Simple exit encouragement does not always enhance cooperation

Voluntary participation is a well-established mechanism for sustaining cooperation, often operating through cyclic dominance among strategies. Here, we extend this framework by studying spatially structured Prisoner’s Dilemma games in which non-participants not only secure their own payoff but also impose externalities on others. Evolutionary simulations on regular lattices reveal that simply encouraging exit does not necessarily enhance cooperation. While appropriate incentives to exit can promote cooperation, their effectiveness is highly context dependent. When exit inflicts substantial harm on others, discouraging exit becomes essential. Conversely, when exit generates strong benefits for others, moderate incentives can paradoxically undermine cooperation by empowering defectors. These findings demonstrate that fostering cooperation through voluntary participation requires carefully tuning both the attractiveness and the social consequences of exit behavior, balancing when to encourage or discourage non-participation depending on the externalities involved.