# [EM] Party lists and candidate multiwinner elections

Kristofer Munsterhjelm km_elmet at t-online.de
Mon Oct 20 11:31:37 PDT 2014

```On 10/20/2014 06:29 PM, Toby Pereira wrote:
> From: Kristofer Munsterhjelm <km_elmet at t-online.de>
>
>  >So with that in mind, let me alter the example.
>  >You have a set of ballots that rank (or rate, or approve, etc) n
>  >candidates. First, consider the candidates to be parties and run a party
>  >list election with, say, 500 seats. k parties will be elected. Then run
>  >a k-seat multiwinner election using the same ballots, but let the
>  >candidates be individuals.
>
>  >Would there then be any situation where the set of parties that got at
>  >least one seat in the assembly would ideally differ from the set of
>  >candidates that got elected in the k-seat multiwinner election? If so,
>  >when and why?
>
> In the case with 500 seats, there will be far less rounding effects, and
> parties will be able to win seats very accurately according to their
> proportion of the support.
>
> But when you just have k seats (say, 10), the top ten won't all have
> exactly 10% of the support. Someone might have 50%, someone might have
> 2%. In this case, depending on the voting system used, the second
> choices of the voters of the most popular candidates will become more
> significant. There might be a candidate who has no first choices but is
> the second choice for all of the 50% who support the most popular
> candidate. So in this case, they are very likely to be elected. But in
> the 500-seat example, they might not be, particularly if we are using a
> ranked system.

I think I see what you're saying. In party list, voters who foremost
support A can make their support fully count towards A, but in a
multiwinner election, they can't do so. So in the latter, one might
expect the surplus, as it were, to go elsewhere instead and thus to
elect someone who had no direct chance.

That suggests that an example to show this property is undesirable
should have clone parties. Let some party A be turned into A and B, and
where everybody votes these in sequence, A first. Then the equivalence
would imply that either the A-first voters can't get B elected by doing
so (in the multiwinner case), or that in the party list case, B will get
some support at the cost of A.

If the former, the surplus would be wasted in the multiwinner case (if
you make A large enough); and if the latter, that means that ideal party
distribution would include some members of B even when everybody who
votes for B prefers A, and A can simply be given more seats.

> So yes, I'd say there will be cases where they would give different
> results. If I understand correctly, in the k-seat example, it's
> effectively a party election but with each party only fielding one
> candidate. Since there is likely to be a lot of "over-support", which
> won't exist in the 500-seat example, there would and should be different
> results in some scenarios.

It is strange, however, because I think my "Statistical Condorcet"
method passes this criterion (in a sense), and I wouldn't imagine it to
be a kind of undercompromising Plurality-like method.

Basically, it's Schulze STV with a multinomial. In a contest of [XXYZ]
vs [XXYW], where X are candidates from party X (same with Y, Z, and W),
it distributes voters who prefer any of parties {X, Y, Z} to W in such a
way as to maximize the multinomial probability of seeing [XXYW] given
probabilities for drawing X (Y, W respectively) according to the
fraction of voters who were assigned to X (or Y or W). The number of
voters in play, times the value from the multinomial pmf, is then the
score for the contest on the [XXYZ] side.

Now, say that the number of seats, n, is very large. Then the
multinomial pmf returns very low values except when the seats are
exactly proportionally allocated. So the compromising capacity of
Condorcet is removed (as we'd expect). However, this means that we can
calculate the winning set directly based only on which coalitions are
pitted against each other, i.e. only on the parties. So the contest
function in the limit takes the parties "for" and "against", where only
one party can be different from "for" and "against"; and the score is
the number of votes one can redistribute so that all prefers the "for"
bloc to the opposing party (the one that's in the "against" one but not
in the "for" one) while keeping perfect proportionality. But that sounds
a lot like Schulze STV (same kind of logic, if not the same minmax
linear programming system), which is a candidate method!

Hm. This might be too opaque; I can feel that I'm using my internal
intuition of the method even as I'm writing this, so it might be
difficult to understand.
And the contest function for very large numbers of seats might be
sufficiently different from Schulze STV that it isn't really like a
candidate method at all.
```