# [EM] Simulating multiwinner goodness

Jonathan Lundell jlundell at pobox.com
Thu Mar 11 08:29:47 PST 2010

```On Mar 11, 2010, at 4:35 AM, 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.
>
> 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.

As with any choice system based on cardinal utility, there end up being two problems that are not, I think, amenable to solution. One is the incomparability of individual utility measures from voter to voter (and here we're talking about utility deltas, since the utilities are normalized to max=1.0). The other is that, even if comparability were solved, we don't have a means of, in the individual case, determining what they are.

In particular, reported utility isn't very useful, since for the system to work, we need sincere utility, and a utility-based system provides every incentive to strategize. And, as Terry suggests, it's not clear what we *mean* by utility here. Happiness with what? The outcome of the individual election? The makeup of the resulting legislature? The legislation resulting from that legislature?

And even if we could somehow measure the voter's ultimate happiness as a function of legislative outcome and come back in time and cast a vote, we don't have utilities for the counterfactual alternatives.

However attractive it might be to fantasize about functions from cardinal utility to social choice, it comes down to an attempt to square a circle or invent a perpetual motion machine. The attemp might be fun, but we know a priori that it will fail.
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