  Existing studies have revealed a paradoxical phenomenon in public goods
games, wherein destructive agents, harming both cooperators and defectors, can
unexpectedly bolster cooperation. Building upon this intriguing premise, our
paper introduces a novel concept: constructive agents, which confer additional
benefits to both cooperators and defectors. We investigate the impact of these
agents on cooperation dynamics within the framework of public goods games.
Employing replicator dynamics, we find that unlike destructive agents, the mere
presence of constructive agents does not significantly alter the defective
equilibrium. However, when the benefits from constructive agents are outweighed
by the damage inflicted by destructive agents, the addition of constructive
agents does not affect the ability of destructive agents to sustain
cooperation. In this scenario, cooperators can be maintained through a cyclic
dominance between cooperators, defectors, and destructive agents, with
constructive agents adding complexity but not fundamentally changing the
equilibrium. Conversely, if the benefits from constructive agents surpass the
harm caused by destructive agents, the presence of constructive agents
nullifies the ability of destructive agents to foster cooperation. Our results
highlight the nuanced role of constructive agents in cooperation dynamics,
emphasizing the necessity of carefully assessing incentive balances when
encouraging cooperation.
