Cardinal proportional representation
Cardinal PR (Proportional Representation with rated methods) is a class of proportional rated voting methods.
It should be noted that these methods follow different types and philosophies of proportionality than most other PR methods; they all fail PSC (though Sequential Monroe voting comes closest).
Because of the nature of rated ballots, it is possible to make assumptions that allow us to examine many different variations of what it means to "represent" voters in the multi-winner context, and to observe to what degree they are all represented.
Categories
When investigating cardinal PR, it is often categorized into optimal PR methods, which generally work by assigning every possible winner set a score based on how good it is, and picking the best winner set out of all possible winner sets, and sequential PR methods, which elect one candidate at a time. Optimal PR has the issue of being non-hand-countable and very computationally expensive and complex (in fact, with large committees, they may be both completely impossible to compute and very, very vulnerable to strategic voting[1]), so in practice, many sequential cardinal PR methods are designed to approximate certain optimal PR methods. When simulating the quality of various cardinal PR methods, sometimes it's common to use optimal PR methods more as "benchmarks" of how good the winner set chosen by the sequential method is, rather than as an actual way of running an election.
The KP transform can be very useful in allowing Approval voting-based cardinal PR methods to work with rated ballots with more than two allowed scores.
Optimal methods
There is a certain parametrization of optimal PR methods that are in effect based on something like a highest averages method. Psi voting and harmonic voting are two voting methods that fall in this category. Psi voting becomes harmonic voting using the KP transform. Logarithms and the harmonic function appear prominently when discussing these methods.
Monroe's method is based on the theory that each voter should only have one representative.
Sequential methods
Sequential Monroe voting, Sequentially Spent Score, and Reweighted Range Voting are the most common such methods. They are each based on different philosophies of what PR should be and differ in the details of how best to optimize for their particular philosophies.
Often-discussed properties
Universally liked candidate criterion: Candidates who are given maximal support by all voters shouldn't affect the proportionality of the remaining candidates in the winner set when elected.
Notes
Because rated voting methods allow a voter to give no candidate the highest score, it is possible to give some voters less power to their ballots if they choose it.
Just about all cardinal PR methods are immune to Woodall free riding, though their vulnerability to Hylland free riding varies. Some, like Sequentially Shrinking Quota, are maximally resistant.
See also
Monotonicity appears very often in discussions of cardinal PR; it is a point of pride that Score voting passes every imaginable generalized form of monotonicity (which practically no other voting methods can), and cardinal PR advocates actively search for PR methods that imitate and extend that feat as much as possible in the multi-winner context.