Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies

TL;DR

This paper investigates how different benefit functions influence the evolution of cooperation in spatial public goods games with continuous strategies, revealing optimal intermediate contributions and the effects of benefit function parameters.

Contribution

It introduces a generalized benefit function framework with parameters for steepness and threshold, showing their impact on cooperation levels in structured populations.

Findings

Intermediate benefit parameters maximize collective contributions.

Higher cost-to-benefit ratios lower the optimal threshold.

Results hold for complex sigmoid benefit functions.

Abstract

Cooperation and defection may be considered as two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases the characteristic threshold decreases, while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation.