[EM] Two election method observations

[Forest Simmons]

Cool ... I think you're onto something good!

On Mon, Feb 20, 2023, 6:08 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> First: Let's say that method B is a more decisive version of method A if
> you can get B by breaking ties in the outcomes produced by A. (I.e. B's
> outcome never ranks X ahead of Y whn A ranks Y ahead of X, but sometimes
> ranks X ahead of Y when A ranks X equal to Y.)
>
> Then it's possible for B to be summable even when A isn't. So
> tiebreaking can make matters *easier*. (This unfortunately makes
> election method design harder.)
>
> Second: Suppose you have some black box method that only does "n-1 out
> of n" multiwinner elections, i.e. where the number of seats is one less
> than the number of candidates. Suppose this method is Droop
> proportional. Then the following multiwinner method is Droop
> proportional in general and extends the method to any-seat elections:
>
>         1. Do an n-1 out of n election.
>         2. If n-1 is equal to the number of seats, you're done and that's
> the
> outcome.
>         3. Otherwise, eliminate whoever was not elected and go to 1.
>
> My proof idea is like this: Suppose that we're trying to elect k seats
> and there are n candidates, and |V| voters. Then the Droop quota for k
> seats is |V|/(k+1). So suppose that the Droop proportionality criterion
> states that q candidates from some coalition must be elected because
> q|V|/(k+1) voters ranked them first. Then since the Droop quota for n-1
> seats is lower than that for k seats, at least q candidates from the
> coalition must also be elected in the (n-1) seat election. Hence the nth
> candidate (the one who's not elected when the (n-1) seat election is
> performed) can't be any of the q. So eliminating him can't violate any
> future Droop constraints.
>
> The rest follows pretty straightforwardly from induction.
>
> So in a sense, the hard part of a proportional multiwinner method is the
> (n-1) out of n election. Solve that and you can get Droop
> proportionality in general -- although it would probably be neither
> monotone nor summable.
>
> The funny/interesting part is that no actual multiwinner election method
> I know of is constructed this way. Perhaps something notable could be
> found in the space of election methods that look like this?
>
> -km
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