A Unified Model of Direct and Indirect Reciprocity in Multichannel Games

Reciprocity plays a crucial role in maintaining cooperation in human societies and AI systems. In this paper, we focus on reciprocity within multichannel games and examine how cooperation evolves in this context. We propose a unified framework that allows us to evaluate the reputations of interdependent actions across multiple channels while simultaneously exploring both direct and indirect reciprocity mechanisms. We identify partner and semi-partner strategies under both forms of reciprocity, with the former leading to full cooperation and the latter resulting in partial cooperation. Through equilibrium analysis, we characterize the conditions under which full cooperation and partial cooperation emerge. Moreover, we show that when players can link multiple interactions, they learn to coordinate their behavior across different games to maximize overall cooperation. Our findings provide new insights into the maintenance of cooperation across various reciprocity mechanisms and interaction patterns.