General revision protocols in best response algorithms for potential games

In this paper, we characterize the revision sets in different variants of the best response algorithm that guarantee convergence to pure Nash Equilibria in potential games. We prove that if the revision protocol is separable (to be defined in the paper), then the greedy version as well as smoothed versions of the algorithm converge to pure Nash equilibria. If the revision protocol is not separable, then convergence to Nash Equilibria may fail in both cases. For smoothed best response, we further show convergence to Nash Equilibria with optimal potential when players can only play one by one. Again this may fail as soon as simultaneous play is allowed, unless the number of players is two. We also provide several examples/counter-examples testing the domain of validity of these results.