Pure strategy Nash equilibrium points and the Lefschetz fixed point theorem

A pure strategy Nash equilibrium point existence theorem is established for a class ofn-person games with possibly nonacyclic (e.g. disconnected) strategy sets. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps, due to Eilenberg and Montgomery, which extends their better known. Eilenberg-Montgomery fixed point theorem (EMT) Eilenberg/Montgomery, Theorem 1, p. 215] to nonacyclic spaces. Special cases of the existence theorem are also discussed.