STRATEGIC NETWORK FORMATION WITH MANY AGENTS
KONRAD MENZEL
NEW YORK UNIVERSITY
Abstract. We consider a random utility model of strategic network formation, where we derive a
tractable approximation to the distribution of network links using many-player asymptotics. Our
framework assumes that agents have heterogeneous tastes over links, and allows for anonymous and
non-anonymous interaction effects among links. The observed network is assumed to be pairwise
stable, and we impose no restrictions regarding selection among multiple stable outcomes. Our main
results concern convergence of the link frequency distribution from finite pairwise stable networks
to the many-player limiting distribution. The possible limiting distributions are shown to have
a fairly simple form and are characterized through aggregate equilibrium conditions, which may
permit multiple solutions. We analyze identification of link preferences and propose a method for
estimation of preference parameters.
PRELIMINARY - PLEASE DO NOT CIRCULATE.
JEL Classification: C12, C13, C31, C35, C78
Keywords: Networks, Large Games, Pairwise Stability, Discrete Choice, Multiple Equilibria
1. Introduction
Network models can be used to describe the structure and the effects of systems of contracts,
transactions, and other formal or informal relationships between economic agents. For example,
peer effects in educational outcomes (see e.g. Calvó-Armengol, Patacchini, and Zenou (2009)),
health (Christakis and Fowler (2007)), or crime (Calvó-Armengol and Zenou (2004)) often act
through friendships, proximity, or network interactions of various kinds. Social networks can
spread information about consumer products (Campbell (2013)) or social programs (Banerjee,
Chandrasekhar, Duflo, and Jackson (2013)), or be used for job search and referrals (Calvó-Armengol
and Jackson (2004), Beaman and Magruder (2012)). Networks of financial transactions allow economic agents to share risks, or access liquidity (Kinnan and Townsend (2012), Bonaldi, Hortaçsu,
and Kastl (2013)), mergers and business partnerships affect market structure (Fox (2010) and Lee
and Fong (2013)), and mergers between political entities can affect revenue and public good provision (Weese (2008)). Over the recent years, the possibility of collecting network data and new
theoretical developments on network and random graph models1 have led to a growing interest in
empirical work in economics and other social sciences that explicitly accounts for the structure of
the relevant networks.
Date: January, 2015.
1
See Jackson (2008) for an summary of recent advances that are most relevant to economic applications.
1
A better understanding of the structure of a social or economic network, and the forces and
incentives determining that structure, is often crucial for a proper evaluation of the policy consequences. In the presence of social interactions, direct effects of agents’ and peers’ attributes have
different policy implications from indirect effects through peers’ choices or outcomes. For example
if network links exhibit homophily with respect to some agent characteristic that is imperfectly
observed or not adequately controlled for, the analysis may falsely attribute the effect of that covariate to a systematic interaction effect. Manski (1993) coined the term “reflection problem” for
this difficulty, and in general, information about the interaction structure is needed to identify
those effects separately. A model for the formation of links can be used to address endogeneity of
the network of interactions (see Goldsmith-Pinkham and Imbens (2012)), or aspects of the network
structure may be an outcome of interest in itself. Similar problems arise in determining which
agents to target in an information campaign for a new consumer product or social program, see
also Banerjee, Chandrasekhar, Duflo, and Jackson (2013) and Jackson (2014) for a more detailed
discussion.
This paper develops new formal tools for models of strategic network formation, where we focus
on the incentives for forming or severing links at the micro-level rather than a global description of
the network structure. The main theoretical result is a tractable approximation to the outcome of
the network formation game, assuming that the number of agents (nodes) of the network is large.
Specifically, we derive a sharp characterization of the set of link distributions that can be generated
by pairwise stable outcomes. With strategic interaction effects between links, this set is in general
not a singleton. As a second step, we then propose strategies for estimation and inference based on
this approximation. This is not a trivial task due to multiplicity of the limiting distribution, and
we will develop tools for asymptotically valid inference that is conditional on the selected pairwise
stable distribution.
Broadly speaking, the researcher may be interested in a structural, rather than descriptive,
approach to characterizing and estimating network models when (a) link preferences are of primary
empirical interest,2 (b) the researcher is interested in in-sample or counterfactual predictions of
network outcomes,3 or (c) a model for a game played on a network explicitly needs to account for
endogeneity of network structure with respect to unobserved heterogeneity.4 Furthermore given the
modeling assumptions, (d) structural models identify objects and parameters that are invariant with
respect to changes in the marginal distribution of types or equilibrium selection. Hence structural
models may also give more stable results for prediction than a purely descriptive model, especially
if the network formation game has multiple equilibria, or the distribution of agent characteristics
change while preferences can be assumed to be stable.
Strategic models of network formation typically do not result in exponential random graph models
that treat links as conditionally independent random variables, for which some results on estimation
2
For example, several recent studies are interested in homophily in preferences and resulting macro-features such as
network segregation, see e.g. Christakis, Fowler, Imbens, and Kalyanaraman (2010), Mele (2012), or Graham (2014)
3
For example, Chandrasekhar and Lewis (2011) use a network model to correct for measurement error in network
statics that are computed from a partial sample of nodes.
4
See e.g. Goldsmith-Pinkham and Imbens (2012) and discussion.
2
and large-sample theory are already available.5 Furthermore, there need not be an unambiguous
relationship between parameters in random graph models to stable, “structural” features of an
underlying population, especially if the network formation model admits multiple stable outcomes.
We are not aware of any existing work that derives tractable distributions of network outcomes
for a random utility model with an economic equilibrium concept. Most existing approaches rely
heavily on simulation methods, this includes Hoff, Raftery, and Handcock (2002),Christakis, Fowler,
Imbens, and Kalyanaraman (2010), Mele (2012), and Sheng (2014). Instead of considering the joint
distribution of the adjacency matrix or local “neighborhoods” inside the network (as considered
by Sheng (2014), De Paula, Richards-Shubik, and Tamer (2014) or Graham (2014)), we show that
it is sufficient for estimation to consider the frequencies of links between pairs of nodes with a
given combination of exogenous attributes and endogenous network characteristics. Our analysis
differs from De Paula, Richards-Shubik, and Tamer (2014) in that we model link preferences as
non-anonymous, and therefore have to characterize explicitly how subnetworks interact with the
full network through link availability and strategic interaction effects with neighboring nodes. Our
asymptotic approximations allows to characterize that dependence using aggregate state variables
that satisfy certain equilibrium conditions in order for the network to be pairwise stable. Boucher
and Mourifié (2012) give conditions for weak dependence of network links under increasing domain
asymptotics, whereas our approach can be conceived as “infill” asymptotics where link frequencies
between distant nodes are non-trivial under any metric on the space of node characteristics.
The asymptotic approximation is obtained by embedding the finite-player network corresponding
to the observable data into a sequence of network formation models with an increasing number
of agents. Using statistical large-sample theory, we derive the limit for the distribution of links
along that sequence. The primary motivation for many-agent asymptotics in the network model
is to arrive at a tractable model that does not require an explicit account for certain strategic
considerations that are not of first order in the limiting experiment. This approach extends limiting
results for two-sided matching markets developed by Dagsvik (2000) and Menzel (2013).
Our approach regarding estimation and inference also differs from existing literature in how it
deals with multiplicity of stable outcomes: Existing methods either formulate worst-case bounds
with respect to distributions over stable networks6, or assume a specific mixture distribution or
sequential protocol for generating the observable network outcome.7 In contrast, we argue that
in many relevant cases, it is possible to identify aggregate state variables that are sufficient statistics for the selected stable network and can be estimated consistently from a large network.
We then propose conditional estimation or inference given those state variables, similar to Menzel
(2012)’s analysis of discrete games, where in some relevant cases the structural parameters are
point-identified from the conditional limiting distribution.
5
See e.g. Frank and Strauss (1986), Wasserman and Pattison (1996), Bickel, Chen, and Levina (2011), Chandrasekhar
and Jackson (2011), or Snijders (2011) for a survey.
6
This includes the approaches in Baccara, Imrohoroglu, Wilson, and Yariv (2012), De Paula, Richards-Shubik, and
Tamer (2014), Miyauchi (2012), Sheng (2014)
7
See e.g. Hoff, Raftery, and Handcock (2002),Christakis, Fowler, Imbens, and Kalyanaraman (2010), Mele (2012)
and the discussion of pairwise stability as a solution concept below.
3
The remainder of the paper is organized as follows: we first describe the economic model, including alternative solution concepts. Section 3 then give a characterization of the limiting model,
and section 4 gives our main asymptotic result establishing convergence of the finite network to
that limit. Section 5 discusses strategies for identification and estimation, and section 6 concludes.
2. Model Description
The network consists of a set of n agents (“nodes” or “vertices”), which we denote with N =
{1, . . . , n}. We assume that each agent is associated with a vector of exogenous attributes (types)
xi ∈ X , where the type space X is some (continuous or discrete) subset of a Euclidean space, and
the marginal distribution of types is given by the p.d.f. w(x).
Using standard notation (see Jackson (2008)), we identify the network graph with the adjacency
matrix L, where the element
(
1
if there is a direct link from node i to node j
Lij =
0
otherwise
As a convention, we do not allow for any node to be linked to itself, Lii = 0 for all i = 1, . . . , N .
For our results, we assume that all links are undirected, Lij = Lji , i.e. the adjacency matrix L is
symmetric. We also let L − {ij} be the network resulting from setting Lij = 0, that is from deleting
the edge ij from L. Similarly, L + {ij} denotes the network resulting from setting Lij = 1, that is
from adding the edge ij to L.
2.1. Payoffs. Player i’s payoffs are of the form
Πi (L) = Bi (L) − Ci (L)
where Bi (L) denotes the gross benefits from the network structure, and Ci (L) the cost of maintaining links. We will see below that identification of costs and benefits entails a location normalization
of some kind. Hence, we will generally assume that the cost Ci (L) is only a function of the number
of outlinks for player i, but not the identities or characteristics of the individuals that i is directly
connected to under the network structure L.
We specify the model in terms of the incremental benefit of adding a link ij to the network L,
Uij (L) := Bi (L + {ij}) − Bi (L − {ij})
and the cost increment of adding that link,
M Cij (L) := Ci (L + {ij}) − Ci (L − {ij})
With a slight departure from common usage of those terms, we refer to Uij (L) and Mij (L) as the
marginal benefit and marginal cost (to player i), respectively, of adding the link ij to the network.
Throughout our analysis we model the marginal benefit function as
Uij (L) = Uij∗ (L) + σηij
(2.1)
where Uij∗ (L) is a deterministic function of attributes x1 , . . . , xn and the adjacency matrix L, and
will be referred to as the systematic part of the marginal benefit function. The idiosyncratic taste
shifter ηij is assumed to be independent of xi and xj and distributed according to a continuous
4
c.d.f. G(·), and σ > 0 is a scale parameter. Also, marginal costs are given by
M Ci := max σηi0,j
j=1,...,J
(2.2)
where ηi0,j are independent of xi and across draws j = 1, 2, . . . , and the choice of the number of
draws J will be discussed further below. In particular, we let J to grow as n increases in order
for the resulting network to be sparse. In this formulation, marginal costs do not depend on the
network structure. Note that in the absence of further restrictions on the systematic parts of
benefits Uij∗ (L), this is only a normalization.
2.2. Solution Concept. Our formal analysis assumes pairwise stability as a solution concept.
The following definition of pairwise stability was first introduced by Jackson and Wolinsky (1996),
and we will refer to the solution concept as PW1 for our baseline case.
Definition 2.1. (Pairwise Stability, PW1) The undirected network L is a pairwise stable
network according to PW1 if for any link ij with Lij = 1,
Πi (L) ≥ Πi (L − {ij}), and Πj (L) ≥ Πj (L − {ij})
and any link ij with Lij = 0,
Πi (L + {ij}) < Πi (L), or Πj (L + {ij}) < Πj (L)
One central feature of pairwise stability as a solution concept is the premise that agents consider
each link separately. The pairwise stability conditions are only necessary for individually optimal
simultaneous choice over possible links, but not sufficient. Specifically, in a pairwise stable outcome
there may well be a player who can increase her payoff by reconfiguring two or more links unilaterally. In the following, we also consider refinements of this rather weak requirement of pairwise
stability.
Pairwise stability can be a suitable solution concept in an empirical model for network formation if
we can guarantee existence of a stable outcome, and if a stable outcome can be reached through some
mechanism that is plausible given the participating agents’ knowledge and strategic sophistication.
While I am not aware of any fully general existence results, there are some relevant special cases for
which existence of a PSN is not problematic. Miyauchi (2012) considers the case of non-negative
link externalities, in which case pairwise stability can be represented as Nash equilibrium in a
finite game with strategic complementarities. Hence, existence and achievability through a myopic
tâtonnement mechanism follow from general results by Milgrom and Roberts (1990).
Our approach builds on local stability conditions for each link in isolation, and therefore only
requires that any given link satisfies the pairwise stability conditions with probability approaching
1. Hence in the context of tâtonnement dynamics, existence of a pairwise stable matching will not
be strictly necessary for our approach to work as long as the share of links that are not pairwise
stable at an advanced stage of the adjustment process goes to zero as n grows large.Specifically,
Lemma 1 in Jackson and Watts (2002) establishes that either a PSN exists, or we can find closed
cycles of sequential formations or deletions of individual links. Our asymptotic results may have
implications as to whether the proportion of links violating the stability conditions along such a
cycle vanishes as n grows large. We plan to investigate this further in a future version of this paper.
5
It is also interesting to contrast our use of an essentially static solution concept to the approaches
in Christakis, Fowler, Imbens, and Kalyanaraman (2010) and Mele (2012) who consider link distributions resulting from myopic random revisions of past link formation decisions, where agents
are not assumed to be forward-looking regarding future stages of the formation game. Christakis,
Fowler, Imbens, and Kalyanaraman (2010) assume a stochastic initial condition for the link formation process, so that (in the absence of further shocks to the process) further iterations of the
tâtonnement process would generate a distribution over pairwise stable outcomes or cycles with
mixing weights depending on that initial condition. The revision process in Mele (2012) is represented by a potential function, favoring formation of links that lead to larger cardinal utility
improvements, and networks generating a large “systematic” surplus. Our approach allows for any
pairwise stable outcome and does not take an implicit stand on equilibrium selection.
For a revealed-preference analysis it is useful to represent the pairwise stability conditions as a
discrete choice problem, where preferences are given by the random utility model described above,
and the set of available “alternatives” for links arises endogenously from the equilibrium outcome.
Specifically, given the network L we define the link opportunity set Wi (L) ⊂ N as the set of nodes
who would prefer to add a link to i,
Wi (L) := {j ∈ N \{i} : Uji (L) ≥ M Cji (L)}
and denote the corresponding “availability indicators” with
Dji (L) := 1l{j ∈ Wi (L)}
Using this notation, we can rewrite the pairwise stability condition in terms of individually optimal
choices from the opportunity sets arising from a network L.
Lemma 2.1. Assuming that all preferences are strict, a network L∗ is pairwise stable according to
PW1 if and only if
(
1
if Uij (L∗ ) ≥ M Cij (L∗ )
∗
Lij =
(2.3)
0
if Uij (L∗ ) < M Cij (L∗ )
for all j ∈ Wi [L∗ ], and i = 1, . . . , n.
The proof for this lemma is identical to that of Lemma 2.1 in Menzel (2013).
We now turn to some potential refinements of pairwise stability that may greatly reduce the
number of distributional outcomes without imposing unreasonable constraints on individual choices,
as we will argue below. For a first modification of PW1, notice that the notion of pairwise stability
in matching models (see Gale and Shapley (1962) and Roth and Sotomayor (1990)) allows for richer
deviations from a status quo than PW1. Specifically, a proposed matching is blocked by a pair if
at least one agent would prefer to reject her current match (i.e. break a current link) in favor of
another available matching partner (i.e. simultaneously form a link to a new available node). We
can define PW2 as stability of a network with regard to these slightly more complex deviations:
Definition 2.2. (Pairwise Stability, PW2) The undirected network L is a pairwise stable
network according to PW2 if for any link ij with Lij = 1,
Uij (L) ≥ max{M Ci (L), Uik (L − {ij})}, and Uji (L) ≥ max{M Cj (L), Ujl (L − {ij})}
6
and for any link ij with Lij = 0,
Uij (L) < min{M Ci (L), Uik (L − {ij})}, , or Uji (L) < min{M Cj (L), Ujl (L − {ij})}
for any k such that Uki (L) ≥ M Ck (L) and l such that Ulj (L) ≥ M Cl (L).
Note that for simplicity we formulate the stability conditions only in terms of marginal utilities,
in analogy with the characterization of PW1 in Lemma 2.1. Notice that if there is no externality
among links (as e.g. the capacity constraint in the classical matching problem), then the additional deviations considered for PW2 do not make a difference. Otherwise the added requirement
stipulates that at the margin, each agent selects the “best” link opportunity over alternatives with
lower marginal utility, thereby removing one major source of multiplicity in the agent’s individual
response.
Finally, we can strengthen this stability notion further by allowing for any sequence of successive
unilateral readjustments by any node.
Definition 2.3. (Pairwise Stability, PW3) The undirected network L is a pairwise stable
network according to PW3 if for any node i,
Πi (L) ≥ max Πi (L + (dj Dji − Lij ){ij})
d1 ,...,dn
This last stability criterion allows each agent to optimize over all available links simultaneously
and without any further constraints. Note that by the definition of marginal utilities and marginal
costs, agent i’s individual response under PW3 corresponds to a maximum of Πi (L) over all combinations of links that are available to i. Therefore, if the support of marginal utilities is an open
set in Rn , the individual response under PW3 is unique with probability 1.
It is immediate that PW2 and PW3 are successive refinements of the standard notion of pairwise
stability, PW1. Specifically, PW3 implies PW2, which in turn implies PW1. Notice that either
refinement imposes additional restrictions only regarding unilateral deviations by any given agent.
This is fundamentally different from equilibrium selection in multi-agent (game-theoretic) decision
problems, where any comparable refinement would require strong strategic assumptions on how
different agents coordinate their actions.
There are few general results regarding existence of pairwise stable networks, and that issue will
be investigated further in a future version of this paper. Specifically, in addition to existence of a
pairwise stable network, we may formulate conditions under which the share of “badly matched”
agents converges to zero as n increases, a condition that would be sufficient for our approximations
to work.
2.3. Reference Model. We next give a reference specification for payoffs to illustrate the main
types of interaction effects that can be included in our framework. We let the marginal benefit
from adding the link from i to j depend on agent i and j’s type and the structure of the network
through vector-valued statistics Rij , Si , Sj , Tij that summarize the payoff-relevant features,
′
Uij∗ (L) ≡ U ∗ (xi , xj ; Si , Sj , Tij ) = Rij
βR + Si′ βS1 + Sj′ βS2 + Tij′ βT
7
(2.4)
where the statistic
Rij := R(xi , xj )
depends only on attributes of agents i and j. This component allows to model type-specific propensities to form links, and model homophily or assortativity through terms that measure similarity
or distance of i and j in the type space X .
The propensity of agent i to form an additional link, and the attractiveness of a link to agent j
may both depend on the position of either node in the network. To account for effects of this type,
we can include features of the form
Si := S(L, xi , i)
with our payoff specification. For now, we assume that the statistic has a recursive representation
S(L + {ij}, xi , i) = S+1 (xi , xj ; S(L − {ij}, xi , i), S(L − {ij}, xj , j))
for some function S+ (·). Network statistics of this type include the network degree (number of
direct neighbors),
X
S(L, xi , i) :=
Lij
j6=i
or other measures of network centrality. Here the explicit dependence of the function S(·) on the
index i is needed to capture the relative position of the agent in the network L.
Payoffs may also depend on both agents’ position in the network relative to each other in the
absence of a direct link. Specifically, the researcher may also want to include measures of the form
Tij := T (L, xi , xj , i, j)
For example, we can model a preference for closure of “triads” or “cliques” of larger sizes using
statistics of the form
X
T1 (L, xi , xj , i, j) =
Lik Ljk , or T2 (L, xi , xj , i, j) = max {Lik Ljk : k =
6 i, j}
k6=i,j
Here, T1ij counts the number of immediate neighbors that both i and j have in common, and T2ij
is an indicator whether i and j have any common neighbor. More generally, Tij could include other
measures of the distance between agents i and j in the absence of a direct link.
In contrast to the index Rij , the variables Si , Sj , and Tij are endogenous to the network formation
process, and the characterization of the limiting model includes equilibrium conditions for the joint
distribution of types xi and network statistics Si and Tij .
2.4. Link Frequency Distribution. We derive our limiting results in terms of the link frequency
distribution, which we define as
n
1 XX
Fn (x1 , x2 ; s1 , s2 , t) :=
P (Lij = 1, xi ≤ x1 , xj ≤ x2 , Si ≤ s1 , Sj ≤ s2 , Tij ≤ t)
n
i=1 j6=i
The link frequency distribution is not a proper probability distribution but integrates to a nonnegative value (“mass”) equal to the average degree across nodes, which is in general different from
one. We also denote the corresponding density with f (x1 , x2 ; s1 , s2 , t).
8
Most previous approaches are based on the distribution of the entire adjacency matrix (see e.g.
Christakis, Fowler, Imbens, and Kalyanaraman (2010), Chandrasekhar and Jackson (2011), or Mele
(2012)), which typically requires simulation of the entire network at a substantial computational
cost. We argue that for typical specifications of a network formation model, the link frequency
distribution is a sufficient statistic for the preference parameters: specifically we can encode the
adjacency matrix - which will be sparse under our specification - more efficiently (i.e. requiring less
memory) as a list of pairs of nodes that are connected by a direct link. Then a parameterization of
the model as in (2.4) allows to treat the nodes as exchangeable units so that instead of recording
the label of any node, and it is sufficient to record the corresponding exogenous attributes xi , xj ,
and endogenous network characteristics Si , Sj and Tij . While in principle knowing which links
emanate from a common node could also be informative, we find that in the limit, stability of the
link Lij and any other link Lik are conditionally independent events given the characteristics of
nodes i, j, k. Hence the representation of the adjacency matrix as a sample of links does not result
in any loss of statistical information regarding the preference parameters in the structural model.
2.5. Choquet Capacities. An important complication arises from the fact that in the presence
of interaction effects among links, individual responses are in general not unique. Hence, for a
given value of the aggregates Γ and w, the resulting distributions of the network statistics si that
correspond to mixtures over the values of si supported by the individual response. Following the
approach in Galichon and Henry (2011), we characterize the set of distributions generated by nonunique stable outcomes using capacities (see Choquet (1954), Molchanov (2005)). We next give
main definitions and examples, and then state the main formal assumptions and results regarding
the equilibrium conditions on the reference distribution.
Let 2S denotes the set of all subsets of S, and ∆S the probability simplex of distributions over
elements of S. We also say that a sequence of sets (An )n≥0 is increasing (with respect to set
inclusion) if An ⊂ An+1 for all n, and we say that the sequence is decreasing if An+1 ⊂ An for all
n.
Definition 2.4. (Choquet capacity) A mapping Ω : 2S → [0, 1] is called a Choquet capacity
(upper probability) on the set S if (a) Ω(∅) = 0, Ω(S) = 1, (b) Ω is monotone with respect to set
inclusion, i.e. Ω(S ′ ) ≤ Ω(S) whenever
S′ ⊂ S ⊂ S, and (c) for any increasing sequence of subsets
S
(Sn )n≥0 of S, limn Ω(Sn ) = Ω
n≥0 Sn , whereas for any decreasing sequence of subsets (Sn )n≥0 ,
T
limn Ω(Sn ) = Ω
n≥0 Sn .
The normalization of the values of the capacity in part (a) is not part of the usual (i.e. more
general) definition of a Choquet capacity, but is assumed throughout in this paper, so that a
capacity can be interpreted as representing a set of proper probability distributions, as we discuss
below. Note that in order to characterize the capacity fully, it is in general not sufficient to find
the upper bounds for the elementary events of the form {si = s}, but we also need to account for
any composite events of the form si ∈ S, for arbitrary subsets S ⊂ S.
In the present context, we construct a capacity Ω0 to characterize the set of distributions of node
attributes zi := (x′i , s′i )′ that can be generated by a individual response consistent with pairwise
stability. Specifically, let Q(s1 , s2 |x) denote the conditional probability that both si = s1 and si = s2
9
are supported by the individual response given xi = x. Furthermore, for two sets S1 , S2 ⊂ S, let
Q(S1 , S2 |x) denote the conditional probability that the individual response supports at least one
outcome si ∈ S2 and one outcome sj ∈ S2 . Then the mapping
Ω(x, S) := Q(S, S|x) for any S ⊂ S
corresponds to the conditional probability that at least one outcome in S is supported by the
individual response given xi = x and Γ and w, and it is straightforward to verify that Ω0 is indeed
a Choquet capacity for all values of x.
The resulting capacity is subadditive: for any two sets S1 , S2 ⊂ S, we have Ω(x, S1 ) + Ω(x, S2 ) =
Ω(x, S1 ∪ S2 ) + Q(S1 , S2 |x) where the second term on the right-hand side need not be zero even if
S1 ∩ S2 = ∅ if some possible values for payoffs support individual responses in both sets.
For our estimation approach, Ω0 does not have to be constructed explicitly by the researcher as
long as the regularity conditions in Assumption 4.5 below are known to hold. However, when the
primary objective is model prediction rather than inference, derivation of the equilibrium reference
distributions requires knowledge of Ω0 .
If the set S is finite, the capacity can be constructed from primitives recursively using
Ω(x, {sk } ∪ Sk−1 ) = Ω(x, {sk }) + Ω(x, Sk−1 ) − Q(sk , Sk−1 |x)
In the context of our model, the probability Q(sk , Sk−1 |x) is an integral of coordinate-wise independent extreme-value distributions over the intersection of a rectangular set with a union of k − 1
rectangular sets of values for the taste shifters ηij . That probability can in turn be rewritten in
terms of sums and differences of probabilities of elementary events Q(sk , sl |x) corresponding to
rectangular sets in the space of taste shifters.
Choquet capacities can be used to represent sets of probability distributions, where that set is
referred to as the core of the capacity:
Definition 2.5. (Core) The core of the capacity Ω is the set of all probability distributions F (s)
over S such that
Z
F (ds) ≤ Ω(S) for all subsets S ⊂ S
S
In that event, we also write F ∈ core(Ω).
Hence, the core of the capacity Ω is a subset of the probability simplex ∆S. Clearly, the
core of Ω is convex: if F1 and F2 are in the core, then we also have that for any λ ∈ [0, 1]
R
S λF1 (ds) + (1 − λ)F2 (ds) ≤ Ω(S) for all S ⊂ S. Moreover, if the rule for selecting from the
individual response is unrestricted, then every distribution in the core can be attained by pairwise
stable network. Consider any two points in Ω0 that are supported by selection rules corresponding
to mixture weights α, α′ . Then any convex combination of the two distributions can be generated
by the mixture λα + (1 − λ)α′ as λ varies on the unit interval. Hence, in the absence of additional
constraints on the selection mechanism we can represent the set of reference distributions consistent
with pairwise stability using the capacity Ω0 : T × ∆Z × X ⇒ ∆S, a mapping to convex subsets of
the probability simplex ∆S. We illustrate the construction of the capacity Ω0 (x; s) describing the
possible distributions of endogenous network statistics with several examples in the next section.
10
3. Limiting Model
While individual decisions about whether to form or destroy a link are interrelated, the asymptotic approximation developed in this paper allows to characterize the link frequency distribution
in terms of strategic aggregates at the network level, and a node-level “best response” to those
aggregate states.
Specifically, we use the term individual response for any conditional probability of the form
H(s1 , s2 , t12 |x1 , x2 ) := lim nP (L∗ij = 1, Si = s1 , Sj = s2 , Tij = t12 |xi = x1 , xj = x2 ; Γ)
n
(3.1)
for links L∗ij corresponding to a pairwise stable network. Also, following Chandrasekhar and Jackson
(2011), the cross-sectional distribution of exogenous attributes xi and the payoff-relevant network
statistics si resulting from a given network configuration will be called the reference distribution
of (x′i , s′i )′ , where we denote its p.d.f. with w(x; s). The inclusive value function Γ(x; s) is a
nonnegative function depending on the distribution of link opportunities for an agent with attributes
xi = x and si = s. We show that in the limit, Γ(x; s) is a sufficient statistic for that agent’s link
opportunity set with respect to the probability that for a given combination of links the pairwise
stability conditions with respect to agent i’s payoffs are satisfied. We derive equilibrium conditions
for the reference distribution and inclusive value function that are necessary and sufficient for
w(x; s) and Γ(x; s) to be supported by a pairwise stable network.
3.1. Characterization of the Limiting Distribution. In the presence of nontrivial network
externalities there are in general multiple individual responses that are supported by stable network
outcomes, and we show below that pairwise stability is consistent with a convex set of conditional
probabilities of this form. In order to describe the set of possible distributions satisfying the stability
requirements, we also define the bounds
H̄(S1 , S2 , T12 |x1 , x2 ) := limn nP L∗ij = 1, S(L∗ , xi , i) ∈ S1 , S(L∗ , xj , j) ∈ S2 ,
T (L∗ , xi , xj , i, j) ∈ T12 for some network L∗
satisfying PW xi = x1 , xj = x2
(3.2)
for any sets S1 , S2 ⊂ S and T12 ⊂ T . The limiting probabilities that outcomes of the form
{L∗ij = 1, S(L∗ , xi , i) ∈ S1 , S(L∗ , xj , j) ∈ S2 , T (L∗ , xi , xj , i, j)} are supported by a payoff draw define a capacity on S 2 ×T that will be used to characterize the set of individual responses compatible
with a pairwise stable outcome. Note that this part of the model has a structure that is completely
analogous to that of static game-theoretic models and therefore amenable to the techniques developed in Beresteanu, Molchanov, and Molinari (2011) and Galichon and Henry (2011). We also
show that the values of the capacity H̄(S1 , S2 , T12 |x1 , x2 ) can be obtained from probabilities for
rectangular events with respect to independent extreme-value type-I random variables.
We derive an asymptotic representation to the link distribution that consists exclusively of these
three components. Specifically, we find that the limit of the link frequency distribution is of the form
f (x1 , x2 ; s1 , s2 , t12 ) = H(s1 , s2 , t12 |x1 , x2 )w(x1 )w(x2 ), where the individual response H(·|x1 , x2 ) is
11
characterized by
H(·|x1 , x2 ) ∈ core H̄(·|x1 , x2 ) ≡ core H̄(·|x1 , x2 ; Γ∗ , w∗ )
(3.3)
where the limiting capacity H(S1 , S2 , T12 |x1 , x2 ; Γ∗ , w∗ ) is available as a closed-form expression
given Γ∗ , w∗ that is based on an extreme-value approximation. Furthermore, the inclusive value
function Γ∗ satisfies the fixed-point condition
Γ∗ (x; s) = Ψ0 [Γ∗ , w∗ ](x; s)
(3.4)
for all values of x, s, and the operator
Z
(s2 + 1) exp{U ∗ (x, x2 ; S+1 (x, x2 ; s, s2 ), s2 ) + U ∗ (x2 , x; s2 , s)}
Ψ0 [Γ, w](x; s) :=
w(x2 ; s2 )ds2 dx2
1 + Γ(x2 ; s2 )
X ×S
Furthermore, the equilibrium conditions for the reference distribution are fully characterized by the
fixed-point condition
w∗ (x; s)
∈ core Ω0 [Γ∗ , w∗ ](x; s)
(3.5)
w(x)
for all values of x, s, where the exact form of Ω0 (·) depends on the functions S(·) in the construction
of the network characteristics. We discuss the construction of Ω0 for relevant examples in more
detail below.
Most importantly, the probability that a given link {ij} is established depends on the structure
of the larger network only through Γ and w in addition to “local” characteristics of the two nodes
i and j.
Example 3.1. (Single-Valued Individual Response) In order to illustrate the usefulness of
the general result, we can state the limiting distribution for the special case in which interaction
effects are only anonymous, and the individual response H(S1 , S2 |x1 , x2 ) is unique with probability
one. In that event, we find that the resulting limiting link frequency distribution has p.d.f.
f (x1 , x2 ; s1 , s2 ) =
s1 s2 exp{U ∗ (x1 , x2 ; s1 , s2 ) + U ∗ (x2 , x1 ; s2 , s1 )}w∗ (x1 ; s1 )w∗ (x2 ; s2 )
(1 + Γ∗ (x1 , s1 ))(1 + Γ∗ (x2 , s2 ))
where the fixed-point condition for the the inclusive value function Γ∗ is the same as above. Finally,
since by assumption si is uniquely defined from the individual response, the core-determining class
for the Ω0 are the singleton sets {s} ⊂ S, so that the equilibrium conditions for the reference
distribution are fully characterized by
w∗ (x; s)
= Ω0 [Γ∗ , w∗ ](x; s)
w(x)
for all values of x, s. The resulting link frequency distribution need not be unique if the fixed-point
equation for the reference distribution admits multiple solutions.
For the general case of a set-valued individual response, we next illustrate how to use the asymptotic results to represent the core of the individual response with its support function on the |S|2 |T |dimensional probability simplex. We start by giving a general formulation of the individual response
using the conceptual framework in Beresteanu, Molchanov, and Molinari (2011). We then proceed
12
to give a more specific characterization of the resulting set of distributions for the alternative solution concepts PW1-PW3. We assume throughout that S = {s(1) , s(2) , . . . } and T = {t(1) , t(2) , . . . }
are discrete, where we identify (s(p) , s(q) , t(r) ) ∈ S 2 × T with the (p + |S|q + |S|2 r)th unit vector in
2
R|S| |T | .
Assuming pairwise stability PW1 as a solution concept, let the random set
QP W 1 := (s1 , s2 , t12 ) ∈ S 2 L∗ij = l, S(L∗ , xi , i) = s1 , S(L∗ , xj , j) = s2 , T (L∗ , xi , xj ; s1 , s2 ) = t12
for some network L∗ satisfying PW1 }
where we define QP W 2 and QP W 3 analogously for the corresponding refinements of pairwise stability. Generally, QP W 1 depends on the joint distribution of agent i and j’s taste shifters, as well
as the availability indicators Dki (L∗ ), Dkj (L∗ ) for all k 6= i, j. Hence, an exact characterization of
the distribution of QP W 1 for the general case is infeasible. Instead, we give an asymptotic approximation which can be parameterized with the inclusive value function Γ∗ and reference distribution
w∗ (x; s).
The convex hull of the random set QP W 1 can be characterized by its support function
̺ (u |QP W 1 ) :=
sup u′ q
q∈QP W 1
2
where the argument u ∈ R|S| |T | . We consider distribution of the set QP W 1 for a randomly selected
pair i, j. Since the solution concept doesn’t restrict selection from QP W 1 across different players in
the limit, any convex combinations/mixtures of points in the set can be attained for the distribution
in the cross-section. This “purification” of mixtures of the individual response with a large number
of players is perfectly analogous to purification of mixed-strategy equilibria in many player games
analyzed by Kalai (2004).
It is known that the selection expectation of a random set Q is convex (see e.g. Beresteanu,
Molchanov, and Molinari (2011)), and that its support function equals the expectation of the
support function of Q,
E [ ̺ (u |QP W 1 )| xi = x1 , xj = x2 ] = ̺ (u |E [ QP W 1 | xi = x1 , xj = x2 ] )
Also, by construction the expectation of QP W 1 is a convex polyhedron in the |S|2 × |T |-dimensional
probability simplex, with vertices whose coordinates correspond to the maximal probability for
subsets S1 , S2 ⊂ S and T12 ∈ T given by the capacity H̄(S1 , S2 , T12 |x1 , x2 ) defined in (3.2). It is
also immediate that for any direction u, the supremum in the definition of the support function is
attained at at one of the vertices of that polyhedron.
Following Beresteanu, Molchanov, and Molinari (2011), the set of limiting distributions for the
individual response H(s1 , s2 |x1 , x2 ) defined in (3.1) has to satisfy
u′ H(s1 , s2 , t12 |x1 , x2 ) − lim nE [̺ (u |QP W 1 )| xi = x1 , xj = x2 ] ≤ 0
n
13
for all u ∈ R|S|
2 ×|T
|
(3.6)
In addition, our identification analysis and approach to estimation entails “conditioning” on the
reference distribution w(x; s), so that we can impose a marginal constraints of the form
Z
w(x; s)
=
H(s − 1, s2 , t12 |x, x2 )w(x2 )ds2 dx2 for all x ∈ X , s ∈ S
(3.7)
w(x)
Z
w(x; s)
=
H(s1 , s − 1, t12 |x, x2 )w(x1 )ds1 dx1 for all x ∈ X , s ∈ S
w(x)
We find that the conditions (3.6) and (3.7) give a sharp characterization of the set of distributions
for the individual response conditional on w(x; s). This statement will be made more precise below.
3.1.1. Baseline Case: No Interaction Effects. As a benchmark, we first consider the case in which
there are no externalities or interaction effects across different links, i.e. we consider a specification
of the reference model (2.4) with payoffs
Uij ≡ U ∗ (xi , xj ) + σηij
In this case, conditional link formation rates depend only on the respective network degree Si :=
P
j6=i Lij of each node, so that the relevant reference distribution of types and network character∗
(x;s)
istics is the conditional degree distribution ww(x)
.
Theorem 4.2 below then implies that the limiting link frequency distribution is fully characterized
by its p.d.f.
f (x1 , x2 ; s1 , s2 ) =
s1 s2 exp{U ∗ (x1 , x2 ) + U ∗ (x2 , x1 )}w∗ (x1 ; s1 )w∗ (x2 ; s2 )
(1 + Γ∗ (x1 ))(1 + Γ∗ (x2 ))
where the inclusive value functions Γ(x) satisfy the fixed-point condition
Z
exp{U ∗ (x, x2 ) + U ∗ (x2 , x)}
(s + 1)
w(x2 ; s)dsdx2
Ψ0 [Γ, w](x) :=
1 + Γ(x2 )
X ×S
and the degree distribution w∗ (x; s) is given by
w∗ (x; s)
sΓ(x)s
=
w(x)
(1 + Γ(x))s+1
Furthermore, Corollary 4.1 below establishes that under the assumptions of the model, the resulting
link distribution is unique. This problem is in fact very similar to the one-to-one matching problem
except that we now allow for multiple links for each node.
3.1.2. Case 2: Anonymous Interactions, Degree Centrality. In order to illustrate the role of the
equilibrium condition (3.5), we show how to derive the mapping Ω0 (·) for the case of preferences
over the degree (i.e. the number of direct links) of an agent. The degree of node i is defined as the
network statistic
X
Si = S(L; xi , i) :=
Lij
j6=i
In terms of the latent random utility model, Si = s corresponds to the event that M Ci is the
(s + 1)st highest order statistic of the sample {M Ci } ∪ {Uij }j∈Wi (L∗ ) . Given the scalar network
characteristics Si , Sj we can consider a version of the reference model (2.4) with payoffs
Uij ≡ U ∗ (xi , xj ; Si , Sj , Tij ) + σηij
14
where we also assume that U ∗ (x1 , x2 ; s1 , s2 ) is nondecreasing in s1 . Let
Uij (s) := U ∗ (xi , xj ; s1 , sj ) + σηij ,
and define the events Aij (s) := {Uij (s) ≥ M Ci }, Bij (s) := {M Ci > Uij (s)}, and Cij (s1 , s2 ) :=
{Uij (s2 ) ≥ M Ci > Uij (s1 )}, noting that Aij (s1 ), Cij (s1 , s2 ), and Bij (s2 ) are mutually exclusive
for any s2 ≥ s1 . Then {si = s1 } is supported by the individual response satisfying PW1 if and
only if Aij (s1 ) is true for exactly s1 distinct nodes j ∈ {j1 , . . . , js1 } in agent i’s link opportunity
set. Similarly, the event {si ∈ {s1 , s2 }} is supported for s1 ≤ s2 if and only if Aij (s1 ) is true for
exactly s1 nodes, and Cik (s1 , s2 ) holds for exactly s2 − s1 nodes k ∈ {k1 , . . . , ks2 −s1 }. In general,
for S1 ⊂ S consisting of r1 elements s11 < s12 · · · < s1r1 , si ∈ S1 is supported by an individual
response satisfying PW1 if there are s11 nodes in i’s opportunity set for which Aij (s1 ) is true, and
for any k = 1, . . . , r − 1 there are exactly s1(k+1) − s1k nodes for which Cij (s1k , s1(k+1) ) holds. The
analogous characterization holds for the event sj ∈ S2 , consisting of the values s21 < s22 · · · < s2r2 .
Hence, the support function evaluated at uS1 ,S2 corresponds to the probability of the event that
Lij = 1 and all values of si ∈ S1 and sj ∈ S2 are supported by an individual response satisfying
PW1. In particular since L∗ij = 1 for each s ∈ S1 , we need Aij (s1 ) to hold true. Hence, from
Lemma 4.2 and elementary calculations we obtain
H̄(S1 , S2 |x1 , x2 ) =
s11 s21 exp{U ∗ (xi , xj ; s11 , s21 ) + U ∗ (xj , xi ; s21 , s11 )}
(1 + Γ(xi ; s1r1 ))(1 + Γ(xj ; s2r2 ))
Γ(xi ; s11 )s11 −1 Γ(xj ; s21 )s21 −1
(1 + Γ(xi ; s11 ))s11 (1 + Γ(xj ; s21 ))s21
!
rY
1 −1 Γ(xi ; s1(k+1) ) − Γ(x1 ; s1k ) s1(k+1) −s1k
×
1 + Γ(x1 ; s1(k+1) )
k=1
!
rY
2 −1 Γ(xj ; s2(k+1) ) − Γ(xj ; s2k ) s2(k+1) −s2k
×
1 + Γ(xj ; s2(k+1) )
×
k=1
Specifically, given values for Γ(x; s), we can characterize the capacity for Lij , si , sj in closed form,
which is in turn sufficient to characterize the core of link distributions generated by pairwise stable
networks. The same principle can be applied to other network characteristics of node i. For
example, “deeper” network characteristics that depend on a wider network neighborhood of a
given node can be characterized recursively in this manner by defining S(L; x, i) as a function of
network characteristics of i’s neighbors.
3.1.3. Case 3: Individual-Specific Interaction Effects, Transitive Triads. Another important type
of interaction effects are non-anonymous in that agents may have preferences regarding network
connections among other players in their network neighborhood. A leading example for a complementarity of this type are triangles (transitive triples), which are defined as a combinations of three
distinct nodes i, j, k such that each pair of nodes is connected by a direct link, Lij = Lik = Ljk = 1.
Transitive triples are the basis for commonly used measures of clustering or “cliquishness” (see
Jackson (2008)).
15
In order to account for a motive to “complete” such a transitive triangle, we can make payoffs
dependent on the network statistic
Tij = max{Lik Ljk }
k6=i,j
which is equal to one if i and j have at least one common network neighbor. To this end, we
consider the reference model (2.4) with payoffs
Uij ≡ U ∗ (xi , xj ; Si , Sj , Tij ) + σηij = R(xi , xj )′ βR + Si′ βS1 + Sj′ βS2 + Tij βT + σηij
where we assume for simplicity that βS1 = βS2 = 0 and βT ≥ 0. Preferences of this kind allow
for a propensity to complete transitive triads and may result in a non-trivial amount of clustering/cliquishness in the network.
In the remainder of this subsection, we describe the set of limiting distributions for the individual
response H(t|x1 , x2 ) for each t ∈ {0, 1}, where for simplicity. Specifically, we denote
Uij (t) = R(xi , xj )′ βR + βT t + σηij
Similar to the previous example, we define the events Aij (t) := {Uij (t) ≥ M Ci }, Bij (t) := {M Ci >
Uij (t)}, and Cij (t1 , t2 ) := {Uij (t2 ) ≥ M Ci > Uij (t1 )}. We then have that the transitive triad ijk is
supported by individual responses satisfying PW1 if and only if Alm (1) holds for all l, m ∈ {ijk}.
We can then use the limiting approximation in Lemma 4.2 to obtain the limit
H1,ijk := lim n2 P (Aik (1), Aki (1), Ajk (1), Akj (1)|xi , xj , xk )
n
=
exp{U ∗ (xi , xk ) + U ∗ (xk , xi ) + U ∗ (xj , xk ) + U ∗ (xk , xj ) + 4βT }
(1 + Γ(xi ))(1 + Γ(xj ))(1 + Γ(xk ))2
Furthermore by the same Lemma, events of the form Aij (t1 ), Bjk (t2 ), Cki (t3 ) are asymptotically
independent for every fixed values t1 , t2 , t3 . In order to aggregate these triad-level probabilities,
note we can then approximate
P
(Aik (1), Aki (1), Ajk (1), Akj (1) for some k 6= i, j|xi = x1 , xj = x2 )
= 1 − P (Bik (1), Bki (1), Bjk (1), or Bkj (1) for each k 6= i, j|xi = x1 , xj = x2 )
X
Y
H1,ijk
≈ 1−
(1 − n−2 H1,ijk ) ≈ n−2
k6=i,j
k6=i,j
where the last step uses that n−2 H1,ijk vanishes fast enough for the sum of the remaining crossterms in the product to be negligible as n grows large. The precise argument will be included in a
future version of this paper.
Hence, we obtain the sharp upper bound on the probability of {Lij = 1, Tij = 1},
H̄(Tij = 1|xi , xj ) =
exp{U ∗ (x1 , x2 ) + U ∗ (x2 , x1 ) + 2βT }
(1 + Γ(x1 ))2 (1 + Γ(x2 ))2
X exp{U ∗ (x1 , xk ) + U ∗ (xk , x1 ) + U ∗ (x2 , xk ) + U ∗ (xk , x2 ) + 4βT }
×
(1 + Γ(xk ))2
k6=i,j
For the outcome {Lij = 1, Tij = 0}, notice that if Lij = Ljk = 1, the triad {ijk} is not completed
if Lik = 0 or Ljk = 0. Hence we can partition the event into {Lij = 1, Lik = 1, Ljk = 0} ∪ {Lij =
16
(1)
1, Lik = 0, Ljk = 0} ∪ {Lij = 1, Lik = 0, Ljk = 1}. The first outcome Eijk := {Lij = 1, Lik =
1, Ljk = 0} is supported if Aij (0), Aji (0), Aik (t̃ik ), Aki (t̃ik ) and Bjk (1), Bkj (1) hold, where t̃ik de(2)
notes the value of Tik in the absence of a direct link between i and j. The second outcome Eijk :=
{Lij = 1, Lik = 0, Ljk = 0} requires Aij (0), Aji (0) and Bik (t̃ik ), Bki (t̃ik ), Bjk (t̃jk ), Bkj (t̃jk ), and the
(3)
third outcome Eijk := {Lij = 1, Lik = 0, Ljk = 1} is supported by Aij (0), Aji (0), Ajk (t̃jk ), Akj (t̃jk )
and Bik (1), Bki (1). We find that in the limit, we can treat t̃jk and t̃ik as independent draws from
the equilibrium distribution h(t|xj , xk ) and h(t|xi , xk ), respectively. We can then obtain a limiting
(1)
(2)
(3)
expression H0,ijk := limn n3 (P (Eijk ) + P (Eijk ) + P (Eijk )) based on Lemma 4.2. Hence an upper
bound on lim nP (Lij = 1, Tij = 0|xi = x1 , xj = x2 ) can be obtained following the same sequence
of steps as before.
A simple calculation based on the expressions for H(t|x1 , x2 ) shows that, in order to achieve
a non-trivial clustering coefficient in the limiting model, we need to choose the rate exp{βT } =
O(n1/3 ) for the asymptotic sequence.
4. Convergence to the Limiting Distribution
This section develops the main convergence results for the link frequency distribution. We first
state the main formal assumptions, followed by the main intermediate formal steps. The main
result in this section is contained in Theorem 4.2. Noting that for some relevant aspects of the
model, only the sum of the systematic part of marginal utilities between the two nodes at either
end of the link matters, we also define the pseudo-surplus for a link {ij} as
Vij∗ := V ∗ (xi , xj ; si , sj ) := U ∗ (xi , xj ; si , sj ) + U ∗ (xj , xi ; sj , si )
Clearly Vij∗ = Vji∗ , so pseudo-surplus is symmetric with respect to the identities of the two nodes.
4.1. Formal Assumptions. The main formal assumptions regarding the random utility model
are similar to those in Menzel (2013). For one, we will maintain that the deterministic parts of
random payoffs satisfy certain uniform bounds and smoothness restrictions:
Assumption 4.1. (Systematic Part of Payoffs) (i) The systematic parts of payoffs |U ∗ (x, x′ , s, s′ , t0 )| ≤
Ū < ∞ are uniformly bounded in absolute value for some value of t = t0 and continuous in X and
Z, respectively. Furthermore, (ii) at all values of s, s′ , the function U ∗ (x, x′ , s, s′ , t0 ) is p ≥ 1 times
differentiable in x with uniformly bounded partial derivatives.
We next state our assumptions on the distribution of unobserved taste shifters. Most importantly,
we impose sufficient conditions for the distribution of maxj ηij to belong to the domain of attraction
of the extreme-value type I (Gumbel) distribution. Following Resnick (1987), we say that the upper
tail of the distribution G(η) is of type I if there exists an auxiliary function a(s) ≥ 0 such that the
c.d.f. satisfies
1 − G(s + a(s)v)
lim
= e−v
s→∞
1 − G(s)
for all v ∈ R. We are furthermore going to restrict our attention to distributions for which the
auxiliary function can be chosen as a(s) := 1−G(s)
g(s) . This property is shared for most standard
specifications of discrete choice models, e.g. if ηij follows the extreme-value type I, normal, or
17
Gamma distribution, see Resnick (1987). We can now state our main assumption on the distribution
of the idiosyncratic part of payoffs:
Assumption 4.2. (Idiosyncratic Part of Payoffs) ηij are i.i.d. draws from the distribution
G(s), and are independent of xi , xj , where (i) the c.d.f. G(s) is absolutely continuous with density
g(s), and (ii) the upper tail of the distribution G(s) is of type I with auxiliary function a(s) :=
1−G(s)
g(s) .
We also need to specify the approximating sequence of networks. Here it is important to emphasize that the main purpose of the limiting analysis is a reliable approximation to the (finite)
n-agent version of the network rather than a counterfactual description of network outcomes were
the size of the real-world network to increase. Hence our approach is to embed the n-agent model
into an asymptotic sequence whose limit preserves the main qualitative features of the finite-agent
model.
Specifically, the network should remain sparse, in that the number of equilibrium links remains
finite as the size of the market grows - in particular the model should allow for agents to remain
isolated and not form any links with non-zero probability even in the limit. The limiting conditional
link formation frequencies given node-level attributes should be non-degenerate, and depend nontrivially on the systematic parts of payoffs. Finally, the limiting approximation should retain
network features at a nontrivial frequency that are deemed important by the researcher, e.g. closed
triangles or other forms of clustering among links.
For the first requirement, it is necessary to increase the magnitude of marginal costs M Ci as the
number of available alternatives grows, whereas to balance the relative scales of the systematic and
idiosyncratic parts we have to choose the scale parameter σ ≡ σn at an appropriate rate. For the
last requirement we have to scale other components of the payoff functions at an appropriate rate,
which we discuss for specific cases below.
Specifically we are going to assume the following in the context of the reference model:
Assumption 4.3. (Network Size)(i) The number n of agents in the network grows to infinity,
and (ii) the random draws for marginal costs M Ci are governed by the sequence J = n1/2 , where
[x] denotes the value of x rounded to the closest integer. (iii) The scale parameter for the taste
shifters σ ≡ σn = a(b1n ) , where bn = G−1 1 − √1n , and a(s) is the auxiliary function specified in
Assumption 4.2 (ii). Furthermore, (iv) the components of the parameter βT , βT,k , grow at respective
rates rn,k which depend on the corresponding statistics Tk (·).
The rate for J in part (ii) is chosen to ensure that the share of unmatched agents is bounded
away from zero and one along the sequence.8
The construction of the sequence σn in part (iii) implies a scale normalization for the systematic
parts Uij∗ = Uij∗ (L), and is chosen as to balance the relative magnitude for the respective effects of
observed and unobserved taste shifters on choices as n grows large. Specifically, for an alternative
∗ , . . . , U ∗ become perfect predictors
rate σ̃n such that σ̃n a(bn ) → 0, the systematic parts of payoffs Ui1
in
8Note also that choosing J˜ = [αn1/2 ] for a given choice of α > 0 would be equivalent to the original rate J = [√n]
n
n
with a different value for the intercept of the random utility from the outside option, so that our implicit choice α = 1
is only a location normalization.
18
for choices as n grows large, whereas if σ̃n a(bn ) → ∞, the systematic parts become uninformative
in the limit. For example, if G(η) has very thin tails, the distribution of the maximum of J
i.i.d. draws from G(·) becomes degenerate at a deterministic drifting sequence as J grows, and
it is therefore necessary to increase the scale parameter σ in order for the scale of the maximum
of idiosyncratic taste shifters to remain of the same order as differences in the systematic part.
Specifically, if ηij ∼ Λ(η), the extreme-value type I (or Gumbel) distribution, then we choose
bn ≍ 12 log n and σn = 1. For ηij ∼ N (0, σ 2 ), it follows from known results from extreme value
q
√
n
theory that the constants can be chosen as bn ≍ σ W 2π
≍ σ log n and σn ≍ σbn2 , where W (x)
is the Lambert-W (product log) function, and for Gamma-distributed ηij , bn ≍ log n and σn = 1.9
For asymptotics it is generally necessary to distinguish between anonymous interaction effects
(i.e. dependence on statistics Si ) and onymous, or individual-specific interaction effects through
statistics of the form Tij . For example, a constant coefficients model may not be able to generate a
network that is sufficiently sparse, or that exhibits a certain patterns of clustering. Specifically, for
the case of transitive-triads based on the statistic T2 (·), we can show that the corresponding rate
is exp{βT } = O(n1/3 ).
4.2. Convergence of the Link Frequency Distribution. The derivation of the limiting link
frequency distribution follows a similar line of argument as the convergence proof in Menzel (2013).
However, the generalizations of the steps are in some cases not trivial, and we will highlight the main
differences to the original proof. The argument relies on the use of inclusive values to parameterize
the endogenous and unobserved link opportunity sets.
The central difficulty for a revealed-preference analysis of the network formation game is that
the link opportunity sets Wi are unobserved and endogenous. We show that for a broad class of
commonly used specifications for random utility models, the conditional choice probabilities converge to those implied by the conditional logit (extreme-value type I) model in the limit. In that
case, the composition and size of the set of link opportunities affects the conditional choice probabilities only through the inclusive value, which is a scalar parameter summarizing the systematic
components of payoffs for the available options, see Luce (1959), McFadden (1974), and Dagsvik
(1994). Specifically, we define the inclusive value of agent i’s link opportunity set W as
1 X
Ii [W ] := 1/2
exp{Uij∗ (L + {i, j})}
n j∈W
Inclusive values have previously been used to simplify the state space for a problem involving
endogenous choice sets by Hendel and Nevo (2006) in a model of dynamic product competition.
Our use of inclusive values for an asymptotic representation of the link distribution extends the
argument in Menzel (2013) for a model for matching markets.
Notation. As an intermediate step, we consider conditional link formation probabilities, assuming
that i’s link opportunity set is fixed exogenously at Wi (L). Specifically, for a fixed network L, we
9See Resnick (1987), pages 71-72.
19
define the event
Wi := xi , xj , Dji (L̃), Dji (L̃)Sj (L̃), Dji (L̃)Tji (L̃)
j≤n
: L̃kl = Lkl
for all k, l 6= i
The conditioning set Wi includes the potential values for the network statistics under any alternative
network configuration that coincides with L except for the links to and from node i. In the following,
we also let L−i := L − {i1, . . . , in} denote the network after deleting all links to i. In order to
characterize links formed in a given pairwise stable network, we define the event
Wi∗ := xi , xj , Dji (L̃), Dji (L̃)Sj (L̃), Dji (L̃)Tji (L̃)
: L̃kl = L∗kl for all k, l 6= i
j≤n
Dependence. In a first step, we show that dependence between idiosyncratic taste shifters ηij and
the link opportunity set Wi (L∗ ) vanishes as n grows large: In pairwise stable networks, establishing
a link ij may affect subsequent decisions by other nodes that are available to i or j, which may in
turn affect link choices by other agents that need not be directly linked to i or j. Such a chain of
adjustments may eventually link back to either i or j’s link opportunity sets. Preference cycles of
this type generally lead to dependence between taste shocks ηij and link opportunity sets Wi . We
therefore need to formulate conditions on the strength of interaction effects among links that limit
that type of dependence as the network grows in size. These conditions can alternatively be viewed
as ensuring local stability of the resulting stable network.
In order to formulate that stability requirement, we say that a node i is well-matched according
to a given solution concept (e.g. PW1) if the corresponding stability conditions with respect to
node i’s payoffs are satisfied. Specifically, node i is well-matched in the network L if and only if
Lij = 1l{Uij (L) ≥ M Ci (L) and Uji (L) ≥ M Cj (L)}.10 For the next definition consider a setting in
which node i is well-matched, and the link ij is changed from its initial state lij to 1 − lij . Suppose
then that there is a sequence of adjustments of links ik1 , . . . , ikr such that after these changes node
i is well-matched, and each link ik1 , . . . , ikr is in the link opportunity set Wi after the adjustment.
If there is no other adjustment of that type that requires a smaller number of changes, then say
that the links ik1 , . . . , ikr are successors of the link ij.
Note that if the typical number of successors to changing a link is sufficiently small, a local
perturbation of a given network will only trigger a short chain of adjustments that dies off after a
few iterations. On the other hand with a large number of successors at each stage, the number of
badly matched nodes grows exponentially at subsequent iterations, so that a chain of adjustments
is likely to be long-lived and eventually reach back to one of the nodes affected by the initial change.
We state the key requirement for low dependence as a high-level condition, and discuss primitive
conditions on the systematic payoff functions for some special cases in detail further below in this
paper.
Assumption 4.4. (Local Stability) Either of the following hold
(a) The expected number of successors to a link ij is bounded by λ̄ < 1.
10Here we adopt the terminology from Pȩski (2014)’s analysis of the stable roommate problem, where Tan (1991)
and Chung (1997) showed that the existence of “odd rings” plays a role for existence of a stable solution that is
analogous to Jackson and Watts (2002)’s characterization of pairwise stable networks.
20
(b) For any sequence of networks such that for n large enough we have n−1/2 |Wi | ≤ C < ∞ for
all nodes i, and the degree distribution is bounded, the expected number of successors to a
link ij is bounded by λ̄ < 1 for n sufficiently large.
We show below that the conditions on the network in part (b) are generally met by networks
along the asymptotic sequence given in Assumption 4.3. Clearly part (b) is weaker than part (a)
which assumes that the number of followers is bounded for any possible network L. While only the
weaker requirement in part (b) for our formal results, the requirement in part (a) is more readily
interpretable. We also note that the only technical result using Assumption 4.4 is Lemma 4.1
below, all other formal steps in the derivation of our main result do not make direct use of that
assumption.
In order to illustrate the scope of this high-level assumption, we next state primitive conditions for
Assumption 4.4 for a few basic versions of the network formation model in the following proposition:
Proposition 4.1. Suppose Assumptions 4.1-4.3 hold. Then
(a) if payoffs are Uij∗ = U ∗ (xi , xj ), then part (a) of Assumption 4.4 holds with λ̄ = 0.
(b) if payoffs are Uij∗ = U ∗ (xi , xj ) if the network degree si < s̄, and Uij∗ = −∞ for si ≥ s̄, then
s̄
exp{Ū }
part (a) of Assumption 4.4 holds with λ̄ = 1+exp{
.
Ū }
∗
∗
(c) if payoffs are Uij = U (xi , xj ) + Tij βT , where Tij is an indicator whether i and j have a
common network neighbor, and n exp{2βT } → 0, then part (b) of Assumption 4.4 holds
with λ̄ = 0.
(d) if the surplus effect of an added link ik on the link ij is non-positive,
V ∗ (x, x′ , S(L + {ik}, i), s′ ) − V ∗ (x, x′ , S(L − {ik}, i), s′ ) ≤ 0
then part (b) of Assumption 4.4 holds with a value of λ̄ < 1 depending on the bound Ū .
The case of no endogenous interaction effects corresponds to the “pure-homophily” model in
Calvó-Armengol and Jackson (2004) and Graham (2014), and is also the leading case developed
in De Paula, Richards-Shubik, and Tamer (2014). The “capacity constraint” model in part (b)
generalizes one-to-one and one-to-many matching models without peer effects, where any node is
only permitted to form a limited number of links, the one-to-one matching model in Menzel (2013)
is a special case with s̄ = 1. Model (c) allows for a preference for completion of “transitive triad,”
a non-anonymous interaction effect, and model (d) nests some models of anonymous interaction
effects, where the negative sign of the (combined) interaction effect could arise e.g. from a “congestion externality” at node i. This list is clearly not exhaustive, and future work will aim at
deriving more general primitive conditions on payoff functions, as well as combining bounds for
several specific types of interaction effects.
In the following, let ηi = (ηi0 , ηi1 , . . . , ηin )′ , where we denote the conditional distribution of ηi
given Wi∗ with
Gη|W (ηi |W) := P (ηi ≤ η|Wi∗ = W)
and the associated p.d.f. with gη|W (η|W), and the unconditional distribution functions with Gη (η)
and gη (η), respectively. The following lemma summarizes the main finding regarding dependence
between taste shifters ηi and link opportunities represented by Wi∗ .
21
Lemma 4.1. Suppose Assumptions 4.1, 4.2, and 4.3 hold. Then, for any pairwise stable network,
gη|W (η|Wi∗ )
= oP (1)
−
1
gη (η)
with probability approaching 1 as n → ∞.
The proof of this result parallels that of Lemma B.3 in Menzel (2013). However there is an added
complication in that with general link externalities, any adjustments to the network position for a
given node may affect a larger number of “followers.” Hence, Assumption 4.1 imposes additional
requirements to limit the degree of interdependence of individual link decisions.
Conditional Choice Probabilities. The second step takes the limit of the conditional probability
that agent i is willing to form a link to agent j given xi , zj , and her other links. As in the case of
the matching model (Lemma 3.1 in Menzel (2013)), we find that given our specification the number
of links “accepted” by agent i is substantially smaller than the number of “proposals” j ∈ Wi (L∗ ),
so that the conditional probability of proposing or accepting a link depends only on the upper tail
of G(·), the distribution for the taste shifters ηij . The assumption that G(·) has tails of type I can
then be used to establish that conditional choice probabilities can be approximated by those implied
by the Logit model with taste shifters generated by an extreme-value type-I distribution. Here a
complication arises from the fact that all links (and availability to j) are determined simultaneously,
so that it is necessary to consider joint probabilities of the form
Φ(i, j1 , . . . , jr |Wi ) = P Uij1 , . . . , Uijr ≥ M Ci > Uij ′ all other j ′ ∈ Wi |Wi
Notice that marginal benefits Uij depends on Si and Tij , so that Φ(i, j1 , . . . , jr |Wi ) cannot be
directly interpreted as a conditional choice probability, but equals the probability that the configuration Lij1 = · · · = Lijr = 1 and Lij ′ = 0 for all other j ′ ∈ Wi satisfies the pairwise stability
conditions regarding player i’s payoffs. Such a configuration is not necessarily unique, but externalities among links emanating from i may support several stable outcomes for a given realization
of random payoffs.
We find that under our assumptions, we can approximate the conditional probability Φ(i, j1 , . . . , jr |Wi )
with its analog under the assumption of independent extreme-value type-I taste shifters.
Lemma 4.2. Suppose that Assumptions 4.1-4.3 hold, and that the marginal benefit functions
Ui1 , . . . , UiJ are J i.i.d. draws from the model (2.1), and marginal cost M Ci is an independent
draw from (2.2). Then as J → ∞,
Q
r! rs=1 exp{U ∗ (xi , xjs ; si , sjs , tijs )}
r/2
→ 0 (4.1)
n Φ(i, j1 , . . . , jr |Wi ) − r+1
P
1 + J1 Jj=1 exp{U ∗ (xi , xj ; S+1 (xi , xj ; si , sj ), sj , tij )}
for any r = 0, 1, 2, . . . , where si := S(L∗−i + {ij1 , . . . , ijr }, xi , i), sj = S(L∗−i + {ij1 , . . . , ijr }, xj , j),
and tij := T (L∗−i + {ij1 , . . . , ijr }, xi , xj , i, j).
Note that by construction, si , sj , tij are all fixed conditional on Wi even though L∗−i generally
isn’t. This approximation allows the use of inclusive values for the link opportunity sets to reparameterize conditional choice probabilities even if the distribution of taste shifters ηij is not
22
extreme-value type-I, but belongs to its domain of attraction. We also find that we can take joint
limits for any finite set of nodes, i1 , . . . , is conditional on Wi1 , . . . , Wis in an analogous fashion.
Law of Large Numbers. It follows from the previous two steps that we can approximate the
distribution of the individual response using the inclusive value of agent i’s link opportunity set
W , which we defined as
1 X
Ii [W ] := 1/2
exp{Uij∗ (L + {i, j})}
n j∈W
The third step of the argument establishes a conditional law of large numbers for the inclusive
∗ := I [W (L∗ )] which are sample averages over the characteristics of agents in the link
values Iin
i
i
√
opportunity set Wi (L∗ ), where the size of the set |Wi (L∗ )| grows at a rate n for any PSN.
Lemma 4.3. Suppose Assumptions 4.1, 4.2, and 4.3 hold. Then, (a) there exists a random function
Γ̂n (x, s) such that the inclusive values resulting from a pairwise stable network satisfy
Ii∗ = Γ̂n (xi , si ) + op (1)
for all i = 1, . . . , nw and j = 1, . . . , nm . Furthermore, (b), if the weight functions ω(x, x′ , s, s′ ) ≥ 0
are bounded and form a Glivenko-Cantelli class in (x, s), then
n
sup
x∈X s∈S
1X
ω(x, xj , s, sj )(Imj − Γ̂(xj , sj )) = op (1)
n
j=1
Given our results in Lemmata 4.1-4.2, the proof of this result is completely analogous to that for
Lemma B.5 in Menzel (2013). We refer to the limit Γ(x; s) as the inclusive value function. This
implies that up to sampling error, inclusive values only depend on agents’ own characteristics xi , Si ,
so that we do not need to keep track of individual-specific link opportunity sets separately as we
take limits.
Fixed-Point Mapping for Inclusive Value Functions. Next, we derive an (approximate)
fixed-point condition for the inclusive value function Γ(x; s) resulting from the law of large numbers
in the previous step. In the following, we partition the vector of node i’s network characteristics
P
into si = (s1i , s′2i )′ , where s1i := nj=1 Lij denotes the network degree of node i, and s2i a vector
of other payoff-relevant network statistics.
We now consider the link proposal indicator Dji = 1l{Uji ≥ M Cj }. Suppose first that sj = s
and i ∈
/ Wj . Note that in that case, a link proposal to i does not result in a new link, and therefore
Dji does not affect the network structure. Hence, for a given value of si , sj , tij , the link proposal
indicator Dji is uniquely determined from the realized payoffs. From the asymptotic approximation
23
to the individual response in Lemma 4.2, we can now obtain
nP (Dji = 1|Wi )
nE[Φ(i, j1 , . . . , jr |Wi )|Wi ]
X X
n
Φ(i, j1 , . . . , jr |Wi )
=
=
r≥0 j1 ,...,jr
X (r + 1)! exp{U (xi , xj ; r, sj ) + U (x, xi ; sj , r)}Γ(xi ; r)r
→
Z
=
(1 + Γ(xi ; r))r+2
r!
r≥0
(s1j + 1) exp{U (xi , xj ; Si , sj ) + U (x, xi ; sj , Si )}w∗ (x; sj )dsj
Hence, aggregating over j 6= i, we obtain
X
Γ̂n (xi ; si ) =
exp{U (xi , xj ; si , sj )}P (Dji = 1|Wi )
j6=i
=
X
exp{U (xi , xj ; si , sj )}P (Dji = 1|Wi ) +
j∈Wi
→
X
exp{U (xi , xj ; si , sj )}P (Dji = 1|Wi )
j ∈W
/ i
Z
1 X
(s1j + 1) exp{U (xi , xj ; Si , sj ) + U (x, xi ; sj , Si )}w∗ (x; sj )dsj
n
j ∈W
/ i
using the approximation from the previous display, and noting that by Lemma B.1, |Wi |/n → 0
almost surely, so that the contribution of individuals j ′ ∈ Wi is dominated by the contribution of
individuals j ∈
/ Wi .
From the definition of Ii and convergence of Ij to Γ(xj ; sj ), we can now write
n
Γ̂n (xi ; si ) =
1 X (s1j + 1) exp{U (xi , xj ; si , sj ) + U (xj , xi ; sj , si )}
+ op (1)
n
1 + Γ̂n (xj ; sj )
j=1
Hence, defining the mapping
n
Ψ̂n [Γ](x; s) :=
1 X (s1j + 1) exp{U (x, xj ; s, sj ) + U (xj , x; sj , s)}
n
1 + Γ(xj ; sj )
(4.2)
j=1
the inclusive value functions Γ∗ (x, s) resulting from a PSN satisfy the approximate fixed-point
condition
Γ̂n (x; s) = Ψ̂n [Γ̂n ](x; s) + op (1)
(4.3)
where, noting that Γ ≥ 0, the remainder converges in probability uniformly in the argument Γ.
Existence and Uniqueness of Fixed Point for Inclusive Value Functions. Next we can
characterize the limit for Γ̂n for a given reference distribution w(x; s). To this end, we define the
population analog of the fixed-point operator in (4.2),
Z
(s1j + 1) exp{U (x, xj ; s, sj ) + U (xj , x; sj , s)}
Ψ0 [Γ, w](x; s) :=
w(xj ; sj )dxj dsj
(4.4)
1 + Γ(xj ; sj )
Given the reference distribution w(x; s), we then let Γ∗ (x; s) be a solution of the fixed-point problem
Γ∗ = Ψ0 [Γ∗ , w]
24
(4.5)
We next give conditions under which for any given reference distribution, the fixed point exists and
is unique:
Proposition 4.2. Suppose that Assumptions 4.1-4.3 hold. Then (i) for any given distribution
w(x; s) for which the network degree s1i satisfies E[s1i |xi ] + 1 < Bs < ∞ almost surely, the mapping
Bs exp{2Ū }
log Γ 7→ log Ψ[Γ] is a contraction mapping with contraction constant λ < 1+B
. Moreover,
s exp{2Ū }
(ii) the fixed points in (4.5) are continuous functions that have bounded partial derivatives at least
up to order p.
The formal argument for this result closely parallels the proof of Theorem 3.1 in Menzel (2013)
Bs exp{2Ū }
with contraction constant equal to 1+B
, a separate proof is therefore omitted.
s exp{2Ū }
One case of particular interest for which the contraction property holds without additional assumptions is that of no endogenous interaction effects, as shown by the following corollary:
Corollary 4.1. Suppose Assumptions 4.1-4.3 hold, and U (x1 , x2 ; s1 , s2 , t12 ) = U (x1 , x2 ). Then the
solution Γ∗ (x; s) = Γ∗ (x) to the fixed point problem (4.5) is unique.
The proof of this corollary is given in the appendix.
Fixed-Point Conditions for Reference Distributions. We next discuss convergence of the
fixed-point problem characterizing w∗ (x; s), where a reference distribution resulting from a pairwise
stable network has to be an element of the core of the capacity Ω0 [Γ∗ , w∗ ]. While a capacity is
defined on all possible subsets of S, for a full characterization of the resulting core it is often
sufficient to focus at a much smaller class of sets. We say that the collection R ⊂ 2S is coredetermining if Ω1 (S) = Ω2 (S) for all S ∈ R implies core(Ω1 ) = core(Ω2 ), see Galichon and Henry
(2011). For example, if the core of the capacity is a single distribution, then the singleton sets
{{s} ∈ S} are core-determining.
We can now describe the set of possible distributions for the network statistics si that result from
the asymptotic approximation to the individual response for a given distribution w and inclusive
value function Γ: Let Ω̂n [Γ, w] denote the capacity describing the set of reference distributions
w(x; s) consistent with pairwise stability. Then the reference distribution ŵn (x; s) in the n-agent
network has to satisfy the equilibrium conditions
Z
ŵn (x; s)
ds ≤ Ω̂n [Γ̂n , ŵn ](x; S) + op (1) for all S ∈ R
(4.6)
S w(x)
where R is a core-determining class of sets for Ω0 and Ω̂n . We can also formulate a population
analog of these equilibrium conditions
Z
w∗ (x; s)
ds ≤ Ω0 [Γ∗ , w∗ ](x; S) for all S ∈ R
(4.7)
w(x)
S
where Ω0 is a population analog of the capacity Ω̂n .
If the individual response with respect to si is almost surely unique, then Ω̂n and Ω0 are proper
probability distributions, so that the singleton sets {{s} : s ∈ S} are a core-determining class. In
that case, the equilibrium conditions in (4.7) simplify to
w∗ (x; s)
= Ω0 [Γ∗ , w∗ ](x; s)
w(x)
25
with an analogous condition for the finite-sample problem (4.6).
Since the selection mechanism on the individual response may vary discontinuously in x, the
core of Ω0 (x; S) can be a fairly rich set of probability distributions, many of which do not meet
any useful smoothness criteria. However for our analysis of convergence of the set of equilibrium
reference distributions, it is sufficient to restrict our attention to distributions that determine the
boundary of the core on the probability simplex ∆(X ×S). Specifically, we say that the distribution
w(x; s)
R is on the boundary of the core of Ω0 if at all values of x, there exists a set S(x) ∈ R such
that
w(x;s)
w(x)
S(x)
= Ω0 (x, S(x)) with equality.
We can now formulate the main assumptions on the fixed-point mappings Ω̂n and Ω0 for the
reference distributions in the finite network and the limiting economy, respectively:
Assumption 4.5. (i) The mapping is upper hemi-continuous and compact in Γ, w for all x ∈ X
and S ∈ R, and (ii) the core of Ω0 [Γ, w] is nonempty, where the boundary
of the core of Ω0 [Γ,w] is
in some subset U ⊂ ∆(X × S) for all values of Γ, w. (iii) supx,Z∈R Ω̂n [Γ, w](Z) − Ω0 [Γ, w](Z) → 0
uniformly in Γ ∈ T and distributions w ∈ U , where R is a core-determining class for Ω0 . (iv) The
set of fixed points of (Γ, w) ⇒ (Ψ0 , Ω0 )[Γ, w] is compact.
These high-level assumptions on the equilibrium mapping Ω0 have to be verified on a case by
case basis. Since in standard applications, the distributions in the core of Ω0 [Γ, w] are generated
by mixtures of individual responses to the aggregate state variables, convexity in part (ii) is always
satisfied in the absence of restrictions on the statistical mechanism for selecting pairwise stable
outcomes.
Uniform convergence of Ω̂n with respect to w in part (iii) is only stated only as a high-level
condition in order to keep the result as general as possible.11 As Example ?? illustrates, for some
cases of applied interest the mapping Ω̂n need not depend on the sampling distribution of types,
in which case uniform convergence as in part (iii) may trivially hold.
It is important to notice that the assumptions on the fixed-point mapping Ω0 are sufficient to
guarantee existence of an equilibrium reference distribution, as stated in the following proposition,
which is proven in the appendix.
Proposition 4.3. (Fixed Point Existence) Suppose that the conditions of Proposition 4.2 hold
and that Ω0 satisfies Assumption 4.5 (i)-(ii). Then the mapping (Γ, w) ⇒ (Ψ0 , Ω0 )[Γ, w] has a fixed
point.
Convergence of the link frequency distribution. We can now combine the previous steps to
show joint convergence for the reference distribution ŵ(x; s) and the inclusive value function Γ̂n (x; s)
to solutions of the population fixed-point problem (4.5) and (4.7). Specifically, Lemmata 4.2 and 4.3
11Note that the boundary distributions are pointwise minima of a selection of
n
o
Ω(x; S) : S ∈ 2|S|
and
n
o
Ω̂n (x; S) : S ∈ 2|S| , respectively. Hence, if S is finite, and Ω0 (x; S) and Ω̂n (x; S) have bounded partial derivatives
of order p ≥ 1, then the boundary distributions can be represented using a finite intersection of Glivenko-Cantelli
classes, which is also Glivenko-Cantelli. In that case, uniform laws of large numbers with respect to potential boundary distributions of the core can be established under otherwise standard regularity conditions, which can be used to
establish uniform convergence if Ω̂n (x; S) depends on sample averages with respect to the sampling distribution of
xi .
26
imply that link opportunity sets can be parameterized with the inclusive value functions, whereas
the fixed-point conditions for the inclusive value function and reference distribution converge to
their respective population limits.
Theorem 4.1. Suppose the conditions for Proposition 4.2 and Assumption 4.5 hold. Then for any
stable network, the inclusive value function Γ̂n (x; s) and reference distribution w∗ (x; s) satisfy the
fixed-point conditions in (4.3) and (4.6). Moreover, there exist Γ∗ , w∗ satisfying the population fixedpoint conditions in (4.5) and (4.7) such that kΓ̂n − Γ∗ k = op (1) and kŵn (x; s) − w∗ (x; s)k = op (1).
Since in the limit, the link frequency distribution can be parameterized in terms of Γ∗ and w∗ ,
we then also obtain convergence of the link frequency distribution to the model specified in (3.3).
Formally, we can combine the result in Theorem 4.1 with Lemma 4.2 to obtain our main result in
this section:
Theorem 4.2. Under the assumptions made before, there exists a distribution F (x1 , x2 ; s1 , s2 ) with
the p.d.f. f (x1 , x2 ; s1 , s2 ) specified in (3.3), such that the link frequency distribution
sup
x1 ,x2 ,s1 ,s2
|F̂n (x1 , x2 ; s1 , s2 ) − F (x1 , x2 ; s1 , s2 )| = op (1)
where the inclusive value function Γ∗ and distribution of network characteristics w∗ satisfy the
fixed-point conditions (3.4) and (3.5).
This limiting model gives a tractable characterization of the link distribution. By considering
only the distribution of links rather than the full adjacency matrix, we do not need to characterize
the structure of the full network explicitly, but the model is closed via equilibrium conditions on the
distribution w∗ (x; s). In contrast, the expressions in Chandrasekhar and Jackson (2011) and Mele
(2012) can only be approximated by simulation over all possible networks, the number of which
grows very fast as n increases.
Summing up, the limiting sequence considered has the following qualitative features: (1) each
agent can choose from a large number of possible link formation opportunities, and (2) similar agents
face similar choices, at least as measured by the inclusive values corresponding to link opportunity sets. (3) By construction, additional links become increasingly costly along the asymptotic
sequence, so that the resulting network remains sparse. (4) The resulting limiting distribution of
links for a pairwise stable network need in general not be unique due to the possible multiplicity
of reference distributions and indeterminacy of the individual response at the individual level.
5. Estimation of Preference Parameters
We next outline a strategy for estimating structural payoff parameters from network data, where
we assume that all payoff-relevant attributes xi and network characteristics si are observed for
a random sample of nodes i = 1, . . . , K included in the sample. The arguments below could be
extended to different sampling protocols and certain cases in which some components of xi are
not directly observed but generated from a distribution that is known up to a parameter to be
estimated. We plan to pursue some of those generalizations as part of the proposed activity.
27
5.1. Identification. The researcher may either have a complete data set of all links in the population, or a sample of links from the network.12 The link frequency distribution f (x1 , x2 ; s1 , s2 , t)
can then be used to derive sampling distributions under various protocols for sampling individuals
or links from the population, see the discussion in section 2.5 in Menzel (2013).
In the case of perfectly observable attributes xi and knowledge of the complete network L, the
network statistics Si and Tij can be computed from the available data. Moreover the equilibrium
distributions w∗ (x; s) and h∗ (t|x1 , x2 ; s1 , s2 ) can be estimated consistently from the observed sample. The inclusive value function Γ∗ (x; s) is only implicitly defined through the fixed-point condition
(3.4) which is known to have a unique solution.
We now outline preliminary nonparametric identification results based on the asymptotic approximation, which do not yet address the case of a set-valued individual response, but are otherwise
fully general. Define
V ∗ (x1 , x2 ; s1 , s2 ) := U ∗ (x1 , x2 ; s1 , s2 ) + U ∗ (x2 , x1 ; s2 , s1 )
The conditional probability of a link forming between agent i and j given node attributes xi , xj ,
and network characteristics si , sj is given by
H(1, xi , xj ; si , sj ) := nP (Lij = 1|xi , xj ; si , sj ) =
f (xi , xj ; si , sj )
∗
i ; si )w (xj ; sj )
w∗ (x
noting that all objects on the left-hand side are identified from the data. Substituting the expression
from (3.3) and taking logs,
log H(1, xi , xj ; si , sj ) = V ∗ (xi , xj ; si , sj ) − log(1 + Γ∗ (xi , si )) − log(1 + Γ(xj , sj ))
On the other hand, we can show that the probability of no link between i and j satisfies
H(0, xi , xj ; si , sj ) := n(1 − P (Lij = 0|xi , xj ; si , sj )) =
1
(1 + Γ∗ (xi , si ))(1 + Γ∗ (xj , sj ))
Hence, we can point-identify V ∗ (x1 , x2 ; s1 , s2 ) from the log of the likelihood ratio,
V ∗ (x1 , x2 ; s1 , s2 ) = log H(1, x1 , x2 ; s1 , s2 ) − log H(0, x1 , x2 ; s1 , s2 )
The case of individual-specific network statistics is less straightforward since we can only construct
asymptotic bounds for the conditional choice probabilities.
5.2. Estimation and Inference. Estimation and inference for the network model is complicated
by the presence of multiple stable outcomes. However, while the fixed-point conditions in (3.5) may
admit multiple solutions, the distribution w∗ (x, s) resulting from the equilibrium chosen in the data
can be estimated consistently from the observed network. Our approach is therefore conditional on
the non-unique equilibrium distribution w∗ (x, s), which we replace by a consistent estimate. This
strategy for dealing with multiple equilibria is analogous to Menzel (2012)’s approach for the case
of discrete action games.
12For example, the researcher may sample nodes at random and eliciting all links emanating from each node (“induced
subgraph”), or only the links among the nodes included in the survey (“star subgraph”), see Chandrasekhar and Lewis
(2011) for a discussion.
28
The other potential difficulty is that the limiting distribution in (3.3) depends on the (unobserved)
inclusive value function. Following the approach in Menzel (2013) for the case of matching markets,
we suggest to treat Γ∗ (x, s) as an auxiliary parameter in maximum likelihood estimation of the
surplus function V ∗ (x1 , x2 ; s1 , s2 ) satisfying the fixed-point condition (3.4). Specifically, we propose
the maximum likelihood estimator θ̂ solving
max LK (θ, Γ)
θ,Γ
s.t. Γ = Ψ̂K (Γ)
(5.1)
where LK (·) is the log-likelihood function for the sample based on the approximation in (3.3), and
Ψ̂K (·) is the sample analog of the fixed-point mapping Ψ0 (·), where expectations are replaced with
a sample average over observed nodes i = 1, . . . , K. For other sampling protocols with uniform
qualification probabilities, the formulae for LK (·) and ΨK (·) can be adjusted using weights.
6. Discussion
This paper develops an asymptotic representation of the link frequency distribution resulting
from a network formation game. The most important further steps envisioned for this project
involve extensions of the main theoretical result, and design and implementation of estimation
procedures based on that limiting approximation.
A future version of this paper will discuss estimation techniques in greater detail, including
their asymptotic properties with a focus on a computationally attractive implementation for the
empirically relevant case of continuous attributes xi and unobserved effects models. The proposed
approach towards estimation and inference is conditional on the equilibrium distributions w∗ (x; s)
and h∗ (t12 |x1 , x2 ; s1 , s2 ), which can be estimated consistently if the relevant node attributes and
network variables are observed. While the conceptual arguments justifying this approach follow
closely the approach in Menzel (2012), the main challenge is construction of a coupling between
network outcomes for different values of n.
One important theoretical extension concerns identification and estimation when player attributes are only partially observed. Specifically we aim at developing a version of the model
in which an unobserved component of xi simultaneously affects another node-level outcome variable in a network model of peer effects, similar to the framework in Goldsmith-Pinkham and
Imbens (2012). This involves a characterization of equilibrium conditions for the joint distribution
of the outcome of the network formation game and the response variable in a large network, and
investigate identification and estimation of treatment effects.
Appendix A. Proofs for Results from Section 2
A.1. Proof of Lemma 2.1. To verify that the statement in Lemma 2.1 is indeed equivalent to
the usual definition of pairwise stability, notice that if L∗ is not pairwise stable, there exists two
nodes i, j with L∗ij = 0 such that Uij (L∗ ) > M Cij (L∗ ) and Uji (L∗ ) > M Cji (L∗ ). In particular, j
is available to i under L∗ , i.e. j ∈ Ni [L∗ ], violating (2.3). Conversely, if (2.3) does not hold for
node i, then there exists j ∈ Ni [L∗ ] such that Uij (L∗ ) ≥ M Cij (L∗ ). On the other hand, j ∈ Ni [L∗ ]
implies that Uji (L∗ ) ≥ M Cji (L∗ ), where all inequalities are strict in the absence of ties.
29
Appendix B. Proofs for Results from Section 3
B.1. Proof of Lemma 4.2. This result is a generalization of Lemma B.1 in Menzel (2013). We
therefore refer to the proof of that result for some of the intermediate technical steps below. Define Ũij := U ∗ (xi , xj ; si , sj , tij ) for j = j1 , . . . , jr , and Ũij := U ∗ (xi , xj ; S+1 (xi , xj ; si , sj ), sj , tij )
otherwise. Then by independence of ηi1 , . . . , ηiN ,
J r Φ(i, j1 , . . . , jr |Wi ) = J r P (Uij1 ≥ M Ci , . . . , Uijr ≥ M Ci , Uijr+1 < M Ci , . . . , UijJ < M Ci )



Z
r
J
Y
Y
= Jr 
P (Uijq ≥ σs) 
P (Uijq < σs) JG(s)J−1 g(s)ds
q=1
= Jr
=
Z

Z

r
Y
q=r+1

(1 − G(s − σ −1 Ũijq )) 
q=1


r
Y
q=1

J
Y
q=r+1
g(s)
J(1 − G(s − σ −1 Ũijq )) J
G(s)

G(s − σ −1 Ũijq ) JG(s)J−1 g(s)ds

J

X
1
J log G(s − σ −1 Ũijq ) ds
× exp J log G(s) +


J q=r+1


Limit for joint p.d.f. of highest order statistics. Now let bJ := G−1 1 − J1 and aJ = a(bJ ), where
a(·) is the auxiliary function in Assumption 4.2 (ii). By Assumption 4.3 (iii), σ = a(b1J ) , so that a
change of variables s = aJ t + bJ yields


Z
r
Y
aJ g(bJ + aJ t)

J(1 − G(bJ + aJ (t − Ũijq ))) J
J r Φ(i, j1 , . . . , jr |Wi ) =
G(bJ + aJ t)
q=1


J


1 X
× exp J log G(bJ + aJ t) +
J log G(bJ + aJ (t − Ũijq )) dt


J
q=r+1
By Assumption 4.2 (ii), J(1 − G(bJ + aJ t)) → e−t and
JaJ g(bJ + aJ t) = Ja(bJ )g(bJ + a(bJ )t) = a(bJ )
1 − G(bJ + aJ t)
→ e−t
a(bJ + aJ t)(1 − G(bJ ))
where the last step uses Lemma 1.3 in Resnick (1987). Also, following steps analogous to the proof
of Lemma B.1 in Menzel (2013), we can take limits and obtain


r
r


X
Y
J(1 − G(bJ + aJ (t − Ũijq ))) → exp −rt +
Ũijq


q=1
q=1
J log G(bJ + aJ (t − Ũijq )) → −e−t exp{Ũijq }
30
Combining the different components, we can take the limit of the integrand in (B.1),


r
Y
aJ g(bJ + aJ t)
J(1 − G(bJ + aJ (t − Ũijq ))) J
RJ (t) := 
G(bJ + aJ t)
q=1


J


1 X
× exp J log G(bJ + aJ t) +
J log G(bJ + aJ (t − Ũijq ))


J q=r+1




r
J


X
X
1
Ũijq + o(1)
exp{Ũijq } − (r + 1)t +
= exp −e−t 1 +


J
q=1
q=r+1
for all t ∈ R. Using the same argument as in the proof of Lemma B.1 in Menzel (2013), pointwise
convergence and boundedness of the integrand imply convergence of the integral by dominated
convergence, so that we obtain




Z ∞
J
r


X
X
1
exp −e−t 1 +
J r Φ(i, j1 , . . . , jr |Wi ) →
exp{Ũijq } − (r + 1)t +
Ũijq dt


J q=r+1
−∞
q=1

 

Z 0
r
J


X
X
1
Ũijq sr ds
exp{Ũijq } +
exp s 1 +
=


J
−∞
q=1
q=r+1
=
P
r! exp{ rq=1 Ũikq }
or+1
n
P
1 + J1 Jq=r+1 exp Ũikq
where the first step uses a change of variables s = −e−t , and the last step can be obtained recursively via integration by parts. Furthermore, if Jr → 0, boundedness of the systematic parts from
Assumption 4.1 implies that
J
J
o
n o 1 X
n
1 X
exp Ũij −
exp Ũikq → 0
J
J q=r+1
j=1
so that
r
J Φ(i, j1 , . . . , jr |Wi ) → 1+
which completes the proof
r!
1
J
Qr
q=0 exp{Ũikq }
n or+1
exp
Ũij
j=1
PJ
Size of Opportunity Sets. In order to prove Theorem ??, we start by establishing the main
technical steps separately as Lemmata B.1-??. The first result concerns the rate at which the
number of available potential spouses increases for each individual in the market. For a given PSN
L∗ , we let
n
X
∗
∗
1l {Uji (L∗ ) ≥ M Cj }
Ji := Ji [L ] :=
j=1
31
denote the size of the link opportunity set available to agent i. Similarly, we let
Ki∗ =
n
X
j=1
1l {Uij (L∗ ) ≥ M Ci }
so that Ki∗ is the number of nodes to whom i is available.
Lemma B.1 below establishes that in our setup, the number of available potential matches grows
at a root-n rate as the size of the market grows.
Lemma B.1. Suppose Assumptions 4.1, 4.2, and 4.3 hold. Then for any pairwise stable network,
n1/2 exp{−Ū } ≤
Ji∗
≤ n1/2 exp{Ū }
n1/2 exp{−Ū } ≤ Ki∗ ≤ n1/2 exp{Ū }
for each i = 1, . . . , n with probability approaching 1 as n → ∞.
Proof of Lemma B.1: Notice that in the absence of interaction effects across links, Dji does
not depend on the number of “proposals” that can be reciprocated, but only the magnitude of
M Ci . Furthermore, by Assumption 4.1, the systematic parts of payoffs are uniformly bounded
for all values of si , sj . Hence the proof closely parallels the argument for the matching case. We
therefore only demonstrate that externalities across links do not alter that conclusion, for the
remaining technical steps we refer the reader to the proof of Lemma B.2 in Menzel (2013), which
is the analogous result for the two-sided matching problem.
Fix i, j ≤ n, and let Ũij := U ∗ (xi , xj , si , sj , t0 ). By Assumption 4.1, |Ũij | ≤ Ū , so that, following
a similar series of steps as in the proof of Lemma 4.2, the marginal probability
Z ∞
Z ∞
J
GJ (Ū + s)g(s)ds → exp{Ū }
G (Ũij + s)g(s)ds ≤ J
JP (Uij ≥ M Ci ) = J
−∞
−∞
Similarly, we find that
JP (Uij ≥ M Ci ) ≥ J
Since
Ki∗
:=
Pn
j=1 1l{Uij
Z
∞
−∞
GJ (−Ū + s)g(s)ds → exp{−Ū }
≥ M Ci }, we can bound the expectation,
n1/2 exp{−Ū } ≤ E[Ki∗ ] ≤ n1/2 exp{−Ū }
P
as n grows large. Similarly, Ji∗ := nj=1 1l{Uji ≥ M Cj } so that for n sufficiently large,
n1/2 exp{−Ū } ≤ E[Ji∗ ] ≤ n1/2 exp{−Ū }
These bounds are uniform for i = 1, 2, . . . . Given these rates for the expectation of the upper and
lower bounds for Ji∗ and Ki∗ , the conclusion of this lemma follows the same sequence of steps as in
the proof of Lemma B.2 in Menzel (2013)
B.2. Proof of Corollary 4.1: Given part (i) of Proposition 4.2, it is sufficient to show that
E[s1i |xi = x] is uniformly bounded for x ∈ X . To this end, notice that for payoffs of the form
U (x1 , x2 ; s1 , s2 ) = U (x1 , x2 ), the inclusive value function only depends on x, i.e. Γ∗ (x; s) = Γ∗ (x).
Furthermore, the individual response is unique so that the conditional degree distribution given
Γ∗ (x)s
xi = x has p.d.f. P (s1i = s|xi = x) = (1+Γ
∗ (x))s+1 . Hence, the conditional expectation of s1i is
32
given by
E[s1i |xi = x]
=
∞
X
s=0
=:
1
1 + Γ∗ (x)
Γ∗ (x)
∞
X
Γ∗ (x)s
1
s
s
=
(1 + Γ∗ (x))s+1
1 + Γ∗ (x)
s=0
∞
X
sδs =
s=0
Γ∗ (x)
1 + Γ∗ (x)
s
δ
1
= Γ∗ (x)
∗
1 + Γ (x) (1 − δ)2
where δ := 1+Γ∗ (x) . Finally, it remains to be shown that Γ∗ (x) is uniformly bounded: from the
fixed-point condition (4.5),
Z
(s1j + 1) exp{U (x, xj ; s, sj ) + U (xj , x; sj , s)}
Ψ[Γ, w](x) =
w(xj ; sj )dxj dsj
1 + Γ(xj )
Z
(Γ∗ (xj ) + 1) exp{U (x, xj ; s, sj ) + U (xj , x; sj , s)}
=
w(xj ; sj )dxj
1 + Γ(xj )
≤ exp{2Ū }
where Ū < ∞ is the bound in Assumption 4.1. Hence the range of Ψ0 is uniformly bounded, so
that the fixed point Γ∗ also has to satisfy this bound
B.3. Proof of Proposition 4.3. Notice that under the conditions of Proposition 4.2, Ψ0 is a
continuous, single-valued compact mapping.
R
wj (x;s)ds
Next, notice that for any two distributions w1 (x; s), w2 (x; s) satisfying S w(x)
≤ Ω0 (x, S) for
all core-determining sets S ⊂ S, the convex combination λw1 (x; s) + (1 − λ)w2 (x; s) satisfies the
same inequality constraints. Hence, the core of the capacity Ω0 is a convex subset of the probability
simplex ∆(X × S).
Furthermore,
if w3 (x; s) is in the complement of the core, there exists at least one set S ∈ R such
R
w3 (x;s)ds
> Ω0 (x, S) + ε, where ε > 0. Then for any distribution w′ with kw′ − w3 k∞ ≤ ε/2,
w(x)
R
w ′ (x;s)ds
> Ω0 (x, S) + ε/2. Hence the complement of the core is open, implying that the
have S w(x)
that
S
we
core is also a closed subset of ∆(X × S) with respect to the L∞ -norm on the probability simplex.
Hence, given the conditions on Ω0 in Assumption 4.5 (i)-(ii), existence of a fixed point is a direct
consequence of the Kakutani-Fan fixed point theorem for Banach spaces (Theorem 3.2.3 in Aubin
and Frankowska (1990))
References
Aubin, J., and H. Frankowska (1990): Set-Valued Analysis. Birkhäuser.
Baccara, M., A. Imrohoroglu, A. Wilson, and L. Yariv (2012): “A Field Study on Matching with Network Externalities,” American Economic Review, 102(5).
Banerjee, A., A. Chandrasekhar, E. Duflo, and M. Jackson (2013): “The Diffusion of
Microfinance,” Science, 341, DOI: 10.1126/science.1236498.
Beaman, L., and J. Magruder (2012): “Who gets the Job Referral? Evidence from a Social
Networks Experiment,” American Economic Review, 102(7), 3574–3593.
Beresteanu, A., I. Molchanov, and F. Molinari (2011): “Sharp Identification Regions in
Models with Convex Moment Predictions,” Econometrica, 79(6), 1785–1821.
33
Bickel, P., A. Chen, and E. Levina (2011): “The Method of Moments and Degree Distributions
for Network Models,” Annals of Statistics, 39(5), 2280–2301.
Bonaldi, P., A. Hortaçsu, and J. Kastl (2013): “An Empirical Analysis of Systemic Risk in
the Euro-Zone,” working paper, Chicago and Stanford.
Boucher, V., and I. Mourifié (2012): “My Friend Far Far Away: Asymptotic Properties of
Pairwise Stable Networks,” working paper, University of Toronto.
Calvó-Armengol, A., and M. Jackson (2004): “The Effects of Social Networks on Employment
and Inequality,” American Economic Review, 94(3), 426–454.
Calvó-Armengol, A., E. Patacchini, and Y. Zenou (2009): “Peer Effects and Social Networks
in Education,” Review of Economic Studies, 76, 1239–1267.
Calvó-Armengol, A., and Y. Zenou (2004): “Social Networks and Crime Decisons: The Role
of Social Structure in Facilitating Delinquent Behavior,” International Economic Review, 45(3),
939–958.
Campbell, A. (2013): “Word-of-Mouth Communication and Percolation in Social Networks,”
American Economic Review, 103(6), 2466–2498.
Chandrasekhar, A., and M. Jackson (2011): “Tractable and Consistent Random Graph Models,” working paper, Stanford.
Chandrasekhar, A., and R. Lewis (2011): “The Econometrics of Sampled Networks,” working
paper, Stanford and Google Research.
Choquet, G. (1954): “Theory of Capacities,” Annales de l’Institut Fourier, 5, 131–295.
Christakis, N., and J. Fowler (2007): “The Spread of Obesity in a large Social Network over
32 Years,” The New England Journal of Medicine, 357, 370–379.
Christakis, N., J. Fowler, G. Imbens, and K. Kalyanaraman (2010): “An Empirical Model
for Strategic Network Formation,” working paper, Harvard University.
Chung, K. (1997): “On the Existence of Stable Roommate Machings,” Games and Economic
Behavior, 33, 206–230.
Dagsvik, J. (1994): “Discrete and Continous Choice, Max-Stable Processes, and Independence
from Irrelevant Attributes,” Econometrica, pp. 1179–1205.
(2000): “Aggregation in Matching Markets,” International Economic Review, 41(1), 27–
57.
De Paula, A., S. Richards-Shubik, and E. Tamer (2014): “Identification of Preferences in
Network Formation Games,” working paper, UCL, CMU, and Harvard.
Fox, J. (2010): “Identification in Matching Games,” Quantitative Economics, 1, 203–254.
Frank, O., and D. Strauss (1986): “Markov Graphs,” Journal of the American Statistical
Association, 81, 832–842.
Gale, D., and L. Shapley (1962): “College Admissions and the Stability of Marriage,” The
American Mathematical Monthly, 69(1), 9–15.
Galichon, A., and M. Henry (2011): “Set Identification in Models with Multiple Equilibria,”
Review of Economic Studies, 78, 1264–1298.
Goldsmith-Pinkham, P., and G. Imbens (2012): “Social Networks and the Identification of
Peer Effects,” Journal of Business and Economic Statistics, 31, 253–264.
34
Graham, B. (2014): “An Empirical Model of Network Formation: Detecting Homophily when
Agenst are Heterogeneous,” working paper, UC Berkeley.
Hendel, I., and A. Nevo (2006): “Measuring the Implications of Sales and Consumer Inventory
Behavior,” Econometrica, 74(6), 1637–1673.
Hoff, P., A. Raftery, and M. Handcock (2002): “Latent Space Approaches to Social Network
Analysis,” Journal of the American Statistical Association, 97, 1090–1098.
Jackson, M. (2008): Social and Economic Networks. Princeton University Press.
(2014): “Networks in the Understanding of Economic Behaviors,” Journal of Economic
Perspectives, 28(4), 3–22.
Jackson, M., and A. Watts (2002): “The Evolution of Social and Economic Networks,” Journal
of Eonomic Theory, 106, 265–295.
Jackson, M., and A. Wolinsky (1996): “A Strategic Model of Social and Economic Networks,”
Journal of Economic Theory, 71, 44–74.
Kalai, E. (2004): “Large Robust Games,” Econometrica, 72(6), 1631–1665.
Kinnan, C., and R. Townsend (2012): “Kinship and Financial Networks, Formal Financial
Access, and Risk Reduction,” American Economic Review: P&P, 102(3), 289–293.
Lee, R., and K. Fong (2013): “An Applied Framework for Bilateral Oligopoly and Bargaining
in Buyer-Seller Networks,” working paper, Harvard and Stanford.
Luce, R. (1959): Individual Choice Behavior. Wiley, New York.
Manski, C. (1993): “Identification of Endogenous Social Effects: The Reflection Problem,” Review
of Economic Studies, 60, 531–542.
McFadden, D. (1974): Conditional Logit Analysis of Qualitative Choice Behavior pp. 105–142.
Academic Press, New York, in Zarembka (ed.): Frontiers in Econometrics.
Mele, A. (2012): “A Structural Model of Segregation in Social Networks,” working paper, Johns
Hopkins University.
Menzel, K. (2012): “Inference for Games with Many Players,” working paper, New York University.
(2013): “Large Matching Markets as Two-Sided Demand Systems,” working paper, New
York University.
Milgrom, P., and J. Roberts (1990): “Rationalizability, Learning, and Equilibrium in Games
with Strategic Complementarities,” Econometrica, 58(6), 1255–1277.
Miyauchi, Y. (2012): “Structural Estimation of a Pairwise Stable Network with Nonnegative
Externality,” working paper, MIT.
Molchanov, I. (2005): Theory of Random Sets. Springer, London.
Pȩski, M. (2014): “Large Roommate Problem with Non-Transferable Utility,” .
Resnick, S. (1987): Extreme Values, Regular Variation, and Point Processes. Springer.
Roth, A., and M. Sotomayor (1990): Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press.
Sheng, S. (2014): “A Structural Econometric Analysis of Network Formation Games,” working
paper, UCLA.
35
Snijders, T. (2011): “Statistical Models for Social Networks,” Annual Review of Sociology, 37,
131–153.
Tan, J. (1991): “A Necessary and Sufficient Condition for the Existence of a Complete Stable
Matching,” Journal of Algorithms, 12, 154–178.
Wasserman, S., and P. Pattison (1996): “Logit Models and Logistic Regressions for Social
Networks: I. An Introduction to Markov Graphs and p∗ ,” Psychometrika, 61(3), 401–425.
Weese, E. (2008): “Political Mergers as Coalition Formation: Evidence from Japanese Municipal
Amalgamations,” working paper, Yale University.
36