Forest Simmons and I finally found the long-sought "holy grail" combination of properties for a multiwinner election method. (Unless we didn't, because my proof has some bug...) I had previously conjectured the holy grail was impossible. If anybody wants to read preliminary manuscript explaining: http://rangevoting.org/HolyGrailPR.html Unfortunately, our holy grail election method is computationally hard, large computer runtime. I do not know whether that is necessary. I proved that PR (proportional representation multiwinner) elections, to assure good quality results, inherently must solve NP-hard problems. That kind of hardness does not bother me since it is unavoidable. But the holy grail method we discovered is worse than that, in the sense that the "optimum" parliament which it finds, i.e. the one with maximum quality... well, now the quality function ITSELF seems hard to evaluate. I may be able (although so far I haven't) to prove my quality function is NP-hard to evaluate. Is there a way to attain the holy grail with a polytime quality function? We presently do not know, but that would make me a lot happier. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)