Strict Nash Equilibria in Non-Atomic Games with Strict Single Crossing in Players (or Types) and Actions

In this paper we study games where the space of player types is atomless and payoff functions satisfy the property of strict single crossing in types and actions. Under an additional assumption of quasisupermodularity in actions of payoff functions and mild assumptions on the type space - partially ordered and with sets of uncomparable types having negligible size - and on the action space - lattice, second countable and satisfying a separation property with respect to the ordering of actions - we prove that every Nash equilibrium is essentially strict. Further, by building on McAdams (2003, Theorem 1), we prove existence of a strict Nash equilibrium and an evolutionarily stable strategy in a general class of incomplete information games satisfying strict single crossing in types and actions.