A Structural Econometric Analysis of Network Formation Games

Transcription

1 A Structural Econometric Analysis of Network Formation Games Shuyang Sheng June 7, 2014 Abstract The objective of this paper is to identify and estimate network formation models using observed data on network structure. We characterize network formation as a simultaneous-move game, where the utility from forming a link depends on the structure of the network, thereby generating strategic interactions between links. Because a unique equilibrium may not exist, the parameters are not point identified. We leave the equilibrium selection unrestricted and propose a partial identification approach. We derive bounds on the probability of observing a subnetwork, where a subnetwork is the restriction of a network to a subset of the individuals. Unlike the standard bounds as in Ciliberto and Tamer (2009, these subnetwork bounds are computationally tractable even for large networks. The identified set defined by these bounds can be consistently estimated following the moment inequality literature, with the bounds computed by simulation. We provide Monte Carlo evidence that the subnetwork bounds are informative. JEL Classifications: C13, C31, C57, D85 KEYWORDS: Network formation, simultaneous-move games, multiple equilibria, subnetworks, partial identification, moment inequalities, simulation. This paper is a revision of Chapter 2 of my dissertation. I am very grateful to my advisor Geert Ridder for his enormously valuable advice and guidance. I also thank Aureo de Paula, Bryan Graham, Jinyong Hahn, Matthew Jackson, Max Kasy, Rosa Matzkin, Roger Moon, Hashem Pesaran, Matthew Shum, Martin Weidner, Simon Wilkie, seminar participants at USC, UCLA, UCSD, JHU, Pittsburgh, Tilburg, CORE, California Econometrics Conference, NASM, NAWM, EMES for helpful discussions and comments. Financial support from the USC Graduate School Dissertation Completion Fellowship is acknowledged. All the errors are mine. Department of Economics, UCLA, Los Angeles, CA ssheng@econ.ucla.edu. 1

2 1 Introduction Social and economic networks influence a variety of individual behaviors and outcomes, including educational achievement (Calvó-Armengol, Patacchini, and Zenou (2009, employment (Calvó-Armengol and Jackson (2004, technology adoption (Conley and Udry (2010, consumption (Moretti (2011, and smoking (Nakajima (2007. As networks are often the result of individual decisions, understanding the formation of networks is important for the investigation of network effects. Despite that the theoretical literature on network formation has flourished in the past decades (see Jackson (2008 and Goyal (2007 for a survey, econometric studies on the identification and estimation of network formation models are still at an infant stage. The objective of this paper is to provide insight into this latter area. More precisely, assume that we observe the network structure, i.e., who is linked with whom. We propose new methods to identify and estimate the structural parameters in the model explaining how links are formed. The statistical analysis of network formation dates back to the seminal work of Erdős and Rényi (1959, who proposed a random graph model where links are formed independently with a fixed probability. Statisticians later extended the Erdős-Rényi model to allow for dependence between links and developed a large class of exponential random graph models (ERGM (e.g., Snijder (2002. While ERGMs may well fit the observed network statistics, they usually lack microfoundations which are essential for counterfactual analysis. Alternatively, economists view network formation as the optimal choices of individuals that maximize their utilities. A simple and widely used empirical approach in this spirit is to employ a pairwise regression, where the formation of a link is modeled by a binary choice model of the pair involved (e.g., Fafchamps and Gubert (2007, Mayer and Puller (2008. In order to treat links in a network as independent observations, this approach needs to assume that there is no influence from indirect friends (e.g., friends of friends, which could be restrictive in many applications given the prevalence of clustering (e.g., Jackson and Rogers (2007, Jackson, Barraquer and Tan (2012. Recognizing the weakness, econometricians have recently started to develop empirical network formation models that allow for utility externality, thereby introducing strategic interactions between links (Christakis, Fowler, Imbens, and Kalyanaraman (2010, Mele (2011, Boucher and Mourifié (2013, Miyauchi (2013, Leung (2013. A contribution of this paper is 2

3 to provide a different approach for the identification and estimation of such models. A crucial problem in the identification of network formation models with strategic interactions is the presence of multiple equilibria. Bouncher and Mourifié (2013 get around this problem by assuming there is a unique equilibrium in the observed data. Leung (2013 considers a network formation model with incomplete information. In his model, independence of private information creates conditional independence of links in a network under the assumption of a separable utility function, thereby enabling consistent estimation from a single large network. This approach does not need to make assumptions about the equilibrium selection if the interest is the parameters rather than counterfacturals. Christakis et al. (2010 and Mele (2011 circumvent the multiplicity issue by considering a sequential model where each link is formed in a random sequence and myopically. The Markov chain of networks achieved in each period would converge to a unique stationary distribution over the collection of equilibrium networks. Employing the stationary distribution to construct the data likelihood is equivalent to imposing implicitly an equilibrium selection mechanism in the corresponding static model (Young (1993, Jackson and Watts (2002. Unlike these studies, we admit multiple equilibria and do not impose any restrictions on the utility function or equilibrium selection. Since a unique equilibrium may not exist in this general case, the parameters are not necessarily point identified. We adopt the partial identification approach and examine what we can learn about the parameters from bounds on conditional choice probabilities. The study close to ours is by Miyauchi (2013, who considers partial identification as well. Miyauchi derives his bounds from a partial ordering of equilibrium networks under a nonnegative externality assumption, while our bounds hold for more general utility functions. The estimation of network formation models is computationally challenging because the number of possible networks is enormous: for n individuals the number of possible undirected networks is 2 n(n 1/2. In ERGMs, parameter estimation relies crucially on sampling networks from exponential family distributions. Given the huge space of possible networks, the sampling is typically carried out using Markov Chain Monte Carlo (MCMC methods. However, the mixing time of MCMC is O(e n unless links are approximately independent, in which case the model is not appreciably different from the Erdős-Rényi model (Bhamidi, Bresler, and Sly (2011. Chandrasekhar and Jackson (2013 provide Monte Carlo evidence that slow convergence of MCMC leads to poor performance of ERGMs. In sequential models of network 3

4 formation, likelihoods constructed using stationary distributions may be computationally intractable because such likelihoods typically include a sum over all possible networks (e.g., Mele (2011. While MCMC methods can be used to avoid computing intractable likelihoods, they need simulating networks from the stationary distributions where the mixing rate can be as slow as O(e n. Hence, sequential models suffer from the same computational problem as in ERGMs. In our model, the computation of the bounds may be intractable as well because it requires checking equilibrium conditions for all possible networks. We propose a completely new approach to tackle this computational problem. The idea is to make use of subnetworks. A subnetwork is the restriction of a network to a subset of the individuals. Under the equilibrium concept we consider (i.e., pairwise stability proposed by Jackson and Wolinsky (1996, if a network is in equilibrium, any subnetwork in it must also be in equilibrium. Thus, we can derive bounds on the probability of observing a subnetwork, taking into account only the equilibrium within the subnetwork. Under an assumption of local utility externality, these subnetwork bounds are computationally tractable no matter how large the networks are, as long as we only consider small subnetworks. This approach only needs choice probabilities within subnetworks, so it is still applicable if we do not observe an entire network, but links in subnetworks. Our paper is related to the econometric literature on static games of complete information (e.g., Bresnahan and Reiss (1991, Tamer (2003, Ciliberto and Tamer (2009, Bajari, Hong, and Ryan (2010, Bajari, Hahn, Hong, and Ridder (2011. Such games often face the identification problem due to the prevalence of multiple equilibria. To avoid imposing restrictions on equilibrium selection, econometricians have applied partial identification to such games (e.g., Andrews, Berry and Jia (2004, Pakes, Porter, Ho and Ishii (2006, Berry and Tamer (2006, Ciliberto and Tamer (2009, Beresteanu, Molchanov, and Molinari (2011. However, most studies look at simple entry games where the number of agents is small. We contribute to this literature by providing an application of partial identification to network formation games where the number of agents can be large, so standard probability bounds are computationally intractable. By trading off sharpness of bounds, we can achieve computational feasibility. This idea may shed light on the analysis of other games with a large number of agents (e.g., matching games and provide a new perspective on tackling computational problems in those models. 4

5 The estimation and inference of the identified set defined by the moment inequalities derived from the subnetwork bounds is a straightforward application of the works on partially identified models (e.g., Chernozhukov, Hong and Tamer (hereafter CHT, 2007, Andrews and Soares (2010, Romano and Shaikh (2010, Andrews and Jia (2012. While the distributions of subnetworks involved in the moment inequalities may depend on the choice of subnetworks, we demonstrate that subnetworks from a network with observationally identical individuals will follow the same distribution, so conditional on the observed characteristics it does not matter which subnetwork we choose. The bounds may not have closed forms. We propose how to compute them by simulation. The remainder of the paper is organized as follows. Section 2 develops the model. Section 3 presents the identification results. We address the multiple equilibrium problem in Section 3.1 and propose the partial identification approach in Section 3.2. The subnetwork bounds are derived in Section 3.3. Section 4 describes the estimation and inference methods. We discuss how to choose subnetworks in Section 4.1, how to estimate the identified set and construct confidence regions in Section 4.2, and how to compute the bounds in Section 4.3. Section 5 performs a Monte Carlo study, and Section 6 concludes. 2 A Model of Network Formation In this section, we develop the network formation model. Let V = {1, 2,..., N} be the set of individuals who can form links. The links are undirected in the sense that forming a link requires the consent of both individuals involved in the link, but severing a link can be unilateral. This is the natural setting in the context of friendship networks, and for that reason we call linked individuals friends. The links form a network, which we denote by G G. It is an N N binary matrix, where G ij = 1 if i and j are friends, and 0 otherwise for all i j. Since we consider undirected links, G is a symmetric matrix. We normalize G ii = 0 for all i V. Utility Each i V has a K 1 vector of observed characteristics X i (e.g., gender, age, race and an (N 1 1 vector of unobserved (to researchers preferences ε i = (ε i1,..., ε i,i 1, ε i,i+1,..., ε in, where ε ij is i s preference for link ij. Let 5

6 X = (X 1,..., X N and ε = (ε 1,..., ε N. The utility of i in a network in general depends on the network configuration G, the observed characteristics X, and i s unobserved preferences ε i, i.e., U i (G, X, ε i. For any i j, we decompose G into (G ij, G ij, where G ij G ij is the network obtained from G by removing link ij. Then the marginal utility of i from forming a link with j is ij U i (G ij, X, ε i = U i (1, G ij, X, ε i U i (0, G ij, X, ε i. (1 In this paper, we consider the utility specification U i (G, X, ε i = N G ij u(x i, X j, ε ij + N j=1 j=1 N G ij G jk γ 1 + N k=1 k i j=1 k>j ( N N G ij G ik G jk γ 2 C i G ij, with u(x i, X j, ε ij = β 0 + β 1X i + β 2 X i X j + ε ij, where we assume that γ 1, γ 2 R are constants, while C i (. : R + R + can depend on X i. 1 In this specification, the first term is the utility from direct friends, where X i X j is to capture the homophily effect, which says that people tend to make friends with those who are similar to them (Currarini, Jackson and Pin (2009, Christakis et al. (2010. In addition to the direct-friend effects, (2 also allows for the effects of indirect friends. The second term in (2 captures the utility from i s friends of friends, and the third term captures the additional utility if i and i s friend have friends in common. 2 last term is the cost of direct friends. Hence, if we consider the marginal utility of i from forming a link with j, which is given by ij U i (G ij, X i, X j, ε ij = u(x i, X j, ε ij + N G jk γ 1 + N G ik G jk γ 2 C i N k=1 k i 1 In the econometric analysis we consider a linear cost function, but let C i (n = if n > b for some b < so that the number of one s direct friends is bounded by an upper limit. See Section for details. 2 The latter is motivated by the clustering hypothesis, which says that if two individuals have friends in common, they are more likely to be friends than if links are formed randomly (Jackson and Rogers (2007, Jackson (2008, Christakis et al. (2010, Jackson et al. (2012. k=1 k=1 k j j=1 G ik (2 The, (3 6

7 where C i (n = C i (n + 1 C i (n, then it consists of not only the direct utility from j, but also the indirect utility from i s other friends, j s other friends, and i, j s friends in common. This utility function follows closely the specification in Christakis et al. ( It is also related to the specifications in Mele (2011 and Goyal and Joshi (2006, but is more general than both. 4 Equilibrium Given the utilities, individuals choose friends simultaneously as in the link-announcement game (Myerson (1991, Jackson (2008. We assume that individuals observe X and ε, so it is a complete information game. Depending whether transfers are allowed for, each individual announces a set of intended links or intended transfers. Under nontransferable utility (NTU, a link is formed if both individuals intend to form it, while under transferable utility (TU a link is formed if the sum of the two transfers for it is nonnegative. The network formed is an equilibrium. The equilibrium concept we consider in this paper is pairwise stability (Jackson and Wolinsky (1996 for NTU, Bloch and Jackson (2006, 2007 for TU. We say a network is pairwise stable if no pair of individuals wants to create a new link, and no individual wants to sever an existing link. Formally, Definition 1 A network G is NTU-pairwise stable (NTU-PS if 1. for all G ij = 1, ij U i (G ij, X i, X j, ε ij 0 and ij U j (G ij, X j, X i, ε ji 0; 2. for all G ij = 0, ij U i (G ij, X i, X j, ε ij > 0 = ij U j (G ij, X j, X i, ε ji < 0. Definition 2 A network G is TU-pairwise stable (TU-PS if 1. for all G ij = 1, ij U i (G ij, X i, X j, ε ij + ij U j (G ij, X j, X i, ε ji 0; 2. for all G ij = 0, ij U i (G ij, X i, X j, ε ij + ij U j (G ij, X j, X i, ε ji 0. 3 Christakis et al. (2010 allow for nonlinear effects from friends of friends and friends in common. Our specification is a linear version of theirs. However, with linearity we can establish the existence of equilibrium (see Appendix 7.1, which is an open question for the specification they use. 4 Mele (2011 considers a linear utility function which does not allow for the effects of friends in common. Goyal and Joshi (2006 assumes that the direct-friend effects are homogeneous across individuals. 7

8 Since we allow for utility interdependence, the pairwise stability condition leads to a simultaneous discrete choice model, i.e., G ij = 1 { ij U i (G ij, X i, X j, ε ij 0, ij U j (G ij, X j, X i, ε ji 0}, i j V, (4 under NTU and G ij = 1 { ij U i (G ij, X i, X j, ε ij + ij U j (G ij, X j, X i, ε ji 0}, i j V, (5 under TU, 5 where the choice of a link G ij depends on the choices of others, G ij. This implies that we cannot treat each link as a single observation and use a pairwise regression because G ij is endogenous in the model, so could be correlated with (ε ij, ε ji. What further complicates the statistical inference of (4 and (5 is that there may be multiple equilibria, which will affect the identification of the parameters. The analysis for NTU and TU is almost identical except that the existence of equilibrium holds under weaker conditions for TU compared to NTU (see Appendix 7.1 for discussion on the existence issue. In sequel we consider TU only and abbreviate TU-PS to PS for convenience. Remark 2.1 There are other equilibrium concepts in the network literature, and they differ mainly in the coordination that individuals are assumed to have. The simplest concept is Nash equilibrium, which allows for no coordination. In the mutual-consent setting, Nash equilibrium is not appropriate because even if forming a link is beneficial for both individuals involved, it can still be optimal in the Nash sense that they do not form the link, merely due to coordination failure. 6 This is why Jackson and Wolinsky proposed pairwise stability, which allows two individuals to coordinate so they do not fail to form a link if that is beneficial for both. Pairwise stability only allows for the coordination of a pair on one link. There are other equilibrium concepts that allow for higher level coordination. For example, bilateral equilibrium allows for the coordination of a pair on more than one link (Goyal and Vega-Redondo (2007, and strong stability allows for the coordination of a coalition (Dutta and Mutuswami 5 Equations (4 and (5 differ slightly from Definitions 1 and 2 in the indifference case, but the discrepency is negligible when ε follows a continuous distribution. 6 This is because if i rejects the link, it does not matter whether or not j rejects it. Then rejection is a (weakly optimal choice for j. Moreover, given j s rejection, it is also (weakly optimal for i to reject the link. 8

9 (1997, Jackson and van den Nouweland (2005. These concepts refine pairwise stability with further restrictions. In this paper, we want to keep the assumptions as weak as possible, so we only assume pairwise stability. 3 Identification In this section, we examine the framework that we use to identify the model. We first discuss multiple equilibria, the main problem in identification. Then we show how much we can learn without imposing any restrictions on the equilibrium selection. We consider the following data generating process. Suppose there is an infinite population of individuals. We first generate an integer N from a distribution on {2, 3,...}. Given N, we then draw at random N individuals. Each individual i is associated with a vector of characteristics X i and a vector of preferences ε i. let these individuals form links, and a PS network emerges. We observe G (N and X (N = (X 1,..., X N, but not ε (N = (ε 1,..., ε N. 7 We This procedure is repeated independently T times, and we obtain a sample (G t (N t, X t (N t, N t, t = 1,..., T. Let V t = {1, 2,..., N t } and ε t (N t the preferences of the individuals in observation t. For simplicity, we suppress N in G (N, X (N, and ε (N whenever possible. Throughout the paper we make the following assumptions. Assumption 1 (Data generating process (i We have an i.i.d. sample of (G t, X t, N t, t = 1,..., T. Let T. (ii X t and ε t are independent, t = 1,..., T. (iii ε t,ij, i j V t, t = 1,..., T, are i.i.d. with an absolutely continuous distribution (with respect to the Lebesgue measure, which is known up to finite dimensional θ ε Θ ε. 8 Denote the distribution of ε t, t = 1,..., T, by F ε (ε; θ ε. Assumption 2 (Utility The marginal utility of i from forming a link with j takes the form ij U i (G ij, X i, X j, ε ij ; θ u, which is known up to finite dimensional θ u Θ u and is continuous in θ u. Moreover, the marginal utility function satisfies that for any (X, ε and θ u Θ u there exists a PS network. The parameter of interest is θ = (θ u, θ ε Θ u Θ ε = Θ. 7 Due to the sampling scheme X i are i.i.d. for i = 1,..., N. 8 The i.i.d. assumption is not crucial to our analysis. In fact, we can introduce dependence by letting ε ij = α i + ξ ij, where α i is the individual random effects, α i and ξ ij are independent, and ( ξij, ξ ji are i.i.d.. In this case, εij, j = 1,..., N are dependent through α i. 9

10 3.1 Multiple Equilibria and No Point Identification For a given X and ε, the model yields a collection of PS networks, denoted by PS ( U (X, ε, where U (X, ε = {{ ij U i (G ij, X i, X j, ε ij } G ij G ij } i,j V,i j R N(N 1 G /2 is the marginal-utility profile. To complete the model, suppose there is an equilibrium selection mechanism that picks one network from the collection of PS networks. Denote by Pr (G PS ( U (X, ε, X, ε the selection probability. Then the probability that we observe network G conditional on X is Pr (G X = Pr (G PS ( U (X, ε, X, ε df ε (ε, (6 Equation (6 is similar to what Ciliberto and Tamer (2009 establish in entry games and Bajari, Hong, and Ryan (2010 in discrete games with complete information. Since the equilibrium selection probability in (6 is unknown when there are multiple equilibria (i.e., PS ( U (X, ε 2, whether the true parameter value θ 0 can be point identified from condition (6 depends on whether there is an unique equilibrium (i.e., PS ( U (X, ε = 1. If for any θ Θ there is a network that can only be a unique equilibrium, then under certain conditions the unique equilibrium may provide moment restrictions to point identify θ 0. However, if for some θ Θ all the networks are part of multiple equilibria, then θ 0 cannot be point identified without additional restrictions on the equilibrium selection. In this case, we encounter the incoherency problem addressed in the literature (Bresnahan and Reiss (1991, Tamer (2003. For the game described in Section 2, the presence of multiple equilibria is prevalent because of the interdependence of marginal utilities across links. 9 multiple equilibria in the example below. We illustrate Example 1 Consider networks of N = 3. Suppose the utility function is as in (2 with u ij = ε ij, γ 1 < 0, γ 2 > 0, γ 1 + γ 2 > 0, C i ( = 0. For simplicity we assume ε ij = ε ji, i, j = 1, 2, 3, so ε = (ε 12, ε 23, ε 13. Figure 1 shows the 8 possible networks. Given the utility specification, we can calculate all possible collections of PS networks and the corresponding regions of (ε 12, ε 23, ε 13, as collected in Table 1, where θ 1 = γ 1 γ 2, θ 2 = γ Under the assumed parameter values, we find both multiple 9 If there is no utility interdependence, i.e., ij U i (G ij, X i, X j, ε ij = ij U i (X i, X j, ε ij, then a PS network must be unique. 10 Note that a collection of PS networks must contain only nonadjacent networks, i.e., networks 10

11 Figure 1: Networks of Three Individuals equilibria and unique equilibria. The regions of (ε 12, ε 23, ε 13 corresponding to these equilibrium collections form a partition of R In Example 1, there are networks that can only be a unique equilibrium (i.e., g 5, g 6, g 7 under the assumed parameter values. For other parameter values, however, an unique equilibrium may not exist, as illustrated in Example 2. This implies that the existence of an unique equilibrium is not guaranteed. Hence, the true parameter value is generally not point identified unless we make additional assumptions about the equilibrium selection rule. Example 2 Consider the same setting as in Example 1 except that γ 1 > 0, γ 2 > 0. Then no network can be a unique equilibrium. All the eight networks belong to certain multiple equilibria. 3.2 Partial Identification One can achieve point identification by making certain assumptions about the equilibrium selection. See Remark 3.1 for detailed discussion. In this paper, we do not want to impose any restrictions on the equilibrium selection, so we get around the non-identifiability issue using partial identification. This approach has been widely that differ by at least two links. This is because if two networks differ by only one link, the pair involved in the link must prefer one network over the other, and the latter cannot be PS. Hence, to find all possible equilibria, we only need to look at the collections of nonadjacent networks. 11 They form a partition of R 3 because a PS network exists for any ε. 11

12 Table 1: All Possible Equilibria and the Partition of the ɛ space Equilibrium collections Regions of (ε 12, ε 23, ε 13 Multiple equilibria {g 1, g 8 } [θ 1, 0 [θ 1, 0 [θ 1, 0 {g 2, g 3 } [0, θ 2 (, θ 1 [0, θ 2 {g 2, g 4 } (, θ 1 [0, θ 2 [0, θ 2 {g 3, g 4 } [0, θ 2 [0, θ 2 (, θ 1 {g 2, g 8 } {[θ 1, 0 [θ 1, 0 [0, θ 2 } {[θ 1, θ 2 [θ 1, θ 2 [θ 2, } {g 3, g 8 } {[0, θ 2 [θ 1, 0 [θ 1, 0} {[θ 2, [θ 1, θ 2 [θ 1, θ 2 } {g 4, g 8 } {[θ 1, 0 [0, θ 2 [θ 1, 0} {[θ 1, θ 2 [θ 2, [θ 1, θ 2 } {g 2, g 3, g 8 } [0, θ 2 [θ 1, 0 [0, θ 2 {g 3, g 4, g 8 } [0, θ 2 [0, θ 2 [θ 1, 0 {g 2, g 4, g 8 } [θ 1, 0 [0, θ 2 [0, θ 2 {g 2, g 3, g 4, g 8 } [0, θ 2 [0, θ 2 [0, θ 2 Unique equilibria {g 1 } {(, 0 (, θ 1 [θ 1, 0} {[θ 1, 0 (, 0 (, θ 1 } {(, θ 1 [θ 1, 0 (, 0} {(, θ 1 (, θ 1 (, θ 1 } {g 2 } {(, θ 1 (, 0 [0, θ 2 } {(, θ 1 (, θ 2 [θ 2, } {[θ 1, 0 (, θ 1 [0, θ 2 } {[θ 1, θ 2 (, θ 1 [θ 2, } {g 3 } {[0, θ 2 (, 0 (, θ 1 } {[θ 2, (, θ 2 (, θ 1 } {[0, θ 2 (, θ 1 [θ 1, 0} {[θ 2, (, θ 1 [θ 1, θ 2 } {g 4 } {(, 0 [0, θ 2 (, θ 1 } {(, θ 2 [θ 2, (, θ 1 } {(, θ 1 [0, θ 2 [θ 1, 0} {(, θ 1 [θ 2, [θ 1, θ 2 } {g 5 } [θ 2, (, θ 1 [θ 2, {g 6 } (, θ 1 [θ 2, [θ 2, {g 7 } [θ 2, [θ 2, (, θ 1 {g 8 } {[θ 2, [θ 1, [θ 2, } {[θ 2, [θ 2, [θ 1, θ 2 } {[θ 1, θ 2 [θ 2, [θ 2, } 12

13 applied to game-theoretic models with multiple equilibria (Andrews, Berry and Jia (2004, Pakes, Porter, Ho and Ishii (2006, Berry and Tamer (2006, Ciliberto and Tamer (2009. Following closely Ciliberto and Tamer (2009, we divide the integral in (6 into two parts Pr (G X = + G PS( U(X,ε& PS( U(X,ε =1 G PS( U(X,ε& PS( U(X,ε 2 df ε (ε Pr (G PS ( U (X, ε, X, ε df ε (ε, (7 depending on whether there is a unique equilibrium or multiple equilibria. Note that the selection probability is trivially 1 when a network is a unique equilibrium. When there are multiple equilibria, the selection probability, though unknown, lies between 0 and 1. Replacing the selection probability with these bounds, we derive an upper and lower bound for Pr(G X, i.e., and Pr (G X Pr (G X G PS( U(X,ε G PS( U(X,ε& PS( U(X,ε =1 df ε (ε, (8 df ε (ε. (9 The upper bound is the probability that network G is PS, and the lower bound is the probability that G is uniquely PS. These are the best possible bounds because the selection probability in (7 can be any number between 0 and 1. Unfortunately, the lower bound in (9 is computationally infeasible if the network size N is large. This is because to compute the lower bound, we need to check pairwise stability for 2 N(N 1/2 possible networks. 12 This is computationally intractable even for a moderate value N. For example, in the case of 20 people, the number of possible networks is Remark 3.1 An alternative approach is to achieve point identification by making additional assumptions about the equilibrium selection. In network formation, one way to do this is to consider a sequential model as in Jackson and Watts (2002 (see 12 Unlike the upper bound, the lower bound has no closed form and needs to be computed by simulation. For each simulated ε, we need to check whether a network is uniquely pairwise stable, which amounts to checking pairwise stability for all possible networks. 13

14 also Christakis et al. (2010 and Mele (2011. This sequential model assumes that individuals are myopic and form links in a random sequence: in each period only pair of individuals is randomly selected and only that pair can update their relationship. The sequence of networks realized in each period form a Markov chain with states corresponding to the networks. Under certain conditions 13 the Markov chain converges to a unique stationary distribution, which typically assigns probability one to a single PS network. 14 Hence the stationary distribution amounts to a particular selection rule. Alternatively, one can assume a more general equilibrium selection mechanism, for example, by specifying a parametric form (Bajari, Hong, and Ryan (2010 or considering a nonparametric equilibrium selection (Bajari, Hahn, Hong, and Ridder (2011. Note that in the game we consider a fully nonparametric equilibrium selection is not identified. Certain restrictions must be imposed on it (e.g., independence of ε in order to achieve identification. 3.3 Subnetworks We propose a novel method to tackle the computational problem. The idea is to use the pairwise stability of certain parts of a network, called subnetworks. A subnetwork is the restriction of a network to a subset of the individuals. To be precise, denote a network by G = (V, E, where V is the vertex set and E is the edge set. For A V, we say G A = (A, E A is the subnetwork of G in A if E A E contains the links in G that connect two individuals in A. Moreover, let G A = (V, E\E A be the complement of G A in G, i.e., the remainder of G after the links in E A are deleted. It contains links in G that connect either two individuals in V \A or an individual in A to an individual in V \A. The former type of links form the subnetwork in V \A, G V \A, and the latter type of links form the neighborhood of A in G, denoted by B A (G = (V, E\ ( E A E V \A. Clearly, we have the decomposition G = (G A, G A = ( G A, B A (G, G V \A. These definitions are based on graph notation. In matrix notation, subnetwork G A (neighborhood B A corresponds to the submatrix of G with rows and columns in A (rows in A and columns in V \A. For 13 An example of such conditions would be (i the individuals are assumed to make mistakes (i.e., forming or deleting a link randomly rather than based on utility maximization and (ii the probability of making a mistake is suffi ciently small. 14 This network is essentially the most "stable" one among all the PS networks, or more precisely, the network that has the minimum resistance (Young (1993, Jackson and Watts (

15 Figure 2: An Example of a Subnetwork brevity, we suppress G in B A (G whenever possible and abbreviate G {i,j}, G {i,j} and B {i,j} to G ij, G ij, and B ij. The sets of all possible G A, G A and B A are denoted by G A, G A and B A. Figure 2 illustrates a subnetwork. From G = (G A, G A, the distribution of subnetwork G A is simply a marginal distribution of the distribution of G, i.e., Pr (G A X = Pr (G A, G A X G A = G A Pr (G A, G A PS ( U (X, ε, X, ε df ε (ε. (10 The summed equilibrium selection probability in (10 is unknown unless all the networks in PS ( U (X, ε have the same subnetwork in A. Following the same idea as in (6, we can derive an upper and lower bound for Pr (G A X. Specifically, divide the integral in (10 into two parts Pr (G A X = df ε (ε + G A PS A ( U(X,ε & PS A ( U(X,ε =1 G A PS A ( U(X,ε & PS A ( U(X,ε 2 G A Pr (G A, G A PS ( U (X, ε, X, ε df ε (ε, (11 { where PS A ( U (X, ε = GA G A : G A G A, ( G A, G A PS ( U (X, ε} 15

16 is the set of subnetworks in A that are part of a network in PS ( U (X, ε. Replacing the sum term in (11 by 0 and 1 yields and Pr (G A X Pr (G A X G A PS A ( U(X,ε G A PS A ( U(X,ε& PS A ( U(X,ε =1 df ε (ε, (12 df ε (ε, (13 The bounds are analogous to those in (8 and (9: the upper bound is the probability that G A is part of a PS network, and the lower bound is the probability that only G A can be part of a PS network. These bounds are the best possible because the sum term in (11 can be any number between 0 and 1. However, they are still infeasible: the upper bound requires checking the pairwise stability of (G A, G A for all G A and the lower bound requires checking that of (G A, G A for all G A and all G A G A, so the computational complexity of them is exactly the same as that of the lower bound in (9. In the next step, we will derive from these bounds inequalities that are tractable Main idea The main idea is to use the pairwise stability of subnetworks rather than networks. We say a subnetwork G A is pairwise stable (PS for a given G A if condition (5 is satisfied for all i j A. More generally, we say a collection of links is pairwise stable for a given complement of the collection if all the links in the collection satisfy condition (5. 15 By definition, if network G is pairwise stable, subnetwork G A is pairwise stable for G A. From this simple property, we can derive inequalities that involve only the pairwise stability of subnetworks. To proceed, for i j A write G A = (G ij, G A ij, where G A ij G A ij is the remainder of G A after link ij is deleted. It is clear that G ij = (G A ij, G A. Let X A = (X i i A and ε A = (ε ij i j A be the vectors of observed characteristics and unobserved preferences of the individuals in A. Denote the distribution of ε A by F εa (ε A. For a given (G A, X A, ε A, there is a collection of PS subnetworks in A, denoted by PS ( U A (G A, X A, ε A, where U A (G A, X A, ε A = {{ ij U i (G A ij, G A, X i, X j, ε ij } GA ij G A ij } i j is the marginal-utility profile of the individuals in A. Then the aforementioned prop- 15 This definition is useful when we consider the pairwise stability of G A for a given G A. 16

17 erty yields the lemma below. Lemma 3.1 Under Assumptions 1-2, for any A V, Pr (G A X A df εa (ε A, (14 G A,G A PS( U A (G A,X A,ε A and Pr (G A X A df εa (ε A. (15 G A,G A PS( U A (G A,X A,ε A & PS( U A (G A,X A,ε A =1 Proof. See the appendix. The upper bound for Pr (G A X A is the probability that subnetwork G A is PS for some G A, and the lower bound is the probability that for any G A only subnetwork G A can be PS. These bounds are wider than those in (12 and (13 and thus not sharp, but the loss of sharpness is exactly the source from which we reduce the computational burden. Because the new bounds take into account only the pairwise stability of subnetworks, ignoring the pairwise stability of the rest of the networks, we only need to check pairwise stability for all possible subnetworks rather than all possible networks. The bounds in (12 and (13 also make use of subnetworks, but because the links inside a subnetwork are generally correlated with the links outside, while those bounds take into consideration the correlation and internalize it by employing the pairwise stability of the entire network, their computational complexity is not reduced. In contrast, the bounds in Lemma 3.1 only consider the maximum and minimum of the correlation, so their computational complexity can be lower. In fact, this computational problem arises in many applications where there is a intractably large set of choices/agents. Researchers tend to use a subset of the choices/agents to reduce the computational complexity, but how to handle the correlation between choices inside the subset and choices outside is essential. In multinomial choice models, for example, people assume that the choices follow a block-additive GEV model, so using random subsets of the full choice set would yield consistent estimates (Bierlaire, Bolduc and McFadden, Our approach is different in the sense that we do not need additional assumptions to restrict the correlation, but use bounds for it. In practice, because G A is large even for a moderate N, the computation of the bounds in Lemma 3.1 is still intractable. However, because G A plays a role only 17

18 through U A (G A, X A, ε A, if we can restrict the effect of G A in the utility function, we may avoid checking for all G A. In particular, we assume that the marginal utility ij U i (G ij, X i, X j, ε ij depends on G ij only through the neighborhood of i and j, B ij. This local externality property holds for the utility function in (2. Assumption 3 (Local externality For any i j V, ij U i (G ij, X i, X j, ε ij ; θ u = ij U i (B ij, X i, X j, ε ij ; θ u. Because B ij is determined by G A ij and B A, local externality implies that for A V, the marginal-utility profile in A satisfies U A (G A, X A, ε A = U A (B A, X A, ε A = {{ ij U i (B ij (G A ij, B A, X i, X j, ε ij } GA ij G A ij } i j A. Hence, Proposition 3.1 Under Assumptions 1-3, for any A V, Pr (G A X A df εa (ε A, (16 B A,G A PS( U A (B A,X A,ε A and Pr (G A X A df εa (ε A. (17 B A,G A PS( U A (B A,X A,ε A & PS( U A (B A,X A,ε A =1 The bounds in Proposition 3.1 consider all possible B A. There is an upper bound for Pr (G A, B A X A that considers the realized B A. Proposition 3.2 Under Assumptions 1-3, for any A V, Pr (G A, B A X A df εa (ε A. (18 G A PS( U A (B A,X A,ε A Proof. See the appendix. When we compute the bounds in (16 and (17, the total number of subnetworks that need to be checked for pairwise stability is equal to 2 A ( A 1/2 times the number of B A B A that yield distinct U A (B A. The latter number could still be large because B A increases exponentially in N. In practice, we need additional restrictions on the utility specification to keep that number at a tractable level. In this paper, we consider the utility function as in (2, which implicitly impose several assumptions to reduce the number of distinct U A (B A. First, the effects of each link in B ij are 18

19 Table 2: Regions of ɛ 13 B 13 = (0, 0 B 13 = (1, 0 B 13 = (0, 1 B 13 = (1, 1 G 13 = 1 [0, [ γ 1, [ γ 1, [ γ 1 γ 2, G 13 = 0 (, 0 (, γ 1 (, γ 1 (, γ 1 γ 2 assumed to be identical, i.e., they do not depend on the characteristics of k i, j, so B A can be captured by numbers of friends. 16 Second, (2 only consider two types of friends, i.e., friends of each individual and common friends of each pair. Third, we choose a cost function C i (n such that C i (n = if n > b for some integer b, so in equilibrium nobody has more than b friends. Under these assumptions, B A can be captured by ( ( A 1 + A { 2 integers, whose values are in 0, 1,..., b}. Thus, the number of distinct U A (B A is bounded above by ( b A ( A +1/2, + 1 which is independent of N. 17 Therefore, the computation of the bounds in (16 and (17 is tractable no 18 matter how large N is, provided that we choose small A and b. As for the upper bound in (18, it is easy to compute because it only needs the realized B A. Example 3 (Example 1 continued Assume the same setting as in Example 1. We want to derive (16 and (17 for subnetwork G 13 = 1. In this case, B 13 = (G 12, G 23 takes four possible values, i.e., B 13 = {(0, 0, (1, 0, (0, 1, (1, 1}. Because the pairwise stability of G 13 for a given B 13 is affected by ε only through ε 13, we find the regions of ε 13 in which G 13 = 1 or 0 is PS for each B 13 B 13. All the regions are collected in Table 2. By taking the union of the regions in the first row and in the second row respectively, we obtain (16 and (17 as follows: Pr (G 13 = 1 [ γ 1 γ 2, df ε13 (ε 13, (19 16 To be specific, B A can be captured by a vector of integers, whose components are the numbers of friends in V \A that each i in A has, that each pair (i, j in A has in common, that each triplet (i, j, k in A has in common,..., and that all individuals in A have in common. 17 In fact, the number of distinct U A (B A is substantially smaller than ( b + 1 A ( A +1/2 because the upper limit b is imposed on the total number of i s own friends and the friends that i shares with others. 18 The computational complexity will not increase for other utility specifications that satisfy Assumption 3, provided that the number of friends is bounded above by b and only pairs friends in common are considered. In particular, the computational complexity remains the same if the effects of own s friends or common friends are allowed to be nonlinear or interact with the characteristics. 19

20 and Pr (G 13 = 1 df ε13 (ε 13. (20 [ γ 1, The identified set is defined as the collection of θ Θ that satisfy inequalities (16-(18. Definition 3 The identified set is Θ I = {θ Θ : (16-(18 hold, A s.t. 2 A ā, X A } (21 for some integer ā. The trade-off in choosing ā is between the computational complexity and the tightness of the bounds. Bounds from smaller subnetworks are easier to compute, but they are also wider. We suggest to start with the smallest subnetworks, i.e., pairs, and choose the maximum size ā based on the computational capability. Note that if the networks in data happen to be small and we are able to choose ā = N, then the bounds in (16 and (17 reduce to those in (8 and (9. 4 Estimation and Inference In this section, we discuss the estimation and inference of the identified set in (21. This set is defined by conditional moment inequalities (16-(18. We assume that X is discrete and transform the conditional moment inequalities into their unconditional counterparts Pr (G A = g A, X A = x A H 1 (g A, x A ; θ Pr (X A = x A 0 H 2 (g A, x A ; θ Pr (X A = x A Pr (G A = g A, X A = x A 0 Pr (G A = g A, B A = b A, X A = x A H 3 (g A, b A, x A ; θ Pr (X A = x A 0, (22 where H 1 (g A, x A ; θ = H 2 (g A, x A ; θ = max 1 {g A PS ( U A (b A, x A, ε A ; θ u } df εa (ε A θ ε b A { } g A PS ( U A (b A, x A, ε A ; θ u & min 1 df εa (ε A θ ε b A PS ( U A (b A, x A, ε A ; θ u = 1 20

21 Figure 3: An Example of Multiple Subnetworks H 3 (g A, b A, x A ; θ = 1 {g A PS ( U A (b A, x A, ε A ; θ u } df εa (ε A θ ε. (23 What we need is to estimate the probabilities in (22 and compute the bounds in (23. The former is discussed in Section 4.1. We then discuss in Section 4.2 the estimation and inference of the identified set, assuming that the bounds are known. The computation of the bounds is discussed in Section Estimation of the Probabilities To estimate the probabilities in (22, we need to pick subnetworks in A with certain characteristics. In most network data (such as Add Health individuals do not have identities, so those individuals with the same characteristics are viewed as identical and labeled arbitrarily. 19 This implies that subset A may not be uniquely determined and there may be more than one way to choose a subnetwork in A with certain characteristics. For example, if we consider the subnetworks of a boy and a girl in the network in Figure 3, we will find four such subnetworks. For discrete X, this indeterminacy occurs with positive probability. Nevertheless, because pairwise stability and equilibrium selection do not depend on how we label the individuals, we can show that subnetworks with the same characteristics follow the same distribution, so it does not matter which one we choose. To see this, fix n and let π be a permutation over V = {1,..., n}. Express the characteristics explicitly, and write a subnetwork as (g A, x A and a subnetwork with 19 Network data with identities are those that consist of repeated observations for a given group of individuals. In such data, each individual is labeled by a fixed number across networks. The difference between with and without identities is analogous to the difference between panel data and repeated cross-sectional data. 21

22 a neighborhood as (g A, b A, x A. We define two subnetworks (or subnetworks with neighborhoods to be isomorphic if they are identical except for the labels. Definition 4 (i For A, A V and subnetworks (g A, x A and (g A, x A, if there is a permutation π over V with π (A = A such that (a g ij = g π(iπ(j, i, j A, i j, and (b x i = x π(i, i A, then (g A, x A and (g A, x A are isomorphic, and we write (g A, x A = (g A, x A. (ii For A, A V and subnetworks with neighborhoods (g A, b A, x A and (g A, b A, x A, if there is a permutation π over V with π (A = A such that (a g ij = g π(iπ(j, i A, j V, i j, and (b x i = x π(i, i A, then (g A, b A, x A and (g A, b A, x A are isomorphic, and we write (g A, b A, x A = (g A, b A, x A. Proposition 4.1 Suppose Assumptions 1-3 are satisfied. (i For A, A subnetworks (g A, x A and (g A, x A, if (g A, x A = (g A, x A, then V and Pr (G A = g A, X A = x A = Pr (G A = g A, X A = x A, (24 H 1 (g A, x A ; θ = H 1 (g A, x A ; θ, (25 H 2 (g A, x A ; θ = H 2 (g A, x A ; θ. (26 (ii For A, A V and subnetworks with neighborhoods (g A, b A, x A and (g A, b A, x A, if (g A, b A, x A = (g A, b A, x A, then Pr (G A = g A, B A = b A, X A = x A = Pr (G A = g A, B A = b A, X A = x A, (27 Proof. See the appendix. H 3 (g A, b A, x A ; θ = H 3 (g A, b A, x A ; θ. (28 Proposition 4.1 shows that if we treat isomorphic subnetworks (g A, x A and (g A, x A as identical realizations, (G A, X A and (G A, X A follow the same distribution. This can be satisfied for any A, A V such that A = A, so we can choose any subnetworks of size A to estimate Pr (G A = g A, X A = x A and similar for Pr(G A = g A, B A = b A, X A = x A. Another implication of Proposition 4.1 is that, letting A = A, we can treat isomorphic subnetworks (g A, x A and (g A, x A as the same realization of (G A, X A. Therefore, the distribution of (G A, X A is considered as over the equivalence classes of subnetworks, defined by the isomorphism. This resolves the problem that different labeling may lead to different matrix representations of 22

23 subnetworks. 20 Although picking from each network any subnetwork of size A provides consistent estimators as the number of networks goes to infinity, we exploit the data more effi ciently by picking from each network all subnetworks of size A and taking the average over them. 21 To be precise, let a = A and A 1,..., A ( n be the subsets of a {1,..., n} that have a elements. Denote by (g a, x a and (g a, b a, x a the realizations of (G A, X A and (G A, B A, X A. Then the moment functions for observation t in (22 take the form where m 1t (g a, x a ; θ = p t (g a, x a H 1t (g a, x a ; θ p t (x a, m 2t (g a, x a ; θ = H 2t (g a, x a ; θ p t (x a p t (g a, x a, m 3t (g a, b a, x a ; θ = p t (g a, b a, x a H 3t (g a, b a, x a ; θ p t (x a, (29 p t (g a, b a, x a = 1 1 { } G t,aj = g a, B t,aj = b a, X t,aj = x a, ( nt a ( n t a j=1 p t (g a, x a = 1 1 { } G t,aj = g a, X t,aj = x a, ( nt a ( n t a j=1 p t (x a = 1 1 { } X t,aj = x a. (30 ( nt a ( n t a j=1 and the bounds are those in (23 multiplied by ( n t a, e.g., H1t (g a, x a ; θ = ( n t a H 1 (g A, x A ; θ. Clearly, E p t (g a, x a = Pr (G A = g a, X A = x a, and V p t (g a, x a < V (1 {G t,a1 = g a, X t,a1 = x a }. 22 Thus taking the average over all eligible subnetworks would lower the variance of the estimator of the identified set. 20 In practice, we need a graph isomorphism algorithm to find which subnetworks are isomorphic. In the Monte Carlo study we use the algorithm Nauty ( More details about the implementation of Nauty can be found in the appendix. 21 It is true that subnetworks from the same network are dependent because they are part of a single PS network. However, the dependence does not cause any additional problems if we consider the asymptotics as the number of networks goes to infinity. 22 This is from the Cauchy-Schwarz inequality and Pr ( G Aj = G Ak, X Aj = X Ak, N = n 1, j k. 23

24 4.2 Estimation and Inference of the Identified Set Now we discuss the estimation and inference of the identified set, assuming the bounds in (23 are known. Suppose (g a, x a and (g a, b a, x a take κ g and κ b distinct values respectively, denoted by (g a, x a 1,..., (g a, x a κg, and (g a, b a, x a 1,..., (g a, b a, x a κ b. 23 Stack the moment functions in (29 into a vector m t (θ = ( m 2 t (θ,, māt (θ where m a t (θ = (m a 1t (θ, m a 2t (θ, m a 3t (θ, a = 2,..., ā, m a jt (θ = ( m jt ((g a, x a 1 ; θ,..., m jt ((g a, x a κg ; θ, j = 1, 2, m a 3t (θ = ( m 3t ((g a, b a, x a 1 ; θ,..., m 3t ((g a, b a, x a κ b ; θ. Then the moment inequalities in (22 can be written as Em t (θ 0. (31 Following the set inference literature (CHT (2007, Romano and Shaikh (2010, Andrews and Soares (2010 among others, we estimate the identified set by minimizing the sample analog of a criterion function based on (31. We use the criterion function as in CHT (2007, Q (θ = (Em t (θ 2 +, where (x+ = max (x, 0 and is the Euclidean norm. The identified set is given by Θ I = {θ Θ : Q (θ = 0}. (32 Let Q T (θ = (ET m t (θ + 2 be the sample analog of Q (θ, where ET m t (θ = 1 T T t=1 m t (θ. In practice we use the normalized sample criterion Q T (θ = Q T (θ inf θ Θ Q T (θ to account for misspecification (CHT (2007, Ciliberto and Tamer (2009 and propose the estimator ˆΘ I = {θ Θ : T Q T (θ c T }, (33 23 That is, each (g a, x a i represents an equivalence class of subnetworks, i = 1,..., κ g, so do (g a, b a, x a j, j = 1,..., κ b. The numbers κ g and κ b are determined by a and the cardinality of the support of X. 24

25 where c T is chosen to be c T and c T /T 0, e.g. c T ln T. Let d (A, B = max {sup a A d (a, B, sup b B d (b, A} be the Hausdorff distance between sets A and B, where d (a, B = inf b B a b. It can be shown that ˆΘ I is consistent under Hausdorff distance, i.e., d ( ˆΘI, Θ I p 0 as T. Theorem 4.1 Assume Θ is a compact set. (a If Q (θ and Q T (θ defined as above satisfy (i Q (θ is continuous in θ, (ii sup θ Θ Q (θ Q T (θ = O p (1/ T, and (iii sup θ ΘI Q T (θ = O p (1/T, then ˆΘ I in (33 is a consistent estimator of Θ I, i.e., d ( ˆΘI, Θ I p 0, as T. (b Under Assumptions 1-3, Conditions (i-(iii in part (a hold. Proof. See the appendix. The confidence region for the true θ 0 can be constructed by inverting the acceptance region of a test (e.g. CHT (2007, Andrews and Soares (2010, Andrews and Jia (2012. We use the confidence region proposed by CHT (2007 C T = {θ Θ : T Q T (θ c 1 α (θ}, (34 where c 1 α (θ = min (ĉ 1 α (θ, ĉ 1 α, ĉ 1 α (θ is a consistent estimator of c 1 α (θ, the 1 α quantile of the limiting distribution of T Q T (θ, and ĉ 1 α is a data-dependent variable that is O p (1 and larger than sup θ ΘI c 1 α (θ. It can be shown that for any θ Θ I, C T is asymptotically correct, i.e., lim inf Pr (θ C T 1 α. The limiting T distribution of T Q T (θ is data dependent. We can use the subsampling method in CHT (2007 to obtain c 1 α (θ. 4.3 Computation of the Bounds The discussion in the last section assumes that the bounds in (23 are known. In practice, we need to compute the unknown bounds first. The bound H 3 has a closed form H 3 (g A, b A, x A = i,j A g ij =1 Pr ( ij U i (b ij, x i, x j, ε ij + ij U j (b ij, x j, x i, ε ji 0 25