Rated voting

There are several voting systems that allow independent ratings of each candidate. 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.
  • Graduated majority judgment, the most common such rule.
• 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 or Thiele's voting rules. Examples include:

• Proportional approval voting
• Fair Majority Voting
• Method of Equal Shares

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

Cardinal voting methods are not subject to Arrow's impossibility theorem,^[9] which proves that ranked-choice voting methods can be manipulated by strategic nominations.^[10] 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.^[11][10]

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 stated that cardinal methods are "probably the best."^[14]

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.^{[15][16][17][18]}

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,^[19] 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).

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

• Ranked-choice 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