Paper detail

Integrating upstream and downstream reciprocity stabilizes cooperator-defector coexistence in others-only public goods games

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. We study public goods games without self-return (often called "others-only" PGGs) with benefit b and cost c and analyze evolutionary dynamics for three strategies: unconditional cooperation (ALLC), unconditional defection (ALLD), and an integrated reciprocity strategy combining unconditional forwarding with reputation-based discrimination. We show that integrating upstream and downstream reciprocity can yield a globally asymptotically stable mixed equilibrium of ALLD and integrated reciprocators when 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 ALLC but also alternative conditional strategies from invading. Rather than requiring uniformity, our model reveals one pathway to stable cooperation through strategic diversity. ALLD serves 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.