High thresholds encouraging the evolution of cooperation in threshold public-good games

Kris De Jaegher

doi:10.1038/s41598-020-62626-3

Abstract

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. We investigate the effect of an increase in the threshold on the level of cooperation that evolves. It is shown that for sufficiently large participation costs, the level of cooperation is higher for low and for high thresholds, than it is for intermediate thresholds – where in the latter case cooperation may not evolve at all. The counterintuitive effect where an increase in the threshold from an intermediate to a high one decreases the probability of cooperation, is related to the so-called common-enemy hypothesis of the evolution of cooperation. We further apply our analysis to assess the relative weight of different game types across the parameter space, and show that game types where either a small, or a large fraction of the population evolves as cooperators, receive more weight compared to game types where an intermediate fraction of cooperators evolves.

Highlights

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates
Taking participation costs sufficiently large such that cooperation cannot evolve with an intermediate threshold, lowering the threshold enables the evolution of cooperation, fitting the initial intuition
How often do we expect a combination of polymorphism and bistability, compared to situations with only polymorphism, only bistability, or situations that fit the Prisoner’s Dilemma? Our analysis answers this question by assessing the relative frequency of several game types across the parameter space, where the parameters we vary are the threshold and the participation costs

Summary

Introduction

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. This paper fits into a theoretical literature that explains the evolution of cooperation by considering non-linear as well as linear impact functions[21,22,23,24,25,26,27,28] (where the impact function relates the level of the public good produced to the number of cooperating players in a group); for an overview, see[29] In this way, cooperative games can take the form of an n-person version of the Prisoner’s Dilemma[7,30] (where joint defection is the only stable fixed point), but can be characterized by polymorphism (a stable fixed point exists where cooperators and defectors coexist), bistability (both a cooperative stable fixed point and a joint-defection stable fixed point exists), or a combination of both. Polymorphism without bistability is only allowed for the minimal threshold[31,32], and bistability without polymorphism is only allowed for the maximal threshold[5]

Methods

Results

Conclusion