The geometrical pattern of the evolution of cooperation in the Spatial Prisoner's Dilemma: an intra-group model

TL;DR

This paper investigates the evolution of cooperation in a spatial Prisoner's Dilemma model with intra-group interactions, revealing distinct geometrical patterns and dynamics compared to inter-group models, with implications for understanding cooperative behavior.

Contribution

It introduces an intra-group interaction model for the spatial Prisoner's Dilemma, contrasting it with traditional inter-group models, and analyzes the resulting geometrical and dynamical differences.

Findings

Intra-group model shows similar asymptotic behavior to inter-group but differs in intermediate dynamics.

Oscillations are present in intra-group interactions, preventing data collapse.

Fluctuations are smaller in the intra-group model, affecting cluster geometries.

Abstract

The Prisoner's Dilemma (PD) deals with the cooperation/defection conflict between two agents. The agents are represented by a cell of $L \times L$ square lattice. The agents are initially randomly distributed according to a certain proportion $\rho_c(0)$ of cooperators. Each agent does not have memory of previous behaviors and plays the PD with eight nearest neighbors and then copies the behavior of who had the greatest payoff for next generation. This system shows that, when the conflict is established, cooperation among agents may emerge even for reasonably high defection temptation values. Contrary to previous studies, which treat mean inter-group interaction, here a model where the agents are not allowed to self-interact, representing intra-group interaction, is proposed. This leads to short time and asymptotic behaviors similar to the one found when self-interaction is considered. Nevertheless, the intermediate behavior is different, with no possible data collapse since oscillations are present. Also, the fluctuations are much smaller in the intra-group model. The geometrical configurations of cooperative clusters are distinct and explain the $\rho_c(t)$ differences between inter and intra-group models. The boundary conditions do not affect the results.