Schedulers, potentials and weak potentials in weakly acyclic games

In a number of large, important families of finite games, not only do pure-strategy Nash equilibria always exist but they are also reachable from any initial strategy profile by some sequence of myopic single-player moves to a better or best-response strategy. This weak acyclicity property is shared, for example, by all perfect-information extensive-form games, which are generally not acyclic since even sequences of best-improvement steps may cycle. Weak acyclicity is equivalent to the existence of weak potential, which unlike a potential increases along some rather than every sequence as above, as well as to the existence of an acyclic scheduler, which guarantees convergence to equilibrium by disallowing certain (improvement) moves. A number of sufficient conditions for acyclicity and weak acyclicity are known.