  Cooperators that refuse to participate in sanctioning defectors create the
second-order free-rider problem. Such cooperators will not be punished because
they contribute to the public good, but they also eschew the costs associated
with punishing defectors. Altruistic punishers - those that cooperate and
punish - are at a disadvantage, and it is puzzling how such behaviour has
evolved. We show that sharing the responsibility to sanction defectors rather
than relying on certain individuals to do so permanently can solve the problem
of costly punishment. Inspired by the fact that humans have strong but also
emotional tendencies for fair play, we consider probabilistic sanctioning as
the simplest way of distributing the duty. In well-mixed populations the public
goods game is transformed into a coordination game with full cooperation and
defection as the two stable equilibria, while in structured populations pattern
formation supports additional counterintuitive solutions that are reminiscent
of Parrondo's paradox.
