  A large body of empirical evidence suggests that humans are willing to engage
in costly punishment of defectors in public goods games. Based on such pieces
of evidence, it is suggested that punishment serves an important role in
promoting cooperation in humans, and possibly other species. Nevertheless,
theoretical work has been unable to show how this is possible. The problem
originates from the fact that punishment, being costly, is an altruistic act
and its evolution is subject to the same problem that it tries to address. To
suppress this so-called second-order free-rider problem, known theoretical
models on the evolution of punishment resort to one of the few established
mechanisms for the evolution of cooperation. This leaves the question that
whether altruistic punishment can evolve and give rise to the evolution of
cooperation, unaddressed. Here, by considering a population of individuals who
play a public goods game, followed by a public punishing game, introduced here,
we show that altruistic punishment indeed evolves and promotes cooperation, in
a general environment and in the absence of a cooperation favoring mechanism.
Besides, our analysis shows, being close to a physical phase transition
facilitates the evolution of altruistic punishment.
