[EM] Simulating multiwinner goodness

[Brian Olson]

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.

Here's a simple way to do it in simulator:
Each voter has a preference [0.0 ... 1.0] for each candidate. Measuring the desirability of a winning set of candidates is simply summing up those preferences, but capping each voter at a maximum of 1.0 satisfaction.

Unfortunately, this won't show proportionality. If 3/4 of the population have a 1.0 preference for a slate of 3/4 of the choices, we would measure electing one of them as being just as good as electing the whole set.

So, we could apply the quota. If a candidate is elected by 3 times the quota, only apply 1/3 of each voter's preference for that candidate to their happiness sum.

Now the huge coalition with their slate elected should each add up to about 1.0 happiness, and smaller coalitions should get theirs too.

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.

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.