On Mon, Jul 18, 2011 at 6:00 PM, <[email protected]> wrote: > Andy and I were thinking mostly of Party Lists via RRV. His question was > that if we used RRV, either > sequential or not, would we get the same result as the Ultimate Lottery > Maximization. I was able to > show to our satisfaction, that at least in the non-sequential RRV version, > the results would be the > same. It seems like the initial differences between sequential and > non-sequential RRV would disappear > in the limit as the number of candidates to be seated approached infinity. > > Would that imply P=NP? In other words, sequential RRV might be an > efficient method of > approximating a solution (for large n) of non-sequential RRV (which is > undoubtedly NP hard). What > would be analogous in the Traveling Salesman Problem? Don't hold your > breath, but it would be > interesting to sort out the analogy, if possible. >
I am still hopeful that sequential RRV with a large number of seats, leaving each candidate in as if they were their own party, would be a good and tractable way to choose legislators and give them each a different amount of "voting power". I'm hoping it would be possible to calculate the proportions in the limit as n goes to infinity. But sequential RRV is completely ignorant about how many seats need to be filled, so it's not really going to find the globally optimum N-winner representative body like ULM and non-sequential RRV aim to do. This "infinite sequential RRV" might be good when there is no pre-determined number of seats to fill but instead we want the method to choose the number of winners. For real elections, however, I suspect that it will give some voting power to every candidate, so maybe it's not that good for choosing a representative body. Here's an example, on the other hand, where this method chooses too few winners: 10 voters approve A and C 10 voters approve A and D 10 voters approve A and E 10 voters approve B and C 10 voters approve B and D 10 voters approve B and E If you're choosing two winners, I think the obvious winners are A and B. But if you want to choose three winners, I think the obvious choice is C, D, and E. Only a method that knows how many winners you're going to choose can make the correct decision here. In this case, RRV will choose A and B. If A and B are "left in" (pretending they are parties even if they are candidates) then RRV will continue to alternate between A and B. In the limit, it will give half of the voting power to A and half to B. This is just not helpful if you wanted to choose three winners. ULM and non-sequential RRV evaluate each possible combination of winners and can do the right thing in the three winner case. Andy
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