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Cardinal voting refers to any electoral system which allows the voter to give each candidate an independent evaluation, typically a rating or grade.[1] These are also referred to as "rated" (ratings ballot), "evaluative", "graded", or "absolute" voting systems.[2][3] 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).[4][5][6]
There are several voting systems that allow independent ratings of each candidate. For example:
In addition, every cardinal system can be converted into a proportional or semi-proportional system by using Phragmen's voting rules. Examples include:
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.
Cardinal voting methods are not subject to Arrow's impossibility theorem,[23] which proves that ranked-choice voting methods can be manipulated by strategic nominations,[24] 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.[25][24]
Others, however, argue that ratings are fundamentally invalid, because meaningful interpersonal comparisons of utility are impossible.[26] This was Arrow's original justification for only considering ranked systems,[27] but later in life he stated that cardinal methods are "probably the best".[28]
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.[29][30][31][32]
Cardinal methods can satisfy the Condorcet winner criterion.[citation needed]
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.[33] 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.