Equivalence of efficiency notions for ordinal assignment problems

•We study four efficiency notions in ordinal assignment problems.•We show that sd-, dl-, and ul-efficiency are equivalent.•We provide conditions for equivalence of the three notions and ex post efficiency.•Our conditions are sufficient and necessary for the equivalence.

In ordinal (probabilistic) assignment problems, each agent reports his preference rankings over objects and receives a lottery defined over those objects. A common efficiency notion, sd-efficiency, is obtained by extending the preference rankings to preferences over lotteries by means of (first-order) stochastic dominance. Two alternative efficiency notions, which we call dl- and ul-efficiency, are based on downward and upward lexicographic dominance, respectively. We show that sd-, dl-, and ul-efficiency are all equivalent. Noting that the three efficiency notions are a refinement of ex post efficiency-another well-known efficiency notion-we also identify sufficient and necessary conditions on preference profiles under which ex post efficiency is equivalent to the three notions.