[EM] Preferential Party-List Proportional Representation (PPLPR)

[Kristofer Munsterhjelm]

On 11/09/2014 07:01 PM, Vidar Wahlberg wrote:
> On Sun, Nov 09, 2014 at 02:57:57PM +0000, Toby Pereira wrote:
>> The problem is how you want to define a proportional system. As you
>> say, cardinal systems handle this better because you know how much
>> support B really has, and you can allocate accordingly. But when you
>> have ranks, it would go against how most people would define
>> proportionality. I would say that if a certain proportion of people
>> rank a certain party top, then that party should get that proportion
>> of seats, subject to rounding errors.
>
> If the definition of PR systems is that parties should receive as much
> proportion of the seats as proportion of voters who rank that party at
> top, then I agree, PPLPR obviously does not do that.
> Interestingly, that more or less exclude every single other system than
> plurality systems, with the possible exception of systems where voters
> can split their vote. Even STV would not meet this definition; A
> party/candidate could receive votes from later preferences and gain
> representation beyond rounding errors. Using my previous example, B
> could, depending on the STV implemention, win a seat in a 6-seat
> election (16.7%) with only 1 of 401 votes (0.25%).

A more general proportionality criterion (for candidate multiwinner 
elections) is the Droop Proportionality criterion: if more than k Droop 
quotas vote a set of n candidates ahead of everybody else (but not 
necessarily in the same order), then min(k, n) of the members of that 
set should be in the outcome.

If the election is a p-seat one and there are v voters in total, then a 
Droop quota is v/(p+1).

This particular criterion reduces party list to LR-Droop (and this fails 
population pair monotonicity), so it's not perfect. However, it does 
give a more general definition of one kind of proportionality.

If we extend Toby's party list PR example to a candidate multiwinner 
election, it would be something like:

50: L1>L2>L3>...>Ln>C1>...>Cn
50: R1>...>Rn

Let the number of seats be suitably large, say 1000. Then a Droop quota 
is 100/1001 = 0.099900... The first group represents more than 500 Droop 
quotas (but not more than 501) and the same goes for the second group. 
So 500 of the seats should be given to L and C, while the other 500 
should be given to R. If n is greater than 500, then L will get half of 
the assembly and R will get the other half.

The single-winner special case of this criterion is the mutual majority 
criterion, which some Condorcet methods (like Schulze and Ranked Pairs) 
pass.

STV passes the Droop proportionality criterion. My most recent 
multiwinner method probably does not, since it's based on divisor method 
logic, but I haven't found any good analogous divisor-method based criteria.