Uncovered set: Difference between revisions
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{{Wikipedia|Landau set}}
The '''minimal''' '''uncovered set''' (sometimes referred to as the "'''[[Lev Landau|Landau]] set'''" or "'''[[Peter Fishburn|Fishburn]] set'''") is defined for a set of [[preferential voting|rank-order]] preferences
A Landau candidate will beat every non-Landau candidate one-on-one, and cannot be replaced by a "strictly better" candidate. "Strictly better" means a candidate that would win every pairwise matchup won by the Landau candidate (and some additional matchups).
== Definition ==
We assume here that there are no pairwise-ties. Let some set be called the Fishburn set, and the candidates outside the set are called the '''Fishburn losers'''. A Fishburn loser is a candidate who is '''dominated''' or '''covered''' by some other candidate: the dominating candidate wins every pairwise matchup that the other candidate would win. The uncovered set is therefore equivalent to the set of '''Fishburn winners''':
{{definition|Select the candidate or candidates that are not Fishburn losers. A candidate ''i'' is a Fishburn loser if there is some other candidate ''j'' such that every candidate that pairwise beats ''j'' also pairwise beats ''i'' and there is at least one candidate that pairwise beats ''i'' but does not pairwise beat ''j''.}}▼
{{definition|Select the candidate or candidates that are not Fishburn losers. A candidate ''loser'' is a Fishburn loser if there exists some other candidate ''cover'' satisfying:
# Every candidate that beats ''cover'' one-on-one also beats ''loser'' one-on-one, and
# At least one candidate beats ''loser'' one-on-one but does not beat ''cover'' one-on-one.
}}
The Fishburn winners are a kind of Pareto frontier for the set of candidates, where the frontier is measured by the pairwise-victories. It is impossible to gain some extra pairwise victories, but no pairwise losses, by switching from a candidate in the Landau set to a candidate outside the Landau set.
The Landau set is a nonempty subset of the [[Smith set]]. It was discovered by Nicholas Miller.
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{{definition|An alternative a is said to cover alternative b whenever every alternative dominated by b is also dominated by a.}}
Yet another definition:<ref name="Laffond Laslier 1991 pp. 365–369">{{cite journal | last=Laffond | first=Gilbert | last2=Laslier | first2=Jean-François | title=Slaters's winners of a tournament may not be in the Banks set | journal=Social Choice and Welfare | publisher=Springer | volume=8 | issue=4 | year=1991 | issn=01761714 | jstor=41105997 | pages=365–369 | url=http://www.jstor.org/stable/41105997 | access-date=2022-09-11}}</ref> {{definition|The ''uncovered set'' is the set of all outcomes ''x'' such that there is no outcome beating ''x'' and all the outcomes that ''x'' beats.}}
▲{{definition|Select the candidate or candidates that are not Fishburn losers. A candidate ''i'' is a Fishburn loser if there is some other candidate ''j'' such that every candidate that pairwise beats ''j'' also pairwise beats ''i'' and there is at least one candidate that pairwise beats ''i'' but does not pairwise beat ''j''.}}
When there are pairwise ties, many generalizations are possible, all of which are equivalent when there are no pairwise ties.<ref name="Miller">{{cite web | last=Miller | first=Nicholas M. |title=Alternate definitions of the covering relation: an extended tour |url=https://userpages.umbc.edu/~nmiller/RESEARCH/COVERING.REV3.pdf}}</ref> One generalization by Fishburn is:
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In a game where two players choose candidates and then the player who chose the candidate who beats the other candidate pairwise wins, there's a randomized strategy (a [[Nash equilibrium]]) where no other strategy can be used against it to consistently win at this game. The '''essential set''', a subset of the Dutta set, is the set of all candidates who are chosen some of the time when using a Nash equilibrium strategy.<ref name="Brandt Fischer 2008" />
=== Minimal extending set ===
{{Expand section|date=April 2024}}
The minimal extending set is a subset of the Banks set. It's relevant to strategic voting: narrowing the set of winners to this set when there is no Condorcet winner has not been shown to introduce an incentive to strategically create a cycle when a sincere [[Condorcet winner]] exists.<ref name="Botan Endriss 2021 pp. 5202–5210">{{cite journal | last=Botan | first=Sirin | last2=Endriss | first2=Ulle | title=Preserving Condorcet Winners under Strategic Manipulation | journal=Proceedings of the AAAI Conference on Artificial Intelligence | publisher=Association for the Advancement of Artificial Intelligence (AAAI) | volume=35 | issue=6 | date=2021-05-18 | issn=2374-3468 | doi=10.1609/aaai.v35i6.16657 | pages=5202–5210}}</ref>
A method electing from this set must fail [[monotonicity]].<ref name="Brandt Harrenstein Seedig 2017 pp. 55–63">{{cite journal | last=Brandt | first=Felix | last2=Harrenstein | first2=Paul | last3=Seedig | first3=Hans Georg | title=Minimal extending sets in tournaments | journal=Mathematical Social Sciences | publisher=Elsevier BV | volume=87 | year=2017 | issn=0165-4896 | doi=10.1016/j.mathsocsci.2016.12.007 | pages=55–63}}</ref> However, the proof is nonconstructive and no concrete nonmonotonicity examples have been found so far.
=== Schattschneider set ===
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