As an example, consider an economy with three items and two agents, with the following rankings: Consider the allocation [Alice: x, George: y,z].
An allocation X = (X1,...,Xn) Pareto-dominates another allocation Y = (Y1,...,Yn), if every agent i weakly prefers the bundle Xi to the bundle Yi, and at least one agent j strictly prefers Xj to Yj.
The above definitions depend on the agents' ranking of bundles (sets of items).
The two definitions are logically equivalent: The NecPE condition remains the same whether we allow all additive bundle rankings, or we allow only rankings that are based on additive valuations with diminishing differences.
[clarification needed] The Pareto-possible condition remains the same whether we allow all additive bundle rankings, or we allow only rankings that are based on additive valuations with diminishing differences.
The sd relation has several equivalent definitions; see responsive set extension.
This is similar to PosPE, but emphasizes that the bundle rankings must be based on additive utility functions, and the allocations may be fractional.
As noted above, Pareto-possible implies PosPE, but the other direction is not logically true.
McLennan[7] proves that they are equivalent in the fair random assignment problem (with strict or weak item rankings).
[7] Bogomolnaia and Moulin[4]: Lem.3 prove another useful characterization of sd-efficiency, for the same fair random assignment setting but with strict item rankings.
Define the exchange graph of a given fractional allocation as a directed graph in which the nodes are the items, and there is an arc x→y iff there exists an agent i that prefers x and receives a positive fraction of y.
Define an allocation as acyclic if its exchange graph has no directed cycles.
Fishburn proved the following equivalence on dominance relations of discrete bundles, with responsive bundle rankings:[8][1]: Lem.2.1 Therefore, the following holds for dominance relations of discrete allocations: X >> Y iff X necessarily Pareto-dominates Y.
Cho presents two other efficiency notions for the setting of fair random assignment, based on lexicographic dominance.
Cho and Dogan[6] prove that, in this particular case, dl-efficiency and ul-efficiency are equivalent to sd-efficiency.