Zealots can both promote and undermine cooperation in the generalized optional public goods game

Previous theoretical and empirical studies have shown that zealots-individuals who consistently cooperate and never adapt-generally enhance collective cooperation. However, this conclusion largely relies on the assumption that participation is compulsory. Here, we examine how zealots affect the evolution of cooperation in a generalized optional public goods game, where individuals can cooperate, defect, or opt out. Non-participants receive an external payoff and may also exert external effects on the public good. Using pairwise comparison dynamics in an infinite well-mixed population, we show that zealots do not necessarily promote cooperation. Cooperation is enhanced only at moderate zealot densities when non-participants negatively affect the public good. Otherwise, higher zealot densities or non-negative effects from non-participants lead zealots to undermine cooperation. These findings reveal a dual role of zealots, determined by their density and the external influence of non-participants on the public good.