Towards effective payoffs in the prisoner's dilemma game on scale-free networks

TL;DR

This paper investigates how normalizing payoffs in the prisoner's dilemma game on scale-free networks affects cooperation, revealing a gradual loss of heterogeneity benefits and a reversal in strategy flow.

Contribution

It introduces a normalization parameter that transitions from accumulated to normalized payoffs, analyzing its impact on cooperation dynamics in scale-free networks.

Findings

Heterogeneity-based cooperation facilitation deteriorates with normalization.

Strategy flow from high-degree to low-degree nodes reverses at full normalization.

Hubs tend to remain occupied by cooperators even with moderate normalized payoffs.

Abstract

We study the transition towards effective payoffs in the prisoner's dilemma game on scale-free networks by introducing a normalization parameter guiding the system from accumulated payoffs to payoffs normalized with the connectivity of each agent. We show that during this transition the heterogeneity-based ability of scale-free networks to facilitate cooperative behavior deteriorates continuously, eventually collapsing with the results obtained on regular graphs. The strategy donations and adaptation probabilities of agents with different connectivities are studied. Results reveal that strategies generally spread from agents with larger towards agents with smaller degree. However, this strategy adoption flow reverses sharply in the fully normalized payoff limit. Surprisingly, cooperators occupy the hubs even if the averaged cooperation level due to partly normalized payoffs is moderate.