The comparative statics of dominance

In finite problems comprising objects, cases, and an object- and case-contingent payoff function, we study the comparative statics of the set of undominated objects, meaning those for which there exists no mixture over objects that is superior in every case. We consider both weak and strict dominance (corresponding to different degrees of ‘strictness’ in the definition of superiority). Our main theorem characterises those payoff transformations which robustly expand the not-weakly-dominated and not-strictly-dominated sets: the necessary and sufficient condition is that payoffs be transformed separately across cases, in either a monotone–concave or a constant manner. We apply our results to Pareto frontiers and games.