  The multi-population replicator dynamics (RD) can be considered a dynamic
approach to the study of multi-player games, where it was shown to be related
to Cross' learning, as well as of systems of coevolving populations. However,
not all of its equilibria are Nash equilibria (NE) of the underlying game, and
neither convergence to an NE nor convergence in general are guaranteed.
Although interior equilibria are guaranteed to be NE, no interior equilibrium
can be asymptotically stable in the multi-population RD, resulting, e.g., in
cyclic orbits around a single interior NE. We introduce a new notion of
equilibria of RD, called mutation limits, which is based on the inclusion of a
naturally arising, simple form of mutation, but is invariant under the specific
choice of mutation parameters. We prove the existence of such mutation limits
for a large range of games, and consider a subclass of particular interest,
that of attracting mutation limits. Attracting mutation limits are approximated
by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence,
offer an approximate dynamic solution of the underlying game, especially if the
original dynamic has no asymptotically stable equilibria. In this sense,
mutation stabilises the system in certain cases and makes attracting mutation
limits near-attainable. Furthermore, the relevance of attracting mutation
limits as a game theoretic equilibrium concept is emphasised by a similarity of
(mutation-)perturbed RD to the Q-learning algorithm in the context of
multi-agent reinforcement learning. In contrast to the guaranteed existence of
mutation limits, attracting mutation limits do not exist in all games, raising
the question of their characterization.
