View source for Social:Cardinal voting

{{Short description|Electoral systems with independent candidate ratings}}
[[Image:Rated voting.png|thumb|upright|alt= A theoretical ballot with the instructions "Rate each between negative ten and ten." There are five options, each one with a number corresponding to it. The numbers, from top to bottom, are seven, ten, negative three, zero, and ten.|On a rated ballot, the voter may rate each choice independently.]]
[[Image:Approval ballot.svg|thumb|upright|alt= A theoretical ballot with the instructions "Vote for any number of options." Two choices are marked, three are not. There is no difference between the markings.|An approval voting ballot does not require ranking or exclusivity.]]

'''Cardinal voting''' refers to any [[Social:Electoral system|electoral system]] which allows the voter to give each candidate an independent evaluation, typically a rating or grade.<ref>{{Cite journal|last1=Baujard|first1=Antoinette|last2=Gavrel|first2=Frédéric|last3=Igersheim|first3=Herrade|last4=Laslier|first4=Jean-François|last5=Lebon|first5=Isabelle|date=September 2017|title=How voters use grade scales in evaluative voting|journal=European Journal of Political Economy|volume=55|pages=14–28|doi=10.1016/j.ejpoleco.2017.09.006|issn=0176-2680|quote=A key feature of evaluative voting is a form of independence: the voter can evaluate all the candidates in turn ... another feature of evaluative voting ... is that voters can express some degree of preference.|url=https://halshs.archives-ouvertes.fr/halshs-01618039/file/1729.pdf}}
</ref> These are also referred to as "rated" ('''ratings ballot'''), "evaluative", "graded", or "absolute" voting systems.<ref>{{Cite web|url=https://electowiki.org/wiki/Cardinal_voting_systems|title=Cardinal voting systems—Electowiki|website=electowiki.org|language=en|access-date=31 January 2017}}</ref><ref>{{Cite web|url=https://electowiki.org/wiki/Voting_system#Rated_Voting_methods|title=Voting system - Electowiki|website=electowiki.org|language=en|access-date=31 January 2017}}
</ref> [[Cardinal number|''Cardinal'']] methods (based on cardinal utility) and ''[[Social:Ranked voting|ordinal methods]]'' (based on ''ordinal utility'') are two main categories of modern voting systems, along with [[Social:Plurality voting|plurality voting]].<ref>{{Cite book|title=Liberalism against populism : a confrontation between the theory of democracy and the theory of social choice|last=Riker, William Harrison.|date=1982|publisher=Waveland Pr|isbn=0881333670|pages=29–30|oclc=316034736|quote=''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.}}
</ref><ref name="url(PDF) Ordinal Versus Cardinal Voting Rules: A Mechanism Design Approach">{{cite web |url=https://www.researchgate.net/publication/316805816 |title=Ordinal Versus Cardinal Voting Rules: A Mechanism Design Approach }}
</ref><ref name="urlCardinal Voting: The Way to Escape the Social Choice Impossibility by Sergei Vasiljev :: SSRN">{{cite journal |title=Cardinal Voting: The Way to Escape the Social Choice Impossibility by Sergei Vasiljev :: SSRN |ssrn = 1116545|date= April 2008|last1= Vasiljev|first1= Sergei}}
</ref>

== Variants ==
[[Image:Sample_ballot_for_Majority_Judgment_(SF).png|thumb|alt= A scan of a real ballot that was already marked, with instructions to mark each candidate from A to F, where A is best. Spaces left blank are considered as F. The options from top to bottom are Eleanor Roosevelt, graded C, Cesar Chavez, graded B, Walter Lum, graded C, John Hancock, graded F, Martin Luther King Jr, graded B, and Nancy Reagan, graded A.|A majority judgment ballot is based on grades like those used in schools.]]

There are several voting systems that allow independent ratings of each candidate. For example:

* [[Social:Approval voting|Approval voting]] (AV) is the simplest possible method, which allows only the two grades (0, 1): "approved" or "unapproved".<ref name=":1">{{Cite journal|url=https://epub.ub.uni-muenchen.de/653/|title=The Case for Utilitarian Voting|last=Hillinger|first=Claude|date=1 May 2005|website=Open Access LMU|location=Munich|doi=10.5282/ubm/epub.653 |language=en|access-date=15 May 2018|quote=Specific UV rules that have been proposed are approval voting, allowing the scores 0, 1; range voting, allowing all numbers in an interval as scores; evaluative voting, allowing the scores −1, 0, 1.}}</ref>
* Evaluative voting (EV) or [[Social:Combined approval voting|combined approval voting]] (CAV) uses 3 grades (−1, 0, +1): "against", "abstain", or "for".<ref name=":1" /><ref>{{Cite journal|last=Hillinger|first=Claude|date=1 October 2004|title=On the Possibility of Democracy and Rational Collective Choice|language=en|location=Rochester, NY|quote=I favor 'evaluative voting' under which a voter can vote for or against any alternative, or abstain.|ssrn=608821}}</ref><ref>{{Cite journal|last=Felsenthal|first=Dan S.|date=January 1989|title=On combining approval with disapproval voting|journal=Behavioral Science|language=en|volume=34|issue=1|pages=53–60|doi=10.1002/bs.3830340105|issn=0005-7940|quote=under CAV he has three options—cast one vote in favor, abstain, or cast one vote against.}}</ref>
* Score voting or range voting, in which ratings are numerical and the candidate with the highest ''average'' (or total<ref>{{Cite web|url=http://www.socialchoiceandbeyond.com/scabpage38.html|title=Range Voting|website=Social Choice and Beyond|access-date=10 December 2016|quote=with the winner being the one with the largest point total. Or, alternatively, the average may be computed and the one with the highest average wins}}</ref><ref>{{Cite news|url=https://electology.org/score-voting|title=Score Voting|date=21 May 2015|newspaper=The Center for Election Science|quote=Simplified forms of score voting automatically give skipped candidates the lowest possible score for the ballot they were skipped. Other forms have those ballots not affect the candidate’s rating at all. Those forms not affecting the candidates rating frequently make use of quotas. Quotas demand a minimum proportion of voters rate that candidate in some way before that candidate is eligible to win.|access-date=10 December 2016}}</ref>) rating wins.
** Score voting uses a discrete integer scale, typically from 0 to 5 or 0 to 9.<ref name=":2">{{Cite web|url=https://voteupapp.com/blog/expressive-voting|title=Should you be using a more expressive voting system?|website=VoteUp app|access-date=15 May 2018|quote=Score Voting—it's just like range voting except the scores are discrete instead of spanning a continuous range.}}</ref><ref>{{Cite web|url=http://rangevoting.org/RateScaleResearch.html|title=Rating Scale Research|website=RangeVoting.org|access-date=15 May 2018|quote=The present page seems to conclude 0-9 is the best scale.}}</ref>
** Range voting uses a continuous scale from 0 to 1.<ref name=":1" /><ref name=":2" /><ref>{{Cite web|url=http://rangevoting.org/PreEmProp.html|title=Good criteria support range voting|website=RangeVoting.org|access-date=15 May 2018|quote=Definition 1: For us "Range voting" shall mean the following voting method. Each voter provides as her vote, a set of real number scores, each in [0,1], one for each candidate. The candidate with greatest score-sum, is elected.}}</ref><ref>{{Cite web|url=http://scorevoting.net/WarrenSmithPages/homepage/rangevote.pdf|title=Range Voting|last=Smith|first=Warren D.|date=December 2000|quote=The “range voting” system is as follows. In a c-candidate election, you select a vector of c real numbers, each of absolute value ≤1, as your vote. E.g. you could vote (+1, −1, +.3, −.9, +1) in a five-candidate election. The vote-vectors are summed to get a c-vector x and the winner is the i such that x<sub>i</sub> is maximum.}}</ref>
* Highest median rules, which elect the candidate with the highest ''median'' grade. The various highest median rules differ in their tie-breaking methods. The majority judgment, in which the grades are associated to expressions (such as "Excellent", to "Poor"), is the most common example as it is the first such rule that has been studied, but other rules have since been proposed, e.g. the typical judgment or the [[Social:Usual judgment|usual judgment]].
*[[Social:STAR voting|STAR voting]], in which scores are from 0 to 5, and the most-preferred of the top-two highest-scoring candidates wins.<ref>{{Cite news|url=https://www.equal.vote/starvoting|title=STAR Voting|work=Equal Vote Coalition|access-date=14 July 2018|archive-date=1 July 2020|archive-url=https://web.archive.org/web/20200701214017/https://www.equal.vote/starvoting|url-status=dead}}</ref><ref>{{Cite news|url=http://www.registerguard.com/opinion/20180610/star-voting-intriguing-innovation|title=STAR voting an intriguing innovation|work=The Register Guard|access-date=14 July 2018|language=en}}</ref><ref>{{Cite news|url=https://ivn.us/2018/02/01/cutting-edge-voting-reform/|title=Are We Witnessing the Cutting Edge of Voting Reform?|date=1 February 2018|work=IVN.us|access-date=14 July 2018|language=en-US}}</ref>
* Majority Approval Voting, a scored variant of Bucklin voting, typically using letter grades (such as "A" through "F").<ref>{{Cite web|url=https://electowiki.org/wiki/Majority_Approval_Voting|title=Majority Approval Voting|website=Electowiki|language=en|access-date=26 August 2018}}</ref>
* 3-2-1 voting, in which voters rate each candidate "Good", "OK", or "Bad", and there are three automatic elimination steps to tally them: first step selects the three candidates with the most "Good" ratings, second the two with the least "Bad", and out of these the one preferred by the majority wins.<ref>{{Cite web|url=https://electowiki.org/wiki/3-2-1_voting|title=3-2-1 voting|website=Electowiki|date=30 June 2021 }}</ref><ref>{{Cite web|url=https://medium.com/@jameson.quinn/make-all-votes-count-part-2-single-winner-5a2fb47123d5|title=Make. All. Votes. Count. (Part II: single-winner)|last=Quinn|first=Jameson|date=28 May 2017|website=Jameson Quinn|access-date=14 July 2018}}</ref>

Additionally, several cardinal systems have variants for multi-winner elections, typically meant to produce [[Social:Proportional representation|proportional representation]], such as:

* [[Social:Proportional approval voting|Proportional approval voting]]
* [[Social:Sequential proportional approval voting|Sequential proportional approval voting]]
* [[Social:Satisfaction approval voting|Satisfaction approval voting]]
* Re-weighted range voting<ref>{{Cite web|url=http://www.rangevoting.org/RRV.html|title=Reweighted Range Voting - a PR voting method that feels like range voting|website=RangeVoting.org|access-date=24 March 2018}}</ref>

== Relationship to rankings ==
Ratings ballots can be converted to ranked/preferential ballots. For example:
{| class="wikitable" border="1"
|-
!  
!  Rating (0 to 99)
!  Preference order
|-
|  Candidate A
|  99
|  First
|-
|  Candidate B
|  20
|  Third
|-
|  Candidate C
|  20
|  Third
|-
|  Candidate D
|  55
|  Second
|}

This requires the voting system to accommodate a voter's indifference between two candidates (as in Ranked Pairs or [[Social:Schulze method|Schulze method]]).

The opposite is not true: Rankings cannot be converted to ratings, since ratings carry more information about strength of preference, which is destroyed when converting to rankings.

== Analysis ==
By avoiding ranking (and its implication of a monotonic approval reduction from most- to least-preferred candidate) cardinal voting methods may solve a very difficult problem:

A foundational result in [[Philosophy:Social choice theory|social choice theory]] (the study of voting methods) is [[Arrow's impossibility theorem]], which states that no method can comply with all of a simple set of desirable criteria. However, since one of these criteria (called "universality") implicitly requires that a method be ordinal, not cardinal, Arrow's theorem does not apply to cardinal methods.<ref>{{Cite journal|last=Vasiljev|first=Sergei|date=1 April 2008|title=Cardinal Voting: The Way to Escape the Social Choice Impossibility|ssrn=1116545|location=Rochester, NY|publisher=Social Science Research Network}}</ref><ref name=":0">{{cite web|url=https://electology.org/podcasts/2012-10-06_kenneth_arrow|title=Interview with Dr. Kenneth Arrow|date=6 October 2012|website=The Center for Election Science|quote=''CES:'' you mention that your theorem applies to preferential systems or ranking systems. ... But the system that you're just referring to, Approval Voting, falls within a class called cardinal systems. ... ''Dr. Arrow:'' And as I said, that in effect implies more information. ... 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.|access-date=2016-12-10|archive-date=2018-10-27|archive-url=https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow|url-status=dead}}</ref><ref>{{Cite web|url=http://rangevoting.org/ArrowThm.html|title=RangeVoting.org - Arrow's theorem|website=rangevoting.org|access-date=10 December 2016|quote=according to Arrow's definition, range voting is "not" a voting system at all}}</ref><ref>{{Cite news|url=https://electology.org/blog/how-i-came-care-about-voting-systems|title=How I Came to Care About Voting Systems|date=21 December 2011|newspaper=The Center for Election Science|quote=But Arrow only intended his criteria to apply to ranking systems.|access-date=10 December 2016}}</ref>

Others, however, argue that ratings are fundamentally invalid, because meaningful interpersonal comparisons of utility are impossible.<ref>{{Cite news|url=https://electology.org/blog/why-not-ranking|title=Why Not Ranking?|date=31 May 2016|newspaper=The Center for Election Science|language=en|quote=Many voting theorists have resisted asking for more than a ranking, with economics-based reasoning: utilities are not comparable between people. ... But no economist would bat an eye at asking one of the A voters above whether they'd prefer a coin flip between A and B winning or C winning outright...|access-date=22 January 2017}}</ref> This was Arrow's original justification for only considering ranked systems,<ref>"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the [[Philosophy:Identity of indiscernibles|identity of indiscernibles]] demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on [https://books.google.com/books?id=7ECXDjlCpB0C&pg=PA33 p.&nbsp;33] by {{citation|last=Racnchetti|first=Fabio|title=The Active Consumer: Novelty and Surprise in Consumer Choice|year=2002|series=Routledge Frontiers of Political Economy|volume=20|pages=21–45|contribution=Choice without utility? Some reflections on the loose foundations of standard consumer theory|publisher=Routledge|editor-last=Bianchi|editor-first=Marina}}</ref> but later in life he stated that cardinal methods are "probably the best".<ref name=":02">{{cite web|url=https://electology.org/podcasts/2012-10-06_kenneth_arrow|title=Interview with Dr. Kenneth Arrow|date=6 October 2012|website=The Center for Election Science|quote=''CES:'' you mention that your theorem applies to preferential systems or ranking systems. ... But ... Approval Voting, falls within a class called cardinal systems. ... ''Dr. Arrow:'' And as I said, that in effect implies more information. ... I'm a little inclined to think that score systems where you categorize in maybe three or four classes ... is probably the best.|access-date=2016-12-10|archive-date=2018-10-27|archive-url=https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow|url-status=dead}}</ref>  Cardinal methods inherently impose a tactical concern that any voter has regarding their second-favorite candidate, in the case that there are 3 or more candidates.  Score too high (or Approve) and the voter harms their favorite candidate's chance to win.  Score too low (or not Approve) and the voter helps the candidate they least desire to beat their second-favorite and perhaps win.

Psychological research has shown that cardinal [[Social:Rating scale|ratings]] (on a numerical or [[Social:Likert scale|Likert]] scale, for instance) are more valid and convey more information than ordinal rankings in measuring human opinion.<ref>{{Cite journal|last1=Conklin|first1=E. S.|last2=Sutherland|first2=J. W.|date=1 February 1923|title=A Comparison of the Scale of Values Method with the Order-of-Merit Method.|url=http://content.apa.org/journals/xge/6/1/44|journal=Journal of Experimental Psychology|language=en|volume=6|issue=1|pages=44–57|doi=10.1037/h0074763|issn=0022-1015|quote=the scale-of-values method can be used for approximately the same purposes as the order-of-merit method, but that the scale-of-values method is a better means of obtaining a record of judgments}}</ref><ref>{{Cite journal|last=Moore|first=Michael|date=1 July 1975|title=Rating versus ranking in the Rokeach Value Survey: An Israeli comparison|journal=European Journal of Social Psychology|language=en|volume=5|issue=3|pages=405–408|doi=10.1002/ejsp.2420050313|issn=1099-0992|quote=The extremely high degree of correspondence found between ranking and rating averages ... does not leave any doubt about the preferability of the rating method for group description purposes. The obvious advantage of rating is that while its results are virtually identical to what is obtained by ranking, it supplies more information than ranking does.}}</ref><ref>{{Cite journal|last1=Maio|first1=Gregory R.|last2=Roese|first2=Neal J.|last3=Seligman|first3=Clive|last4=Katz|first4=Albert|date=1 June 1996|title=Rankings, Ratings, and the Measurement of Values: Evidence for the Superior Validity of Ratings|journal=Basic and Applied Social Psychology|volume=18|issue=2|pages=171–181|doi=10.1207/s15324834basp1802_4|issn=0197-3533|quote=Many value researchers have assumed that rankings of values are more valid than ratings of values because rankings force participants to differentiate more incisively between similarly regarded values ... Results indicated that ratings tended to evidence greater validity than rankings within moderate and low-differentiating participants. In addition, the validity of ratings was greater than rankings overall.}}</ref><ref>{{Cite journal|last1=Johnson|first1=Marilyn F.|last2=Sallis|first2=James F.|last3=Hovell|first3=Melbourne F.|date=1 September 1999|title=Comparison of Rated and Ranked Health and Lifestyle Values|journal=American Journal of Health Behavior|volume=23|issue=5|pages=356–367|doi=10.5993/AJHB.23.5.5|quote=the test-retest reliabilities of the ranking items were slightly higher than were those of the rating items, but construct validities were lower. Because validity is the most important consideration ... the findings of the present research support the use of the rating format in assessing health values. ... added benefit of item independence, which allows for greater flexibility in statistical analyses. ... also easier than ranking items for respondents to complete.}}</ref>

Cardinal methods can, but don't have to satisfy the Condorcet winner criterion.

=== Strategic voting ===
The weighted mean utility theorem gives the optimal strategy for cardinal voting, which is to give maximum score for all options above expected value of the winning option and minimum score for all other options.<ref>Approval Voting, Steven J. Brams, Peter C. Fishburn, 1983</ref>

== See also ==

* Borda count

== References ==
{{reflist|30em}}

[[Category:Electoral systems]]
[[Category:Social choice theory]]

{{Sourceattribution|Cardinal voting}}