V voters score each of C candidates on an 0-to-1 real scale. Those are the ballots. The goal is to use them to elect W winners, 0<W<C. The question is how. I've been working on this for some time and every system I know of seems unsatisfactory in some easily identifiable way or another. Maybe one of you will come up with a new idea that will be enough to break the logjam; or perhaps it just is impossible to satisfy all of my desiderata. Desiderata include * PR: if all voters & candidates are "colored," each voter scores same-color candidates 1 and other-color candidates 0, then the parliament should have same color composition as electorate (up to unavoidable errors caused by integer roundoff, or if too few candidates of some color run) * Stronger PR: if there also are uncolored candidates, who each receive scores from voters that depend only on the candidate, not on the voter, then the colored part of the parliament should have same color-composition as electorate. * Monotonicity: if voters raise a candidate's score, that should help him win. In particular, if parliament A gets the same scores or better than parliament B in all cases, with at least some "better," then B should not be the winner-set. * Simplicity of description of the system. * Single-winner case (W=1) should just be "highest average score wins." * Scaling invariance: if all input scores are multiplied by some C>0, the winning parliament should be unaltered. * I also have a notion of "continuity" that I want, but won't formulate it here. I've got systems that satisfy any of these desiderata, but none satisfying all of them simultaneously. The best I've been able to do so far is satisfy all except "stronger PR." I can prove any system that does all that must solve NP-hard problems. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)