[EM] Nondeterminism in Multiwinner Methods

[Raph Frank]

On Wed, Oct 29, 2008 at 8:42 PM, Kristofer Munsterhjelm
<km-elmet at broadpark.no> wrote:
> There's another problem. If you pick n ballots, with some probability more
> than one ballot is going to have the same first place candidate. This might
> be solvable by picking the first place candidate of the first of the n
> ballots, then eliminating it, and so on. Would that be cloneproof?

It depends on how you handle the elimination.

If each voter submits a ranked ballot and the highest ranked
(non-elected) candidate on the ballot is elected, then I think it is
clone proof.  The order of the ballots being drawn matters, but that
is just randomness.

A faction cannot increase its probability of winning by fielding many
candidates.

> I think Woodall wrote that Clone-Winner is incompatible with Droop
> proportionality; but what of Clone-Loser?

Dunno.  However, the above system is not guaranteed to be Droop
proportional, it is just proportional on average.

> It should pass Clone loser, since
> adding a clone never splits the vote, since the worst thing that could
> happen is that one of the clones are eliminated, after which another clone
> may be in first place on the other ballots.

Sounds reasonable.

> It might be nonmonotonic. The monotonicity criterion for nondeterministic
> methods would be "raising A on a ballot shouldn't decrease A's probability
> of winning". The single-winner equivalent would be "IRV with random ballot
> elimination".

I think it is probably still non-monotonic.  If you decrease your
favourite's chance of being eliminated, he might end up making it to
the final round and then losing.  However, if you don't boost your
favourite, your 2nd choice might be able to beat your least favourite.

I don't think elimination order would help here.