Best response dynamics on random graphs

TL;DR

This paper studies how best response dynamics in symmetric 2-player games on random graphs lead to rapid consensus or persistent discord, depending on the graph's connectivity and payoff structure.

Contribution

It characterizes the conditions under which the system converges to unanimity and identifies critical thresholds and persistent substructures in random graph populations.

Findings

Rapid convergence to unanimity for certain edge probabilities p.

Existence of thresholds on p for reaching unanimity with high probability.

Identification of discordant substructures when unanimity is not achieved.

Abstract

We consider evolutionary games on a population whose underlying topology of interactions is determined by a binomial random graph $G(n,p)$. Our focus is on 2-player symmetric games with 2 strategies played between the incident members of such a population. Players update their strategies synchronously. At each round, each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for $p$ in a range that depends on a certain characteristic of the payoff matrix. In the presence of a bias among the pure Nash equilibria of the game, we determine a sharp threshold on $p$ above which the largest connected component reaches unanimity with high probability. For $p$ below this critical value, where this does not happen, we identify those substructures inside the largest component that remain discordant throughout the evolution of the system.