Network Formation Models

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Strategic models of network formation generalize the frameworks of single agents and multiple agents discrete choice models reviewed in Sections Discrete Choice in Single Agent Random Utility Models and Static, Simultaneous-Move Finite Games with Multiple Equilibria. They posit that pairs of agents (nodes) form, maintain, or sever connections (links) according to an explicit equilibrium notion and utility structure. Each individual's utility depends on the links formed by others (the network) and on utility shifters that may be pair-specific. One may conjecture that the results reported in Sections Discrete Choice in Single Agent Random Utility Models-Static, Simultaneous-Move Finite Games with Multiple Equilibria apply in this more general context too. While of course lessons can be carried over, network formation models present challenges that combined cannot be overcome without the development of new tools. These include the issue of equilibrium existence and the possibility of multiple equilibria when they exist, due to the interdependence in agents' choices (this problem was already discussed in Section Static, Simultaneous-Move Finite Games with Multiple Equilibria). Another challenge is the degree of correlation between linking decisions, which interacts with how the observable data is generated: one may observe a growing number of independent networks, or a growing number of agents on a single network. Yet another challenge, which substantially increases the difficulties associated with the previous two, is the combinatoric complexity of network formation problems. The purpose of this section is exclusively to discuss some recent papers that have made important progress to address these specific challenges and carry out partial identification analysis. For a thorough treatment of the literature on network formation, I refer to the reviews in , , , and . Depending on whether the researcher observes data from a single network or multiple independent networks, the underlying population of agents may be represented as a continuum or as a countably infinite set in the first case, or as a finite set in the second case. Henceforth, I denote generic agents as [math]i[/math], [math]j[/math], [math]k[/math], and [math]m[/math]. I consider static models of undirected network formation with non-transferable utility. The collection of all links among nodes forms the network, denoted [math]\ey[/math]. For any pair [math](i,j)[/math] with [math]i\neq j[/math], [math]\ey_{ij}=1[/math] if they are linked, and [math]\ey_{ij}=0[/math] otherwise ([math]\ey_{ii}=0[/math] for all [math]i[/math] by convention). The notation [math]\ey-\{ij\}[/math] denotes the network that results if a link present between nodes [math]i[/math] and [math]j[/math] is deleted, while [math]\ey+\{ij\}[/math] denotes the network that results if a link absent between nodes [math]i[/math] and [math]j[/math] is added. Denote agent [math]i[/math]'s payoff by [math]\bu_i(\ey,\ex,\epsilon)[/math]. This payoff depends on the network [math]\ey[/math] and the payoff shifters [math](\ex,\epsilon)[/math], with [math]\ex[/math] observable both to the agents and to the researcher, [math]\epsilon[/math] only to the agents, and [math](\ex,\epsilon)[/math] collecting [math](\ex_{ij},\epsilon_{ij})[/math] for all [math]i[/math] and [math]j[/math]. Following much of the literature, I employ pairwise stability as equilibrium notion: [math]\ey[/math] is a pairwise stable network if all linked agents prefer not to sever their links, and all non-existing links are damaging to at least one agent. Formally,

[[math]] \begin{align*} \forall(i,j):\ey_{ij}&=1,\bu_i(\ey,\ex,\epsilon)\ge \bu_i(\ey-\{ij\},\ex,\epsilon)\mathrm{and}\bu_j(\ey,\ex,\epsilon)\ge \bu_j(\ey-\{ij\},\ex,\epsilon),\\ \forall(i,j):\ey_{ij}&=0,\mathrm{if}\bu_i(\ey+\{ij\},\ex,\epsilon) \gt \bu_i(\ey,\ex,\epsilon)\mathrm{then}\bu_j(\ey+\{ij\},\ex,\epsilon) \lt \bu_j(\ey,\ex,\epsilon). \end{align*} [[/math]]

Under this equilibrium notion, if equilibria exist multiplicity is likely; see, among others, the examples in , , and . The model is therefore incomplete, because it does not specify how an equilibrium is selected in the region of multiplicity. For the same reasons as discussed in the context of finite games in Section Static, Simultaneous-Move Finite Games with Multiple Equilibria, partial identification results (unless one is willing to impose restrictions on the equilibrium selection mechanism). However, as I explain below, an immediate application of the identification analysis carried out there presents enormous practical challenges because there are [math]2^{n(n-1)/2}[/math] possible network configurations to be checked for stability (and the dimensionality of the space of unobservables is also very large). In what follows I consider two distinct frameworks that make different assumptions about the utility function and how the data is generated, and discuss what can be learned about the parameters of interest in these cases.

Data from Multiple Independent Networks

I first consider the case that the researcher observes data from multiple independent networks. I follow the set-up put forward by .

Identification Problem (Network Formation Model with Multiple Independent Networks)

Let there be [math]n\in\{2,3,\dots\},n \lt \infty[/math] agents, and let [math](\ex,\ey)\sim\sP[/math] be observable random variables in [math]\times_{j=1}^n\R^d\times\{0,1\}^{n(n-1)/2}[/math], [math]d \lt \infty[/math]. Suppose that [math]\ey[/math] is a pairwise stable network. For each agent [math]i[/math], let the utility function be known up to finite dimensional parameter vector [math]\delta\in\Delta\subset\R^p[/math], and given by

[[math]] \begin{multline} \bu_i(\ey,\ex,\epsilon;\delta)=\sum_{j=1}^n \ey_{ij}(f(\ex_i,\ex_j;\delta_1)+\epsilon_{ij})\\ +\delta_2\frac{\sum_{j=1}^n\sum_{k\neq i,k=1}^n\ey_{ij}\ey_{jk}}{n-2}+\delta_3\frac{\sum_{j=1}^n\sum_{k=j+1}^n\ey_{ij}\ey_{ik}\ey_{jk}}{n-2}\label{eq:utility:network:1} \end{multline} [[/math]]

with [math]f(\cdot,\cdot;\cdot)[/math] a continuous function of its arguments. Suppose that [math]\epsilon_{ij}[/math] are independent for all [math]i\neq j[/math] and identically distributed with CDF known up to parameter vector [math]\gamma\in\Gamma\subset\R^m[/math], denoted [math]\sF_\gamma[/math]. Assume that the support of [math]\sF_\gamma[/math] is [math]\R[/math], that [math]\sF_\gamma[/math] is absolutely continuous with respect to Lebesgue measure, and continuously differentiable with respect to [math]\gamma\in\Gamma[/math]. Let [math]\Theta=\Delta\times\Gamma[/math]. Assume that the researcher observes a random sample of networks and observable payoff shifters drawn from [math]\sP[/math]. In the absence of additional information, what can the researcher learn about [math]\theta\equiv[\delta_1\delta_2\delta_3\gamma][/math]?

analyzes this problem. She establishes equilibrium existence provided that [math]\delta_2\ge 0[/math] and [math]\delta_3\ge 0[/math] . Given payoff shifters [math](\ex,\epsilon)[/math] and parameters [math]\vartheta\equiv[\tilde\delta_1\tilde\delta_2\tilde\delta_3\tilde\gamma]\in\Theta[/math], let [math]\eY_\vartheta(\ex,\epsilon)[/math] denote the collection of pairwise stable networks implied by the model. It is easy to show that [math]\eY_\vartheta(\ex,\epsilon)[/math] is a random closed set as in Definition. The networks in [math]\eY_\vartheta(\ex,\epsilon)[/math] are [math]n\times n[/math] symmetric adjacency matrices with diagonal elements equal to zero and off diagonal elements in [math]\{0,1\}[/math]. To ease notation, I omit [math]\eY_\vartheta[/math]'s dependence on [math](\ex,\epsilon)[/math] in what follows. Under the assumption that [math]\ey[/math] is a pairwise stable network, at the true data generating value of [math]\theta\in\Theta[/math], one has

[[math]] \begin{align} \ey\in\eY_\theta\mathrm{a.s.} \label{eq:y_in_Y_network_multiple} \end{align} [[/math]]

Equation \eqref{eq:y_in_Y_network_multiple} exhausts the modeling content of Identification Problem. Theorem can be leveraged to extract its empirical content from the observed distribution [math]\sP(\ey,\ex)[/math]. Let [math]\cY[/math] be the collection of [math]n\times n[/math] symmetric matrices with diagonal elements equal to zero and all other entries in [math]\{0,1\}[/math], so that [math]|\cY|=2^{n(n-1)/2}[/math]. For a given set [math]K\subset\cY[/math], let [math]\sT_{\eY_{\vartheta}}(K;\sF_\gamma)[/math] denote the probability of the event [math]\{\eY_\vartheta\cap K\neq \emptyset\}[/math] implied when [math]\epsilon\sim\sF_\gamma[/math], [math]\ex[/math]-a.s.

Theorem (Structural Parameters in Network Formation Models with Multiple Independent Networks)

Under the assumptions of Identification Problem, the sharp identification region for [math]\theta[/math] is

[[math]] \begin{align} \idr{\theta}=\{\vartheta\in\Theta:\sP(\ey\in K|\ex)\le \sT_{\eY_{\vartheta}}(K;\sF_{\tilde\gamma})\,\forall K\subset\cY, \, \ex\text{-a.s.}\}.\label{eq:SIR:networks:1} \end{align} [[/math]]

Show Proof

Follows from similar arguments as for the proof of Theorem.

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The characterization of [math]\idr{\theta}[/math] in Theorem SIR- is new to this chapter. While technically it entails a finite number of conditional moment inequalities, in practice their number can be prohibitive as it can be as large as [math]2^{2^{n(n-1)/2}}-2[/math]. Even using only a subset of the inequalities in \eqref{eq:SIR:networks:1} to obtain an outer region, for example applying the insights in , may not be practical (with [math]n=20[/math], [math]|\cY|\approx 10^{57}[/math]). Moreover, computation of [math]\sT_{\eY_{\vartheta}}(K;\sF_\gamma)[/math] may require (depending on the set [math]K[/math]) evaluation of rather complex integrals. To circumvent these challenges, proposes to analyze network formation through subnetworks. A subnetwork is the restriction of a network to a subset of the agents (i.e., a subset of nodes and the links between them). For given [math]A\subseteq\{1,2,\dots,n\}[/math], let [math]\ey^A=\{\ey_{ij}\}_{i,j\in A, i\neq j}[/math] be the submatrix in [math]\ey[/math] with rows and columns in [math]A[/math], and let [math]\ey^{-A}[/math] be the remaining elements of [math]\ey[/math] after [math]\ey^A[/math] is deleted. With some abuse of notation, let [math](\ey^A,\ey^{-A})[/math] denote the composition of [math]\ey^A[/math] and [math]\ey^{-A}[/math] that returns [math]\ey[/math]. Recall that [math]\eY_\vartheta\equiv\eY_\vartheta(\ex,\epsilon)[/math], and let

[[math]] \begin{align*} \eY_{\vartheta}^A=\{\ey^A\in\{0,1\}^{|A|}:\exists \ey^{-A}\in\{0,1\}^{|-A|}\mathrm{suchthat}(\ey^A,\ey^{-A})\in\eY_{\vartheta}\} \end{align*} [[/math]]

be the collection of subnetworks with rows and columns in [math]A[/math] that can be part of a pairwise stable network in [math]\eY_\vartheta[/math]. Let [math]\ex^A[/math] denote the subset of [math]\ex[/math] collecting [math]\ex_{ij}[/math] for [math]i,j\in A[/math]. For a given [math]y^A\in\{0,1\}^{|A|}[/math], let [math]\sC_{\eY_{\vartheta}^A}(y^A;\sF_\gamma)[/math] and [math]\sT_{\eY_{\vartheta}^A}(y^A;\sF_\gamma)[/math] denote, respectively, the probability of the events [math]\{\eY_\vartheta^A=\{y^A\}\}[/math] and [math]\{\{y^A\}\in\eY_\vartheta^A\}[/math] implied when [math]\epsilon\sim\sF_\gamma[/math], [math]\ex[/math]-a.s. The first event means that only the subnetwork [math]y^A[/math] is part of a pairwise stable network, while the second event means that [math]y^A[/math] is a possible subnetwork that is part of a pairwise stable network but other subnetworks may be part of it too. provides the following outer region for [math]\theta[/math] by adapting the insight in to subnetworks. In the theorem I abuse notation compared to Table by introducing a superscript, [math]A[/math], to make explicit the dependence of the outer region on it.

Theorem (Subnetworks-based Outer Region on Structural Parameters in Network Formation Models with Multiple Independent Networks)

Under the assumptions of Identification Problem, for any [math]A\subseteq\{1,2,\dots,n\}[/math], an [math]A[/math]-dependent outer region for [math]\theta[/math] is

[[math]] \begin{align} \mathcal{O}^A_\sP[\theta]=\{\vartheta\in\Theta:\sC_{\eY_{\vartheta}^A}(y^A;\sF_{\tilde\gamma})\le\sP(\ey^A=y^A|\ex^A)\le \sT_{\eY_{\vartheta}^A}(y^A;\sF_{\tilde\gamma})\,\forall y^A\subset\cY^A, \, \ex^A\text{-a.s.}\},\label{eq:OR:networks:1} \end{align} [[/math]]

where [math]\cY^A[/math] is the collection of [math]|A|\times|A|[/math] symmetric matrices with diagonal elements equal to zero and all other elements in [math]\{0,1\}[/math] so that [math]|\cY^A|=2^{|A|(|A|-1)/2}[/math].

Show Proof

Let [math]\eu(\tilde\ey|\eY_\vartheta)[/math] be a random variable in the unit simplex in [math]\R^{n(n-1)/2}[/math] which assigns to each possible pairwise stable network [math]\tilde\ey[/math] that may realize given [math](\ex,\epsilon)[/math] and [math]\vartheta\in\Theta[/math] the probability that it is selected from [math]\eY_\vartheta[/math]. Given [math]y\in\cY[/math], denote by [math]\sM(y|\ex)[/math] the model predicted probability that the network realizes equal to [math]y[/math]. Then the model yields

[[math]] \begin{align} \sM(y|\ex)&=\int\eu(y| Y_\vartheta)d\sF_\gamma=\int_{y\in Y_\vartheta,| Y_\vartheta|=1}d\sF_\gamma+\int_{y\in Y_\vartheta,| Y_\vartheta|\ge 2}\eu( y| Y_\vartheta)d\sF_\gamma.\label{eq:model:distrib:network:1} \end{align} [[/math]]

The model implied distribution for subnetwork [math]\tilde\ey^A[/math] is obtained by taking the marginal of expression \eqref{eq:model:distrib:network:1} with respect to [math]\tilde\ey^{-A}[/math]

[[math]] \begin{align} \sM(y^A|\ex)&=\sum_{y^{-A}}\sM((y^A,y^{-A})|\ex)= \int_{y^A\in Y_\vartheta^A,| Y_\vartheta^A|=1}d\sF_\gamma+\int_{y^A\in Y_\vartheta^A,| Y_\vartheta^A|\ge 2}\sum_{y^{-A}}\eu((y^A,y^{-A})| Y_\vartheta)d\sF_\gamma.\label{eq:model:distrib:subnetwork:1} \end{align} [[/math]]

Replacing [math]\eu[/math] in \eqref{eq:model:distrib:subnetwork:1} with zero and one yields the bounds in \eqref{eq:OR:networks:1}.

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further assumes that the selection mechanism [math]\eu(\tilde\ey|\eY_\vartheta)[/math] is invariant to permutations of the labels of the players. Under this condition and the maintained assumptions on [math]\epsilon[/math], she shows that the inequalities in \eqref{eq:OR:networks:1} are invariant under permutations of labels, so subnetworks in any two subsets [math]A,A'\subseteq\{1,2,\dots,n\}[/math] with [math]|A|=|A'|[/math] and [math]\ex^A=\ex^{A'}[/math] yield the same inequalities for all [math]y^A=y^{A'}[/math]. It is therefore sufficient to consider subnetwork [math]A[/math] and the inequalities in \eqref{eq:OR:networks:1} associated with it. Leveraging this result, proposes an outer region obtained by looking at unlabeled subnetworks of size [math]|A|\le\bar{a}[/math] and given by

[[math]] \begin{align*} \outr{\theta}=\bigcap_{|A|\le\bar{a}}\mathcal{O}^A_\sP[\theta]. \end{align*} [[/math]]

As long as the subnetworks are chosen to be small, e.g., [math]|A|\le 2,3,4[/math], the inequalities in \eqref{eq:OR:networks:1} can be computed even if the network is large. shows that the inequalities in \eqref{eq:OR:networks:1} remain informative as [math]n[/math] grows. This fact highlights the importance of working with subnetworks. One could have applied the insight of directly to the full network by setting [math]\eu[/math] equal to zero and to one in \eqref{eq:model:distrib:network:1}. The resulting bounds, however, would vanish to zero as [math]n[/math] grows and become uninformative for [math]\theta[/math]. The characterization in Theorem can be refined to obtain a smaller region, adapting the results in to subnetworks. The size of this refined region is weakly decreasing in [math]|A|[/math]. However, the refinement does not yield [math]\idr{\theta}[/math] because it is applied only to subnetworks.

Key Insight: At the beginning of this section I highlighted some key challenges to inference in network formation models. Identification Problem bypasses the concern on the dependence among linking decisions through the independence assumption on [math]\epsilon_{ij}[/math] and the presumption that the researcher observes data from multiple independent networks, which allows for identification of [math]\sP(\ey,\ex)[/math]. takes on the remaining challenges by formally establishing equilibrium existence and allowing for unrestricted selection among multiple equilibria. In order to overcome the computational complexity of the problem, she puts forward the important idea of inference based on subnetworks. While of course information is left on the table, the approach remains feasible even with large networks.

considers a framework similar to the one laid out in Identification Problem. He assumes non-negative externalities, and shows that in this case the set of pairwise stable equilibria is a complete lattice with a smallest and a largest equilibrium. He then uses moment functions that are monotone in the pairwise stable network (so that they take their extreme values at the smallest and largest equilibria), to obtain moment conditions that restrict [math]\theta[/math]. Examples of the moment functions used include the proportion of pairs with a link, the proportion of links belonging to traingles, and many more (see ). considers unilateral and bilateral directed network formation games, still under a sampling framework where the researcher observes many independent networks. The equilibrium notion that she uses is pure strategy Nash. She assumes that the payoff that player [math]i[/math] receives from forming link [math]ij[/math] is allowed to depend on the number of additional players forming a link pointing to [math]j[/math], but rules out other spillover effects. Under this assumption and some regularity conditions, shows that the network formation game can be decomposed into local games (i.e., games whose sets of players and strategy profiles are subsets of the network formation game's ones), so that the network formation game is in equilibrium if and only if each local game is in equilibrium. She then obtains a characterization of [math]\idr{\theta}[/math] using elements of random set theory.

Data From a Single Network

When the researcher observes data from a single network, extra care has to be taken to restrict the dependence among linking decisions. This can be done in various ways (see, e.g., ). Here I consider a framework proposed by .

Identification Problem (Network Formation Model with a Single Network)

Let there be a continuum of agents [math]j\in\cI=[0,\mu][/math], with [math]\mu \gt 0[/math] their total measure, who choose whom to link to based on a utility function specified below. Let [math]y:\cI\times\cI\to\{0,1\}[/math] be an adjacency mapping with [math]y_{jk}=1[/math] if nodes [math]j[/math] and [math]k[/math] are linked, and [math]y_{jk}=0[/math] otherwise. Assume that only connections up to distance [math]\bar{d}[/math] affect utility and that preferences are such that agents never choose to form more than a total of [math]\bar{l}[/math] links. To simplify exposition, let [math]\bar{d}=2[/math]. Let each agent [math]j[/math] be endowed with characteristics [math]\ex_j\in\cX[/math], with [math]\cX[/math] a finite set in [math]\R^p[/math], that are observable to the researcher. Additionally, let each agent [math]j[/math] be endowed with [math]\bar{l}\times|\cX|[/math] preference shocks [math]\epsilon_{j\ell}(x)\in\R,\ell=1,\dots,\bar{l},x\in\cX[/math], that are unobservable to the researcher and correspond to the possible direct connections and their characteristics. Suppose that the vector of preference shocks is independent of [math]\ex[/math] and has a distribution known up to parameter vector [math]\gamma\in\Gamma\subset\R^m[/math], denoted [math]\sQ_\gamma[/math]. Let [math]\cI(j)=\{k:y_{jk}=1\}[/math]. Assume that agents with characteristics and preference shocks [math](x,e)[/math] value links according to the utility function

[[math]] \begin{multline} \bu_j(y,x,e)=\sum_{k\in\cI(j)}(f(x_j,x_k)+e_{j\ell(k)}(x_k))\\ +\delta_1\left|\bigcup_{k\in\cI(j)}\cI(k)-\cI(j)-\{j\}\right| +\delta_2\sum_{k\in\cI(j)}\sum_{m\in\cI(j):m \gt k}y_{km}-\infty\one(|\cI(k)| \gt \bar{l})\label{eq:utility:network:2} \end{multline} [[/math]]

Assume that the network [math]\ey[/math] formed by agents with characteristics and shocks [math](\ex,\epsilon)[/math] is pairwise stable. Let [math]\Theta\equiv\Upsilon\times\Delta\times\Gamma[/math], with [math]\Upsilon[/math] the parameter space for [math]\cf\equiv\{f(x,w):x\in\cX,w\in\cX\}[/math]. In the absence of additional information, what can the researcher learn about [math]\theta\equiv[\cf\delta_1\delta_2\gamma][/math]?

Identification Problem enforces dimension reduction through the restrictions on depth and degree (the bounds [math]\bar{d}[/math] and [math]\bar{l}[/math]), so that it is applicable to frameworks with networks that have limited degree distribution (e.g., close friendships network, but not Facebook network). It also requires that individual identities are irrelevant. This substantially reduces the richness of unobserved heterogeneity allowed for and the dimensionality of the space of unobservables. While the latter feature narrows the domain of applicability of the model, it is very beneficial to obtain a tractable characterization of what can be learned about [math]\theta[/math], and yields equilibria that may include isolated nodes, a feature often encountered in networks data.

study Identification Problem focusing on the payoff-relevant local subnetworks that result from the maintained assumptions. These are distinct from the subnetworks used by : whereas looks at subnetworks formed by arbitrary individuals and whose size is chosen by the researcher on the base of computational tractability, look at subnetworks among individuals that are within a certain distance of each other, as determined by the structure of the preferences. On the other hand, analysis does not require that agents have a finite number of types nor bounds the number of links that they may form.

To characterize the local subnetworks relevant for identification analysis in their framework, propose the concepts of network type and preference class. A network type [math]t=(a,v)[/math] describes the local network up to distance [math]\bar{d}[/math] from the reference node. Here [math]a[/math] is a square matrix of size [math]1+\bar{l}\sum_{d=1}^{\bar{d}}(\bar{l}-1)^{d-1}[/math] that describes the local subnetwork that is utility relevant for an agent of type [math]t[/math]. It consists of the reference node, its direct potential neighbors ([math]\bar{l}[/math] elements), its second order neighbors ([math]\bar{l}(\bar{l}-1)[/math] elements), through its [math]\bar{d}[/math]-th order neighbors ([math]\bar{l}(\bar{l}-1)^{\bar{d}-1}[/math] elements). The other component of the type, [math]v[/math], is a vector of length equal to the size of [math]a[/math] that contains the observable characteristics of the reference node and her alters. The bounds [math]\bar{d}[/math] and [math]\bar{l}[/math] enforce dimension reduction by bounding the number of network types. The partial identification approach of depends on this number, rather than on the number of agents. For example, the number of moment inequalities is determined by the number of network types, not by the number of agents. As such, the approach yields its highest dividends for dimension reduction in large networks. Let [math]\cT[/math] denote the collection of network types generated from a preference structure [math]\bu[/math] and set of characteristics [math]\cX[/math]. For given realization [math](x,e)[/math] of the observable characteristics and preference shocks of a reference agent, and for given [math]\vartheta\in\Theta[/math], define the collection of network types for which no agent wants to drop a link by

[[math]] \begin{align*} H_\vartheta(x,e)=\{(a,v)\in\cT:v_1=x\mathrm{and}\bu(a,v,e)\ge \bu(a_{-\ell},v,e)\forall\ell=1,\dots,\bar{l}\}, \end{align*} [[/math]]

where [math]a_{-\ell}[/math] is equal to the local adjacency matrix [math]a[/math] but with the [math]\ell[/math]-th link removed (that is, it sets the [math](1,\ell+1)[/math] and [math](\ell+1,1)[/math] elements of [math]a[/math] equal to zero). Because [math](\ex,\epsilon)[/math] are random vectors, [math]\eH_\vartheta\equiv H_\vartheta(\ex,\epsilon)[/math] is a random closed set as per Definition. This random set takes on a finite number of realizations (equal to the possible subsets of [math]\cT[/math]), so that its distribution is completely determined by the probability with which it takes on each of these realizations. A preference class [math]H\subset\cT[/math] is one of the possible realizations of [math]\eH_\vartheta[/math] for some [math]\vartheta\in\Theta[/math]. The model implied probability that [math]\eH_\vartheta=H[/math] is given by

[[math]] \begin{align} \sM(H|\ex;\vartheta)\equiv\sQ_{\tilde\gamma}(\epsilon:\eH_\vartheta=H|\ex).\label{eq:model:prediction:network:class} \end{align} [[/math]]

Observation of data from one network allows the researcher, under suitable restrictions on the sampling process, to learn the distribution of network types in the data (type shares), denoted [math]\sP(t)[/math]. For example, in a network of best friends with [math]\bar{l}=1[/math] and [math]\bar{d}=2[/math], and [math]\cX=\{x^1,x^2\}[/math] (e.g., a simplified framework with only two possible races), agents are either isolated or in a pair. Network types are pairs for the agents' race and the best friend's race (with second element equal zero if the agent is isolated). Type shares are the fraction of isolated blacks, the fraction of isolated whites, the fraction of blacks with a black best friend, the fraction of whites with a black best friend, and the fraction of whites with a white best friend. The preference classes for a black agent are [math]H^1(b,e)=\{(b,0)\}[/math], [math]H^2(b,e)=\{(b,0),(b,b)\}[/math], [math]H^3(b,e)=\{(b,0),(b,w)\}[/math], [math]H^4(b,e)=\{(b,0),(b,w),(b,b)\}[/math] (and similarly for whites). In each case, being alone is part of the preference class, as there are no links to sever. In the second class the agent has a preference for having a black friend, in the third class for a white friend, and in the last class for a friend of either race. It is easy to see that the model is incomplete, as for a given realization of [math]\epsilon[/math] it makes multiple predictions on the agent's preference type. propose to map the distribution of preference classes into the observed distribution of preference types in the data through the use of allocation parameters, denoted [math]\alpha_H(t)\in[0,1][/math]. These are distinct from but play the same role as a selection mechanism, and they represent a candidate distribution for [math]t[/math] given [math]\eH_\vartheta=H[/math]. The model, augmented with them, implies a probability that an agent is of network type [math]t[/math]:

[[math]] \begin{align} \sM(t;\vartheta,\alpha)=\frac{1}{\mu}\sum_{H\subset\cT}\mu_{v_1(t)}\sM(H|v_1(t);\vartheta)\alpha_H(t),\label{eq:model:prediction:network:2} \end{align} [[/math]]

where [math]\mu_{v_1(t)}[/math] is the measure of reference agents with characteristics equal to the second component of the preference type [math]t[/math], [math]\ex=v_1(t)[/math], and [math]\alpha\equiv\{\alpha_H(t):t\in \cT, H\subset\cT\}[/math].

provide a characterization of an outer region for [math]\theta[/math] based on two key implications of pairwise stability that deliver restrictions on [math]\alpha[/math]. They also show that under some additional assumptions, this characterization yields [math]\idr{\theta}[/math] . Here I focus on their more general result. The first implication that they use is that existing links should not be dropped:

[[math]] \begin{align} t\notin H\Rightarrow\alpha_H(t)=0.\label{eq:networks:2:PS1} \end{align} [[/math]]

The condition in \eqref{eq:networks:2:PS1} is embodied in [math]\bar\alpha\equiv\{\alpha_H(t):t\in H, H\subset\cT\}[/math]. The second implication is that it should not be possible to establish mutually beneficial links among nodes that are far from each other. Let [math]t^\prime[/math] and [math]s^\prime[/math] denote the network types that are generated if one adds a link in networks of types [math]t[/math] and [math]s[/math] among two nodes that are at distance at least [math]2\bar{d}[/math] from each other and each have less than [math]\bar{l}[/math] links. Then the requirement is

[[math]] \begin{align} \left(\sum_{H\subset\cT}\mu_{v_1(t)}\sM(H|v_1(t);\vartheta)\alpha_H(t)\one(t^\prime\in H)\right)\left(\sum_{H\subset\cT}\mu_{v_1(s)}\sM(H|v_1(s);\vartheta)\alpha_H(s)\one(s^\prime\in H)\right)=0\label{eq:networks:2:PS2} \end{align} [[/math]]

In words, if a positive measure of agents of type [math]t[/math] prefer [math]t^\prime[/math] (i.e., [math]\alpha_H(t) \gt 0[/math] for some [math]H[/math] such that [math]t^\prime\in H[/math]), there must be zero measure of type [math]s[/math] individuals who prefer [math]s^\prime[/math], because otherwise the network is unstable. show that the conditions in \eqref{eq:networks:2:PS2} can be embodied in a square matrix [math]q[/math] of size equal to the length of [math]\bar{\alpha}[/math]. The entries of [math]q[/math] are constructed as follows. Let [math]H[/math] and [math]\tilde{H}[/math] be two preference classes with [math]t\in H[/math] and [math]s\in\tilde{H}[/math]. With some abuse of notation, let [math]q_{\alpha_H(t),\alpha_{\tilde{H}}(s)}[/math] denote the element of [math]q[/math] corresponding to the index of the entry in [math]\bar\alpha[/math] equal to [math]\alpha_H(t)[/math] for the row, and to [math]\alpha_{\tilde{H}}(s)[/math] for the column. Then set [math]q_{\alpha_H(t),\alpha_{\tilde{H}}(s)}(\vartheta)=\one(t^\prime\in H)\one(s^\prime\in\tilde{H})[/math]. It follows that this element yields the term [math]\big(\alpha_H(t)\one(t^\prime\in H)\big)\big(\alpha_{\tilde{H}}(s)\one(s^\prime\in \tilde{H})\big)[/math] in the quadratic form [math]\bar{\alpha}^\top q \bar{\alpha}[/math]. As long as [math]\mu_{v_1(\cdot)}[/math] and [math]\sM(\cdot|\ex;\vartheta)[/math] in \eqref{eq:model:prediction:network:class} are strictly positive, this term is equal to zero if and only if condition \eqref{eq:networks:2:PS2} holds for types [math]t[/math] and [math]s[/math]. With this background, Theorem OR- below provides an outer region for [math]\theta[/math]. The proof of this result follows from the arguments laid out above (see ).

Theorem (Outer Region on Parameters of a Network Formation Model with a Single Network)

Under the assumptions of Identification Problem,

[[math]] \begin{align} \outr{\theta}=\left\{\vartheta\in\Theta: \left(\begin{array}{[rl]} \min_{\bar{\alpha}} \bar{\alpha}^\top q \bar{\alpha} & \\ s.t. & \sM(t;\vartheta,\bar{\alpha})=\sP(t) \forall t\in\cT \\ & \sum_{t\in H}\bar\alpha_H(t)=1\forall H\subset \cT \\ & \alpha_H(t)\ge 0\forall t\in H,\forall H\subset \cT \end{array} \right){{=}}0 \right\}.\label{eq:OR:networks:2} \end{align} [[/math]]

The set in \eqref{eq:OR:networks:2} does not equal [math]\idr{\theta}[/math] in all models allowed for in Identification Problem because condition \eqref{eq:networks:2:PS2} does not embody all implications of pairwise stability on non-existing links. While the optimization problem in \eqref{eq:OR:networks:2} is quadratic, it is not necessarily convex because [math]q[/math] may not be positive definite. Nonetheless, the simulations reported by suggest that [math]\outr{\theta}[/math] can be computed rapidly, as least for the examples they considered.

Key Insight: At the beginning of this section I highlighted some key challenges to inference in network formation models. When data is observed from a single network, as in Identification Problem, proposal to base inference on local networks achieves two main benefits. First, it delivers consistently estimable features of the game, namely the probability that an agent belongs to one of a finite collection of network types. Second, it achieves dimension reduction, so that computation of outer regions on [math]\theta[/math] remains feasible even with large networks and allowing for unrestricted selection among multiple equilibria.

Further Theoretical Advances and Empirical Applications

In order to discuss the partial identification approach to learning structural parameters of economic models in some level of detail while keeping this chapter to a manageable length, I have focused on a selection of papers. In this section I briefly mention several other excellent theoretical contributions that could be discussed more closely, as well as several empirical papers that have applied partial identification analysis of structural models to answer a wide array of questions of substantive economic importance.

and propose to embed revealed preference-based inequalities into structural models of both demand and supply in markets where firms face discrete choices of product configuration or of location. Revealed preference arguments are a trademark of the literature on discrete choice analysis. and use these arguments to leverage a subset of the model's implications to obtain easy-to-compute moment inequalities. For example, in the context of entry games such as the ones discussed in Section Static, Simultaneous-Move Finite Games with Multiple Equilibria, they propose to base inference on the implication that a player enters the market if and only if (s)he expects to make non-negative profits. This condition can be exploited even when players have heterogeneous (unobserved to the researcher) information sets, and it implies that the expected profits for entrants should be non-negative. Nonetheless, the condition does not suffice to obtain moment inequalities that include only observed payoff shifters and preference parameters. This is because the expected value of unobserved payoff shifters for entrants is not equal to zero, as the group of entrants is selected. The authors require the availability of valid (monotone) instrumental variables to solve this problem (see Section Treatment Effects with and without Instrumental Variables for uses of instrumental variables and monotone instrumental variables in partial identification analysis of treatment effects). Interesting features of their approach include that the researcher does not need to solve for the set of equilibria, nor to require that the distribution of unobservable payoff shifters is known up to finite dimensional parameter vector. Moreover, the same basic ideas can be applied to single agent models (with or without heterogeneous information sets). A shortcoming of the method is that the set of parameter vectors satisfying the moment inequalities may be wider than the sharp identification region under the maintained assumptions.

The breadth of applications of the approach proposed by and is vast. For example, uses it to model the formation of the hospital networks offered by US health insurers, and and use it to obtain bounds on firm fixed costs as an input to modeling product choices in the movie industry and in the US video game industry, respectively. estimates the effects of Wal-Mart's strategy of creating a high density network of stores. While the close proximity of stores implies cannibalization in sales, Wal-Mart is willing to bear it to achieve density economies, which in turn yield savings in distribution costs. His results suggest that Wal-Mart substantially benefits from high store density. measure the effects of chain economies, business stealing, and heterogeneous firms' comparative advantages in the discount retail industry. estimate a model of strategic voting and quantify the impact it has on election outcomes. As in other models analyzed in this section, the one they study yields multiple predicted outcomes, so that partial identification methods are required to carry out the empirical analysis if one does not assume a specific selection mechanism to resolve the multiplicity. They estimate their model on Japanese general-election data, and uncover a sizable fraction of strategic voters. They also estimate that only a small fraction of voters are misaligned (voting for a candidate other than their most preferred one). studies whether the rapid removal from the market for personal computers of existing central processing units upon creation of new ones through innovation reduces surplus. He finds that a limited group of price-insensitive consumers enjoys the largest share of the welfare gains from innovation. A policy that kept older technologies on the shelf would allow for the benefits from innovation to reach price-sensitive consumers thanks to improved access to mobile computing, but total welfare would not increase because consumer welfare gains would be largely offset by producer losses. analyze hospital referrals for labor and birth episodes in California in 2003, for patients enrolled with six health insurers that use, to a different extent, incentives to referring physicians groups to reduce hospital costs (capitation contracts). The aim is to learn whether enrollees with high-capitation insurers tend to be referred to lower-priced hospitals (ceteris paribus) compared to other patients with same-severity conditions, and whether quality of care was affected. Their model allows for an insurer-specific preference function that is additively separable in the hospital price paid by the insurer (which is allowed to be measured with error), the distance traveled, and plan and severity-specific hospital fixed effects. Importantly, unobserved heterogeneity entering the preference function is not assumed to be drawn from a distribution known up to finite dimensional parameter vector. The results of the empirical analysis indicate that the price paid by insurers to hospitals has an impact on referrals, with higher elasticity to price for insurers whose physicians groups are more highly capitated. study how the information that potential exporters have to predict the profits they will earn when serving a foreign market influences their decisions to export. They propose a model where the researcher specifies and observes a subset of the variables that agents use to form their expectations, but may not observe other variables that affect firms' expectations heterogeneously (across firms and markets, and over time). Because only a subset of the variables entering the firms' information set is observed, partial identification results. They show that, under rational expectations, they can test whether potential exporters know and use specific variables to predict their export profits. They also use their model's estimates to quantify the value of information. studies the implications of the \$85 billion automotive industry bailout in 2009 on the commercial vehicle segment. He finds that had Chrysler and GM been liquidated (or aquired by a major competitor) rather than bailed out, the surviving firms would have experienced a rise in profits high enough to induce them to introduce new products.

A different use of revealed preference arguments appears in the contributions of , , , , , , , and many others. For example, proposes a method to partially identify income-leisure preferences and to evaluate the associated effects of tax policies. He starts from basic revealed-preference analysis performed under the assumption that individuals prefer more income and leisure, and no other restriction. The analysis shows that observing an individual's time allocation under a status quo tax policy yields bounds on his allocation that may or may not be informative, depending on how the person allocates his time under the status quo policy and on the tax schedules. He then explores what more can be learned if one additionally imposes restrictions on the distribution of income-leisure preferences, using the method put forward by . One assumption restricts groups of individuals facing different choice sets to have the same distribution of preferences. The other assumption restricts this distribution to a parametric family. build on and expand 's framework to evaluate the effect of Connecticut's Jobs First welfare reform experiment on women' labor supply and welfare participation decisions.

propose a method to learn features of households' risk preferences in a random utility model that nests expected utility theory plus a range of non-expected utility models. They allow for unobserved heterogeneity in preferences (that may enter the utility function non-separably) and leave completely unspecified their distribution. The authors use revealed preference arguments to infer, for each household, a set of values for its unobserved heterogeneity terms that are consistent with the household's choices in the three lines of insurance coverage. As their core restriction, they assume that each household's preferences are stable across contexts: the household's utility function is the same when facing distinct but closely related choice problems. This allows them to use the inferred set valued data to partially identify features of the distribution of preferences, and to classify households into preference types. They apply their proposed method to analyze data on households' deductible choices across three lines of insurance coverage (home all perils, auto collision, and auto comprehensive). Their results show that between 70 and 80 percent of the households make choices that can be rationalized by a model with linear utility and monotone, quadratic, or even linear probability distortions. These probability distortions substantially overweight small probabilities. By contrast, fewer than 40 percent can be rationalized by a model with concave utility but no probability distortions.

propose a method to carry out demand analysis while allowing for general forms of unobserved heterogeneity. Preferences and linear budget sets are assumed to be statistically independent (conditional on covariates and control functions). show that for continuous demand, average surplus is generally not identified from the distribution of demand for a given price and income, and therefore propose a partial identification approach. They use bounds on income effects to derive bounds on average surplus. They apply the bounds to gasoline demand, using data from the 2001 U.S. National Household Transportation Survey.

Another strand of empirical applications pertains to the analysis of discrete games. use the method they develop, described in Section An Inference Approach Robust to the Presence of Multiple Equilibria, to study market structure in the US airline industry and the role that firm heterogeneity plays in shaping it. Their findings suggest that the competitive effects of each carrier increase in that carrier's airport presence, but also that the competitive effects of large carriers (American, Delta, United) are different from those of low cost ones (Southwest). They also evaluate the effect of a counterfactual policy repealing the Wright Amendment, and find that doing so would see an increase in the number of markets served out of Dallas Love.

proposes a model of static entry that extends the one in Section Static, Simultaneous-Move Finite Games with Multiple Equilibria by allowing individuals to have flexible information structures, where players's payoffs depend on both a common-knowledge unobservable payoff shifter, and a private-information one. His characterization of [math]\idr{\theta}[/math] is based on using an unrestricted selection mechanism, as in and . He applies the model to study the impact of supercenters such as Wal-Mart, that sell both food and groceries, on the profitability of rural grocery stores. He finds that entry by a supercenter outside, but within 20 miles, of a local monopolist's market has a smaller impact on firm profits than entry by a local grocer. Their entrance has a small negative effect on the number of grocery stores in surrounding markets as well as on their profits. The results suggest that location and format-based differentiation partially insulate rural stores from competition with supercenters.

A larger class of information structures is considered in the analysis of static discrete games carried out by . They allow for all information structures consistent with the players knowing their own payoffs and the distribution of opponents' payoffs. As solution concept they adopt the Bayes Correlated Equilibrium recently developed by . Also with this solution concept multiple equilibria are possible. The authors leave completely unspecified the selection mechanism picking the equilibrium played in the regions of multiplicity, so that partial identification attains. use the random sets approach to characterize [math]\idr{\theta}[/math]. They apply the method to estimate a model of entry in the Italian supermarket industry and quantify the effect of large malls on local grocery stores. provide partial identification results (and Bayesian inference methods) for semiparametric dynamic binary choice models without imposing distributional assumptions on the unobserved state variables. They carry out an empirical application using 's model of bus engine replacement. Their results suggest that parametric assumptions about the distribution of the unobserved states can have a considerable effect on the estimates of per-period payoffs, but not a noticeable one on the counterfactual conditional choice probabilities. use the random sets approach to partially identify and estimate dynamic discrete choice models with serially correlated unobservables, under instrumental variables restrictions. They extend two-step dynamic estimation methods to characterize a set of structural parameters that are consistent with the dynamic model, the instrumental variables restrictions, and the data. uses the random sets approach and a network formation model, to learn about Italian firms' incentives for having their executive directors sitting on the board of their competitors.

use the method described in Section Unobserved Heterogeneity in Choice Sets and/or Consideration Sets to partially identify the distribution of risk preferences using data on deductible choices in auto collision insurance. They posit an expected utility theory model and allow for unobserved heterogeneity in households' risk aversion and choice sets, with unrestricted dependence between them. Motivation for why unobserved heterogeneity in choice sets might be an important factor in this empirical framework comes from the earlier analysis of and novel findings that are part of contribution. They show that commonly used models that make strong assumptions about choice sets (e.g., the mixed logit model with each individual's choice set assumed equal to the feasible set, and various models of choice set formation) can be rejected in their data. With regard to risk aversion, their key finding is that their estimated lower bounds are significantly smaller than the point estimates obtained in the related literature. This suggests that the data can be explained by expected utility theory with lower and more homogeneous levels of risk aversion than it had been uncovered before. This provides new evidence on the importance of developing models that differ in their specification of which alternatives agents evaluate (rather than or in addition to models focusing on how they evaluate them), and to data collection efforts that seek to directly measure agents' heterogeneous choice sets .

study the effect of pre-vote deliberation on the decisions of US appellate courts. The question of interest is weather deliberation increases or reduces the probability of an incorrect decision. They use a model where communication equilibrium is the solution concept, and only observed heterogeneity in payoffs is allowed for. In the model, multiple equilibria are again possible, and the authors leave the selection mechanism completely unspecified. They characterize [math]\idr{\theta}[/math] through an optimization problem, and structurally estimate the model on US Courts of Appeal data. compare the probability of making incorrect decisions under the pre-vote deliberation mechanism, to that in a counterfactual environment where no deliberation occurs. The results suggest that there is a range of parameters in [math]\idr{\theta}[/math], for which judges have ex-ante disagreement of imprecise prior information, for which deliberation is beneficial. Otherwise deliberation leads to lower effectiveness for the court.

propose a test for the hypothesis of rational expectations for the case that one observes only the marginal distributions of realizations and subjective beliefs, but not their joint distribution (e.g., when subjective beliefs are observed in one dataset, and realizations in a different one, and the two cannot be matched). They establish that the hypothesis of rational expectations can be expressed as testing that a continuum of moment inequalities is satisfied, and they leverage the results in to provide a simple-to-compute test for this hypothesis. They apply their method to test for and quantify deviations from rational expectations about future earnings, and examine the consequences of such departures in the context of a life-cycle model of consumption.

estimate the demand for health insurance under the Affordable Care Act using data from California. Methodologically, they use a discrete choice model that allows for endogeneity in insurance premiums (which enter as explanatory variables in the model) and dispenses with parametric assumptions about the unobserved components of utility leveraging the availability of instrumental variables, similarly to the framework presented in Section Endogenous Explanatory Variables. The authors provide a characterization of sharp bounds on the effects of changing premium subsidies on coverage choices, consumer surplus, and government spending, as solutions to linear programming problems, rendering their method computationally attractive.

Another important strand of theoretical literature is concerned with partial identification of panel data models. consider a dynamic random effects probit model, and use partial identification analysis to obtain bounds on the model parameters that circumvent the initial conditions problem. considers a fixed effect panel data model where he imposes a conditional quantile restriction on time varying unobserved heterogeneity. Differencing out inequalities resulting from the conditional quantile restriction delivers inequalities that depend only on observable variables and parameters to be estimated, but not on the fixed effects, so that they can be used for estimation. obtain bounds on average and quantile treatment effects in nonparametric and semiparametric nonseparable panel data models. provide partial identification results in linear panel data models when censored outcomes, with unrestricted dependence between censoring and observable and unobservable variables. Their results are derived for two classes of models, one where the unobserved heterogeneity terms satisfy a stationarity restriction, and one where they are nonstationary but satisfy a conditional independence restriction. provides a method to partially identify state dependence in panel data models where individual unobserved heterogeneity needs not be time invariant. study semiparametric multinomial choice panel models with fixed effects where the random utility function is assumed additively separable in unobserved heterogeneity, fixed effects, and a linear covariate index. The key semiparametric assumption is a group stationarity condition on the disturbances which places no restrictions on either the joint distribution of the disturbances across choices or the correlation of disturbances across time. propose a within-group comparison that delivers a collection of conditional moment inequalities that they use to provide point and partial identification results. proposes a related method, where partial identification relies on the observation of individuals whose outcome changes in two consecutive time periods, and leverages shape restrictions to reduce the number of between alternatives comparisons needed to determine the optimal choice.

General references

Molinari, Francesca (2020). "Microeconometrics with Partial Identification". arXiv:2004.11751 [econ.EM].