More on PR.

http://scorevoting.net/PropRep.html

Regarding single-winner methods, Score Voting is best. http://scorevoting.net/BayRegsFig.html

Score/range voting and approval voting are particularly bad for single winner elections in ways that typical mathematical simulations fail to account for because mapping from sentiment to binary approve/disapprove or score ratings is inconsistent between individuals (and, as has been studied fairly extensively in the case of score rankings in non-bullying contexts, have particularly strong ethnic/cultural variation.)

If you could directly measure internal sentiment and use a consistent function to map it it to scores, score/range voting would be a good method, and mathematical simulations tend to model this case rather than reality.

(Approval voting is good for group decisions with non-secret ballots where people can opt-out of the activity selected, and an "approve" vote is also a commitment to opt-in, because then there is a consistent meaning; range/score voting similarly can be good in group decision-making what shows are tied to a concrete price that is to be paid for or to avoid an outcome, which again makes the ballot marking have a consistent meaning.)