I wrote 1.7 papers on this topic. Part I: http://rangevoting.org/QualityMulti.html Part II: http://rangevoting.org/NonlinQuality.html The "holy grail" proportional representation voting method is defined http://rangevoting.org/NonlinQuality.html#holygrail and I have been unable to find a method that satisfies those demands. This leads to the conjecture no such voting method is possible. But I have been unable to prove impossibility. The impossibility (if it is impossible) may be topological in nature, so topologists might want to think about it. [People have also asked me about Arrow and Gibbard-Satterthwaite impossibility theorems. Those are not related in the sense they are about single-winner, not PR (multiwinner) voting. But you can learn about them here: http://rangevoting.org/ArrowThm.html http://rangevoting.org/GibbSat.html as well as my own impossibility theorem http://www.rangevoting.org/SimmonsSmithPf.html and there are several more impossibility theorems.] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)