[EM] Fair and Democratic versus Majority Rules

[Andy Jennings]

On Tue, Nov 16, 2010 at 1:54 PM, Kristofer Munsterhjelm <
km-elmet at broadpark.no> wrote:

> Jonathan Lundell wrote:
>
>> On Nov 16, 2010, at 5:57 AM, Kristofer Munsterhjelm wrote:
>>
>>> I suspect that one can't have both quota proportionality and
>>> monotonicity, so I've been considering divisor-based proportional
>>> methods, but it's not clear how to generalize something like
>>> Webster to ranked ballots. I did try (with my M-Set Webster
>>> method), and it is, to my knowledge, monotone, but it's not very
>>> good in the single-winner instance.
>>>
>>
>> Woodall in 2003 formalized a method he called "Quota-Preferential by
>> Quotient" (QPQ), based on a suggestion by Olli Salmi on this list.
>> Woodall demonstrates that it satisfies DPC, but doesn't say much
>> about other criteria.
>>
>
> I know about QPQ, but I don't think it's monotone.
>
> The kind of "ranked divisor method" I'm thinking about would probably need
> another criterion. The reason I suspect this (like I suspect the
> incompatibility of monotonicity with the DPC) is based on party list
> apportionment methods.
>
> For apportionment methods, no method can meet both population-pair
> monotonicity (moving votes from one party to another won't lead the former
> to gain seats and the latter to lose them) and quota. Divisor methods can
> meet population-pair monotonicity, but they do so by some times failing
> quota, with Webster failing quota the least.
>
> If one can link ranked vote monotonicity and population-pair monotonicity,
> and quota and DPC, then that would suggest that:
>
> 1. you can't have both monotonicity and the DPC.
> 2. divisor methods can be monotone, but they will fail the DPC.
>
> This might be doable by "emulating" party list PR inside a ranked ballot
> method by having every voter vote only for the candidates of some party, in
> a predetermined order, with different voters in the electorate voting for
> different parties that way. I think STV reduces to largest-remainder with a
> Droop quota if you do that, but I am not sure.
>
> I think it would be harder to link population-pair monotonicity to
> ballot-based monotonicity than quota to the DPC. The quota restriction would
> be stricter than the DPC: even if you had, say, a "Imperiali quota
> criterion", you couldn't have both it and monotonicity -- if you can link
> the criteria in the way I mentioned.
>
>
> My M-Set Webster method replaces the DPC with a constraint set that every
> solid coalition of k candidates preferred by v voters should be entitled to
> at least min(k, round(v/q)) of the seats, where q is set to the least value
> where the combined constraints thus produced can all be met.
>
> It seems to work (there's also a margins phase to make monotonicity work in
> certain ambiguous cases), but it's very specific and, as the single-winner
> version shows, not very good at finding compromises.
>
> A full description can be found at
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-March/025641.htmlif you're interested.
>


Have you looked into Monroe's method?  (The American Political Science
Review, Vol. 89, No. 4 (Dec., 1995), pp. 925-940)

Every voter submits a grade for every candidate.  Say there are N voters, M
candidates and S seats to be filled.  A valid election outcome consists of
choosing the S winners out of the M candidates as well as assigning the
voters, evenly, to the winning candidates.  That is, every winning candidate
must be assigned either floor(N/S) or ceil(N/S) constituents.  We define the
quality of the election outcome to be the sum of each voter's grade for his
assigned candidate.  Ideally, we would find the election outcome that
maximizes quality, but that problem is non-polynomial in the size of M and
S.  Here is Warren Smith's complexity analysis:
http://rangevoting.org/MonroeMW.html

In my opinion, Monroe's method works most naturally with score voting
inputs.  It could also be used with approval voting inputs.

It could even be used with ranked ballots, using a positional vector to
translate each candidate's ranking position into a grade.

Ignoring the algorithmic difficulty, what criteria does this method satisfy?
Monotonicity?
Proportionality?
Quota?
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