Uncovered set: Difference between revisions
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'''Independence of covered alternatives''' says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[uncovered set]]. Independence of covered alternatives implies [[Independence of Smith-dominated Alternatives]] (since independence of covered alternatives implies that one can eliminate everyone outside of the uncovered set without changing the winner, and the uncovered set is a subset of the Smith set, therefore eliminating everyone outside of the Smith set also can't change the winner), which further implies [[Smith criterion|Smith]] and thus [[Condorcet criterion|Condorcet]]. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first.
The uncovered set implies [[Pareto]], because Pareto implies that the Pareto-dominant candidate pairwise beats any candidates the Laredo-inferior candidate beat or tied. This is because all voters rank the Pareto candidate equal to or better than the Pareto-inferior candidate. <ref>{{Cite web|url=https://www.researchgate.net/publication/225729286_Alternate_Definitions_of_the_Uncovered_Set_and_Their_Implications|title=Alternate Definitions of the Uncovered Set and Their Implications|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref>
The Banks set<ref>http://spia.uga.edu/faculty_pages/dougherk/svt_13_multi_dimensions2.pdf "The Banks set (BS) is the set of alternatives resulting from strategic voting in a successive elimination procedure"</ref> (the set of candidates who could win a [[:Category:Sequential comparison Condorcet methods|sequential comparison]] contest for at least one ordering of candidates when voters are strategic), [[Copeland]] set (set of candidates with the highest Copeland score), and Schattschneider set are all subsets of the uncovered set. <ref name="Seising 2009 p. ">{{cite book | last=Seising | first=R. | title=Views on Fuzzy Sets and Systems from Different Perspectives: Philosophy and Logic, Criticisms and Applications | publisher=Springer Berlin Heidelberg | series=Studies in Fuzziness and Soft Computing | year=2009 | isbn=978-3-540-93802-6 | url=https://books.google.com/books?id=yCBqCQAAQBAJ | access-date=2020-03-13 | page=350}}</ref>▼
▲The Banks set<ref>http://spia.uga.edu/faculty_pages/dougherk/svt_13_multi_dimensions2.pdf "The Banks set (BS) is the set of alternatives resulting from strategic voting in a successive elimination procedure"</ref> (the set of candidates who could win a [[:Category:Sequential comparison Condorcet methods|sequential comparison]] contest for at least one ordering of candidates when voters are strategic), [[Copeland]] set (set of candidates with the highest Copeland score), and Schattschneider set are all subsets of the uncovered set. <ref name="Seising 2009 p.
The '''Banks set''' is a subset of the Smith set because when all but one candidate in the Smith set has been eliminated in a sequential comparison election, the remaining Smith candidate is guaranteed to pairwise beat all other remaining candidates, since they are all non-Smith candidates, and thus can't be eliminated from that point onwards, meaning they will be the final remaining candidate and thus win.
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