Pure Strategy Nash Equilibrium Points and the Lefschetz Fixed Point Theorem

A pure strategy Nash equilibrium point existence theorem is established for a class of n-person games with possibly nonacyclic (e.g., disconnected) strategy spaces. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps which extends the well known "Eilenberg-Montgomery fixed point theorem" to nonacyclic spaces. Special cases of the existence theorem are also discussed