Condorcet winner criterion

In an election, a candidate is called a majority winner or majority-preferred candidate if more than half of all voters would support them in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the majority-rule principle, because they extend the principle of majority rule to elections with multiple candidates.

In situations where equal or tied ranks are allowed, a candidate who wins a simple or relative majority—more votes for than against, ignoring abstentions—is called a Condorcet , beats-all, or tournament winner (by analogy with round-robin tournaments). However, precise terminology on the topic is inconsistent. Surprisingly, an election may not have a beats-all winner: it is possible to have a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox, and is analogous to the counterintuitive intransitive dice phenomenon known in probability.

However, if voters are arranged on a left-right political spectrum and prefer candidates who are more similar to themselves, a majority-rule winner always exists and is the candidate whose ideology is most representative of the electorate, a result known as the median voter theorem. However, if political candidates differ substantially in ways unrelated to left-right ideology or overall competence, this can lead to voting paradoxes. Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races.

Systems that elect majority winners include Ranked Pairs, Schulze's method, and the Tideman alternative method. Methods that do not include instant-runoff voting (often called ranked-choice in the United States), first preference plurality, and the two-round system. Most rated systems, like score voting and highest median, fail the majority winner criterion intentionally (see tyranny of the majority).

History
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. Because his manuscript Ars Electionis was lost soon after his death, his ideas were overlooked for the next 500 years.

The first revolution in voting theory coincided with the rediscovery of these ideas during the Age of Enlightenment by Nicolas de Caritat, Marquis de Condorcet, a mathematician and political philosopher.

Example
Suppose the government comes across a windfall source of funds. There are three options for what to do with the money. The government can spend it, use it to cut taxes, or use it to pay off the debt. The government holds a vote where it asks citizens which of two options they would prefer, and tabulates the results as follows: In this case, the option of paying off the debt is the beats-all winner, because repaying debt is more popular than the other two options. But, it is worth noting 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.

Majority-rule winners can be determined from rankings by counting the number of voters who rated each candidate higher than another.

Desirable properties
The Condorcet criterion is related to several other voting system criteria.

Stability (no-weak-spoilers)
Condorcet methods are highly resistant to spoiler effects. Intuitively, this is because the only way to dislodge a Condorcet winner is by beating them, implying spoilers can exist only if there is no majority-rule winner.

Participation
One disadvantage of majority-rule methods is they can all theoretically fail the participation criterion in constructed examples. However, studies suggest this is empirically rare for modern majority-rule systems, like ranked pairs. One study surveying 306 publicly-available election datasets found no examples of participation failures for methods in the ranked pairs-minimax family.

Stronger criteria
The top-cycle criterion guarantees an even stronger kind of majority rule. It says that if there is no majority-rule winner, the winner must be in the top cycle, which includes all the candidates who can beat every other candidate, either directly or indirectly. Most, but not all, Condorcet systems satisfy the top-cycle criterion.

Pass
Most sensible tournament solutions satisfy the Condorcet criterion. Other methods satisfying the criterion are:


 * Black
 * Kemeny-Young
 * Minimax
 * Ranked pairs
 * Schulze
 * Total Vote Runoff
 * Tideman's alternative method

See for more.

Fail
The following ordinal voting methods do not satisfy the Condorcet criterion.


 * Instant-runoff voting
 * Coombs' rule
 * Bucklin voting (and the closely related median voting)
 * All kinds of positional voting, including:
 * Borda count
 * Choose-one voting

Rated voting
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.

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 beats-all 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.

Instant-runoff voting
In instant-runoff voting (IRV) voters rank candidates from first to last. The last-place candidate (the one with the fewest first-place votes) is eliminated; the votes are then reassigned to the non-eliminated candidate the voter would have chosen had the candidate not been present.

Instant-runoff 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}:


 * A > B > C: 35
 * C > B > A: 34
 * B > C > A: 31

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. Under IRV, B is ranked first by the fewest voters and is eliminated, and then C wins with the transferred votes from B.

Bucklin/Median
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.


 * 35 voters rate candidate A "excellent", B "fair", and C "poor",
 * 34 voters rate candidate C "excellent", B "fair", and A "poor", and
 * 31 voters rate candidate B "excellent", C "good", and A "poor".

B is preferred to A by 65 votes to 35, and B is preferred to C by 66 to 34. Hence, B is the beats-all 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.

Plurality voting
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 vote-splitting 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.

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 majority-Condorcet criterion. For example: 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 top-two according to score).