Integrating upstream and downstream reciprocity stabilizes cooperator-defector coexistence in N-player giving games

Abstract: Human cooperation persists among strangers despite theoretical predictions of difficulties in large, well-mixed populations, 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 N-player giving games 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. We show that integrating upstream and downstream reciprocity can yield a globally asymptotically stable mixed equilibrium of unconditional defectors and integrated reciprocators whenever the benefit-to-cost ratio exceeds a threshold (b/c > 2). Counterintuitively, introducing small complexity costs, rather than destabilizing, stabilizes the equilibrium by preventing not only unconditional cooperators (viewed as second-order freeloaders) but also alternative conditional strategies from invading. While the equilibrium frequency of integrated reciprocators decreases with group size N, it remains positive for any finite N. Rather than requiring uniformity, our model reveals one pathway to stable cooperation through strategic diversity. Defectors serve as "evolutionary shields" preventing system collapse while integrated reciprocators flexibly combine open and discriminative responses. This framework demonstrates how pay-it-forward chains 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.