Random matching in adaptive dynamics

This paper studies the effect of randomness in per-period matching on the long-run outcome of non-equilibrium adaptive processes. If there are many matchings between each strategy revision, the randomness due to matching will be small; our question is when a very small noise due to matching has a negligible effect. We study two different senses of this idea, and provide sufficient conditions for each. The less demanding sense corresponds to sending the matching noise to zero while holding fixed all other aspects of the adaptive process. The second sense in which matching noise can be negligible is that it doesn’t alter the limit distribution obtained as the limit of the invariant distributions as an exogeneous “mutation rate” goes to zero. When applied to a model with mutations, the difference between these two senses is in the order of limits: the first sense asks for continuity of e.g. the ergodic distribution in the matching noise holding the mutation rate fixed, whereas the second sense asks for continuity of the limit distribution in the matching noise.