[EM] How to make summable any method based on range ballots.

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:
> How to make summable any method based on range ballots.
> 
> If no range ballot rates more than one alternative at the top range
> value, then replace each ballot with the average of all of the
> ballots that have the same favorite.
> 
> Otherwise, first split each ballot into n ballots, one for each equal
> top alternative on the ballot, and assign a weight of 1/n to each
> such ballot for use in the averaging.

I'm not sure, but I don't think that could be used in a multiwinner 
context. The idea would be to design a set of Approval ballots (special 
case of Range) and a method passing Jameson's "approval DPC" criterion, 
so that when averaged the way you mention, the reconstructed ballots 
would elect a different council than that imposed by the DPC criterion 
constraints on the original ballots.

I know it's possible with ranked ballots and STV. Just make everybody 
unanimously rank X top, then that preference will average out the 
second, third, etc, ranks and cloak the rest of the information required 
to find the DPC constraints.
You could do something similar in a multiwinner method based on 
Range-100 by having everybody rate X at 100 and rate the rest of the 
candidates as Range-99, but Jameson's approval DPC criterion doesn't 
technically apply to that situation. By increasing the range of the 
Range multiwinner method, you could get arbitrarily close, however, and 
it would be hard to argue that a method should pass the criterion if 
everybody only max- and min-rates, but if they instead rate all but a 
single of the approved candidates ever so slightly lower, it should 
provide a much less proportional result