Folk theorem on equilibrium selection in coordination games

This paper studies equilibrium selection in a 2x2 coordination game. I focus on a class of synchronizable systems with compliant rules. I find that for any given such system, a folk theorem holds if we do not impose any other conditions on introducing rules into the system: we can always add a rule into the system such that any arbitrarily prescribed stage game equilibrium can be supported in the long run. Therefore, it is not necessary that a risk dominant equilibrium or a payoff dominant one will be selected by an evolutionary system in the long run.