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We present an approach to network formation based on the notion that social
networks are formed by individual decisions that trade off the costs of forming and
maintaining links against the potential rewards from doing so. We suppose that a link with
another agent allows access, in part and in due course, to the benefits available to the
latter via his own links. Thus individual links generate externalities whose value depends
on the level of decay delay associated with indirect links. A distinctive aspect of our
approach is that the costs of link formation are incurred only by the person who initiates
the link. This allows us to formulate the network formation process as a noncooperative
game.
We first provide a characterization of the architecture of equilibrium networks. We
then study the dynamics of network formation. We find that individual efforts to access
benefits offered by others lead, rapidly, to the emergence of an equilibrium social
network, under a variety of circumstances. The limiting networks have simple architec
tures, e.g., the wheel, the star, or generalizations of these networks. In many cases, such
networks are also socially efficient.
KEYWORDS: Coordination, learning dynamics, networks, noncooperative games.

INTRODUCTION
THE IMPORTANCE OF SOCIAL AND ECONOMIC networks has been extensively
documented in empirical work. In recent years, theoretical models have high
lighted their role in explaining phenomena such as stock market volatility,
collective action, the career profiles of managers, and the diffusion of new
products, technologies and conventions.2 These findings motivate an examina
tion of the process of network formation.
We consider a setting in which each individual is a source of benefits that
others can tap via the formation of costly pairwise links. Our focus is on benefits
1 A substantial portion of this research was conducted when the first author was visiting Columbia
University and New York University, while the second author was visiting Yale University. The
authors thank these institutions for their generous hospitality.
We are indebted to the editor and three anonymous referees for detailed comments on earlier
versions of the paper. We thank Arun Agrawal, Sandeep Baliga, Alberto Bisin, Francis Bloch,
Patrick Bolton, Eric van Damme, Prajit Dutta, David Easley, Yossi Greenberg, Matt Jackson,
Maarten Janssen, Ganga Krishnamurthy, Thomas Marschak, Andy McLennan, Dilip Mookherjee,
Yaw Nyarko, Hans Peters, Ben Polak, Roy Radner, Ashvin Rajan, Ariel Rubinstein, Pauline
Rutsaert, and Rajeev Sarin for helpful comments. Financial support from SSHRC and Tinbergen
Institute is acknowledged. Previous versions of this paper, dating from October 1996, were circulated
under the title, ‘‘Self-Organization in Communication Networks.’’
2
For empirical work see Burt 1992 , Coleman 1966 , Frenzen and Davis 1990 , Granovetter
Ž . Ž . Ž .
Ž .
Ž .
1974 , and Rogers and Kincaid 1981 . The theoretical work includes Allen 1982 , Anderlini and
Ž .
Ž . Ž . Ž . Ž . Ž .
Ianni 1996 , Baker and Iyer 1992 , Bala and Goyal 1998 , Chwe 1998 , Ellison 1993 , Ellison and
Ž .
Fudenberg 1993 , Goyal and Janssen 1997 , and Kirman 1997 .
Ž . Ž .
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. BALA AND S. GOYAL
V
that are nonri al.3 We suppose that a link with another agent allows access, in
part and in due course, to the benefits available to the latter via his own links.
Thus individual links generate externalities whose value depends on the level of
decay delay associated with indirect links. A distinctive aspect of our approach
is that the costs of link formation are incurred only by the person who initiates
the link. This allows us to model the network formation process as a noncooper
ative game, where an agent’s strategy is a specification of the set of agents with
whom he forms links. The links formed by agents define a social network.4
We study both one-way and two-way flow of benefits. In the former case, the
link that agent i forms with agent j yields benefits solely to agent i, while in the
latter case, the benefits accrue to both agents. In the benchmark model, the
benefit flow across persons is assumed to be frictionless: ifanagent i is linked
4
with some other agent j via a sequence of intermediaries, j ,..., j , then the
1
s
benefit that i derives from j is insensitive to the number of intermediaries.
Apart from this, we allow for a general class of individual payoff functions: the
Ž
payoff is strictly increasing in the number of other people accessed directly or
.
indirectly and strictly decreasing in the number of links formed.
Our first result is that Nash networks are either connected or empty.5 Connect
edness is, however, a permissive requirement: for example, with one-way flows a
society with 6 agents can have upwards of 20,000 Nash networks representing
more than 30 different architectures.6 This multiplicity of Nash equilibria
motivates an examination of a stronger equilibrium concept. If an agent has
multiple best responses to the equilibrium strategies of the others, then this may
make the network less stable as the agent may be tempted to switch to a
payoff-equivalent strategy. This leads us to study the nature of networks that can
be supported in a strict Nash equilibrium.
We find that the refinement of strictness is very effective in our setting: in the
one-way flow model, the only strict Nash architectures are the wheel and the empty
network. Figure 1A depicts a wheel, which is a network where each agent forms
Ž
exactly one link, represented by an arrow pointing to the agent. The arrow also
.
indicates the direction of benefit flow . The empty network is one where there
are no links. In the two-way flow model, the only strict Nash architectures are the
center-sponsored star and the empty network. Figure 1B depicts a center-sponsored
3 Examples include information sharing concerning brands products among consumers, the
opportunities generated by having trade networks, as well as the important advantages arising out of
social favors.
4 The game can be interpreted as saying that agents incur an initial fixed cost of forging links with
others where the cost could be in terms of time, effort, and money. Once in place, the network
yields a flow of benefits to its participants.
5 A network is connected if there is a path between every pair of agents. In recent work on social
learning and local interaction, connectedness of the society is a standard assumption; see, e.g.,
Anderlini and Ianni 1996 , Bala and Goyal 1998 , Ellison 1993 , Ellison and Fudenberg 1993 ,
Ž . Ž . Ž . Ž .
Ž .
Goyal and Janssen 1997 . Our results may be seen as providing a foundation for this assumption.
6 Two networks have the same architecture if one network can be obtained from the other by
permuting the strategies of agents in the other network.
NETWORK FORMATION
1183
FIGURE 1A. Wheel network.
FIGURE 1B. Center-sponsored star.
Ž
star, where one agent forms all the links agent 3 in the figure, as represented by
the filled circles on each link adjacent to this agent .
.
These results exploit the observation that in a network, if two agents i and j
Ž .
have a link with the same agent k, then one of them say i will be indifferent
between forming a link with k or instead forming a link with j.Weknow that
Nash networks are either connected or empty. This argument implies that in the
one-way flow model a nonempty strict Nash network has exactly n links. Since
the wheel is the unique such network, the result follows. In the case of the
two-way model, if agent i has a link with j, then no other agent can have a link
with j.AsaNash network is connected, this implies that i must be the center of
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V
FIGURE 1C. Flower and linked star networks.
a star. A further implication of the above observation is that every link in this
star must be made or ‘‘sponsored’’ by the center.
While these findings restrict the set of networks sharply, the coordination
problem faced by individuals in the network game is not entirely resolved. For
Ž .
example, in the one-way flow model with n agents, there are n 1!networks
corresponding to the wheel architecture; likewise, there are n networks corre
sponding to the star architecture. Thus agents have to choose from among these
different equilibria. This leads us to study the process by which individuals learn
about the network and revise their decisions on link formation, over time.
We use a version of the best-response dynamic to study this issue. The
network formation game is played repeatedly, with individuals making invest
ments in link formation in every period. In particular, when making his decision
an individual chooses a set of links that maximizes his payoffs given the network
of the previous period. Two features of our model are important: one, there is
some probability that an individual exhibits inertia, i.e., chooses the same
strategy as in the previous period. This ensures that agents do not perpetually
miscoordinate. Two, if more than one strategy is optimal for some individual,
then he randomizes across the optimal strategies. This requirement implies, in
particular, that a non-strict Nash network can never be a steady state of the
dynamics. The rules on individual behavior define a Markov chain on the state
space of all networks; moreover, the set of absorbing states of the Markov chain
coincides with the set of strict Nash networks of the one-shot game.7
Our results establish that the dynamic process converges to a limit network.
In the one-way flow model, for any number of agents and starting from any initial
network, the dynamic process conerges to a wheel or to the empty network, with
probability 1. The proof exploits the idea that well-connected people generate
positive externalities. Fix a network g and suppose that there is an agent i who
accesses all people in g, directly or indirectly. Consider an agent j who is not
critical for agent i, i.e., agent i is able to access everyone even if agent j deletes
7 Our rules do not preclude the possibility that the Markov chain cycles permanently without
converging to a strict Nash network. In fact, it is easy to construct examples of two-player games with
a unique strict Nash equilibrium, where the above dynamic cycles.
NETWORK FORMATION
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all his links. Allow agent j to move; he can form a single link with agent i and
access all the different individuals accessed by agent i. Thus if forming links is at
all profitable for agent j, then one best-response strategy is to form a single link
with agent i. This strategy in turn makes agent j well-connected. We now
consider some person k who is not critical for j and apply the same idea.
Repeated application of this argument leads to a network in which everyone
accesses everyone else via a single link, i.e., a wheel network. We observe that in
a large set of cases, in addition to being a limit point of the dynamics, the wheel
is also the unique efficient architecture.
In the two-way flow model, for any number of agents and starting from any
initial network, the dynamic process conerges to a center-sponsored star or to the
empty network, with probability 1. With two-way flows the extent of the externali
ties are even greater than in the one-way case since, in principle, a person can
access others without incurring any costs himself. We start with an agent i who
has the maximum number of direct links. We then show that individual agents
who are not directly linked with this agent i will, with positive probability,
eventually either form a link with i or vice-versa. Thus, in due course, agent i
will become the center of a star.8 In the event that the star is not already
center-sponsored, we show that a certain amount of miscoordination among
‘spoke’ agents leads to such a star. We also find that a star is an efficient network
for a class of payoff functions.
The value of the results on the dynamics would be compromised if conver
gence occurred very slowly. In our setting, there are 2nŽn 1. networks with n
agents. With n 8 agents for example, this amounts to approximately 7 1016
networks, which implies that a slow rate of convergence is a real possibility. Our
simulations, however, suggest that the speed of conergence to a limiting network
is quite rapid.
The above results are obtained for a benchmark model with no frictions. The
introduction of decay delay complicates the model greatly and we are obliged
to work with a linear specification of the payoffs. We suppose that each person
potentially offers benefits V and that the cost of forming a link is c.We
introduce decay in terms of a parameter
0, 1 . We suppose that if the
shortest path from agent j to agent i in a network involves q links, then the
value of agent j’s benefits to i is given by qV. The model without friction
corresponds to
We first show that in the presence of decay, strict Nash networks are connected.
We are, however, unable to provide a characterization of strict Nash and
efficient networks, analogous to the case without decay. The main difficulty lies
in specifying the agents’ best response correspondence. Loosely speaking, in the