Английская Википедия:Cardinal voting

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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.

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 electoral system which allows the voter to give each candidate an independent evaluation, typically a rating or grade. These are also referred to as "rated" (ratings ballot), "evaluative", "graded", or "absolute" voting systems. Cardinal methods (based on cardinal utility) and ordinal methods (based on ordinal utility) are the two modern categories of modern voting systems, along with plurality voting (which is itself an ordinal method).

Variants

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:

• Score voting systems, in which voters assign candidates points on a common numerical scale. The candidate with the highest average (or total) rating wins.
  • Approval voting (AV) is the simplest method, and allows only the two grades (0, 1): "approved" or "unapproved".
  • Evaluative voting (EV) or combined approval voting (CAV) uses 3 grades (−1, 0, +1): "against", "abstain", or "for."
  • Range voting refers to a variant with a continuous scale from 0 to 1.
• Highest median rules, which elect the candidate with the highest median grade. The various highest median rules differ in their tie-breaking methods.
  • Bucklin voting, the earliest such rule.
  • Graduated majority judgment (continuous Bucklin), a modern variant of the system.
• 3-2-1 voting, in which voters rate each candidate "Good," "OK," or "Bad," with a three-step procedure: the first step selects the three candidates with the most "Good" ratings, the second selects two with the least "Bad", and out of these, the one preferred by the majority wins.
• STAR (Score then automatic runoff), which selects the top 2 candidates by score voting system to advance to a runoff round (where the candidate preferred by the majority wins).

In addition, every cardinal system can be converted into a proportional or semi-proportional system by using Phragmen's voting rules. Examples include:

• Proportional approval voting
• Sequential proportional approval voting
• Satisfaction approval voting
• Reweighted range voting

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	20	Third
Candidate C	20	Third
Candidate D	55	Second

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

Analysis

Cardinal voting methods are not subject to Arrow's impossibility theorem, which proves that ranked-choice voting methods can be manipulated by strategic nominations, and all will tend to give logically incoherent results. 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.

Others, however, argue that ratings are fundamentally invalid, because meaningful interpersonal comparisons of utility are impossible. This was Arrow's original justification for only considering ranked systems, but later in life he stated that cardinal methods are "probably the best".

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.

Cardinal methods can satisfy the Condorcet winner criterion.Шаблон:Cn

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. As a result, strategic voting with score voting often results in a (weakly) honest ranking of candidates on the ballot (a property missing from most ranked systems).

Most cardinal methods, including score voting and STAR, pass the Condorcet and Smith criteria if voters behave strategically. As a result, cardinal methods with strategic voters tend to produce results results similar to Condorcet methods with honest voters.

See also

• Ranked-choice voting, the other class of cardinal methods
  • Plurality voting, the degenerate case of ranked-choice voting
  • Arrow's impossibility theorem, a theorem on the limitations of ranked-choice voting