Inertia in spatial public goods games under weak selection

TL;DR

This paper investigates how behavioral inertia affects cooperation in spatial public goods games under weak selection, revealing that inertia generally hinders cooperation and linking different updating protocols through inertia limits.

Contribution

It introduces a spatial public goods game model incorporating inertia during imitation and analytically derives the critical synergy factor for cooperation.

Findings

Inertia hampers cooperation by slowing coarsening processes.

Critical synergy conditions are linked across updating protocols via inertia limits.

Monte Carlo simulations confirm analytical results.

Abstract

Due to limited cognitive skills for perceptual error or other emotional reasons, players may keep their current strategies even if there is a more promising choice. Such behavior inertia has already been studied, but its consequences remained unexplored in the weak selection limit. To fill this gap, we consider a spatial public goods game model where inertia is considered during the imitation process. By using the identity-by-descent method, we present analytical forms of the critical synergy factor $r^\star$, which determines when cooperation is favored. We find that inertia hinders cooperation, which can be explained by the decelerated coarsening process under weak selection. Interestingly, the critical synergy conditions for different updating protocols, including death-birth and birth-death rules, can be formally linked by the extreme limits of the inertia factor. To explore the robustness of our observations, calculations are made for different lattices and group sizes. Monte Carlo simulations also confirm the results.