Condorcet winner criterion

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In an election, a candidate is called a majority winner or majority-preferred candidate [1] [2] [3] 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, [4] [5] because they extend the principle of majority rule to elections with multiple candidates.

Contents

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 (English: /kɒndɔːrˈseɪ/ ), [2] 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, [6] 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. [7] However, if political candidates differ substantially in ways unrelated to left-right ideology or overall competence, this can lead to voting paradoxes. [8] [9] Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races. [10]

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. [11]

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:

... vs. Spend more... vs. Cut taxes
Pay debt403–305496–212 2–0 Yes check.svg
Cut taxes522–186 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 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. [12]

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
Votes
ABC
455/51/50/5
400/51/55/5
152/55/54/5
Average2.551.62.6

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).

Further reading

See also

Related Research Articles

Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.

In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing candidate affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

The Copeland or Llull method is a ranked-choice voting system based on counting each candidate's pairwise wins and losses.

Bucklin voting is a class of voting methods that can be used for single-member and multi-member districts. As in highest median rules like the majority judgment, the Bucklin winner will be one of the candidates with the highest median ranking or rating. It is named after its original promoter, the Georgist politician James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system.

Ranked Pairs (RP) is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.

In political science and social choice theory, Black'smedian voter theorem states that if voters and candidates are distributed along a one-dimensional spectrum and voters have single peaked preferences, any voting method satisfying the Condorcet criterion will elect the candidate preferred by the median voter.

The majority favorite criterion is a voting system criterion that says that, if a candidate would win more than half the vote in a first-preference plurality election, that candidate should win. Equivalently, if only one candidate is ranked first by a over 50% of voters, that candidate must win. It is occasionally referred to simply as the "majority criterion", but this term is more often used to refer to Condorcet's majority-rule principle.

A voting system satisfies join-consistency if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion, which only requires join-consistency when one of the sets of votes unanimously prefers A over B.

The mutual majority criterion is a criterion for evaluating electoral system. It requires that whenever a majority of voters prefer a group of candidates above all others, someone from that group must win. It is the single-winner case of Droop-Proportionality for Solid Coalitions.

The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.

Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.

Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause this higher-ranked candidate to lose.

In social choice theory, the independence of clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a very weak form of the independence of irrelevant alternatives (IIA) criterion.

The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets n − 1 points, where n is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.

Instant-runoff voting (IRV), also known as ranked-choice voting or the alternative vote (AV), combines ranked voting together with a system for choosing winners from these rankings by repeatedly eliminating the candidate with the fewest first-place votes and reassigning their votes until only one candidate is left. It can be seen as a modified form of a runoff election or exhaustive ballot in which, after eliminating some candidates, the choice among the rest is made from already-given voter rankings rather than from a separate election. Many sources conflate this system of choosing winners with ranked-choice voting more generally, for which several other systems of choosing winners have also been used.

<span class="mw-page-title-main">Ranked voting</span>

Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner. For example, Dowdall's method assigns 1, 1⁄2, 1⁄3... points to the 1st, 2nd, 3rd... candidates on each ballot, then elects the candidate with the most points. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives each one very different properties.

A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.

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. 1 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. ISSN   0036-1399.
  6. ↑ Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN   0036-1399.
  7. ↑ 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.
  8. ↑ 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.
  9. ↑ 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
  10. ↑ 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.
  11. ↑ 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.
  12. ↑ 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).
  13. ↑ 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.
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