  Altruistic punishment, where individuals incur personal costs to punish
others who have harmed third parties, presents an evolutionary conundrum as it
undermines individual fitness. Resolving this puzzle is crucial for
understanding the emergence and maintenance of human cooperation. This study
investigates the role of an alternative strategy, the exit option, in
explaining altruistic punishment. We analyze a two-stage prisoner's dilemma
game in well-mixed and networked populations, considering both finite and
infinite scenarios. Our findings reveal that the exit option does not
significantly enhance altruistic punishment in well-mixed populations. However,
in networked populations, the exit option enables the existence of altruistic
punishment and gives rise to complex dynamics, including cyclic dominance and
bi-stable states. This research contributes to our understanding of costly
punishment and sheds light on the effectiveness of different voluntary
participation strategies in addressing the conundrum of punishment.
