Long-run equilibria with dominated strategies

We show that the predictions of long-run behavior under the evolutionary process are highly sensitive to the addition and elimination of strictly dominated strategies. In particular, for the best response dynamics [cf. Kandori, M., Rob, R., 1995. Evolution of equilibria in the long run: A general theory and applications. J. Econ. Theory 65 (2), 383-414] we prove that for any symmetric normal form game, any strict Nash equilibrium can be selected as the unique long-run equilibrium by appropriately adding only one single strategy which is strictly dominated by all original strategies. Moreover, if we further assume instantaneous adjustment, then any convex combination of strict Nash equilibria with rational number weights can be realized as the long-run distribution by appropriately adding strictly dominated strategies.