Analysis of Equilibria and Strategic Interaction in Complex Networks

TL;DR

This paper investigates how the structure of complex networks influences the Nash equilibria in multi-player games with linear best responses, using algebraic graph theory and convex optimization to derive bounds and analyze stability.

Contribution

It introduces new bounds on the smallest eigenvalue of network adjacency matrices based on local structural properties, impacting equilibrium analysis.

Findings

Derived bounds linking network structure to eigenvalues.

Showed how local properties affect equilibrium stability.

Validated results with simulations on social networks.

Abstract

This paper studies $n$-person simultaneous-move games with linear best response function, where individuals interact within a given network structure. This class of games have been used to model various settings, such as, public goods, belief formation, peer effects, and oligopoly. The purpose of this paper is to study the effect of the network structure on Nash equilibrium outcomes of this class of games. Bramoull\'{e} et al. derived conditions for uniqueness and stability of a Nash equilibrium in terms of the smallest eigenvalue of the adjacency matrix representing the network of interactions. Motivated by this result, we study how local structural properties of the network of interactions affect this eigenvalue, influencing game equilibria. In particular, we use algebraic graph theory and convex optimization to derive new bounds on the smallest eigenvalue in terms of the distribution of degrees, cycles, and other relevant substructures. We illustrate our results with numerical simulations involving online social networks.