Direct reciprocity in multi-action repeated games

Direct reciprocity is a fundamental mechanism for sustaining cooperation in repeated interactions, where individuals adjust their behavior based on past experiences. Most previous models have focused on the prisoner's dilemma, in which individuals face a strict choice between full cooperation and complete defection. However, this dichotomy oversimplifies the complexity of real-world reciprocal interactions. To address this, we introduce additional actions between these extremes, thereby increasing action diversity. Our analysis demonstrates that a broader range of available actions fosters cooperation more effectively than a binary choice. Through evolutionary analysis, we identify which types of intermediate actions promote cooperation. Moreover, equilibrium analysis establishes the theoretical conditions underlying this effect. While the increased computational complexity makes it infeasible to simulate scenarios with an arbitrarily large number of actions, our theoretical analysis remains applicable to settings with more actions, offering broader insights into the role of action diversity in promoting cooperation. These findings deepen our understanding of direct reciprocity and highlight the importance of action diversity in shaping cooperative behavior.