[EM] Party lists and candidate multiwinner elections

[Kristofer Munsterhjelm]

On 10/20/2014 05:18 AM, Kathy Dopp wrote:

> Yes.  There is a big difference in those two situations, depending on
> the exact proportions of votes cast per party or candidate. You may
> have fewer than 10 parties winning seats in the 500 seat assembly
> because some parties may have much much larger share of the votes than
> others.  However, in the 10 seat election, each seat must be held by a
> different candidate, so 10 people must be elected if there are at
> least 10 candidates.
>
> Unless you forgot to mention an additional rule for the party list
> elections that there must be exactly 10 parties elected.

That's a good point. The relation I'm trying to investigate would be 
most clear if the party list PR election was forced to have exactly 10 
parties, but that is very unintuitive for a party list election. Since 
I'm asking about what "should be" or whether there would reasonably be a 
difference, rather than what certain methods would do in practice, I 
think it would be good to keep the party list PR elections reasonably 
accurate.

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?

The reason I use a very large number of seats is to keep what one might 
call "quantization effects" to complicate the picture. Such effects 
happen when there aren't enough seats to represent each faction fairly, 
and thus a method might need to elect a compromise candidate instead. 
So, again, a better formal question might be "let the number of seats 
tend to infinity and the number of parties to elect be fixed at k. 
Assuming the party list method stabilizes in the limit, is or should 
then the subset of parties elected in a party list method be the same as 
the corresponding candidates elected by a k-winner multiwinner election 
method with the same ballots?". But again, I admit that it might be hard 
to reason about. Hence my unrestricted party list formulation.