[EM] A Framework for adapting single winner methods to the task of Proportional Representation in multi winner districts

[Kristofer Munsterhjelm]

On 1/22/20 12:05 AM, Forest Simmons wrote:
> 
> The Multiwinner Method I have in mind chooses the winners sequentially.
> It is based on the idea that ballots have an initial weight of one, and 
> that as candidates supported by a ballot are added to the winners' 
> circle, the weight is reduced according to some rule designed to 
> diminish the influence of the voters who have already achieved some 
> level of satisfaction.
> 
> At each stage in the election the new seat is filled by the candidate 
> picked by the single winner method applied to the entire ballot set with 
> the current ballot weights in force.
> 
> How, in general, do we diminish the weight of a ballot? Perhaps the 
> simplest way is to make the current weight 1/(1+S) where S is the 
> current satisfaction obtained by comparing the ballot preferences 
> (whether ratings or rankings) with the winners elected so far. As long 
> as the current satisfaction is zero, the weight remains at one since 
> 1/(1+0) is just one.

A quick reply (been a bit busy lately): Approval methods need to pass a 
weaker proportionality criterion than ranked methods. For Approval, you 
just need to give X a seat if enough voters approve X, but Droop 
proportionality is nested: a vote can contribute to multiple solid 
coalitions at once.

Thus I'm not sure basing a ranked proportional method on Approval will 
lead to a good outcome, at least not if that's not explicitly taken into 
account.

E.g. consider the "D'Hondt without lists" proposal from 2002. It 
combined reweighting with pairwise matrices, but I'm pretty sure it 
fails the DPC.