Rated ballot
A ratings ballot, or cardinal ballot, is a ballot in which, for each candidate, a voter is asked to "rate" the candidate on a scale, for example from 0 to 100 or from −5 to +5. Ratings are usually (though not always) restricted to integer values i.e. if the scale is from 0 to 10, it usually isn't allowed for someone to give a rating of, say, 9.35). It is sometimes proposed that voters be able to give negative ratings (indicate disapproval/opposition), as in Evaluative voting.
An approval ballot can be thought of as a ratings ballot with an integer scale from 0 to 1 (inclusive).
Voting systems that use a ratings ballot include Score voting, instant runoff normalized ratings, and cardinal pairwise.
Comparability of scales
Two rated ballots that use different scales can be converted to each other in many election methods, but care must be taken to ensure the method of counting ballots allows for it.
In simple score methods, this is generally possible. For example, if one voter gave a candidate a 5 out of 10 and another voter gave a candidate a 3 out of 7, the 5 out of 10 can be interpreted as a 3.5 out of 7, and the 3 out of 7 as a 4.2857 out of 10. In general, all rated ballots can be thought of as approximations of (and transformable into) a scale from 0 to 1 (or 0% to 100%), with 0 being no support and 1 being full support.
Rated ballots are often shown as:
Alicia:5 Eileen:4 Brandon:3 Charlie:3 David:3
meaning "I give Alicia a score of 5/I give Alicia 5 points, Eileen 4, Brandon 3, etc."
Note that with rated ballots, one might mention what the scale is by putting, for example, Alicia:5/5 (meaning Alicia is a 5 out of 5).
Use in pairwise elections
The general idea of rating is that a voter's pairwise preferences are connected i.e. if a voter indicates A is maximally better than B (by giving A the max score and B the min score), then they must indicate B is no better than C.