Condorcet winner criterion
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In an election, a candidate is called a Condorcet (English: /kɒndɔːrˈseɪ/), beats-all, or majority-rule winner[1][2][3] when a majority of voters support them against any other 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 majority-rule principle, also known as the Condorcet criterion. Condorcet voting methods 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, where multiple candidates all defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox.[4] 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 majority-rule 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.[5] While political candidates differ in ways other than left-right ideology, which can lead to voting paradoxes,[6][7] such cases tend to be rare in practice.[8]
History[edit]
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, leaving his ideas unnoticed for the next 500 years.[9]
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.
Example[edit]
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.
Majority-rule winners can be determined from rankings by counting the number of voters who rated each candidate higher than another.
Desirable properties[edit]
The Condorcet criterion is related to several other voting system criteria.
Stability (no-weak-spoilers)[edit]
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 majority-rule winner. This property, known as stability for majority-rule winners, is a major advantage of such methods.[10]
Participation[edit]
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[edit]
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.
Further reading[edit]
- 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.
See also[edit]
- Condorcet loser criterion
- Condorcet method
- Multiwinner voting - contains information on some multiwinner variants of the Condorcet criterion.
References[edit]
- ^ 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.
- ^ 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.
- ^ 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) - ^ Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN 0036-1399.
- ^ 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.
- ^ 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.
- ^ 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
- ^ 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.
- ^ 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.
- ^ Schulze, Markus (2024-03-03). "The Schulze Method of Voting". p. 351. arXiv:1804.02973 [cs.GT].
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.
- ^ 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).