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Eigenvalues in microeconomics

Benjamin Golub

TL;DR

The paper surveys spectral methods in microeconomic networks, centering on the $\text{Perron–Frobenius}$ theorem to connect eigenvector centrality and network game outcomes. It derives a closed-form price-of-anarchy bound in a symmetric, quadratic network game and links Nash equilibria to Katz–Bonacich centralities, while also providing a spectral characterization of Pareto efficiency via the matrix $B(x)$. It then demonstrates robust, data-driven intervention strategies that exploit dominant eigen-directions despite noisy measurements, and discusses practical considerations such as eigen-gap requirements. The work highlights how spectral structure informs policy design for social learning, public goods, and market interventions under uncertainty, offering a unifying mathematical framework for current topics in microeconomic theory.

Abstract

Square matrices often arise in microeconomics, particularly in network models addressing applications from opinion dynamics to platform regulation. Spectral theory provides powerful tools for analyzing their properties. We present an accessible overview of several fundamental applications of spectral methods in microeconomics, focusing especially on the Perron-Frobenius Theorem's role and its connection to centrality measures. Applications include social learning, network games, public goods provision, and market intervention under uncertainty. The exposition assumes minimal social science background, using spectral theory as a unifying mathematical thread to introduce interested readers to some exciting current topics in microeconomic theory.

Eigenvalues in microeconomics

TL;DR

The paper surveys spectral methods in microeconomic networks, centering on the theorem to connect eigenvector centrality and network game outcomes. It derives a closed-form price-of-anarchy bound in a symmetric, quadratic network game and links Nash equilibria to Katz–Bonacich centralities, while also providing a spectral characterization of Pareto efficiency via the matrix . It then demonstrates robust, data-driven intervention strategies that exploit dominant eigen-directions despite noisy measurements, and discusses practical considerations such as eigen-gap requirements. The work highlights how spectral structure informs policy design for social learning, public goods, and market interventions under uncertainty, offering a unifying mathematical framework for current topics in microeconomic theory.

Abstract

Square matrices often arise in microeconomics, particularly in network models addressing applications from opinion dynamics to platform regulation. Spectral theory provides powerful tools for analyzing their properties. We present an accessible overview of several fundamental applications of spectral methods in microeconomics, focusing especially on the Perron-Frobenius Theorem's role and its connection to centrality measures. Applications include social learning, network games, public goods provision, and market intervention under uncertainty. The exposition assumes minimal social science background, using spectral theory as a unifying mathematical thread to introduce interested readers to some exciting current topics in microeconomic theory.

Paper Structure

This paper contains 5 sections, 5 theorems, 21 equations, 3 figures.

Table of Contents

  1. The Welfare Theory of Network Games
  2. Toward Open Questions
  3. Essential agents
  4. Spectral Radius in Terms of Cycles
  5. Illustration

Key Result

Theorem 1

Let $n \ge 2$, and let $M$ be an $n\times n$ nonnegative, irreducible matrix. Then:

Figures (3)

  • Figure 1: Nodes sized proportional to (a) degree centrality; (b) Bonacich centrality with $z=\bm{1}$ and $\delta=1/3$; (c) eigenvector centrality, which we see is similar since $\delta M$ has spectral radius not far from $1$.
  • Figure 2: A benefits matrix ${B}({0})$ and its graphical depiction, in which player #4 is essential despite providing smaller benefits than the others.
  • Figure 3: Illustration of true vs. estimated parameters in the example. Blue pixels correspond to negative matrix entries, while the red regions to positive ones. The spectral summand refers to the first term in the spectral decomposition $M=\sum_{\ell=1}^n \lambda_\ell w^\ell (w^\ell)^\mathsf{T}$.

Theorems & Definitions (12)

  • Theorem : Perron--Frobenius
  • Proposition 1
  • Definition 1
  • proof
  • Definition 2
  • Proposition 2
  • proof : Sketch
  • Definition 3
  • Proposition 3
  • proof : Proof sketch
  • ...and 2 more