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Various types of ranked voting ballot

Ranked voting, also known as ranked-choice voting or preferential voting, refers to any voting system in which voters use a ranked (or preferential) ballot to select more than one candidate (or other alternative being voted on) and to rank these choices in a sequence on the ordinal scale of 1st, 2nd, 3rd, etc. Ranked voting is different from cardinal voting, where candidates are independently rated rather than ranked.[1] The most important differences between ranked voting systems lie in the methods used to decide which candidate (or candidates) are elected from a given set of ballots. Some of the most significant methods are described below.

Another (more cosmetic) difference lies in the format of the ballot papers. Some jurisdictions require voters to rank all candidates; some limit the number who may be ranked; and some allow voters to rank as many as they see fit, with the rest being lumped together at the end. Other rules (sometimes entailed by the method of determining a winner) are imposed in different cases.

The subject of this article should not be confused with instant-runoff voting or the single transferable vote, specific forms of ranked voting which has been referred to as 'ranked-choice voting' by the American States of Maine[2] and Alaska,[3] and the American organization FairVote.[4] Ranked voting is used in national or state elections in Australia, Ireland, the United Kingdom (Scottish and Welsh Parliaments),[5] the two aforementioned US states, Malta, Slovenia[6] and Nauru. It is also used for some local elections in New Zealand,[7] Canada and some locations in the United States.

History of ranked voting

The first known discussion of ranked voting is in the writings of the Majorcan Ramon Llull at the end of the 13th century. His meaning is not always clear, but he is understood as having advocated what is now known as Copeland's method (implemented through a sequence of two-way elections rather than ranked choice ballots).

His writings came to the attention of Nicholas of Cusa in the early 15th century. Nicholas seems not to have been much influenced by them, and came up independently with what is now called the Borda count, advocating an implementation through ranked ballots. Llull's and Nicholas's writings were then lost, resurfacing in the twentieth century.

The modern study of the subject began when Jean-Charles de Borda published a paper in 1781 which advocated the method now associated with his name. This method drew criticism from the Marquis de Condorcet, who developed a criterion for recognising a collective preference and observed that Borda's method did not always satisfy it (through an example which remains controversial: see Comparison of electoral systems).[8]

Interest in the topic revived in the nineteenth century when the Dane Carl Andræ invented the STV system, which was immediately adopted in his home country and reinvented by Thomas Hare in the UK in 1857. William Robert Ware proposed STV's single-winner variant IRV around 1870, perhaps unaware that Condorcet had previously mentioned it, but only to condemn it.[9][10] In the following years the English mathematician Charles Lutwidge Dodgson (better known as Lewis Carroll) and the Anglo-Australian Edward Nanson published new voting methods.

Theoretical modelling of electoral processes began with a 1948 paper by Duncan Black,[11] which was quickly followed by Kenneth Arrow's work on the consistency of voting criteria. The topic has received academic attention ever since under the rubric of social choice theory, generally subsumed under economics.

Theoretical properties of ranked voting

The Condorcet criterion

Several of the concepts developed by the Marquis de Condorcet in the eighteenth century still play a central role in the subject.

If there is a candidate who is preferred to every other candidate by the majority of voters in an election, then this candidate is known as the Condorcet winner. A voting method which always elects the Condorcet winner if there is one is said to be Condorcet consistent or (equivalently) to satisfy the Condorcet criterion. Methods with this property are known as Condorcet methods.

If there is no Condorcet winner in an election, then there must be a Condorcet cycle, which can be illustrated by an example. Suppose that there are 3 candidates, A, B and C, and 30 voters such that 10 vote C–B–A, 10 vote B–A–C, and 10 vote A–C–B. Then there is no Condorcet winner. In particular, we see that A cannot be a Condorcet winner because 23 of voters prefer B to A; but B cannot be a Condorcet winner because 23 prefer C to B; and C cannot be a Condorcet winner because 23 prefer A to C. But A cannot be a Condorcet winner... thus the search for a Condorcet winner takes us in circles without ever finding one.

Spatial models

A spatial model is a model of the electoral process developed by Duncan Black and extended by Anthony Downs. Every voter and every candidate is assumed to occupy a location in a space of opinions which may have one or more dimensions, and voters are assumed to prefer the closer of two candidates to the more distant. A political spectrum is a simple spatial model in one dimension.

Ballot Count
A–B–C 36
B–A–C 15
B–C–A 15
C–B–A 34

The diagram shows a simple spatial model in one dimension which will be used to illustrate the voting methods later in this article. A's supporters are assumed to vote A–B–C and C's to vote C–B–A while B's are split equally between having A and C as second preference. If there are 100 voters, then the ballots cast will be determined by voters' and candidates' positions in the spectrum according to the table shown.

Spatial models are important because they are a natural way of visualising voters' opinions and because they lead to an important theorem, the median voter theorem, also due to Black. It asserts that for a wide class of spatial models – including all unidimensional models and all symmetric models in higher dimensions – a Condorcet winner is guaranteed to exist and to be the candidate closest to the median of the voter distribution.

If we apply these ideas to the diagram, we see that there is indeed a Condorcet winner – B – who is preferred to A by 64% and to C by 66%, and that the Condorcet winner is indeed the candidate closest to the median of the voter distribution.

Other theorems

Arrow's impossibility theorem casts a more pessimistic light on ranked voting. While the median voter theorem tells us that for many sets of voter preferences it is easy to devise a voting method which works perfectly, Arrow's theorem says that it is impossible to devise a method which works perfectly in all cases.

Whether Arrow's pessimism or Black's optimism is closer to the truth of electoral behaviour is a matter which needs to be determined empirically. A number of studies, including a paper by Tideman and Plassman,[12] suggest that simple spatial models of the type satisfying the median voter theorem give a close match to observed voter behaviour.

Another pessimistic result, Gibbard's theorem (due to Allan Gibbard), asserts that any voting system must be vulnerable to tactical voting, a topic not further discussed here.

Borda count

Candidate Score
A  87
B 130
C  83

The Borda count assigns a score to each candidate by adding a number of points awarded by each ballot. If there are m candidates, then the first-ranked candidate in a ballot receives m – 1 points, the second receives m – 2, and so on until the last-ranked candidate receives none. In the example B is elected with 130 of the total 300 points.

The Borda count is simple to implement but does not satisfy the Condorcet criterion. It has a particular weakness in that its result can be strongly influenced by the nomination of candidates who do not themselves stand any chance of being elected.

Other positional systems

Voting systems which award points in this way, but possibly using a different formula, are known as positional systems. Where the score vector (m – 1, m – 2,... ,0) corresponds to the Borda count, (1, 1213,... ,1/m ) defines the Dowdall system and (1, 0,... ,0) equates to first-past-the-post.

Alternative vote

Instant-runoff voting

1st round 2nd round 3rd round
36 A–B–C A–C A
15 B–A–C A–C A
15 B–C–A C–A A
34 C–B–A C–A A

Instant-runoff voting (IRV) eliminates candidates in a series of rounds, emulating the effect of separate ballots on shrinking sets of candidates. The first round consists of the ballots as actually cast. Since no one has majority of the vote in the first count, the candidate with fewest first-place preferences is identified (in this case B) and deleted from the ballots for subsequent rounds. His or her votes are transferred as per the next marked preference if any. Thus in the second round the ballots express preferences between just 2 candidates (more generally m – 1). We stop at this point because A is identified as the winner on account of being the preference of the majority of voters.

Elimination systems are relatively clumsy to implement, since each ballot needs to be re-examined on each round, rather than allowing computation from a simple table of derived statistics. IRV does not satisfy the Condorcet criterion. Unlike most ranked voting systems, IRV does not allow tied preferences except sometimes between a voter's least preferred candidates.

Single transferable vote

Single transferable vote (STV) is a multiwinner and proportional version of IRV. Under STV, an elector's vote is initially allocated to his or her most-preferred candidate. After candidates have been either elected (winners) by reaching quota or eliminated (losers), surplus votes are transferred from winners to remaining candidates (hopefuls) according to the voters ordered preferences. It has various methods of determining the transfer of votes.


A 36:64 51:49
B 64:36 66:34
C 49:51 34:66

The minimax system determines a result by constructing a results table in which there is an entry for every pair of distinct candidates showing how often the first is preferred to the second. Thus since 51 voters prefer A to C and 49 have the opposite preference, the (A,C) entry reads '51:49'. In each row we identify the least favourable (i.e. minimal) result for the first candidate (shown in bold), and the winning candidate is the one whose least favourable result is most favourable (i.e. maximal). In the example the winner is B, whose least favourable result is a win while the other candidates' least favourable results are slightly different losses.

Determining the minimax winner from a set of ballots is a particularly simple operation. The method satisfies the Condorcet criterion, and can be seen as electing the Condorcet winner if there is one, and as electing the candidate who comes closest to being a Condorcet winner (under a simple metric) otherwise.

Llull's method / Copeland's method

Candidate Score
A 1
B 2
C 0

Copeland's method assigns each candidate a score derived from the results table as shown above for minimax. The score is simply the number of favourable results in the candidate's row, i.e. the number of other candidates a particular candidate was preferred to by a majority of voters. The candidate with the highest score (in this case B) wins.

Copeland's method is simple and Condorcet-consistent but has the drawback that for certain patterns of voter preferences with no Condorcet winner it will yield a tie however large the electorate. Its advocates therefore generally recommend its use in conjunction with a tie-break. Suitable rules for this purpose include minimax, IRV, and the Borda count, the last of which gives the Dasgupta-Maskin method.

Other methods

  • A Condorcet completion elects the Condorcet winner if there is one, and otherwise falls back on a separate procedure for determining the result. If the Borda count is the fallback we get Black's method; if we use IRV we get Tideman's 'Condorcet-Hare'.[13]
  • Coombs' method is a simple modification of IRV in which the candidate eliminated in each round is the one with most last-place preferences rather than with fewest first-place preferences (so C rather than B is eliminated in the first round of the example and B is the winner). Coombs' method is not Condorcet-consistent but nonetheless satisfies the median voter theorem.[14] It has the drawback that it relies particularly on voters' last-place preferences which may be chosen with less care than their first places.
  • The Kemeny-Young method is complex but Condorcet-consistent.
  • Smith's method reduces the set of candidates to the Smith set, which is a singleton comprising the Condorcet winner if there is one, and is otherwise usually smaller than the original set. It is normally advocated for use in conjunction with a tie-break, with IRV and minimax[15] the commonest. It is computationally simple though not intuitive to most voters.
  • The contingent vote is a 2-round version of IRV, and the supplementary vote is a restricted form of contingent vote.
  • Bucklin's method exists in several forms, some of which are Condorcet-consistent.
  • The ranked pairs method, Schulze method and Split cycle method[16] are Condorcet-consistent methods of medium computational complexity based on analysing the cycle structure of ballots.
  • Dodgson's method is famous chiefly for having been devised by Lewis Carroll. It is Condorcet-consistent but computationally complex.

Comparison of ranked voting methods

The simplest form of comparison is through argument by example. The example in the present article illustrates what many people would consider to be a weakness of IRV; other examples show purported weaknesses in other methods.

Logical voting criteria can be thought of as extrapolations of the salient features of examples into infinite spaces of elections. The consequences are often hard to predict: initially plausible criteria contradict each other and reject otherwise satisfactory voting methods.

Empirical comparisons can be performed using simulated elections. Populations of voters and candidates are constructed under a spatial (or other) model, and the accuracy of each voting method – defined as the frequency with which it elects the candidate closest to the centre of the voter distribution – can be estimated by random trials. Condorcet methods (and Coombs' method) give the best results, followed by the Borda count, with IRV some way behind and first-past-the-post worst of all.

The mathematical properties of a voting method need to be balanced against its pragmatic features, such as its intelligibility to the average voter.

Drawbacks of ranked voting

Ranked voting elicits more information about voter preferences than is revealed through an first-past-the-post ballot, but this comes with certain costs. Voters are confronted with more complicated ballot slips to complete,[15]: §8.1   and the counting procedure – depending on the nature of the voting method – is more complicated and slower, often requiring mechanical support.

See also


  1. ^ Riker, William Harrison (1982). Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice. Waveland Pr. pp. 29–30. ISBN 0881333670. OCLC 316034736. Ordinal utility is a measure of preferences in terms of rank orders—that is, first, second, etc. ... Cardinal utility is a measure of preferences on a scale of cardinal numbers, such as the scale from zero to one or the scale from one to ten.
  2. ^ "Ranked Choice Voting in Maine". Maine State Legislature. Retrieved 21 October 2021.
  3. ^ "Alaska Better Elections Implementation". Alaska Division of Elections. Retrieved 21 October 2021.
  4. ^ "Details about Ranked Choice Voting". FairVote. Retrieved 21 October 2021.
  5. ^ Wiki: Politics of U.K.
  6. ^ Toplak, Jurij (2006). "The parliamentary election in Slovenia, October 2004". Electoral Studies. 25 (4): 825–831. doi:10.1016/j.electstud.2005.12.006.
  7. ^
  8. ^ George G. Szpiro, "Numbers Rule" (2010).
  9. ^ Nanson, E. J. (1882). "Methods of election: Ware's Method". Transactions and Proceedings of the Royal Society of Victoria. 19: 206. The method was, however, mentioned by Condorcet, but only to be condemned.
  10. ^ Condorcet, Jean-Antoine-Nicolas de Caritat (1788). On the Constitution and the Functions of Provincial Assemblies. Complete Works of Condorcet (in French). 13 (published 1804). p. 243. En effet, lorsqu'il y a plus de trois concurrents, le véritable vœu de la pluralité peut être pour un candidat qui n'ait eu aucune des voix dans le premier scrutin.
  11. ^ Duncan Black, "On the Rationale of Group Decision-making" (1948).
  12. ^ T. N. Tideman and F. Plassman, "Modeling the Outcomes of Vote-Casting in Actual Elections" (2012).
  13. ^ J. Green-Armytage, T. N. Tideman and R. Cosman, "Statistical Evaluation of Voting Rules" (2015).
  14. ^ B. Grofman and S. L. Feld, "If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule" (2004).
  15. ^ a b R. B. Darlington, "Are Condorcet and Minimax Voting Systems the Best?" (v8, 2021).
  16. ^ W. H. Holliday and E. Pacuit, "Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers" (2021).
This page was last edited on 18 November 2021, at 17:08
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