Cournot tâtonnement and dominance solvability in finite games

If a finite strategic game is strictly dominance solvable, then every simultaneous best response adjustment path, as well as every non-discriminatory individual best response improvement path, ends at a Nash equilibrium after a finite number of steps. If a game is weakly dominance solvable, then every strategy profile can be connected to a Nash equilibrium with a simultaneous best response path and with an individual best response path (if there are more than two players, switches from one best response to another may be needed). Both statements remain valid if dominance solvability in the usual sense is replaced with “BR-dominance solvability”, where a strategy can be eliminated if it is not among the best responses to anything, or if it is not indispensable for providing the best responses to all contingencies. For a two person game, some implications in the opposite direction are obtained.