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 chain-concave 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 concavity∖chain-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 convex∖chain-convex best reply functions. A final extension to Bayesian games is exhibited.