Brian Olson wrote:
There was a question on the list a while ago, and skimming to catch up I didn't see a resolution, about what the right way to measure multiwinner result goodness is.
[snip]
This is sounding a bit like an election method definition, and I expect that this definition of 'what is a good result' does pretty much imply a method of election. At worst, given ratings ballots that we can treat as the simulator preferences, for not too large a set of winning sets of candidates, get a fast computer and run all the combinatoric possibilities and elect the set with the highest measured sum happiness.
The details of proportional representation isn't well known. Proportional representation itself appears to involve a tradeoff between accuracy - proportionality of what counts - and quality - how highly the individual voters rank a given candidate.
There is something similar for single-winner methods: the question of how much to value what few rank very highly in comparison to what some rank in the middle; but for single-winner methods, we at least have concepts like the "median voter" and desirable-sounding criteria like clone independence and the Condorcet criterion.
What I'm trying to say is that before we can optimize, we must know what it is we're going to optimize -- or proceed in a vague direction using feedback (as is part of my reason for experimenting with multiwinner methods). What would be analogous to the median voter concept for multiwinner elections - accurate reproduction of opinion space? According to what measure? And so on...
Another thing we could measure in multiwinner elections (and possibly single winner) is the Gini inequality measure. If we have a result with both pretty high average happiness and low inequality, that's a good result.
The proportionality scoring part of my election methods program works somewhat like this, according to a very simple model. Every candidate and voter has a binary n-vector of ayes/nays (representing binary opinions). Voters prefer candidates closer to them (Hamming distance wise). Then the proportion of each bit being a "yes" can be measured both for the elected council and for the people in general, and the closer the better.
I use either root mean squared error or the Sainte-Lague index for measuring error, though my program can also use the Gini (or the Loosemore-Hamby index for that matter).
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