Rated voting

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On a rated ballot, the voter may rate each choice independently. Rated voting.png
On a rated ballot, the voter may rate each choice independently.
An approval voting ballot does not require ranking or exclusivity. Approval ballot.svg
An approval voting ballot does not require ranking or exclusivity.

Rated, evaluative, [1] [2] graded, [1] or cardinalvotingrules 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]

Contents

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

A majority judgment ballot is based on grades like those used in schools. Sample ballot for Majority Judgment (SF).png
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 be immune to the spoiler effect given certain types of voter behavior. For example:

However, other rated voting methods have a spoiler effect no matter what scales the voters use:

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

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 A99First
Candidate B55Second
Candidate C20Third
Candidate D20Third

Analysis

Arrow'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

Like 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

Related Research Articles

<span class="mw-page-title-main">Score voting</span> Single-winner rated voting system

Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.

Strategic or tactical voting is voting in consideration of possible ballots cast by other voters in order to maximize one's satisfaction with the election's results.

<span class="mw-page-title-main">Spoiler effect</span> Losing candidate affecting election result

In social choice theory and politics, a spoiler is a losing candidate who affects the results of an election simply by participating, a situation that is called a spoiler effect. If a major candidate is perceived to have lost an election because of a minor candidate, the minor candidate is called a spoiler candidate and the major candidate is said to have been spoiled. Often times the term spoiler will be applied to candidates or situations which do not meet the full definition, typically in real-world scenarios where the introduction of a new candidate can cause voters to change their opinions, either through their campaign or merely by existing. A voting system that is not affected by spoilers is called independent of irrelevant alternatives or spoilerproof.

<span class="mw-page-title-main">Arrow's impossibility theorem</span> Proof all ranked voting rules have spoilers

Arrow's impossibility theorem is a key result in social choice theory, showing that no ranking-based decision rule can satisfy the requirements of rational choice theory. Most notably, Arrow showed that no such rule can satisfy independence of irrelevant alternatives, the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option C.

Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome . There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski-Young theorem, and the money pump arguments.

The Gibbard–Satterthwaite theorem is a theorem in voting theory. It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 and then proved independently by the philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner, and shows that for every voting rule of this form, at least one of the following three things must hold:

  1. The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
  2. The rule limits the possible outcomes to two alternatives only; or
  3. The rule is not straightforward, i.e. there is no single always-best strategy.
<span class="mw-page-title-main">Condorcet winner criterion</span> Property of electoral systems

A Condorcet winner is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.

<span class="mw-page-title-main">No-show paradox</span> When voting for a candidate makes them lose

In social choice, a no-show paradox is a surprising behavior in some voting rules, where a candidate loses an election as a result of having too many supporters. More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. Voting systems without the no-show paradox are said to satisfy the participation criterion.

<span class="mw-page-title-main">Majority winner criterion</span> Property of electoral systems

The majority criterion is a voting system criterion applicable to voting rules over ordinal preferences required that if only one candidate is ranked first by over 50% of voters, that candidate must win.

<span class="mw-page-title-main">Positional voting</span> Class of ranked-choice electoral systems

Positional voting is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will or it may form a mathematical sequence such as an arithmetic progression, a geometric one or a harmonic one. The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes.

<span class="mw-page-title-main">Independence of clones criterion</span> Property of electoral systems

In social choice theory, the independence of (irrelevant) clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a weak form of the independence of irrelevant alternatives (IIA) criterion that nevertheless is failed by a number of voting rules. A method that passes the criterion is said to be clone independent.

<span class="mw-page-title-main">Instant-runoff voting</span> Single-winner ranked-choice electoral system

Instant-runoff voting (IRV), is a single-winner, multi-round elimination rule that uses ranked voting to simulate a series of runoff elections. In each round, the last-place finisher according to a plurality vote is eliminated, and the votes supporting the eliminated choice are transferred to their next available preference until one of the options reaches a majority of the remaining votes. Instant runoff falls under the plurality-with-elimination family of voting methods, and is thus closely related to rules like the exhaustive ballot and two-round runoff system

<span class="mw-page-title-main">Electoral system</span> Method by which voters make a choice between options

An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result. Political electoral systems are defined by constitutions and electoral laws, are typically conducted by election commissions, and can use multiple types of elections for different offices.

<span class="mw-page-title-main">Majority judgment</span> Single-winner cardinal voting system

Majority judgment (MJ) is a single-winner voting system proposed in 2010 by Michel Balinski and Rida Laraki. It is a kind of highest median rule, a cardinal voting system that elects the candidate with the highest median rating.

<span class="mw-page-title-main">Ranked voting</span> Voting systems that use ranked ballots

Ranked voting is any voting system that uses voters' rankings of candidates to choose a single winner or multiple winners. More formally, a ranked system 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.

There are a number of different criteria which can be used for voting systems in an election, including the following

<span class="mw-page-title-main">Comparison of voting rules</span> Comparative politics for electoral systems

This article discusses the methods and results of comparing different electoral systems. There are two broad ways to compare voting systems:

  1. Metrics of voter satisfaction, either through simulation or survey.
  2. Adherence to logical criteria.

Combined approval voting (CAV) is an electoral system where each voter may express approval, disapproval, or indifference toward each candidate. The winner is the candidate with the highest score, which is determined by subtracting the number of approval votes by the number of disapproval votes.

<span class="mw-page-title-main">Sincere favorite criterion</span> Criterion that prevents lesser-evil voting

The sincere favorite or no favorite-betrayal criterion is a property of some voting systems that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in lesser-evil voting or a strategy called "decapitation".

<span class="mw-page-title-main">Highest median voting rules</span>

The highest median voting rules are a class of graded voting rules where the candidate with the highest median rating is elected.

References

  1. 1 2 3 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. 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.
  3. "Ordinal Versus Cardinal Voting Rules: A Mechanism Design Approach".
  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. 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.
  6. 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.
  7. "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.
  8. 1 2 3 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.
  9. 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.
  10. 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.
  11. 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
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. Brams, Steven J.; Fishburn, Peter C. (2007). Approval voting (2. ed.). New York, NY: Springer. pp. 84–90. ISBN   978-0-387-49895-9.