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In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈseɪ/), beatsall, or majorityrule winner^{[1]}^{[2]} if more than half of voters would support them in any oneonone matchup with another candidate. Such a candidate is also called an undefeated, or tournament champion, by analogy with roundrobin tournaments. Voting systems where a majorityrule winner will always win the election are said to satisfy the Condorcet criterion. Condorcet voting methods extend majority rule to elections with more than one candidate.
Surprisingly, an election might not have a beatsall winner, because there can be a rock, paper, scissors cycle with multiple candidates each defeating the other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox.^{[3]} When there is no single best candidate, tournament solutions (like ranked pairs) choose the candidate closest to being an majority winner.
If voters are arranged on a leftright political spectrum and prefer candidates who are more similar to themselves, a beatsall winner always exists, and is also the candidate whose ideology is most representative of the electorate; this result is known as the median voter theorem.^{[4]} Real political candidates differ in ways other than leftright ideology, which can lead to voting paradoxes,^{[5]}^{[6]} but such paradoxes tend to be rare in practice.^{[7]}
Condorcet methods and the closelyrelated family of tournament solutions are one of four major families of voting systems, alongside point systems (like chooseone), sequential loser methods (like instantrunoff), and rated systems.
Condorcet methods were first studied in detail by the Spanish philosopher and theologian Ramon Llull in the 13th century, during his investigations into church governance; however, his manuscript Ars Electionis was lost soon after his death, leading his ideas to go unnoticed for the next 500 years.
The first revolution in voting theory coincided with the rediscovery of these ideas during the Age of Enlightenment by political philosopher and mathematician Nicolas de Caritat, Marquis de Condorcet. Inspired by ideals of the American Revolution, de Condorcet devoted his life to studying the mathematical properties of republican government.
Suppose the government comes across a windfall source of funds. There are three options for what to do with the money—spend the money, use it to cut taxes, or use it to pay off the debt. The government holds a vote to decide, where voters say which candidate they prefer for each pair of options, and tabulates the results as follows:
... vs. Spend more  ... vs. Cut taxes  

Pay debt  403–305  496–212  2–0 
Cut taxes  522–186  1–1  
Spend more  0–2 
In this case, the option of paying off the debt is the beatsall winner, because repaying debt is more popular than the other two options. However, it is worth nothing that such a winner will not always exist. In this case, tournament solutions search for the candidate who is closest to being an undefeated champion.
Condorcet winners can be determined from rankings by counting the number of voters who rated each candidate higher than another.
The Condorcet criterion is related to several other voting system criteria.
Condorcet methods are highly resistant to spoiler effects. Intuitively, this is because the only way to dislodge a beatsall champion is by beating them, implying spoilers can only exist if there is no Condorcet winner. This property, known as stability for Condorcet winners, is a major advantage of such methods.^{[8]}
The Condorcet criterion implies the majority criterion, which says that if one candidate is preferred to every other put together, they must be declared the winner of the election. In other words, if more than half of all voters agree on a single best candidate, that candidate will win.
The topcycle criterion guarantees an even stronger kind of majority rule. It says that if there is no Condorcet winner, the winner must be in the Smith set, which includes all the candidates who can defeat the topcycle directly or indirectly (by beating a candidate who beats the winner). Most, but not all, Condorcet systems satisfy the topcycle criterion.
Rae argued and Taylor proved in 1969 that majority rule maximizes the likelihood that the laws a voter supports will pass.^{[9]} Thus, Condorcet methods tend to maximize the probability that a person's vote will matter.
One disadvantage of Condorcet methods is they can all theoretically fail the participation criterion in constructed examples. However, studies suggest this is empirically rare for modern Condorcet systems, like ranked pairs.
One study surveying 306 publiclyavailable election datasets found no examples of participation failures for methods in the ranked pairsminimax family.^{[10]} This is not the case for other systems, such as instantrunoff voting (see list of pathological elections).
Main article: Condorcet method 
All tournament solutions (such as ranked pairs) satisfy the Condorcet criterion. Other methods satisfying the criterion are:
See Category:Condorcet methods for more.
The following ordinal voting methods do not satisfy the Condorcet criterion.
The applicability of the Condorcet criterion to rated voting methods is unclear. Under the traditional definition of the Condorcet criterion—that if most votes prefer A to B, then A should defeat B (unless this causes a contradiction)—these methods fail Condorcet, because they give voters with stronger preferences a greater say on the outcome of the election. However, advocates argue this behavior is desirable because it allows these methods to avoid a tyranny of the majority.
Some election scientists have proposed a scored version of the Condorcet criterion, which says that if A would defeat every other candidate in a oneonone race, A should win the combined election. In this case, most rated voting methods would pass (by satisfying independence of spoilers).^{[citation needed]}
Score voting tends to have high (but not 100%) Condorcet efficiency, especially with strategic voting under the MyersonWeber model.^{[citation needed]}
Main article: Borda count 
Borda count is a voting system in which voters rank the candidates in an order of preference. Points are given for the position of a candidate in a voter's rank order. The candidate with the most points wins.
The Borda count does not comply with the Condorcet criterion in the following case. Consider an election consisting of five voters and three alternatives, in which three voters prefer A to B and B to C, while two of the voters prefer B to C and C to A. The fact that A is preferred by three of the five voters to all other alternatives makes it a beatsall champion. However the Borda count awards 2 points for 1st choice, 1 point for second and 0 points for third. Thus, from three voters who prefer A, A receives 6 points (3 × 2), and 0 points from the other two voters, for a total of 6 points. B receives 3 points (3 × 1) from the three voters who prefer A to B to C, and 4 points (2 × 2) from the other two voters who prefer B to C to A. With 7 points, B is the Borda winner.
Main article: Instantrunoff voting 
Instantrunoff voting (IRV) uses an elimination process to simulate the behavior of plurality voting with strategic voters. Voters rank candidates from first to last. The lastplace candidate (the one with the fewest firstplace votes) is eliminated; the votes are then reassigned to the candidate the voter would have chosen had the candidate not been present.
Instantrunoff does not comply with the Condorcet criterion, i.e. it does not elect candidates with majority support. For example, the following vote count of preferences with three candidates {A, B, C}:
In this case, B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34, so B is preferred to both A and C. B must then win according to the Condorcet criterion. Using the rules of IRV, B is ranked first by the fewest voters and is eliminated, and then C wins with the transferred votes from B.
Instantrunoff voting has a low Condorcet efficiency, and tends to be highly vulnerable to spoiler effects (see list of pathological elections).
Main articles: Highest median voting rules and Bucklin voting 
Highest medians is a system in which the voter gives all candidates a rating out of a predetermined set (e.g. {"excellent", "good", "fair", "poor"}). The winner of the election would be the candidate with the best median rating. Consider an election with three candidates A, B, C.
B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34. Hence, B is the beatsall champion. But B only gets the median rating "fair", while C has the median rating "good"; as a result, C is chosen as the winner by highest medians.
Main article: Plurality voting system 
Plurality voting is a ranked voting system where voters rank candidates from first to last, and the best candidate gets one point (while later preferences are ignored). Plurality fails the Condorcet criterion because of votesplitting effects. An example would be the 2000 election in Florida, where most voters preferred Al Gore to George Bush, but Bush won as a result of spoiler candidate Ralph Nader.
Main article: Score voting 
Score voting is a system in which the voter gives all candidates a score on a predetermined scale (e.g. from 0 to 5). The winner of the election is the candidate with the highest total score. Score voting fails the majorityCondorcet criterion, because it uses information about whether. For example:
Candidates Votes

A  B  C 

45  5/5  1/5  0/5 
40  0/5  1/5  5/5 
15  3/5  4/5  5/5 
Average  2.7  1.45  2.75 
Here, C is declared winner, even though a majority of voters would prefer B; this is because the supporters of C are much more enthusiastic about their favorite candidate than the supporters of B. The same example also shows that adding a runoff does not always cause score to comply with the criterion (as the Condorcet winner B is not in the toptwo according to score).