Estimation of Large Network Formation Games

Transcription

1 Estimation of Large Network Formation Games Geert Ridder Shuyang Sheng February 8, 207 Preliminary Abstract This paper develops estimation methods for network formation using observed data from a single large network. We characterize network formation as a simultaneousmove game with incomplete information, where we allow for utility externalities from indirect friends such as friends in common, so the expected utility from direct friends can be nonlinear. Nonlinearity poses a challenge in estimation because each individual faces an interdependent multinomial discrete choice problem with 2 n alternatives, which is diffi cult to solve. We propose a novel method to linearize the expected utility using Legendre transform and derive a closed-form expression for the conditional choice probability CCP. Using the closed-form expression, we show that the CCP in n-player games converge to the CCP in a limit game as n approaches infinity. We propose a two-step estimation procedure that requires few assumptions on equilibrium selection, is simple to compute, and provides asymptotically valid estimators for the parameters. Monte Carlo results show that the estimation procedure can provide accurate estimates if networks are large. KEYWORDS: Network formation, Large games, Incomplete information, Two-step estimation, Legendre Transform JEL Codes: C3, C3, C57, D85 We are grateful to Denis Chetverikov, Aureo de Paula, Bryan Graham, Jin Hahn, Bo Honore, Zhipeng Liao, Rosa Matzkin, Kevin Song and participants of seminars at UC Berkeley, UC Davis, UCLA, UC Riverside, Princeton, U Colorado, Northwestern, Econometric Society World Congress 205, Econometric Society Winter Meeting and Summer Meeting 206, INET conferences on econometrics of networks at University of Cambridge 205 and USC 205, Berkeley/CeMMAP conference on networks at Berkeley 206. Department of Economics, University of Southern California, Los Angeles, CA ridder@usc.edu. Department of Economics, University of California at Los Angeles, Los Angeles, CA ssheng@econ.ucla.edu.

2 Introduction This paper contributes to the as yet small literature on the estimation of game-theoretic models of network formation. The purpose of the empirical analysis is to recover the preferences of the members of the network, in particular the preferences that determine whether a member of the network will form a link friendship, business relation or some other type of link with a specific other member of the network. The preference for a link depends in general on the characteristics of the two members, and on their position in the network, e.g., their number of friends, their number of common friends. It is the dependence of the link preference on the position in the network that complicates the analysis. The preference also depends on unobservable features of the members and the link and assumptions on the nature of these unobservables play a key role in the empirical analysis. In principle link formation models are discrete choice models where multiple alternatives links can be chosen. If the agent has a myopic strategy s/he chooses to form a link if the utility of the link is larger than the utility of not forming the link. There are two reasons why such a strategy is suboptimal. First, the agent is better off considering the choice of links with the other members in the network simultaneously. Second all members in the network are making link choices and all these choices have to be consistent. In this paper the consistency is achieved by assuming that the observed network is a Bayesian Nash equilibrium. In general there is no unique Bayesian Nash equilibrium. This implies that full-information methods, either have to specify an equilibrium selection mechanism or have to consider partial instead of point identification of the preference parameters. In this paper we propose a limited-information method that is valid even if we do not know the equilibrium selection mechanism. Assumptions regarding the unobservables in the link preferences play an important role in the empirical analysis. The extreme assumptions are complete information where all members but not the econometrician know the unobservables in the preferences for links with all members and incomplete information where a member only knows its own link specific unobservables. The complete information models are the hardest to estimate and they achieve set and not point identification of the parameters Miyauchi 203 and Sheng 206. Leung 205 considers a model in which an agent only knows the link unobservable when considering to form that link. In this paper we consider a case in which the agent knows her/his own unobserved link preferences, but not those of other agents. So our assumption is between that in Leung 205 and complete information, and our assumption is also in line with the usual assumption in discrete choice models. Jackson 2008 surveys game-theoretic models of network formation. 2

3 A further distinction in the empirical literature regards the data. We can have data on a large number of small networks, e.g., friendship networks in classrooms, or we can have data on a single large network. This paper considers the latter case Menzel 206b, Leung 205, and De Paula, Richards-Shubik and Tamer 205 also consider large networks. An advantage of the large single network case is that for a fairly general utility function under a Bayesian Nash equilibrium the link choices converge to the myopic decision rule of a single agent, because the normalized preferences converge to the preferences for this case. This is true even though the link preferences depend on a non-trivial and non-vanishing way on the position of the agent in the network. This simplification only holds if we add a suffi cient statistic that captures network position. An implication of this result is that we can estimate preference parameters by a two-step procedure. Since this procedure only uses the first-order condition for optimal link choice it does not require an assumption on equilibrium selection beyond the assumption that the equilibrium selection does not change with the sample size, so that the first stage converges. The plan of the paper is as follows. In Section 2 we introduce the model and the specific utility function that we will use. We also discuss the Bayesian Nash equilibrium for the network. In Section 3 we obtain a closed-form expression for the link-formation probability that avoids the solution of an integer program. We discuss in Section 4 the uniform convergence of the choice probabilities if the network size grows without bounds. Section 5 discusses the two-step estimator. Section 6 considers the extension to undirected networks and Section 7 conducts a simulation study. 2 Model Let N n = {,..., n} be a set of individuals who play to form links. The links form a network, which we denote by G G. This is an n n binary matrix. The i, j element G ij = if i,j are linked and 0 otherwise. The diagonal elements G ii are set to be 0. We first consider directed networks, i.e., G ij and G ji may be different. The case of undirected links is discussed later in Section 6. Each individual i has a vector of observed characteristics X i X and a vector of unobserved preference shocks ε i = ε ij j Nn R n, where ε ij is i s preference for link ij and ε ii = 0. We assume that the characteristics X = X i i Nn are publicly observed by all the individuals, but the shocks ε i are private information of i. We also assume that the private shocks are i.i.d. and independent of the observables. Assumption i {ε ij, i j N n }, are i.i.d. with CDF F ε θ ε ported on R n that is 3

4 absolutely continuous with respect to Lebesgue measure. F ε θ ε is known up to θ ε Θ ε R dε. ii {X i, i N n } are i.i.d. iii For all i N n, ε i and X are independent. Utility Given the network G, characteristic profile X, and private shocks ε i, individual i has utility U i G, X, ε i ; θ = n G ij v ij G i, X; β ε ij + G ij G ik h i G j, G k ; γ n n 2 k i,j where G i = G ij j Nn is the ith row of G i.e., the links formed by i and G i = G j is the submatrix of G with the ith row deleted i.e., the links formed by individuals other than i. We assume that the utility is known up to θ = β, γ in a compact set Θ u R du. In the utility specification, we use v ij G i, X i, X j ; β to represent the incremental utility from a link that is separable in i s links. A typical example of v ij G i, X i, X j ; β could be v ij G i, X; β = β 0 + X iβ + X i X j β 2 + G ji β 3 + n 2 k i,j G jk w X i, X k β 4 2 where the first four terms capture the direct utility from j, while the last term captures the indirect utility from j s friends, in terms of j s weighted average out-degree, with w X i, X k representing the weights. The specification of separable indirect utility is similar to that in Leung 205. What differentiates our work from his is that in addition to the separable indirect utility, we also allow for utility from indirect friends that are nonseparable in one s links. Such nonseparable indirect utility is represented by h i G j, G k ; γ. In fact, many wellknown network externalities are nonseparable in one s links. For example, if we want to capture the utility from friends in common, we may specify h i G j, G k ; γ = G jk G kj γ + n 3 l i,j,k G jl G kl γ 2 3 where the first term captures the effect of friends in common that are directly connected and the second term captures the effect of friends in common that are indirectly connected. It is essential to be able to allow for such nonseparable externalities if we want to explain clustering, a well-documented feature of social and economic networks Jackson, We normalize the summations appeared above by n or n 2 to ensure that those terms remain bounded when the number of individuals increases to infinity, the asymptotic scenario we will consider for estimation. 4

5 Equilibrium Let G i X, ε i denote individual i s pure strategy, which maps i s information X, ε i to a vector of links in G i = {0, } n. The optimal strategy of individual i maximizes her expected utility E [U i g i, G i, X, ε i X, ε i ] over g i G i, where the expectation is taken with respect to others actions G i given her belief about G i. Because the private shocks are independent by Assumption, individual i s belief about G i depends only on the public information X. Let i g i X = Pr G i X, ε i = g i X be the probability that individual i chooses g i given X. Under Assumption, actions {G i, i N n } are conditional independent given X, so individual i s belief about others actions is i g i X = j g j X. Let X = { i g i X, i N n, g i G i } be the belief profile, a function that maps characteristics X X n to a n2 n -dimensional vector with entry values in [0, ]. Given belief profile, individual i s expected utility is E [U i G i, G i, X, ε i X, ε i, ] = n G ij E [v ij G i, X X, ] ε ij + G ij G ik E [h i G j, G k X, ] n n 2 k i,j = G ij v ij g i, X j g j X ε ij n g i + G ij G ik h i g j, g k j g j X k g k X n n 2 gj,gk k i,j and thus the probability that i chooses g i given X and is 4 Pr G i = g i X, = Pr E [U i g i, G i, X, ε i X, ε i, ] max E [U i g i, G i, X, ε i X, ε i, ] g i G i X,. 5 In a n-player game, a Bayesian Nash equilibrium X is a belief profile that satisfies i g i X = Pr G i = g i X, X 6 for all characteristic profiles X X n, actions g i G i, and individuals i N n. In this paper, we focus on equilibria that are symmetric in individuals observed characteristics. We say an equilibrium X is symmetric if for i and j with X i = X j, we have i X = j X. In network data where individuals do not have identities, it makes sense to assume symmetric equilibria because otherwise the optimal strategies and equilibria may 5

6 depend on how we label the individuals. More importantly, as we assume only one network is observed, the assumption of symmetric equilibria will be crucial for valid estimation and inference. It can be shown that there exists a symmetric equilibrium. We assume that the network observed is a symmetric equilibrium. Proposition For any X X n, there exists a symmetric equilibrium X. Proof. See the appendix. 3 Representation of the Optimal Strategies Analyzing the choice probability in 5, especially its asymptotic behavior when the number of individuals n approaches infinity, is challenging because the expected utility in 4 is nonseparable in G i, while the entries of G i take binary values, so there is no simple solution for the optimal G i. In this section, we propose a novel approach to deriving the optimal G i in a pseudo closed form. This pseudo closed form is simple enough that it enables us to explicitly characterize the dependence between the entries in the optimal G i, derive the limit of the choice probability in 5 as n approaches infinity, and propose estimates for the choice probabilities that are asymptotically valid based on a single observed network. These properties are crucial for the estimation for model parameters. The insight of our approach is to transform the expected utility into a form with auxiliary variables that is linear in G i, so the optimal G i can be solved in closed form. To facilitate the presentation of the idea, we assume that X i has a finite port so the derivation can be presented in simple matrix notation. This assumption is inessential and can be relaxed to allow for continuous X i. Assumption 2 The port of X i has finite distinct values, X = { x,..., x } T. To proceed, first observe that by symmetric assumption the quadratic form in the expected utility in 5 is symmetric in j and k. Moreover, in a symmetric equilibrium X, if two individuals j and k have the same characteristics, i.e., X j = X k, we have j X = k X. This implies that the j, k element in the matrix obtained from the quadratic form, i.e., E [h i G j, G k ] = g j,g k h i g j, g k j g j X k g k X, depends on j, k only through the value of X j, X k. In other words, for two pairs of individuals j, k and l, m with the same characteristics X j, X k = X l, X m, we have E [h i G j, G k ] = E [h i G l, G m ]. Therefore, given X jk = X l l j,k, we can view E [h i G j, G k ] as a function of X j, X k, 6

7 and denote the value of this function at X j, X k = x s, x t, s, t T, by H st X = E [ h i G j, G k X j = x s, X k = x t, X jk ] = g j,g k h i g j, g k j gj X j = x s, X k = x t, X jk k gk X j = x s, X k = x t, X jk Write all such values in a matrix H X = H X H T X. H T X H T T X The above defined H st and H may have subscript i but we abbreviate it for simplicity because whenever we consider them we will fix individual i. Example For the nonseparable utility specification in 3, we have E [h i G j, G k X, ] = jk X kj X γ + n 3 where jk X = Pr G jk = X, j, k N n, so. l i,j,k jl X kl X γ 2 H st X jk = jk x s, x t, X jk kj x s, x t, X jk γ + jl x s, x t, X jk kl x s, x t, X jk γ2 n 3 l i,j,k As discussed H is a real symmetric matrix. It has a real spectral decomposition H = Φ Λ Φ 7 where Λ = diag λ,..., λ T is the diagonal matrix of eigenvalues and Φ = φ,..., φ T is the orthogonal matrix of eigenvectors. Using the eigendecomposition in 7, we can reduce the quadratic form in 4 to a canonical quadratic form that involves only squares of linear functions of G i. Once we have the canonical form, we can linearize those squares of linear functions of G i by applying a special case of Legendre transform Rockafellar, 970 for square functions, namely, { Y 2 = max } 2Y ω ω 2 8 ω R Note that the maximand on the right hand side of 8 is linear in Y. The geometric intuition 7

8 of 8 is that a square function can be approximated by the maximum of its port functions which are linear. Replacing those squares by their counterparts as in the right hand side of 8, we derive the expected utility in a form that are linear in G i. 2 The results are summarized in Proposition 2. Proposition 2 Under Assumptions -2, the expected utility in 4 is equal to E [U i G i, G i, X, ε i X, ε i, ] = n + n n 2 G ij V n,ij X, ε ij λ t t n G ij D jφ t 2 with V n,ij X, = g i v ij g i, X j g j X n 2 D jdiag H D j 9 so by Legendre transform in 8 E [U i G i, G i, X, ε i X, ε i, ] = n Proof. See the appendix. D i = { X i = x },..., { X i = x T } + n n 2 G ij V n,ij X, ε ij t λ t max ω t R { } 2 G ij D n jφ t ω t ω 2 t The optimal G i is solved by maximizing the expected utility. Given the equivalent representation of the expected utility in 0, if we can interchange the inner maximization over ω = ω,..., ω T R T with the outer maximization over G i, then because the maximand is linear in G i, the optimal G i can be solved easily. To achieve this, we first need to move the inner maximization operation outside the entire expected utility function. This is ensured by the assumption that the eigenvalues of H are nonnegative. Lemma then guarantees the validity of the swap of the two maximization operations and the uniqueness of the solution in the swapped problem if the solution to the original problem is unique. Moreover, define ω = ω,..., ω T = Φ ω R T. By 7 and Φ = Φ, we have H ω = Φ Λ ω and ω H ω = ω Λ ω, so the maximization over ω can be replaced by an equivalent maximization over ω. It is easier to work with matrix H than its eigenvalues and eigenvectors 2 We are grateful to Terrence Tao for suggesting the idea of Legendre transform. 0 8

9 because the latter may not be unique. Assumption 3 Positive externality from clustering Any symmetric equilibrium n of 6 has a neighborhood such that any symmetric in the neighborhood satisfies min t λ t H 0. Lemma For any function fx, y : X Y R with x,y f x, y <, we have max y max x f x, y = max x max fx, y y Therefore, if there is a unique x, y such that f x, y = max y max x f x, y, then x, y is also the unique solution to max x max y fx, y. The equivalent transformations are summarized below max max G i ω max E [U i G i, G i, X, ε i X, ε i, ] G i G ij n max max ω G i n max max ω G i n V n,ij X, + n n 2 2D j φ t λ t ω t ε ij G ij V n,ij X, + n n 2 2D jφ Λ ω ε ij G ij V n,ij X, + n n 2 2D jh ω ε ij It is immediate to derive the optimal G i from 2. t n n 2 n n 2 ω Λ ω λ t ω 2 t n n 2 ω H ω 2 Theorem Under Assumptions -3, for any X, ε i and symmetric belief profile in a neighborhood of a symmetric equilibrium n, the optimal strategy G n,i X, ε i, = G n,ij X, ε i, is given by { G n,ij X, ε i, = V n,ij X, + n } n 2 2D jh ω n,i X, ε i, ε ij 0, j i 3 where ω n,i X, ε i, is a maximizer of the problem max ω Π n ω, X, ε i, = n [ V n,ij X, + n ] n 2 2D jh ω ε ij n + n 2 ω H ω 4 with the notation [x] + = max {x, 0}. Moreover, the optimal G n,ij X, ε i, and H ω n,i X, ε i, are unique almost surely. 9 t

10 Proof. It suffi ces to show the uniqueness statement. Under the assumption that ε i has a continuous distribution, the optimal G n,i X, ε i, solved from the original problem in 5 is unique almost surely, so by Lemma the optimal G n,i X, ε i, given in 3 is unique almost surely. The almost sure uniqueness of H ω n,i X, ε i, is immediate. 4 Convergence of the Game In this section, we explore the asymptotic behavior of the n-player game when n approaches infinity. Thanks to the closed form representation in 3-4, we can show that the optimal strategy of an individual and the probability she forms a link converge to some limiting strategy and link formation probability as n goes to infinity. The optimal strategy in the limit has a simple form that conditional on some suffi cient statistics that control for the spillover effects between one s own links, the links of an individual are formed independently. These asymptotic features are crucial for deriving simple enough while asymptotically valid estimators of the model parameters. From 3 the probability that individual i forms a link to j conditional on characteristic profile X and belief profile is P n X i, X j ; X, = Pr G n,ij X, ε i, = X, = Pr V n,ij X, + n n 2 2D jh ω n,i X, ε i, ε ij 0 X, 5 where the probability operation is calculated with respect to ε i. We write X i, X j explicitly in P n X i, X j ; X, to emphasize that by symmetry this link formation probability depends on i, j only through characteristics X i, X j. If the utility specification ensures that V n,ij X, converges to some limit V ij as n, as stated in Assumption 4, we expect that the link formation probability P n X i, X j ; X, converges to a limit given by P X i, X j ; = Pr V ij + 2D jh ω i ε ij 0 Xi, X j, 6 where ω i is a maximizer of the problem max Π ω, X [Vij i, = E + 2D jh ω ε ij X ω ]+ i ω H ω 7 as n. The expectation in 7 is taken with respect to X j and ε ij. The equilibrium 0

11 belief profile is a fixed point solved from ij X i, X j = P X i, X j ; where ij X i, X j = Pr G ij = X i, X j. Assumption 4 For V n,ij X, defined in 9, there is V ij = V X i, X j, such that a given any X i, X j, V n,ij X, V ij 0, and b given any X i, n V p n,ij X, V ij 0. Example 2 Consider the separable utility specification in 2. For any X and symmetric, we have p V n,ij X, = β 0 + X iβ + X i X j β 2 + ji X j, X i β 3 + jk X j, X k w X i, X k β n 2 4 n 2 D jdiag H D j k i,j We verify in the appendix that V ij is given by V ij = β 0 + X iβ + X i X j β 2 + ji X j, X i β 3 +E [ jk X j, X k w X i, X k X i, X j ] β 4 satisfies Assumption 4. We refer to P X i, X j ; defined in 6 the limiting choice probability. It can be understood as the choice probability derived from a "limiting game" with infinite number of players, where each individual i forms a link with j following a limiting strategy given by G ij = { V ij + 2D jh ω i ε ij 0 } In this limiting game, conditional on suffi cient statistic ω i which summarizes the spillover effects from one s other links due to nonseparable utility, each individual makes link decisions myopically and considers each link as a binary choice independent of her other links. Moreover, while statistic ω n,i X, ε i, in the n-player game may be dependent of ε ij, the dependence vanishes as n and in the limit statistic ω i is independent of ε ij. Under Assumption 5 that ensures the maximizer of Π ω, X i, is identified, we can show that the statistic ω n,i and choice probability P n X i, X j ; X, in the finite game converge to the proposed limits given in 7 and 6 as n.

12 Assumption 5 For any X i maximizer H ω i. and symmetric, Π ω, X i, defined in 7 has a unique Proposition 3 Under Assumptions -5, given any X i, H ω n,i X, ε i, H ω p i 0, as n. 8 Moreover, given any X i, X j, we have Proof. See the appendix. p P n X i, X j ; X, P X i, X j ; 0, as n. 9 Remark In Proposition 3 we state the convergence of ω n,i to ω i in terms of Hω n,i and Hω i because H may be singular so ω n,i and ω i may not be unique. Nevertheless, note that only Hω and ω Hω enter 4-5 and 6-7. If two ω ω 2 satisfy Hω = Hω 2, then ω Hω ω 2Hω 2 = ω + ω 2 H ω ω 2 = 0 where the second equality is by symmetry of H. Hence it is enough to consider Hω. 5 Estimation In this section, we discuss the estimation of model parameter θ. We assume that a single large network is observed 3 and propose a simple two step estimator. Unlike the standard literature on incomplete information games which considers many-game asymptotics, we face the challenge that an n-dependent link formation probability needs to be estimated from a single game. Our idea is to use the symmetry feature of an equilibrium, which ensures links between individuals with given characteristics are identically distributed. This idea of doing asymptotics based on symmetry was used by Leung 205 as well. We extend it to the case of nonseparable utility. In our case, links formed by different individuals are independent conditional on observables due to independent private shocks, while links formed by one individual are dependent through statistic ω n,i X, ε i which converges to a constant ω i as n approaches infinity. To fix ideas, we restrict the specification of separable utility v ij G i, X to be also separable in other players. That is, v ij G i, X depends on G i through a linear combination of terms G i j G i2 j 2 G ik j k such that i, i 2,..., i k N n \ {i} are distinct. This restriction 3 If more than one network is observed, we can proceed network by network. That is, we estimate link formation probabilities network by network and pool likelihoods or moments from each network to estimate θ. 2

13 ensures that the expected utility in 4 only involves the probability of forming a link as in 5, so it is enough to propose estimators for such one-link formation probabilities. We see later in Section 6 that in undirected networks the expected utility may involve the probability of forming two links, so there we also need estimators for those two-link formation probabilities. For notational simplicity, let X ij = X i, X j and x st = x s, x t X 2, s, t T. Define p n,st X, to be the probability that a pair with characteristics x st forms a link conditional on X and p n,st X, = Pr G n,ij = X ij = x st, X, It can be estimated by the empirical frequency that such pairs form a link ˆp n,st = G n,ij {X ij = x st } {X ij = x st } i i which we refer to as the link frequency estimator. The following proposition shows that the link frequency estimator is consistent. Proposition 4 Let n be the equilibrium in the data. Under Assumptions -3, as n. Proof. See the appendix. max ˆp n,st p n,st X, n = o p s,t With the first stage estimates ˆp n = ˆp n,st s,t T, we can estimate θ by quasi-mle of GMM. For the former, we construct the quasi log likelihood ln L n θ, p n = i G n,ij ln P n X ij ; X, θ, p n + G n,ij ln P n X ij ; X, θ, p n 20 The first order condition is given by θ P n X ij ; X, θ, p n P n X ij ; X, θ, p n P n X ij ; X, θ, p n G n,ij P n X ij ; X, θ, p n = 0 2 i MLE The quasi MLE estimator ˆθ n is the solution to 2 with p n replaced with ˆp n. For GMM, the model provides the moment conditions E [G n,ij P n X ij ; X, θ, p n X ij, X, p n ] = 0 3

14 Since X is discrete, these conditional moment conditions are equivalent to E [ G n,ij P n X ij ; X, θ, p n { X ij = x st} X, pn ] = 0, s, t T which has the sample analogue m n,st θ, p n = n n G n,ij P n X ij ; X, θ, p n { X ij = x st} i Define the T 2 vector m n θ, p n = m n,st θ, p n s,t T. The GMM estimator given by ˆθGMM n = arg min θ m n θ, ˆp n Ω n θ, ˆp n m n θ, ˆp n where Ω n θ, ˆp n is a positive definite weighting matrix. For example, if we choose ˆθ GMM n is Ω n θ, p n = W n θ, p n W n θ, p n where W n θ, p n = θ P nx ;X,θ,p n P nx ;X,θ,p n P nx ;X,θ,p n θ P nx T T ;X,θ,p n P nx T T ;X,θ,ˆp n P nx T T ;X,θ,p n Then ˆθ GMM n is the solution to θ P n X ij ; X, θ, ˆp n n n P n X ij ; X, θ, ˆp n P n X ij ; X, θ, ˆp n G n,ij P n X ij ; X, θ, ˆp n = 0 i For this optimal weighting matrix, the GMM estimator is equal to the MLE estimator. 6 Undirected Networks In this section we consider the case of undirected networks. We show that the idea in previous sections could also work for undirected networks with mild modifications. To proceed, let G ij denote an undirected link between i and j. Clearly G ij = G ji. It is useful to denote by S ij a directed link from i to j. Under the link announcement framework, S ij represents whether i proposes to form a link with j. The link is formed if both i and j propose to form it, so G ij = S ij S ji. We may write G S to indicate that G is the network induced by proposals S. We consider the same utility specification as in, with G ij representing an undirected 4

15 link. With abuse of notation we use G i to denote the submatrix of G i with the ith row and column deleted. We maintain the same information assumption. The strategy of individual i, denoted by S i X, ε i = S ij X, ε i j Nn with S ii = 0, is a mapping from her information X, ε i to a vector of link proposals in S i = {0, } n. The optimal strategy maximizes her expected utility E [U i G S i, S i, X, ε i X, ε i, ] over s i S i, where the expectation is taken with respect to others proposals S i = S j. The choice probability i s i X and belief profile X are defined similarly as before, with links G i replaced by proposals S i. Given belief profile. individual i s expected utility is now given by E [U i G S i, S i, X, ε i X, ε i, ] = S ij s ji v ij g i s i, X j s j X ji ε ij n s i + S ij S ik s ji s ki h i g j s i, g k s i j s j X 22 n n 2 k i,j s i Let Pr S i = s i X, be the probability that i proposes s i given X and. A Bayesian Nash equilibrium X is a fixed point that solves i s i X = Pr S i = s i X, X. Remark 2 A potential concern with Nash is that in undirected networks players may coordinate. This is reasonable under complete information, where pairwise stability Jackson and Wolinsky 996 and Nash equilibrium are nonnested and neither of them implies the other. However, under incomplete information players won t be able to coordinate even in undirected networks; because i does not observe ε ji, he cannot predict what j proposes and coordinate on that unless in a trivial equilibrium where X 0. In fact, if we define a Bayesian version of the pairwise stability, that is, a network G is Bayesian pairwise stable if G ij = { ij E [U i G S i, S i, X, ε i X, ε i, ] 0 & ji E [U j G S j, S j, X, ε j X, ε j, ] 0} for any i j, where ij E [U i G S i, S i, X, ε i X, ε i, ] is the expected marginal utility if i proposes the link with j, i.e., E [U i G S i : S ij =, S i, X, ε i X, ε i, ] E [U i G S i : S ij = 0, S i, X, ε i X, ε i, ] and similar for ji E [U j G S j, S j, X, ε j X, ε j, ], then any undirected network that is Bayesian Nash must also be Bayesian pairwise stable. This is because for a Bayesian Nash G, G ij = if and only if S ij = S ji = are optimal, so the expected marginal utility from the link must be nonnegative for both i and j. It is thus enough to consider Bayesian Nash 5

16 equilibrium. Unlike in the directed case, the quadratic form in the expected utility 22 may depend on X i, through the proposals S ji and S ki made to i. Nevertheless, it is still symmetric in j and k, so the eigendecomposition and Legendre transform still apply. Define H i,st X jk = E [ S ji S ki h i G j, G k ] X j = x s, X k = x t, X i, X ijk = s ji s ki h i g j s i, g k s i j sj X j = x s, X k = x t, X i, X ijk s i and the matrix H i X = H i, X H i,t X. H i,t X i, X H i,t T X. where the subscript i indicates that they may depend on X i. Following the same idea in previous sections, we can derive the optimal strategies in closed form. Corollary Under Assumptions -3with H replaced with H i, the optimal strategy S n,i X, ε i, = S n,ij X, ε i, is given by { S n,ij X, ε i, = V n,ij X, + n } n 2 2D jh i ω n,i X, ε i, ji ε ij 0, j i where ω n,i X, ε i, is a maximizer of the problem with max ω n [ V n,ij X, + n ] n 2 2D jh i ω ji ε ij n + n 2 ω H i ω V n,ij X, = s i s ji v ij g i s i, X j s j X n 2 D jdiag H i D j Moreover, the optimal S n,ij X i, X j and H i ω n,i X, ε i, are unique almost surely. Given the optimal strategies in the corollary, the probability that individual i proposes 6

17 to link to j and the probability that i proposes to link to both j and k are given by P n X i, X j ; X, = Pr S n,ij X, ε i, = X, = Pr V n,ij X, + n n 2 2D jh i ω n,i X, ε i, ji ε ij 0 X, and Q n X i, X j, X k ; X, = Pr S n,ij X, ε i, =, S n,ik X, ε i, = X, = Pr V n,ij X, + n n 2 2D jh i ω n,i X, ε i, ji ε ij 0 & V n,ik X, + n n 2 2D kh i ω n,i X, ε i, ki ε ik 0 X, The latter is relevant because the expected utility in 22 indicates that individual i may need to form a belief about S n,ji S n,jk. We expect that these choice probabilities converge to limiting probabilities P X i, X j ; = Pr V ij + 2D jh i ω i ji ε ij 0 X i, X j, and QX i, X j, X k ; = P X i, X j ; P X i, X k ; respectively as n, where ω i is a maximizer of the problem max E [Vij + 2D jh i ω ji ε ij X ω ]+ i ω H i ω and V ij is a limit of V n,ij X, that satisfies Assumption 4. That is, P n X i, X j ; X, P X i, X j ; = o p Q n X i, X j, X k ; X, QX i, X j, X k ; = o p 7

18 7 Simulation In this section, we implement the proposed methods in a simulation study. We focus on directed networks and assume the following utility specification U i G, X, ε i ; θ = n G ij β 0 + X i β + X i X j β 2 + n 2 + G ij G ik G jk G kl γ n n 2 k i,j k i,j G jk β 3 ε ij where X i is a binary random variable taking values in {0, } with equal probability and ε ij is standard normal N 0,. The true values of the parameters are β 0, β, β 2, β 3, γ =,, 2,,. The networks are generated according to the n-player incomplete information game described in Section 2, with n taking different values of 0, 25, 50, 00, 250, and 500. For each value of n, we generate a single network and use it to estimate the parameters by two-step MLE or GMM. Each experiment is repeated 00 times. We report the means and standard errors of the estimated parameters. Since the limiting game approximates the finite game asymptotically, we first use the limiting game to estimate the parameters and check how well such estimates could perform. In particular, we construct the likelihood as in 20, with P n X ij ; X, p n replaced by the limiting choice probability P X ij ; p n given in 6. Such approximation has a substantial advantage in computation because P X ij ; p n has a probit-type closed form. The estimates are reported in Table. It is not surprising that the estimates in small networks perform poorly, as the limiting game is only an asymptotic approximation of the finite game. Nevertheless, when the network size gets large e.g. n 00, the estimates become close to the truth. This indicates that the limiting game is a valid approximation of the finite game asymptotically and estimation based on this approximation may yield good estimates for the parameters if networks are suffi ciently large. Next we estimate the parameters by the finite game, and compare the estimates with those from the limiting game. Note that the finite-game choice probability P n X ij ; X, p n does not have a closed form because ω n,i X, ε i depends on ε i. It needs to be computed by simulation. In practice, we simulate P n X ij ; X, p n by a frequency simulator, where in each simulation we generate a vector ε i and for this ε i we solve for the optimal G n,ij X, ε i numerically. We obtain G n,ij X, ε i by solving the integer programming problem directly when n 00 or applying the equivalent binary representation as in 3 and solving for ω n,i X, ε i when n > 00. Table 2 reports the MLE estimates and Tables 3 and 4 report the GMM estimates. By correctly specifying the choice probability, we improve the estimates in 8

19 small networks. The estimation precision is also improved. For example, the 95% quantiles of γ from the finite game are closer to the truth, especially in small networks. These results indicate that we should use the finite game rather than the limiting game for estimation for n relatively small. We want to point out that the small network performance of the finite-game estimates is still unsatisfactory. To deliver satisfactory estimates, we need to have large networks or conduct some bias correction for the estimates. Table : MLE Estimates Using the Limiting Game n β 0 β β 2 β 3 γ DGP 2 Note: Mean estimates and standard errors from 00 repeated samples using the limiting game. 8 Conclusions In this paper, we provide estimation methods for network formation using observed data from a single large network. We model network formation as a simultaneous-move game with private information and extend Leung 205 by allowing for nonseparable utility such as the effect of friends in common. The main innovation is to provide an approach to explicitly represent the pure strategy of an individual in the game. This closed form representation enables us to analyze the asymptotic features of the game as the number of players approaches infinity and thus construct asymptotically valid estimators for the parameters. We propose 9

20 Table 2: MLE Estimates Using the Finite Game n β 0 β β 2 β 3 γ DGP 2 Note: Mean estimates and standard errors from 00 repeated samples using the finite game, where the CCPs are computed from 500 simulations by either solving integer programming for n 00 or applying Legendre transform for n >

21 Table 3: GMM Estimates Using the Finite Game Weights from the Finite Game n β 0 β β 2 β 3 γ DGP 2 Note: Mean estimates and standard errors from 00 repeated samples using the finite game, with the GMM weights simulated independently from the finite game. The CCPs are computed from 500 simulations by either solving integer programming for n 00 or applying Legendre transform for n > 00. 2

22 Table 4: GMM Estimates Using the Finite Game Weights from the Limiting Game n β 0 β β 2 β 3 γ DGP 2 Note: Mean estimates and standard errors from 00 repeated samples using the finite game, with the GMM weights simulated independently from the limiting game. The CCPs are computed from 500 simulations by either solving integer programming for n 00 or applying Legendre transform for n >

23 a two-step estimation procedure which makes little assumption about equilibrium selection and is computationally simple. Our approach can apply to both directed and undirected networks. We focus on discrete observables in this paper, but expect our approach could be extended to continuous observables. 9 Appendix: Proofs Proof of Proposition. Define the set of symmetric X } Σ s X = { X [0, ] n2n : i X = j X if X i = X j It is clear that Σ s X is a convex and compact subset of [0, ] n2n. Equations in 6 forms a mapping from Σ s X to Σ s X, because if Σ s X, then Pr G i = g i X, X = Pr G j = g j X, X for X i = X j and g i = g j with g ii, g ij swapped with g jj, g ji, so Pr G i = g i X, X is also symmetric. The mapping is continuous in because the expected utilities are continuous in X and ε i has a continuous distribution under Assumption. By Brouwer s fixed point theorem there is a fixed point. Proof of Proposition 2. It suffi ces to show the first statement. Denote D it = {X i = x t }. The quadratic form in the expected utility is G ij G ik E [h i G j, G k ] k i,j = G ij G ik D js D kt H st G ij D jt H tt k i s t t = H st G ij D js G ik D kt G ij D jt H tt s t k i t = G ij D j H G ij D j G ij D jdiag H D j 23 23

24 By the real spectral decomposition of H, G ij D j H G ij D j = = G ij D j Φ Λ Φ G ij D j G ij D jφ Λ G ij Φ D j = 2 λ t G ij D jφ t 24 t The desired statement then follows from 23, 24, and simple algebra. Proof of Lemma. Note that max f x, y f x, y, x max y max y max x max x f x, y max f x, y, y f x, y max x x, y X Y max f x, y y x X Similarly, max x max y f x, y max y max f x, y x Hence is proved. If x, y is the unique solution to the LHS of, then by f x, y = max y max x f x, y = max x max y f x, y it is also the unique solution to the RHS of. Proof of Example 2. We verify that V n,ij X, and V ij given in Example 2 satisfy Assumption 4. Note that n 2 D jdiag H D j = op because H is uniformly bounded and that n n 2 D jdiag H D j = n 2 n D j diag H D j = o p by applying uniform law of large numbers to the second term, which is appropriate because the space of symmetric is compact, H is continuous in and is uniformly bounded. It suffi ces to show that for any θ, jk X j, X k w X i, X k β n 2 4 E [ jk X j, X k w X i, X k X i, X j ] β 4 = o p and n k i,j jk X j, X k w X i, X k β n 2 4 E [ jk X j, X k w X i, X k X i, X j ] β 4 = o p k i,j 24

25 The former is satisfied by applying uniform law of large numbers again. As for the latter, write i X j, X k ; = jk X j, X k w X i, X k β 4 E [ jk X j, X k w X i, X k X i, X j ] β 4, which has zero conditional mean E [ i X j, X k ; X i ] = 0. By Cauchy-Schwarz inequality, we have n i y i 2 n i y2 i, so = i X j, X k ; n n 2 k i,j n n 2 2 i X j, X k ; n n 2 2 k i,j i X j, X k ; 2 k i,j + n n 2 2 k i,j l i,j,k 2 2 i X j, X k ; i X j, X l ; The last two terms are U-processes. It remains to show that they are o p uniformly in. For the first term, applying Corollary 7 in Sherman 994 yields n n 2 k i,j i X j, X k ; 2 E [ i X j, X k ; 2 X i ] = O p n so the first terms is O p n uniformly in. As for the second term, note that the product i X j, X k ; i X j, X l ; has zero mean conditional on X i, so by Corollary 7 in Sherman 994 again we obtain n n 2 n 3 k i,j l i,j,k i X j, X k ; i X j, X l ; = O p which proves that the second term is o p uniformly in. The proof is complete. n Proof of Proposition 3. Step : We first prove 8. Because c E [c ε] + = c c ε f ε ε dε = F ε c c the first order condition of the problem 7 is ω Π ω, X i, = 2H E D j F ε Vij + 2D jh ω Xi 2H ω =

26 It is easy to see that any ω i that satisfies the first order condition must be bounded. Hence, without loss of generality we can assume that ω i is in a compact set Ω. Since Π ω, X i, is continuous in, the compactness of Ω implies that the unique maximizer H ω i by Assumption 5 is also well separated. If we can further show that Π n ω, X, ε i, Π ω, X i, = o p 26 ω, then following a standard proof for uniform consistency we can show 8. Specifically, from 26 we have Π n ω n,i, X, ε i, Π n ω i, X, ε i, Π ω i, X i, o p, whence, ω, Π ω i, X i, Π ω n,i, X i, Π n ω n,i, X i, ε i, Π ω n,i, X i, + o p Π n ω, X i, ε i, Π ω, X i, + o p = o p. Well-separateness of H ω i implies that for any ε > 0, there is η > 0 such that, for any, Π ω, X i, < Π ω i, X i, η for every ω with H ω H ω i ε. Therefore, 0 Pr Pr Pr in view of the preceding display 8 is proved. H ω n,i H ω i ε [Π ω n,i, X i, Π ω i, X i, ] < η Π ω n,i, X i, Π ω, X i, < η 26

27 Now we prove 26. The left hand side of 26 satisfies ω, ω, + ω, + ω, [ V n,ij X, + n ] [Vij n n 2 2D jh ω ε ij E + 2D jh ω ε ij X ]+ i + [ ] n n 2 2D jh ω + [ Vn,ij X, + 2D n jh ω ε ij ]+ [ ] V ij + 2D jh ω ε ij + [ Vij + 2D n jh ω ε ij ]+ E [Vij ] + 2D jh ω ε ij X + i 27 For the third term in 27, because [ V ij + 2D jh ω ε ij are i.i.d. conditional on ]+ [Vij X i with conditional mean E + 2D jh ω ε ij X ]+ i, are continuous in ω and, and are bounded by [ [ ] Vij + 2D jh ω ε ij Vij + 2D + jh ω ] ε ij ω, which is absolute integrable because of the continuity of V ij and H and compactness of the spaces of ω and, uniform law of large numbers holds, so ω, [ Vij + 2D n jh ω ε ij ]+ E [Vij ] + 2D jh ω ε ij As for the second term in 27, because [x]+ [y] + x y, we have ω, [ Vn,ij + 2D n jh ω ε ij ]+ [ ] V ij + 2D jh ω ε ij V n,ij V ij = o p n + + X i = o p by Assumption 4ii. Finally, the first term in 27 is o p, again by uniform law of large numbers. Hence 26 is proved. + 27

28 Step 2: Next we prove 9. By definition of P n and P, P n X i, X j ; X, P X i, X j ; { V n,ij X, + n } n 2 2D jh ω n,i X, ε i, ε ij { V ij + 2D jh ω } i ε ij df εi ε i Pr V n,ij X, + n n 2 2D jh ω n,i X, ε i, ε ij > V ij + 2D jh ω i X Pr V n,ij X, + 2D jh ω n,i X, ε i, < ε ij V ij + 2D jh ω i X Since the last two terms are similar, without loss of generality it suffi ces to show that the second last term is o p uniformly in. Define M n X, ε i, = V n,ij X, + n n 2 2D jh ω n,i X, ε i, M = V ij + 2D jh ω i Fix η > 0. Because ε ij M, M n X, ε i, ] implies that ε ij M, M + η] or M n X, ε i, > M + η, we can bound the second last term as follows Pr M n X, ε i, ε ij > M X Pr M + η ε ij > M X + Pr M n X, ε i, > M + η X 2D jh Pr M + η ε ij > M X + Pr { + V n,ij X, V ij > η } 2 n n 2 ω n,i X, ε i, ω i > η 2 X where the last inequality follows because M n X, ε i, M > η implies that V n,ij X, V ij > η or 2 2D jh n ω n 2 n,i X, ε i, ω i > η. It suffi ces to show that the last 2 three terms in the display are o p uniformly in. For any δ > 0, the last term satisfies { Pr V n,ij X, V ij > η } > δ 2 X i, X j Pr V n,ij X, V ij > η 2 X i, X j 0 28

29 as n by Assumption 4i, so it is o p uniformly in. For the second term, we have Pr Pr δ E Pr Pr δ E = δ Pr 2D jh n 2 ω n,i X, ε i, ω i > η 2 X > δ X i, X j 2D jh n 2 ω n,i X, ε i, ω i > η 2 X X i, X j n 2D jh n 2 ω n,i X, ε i, ω i > η 2 X X i, X j n n 2 ω n,i X, ε i, ω i > η 2 X i, X j 0 n 2D jh n as n by Markov inequality, law of iterated expectation, and uniform consistency of H ω n,i X, ε i, proved earlier, so the second term is also o p uniformly in. As for the first term, = = Pr M + η ε ij > M X Fεij M + η F ε ij M f εij M + η η for some η [0, η] where the last equality is from mean value theorem. Since η is arbitrary, choosing η to be o, we can get Pr M + η ε ij > M X, = o p. The proof is complete. Proof of Proposition 4. Let n,st = ˆp n,st p n,st X, n i = G n,ij Pr G n,ij = X ij = x st, X, n {X ij = x st } i {X ij = x st } We want to show that for any δ > 0, lim Pr max n,st > δ n s,t By law of iterated expectation and dominated convergence theorem it suffi ces to show lim Pr n max s,t = 0. n,st > δ X, = o p 29

30 Note Pr max s,t n,st > δ X, E max s,t 2 n,st X, s,t δ 2 E 2 n,st X, T 2 δ 2 max E 2 n,st X, s,t by conditional version of Chebyshev s inequality. It suffi ces to show E 2 n,st X, = op for all s, t. = E 2 n,st X, n 2 n i 2 V ar G n,ij X ij = x st, X, n {X ij = x st } 2 i {X ij = x st } + n 2 n i 2 nn k i,j Cov G n,ij, G n,ik X ij = X ik = x st, X, n {X ij = X ik = x st } 2 i {X ij = x st } nn where we have used the fact that links formed by different individuals are independent, i.e., Cov G n,ij, G n,i j X ij = X i j = xst, X, n = 0 for all i i. Since V ar G n,ij X ij = x st, X, n, the first term is at most 4 { X ij = x st} 4n n n n i As for the second term, because Cov G n,ij, G n,ik X ij = X ik = x st, X, n V ar G n,ij X ij = x st, X, n /2 V ar G n,ik X ik = x st, X, n /2, so the second term is bounded by 4 4n n n n 2 i k i,j δ 2 { X ij = X ik = x st} n n { 2 X ij = x st} The double and triple summations in the last two displays are U-statistics. It is easy to show that they converge to their expectations. Therefore, the sum of the two displayed terms are o p, as desired. The proof is complete. i 30

31 References [] Aradillas-Lopez, A. 200 Semiparametric estimation of a simultaneous game with incomplete infomraiton. Journal of Econometrics, 57, [2] Bajari, P. Hong, H., Krainer, J., & Nekipelov, D. 200 Computing equilibria in static games of incomplete information using the all-solution homotopy, mimeo, University of Minnesota. [3] Bajari, P., Hahn, J., Hong, H., & Ridder, G. 20. A note on semiparametric estimation of finite mixtures of discrete choice models with application to game theoretic models. International Economic Review, 533, [4] De Paula, Richards-Shubik and Tamer 205 Identification of Preferences in Network Formation Games, working paper. [5] Dube, J., Fox, J. T., & C. Su 202 Improving the numerical performance of static and dynamic aggregated discrete choice random coeffi cients demand estimation. Econometrica, 805, [6] Jackson, M. O., & Wolinsky, A. 996 A strategic model of social and economic networks. Journal of Economic Theory, 7, [7] Leung, M. 205 Two-Step Estimation of Network-Formation Models with Incomplete Information, Journal of Econometrics, 88: [8] Miyauchi 203 Structural Estimation of a Pairwise Stable Network with Nonnegative Externality, working paper. [9] Menzel, K. 206a Inference for Games with Many Players, Review of Economic Studies, 83: [0] Menzel, K. 206b Strategic Network Formation with Many Players, NYU, working paper. [] Sheng, S. 206 A Structural Econometric Analysis of Network Formation Games, UCLA, working paper. [2] Sherman, R. 994 Maximal Inequalities for Degenerate U-processes with Applications to Optimization Estimators. Annals of Statistics, 99422, [3] Su, C., & K. L. Judd 202 Constrained optimization approaches to estimation of structural models. Econometrica, 805,