Nash equilibrium uniqueness in nice games with isotone best replies

We prove the existence of a unique pure-strategy Nash equilibrium in nice games with isotone chainconcave best reply functions and compact strategy sets. We show a preliminary fixpoint uniqueness argument which provides sufficient assumptions on the best replies of a nice game for the existence of exactly one Nash equilibrium. Then we examine the necessity and sufficiency of the conditions on the utility functions for such assumptions to be satisfied; in particular, we find necessary and sufficient conditions for the isotonicity and concavitychain-concavity of best reply functions.We extend the results on Nash equilibrium uniqueness to nice games with upper unbounded strategy sets and we present ‘‘dual’’ results for games with isotone convexchain-convex best reply functions. A final extension to Bayesian games is exhibited.