[EM] Fair and Democratic versus Majority Rules

[Kristofer Munsterhjelm]

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.html 
if you're interested.