Exit options sustain altruistic punishment and decrease the second-order free-riders, but it is not a panacea

Why do humans punish free riders at their own expense? This behavior represents an evolutionary puzzle in human societies. This study explores the role of exit strategies in fostering altruistic punishment within evolutionary game theory. We extend the traditional prisoner’s dilemma model by incorporating exiters, players who opt out for a small payoff while nullifying their opponent’s payoff, and altruistic punishers who cooperate and punish non-cooperators. Our findings indicate that in well-mixed populations, exiters destabilize defection but do not promote altruistic punishment. In social networks, however, exiters enable altruistic punishment via cyclic dominance among altruistic punishers, defectors, and exiters. Notably, this cyclic dominance is sensitive to exit payoffs; adjustments can lead to cyclic dominance of non-punishing cooperators, defectors, and exiters, or to a bi-stable state between these two types of cyclic dominance. These results highlight the nuanced impact of exiters on altruistic punishment, emphasizing the need for careful incentivization of exit behavior. While exiters can support altruistic punishment in networked populations, their effectiveness is not a panacea and is highly sensitive to exit payoffs, indicating limits to the voluntary participation mechanism.