Anomalous behavior of Replicator dynamics for the Prisoner's Dilemma on diluted lattices

TL;DR

This paper investigates the unique behavior of replicator dynamics in diluted lattices for the Prisoner's Dilemma, revealing how structural formations hinder cooperation and cause system trapping, with insights into the role of percolation thresholds.

Contribution

It uncovers the anomalous behavior of replicator dynamics in diluted lattices and links it to structural formations and percolation thresholds affecting cooperation.

Findings

Replicator rule exhibits non-trend behavior near percolation threshold.

Structures of holes and defectors trap the system in frozen states.

Percolation threshold influences cluster development and cooperation.

Abstract

In diluted lattices, cooperation is often enhanced at specific densities, particularly near the percolation threshold for stochastic updating rules. However, the Replicator rule, despite being probabilistic, does not follow this trend. We find that this anomalous behavior is caused by structures formed by holes and defectors, which prevent some agents from experiencing fluctuations, thereby restricting the free flow of information across the network. As a result, the system becomes trapped in a frozen state, though this can be disrupted by introducing perturbations. Finally, we provide a more quantitative analysis of the relationship between the percolation threshold and cooperation, tracking its development within clusters of varying sizes and demonstrating how the percolation threshold shapes the fundamental structures of the lattice.