In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈs/), beats-all, or majority-rule winner[1][2] if more than half of voters would support them in any one-on-one matchup with another candidate. Such a candidate is also called an undefeated, or tournament champion, by analogy with round-robin tournaments. Voting systems where a majority-rule 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 beats-all 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 left-right political spectrum and prefer candidates who are more similar to themselves, a beats-all 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 left-right ideology, which can lead to voting paradoxes,[5][6] but such paradoxes tend to be rare in practice.[7]

Condorcet methods and the closely-related family of tournament solutions are one of four major families of voting systems, alongside point systems (like choose-one), sequential loser methods (like instant-runoff), and rated systems.

## 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; 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.

## Example

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 beats-all 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.

## 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 beats-all 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]

### Other kinds of majority rule

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 top-cycle 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 top-cycle directly or indirectly (by beating a candidate who beats the winner). Most, but not all, Condorcet systems satisfy the top-cycle criterion.

### Responsiveness

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.

### Participation

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 publicly-available election datasets found no examples of participation failures for methods in the ranked pairs-minimax family.[10] This is not the case for other systems, such as instant-runoff voting (see list of pathological elections).

## By method

### List

 Main article: Condorcet method

#### Pass

All tournament solutions (such as ranked pairs) satisfy the Condorcet criterion. Other methods satisfying the criterion are:

See Category:Condorcet methods for more.

#### Fail

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

#### 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. 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 one-on-one 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 Myerson-Weber model.[citation needed]

• Approval voting
• Approval arguably passes majority-Condorcet "on a technicality"—voters can only place candidates into 2 tiers, meaning that if more than half of ballots rank A strictly better than B (more than half of ballots approve A but not B), A will win.
• Measured by voters' unstated preferences, approval's Condorcet-depends on whether voters set "approval threshold" at the beats-all winner (i.e. approve the winner and all candidates better than them).
• Score voting (highest averages)
• STAR voting uses score for the first round and a simple majority for the second.
• Highest median voting rules

### Examples

#### Borda count

 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 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

 Main article: Instant-runoff voting

Instant-runoff voting (IRV) uses an elimination process to simulate the behavior of plurality voting with strategic voters. 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 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. 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.

Instant-runoff voting has a low Condorcet efficiency, and tends to be highly vulnerable to spoiler effects (see list of pathological elections).

#### Bucklin/Median

 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.

• 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

 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 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

 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 majority-Condorcet criterion, because it uses information about whether. For example:

Candidates
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 top-two according to score).

• Black, Duncan (1958). The Theory of Committees and Elections. Cambridge University Press.
• Farquharson, Robin (1969). Theory of Voting. Oxford: Blackwell. ISBN 0-631-12460-8.
• Sen, Amartya Kumar (1970). Collective Choice and Social Welfare. Holden-Day. ISBN 978-0-8162-7765-0.

## References

1. ^ Brandl, Florian; Brandt, Felix; Seedig, Hans Georg (2016). "Consistent Probabilistic Social Choice". Econometrica. 84 (5): 1839–1880. arXiv:1503.00694. doi:10.3982/ECTA13337. ISSN 0012-9682.
2. ^ Sen, Amartya (2020). "Majority decision and Condorcet winners". Social Choice and Welfare. 54 (2/3): 211–217. ISSN 0176-1714.
3. ^ Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN 0036-1399.
4. ^ Black, Duncan (1948). "On the Rationale of Group Decision-making". The Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. JSTOR 1825026. S2CID 153953456.
5. ^ Alós-Ferrer, Carlos; Granić, Đura-Georg (2015-09-01). "Political space representations with approval data". Electoral Studies. 39: 56–71. doi:10.1016/j.electstud.2015.04.003. hdl:1765/111247. The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space.
6. ^ Black, Duncan; Newing, R.A. (2013-03-09). McLean, Iain S. [in Welsh]; McMillan, Alistair; Monroe, Burt L. (eds.). The Theory of Committees and Elections by Duncan Black and Committee Decisions with Complementary Valuation by Duncan Black and R.A. Newing. Springer Science & Business Media. ISBN 9789401148603. For instance, if preferences are distributed spatially, there need only be two or more dimensions to the alternative space for cyclic preferences to be almost inevitable
7. ^ Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
8. ^ Schulze, Markus (2024-03-03). "The Schulze Method of Voting". p. 351. The Condorcet criterion for single-winner elections (section 4.7) is important because, when there is a Condorcet winner b ∈ A, then it is still a Condorcet winner when alternatives a1,...,an ∈ A \ {b} are removed. So an alternative b ∈ A doesn't owe his property of being a Condorcet winner to the presence of some other alternatives. Therefore, when we declare a Condorcet winner b ∈ A elected whenever a Condorcet winner exists, we know that no other alternatives a1,...,an ∈ A \ {b} have changed the result of the election without being elected.
9. ^ Anthony J. McGann (2002). "The Tyranny of the Supermajority: How Majority Rule Protects Minorities" (PDF). Center for the Study of Democracy. Retrieved 2008-06-09. `((cite journal))`: Cite journal requires `|journal=` (help)
10. ^ Mohsin, F., Han, Q., Ruan, S., Chen, P. Y., Rossi, F., & Xia, L. (2023, May). Computational Complexity of Verifying the Group No-show Paradox. In Proceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems (pp. 2877-2879).
11. ^ Felsenthal, Dan; Tideman, Nicolaus (2013). "Varieties of failure of monotonicity and participation under five voting methods". Theory and Decision. 75 (1): 59–77.