Rated, evaluative,[1][2] graded,[1] or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate,[3] by giving each one a grade on a separate scale.[1]
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.
Since rated methods allow the voters to express how strongly they support a candidate, these methods are not covered by Arrow's impossibility theorem,[5] and their resistance to the spoiler effect becomes a more complex matter. Some rated methods are immune to the spoiler effect when every voter rates the candidates on an absolute scale, but they are not when the voters' rating scales change based on the candidates who are running.[6]
Variants
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There are several voting systems that allow independent ratings of each candidate, which allow them to be immune to the spoiler effect given certain types of voter behavior. For example:
- Score voting systems, where the candidate with the highest average (or total[7]) rating wins.
- Approval voting (AV) is the simplest method, and allows only the two grades (0, 1): "approved" or "unapproved".[8]
- Combined approval voting (CAV) uses 3 grades (−1, 0, +1): "against", "abstain", or "for."[8][9][10]
- Range voting refers to a variant with a continuous scale from 0 to 1.[8]
- 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, other rated voting methods have a spoiler effect no matter what scales the voters use:
- Quadratic voting is unusual in that it is a cardinal voting system that does not allow independent scoring of candidates.
- Cumulative voting could be classified as a cardinal rule with unconditional spoiler effects.
- 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).
- Any hybrid of a ranked and rated voting system that reduces to majority rule when only two candidates are running fails independence of irrelevant alternatives due to the Condorcet paradox.
In addition, there are many different proportional cardinal rules, often called approval-based committee rules.
- Phragmen's method
- Proportional approval voting (Thiele's method)
- Method of equal shares
Relationship to rankings
editRatings 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
editArrow's impossibility theorem does not apply to cardinal rules.
Psychological research has shown that cardinal ratings (on a numerical or Likert scale, for instance) convey more information than ordinal rankings in measuring human opinion.[11][12][13][14]
Cardinal methods can satisfy the Condorcet winner criterion, usually by combining cardinal voting with a first stage (as in Smith//Score).
Strategic voting
editLike all (deterministic, non-dictatorial, multicandidate) voting methods, rated methods are vulnerable to strategic voting, due to Gibbard's theorem.
Cardinal methods where voters give each candidate a number of points and the points are summed are called additive. Both range voting and cumulative voting are of this type. With a large number of voters, the strategic Myerson-Weber equilibria for such methods are the same as for methods where only extreme ballots are allowed.[15] In this setting, the optimal strategy for Range voting is the same as for approval voting, and the optimal strategy for cumulative voting is the same as for first-past-the-post. For approval voting (and thus Range voting), the optimal strategy involves approving (or rating maximum) everybody above a certain utility threshold, and not approving (or rating minimum) everybody below it.[16]
See also
edit- 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
edit- ^ a b c 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.
- ^ Darmann, Andreas; Grundner, Julia; Klamler, Christian (2019-09-01). "Evaluative voting or classical voting rules: Does it make a difference? Empirical evidence for consensus among voting rules". European Journal of Political Economy. 59: 345–353. doi:10.1016/j.ejpoleco.2019.04.003. ISSN 0176-2680.
- ^ "Ordinal Versus Cardinal Voting Rules: A Mechanism Design Approach".
- ^ 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.
- ^ 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.
- ^ Morreau, Michael (2014-10-13). "Arrow's Theorem". Stanford Encyclopedia of Philosophy. Retrieved 2024-10-09.
One important finding was that having cardinal utilities is not by itself enough to avoid an impossibility result. ... Intuitively speaking, to put information about preference strengths to good use it has to be possible to compare the strengths of different individuals' preferences.
- ^ "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.
- ^ a b c 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.
- ^ Hillinger, Claude (1 October 2004). "On the Possibility of Democracy and Rational Collective Choice". Rochester, NY. doi:10.2139/ssrn.608821. SSRN 608821.
I favor 'evaluative voting' under which a voter can vote for or against any alternative, or abstain.
- ^ 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.
- ^ 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
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Núñez, Matías; Laslier, Jean François (2014). "Preference intensity representation: strategic overstating in large elections". Social Choice and Welfare. 42 (2): 313–340. doi:10.1007/s00355-013-0728-0. ISSN 0176-1714.
- ^ Brams, Steven J.; Fishburn, Peter C. (2007). Approval voting (2. ed.). New York, NY: Springer. pp. 84–90. ISBN 978-0-387-49895-9.