Indirect reciprocity beyond pairwise interactions

Abstract

Cooperation in groups underpins collective responses to challenges from climate governance to public goods provision, yet how moral evaluation sustains it remains poorly understood. Indirect reciprocity -- cooperating to build a good reputation -- is well characterized for pairwise interactions, but real collective action requires individuals to be judged against the reputational profile of an entire group. Here we develop a general framework for multiplayer indirect reciprocity and show that stable group cooperation obeys a simple organizing principle: `all good, help; one bad, halt'. This rule is both necessary and sufficient for cooperation to emerge, and it recovers the classical leading eight norms in the pairwise limit. We further show that group structure fundamentally changes reputation dynamics: unlike pairwise models, which are monostable, multiplayer systems exhibit bistability and hysteresis, with a critical tipping point separating cooperative and defective regimes. Assessment of the latent norms of large language models reveals that they shift toward punitive defection when provided with richer social information, yet fail to follow the full logic of `all good, help; one bad, halt'. Our results establish a unifying principle for reputation-based cooperation in groups and provide a benchmark for evaluating cooperative alignment in artificial intelligence.