  Direct reciprocity facilitates the evolution of cooperation when individuals
interact repeatedly. Most previous studies on direct reciprocity implicitly
assume compulsory interactions. Yet, interactions are often voluntary in human
societies. Here, we consider repeated optional games, where individuals can
freely opt out of each interaction and rejoin later. We find that voluntary
participation greatly promotes cooperation in repeated interactions, even in
harsh situations where repeated compulsory games and one-shot optional games
yield low cooperation rates. Moreover, we theoretically characterize all Nash
equilibria that support cooperation among reactive strategies, and identify
three novel classes of strategies that are error-robust, readily become
equilibria, and dominate in the evolutionary dynamics. The success of these
strategies hinges on the effect of opt-out: it not only avoids trapping in
mutual defection but also poses additional threats to intentional defectors.
Our work highlights that voluntary participation is a simple and effective
mechanism to enhance cooperation in repeated interactions.
