A Phase Transition in Large Network Games

TL;DR

This paper investigates how the Nash equilibrium in large network games undergoes a phase transition influenced by the strength of network perturbations, revealing a critical threshold beyond which societal behavior changes.

Contribution

The study introduces a phase transition framework for understanding how perturbations affect Nash equilibria in large random network games, linking matrix theory with game dynamics.

Findings

NE remains unaffected below the critical perturbation threshold

NE is impacted when perturbation exceeds the critical point

Numerical examples illustrate the phase transition phenomenon

Abstract

In this paper, we use a model of large random network game where the agents plays selfishly and are affected by their neighbors, to explore the conditions under which the Nash equilibrium (NE) of the game is affected by a perturbation in the network. We use a phase transition phenomenon observed in finite rank deformations of large random matrices, to study how the NE changes on crossing critical threshold points. Our main contribution is as follows: when the perturbation strength is greater than a critical point, it impacts the NE of the game, whereas when this perturbation is below this critical point, the NE remains independent of the perturbation parameter. This demonstrates a phase transition in NE which alludes that perturbations can affect the behavior of the society only if their strength is above a critical threshold. We provide numerical examples for this result and present scenarios under which this phenomenon could potentially occur in real world applications.