquire some?possibly
contacts provide new informationwhich they then communicate to theirneighbors and friends.
Here the highly connected individual functions primarily as a connector. See Malcolm Gladwell
(2000) for an engaging discussion of such patterns of information acquisition and social com
munication and Cross and Parker (2004) for a description of social connectors within firms.
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1482
THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
Second, we note that the equilibria presented inFigure 4 are the only outcomes in thepresence
of small heterogeneities across players. In particular, if there is a player with sightly lower costs
of acquiring information and slightly lower costs of being accessed by others, then he becomes
the informationhub, and everyone else forms a linkwith him. If, instead, theplayer with the low
est cost of information acquisition and themost sociable player are different, then in the unique
(strict)equilibrium themost sociable player is a pure connector: he linkswith the low cost infor
mation player, who acquires all the information,while all other players form a link with him.
These claims are formally stated and proved in theOnline Appendix.
Our final remark is about equilibrium in a general model of decay. We note that lack of decay
with respect to information acquired by immediate neighbors is a necessary condition for equi
libriumwhere hubs acquire all information and spokes acquire no information. Indeed, in every
such equilibrium the aggregate information acquisition is y, and there is always a player who
relies at least partly on information acquired by others. If ax < 1, then information available for
this player is strictly less than y, which contradicts Lemma 1.Next observe that lack of decay
with respect to immediate personal contacts is also sufficient: an equilibrium inwhich the net
work is a periphery-sponsored star and the hub acquires all the information exists so long as
? 1 and k <
ax
cy.14We next turn to the nature of influencers under gradual decay. Suppose
1
<
and
>
a2
ax
a3 > 0. A periphery-sponsored star network in which each of the peripheral
?
=
x
chooses
players
y/[I + a2(n
2)] while the central hub player acquires no informationper
n
an
so
as
is
is sufficientlylarge.Hence, in a general model of decay, we
sonally
equilibrium
long
to
connectors.
hubs
the
role
of
pure
expect
play
IV. Discussion
In this section we discuss two aspects of themodel which play a prominent role in our analy
sis: (/) the linear costs of acquiring information and forming links, and (if) the link formation
protocol.
A.
Convex
Costs
The linear costs of acquiring information and forming links has the following implication: for
any player acquiring information the total information accessed is y9 and this level is indepen
dent of the amount of information acquired by the neighbors. However, if the costs of personally
acquiring information are increasing and convex, this is no longer true.This section explores
equilibrium outcomes under convex costs.
=
=
Define
z,
0,
satisfy the following properties: C(0)
*,H-f7/(g)fc,and let C(zi)
=
=
>
on
>
for
and
>
case
for
>
We
0.
focus
the
of
0, C'(zi)
0,
0,
Zi 0,
z,
C"(0)
C"fe)
C'(0)
frictionless information transmission, i.e., an_x= 1. Recall that y^g) > 0 is the information
accessed by player /from the others. The payoffs to player /facing a strategyprofile s = (x, g)
are given by:
=
n,.(s) f(x,+ y,(g)) C{x,+ r?,.(g)*).
(5)
We firstnote that, in equilibrium, a network is either empty orminimally connected.15Moreover,
the lack of decay in information transmission implies that in a minimally connected network
14
We thank an anonymous referee for drawing our attention to this fact.
15
This follows from standard arguments which rely on network externalities
transmission through the network; see, e.g., Bala
and Goyal
and the lack of decay
(2000).
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in information
VOL.
100 NO. 4
GALEOTTIAND GOYAL: THE LAWOF THE FEW
1483
every player accesses the informationacquired by all players. Let y be the aggregate information
acquired in equilibrium. For any player iwho acquires informationpersonally, the following first
order condition must hold:
(6)f'(y)
=
c'(xi+m(s)k).
It then follows thatplayers sponsoring an equal number of linksmust acquire an equal amount
of information, and this personal information acquisition is declining in the number of links
? 1
links, and so at least one player
sponsored. In anyminimally connected network there are n
forms no links. Thus specialization in information acquisition remains an essential aspect of
equilibrium behavior even under convex costs.
We now turn to aggregate information acquisition in equilibrium. Recall that in the basic
model with linear costs, propositions 2, 3, and 5 prove that aggregate information is invariant
with regard to the number of players and the costs of forming links (so long as k < cy). A greater
number of players allows for smaller per capita acquisition of information; under convex costs
this leads to lowermarginal costs. So as we raise the number of players, aggregate information
should increase. Similarly, we expect that link formation costs will affect aggregate information,
as these costs now enter the firstorder conditions of individual optimization (for all positive
informationplayers).
We now develop the implications of these ideas formally.We restrict attention to nonempty
networks and suppose all players acquire information.Define x as the information acquired by
the zero link player. From the optimality of equilibrium actions and equation (6) it follows that
for the zero link player f'(y) = C'(x)
(where y is aggregate information acquired), while for
a player /with r]i links we have that/'(y) = C'(jc,-+ rjfk). So we infer that for any player /,
=
xt+ rjik x. Summing across all players we obtain an expression for aggregate information
?
= nx ?
acquisition y
(n
\)k.
6: Suppose thatpayoffs are given by (5) and that s* = (x*,g*) is a nonempty net
PROPOSITION
work equilibrium inwhich all players acquire information.Aggregate information isdecreasing
in the costs of linking and increasing in the number ofplayers.
The proof of this result is given in theWeb Appendix to the paper. Figure 5 illustrates these
? 4 and different values
findings. Figure 5A presents periphery-sponsored star equilibria for n
=
of k. Figure 5B illustrates periphery-sponsored star equilibria for k
0.1 and different values
of n.
Figure 5 also helps us explore further the role of differentiation and the architecture of social
communication under convex costs. To fix ideas, suppose that every player acquires informa
tion and the network is a periphery-sponsored star. From the above arguments we know that
in such an equilibrium the hub acquires information i, each peripheral player acquires infor
?
?
=
?> 0;
mation xp, where xp + k = x, and f'(nx
(n
l)k)
C'(x). Assume that \imx^ ff(x)
?
=
since C"(0)
0 and is strictly increasing thereafter, itnow follows thatxp
0 and thatx ?? k,
as n gets large. The example in Figure 5B satisfies these hypotheses. Thus we have shown that,
even with convex costs, sharp role differentiationwith a core-periphery communication network
emerges in large societies.
B. The Link Formation Protocol
We have so far assumed that a player can unilaterally form a link with another player. This
is a convenient and simple way tomodel link formation, and the research on network formation
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1484
THE AMERICANECONOMIC REVIEW
A. Periphery-sponsored
star equilibrium, n = 4, k =
=
k
k = 0.15, aggregate
0.1, aggregate
information
1.38
information1.27
Panel
Panel
0.1, 0.15,
the last decade
5. Aggregate
has
shown
Information
that it offers
0.2
k = 0.2, aggregate
1.2
information
B. Periphery-sponsored
star equilibrium, k = 0.1, n = 4, 5, 6
=
n
n = 5, aggregate
n
4, aggregate
information
1.38
information1.55
Figure
over
SEPTEMBER 2010
under Convex
Costs,/(y)
a tractable
framework
=
ln(l +
= 6,
aggregate
information1.66
y),C(z)
to address
=
z2/2
a range
of interest
ing questions. However, from a descriptive point of view, in social contexts it ismore natural to
suppose that social communication requires the active participation of the two players involved
in a link. This in turn suggests that both players in a linkmust bear some costs. This section
examines role differentiation and core-periphery social communication in a model with two
sided network formation.
Suppose a link is formed only when both players agree and the costs of the link are shared
? 1 if and
?
1, and for each link
equally between linked players. Formally, gy
only if g^g^
=
= 1 both
a
to
/
and
The
/
j pay k/2.
payoffs
players
player facing strategy profile s
(x, g)
gy
are given by:
- cXi
(7) n,(s)= /uV + E xj)
?z E?y
jeN
jeNi(g) /
the convention in the network literature,we allow for players to delete links uni
and Asher Wolinsky 1996). In addition, we allow for coordinated changes
Jackson
laterally (see
in information acquisition by players contemplating a new link. These considerations are sum
marized in the following solution concept.
Following
DEFINITION
and
(ii) for
1.A strategyprofile s = (x, g) isa pairwise equilibrium if(i) s isaNash Equilibrium,
= 0,
<
then
ifU^g -h gip x\,xj, x_,y) > n,(s)
gy
ny(g + gijfx[, xj, x_iy)
all
IT,-(s),Vxl,xjex.
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We now provide a partial characterization of pairwise equilibrium.
PROPOSITION
7: Suppose payoffs are given by (7) and k < cy. In a
=
=
rium s*
(x*,g*), x* > k/(2c) for all i G N. Moreover, s*
(x*,g*) with
?
=
v G {1,n
2}, and each player choosing effortx*
y/[v + 1] ispairwise
= x* =
only ifk G [cy/(v+ 1), 2cy/(v + 1)]. The complete network with x*
equilibrium ifand only ifk<
pairwise equilib
a regular network
equilibrium ifand
y/n is a pairwise
2cy/n.
The proof of this result is given in the online Appendix of the paper. If link formation requires
mutual acceptance and if costs of linking are equally shared between two players, then every
individual must acquire some information personally. Indeed, given costs of linking k and costs
of acquiring information c, there is a lower bound to information acquired by each player, and
this is independent of the number of players.Moreover, regular social communication networks
inwhich everyone acquires the same amount of information is an equilibrium. These equilibria
are in sharp contrast to our main results, propositions 1-3 and 5, in themodel with one-sided
link formation.We now examine the sources of this big difference in results in the twomodels.
Since social communication is costly, a player who acquires informationwill agree to form a
linkwith someone else who has no (ormuch less) informationonly ifhe is offered some compen
sation.More generally, an informed person may well ask for some compensation for his efforts
at acquiring the information in the firstplace. This compensation may take the form of social
favors or direct transfers (monetary or in kind). These ideas motivate a model of two-sided link
formationwith transfers.
Suppose each player chooses information acquisition and also proposes a set of transfers to
every other player. The transfersmay be positive (contributing to the costs of communication)
or theymay be negative (asking compensation for information acquired). Transfers proposed
=
where
G H. We assume that g y= 1 if and only if
by player /are denoted by t,
{T^jeN
=
+ T? > k. So, a strategyprofile s
(x, t) specifies effortsx and transferst. A strategypro
file s = (x, t) supports a core-periphery network g(i-) if there are two groups of players Nx(g)
andN2(g) suchthat(/)rtj+ r?< k forall /,j GN{(g), (//)forall /,j GN2(g), rij+ T?^ ^> anc*
(iii) for all i G N2(g) and; G A^g), Ttj4- r? > k.
The payoffs to /given a profile s = (x, t) are
(8)
n/(s)
=
cXi Y^8vkT)Tir
Xj)
1^
jeNi{g{r))
f\Xi + ?
\
J
Building on thework of Francis Bloch and Jackson (2007) we propose the following solution
concept for our game with transfers.
2. A strategy profile s = (x, t) is pairwise equilibrium if and only if (i) s
DEFINITION
> II/(s) then
is a Nash equilibrium and (it) for all rVj+ r? < k, if
II/(s_iy,Ty, r?, x[, xj)
n,.(s_y,
4 Tj,,x\,xj) < n?,
V4 xj e X, andWljyT'yr
The following proposition shows that if individuals ask for compensation to communicate the
information they acquire, the core-periphery communication network inwhich the hub acquires
information is an equilibrium. We assume e > 0.
PROPOSITION
8: Consider themodel with transfers.Suppose thatpayoffs are given by (8) and
assume that k < cy. The profile s* = (x* t*) such that: (i) g*(r*) is a core-periphery network,
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1486
THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
=
every hub acquires informationx*
yj \/(s*) |,and spokes acquire no informationpersonally,
= r* =
hubs
r*
every
for
pair
of
(i,j),
(ii)
k/2, and (Hi)for every hub i and spoke player j,
= ? ecx* and r* = k + ecx* is a
pairwise
equilibrium so long as k + ecy/I(s*) < cy/I(s*).
r*j
The proof of this result is straightforwardand omitted.While core-periphery structureswith
information acquiring hubs are sustainable in equilibrium, regular networks with homogenous
informationacquisition are also equilibrium outcomes. For example, the following profile consti
tutes a pairwise equilibrium in thegame with transfers ifk/c G [y/(y+ 1), 2y/(v +
1)]:For each
such that ? k/2 and ry7= k/2for
i,x* = cy/[v + 1]and there is a set ofplayers {ih i2,iv},
t*j
exactly j G {i{, i2,/v} players. These differentpossibilities naturally raise a question about rel
ative robustness of equilibria. This is an interestingquestion which we leave for future research.16
V.
Conclusions
The determination of information people have when making decisions is a central problem
in the social sciences. Our paper makes two contributions to the study of this problem. One, we
develop a model inwhich players choose investments in personal informationacquisition as well
as in forming links with others to access their information.Two, we show that the law of the
few?the phenomenon where a largemajority of individuals getmost of the informationneeded
for theirdecisions from a very small subset of the group?is a robust equilibrium phenomenon
in such a model.
There are two directions inwhich the analysis of this paper can be extended which appear to
us to be especially promising. In actual practice individuals decide on information acquisition
and linkswith others over time, and it is important to understand these dynamics. A second line
of investigation concerns the relation between personally acquired information and information
acquired by others.We have focused on the case where they are substitutes; itwould be interest
ing to study the case where they are complements.
Appendix
Given a network g, we define a component as a setC(g) of players such thatV/,j G C(g), there
exists a path between them, and there does not exist a path between any /G C(g) and a player
=
j G N\C(g). A component C(g) is non-singleton if C(g)
0,
|
|> 1.A player i is isolated ifgy
Vj GN\{i}.
PROOF OF LEMMA 1:
first prove statement 1 in the lemma. Suppose not and xt+ yt(g) < y for some / in
equilibrium. Under the maintained assumptions ff(xt+ j/(g)) > c, and so player / can
strictly increase his payoffs by increasing personal information acquisition. Next suppose
that Xi> 0 and xt+ y^g) > y. Under our assumptions on /( ) and c, if x( + )\-(g) > y then
f'(xi + J/(g)) < c\ but then / can strictly increase payoffs by lowering personal information
acquisition. This completes the proof of statement 1 in the lemma.
We now prove statement 2 in the lemma. Suppose that s = (x, g) is an equilibrium inwhich
=
xt
y and there is Xj > 0, for some j ^ /.First, since xt > 0, it follows from the firstpart of
We
16
We note that the incentives to acquire information are not affected by transfers as transfers are not conditional
information acquisition. This also limits the ability of transfers to facilitate efficient information acquisition.
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on
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GALEOTTIAND GOYAL: THE LAWOF THE FEW
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1487
1 thatxt+ y^g) = y. This also implies that every player in the neighborhood of imust
=
acquire no information personally. Now consider j, with Xj > 0. This means that ~gtj 0. It
?
=
=
follows fromLemma 1 thatXj + y,(g)
y. If Xj
y then this playermust get payoff/(y)
cy.
If he switched to a linkwith /and reduced personal information acquisition to 0, his payoff is
? k. Since k <
=
cy, Xj
y is clearly not an optimal strategy for player j. So s is not an equi
f(y)
librium. Next suppose thatXj < y. In equilibrium Xj + y;(g) = y, and so there is some player
/^ /such thatgfl= 1 and xt G (0, y). It is clear that ifgjt = 1 then player j can strictly increase
his payoffs by switching the link from / to /.Similarly, ifgtj= 1, then player /gains strictlyby
switching link fromj to /.So s cannot be an equilibrium. A contradiction which completes the
Lemma
proof.
PROOF OF PROPOSITION 1:
The proof for the case k > cy is trivial and thereforeomitted; we focus on k < cy. First suppose
?
that
y. In this case it follows fromLemma 1 that I(s) must be a clique. Furthermore,
^2ieNXi
= 0 for all i G
= 0 for all i,
j ? /(s), and gtj
/(s), j ? /(s). Therefore, each player choosing 0
gij
a
must sponsor linkwith every positive information player. This shows that the networkmust be
a
core-periphery
network.
=
Hereafter, let s
(x, g) be an equilibrium where ? ieNxt > y. The proof for this case is devel
oped in three steps. In the first step,we consider the case inwhich positive informationplayers
choose the same level of information. In the second stepwe consider situations inwhich positive
information players choose different levels of information.The third step uses the observations
derived in the previous two steps to conclude the proof.
Step 1:We prove that ifall players who acquire informationpersonally choose the same level,
then s satisfies ii(a) of Proposition 1. Suppose xt= x, Vi G /(s). If x = y, Lemma 1 implies that
= 1 and therefore
aggregate information is y, a contradiction. Assume x G (0, y); from
I/(s) I
Lemma 1 it follows thatxt 4- y,(g) = y, V/ G /(s). Since, by assumption, xt = x, V/ G /(s), it fol
lows that every player who accesses informationpersonally also gets an equal amount of infor
mation from his neighbors, which immediately implies that every positive information player
has the same number of links with positive informationplayers; letA be this number. Note that
for all i G /(s), xt+ V/(g)= x + Ax = y, which implies thatx = y /(A -f 1). Since aggregate
information is strictlyhigher than y it follows thatA + 1< |/(s) |.Also, from Lemma 1 we
know thatx < y, which implies thatA > 1.Thus, there exist two positive information players
who are neighbors, implying that k < ex. Also, since, by assumption, ^
ieNxt > y, there exist
two positive information players who are not neighbors, implying that k > ex. Hence, k = ex.
= N, the result follows. If not, select
j ? I(s). Clearly, in equilibrium no player
Finally, if I(s)
forms a link with j. So, in equilibrium j must sponsor A + 1 links with positive information
players. This concludes the proof of ii(a) of the proposition.
Step 2: Let g' be the subgraph of g defined on /(s). Let C(g') be a component of g;. By construc
tion each player inC(g') chooses positive information. Suppose that (Al) total sum of informa
tion inC(g') is strictlyhigher than y and (A2) there exists at least a pair of players inC(g') who
choose a different level of information.The following lemma is key.
that (Al) and (A2) hold inC(g'). Then there are two types of players in
C(g/): high informationplayers choose x and low informationplayers choose x<x. Moreover,
every low information player forms rj links with high informationplayers, there are no links
?
? 1<
between low informationplayers, k = ex, x = y
rjx, and {yc/k)
rj < ye/k.
LEMMA
2: Suppose
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THE AMERICANECONOMIC REVIEW
SEPTEMBER 2010
PROOFOF LEMMA 2:
Without loss of generality label players inC(g'), so thaty > xx > x2 > ... > xm. (A2) implies
that there exists /G C(g'), /7^ m, such thatXj = xt = x, for all j < /,and x > xl+l.We startby
proving two claims.
CLAIM
/,gjt =
1:For all j >
1for some i<
I.
PROOFOF CLAIM 1:
?
0, V/ < /.This implies thatj does not sponsor
Suppose that there exists ay'> /such thatg?
links. If, on the contrary, player j sponsors links, then these links are directed to players/ > /,
but thenplayer j could strictlygain by switching a link from/ to some i< I.Note that itmust
also be the case thatj does not receive any links. Suppose j receives a link from a player/. Then
itmust be the case that player 1 is also a neighbor of/, otherwise / strictlygains by switch
ing the link from j to /.But this says that every player who sponsors a link to j is f s neighbor
and since player j only receives links, thismeans that player j accesses from his neighbors at
=
=
most as much information as player /does. Since Xj +
xt + y, (g)
y, this implies that
yy(g)
> xh contradicting our hypothesis that < x = xt. Thus j does not receive links. But then
Xj
Xj
=
Xj + y; (g)
Xj < y, which contradicts Lemma 1.Hence, claim 1 follows.
CLAIM
2: There exists some i, ? <
I such thatgu>= 0.
PROOFOF CLAIM 2:
is a clique. Since aggregate information in component C(g)
Suppose
{1,...,/}
exceeds y, itmust be the case then that for every /< /, there is one player j > I such that
? 0 for all
= 0. Select such a
player j. Clearly, gjj>
/ > /; otherwise j strictly gains by
gij
to
a link from some / > /, then /must
the
/.
link
from
if
receives
switching
/
j
Analogously,
also be a neighbor of/. Therefore, since {1,...,/} is a clique, it follows that every neighbor of
= x. So
= 0,
j is also fs neighbor, and this contradicts the hypothesis thatXj < xt
V/ > /.
~g^
a
note
accesses
that
this
that
/
of
the
accessed
superset
Finally
implies
players
by j, i.e., yf(g)
^ yj(~&)-We know that xt + yf-(g)= y = Xj + yy(g), and so Xj > xh which contradicts the
hypothesis thatj > L Claim 2 follows.
We can now conclude theproof ofLemma 2. From claim 1, thereexists a player j > Iwho spon
sors a link to a player /< l,sok<
ex. Similarly, claim 2 implies that thereexists /,/'< /such that
=
>
ex.
we have k = ex. Next, since k = ex and
this
k
0;
Hence,
implies
gjj'
Xj < x for all j > /,
?
itfollows thatgjj
0 for all j, j' > I.Therefore, every player j > I forms only linkswith players
in {1,..., /}.We now show thatXj = xj+l for all j > I. Select j > I and assume that >
Xj
xj+l > 0.
=
=
and
1
+
+
Lemma
+
+
Then, Xj
implies that
Xj
xj+{
xj+l
rjj+l (g)x.
y7(g)
y,+i(g)
rjj(g)x9
=
=
=
holds
which
whenever
+
+
Since
y,
xj
xj+l
Xj
xj+l
y;(g)
{rjj+{(g)
yj+M
r]j(g))x.
> 1, but then
where
the
last
Xj > Xj+{, then ryy+1(g) ry/g)
xj+h
(^+1(g)
r/y(g))x>x>Xj
inequality follows because, by assumption, Xj < x. Thus, all players j > I choose the same infor
?
mation, say x, and fromLemma 1 it follows thatx + rjj(g)x
y. Thus, every low information
same
number of linkswith high informationplayers, say 77,and x + rjx? y.
player sponsors the
This concludes the proof of Lemma 2.
Step 3:We now conclude the proof of Proposition 1.Recall thatg' is the subgraph of g defined
on /(s).We need to consider two cases: (/)g' is connected, and (//)g' is not connected.
g' is connected: firstobserve that (Al) holds by assumption. If all positive information play
ers choose same action then step 1 applies, and the proof follows. If
(A2) holds then Lemma
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GALEOTTIAND GOYAL: THE LAWOF THE FEW
100 NO. 4
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2 applies. We next observe that in this case every player i G N must choose positive infor
mation. To see this note that since k ? cx every player j ? /(s) will only sponsor links to
high information players. Then, by symmetry, low information players must obtain the same
= x, which
payoffs as players j ? I(s). It is easy to check that this is possible if and only ifx
contradicts (A2).
g' is not connected: Let Cx and C2 be two components in g'.We observe thatx-t< y, and from
Lemma 1 it follows that the components must contain at least two players each. Here, note
that for every
G Cx and j, j' G C2 such thatgiV= gjy = 1,xt>= xy = x > xif Xj and k = ex.
=
= x follows because, if <
Indeed, xt> Xy
xt> xy then player iwould strictlygain by switching
a link from ? to/; for analogous reasons it follows thatxif Xj < x\ Since / sponsors a link to
= cx.
/',k < ex, while /'does not sponsor a link to/, and so k > cx. Thus k
Together, these
observations imply that every player who receives a link in Cx and every player who receives
a link inC2 chooses information x. Thus, if inCx and C2 every positive player receives at least
one link, every player chooses the same information, and the proof follows from Step 1.
Suppose next that there is some player in Cx who does not receive a link, and information
acquisition is not equal across players. If the aggregate information in Cx equals y, thenCx is a
clique, and therefore there is atmost one player who only sponsors links and receives no links.
?
Since Cx is a clique, and aggregate information is y, this player will choose x = y ? ( Cx
|
|
l)x.
Finally, consider the case where aggregate information in Cx exceeds y, and personal infor
mation acquisition is not equal. Then Lemma 2 applies and there are two positive information
acquisition levels, x and x\ with x' < x. We observe that as in the case of connected network
= 0. Since
above, it is possible to rule out j such thatXj
Cx was arbitrary, this completes the proof
of Proposition 1.
PROOF OF PROPOSITION 3:
It is immediate to see that ifxx ? yx then the proposition follows. Next, if jcx=0
thenwe can
use Proposition 2 to show that in a strictequilibrium aggregate information equals y. Note how
ever that ifxx = 0, player 1must access at least yx > y from his neighbors, a contradiction.We
now take up the case of xx G (0, y x).
CLAIM
3: V/, j G 7(s)\{l},
ifgy
=
1 then i and j share the same neighbors;
l G /(s)\{i,7>,/GNiig) ifand onlyifl GN0).
i.e., for every
PROOF OF CLAIM 3:
Let gij = 1, if G 7(s)\{l}, and suppose, without loss of generality, that xt < Xj.We first
prove that for every /G 7(s) \{/, j}, if /G N/(g) then /G Nj(g). Suppose not and there exists a
= 1, then, since
xt < Xj, I (weakly) gains by
player /G 7(s), with /G Af;(g) and /^ A^(g). If gu
=
to
let
1.
the
link
from
/
Since
>
it
fromLemma 1 thatxt
follows
Hence,
0,
ga
jc,
j.
switching
?
?
=
s
to
/
in
an(itne
+ 3^1
cxt
equilibrium
S
payoffs
are/(y)
rji(g)k. Suppose that /deletes
(g)
the link with player / and chooses an information level xt ? xt -f xh then he obtains payoffs
?
?
?
?
cxt
cxt
\)k. Since s is a strict equilibrium this deviation strictlydecreases
f(y)
(rji(g)
f s payoffs,which requires thatk < cxt. Let k < cxtand consider the following two possibilities.
= 0, and since s is a strict
equilibrium, player j must strictly
(/) Xj > xt. In this case, since g?
lose if he forms an additional link with / and chooses information level Xj = Xj ? xt.
?
?
That is,/(j) ? cXj ?
>
which holds if and only
xl)
T}j(g)k f(y)
c(xj
(^(g) -f l)k,
ifk > exf,but this contradicts thatk < cxt.
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THE AMERICANECONOMIC REVIEW
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SEPTEMBER 2010
Here we have two subcases. (2a) Suppose gtj= 1; this implies that the costs for
(ii) Xj < jc/.
/to linkwith j are strictly lower than the costs of information that /accesses fromj, i.e.,
=
k < cxj. Since k < exp ~gtj
0, and, by assumption, xt > Xp then / strictly gains if he
?
links with j and chooses information level xt = xt
Xp So s is not a strict equilibrium.
= 1. Since
access
not
/
he
does
but
sponsors a link to /, it follows
j
(2b) Suppose gji
that Xi > xt. Next note that, by assumption, Xj < xt\ it follows that xt > xt > Xp which
contradicts thatxt < x-}.We have then shown that for every /G /(s) \{/, j}, if /G N,(g)
then /G
Nj(g).
We now showthatif /G /(s)\{/,j} and /GNj(g) then/GN^g). Supposenot; thenplayerj
all positive information players that /accesses plus some other positive information
players. But thiswould contradict thatxt< Xp since yt> y7.This concludes the proof of Claim 3.
accesses
CLAIM
4: Suppose
=
i,j G Nx(g), i,j G /(s), and g y
0, thengu =
gtj
= 0
for all I^
1, /G /(s).
PROOF OF CLAIM 4:
? 0 for all /
Suppose, without loss of generality, xt < XpWe first show that gu
^ 1, /G /(s).
=
some
/^ 1, /G /(s). In view of claim 3, since g y= 0,
1, for
Suppose, on the contrary, thatgn
theng y= 0; this fact and xt < Xjimplies thatgn = 1,and since g^ = 0, then from the strictnessof
=
=
=
^ + j/(g), itmust be
equilibrium, itfollows thatxt > Xp Since xt > Xpg?
l,andxy -fyj; y
the case that there exists some /'G /(s), /'G Nj(g), and /'^ ^(g).
Claim 3 implies that /'^ N^g).
Since jcz> Xj and /'^ A^/(g), then gp>= 1.But g?> = 1 and /^ A^(g) implies thatX/'> xt\ simi
= 1 and /'
a contradiction. Thus, the only neighbor of / is
^ Ni(g) implies thatxt >
larly,gl7
player 1. It is easy to see that the same holds for player j. Indeed, if /G /(s) and /G A^(g), then
claim 3 implies that /^ A^g), but thenplayer j accesses strictlyhigher information than player /,
which contradicts our initial hypothesis thatXj > xh Claim 4 follows.
Final Step in Proof of Proposition 3: We are concerned with the case xx G (0, yx). Since
?
xx + yv
yl9 there exists some /G /(s) such that /G Nx(g). Observe that given such an i, there
exists ay G Nx(g) such thatj ? N^g) and Xj > 0. This is because otherwise xt+ yf(g) > yx> y,
and this contradicts xt > 0 and Lemma 1.From claim 4 above we know thatplayers /and j do not
have any links with players in /(s). This means thatxt+ xx = y ? Xj + xx = y, and so xt ? Xp
Given thatwe are in a strictequilibrium, it follows thatk < cxv This implies that i and j choose
the same information and form a linkwith 1.We observe that this also means thaty > xx> x(
=
xj
>
0.
We now show that player 1 constitutes the hub of his component and that all other players
behave as players i and j identifiedabove. In a path of length of two ormore starting at 1, there
are fourpossible patterns: two players choosing 0, two players choosing positive information,and
twomixed cases. Clearly it is not possible to have two players choosing 0 as the costs of linking
are strictlypositive. Next consider a positive information player followed by a zero information
?
player. Suppose player /^ /(s), and suppose there is a linkwith player m such thatgmX 1. Since
/^ /(s), itmust be the case thatglm= 1 and so m G /(s). However, it is profitable for / to form
this link only ifk < cxm.Moreover, claim 3 implies thatgmi= 0; ifxt> xm, then, since k < exm,
?
player / strictlygains by forming a link with m. So jc,< xm, and since gmi 0 it follows that
k > cxv Putting together these facts we get thatxm> xh and so xm -f ym> xm+ xx > jc,+ xx
=
y. This contradicts Lemma 1.Thus we have ruled out case 2. The case of two positive levels
of information acquisition is ruled out by noting that in that case there is a sequence of players
? 1 is
1, / and T such thatxx+ xx H-xt>< y, but thismeans thatxlf xt<< xh and so a link gu>
not profitable for either /or /'.The last case to consider has a sequence 1, /and /',with jc,= 0
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GALEOTTIAND GOYAL: THE LAWOF THE FEW
100 NO. 4
VOL.
1491
and Xi>> 0. Clearly then gu>= 1 and so CX[>> k. If xx>> xh then it is strictlyprofitable for /' to
lower information acquisition and form a linkwith 1,while ifxt>< xx then it is strictlyprofitable
for 1 to lower information acquisition and form a linkwith /'.
We have thus shown there cannot
exist a path of length of two or more starting at player 1. So player 1 constitutes a hub. Now we
can exploit the fact that there exists a player /such thatgn = 1 and x-t+ xx = y to conclude that
there cannot exist any links between the neighbors of player 1.This proves that player 1 is a
hub of his component and that all other players choose information level x and form a link
with player 1.
The above argument is done for a single component. The connectedness of nonempty strict
equilibrium networks follows from standard arguments; thedetails are omitted. Finally, note that
ifxx = y{, then*, = 0 for all i^ 1 and thereforeproperty (iii) follows. Supposexx G (0, yx), then
we know thateach player /^ 1 chooses xt = x. Since spokes sponsor only a link to the hub, in a
strictequilibrium itmust be the case thatxx > x. Furthermore, for a player i to play x is optimal
?
?
=
=
y xx and, similarly, forplayer 1 to play xx is optimal only ifxx + (n
y\.It
only ifx
l)x
is now
to see
easy
e ?>
that as
then x ??
0,
0.
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