Editing Pareto efficiency (section)

== Variants ==

=== Weak Pareto efficiency{{anchor|weak}} ===
'''Weak Pareto efficiency''' is a situation that cannot be strictly improved for ''every'' individual.<ref>{{Cite book |doi=10.1007/978-1-4020-9160-5_341 |chapter = Pareto Optimality|title = Encyclopedia of Global Justice |pages=808–809 |year = 2011 |last1 = Mock |first1 = William B. T. | isbn=978-1-4020-9159-9}}</ref>

Formally, a '''strong Pareto improvement''' is defined as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-efficient''' if it has no strong Pareto improvements.

Any strong Pareto improvement is also a weak Pareto improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at {10,&nbsp;0}, and George values at {5,&nbsp;5}. Consider the allocation giving all resources to Alice, where the utility profile is (10,&nbsp;0):
* It is a weak PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements). 
* But it is not a strong PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement){{snd}} its utility profile is (10,&nbsp;5).

A market does not require [[local nonsatiation]] to get to a weak Pareto optimum.<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476.</ref>

=== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}}===
'''Constrained Pareto efficiency''' is a weakening of Pareto optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104 p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type ''A'', they pay price ''p''<sub>1</sub>, but if of type ''B'', they pay price ''p''<sub>2</sub>" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price ''p''") or rules based on observable behavior; "if any person chooses ''x'' at price ''p<sub>x</sub>'', then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

=== 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).

=== Ex-ante Pareto efficiency ===
When the decision process is random, such as in [[fair random assignment]] or [[random social choice]] or [[fractional approval voting]], there is a difference between '''ex-post''' and '''ex-ante Pareto efficiency''':
* Ex-post Pareto efficiency means that any outcome of the random process is Pareto-efficient.
* Ex-ante Pareto efficiency means that the ''lottery'' determined by the process is Pareto-efficient with respect to the ''expected'' utilities. That is: no other lottery gives a higher expected utility to one agent and at least as high expected utility to all agents.
If some lottery ''L'' is ex-ante PE, then it is also ex-post PE. ''Proof'': suppose that one of the ex-post outcomes ''x'' of ''L'' is Pareto-dominated by some other outcome ''y''. Then, by moving some probability mass from ''x'' to ''y'', one attains another lottery ''L{{'}}'' that ex-ante Pareto-dominates ''L''.

The opposite is not true: ex-ante PE is stronger that ex-post PE. For example, suppose there are two objects{{snd}} a car and a house. Alice values the car at 2 and the house at 3; George values the car at 2 and the house at 9. Consider the following two lotteries:
# With probability 1/2, give car to Alice and house to George; otherwise, give car to George and house to Alice. The expected utility is {{nobr|(2/2 + 3/2) {{=}} 2.5}} for Alice and {{nobr|(2/2 + 9/2) {{=}} 5.5}} for George. Both allocations are ex-post PE, since the one who got the car cannot be made better-off without harming the one who got the house.
# With probability 1, give car to Alice, then with probability 1/3 give the house to Alice, otherwise give it to George. The expected utility is {{nobr|(2 + 3/3) {{=}} 3}} for Alice and {{nobr|(9 × 2/3) {{=}} 6}} for George. Again, both allocations are ex-post PE.

While both lotteries are ex-post PE, the lottery&nbsp;1 is not ex-ante PE, since it is Pareto-dominated by lottery&nbsp;2.

Another example involves [[dichotomous preferences]].<ref name=":02">{{Cite journal |last1=Bogomolnaia |first1=Anna |last2=Moulin |first2=Hervé |last3=Stong |first3=Richard |date=2005-06-01 |title=Collective choice under dichotomous preferences |url=http://www.ruf.rice.edu/~econ/papers/2003papers/09bogomolnaia.pdf |journal=Journal of Economic Theory |language=en |volume=122 |issue=2 |pages=165–184 |doi=10.1016/j.jet.2004.05.005 |issn=0022-0531}}</ref> There are 5 possible outcomes {{nobr|(''a'', ''b'', ''c'', ''d'', ''e'')}} and 6 voters. The voters' approval sets are {{nobr|(''ac'', ''ad'', ''ae'', ''bc'', ''bd'', ''be'')}}. All five outcomes are PE, so every lottery is ex-post PE. But the lottery selecting ''c'', ''d'', ''e'' with probability 1/3 each is not ex-ante PE, since it gives an expected utility of 1/3 to each voter, while the lottery selecting ''a'', ''b'' with probability 1/2 each gives an expected utility of 1/2 to each voter.

=== Bayesian Pareto efficiency ===
{{Main|Bayesian efficiency}}
'''Bayesian efficiency''' is an adaptation of Pareto efficiency to settings in which players have incomplete information regarding the types of other players.

=== Ordinal Pareto efficiency ===
{{Main|Ordinal Pareto efficiency}}
'''Ordinal Pareto efficiency''' is an adaptation of Pareto efficiency to settings in which players report only rankings on individual items, and we do not know for sure how they rank entire bundles.