The SE allocation satisfies SD-efficiency - a weak ordinal variant of Pareto-efficiency (it means that the allocation is Pareto-efficient for at least one vector of additive utility functions consistent with the agents' item rankings).
If all agents are given the same eating speed, then the SE allocation satisfies SD-envy-freeness - a strong ordinal variant of envy-freeness (it means that the allocation is envy-free for all vectors of additive utility functions consistent with the agents' item rankings).
[1] SE was developed by Hervé Moulin and Anna Bogomolnaia as a solution for the fair random assignment problem, where the fraction that each agent receives of each item is interpreted as a probability.
With equal eating speeds, the lottery is envy-free in expectation (ex-ante) for all vectors of utility functions consistent with the agents' item rankings.
The type of lottery depends on the problem: An important parameter to SE is the eating speed of each agent.
If it is required to give exactly 1 item per agent, then the matrix of probabilities is decomposed into the following two assignment matrices:1 0 0 0 ||| 0 0 1 0 0 0 1 0 ||| 1 0 0 0 0 1 0 0 ||| 0 0 0 1
A fractional assignment is SD-efficient if-and-only-if it is the outcome of SE for some vector of eating-speed functions.
Formally, for every two agents i and j: Note that sd-envy-freeness is guaranteed ex-ante: it is fair only before the lottery takes place.
The algorithm is of course not ex-post fair: after the lottery takes place, the unlucky agents may envy the lucky ones.
As explained above, the allocation determined by PS is fair only ex-ante but not ex-post.
Recently, several algorithms have been suggested, that guarantee both ex-ante fairness and ex-post approximate-fairness.