Ranked voting

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Various types of ranked voting ballot

Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner or multiple winners. More formally, a ranked rule is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.

For example, in the Borda method, the 1st, 2nd, 3rd... candidates on each ballot receive $1, 2, 3\dots$ points, and the candidate with the smallest number of points is elected. In instant-runoff (ranked-choice) voting, lower preferences are used as contingencies (back-up preferences), and are only applied when all higher-ranked preferences on a ballot have been eliminated.

Ranked voting systems are usually contrasted with rated voting methods, which allow voters to indicate how strongly they support different candidates (e.g. on a scale from 0-10).^[1] Ranked vote systems (ordinal systems) produce more information than X voting systems such as first-past-the-post voting. Rated voting systems use more information than ordinal ballots; as a result, they are not subject to many of the problems with ranked voting (including results like Arrow's theorem).^[2]^[3]^[4]

Although not typically described as such, the most common ranked voting system is the well-known plurality rule, where each voter gives a single point to the candidate ranked first and zero points to all others. The most common non-degenerate ranked voting rule is the closely-related instant-runoff (ranked-choice) voting, a staged variant of the plurality system that repeatedly eliminates last-place plurality winners.^[5]

In the United States and Australia, the terms ranked-choice voting and preferential voting are usually used to refer to the alternative or single transferable vote, a misnomer arising by way of conflation. However, these terms have also been used to mean ranked voting systems in general, leading most social choice theorists to recommend the use of more precise terms like instant-runoff voting (IRV).

^ 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.
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
Dr. Arrow: Yes.
CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
Dr. Arrow: And as I said, that in effect implies more information.
^ "Bill Status H.424: An act relating to town, city, and village elections for single-seat offices using ranked-choice voting". legislature.vermont.gov. Retrieved 2024-03-23. Condorcet winner. If a candidate is the winning candidate in every paired comparison, the candidate shall be declared the winner of the election.

[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] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.

[3] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.

[4] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
Dr. Arrow: Yes.
CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
Dr. Arrow: And as I said, that in effect implies more information.

[5] "Bill Status H.424: An act relating to town, city, and village elections for single-seat offices using ranked-choice voting". legislature.vermont.gov. Retrieved 2024-03-23. Condorcet winner. If a candidate is the winning candidate in every paired comparison, the candidate shall be declared the winner of the election.

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