The effect that exogenous mistakes, made by players choosing their strategies, have on the dynamic stability for the replicator dynamic is analyzed for both asymmetric and symmetric normal form games. Through these perturbed games, the dynamic solution concept of limit asymptotic stability is motivated by insisting that such solutions be asymptotically stable for all sufficiently small perturbations (a robustness property). Limit asymptotic stability is then a refinement of the Nash equilibrium. For asymmetric normal form games, it is shown that a strategy pair is limit asymptotically stable if and only if it is a pure strategy pair that weakly dominates alternative best replies. For symmetric normal form games, all evolutionarily stable strategies (ESS's), whether pure or mixed, are limit asymptotically stable. Here, conditions are established for limit asymptotic stability of completely mixed (i.e. interior) strategies as well as strategies on the boundary. Consistency with solutions found by backwards and/or forwards induction is shown for elementary extensive form games. Limit asymptotically stable sets are introduced that generalize other set-valued solution concepts such as the ``strict equilibrium set'' and the ``ES set'' for asymmetric and symmetric normal form games respectively.
