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Altruistic punishment has been shown to invade when rare if individuals are allowed to opt out of cooperative ventures. Individuals that opt out do not contribute to the common enterprise or derive benefits from it. This result is potentially significant because it offers an explanation for the origin of large-scale cooperation in one-shot interactions among unrelated individuals. Here, we show that this result is not a general consequence of optional participation in cooperative activities, but depends on special assumptions about cooperative pay-offs. We extend the pay-off structure of optional participation models to consider the effects of economies and diseconomies of scale in public-goods production, rival and non-rival consumption of goods, and different orderings of the pay-offs of freeriding and opting out. This more general model highlights the kinds of pay-offs for which optional participation favours cooperation, and those in which it does not.

When does optional participation allow the evolution of cooperation?