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Article Excerpt
Summary. The paper extends the Brock-Durlauf social interactions model to richer social structures modelled with arbitrary interaction topologies and examines in detail the star, the wheel and the path. It explores Nash equilibria when agents act on the basis of expectations over, and, alternatively, actual knowledge their neighbors' decisions. It links social interactions with econometrics of systems of simultaneous equations. The local dynamics near steady states combine spectral properties of the adjacency matrix and of the nonlinearities of reaction functions. For regular interaction topologies, adjustment exhibit relative persistence. Cyclical interaction is associated with endogenous spatial oscillations and islands of conformity.
Keywords and Phrases: Social interactions, Dynamics, Spatial oscillations, Interactive discrete choice, Neighborhood effects, Ising model, Random fields, Random networks.
JEL Classification Numbers: C35, C45, C00, D80.
1 Introduction
The paper examines an economy which is populated by individuals whose discrete decisions are influenced by the decisions of other individuals. Interactions among individuals exhibit spatial structure that is modelled as a graph, with individuals as vertices and interaction between individuals as edges, and interpreted as social structure. The paper explores how patterns in the graph topology may affect the properties of the decisions of agents at equilibrium. E.g., what difference does it make if not all individuals are directly connected? What if all individuals are connected only through a common intermediary (neighbor), in which case the topology of interconnections is a star, or if each individual is connected to only two other neighbors, with the topology being a cycle? What if the topology forms an one-dimensional lattice (path)?
The paper extends the Brock-Durlauf model of interactive discrete choice (Brock and Durlauf, 2001) to richer social structures, which range from a number of stylized topologies, like the ones mentioned above, to arbitrary topologies. It explores the properties of Nash equilibria when agents act on the basis of expectations over their neighbors' decisions. It links social interactions theory with the econometric theory of systems of simultaneous equations modelling discrete decisions. The paper obtains general results for the dynamics of adjustment towards steady states and shows that they combine spectral properties of the adjacency matrix of the underlying graph topology with the properties of individuals' nonlinear reaction functions. Multiplicity of equilibria is possible in both static and dynamic settings. Such multiplicity of equilibria is significant because it may lead to permanent effects of initial conditions. That is, if the economy starts with different individuals making different decisions, such differences across the economy may persist, under certain conditions. In contrast, if the economy starts with individuals making identical decisions, social interactions may push the economy away towards heterogeneous outcomes. When all individuals have the same number of neighbors, the dynamics of adjustment exhibit relative persistence. Cyclical interaction is associated with endogenous and generally transient spatial oscillations that form "islands of conformity": groups of adjacent individuals are more likely to be making similar decisions. The paper also analyzes stochastic dynamics for arbitrary interaction topologies, when agents act with knowledge of their neighbors' actual decisions, which involve networked Markov chains in sample spaces of very high dimensionality.
Some results for arbitrary interaction structures are qualitatively similar to the global interaction case, but a richer class of anisotropic equilibria may arise, for some topologies, like the cycle and one-dimensional lattice, in static settings and provided that the interaction coefficients differ. Equilibria with social interactions when individuals react to the actual behavior of their neighbors in the previous period differ qualitatively from those when individuals use the mean of their neighbors' lagged decisions as forecasts of their current ones.
Current interest in the study of social interactions aims inter alia at a better understanding of conditions under which interdependence is responsible for multiplicity in conformist behavior. Ellison (1993) and Young (1998) obtain results that rest on individual agents' behavior being affected by the behavior of a subset of other agents, rather than of all other agents. Ellison (1993) and Glaeser et al. (1996) exploit the intrinsic symmetry of the cyclical interaction topology in modelling, respectively, local matching and by relying on its tractable analytics when the number of agents is large.
The present paper brings different topologies together under the same overarching model. Its single most important result is that unless individuals located in different positions value differently their interactions with others in the social interaction topology, Nash equilibrium outcomes when individuals make decisions based on the expectation of their neighbors' decisions do not differ from the Brock-Durlauf case in static settings. In dynamic settings and regardless of the rule individuals use to forecast their neighbors' contemporaneous decisions, the dynamics of adjustment and the nature of equilibrium outcomes reflect critically the social interaction topology. Our results extend Brock and Durlauf (2001) in a number of ways and complement contributions by others, including in particular, Horst and Scheinkman (2004), who emphasize continuous decisions and allow for random topologies in static models, and Bisin et al. (2004), who also emphasize continuous decisions with fixed one-sided interactions and allow for dynamics with rational expectations but exclude multiple equilibria. Bala and Goyal (2000), Haag and Lagunoff (2001) and Jackson and Wolinsky (1996) have also addressed bringing together different interaction topologies.
2 Interactive discrete choice
Let the elements of a set I represent individuals. Social interactions among individuals I are defined by an undirected graph G(V, E), where: V is the set of vertices, V = {[v.sub.1], [v.sub.2],..., [v.sub.I]}, an one-to-one map of the set of individuals I onto itself, and I = |V| is the number of vertices (nodes), (known as the order of the graph); E is a subset of the collection of unordered pairs of vertices and q = |E| is the number of edges, (known as the size of the graph). We say that agent i interacts with agent j if there is an edge in G(V, E) between nodes i and j. Let [nu](i) define the local neighborhood of agent i: [nu](i) = {j [member of] I|j [not equal to] i, {i, j} [member of] E}. The number of i's neighbors is the degree of node i: [d.sub.i] = |[nu](i)|. Graph G(V, E) may be represented equivalently by its adjacency (a.k.a. acquaintance or sociomatrix) matrix, [GAMMA], an I x I matrix whose element (i, j) is equal to 1, if there exists an edge from agent i and to j, and is equal to 0, otherwise. For undirected graphs, matrix [GAMMA] is symmetric and its spectral properties are both well understood and are used extensively below. We use [N.sup.-1] to denote the diagonal matrix with the inverse of each agent's degree, 1/|[nu](i)|, as its element (i, i).
2.1 The Brock-Durlauf interactive discrete choice model with an arbitrary interaction topology
This section adapts the Brock-Durlauf model of interactive discrete choice (Brock and Durlauf, 2001; Durlauf, 1997) to arbitrary interaction topologies represented by an arbitrary adjacency matrix [GAMMA]. All individuals faces the binary choice set S = {-1, 1}. Let agent i choose [[omega].sub.i], [[omega].sub.i] [member of] S, so as to maximize her utility, which depends on the actions of her neighbors: [U.sub.i] = U([[omega].sub.i]; [~.[omega].sub.[nu](i)]), where [~.[omega].sub.[nu](i)] denotes the vector of dimension [d.sub.i] containing as elements the decisions made by each of agent i's neighbors, j [member of] [nu](i). The I-vector of all agents' decisions, [~.[omega]] = ([[omega].sub.1],..., [[omega].sub.I]), is also known as a configuration, and [~.[omega].sub.[nu](i)] is known as agent i's environment. We assume that an agent's utility function [U.sub.i] is additively separable in a private utility component, which without loss of generality (due to the binary nature of the decision) may be written as h[[omega].sub.i], h > 0, in a social interactions component, which is written in terms of quadratic interactions between her own decision and of the expectation of the decisions of her neighbors, [~.[omega].sub.[nu](i)], [[omega].sub.i][[epsilon].sub.i]{[1/|[nu](i)|] [[summation].sub.j[member of][nu](i)] [J.sub.ij][[omega].sub.j]}, and a random utility component, [epsilon]([[omega].sub.i]), which is observable only by the individual i. That is, [U.sub.i] may be written as:
[U.sub.i]([[omega].sub.i]; [[epsilon].sub.i] [~.[omega].sub.[nu](i)]}) [equivalent to] h[[omega].sub.i] + [[omega].sub.i][[epsilon].sub.i]{[1/|[nu](i)|] [summation over (j[member of][nu](i))] [J.sub.ij][[omega].sub.j]} + [epsilon]([[omega].sub.i]). (1)
The interaction coefficients may be positive, individuals are conformist, or negative, individuals are non-conformist. We define J as the matrix of interaction coefficients, a I x I matrix with element [J.sub.ij], and I as the I x I identity matrix. Also, let the column I-vector [~.[epsilon]] stack the difference of 2I independently and identically type I extreme-value distributed random variables, [[epsilon].sub.i] = [[epsilon].sub.i](1) - [[epsilon].sub.i](-1), written as a column vector, [~.[epsilon]] [equivalent to] [~.[epsilon]] (1) - [~.[epsilon]] (-1), and let 1[R] is a I-vector indicator function of the I vector R, with its ith element equal to 1, if the ith element of R, [R.sub.i] > 0, and is equal to -1, otherwise.
Following Brock and Durlauf (2001) (1) and with [epsilon]([[omega].sub.i]) being independently and identically type I extreme-value distributed (2) across all alternatives and agents i [member of] I, individual i chooses [[omega].sub.i] = 1 with probability
Prob([[omega].sub.i] = 1) = Prob {2h + 2[[epsilon].sub.i]{[1/|[nu](i)|] [summation over (j[member of][nu](i))] [J.sub.ij][[omega].sub.j]} [greater than or equal to] - ([epsilon](1) - [epsilon](-1))}. (2)
In view of the above assumptions, this may be written in terms of the logistic cumulative distribution function:
Prob([[omega].sub.i] = 1) = [exp [[beta] (2h + 2[[epsilon].sub.i] {[1/|[nu](i)|] [[summation].sub.j[member of][nu](i)] [J.sub.ij][[omega].sub.j]})]]/[1 + exp [[beta] (2h + 2[[epsilon].sub.i] {[1/|[nu](i)|] [[summation].sub.j[member of][nu](i)] [J.sub.ij][[omega].sub.j]})]], (3)
where [beta] > is a behavioral parameter that denotes the degree of precision in one response to the random component of private utility, [epsilon]([[omega].sub.i]) in (1). The case of [beta] = implies purely random choice, the two outcomes are equally likely, and of [beta] [right arrow] [infinity] purely deterministic choice. The extreme-value distribution assumption for the [epsilon]'s is both convenient and links with the machinery of the Gibbs distributions theory (Blume, 1997; Brock and Durlauf, 2001).
The state of the economy satisfies the following condition, written in compact notation as:
[~.[omega]] = 1 [2hI + 2[N.sup.-1]J[GAMMA][epsilon]{[~.[omega]]} + [~.[epsilon]]]. (4)
We assume that all agents are identical in terms of preferences but each agent holds expectations of other agents' decisions which are contingent on those agents' position in the social structure. An equilibrium confirms such expectations:
[[epsilon].sub.i]([[omega].sub.j]) = [m.sub.j], [for all]i, j [member of] I. (5)
By writing m for the vector of expectations of decisions, where [m.sub.i] = Prob([[omega].sub.i] = 1) - Prob([[omega].sub.i] = -1), and using the hyperbolic tangent function, tanh(x) [equivalent to] [exp(x)-exp(-x)]/[exp(x)+exp(-x)], -[infinity] < x < [infinity], we have:
[m.sub.i] = tanh [[beta]h + [beta][1/[nu](i)]J[[GAMMA].sub.i]m], i = 1,..., I, (6)
where [[GAMMA].sub.i] denotes the ith row of the adjacency matrix. Succinctly, we now have:
Proposition 1. Under the assumption of location-contingent expectations (5), the system of social interactions with an arbitrary topology (4) admits an equilibrium that satisfies (6).
Proof. This follows readily from Brower's fixed point theorem. The mapping from [-1, 1][.sup.I] into itself, defined by the RHS of (6), has at least one fixed point. [square]
In the mean field theory case, which is equivalent to global interaction and is considered by Brock and Durlauf (2001), each individual's subjective expectations of other agents' decisions are equal, [[epsilon].sub.i]([[omega].sub.j]) = m, [for all]i, j...
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