Smith criterion

The Smith criterion (sometimes generalized Condorcet criterion, but this can have other meanings) is a voting systems criterion defined such that it's satisfied when a voting system always elects a candidate that is in the Smith set, which is the smallest non-empty subset of the candidates such that every candidate in the subset is majority-preferred over every candidate not in the subset. (A candidate X is said to be majority-preferred over another candidate Y if, in a one-on-one competition between X & Y, the number of voters who prefer X over Y exceeds the number of voters who prefer Y over X.) The Smith set is named for mathematician John H Smith, whose version of the Condorcet criterion is actually stronger than that defined above for social welfare functions. Benjamin Ward was probably the first to write about this set, which he called the "majority set". The Smith set is also called the top cycle. The term top cycle may be somewhat misleading, however, since the Smith set can contain candidates that do not cycle. For examples, when there is a Condorcet winner it doesn't cycle with any alternatives, and when the Smith set consists only of two alternatives that tie pairwise, the two do not cycle with any alternatives. The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ(n3) or Kosaraju's algorithm in time Θ(n2). When there is a Condorcet winner—a candidate that is majority-preferred over all other candidates—the Smith set consists of only that candidate. Here is an example in which there is no Condorcet winner: There are four candidates: A, B, C and D. 40% of the voters rank D>A>B>C. 35% of the voters rank B>C>A>D. 25% of the voters rank C>A>B>D. The Smith set is {A,B,C}. All three candidates in the Smith set are majority-preferred over D (since 60% rank each of them over D). The Smith set is not {A,B,C,D} because the definition calls for the smallest subset that meets the other conditions. The Smith set is not {B,C} because B is not majority-preferred over A; 65% rank A over B. (Etc.

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Electoral system

An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result.

Mutual majority criterion

The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. The criterion states that if there is a subset S of the candidates, such that more than half of the voters strictly prefer every member of S to every candidate outside of S, this majority voting sincerely, the winner must come from S. This is similar to but stricter than the majority criterion, where the requirement applies only to the case that S contains a single candidate.

Condorcet loser criterion

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion. A voting system complying with the Condorcet loser criterion will never allow a Condorcet loser to win. A Condorcet loser is a candidate who can be defeated in a head-to-head competition against each other candidate.

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