Prisoner's Dilemma on Graphs with Large Girth

TL;DR

This paper investigates how the structure of a network, specifically large girth, influences the evolution of cooperation in the prisoner's dilemma game, showing that large girth promotes cooperation.

Contribution

It demonstrates that graphs with large girth facilitate cooperation by analyzing the dynamics as a perturbation of the voter model and examining pairwise correlations.

Findings

Large girth graphs promote cooperation.

Graphs with many small cycles favor defection.

Cooperators tend to cluster in large girth graphs.

Abstract

We study the evolution of cooperation in populations where individuals play prisoner's dilemma on a network. Every node of the network corresponds on an individual choosing whether to cooperate or defect in a repeated game. The players revise their actions by imitating those neighbors who have higher payoffs. We show that when the interactions take place on graphs with large girth, cooperation is more likely to emerge. On the flip side, in graphs with many cycles of length 3 and 4, defection spreads more rapidly. One of the key ideas of our analysis is that our dynamics can be seen as a perturbation of the voter model. We write the transition kernel of the corresponding Markov chain in terms of the pairwise correlations in the voter model. We analyze the pairwise correlation and show that in graphs with relatively large girth, cooperators cluster and help each other.