Sasaki, T., Uchida, S., Okada, I.(岡田勇), Yamamoto, H.(山本仁志), & Nakai, Y. (2026).
Integrating upstream and downstream reciprocity stabilizes cooperator-defector coexistence in others-only public goods games.
他者限定公共財ゲームにおける上流・下流互恵性の統合と協力者・非協力者の共存安定化
Frontiers in Behavioral Economics, 4(January), 1–10.
Introduction: Human cooperation persists among strangers in large, well-mixed populations despite theoretical predictions of difficulties, leaving a fundamental evolutionary puzzle. While upstream (pay it forward: helping others because you were helped) and downstream (rewarding reputation: helping those with good reputations) indirect reciprocity have been independently considered as solutions, their joint dynamics in multiplayer contexts remain unexplored. Methods: We study public goods games (PGGs) without self-return (often called “others-only” PGGs) with benefit b and cost c and analyze evolutionary dynamics for three strategies: unconditional cooperation (X), unconditional defection (Y), and an integrated reciprocal strategy (Z) combining unconditional forwarding with reputation-based discrimination. Results: We show that integrating upstream and downstream reciprocity can yield a globally asymptotically stable mixed equilibrium of unconditional defectors and integrated reciprocators when the benefit-to-cost ratio exceeds a threshold (b/c > 2) in the absence of complexity costs. We analytically derive a critical threshold for complexity costs. If cognitive demands exceed this threshold, the stable equilibrium disappears via a saddle-node bifurcation. Otherwise, within the stable regime, complexity costs counterintuitively stabilize the equilibrium by preventing not only unconditional cooperators (viewed as second-order freeloaders) but also alternative conditional strategies from invading. Discussion: Rather than requiring uniformity, our model reveals one pathway to stable cooperation through strategic diversity, where defectors serve as “evolutionary shields” preventing system collapse, while integrated reciprocators flexibly combine open and discriminative responses. This framework demonstrates how pay-it-forward broadcasting and reputation systems can jointly maintain social polymorphism including cooperation despite cognitive limitations and group size challenges, offering a potential evolutionary foundation for behavioral diversity in human societies.