[EM] Simulating multiwinner goodness

[Kristofer Munsterhjelm]

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).