Editing Pareto efficiency (section)

=== Fractional Pareto efficiency{{anchor|fractional}} ===
{{Main|Fractional Pareto efficiency}}

'''Fractional Pareto efficiency''' is a strengthening of Pareto efficiency in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-efficient (fPE or fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto efficiency, which only considers domination by feasible (discrete) allocations.<ref name=":0">Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/abs/1707.04731 "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref><ref>{{cite journal|last1=Sandomirskiy|first1=Fedor|last2=Segal-Halevi|first2=Erel|title=Efficient Fair Division with Minimal Sharing|journal=Operations Research |year=2022 |volume=70 |issue=3 |pages=1762–1782 |doi=10.1287/opre.2022.2279 |arxiv=1908.01669|s2cid=247922344 }}</ref>

As an example, consider an item allocation problem with two items, which Alice values at {3,&nbsp;2} and George values at {4,&nbsp;1}. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,&nbsp;1):
* It is Pareto-efficient, since any other discrete allocation (without splitting items) makes someone worse-off.
* However, it is not fractionally Pareto-efficient, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George{{snd}} its utility profile is (3.5,&nbsp;2).