The comparative statics of dominance
Gregorio Curello, Ludvig Sinander, Mark Whitmeyer
TL;DR
The paper characterises robust comparative statics of the not-weakly-dominated and not-strictly-dominated sets in finite objects-situations problems under payoff transformations. It proves a necessary-and-sufficient condition: transformations are situation-wise and either concave with strict/weak monotonicity (for expansion) or convex with strict monotonicity (for contraction), with a degenerate constant case allowed for expansion. This yields a complete classification of transformations that expand or contract NWD/NSD sets and applies the results to Pareto frontiers and strategic dominance in games, including a mirror theorem for dominance by objects. The work connects to Pratt's theorem and the broader comparative-risk-aversion literature, providing a robust framework for understanding how per-situation risk attitudes shape the geometry of undominated sets and informing analyses in allocation and game-theoretic 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.