Adaptive punishment in public goods games

Jiliang Zhang, Yinzuo Zhou, Jin Wang, Yi-Cheng Zhang and Fanyuan Meng

Chaos, Solitons & Fractals, 2026, vol. 210, issue P1

Abstract: The Public Goods Game (PGG) serves as the foundational paradigm for modeling collective action, yet existing evolutionary models overwhelmingly treat punishment as a static parameter or rely on discrete algorithmic updates. This structural rigidity severs the continuous macroscopic feedback loops inherent to real-world adaptive systems. In this work, we elevate adaptive punishment in PGGs to a rigorous nonlinear dynamical framework by formulating the collective punitive intensity (β) as a continuously coevolving macroscopic state variable within a coupled replicator system. Driven by the environmental synergy factor (R) and modulated by redistributive coupling (η) and self-regulating elasticity (θ), this adaptive mechanism continuously reshapes the effective payoff landscape. Our exact global bifurcation analysis reveals that highly efficient adaptive punishment acts as a non-conservative restoring force. Crucially, this mechanism successfully sustains a stable interior focus of cooperation—even in severely resource-deprived environments (synergy factor R<1) where static PGG models inevitably collapse to a global boundary attractor of absolute defection. Furthermore, we mathematically demonstrate that the elasticity of adaptive punishment strictly governs the system’s topological phase transitions, giving rise to non-dissipative neutral centers, macroscopic heteroclinic cycles, and a Bogdanov–Takens codimension-2 bifurcation hub. Ultimately, these findings provide a purely analytical perspective on how continuous adaptive punishment stabilizes metastable order and drives complex oscillatory dynamics in coevolutionary PGGs, advancing the theoretical taxonomy beyond discrete empirical approximations.

Keywords: Evolutionary game theory; Public goods games; Adaptive punishment; Environmental feedback; Hopf bifurcation (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p1:s0960077926007812

DOI: 10.1016/j.chaos.2026.118640

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