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

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.