Bifurcation and multistability in a collective-risk public goods game with quorum-activated protection

Li, Shulan and Li, Xiaogang and Shi, Lei and He, Jun and Szolnoki, Attila (2026) Bifurcation and multistability in a collective-risk public goods game with quorum-activated protection. CHAOS SOLITONS & FRACTALS, 213. ISSN 0960-0779

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Abstract

Collective-failure risk can promote cooperation when cooperative behavior reduces the probability of a shared loss. Existing studies usually embed risk mitigation in the contribution itself, whereas many real systems rely on a separate costly and non-excludable protective activity that requires a participation quorum. Whether such protection can evolve and persist remains unclear. Here, we formulate an N -player public-goods game with ordinary cooperators, defectors, and protective cooperators. Defectors endogenously increase a saturating probability of collective failure, whereas protective cooperators incur an additional cost and reduce this risk for all group members once their number reaches the quorum. Replicator analysis and numerical bifurcation calculations yield three main findings. First, protective cooperation gains an advantage when the focal participant completes the quorum and the resulting expected loss reduction exceeds its additional cost. This advantage varies non-monotonically with group composition. Second, when a single protective cooperator cannot activate protection, protection effectiveness does not enter its first-order invasion fitness at full defection; greater effectiveness cannot overcome the rare-invasion barrier when full defection is locally asymptotically stable. Third, a boundary saddle–node bifurcation creates a saddle and a locally asymptotically stable coexistence equilibrium of defectors and protective cooperators. When full cooperation and full defection are also locally asymptotically stable, three attractors coexist, and the numerically observed evolutionary outcome depends on the initial population composition. These results distinguish the effectiveness of activated protection from its ability to become established from rarity.

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