Condorcet winner criterion

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In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈs/), beats-all, or majority winner[1][2][3] if a majority of voters would support them in a race against any other candidate. Such a candidate is also called an undefeated or tournament champion (by analogy with round-robin tournaments). Voting systems in which a majority-rule winner will always win the election are said to satisfy the majority-rule principle,[4] also called the Condorcet criterion. Methods that satisfy this criterion extend majority rule to elections with more than one candidate.

Surprisingly, an election may not have a beats-all winner, because there can be a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox.[5]

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. This is the candidate whose ideology is most representative of the electorate. This result is known as the median voter theorem.[6]

While political candidates differ in ways other than left-right ideology, which can lead to voting paradoxes,[7][8] such situations tend to be rare in practice.[9]

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.[10]

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: spend it, use it to cut taxes, or use it to pay off debt. The government holds a vote to decide, in which 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. But, 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.

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.[11]

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.

By method

List

Pass

Most sensible tournament solutions 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.

Examples

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:

Candidates
A B C
45 5/5 1/5 0/5
40 0/5 1/5 5/5
15 2/5 5/5 4/5
Average 2.55 1.6 2.6

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. doi:10.1007/s00355-020-01244-4. ISSN 0176-1714. JSTOR 45286016.
3. ^ Lewyn, Michael (2012), Two Cheers for Instant Runoff Voting (SSRN Scholarly Paper), Rochester, NY, retrieved 2024-04-21`{{citation}}`: CS1 maint: location missing publisher (link)
4. ^ Lepelley, Dominique; Merlin, Vincent (1998). "Choix social positionnel et principe majoritaire". Annales d'Économie et de Statistique (51): 29–48. doi:10.2307/20076136. ISSN 0769-489X.
5. ^ Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN 0036-1399.
6. ^ 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.
7. ^ 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.
8. ^ 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
9. ^ 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.
10. ^ Colomer, Josep M. (February 2013). "Ramon Llull: from 'Ars electionis' to social choice theory". Social Choice and Welfare. doi:10.1007/s00355-011-0598-2.
11. ^ 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).
12. ^ Felsenthal, Dan; Tideman, Nicolaus (2013). "Varieties of failure of monotonicity and participation under five voting methods". Theory and Decision. 75 (1): 59–77. doi:10.1007/s11238-012-9306-7.