Expulsion has been found to promote cooperation in social dilemmas, but only if it does not incur costs or is applied unilaterally.

Here, we show that removing both conditions leads to a spontaneous resolution of the costly expulsion problem. Namely, by studying the public goods game where cooperators and defectors can expel others at a personal cost, we find that public cooperation thrives as expulsion costs increase. This is counterintuitive, as the cost of other-regarding behaviour typically places an additional burden on cooperation, which is in itself costly. Such scenarios are referred to as second-order free-rider problems, and they typically require an additional mechanism, such as network reciprocity, to be resolved.

We perform a mean field analysis of the public goods game with bilateral costly expulsion, showing analytically that the expected payoff difference between cooperators and defectors increases with expulsion costs as long as players with the same strategy have, on average, a higher frequency to interact with each other. As the latter condition is often satisfied in social networks, our results thus reveal a fascinating new path to public cooperation, and they show that the costs of well-intended actions need not be low for them to be effective.

X. Wang, M. Perc, Bilateral costly expulsions resolve the public goods dilemma, Proceedings of the Royal Society A 477 (2254) (2021)