  We show that under some general conditions the finite memory determinacy of a
class of two-player win/lose games played on finite graphs implies the
existence of a Nash equilibrium built from finite memory strategies for the
corresponding class of multi-player multi-outcome games. This generalizes a
previous result by Brihaye, De Pril and Schewe. For most of our conditions we
provide counterexamples showing that they cannot be dispensed with.
  Our proofs are generally constructive, that is, provide upper bounds for the
memory required, as well as algorithms to compute the relevant winning
strategies.
