The comparative statics of dominance

TL;DR

This paper analyzes how the set of undominated objects in finite problems changes under different payoff transformations, providing a characterization of transformations that expand these sets.

Contribution

It offers a main theorem characterizing payoff transformations that robustly expand undominated sets in finite problems, applicable to Pareto frontiers and games.

Findings

Transformations that expand undominated sets are monotone-concave or constant across situations.

Necessary and sufficient conditions for payoff transformations to expand dominance sets.

Application of results to Pareto frontiers and game theory contexts.

Abstract

In finite problems comprising objects, situations, and an object- and situation-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 whatever the situation. 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 situations, in either a monotone-concave or a constant manner. We apply our results to Pareto frontiers and games.