Decelerated invasion and waning moon patterns in public goods games with delayed distribution

TL;DR

This paper investigates how delayed distribution of public goods influences cooperation in spatial games, revealing that intermediate delays promote cooperation while long delays can lead to collapse due to wealth accumulation and waning moon patterns.

Contribution

It introduces the concept of delayed distribution effects in spatial public goods games, showing how timing impacts cooperation dynamics and cluster stability.

Findings

Intermediate delays enhance network reciprocity.

Long delays can cause waning moon patterns that undermine cooperation.

Accumulated wealth can be exploited by defectors when delays are too long.

Abstract

We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a cooperative cluster turns negative, the engulfed defectors can collect the heritage of many generations of cooperators, and by doing so start a waning moon pattern that nullifies the benefits of decelerated invasion. Accidental meeting points of growing cooperative clusters may also act as triggers for the waning moon effect, thus linking the success of cooperators with their propensity to fail in a rather bizarre way. Our results highlight that "investing into the future" is a good idea only if that future is sufficiently near and not likely to be burdened by inflation.