[EM] Median-based Proportional Representation

[Kristofer Munsterhjelm]

Jameson Quinn wrote:
> 
> 
> 2011/7/8 Warren Smith <warren.wds at gmail.com <mailto:warren.wds at gmail.com>>
> 
>     Sorry, as Jameson pointed out, he has invented a voting method he
>     calls AT-TV
>     which (he claims)
>      1. obeys a proportional representation theorem
> 
> 
> Yes. It's instructive to see what PR criterion AT-TV satisfies, and what 
> RRV satisfies.
> 
> The standard Droop criterion, if I'm not mistaken, is: If a group of N 
> droop quotas of voters votes a set of >N candidates above all other 
> candidates, then at least N candidates from that set must win.

It's actually, if *more* than n droop quotas vote for a set of k 
candidates, then min(k,n) must win. So to win exactly the Droop quota is 
not enough, as there may be too many candidates, e.g. for 90 voters to 2 
seats:

30: A>B>C
30: D>E>F
30: G>H>I

Then if it had been "n Droop quotas", not "more than", it would say 
"elect one of {ABC}, and one of {DEF}, and one of {GHI}", which is 
impossible.

> The AT-TV version would be: If a group of N droop quotas of voters votes 
> a set of >N candidates at or above a given rating, and all other 
> candidates below that rating, then at least N candidates from that set 
> will win.
> 
> The RRV version... well, I'm not sure, but my guess is that it would be 
> something like: If a group of N droop quotas of voters votes a set of >N 
> candidates each with at least N times the rating of any candidate 
> outside that set, then at least N candidates from that set must win.
> 
> Note that these are successively weaker criteria on the systems; that 
> is, the coordination of a given party must be successively stronger to 
> ensure PR for that party. Purely on a subjective level, I think that 
> AT-TV criterion is about right, and that the RRV one is too weak.

I agree that the RRV one is too weak.

I further think we could classify proportionality criteria into 
categories such as these:

- very weak: It is possible for a party or group of candidates to get a 
proportional outcome if they coordinate and can control their voters. 
(SNTV, Taiwanese birthday type strategies.)

- weak: It is possible for a group of voters to force a proportional 
outcome if they can coordinate among themselves.

- middle: It is possible for a group of voters to force a proportional 
outcome with regards to a group of candidates they like (but not within 
that group) by employing a common strategy (RRV "totally racist" criterion).

- stronger: If a group of voters reward a group of candidates with some 
desirable relative aspect of their votes (top rank, high scores), then 
the candidates will receive seats in proportion to the size of the voter 
group and the desirable aspect, and this also counts recursively. (Droop 
proportionality, divisor analogs.)

Would that let us find a rated version of the DPC? If proportional 
representation was to be by rating instead of by seat count, it might 
involve that more than a Droop quota can determine the ratio of the 
rating of a candidate they rate highly to the rating of a candidate they 
rate lower... but it seems difficult to get any closer than that.

Perhaps, for the utilitarian "proportionality of ratings" approach, it 
would make more sense to look at party list methods first (since the 
concept of having different ratings for winners makes sense there), then 
generalize.