Rated voting

"Cardinal voting" redirects here. For the voting system used by Cardinals to elect the pope, see Papal conclave § voting.

Rated, evaluative,^[1] graded,^[1] or cardinal voting systems are a class of voting methods which allow voters to state how strongly they support a candidate,^[2] which involves giving each one a grade on a separate scale.^[1] Cardinal methods (based on cardinal utility) and ordinal methods (based on ordinal utility) are the two categories of modern voting systems.^[2][3]

On a rated ballot, the voter may rate each choice independently.

An approval voting ballot does not require ranking or exclusivity.

The distribution of ratings for each candidate—i.e. the percentage of voters who assign them a particular score—is called their merit profile.^[4] For example, if candidates are graded on a 4-point scale, one candidate's merit profile may be 25% on every possible rating (1, 2, 3, and 4), while a perfect candidate would have a merit profile where 100% of voters assign them a score of 4.

Variants

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, which allow them to avoid Arrow's theorem and satisfy spoilerproofness. For example:

• Score voting systems, where the candidate with the highest average (or total^[5]) rating wins.
  • Approval voting (AV) is the simplest method, and allows only the two grades (0, 1): "approved" or "unapproved".^[6]
  • Combined approval voting (CAV) uses 3 grades (−1, 0, +1): "against", "abstain", or "for."^[6][7][8]
  • Range voting refers to a variant with a continuous scale from 0 to 1.^[6]
  • The familiar five-star classification system is a common example, and allows for either 5 grades or 10 (if half-stars are used).
• Highest median rules, where the candidate with the highest median grade wins. The various highest median rules differ in their tie-breaking methods.

However, not all rated voting methods are spoilerproof:

• Quadratic voting is unusual in that it is a cardinal voting system that does not allow independent scoring of candidates.
• Cumulative voting is a technically-cardinal voting system that, in practice, behaves like first-past-the-post voting.
• STAR (score then automatic runoff) is a hybrid of ranked and rated voting systems. It chooses the top 2 candidates by score voting, who then advance to a runoff round (where the candidate is elected by a simple plurality).

In addition, there are many different proportional cardinal rules, often called approval-based committee rules.

• Phragmen's method
• Proportional approval voting (Thiele's method)
• Fair majority voting
• Method of equal shares
• Expanding approvals rule

Relationship to rankings

Ratings ballots can be converted to ranked/preferential ballots, assuming equal ranks are allowed. For example:

	Rating (0 to 99)	Preference order
Candidate A	99	First
Candidate B	55	Second
Candidate C	20	Third
Candidate D	20	Third

Analysis

Cardinal voting methods are not subject to Arrow's impossibility theorem,^[9] which proves that ranked-choice voting methods cannot eliminate the spoiler effect.^{[10][dead link][11]}

Others, however, argue that ratings are fundamentally invalid, because meaningful interpersonal comparisons of utility are impossible.^[12] This was Arrow's original justification for only considering ranked systems,^[13] but later in life he softened his opinion, stating that he is "a little inclined to think that [cardinal methods with three or four classes are] probably the best".^[11]

Psychological research has shown that cardinal ratings (on a numerical or Likert scale, for instance) are more valid and convey more information than ordinal rankings in measuring human opinion.^{[14][15][16][17]}

Cardinal methods can satisfy the Condorcet winner criterion, usually by combining cardinal voting with a first stage (as in Smith//Score).

Strategic voting

The weighted mean utility theorem gives the optimal strategy for cardinal voting under most circumstances, which is to give the maximum score for all options with an above-average expected utility,^[18] which is equivalent to approval voting. As a result, strategic voting with score voting often results in a sincere ranking of candidates on the ballot (a property that is impossible for ranked-choice voting, by the Gibbard–Satterthwaite theorem).

See also

• Ranked voting, the other class of voting methods
  • Plurality voting, the degenerate case of ranked-choice voting
  • Arrow's impossibility theorem, a theorem on the limitations of ranked-choice voting
• Gibbard's theorem, a generalization of the Gibbard-Satterthwaite theorem applicable to rated voting

References

1. Baujard, Antoinette; Gavrel, Frédéric; Igersheim, Herrade; Laslier, Jean-François; Lebon, Isabelle (September 2017). "How voters use grade scales in evaluative voting" (PDF). European Journal of Political Economy. 55: 14–28. doi:10.1016/j.ejpoleco.2017.09.006. ISSN 0176-2680. 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.
2. "Ordinal Versus Cardinal Voting Rules: A Mechanism Design Approach".
3. 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.
4. de Swart, Harrie (2022-06-01). "How to Choose a President, Mayor, Chair: Balinski and Laraki Unpacked". The Mathematical Intelligencer. 44 (2): 99–107. doi:10.1007/s00283-021-10124-3. ISSN 1866-7414.
5. "Score Voting". The Center for Election Science. 21 May 2015. Retrieved 10 December 2016. 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.
6. Hillinger, Claude (1 May 2005). "The Case for Utilitarian Voting". Open Access LMU. Munich. doi:10.5282/ubm/epub.653. Retrieved 15 May 2018. 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.
7. Hillinger, Claude (1 October 2004). "On the Possibility of Democracy and Rational Collective Choice". Rochester, NY. SSRN 608821. I favor 'evaluative voting' under which a voter can vote for or against any alternative, or abstain. {{cite journal}}: Cite journal requires |journal= (help)
8. Felsenthal, Dan S. (January 1989). "On combining approval with disapproval voting". Behavioral Science. 34 (1): 53–60. doi:10.1002/bs.3830340105. ISSN 0005-7940. under CAV he has three options—cast one vote in favor, abstain, or cast one vote against.
9. Vasiljev, Sergei (2008). "Cardinal Voting: The Way to Escape the Social Choice Impossibility". SSRN Electronic Journal. Elsevier BV. doi:10.2139/ssrn.1116545. ISSN 1556-5068.
10. "How I Came to Care About Voting Systems". The Center for Election Science. 21 December 2011. Retrieved 10 December 2016. But Arrow only intended his criteria to apply to ranking systems.
11. "Interview with Dr. Kenneth Arrow". The Center for Election Science. 6 October 2012. Archived from the original on 2018-10-27. Retrieved 2016-12-10. 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.
12. "Why Not Ranking?". The Center for Election Science. 31 May 2016. Retrieved 22 January 2017. 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...
13. "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 identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
14. Conklin, E. S.; Sutherland, J. W. (1 February 1923). "A Comparison of the Scale of Values Method with the Order-of-Merit Method". Journal of Experimental Psychology. 6 (1): 44–57. doi:10.1037/h0074763. ISSN 0022-1015. 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
15. Moore, Michael (1 July 1975). "Rating versus ranking in the Rokeach Value Survey: An Israeli comparison". European Journal of Social Psychology. 5 (3): 405–408. doi:10.1002/ejsp.2420050313. ISSN 1099-0992. 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.
16. Maio, Gregory R.; Roese, Neal J.; Seligman, Clive; Katz, Albert (1 June 1996). "Rankings, Ratings, and the Measurement of Values: Evidence for the Superior Validity of Ratings". Basic and Applied Social Psychology. 18 (2): 171–181. doi:10.1207/s15324834basp1802_4. ISSN 0197-3533. 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.
17. Johnson, Marilyn F.; Sallis, James F.; Hovell, Melbourne F. (1 September 1999). "Comparison of Rated and Ranked Health and Lifestyle Values". American Journal of Health Behavior. 23 (5): 356–367. doi:10.5993/AJHB.23.5.5. 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.
18. Approval Voting, Steven J. Brams, Peter C. Fishburn, 1983