Set theory: Difference between revisions
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'''Set theory''' is the investigation of sets, which can be informally considered groups of objects. Within the context of voting theory, sets are often used to discuss voting method criteria (which sets of candidates should be eligible to win or not under particular circumstances, for example) and certain other things, such as the [https://en.wikipedia.org/wiki/Nakamura_number Nakamura number].
Many voting method criteria can be thought of in terms of sets. For example, the [[Unanimity criterion|unanimity criterion]] requires that if between two candidates, all voters prefer the former over the latter, then the latter candidate must not win; this can be interpreted in set theory (when solely looking at the winner(s) of the election) as "the winner set selected by a voting method must always be a subset of the largest "unanimity-compliant" set of candidates such that there is no candidate who is unanimously preferred over one of the candidates in the unanimity-compliant set."
In the context of ranked methods, several sets have been proposed for the purposes of identifying which candidates or groups of candidates are better than others. One of the most notable of these is the [[Smith set]].
== Definitions ==
'''Subset''': One set is a subset of another set if all elements in the first set can be found in the other set.
'''Proper subset''': between two sets, the former set contains only elements from the latter set, but the latter set contains at least one more element than the former set.
== Condorcet ==
Because of [[Condorcet cycle|Condorcet cycles]], there isn't always a single unambiguously best candidate according to the [[Condorcet criterion]]. Because of this, most [[Condorcet methods]] narrow down their selection to a best set of candidates when selecting a winner. The [[Smith set]] is by far the most common one, with some Condorcet methods choosing from more specific subsets of the Smith set, such as the [[Schwartz set]]. Criteria such as the [[Smith criterion]] show which sets each method chooses from.
Some Condorcet methods pass [[IIA]]-like properties related to the sets they choose from, such as [[Independence of Smith-dominated Alternatives]].
== Multi-winner elections ==
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