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).
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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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