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.