# Glossary

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Voting theory contains many unique terms and symbols. Symbols from set theory, mathematics, and more are used very frequently throughout.

Some articles with further glossary information are Pairwise counting#Terminology and binary relations theory.

## Definitions

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• Minimal pairwise dominant set: Also known as the Smith set, it is the smallest dominating set, which is any group of candidates who beat all candidates not in the group. The pairwise champion will always be the only member of this set when they exist.
• Note that the terms dominating/dominant are often used as shorter versions of pairwise-dominant.
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• pairwise — evaluating two candidates at a time. The following terms are often used when discussing pairwise preferences
• Pairwise matchup: Also known as a head-to-head matchup, it is when voters are asked to indicate their preference between two candidates or winner sets, with the one that voters prefer (i.e. give more votes to) winning. It is usually done on the basis of majority rule (i.e. if more voters prefer one candidate over the other than the number of voters who have the opposing preference, then the candidate preferred by more voters wins the matchup) using choose-one voting, though see the Strength of preference section for alternative ways. Pairwise matchups can be simulated from ranked or rated ballots and then assembled into a table to show all of the matchups simultaneously.
• Pairwise win/beat and pairwise lose/defeated: When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate (is "pairwise preferred" to the latter candidate), and the latter candidate "pairwise loses." Often this is represented by writing "Pairwise winner>Pairwise loser"; this can be extended to show a beatpath by showing, for example, "A>B>C>D", which means A pairwise beats B, B pairwise beats C, and C pairwise beats D (though it may or may not be the case, depending on the context, that, for example, A pairwise beats C).
• Pairwise winner and pairwise loser: The candidate who pairwise wins a matchup is the pairwise winner of the matchup (not to be confused with the pairwise champion; see the definition two spots below). The other candidate is the pairwise loser of the matchup. (Note that sometimes "pairwise loser" is also used to refer to a Condorcet loser, which is a candidate who is pairwise defeated in all of their matchups).
• Pairwise tie: Occurs when two candidates receive the same number of votes in their pairwise matchup. (Note that sometimes it is also called a tie when there is pairwise cycling, though this is different; see the definition two spots below.) Note that some cycles can be symmetrical ties i.e. you can swap the candidates' names without changing the result. (See the Condorcet paradox article for an example, and the neutrality criterion and tie for more information).
• Pairwise champion: Also known as a beats-all winner or Condorcet winner, it is a candidate who pairwise beats every other candidate. Due to pairwise ties (see above) and pairwise cycling (see below), there is not always a pairwise champion.
• Pairwise cycling: Also known as a Condorcet cycle, it is when within a set of candidates, each candidate has at least one pairwise defeat (when looking only at the matchups between the candidates in the set).
• Pairwise order/ranking: Also known as a Condorcet ranking, it is a ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the Condorcet paradox. As a related concept, there is always a Smith ranking that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.
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