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Who Matters to Whom? Identifying Peer Effects with Propagation Geometry

Guy Tchuente

Abstract

This paper develops a unifying theory of peer effects that treats the peer aggregator (the social norm mapping peers' actions into a scalar exposure) as the central behavioral primitive. We formulate peer influence as a norm game in which payoffs depend on own action and an exposure index, and we provide equilibrium existence and uniqueness for a broad class of aggregators. Using economically interpretable axioms, we organize commonly used exposure maps into a small taxonomy that nests linear-in-means, CES (peer-preference) norms, and smooth ``attention-to-salient-peers'' aggregators; rank-based quantile norms are treated as a complementary class. Building on this unification, we show that each aggregator induces an operator that governs how exogenous variation propagates through the network. Linear-in-means corresponds to constant transport (adjacency matrix), recovering the classic (friends-of-friends) instrument families. For nonlinear norms, operator becomes state- and preference-dependent and is characterized by the Jacobian of the exposure map evaluated at an exogenous predictor. This perspective yields geometry-induced instrument that exploit heterogeneity in marginal influence and nonredundant paths, and can remain informative when one-step moments or adjacency-power instruments become weak. Monte Carlo evidence and an application to NetHealth illustrate the practical implications across alternative aggregators and outcomes.

Who Matters to Whom? Identifying Peer Effects with Propagation Geometry

Abstract

This paper develops a unifying theory of peer effects that treats the peer aggregator (the social norm mapping peers' actions into a scalar exposure) as the central behavioral primitive. We formulate peer influence as a norm game in which payoffs depend on own action and an exposure index, and we provide equilibrium existence and uniqueness for a broad class of aggregators. Using economically interpretable axioms, we organize commonly used exposure maps into a small taxonomy that nests linear-in-means, CES (peer-preference) norms, and smooth ``attention-to-salient-peers'' aggregators; rank-based quantile norms are treated as a complementary class. Building on this unification, we show that each aggregator induces an operator that governs how exogenous variation propagates through the network. Linear-in-means corresponds to constant transport (adjacency matrix), recovering the classic (friends-of-friends) instrument families. For nonlinear norms, operator becomes state- and preference-dependent and is characterized by the Jacobian of the exposure map evaluated at an exogenous predictor. This perspective yields geometry-induced instrument that exploit heterogeneity in marginal influence and nonredundant paths, and can remain informative when one-step moments or adjacency-power instruments become weak. Monte Carlo evidence and an application to NetHealth illustrate the practical implications across alternative aggregators and outcomes.
Paper Structure (72 sections, 4 theorems, 71 equations, 4 tables)

This paper contains 72 sections, 4 theorems, 71 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Norm games and a unifying class of peer aggregators
  3. Primitives: actions, network, and a peer aggregator
  4. Peer aggregator (norm/exposure).
  5. Payoffs.
  6. Nash equilibrium.
  7. A taxonomy of peer aggregators
  8. Axioms that deliver a "mean-like" class (Kolmogorov--Nagumo).
  9. From "mean-like" to economically interpretable families.
  10. Example of aggregator: CES peer norms (BRUZ)
  11. Domain.
  12. Existence of a Nash equilibrium (general aggregator)
  13. Interpretation.
  14. Uniqueness of the Nash equilibrium under weak interactions (a contraction condition)
  15. Economic meaning.
  16. ...and 57 more sections

Key Result

Theorem 1

Suppose: Then there exists at least one pure-strategy Nash equilibrium $a^\star$ satisfying eq:nash_def.

Theorems & Definitions (10)

  • Theorem 1: Existence of pure-strategy Nash equilibrium
  • proof
  • Theorem 2: Uniqueness by contraction
  • proof
  • Theorem 3: BDF and BRUZ as special cases of influence-geometry IV
  • proof
  • Theorem 4: Nonparametric identification of peer response with fixed exposure curvature
  • proof
  • Remark 1: On $\beta$
  • Remark 2