[EM] PR methods and Quotas

[Kristofer Munsterhjelm]

Andy Jennings wrote:
> Kristofer Munsterhjelm wrote:
> 
>     Andy Jennings wrote:
> 
>         Like Jameson and Toby, I have spent some time thinking about how
>         to make a median-based PR system.
> 
>         The system I came up with is similar to Jameson's, but simpler,
>         and uses the Hare quota!
> 
> 
>     How about clustering logic? Say you have an electorate of n voters,
>     and you want k seats. The method would be combinatorial: you'd check
>     a prospective slate. Say the slate is {ABC...}. Then that means you
>     make a group of n/k voters and assign A to this gorup, another group
>     of n/k other voters and assign B to that group, and so on.
>     The score of each slate is equal to the sum of the median scores for
>     each assigned candidate, when considering only the voters in the
>     assigned candidate's group. That is, A's median score when
>     considering the voters of the first group, plus B's median score
>     when considering the voters of the second group, and so on. The
>     voters are moved into groups so that this sum is maximized.
> 
> 
> The median is not what you want for clustering like this, because it 
> basically ignores the scores of half the voters assigned to each 
> candidate.  That is, if I'm assigning 11 voters to each candidate, I can 
> assign 6 voters who love that candidate and 5 voters who hate the 
> candidate and still have a very high median.

Well, yes, but the same thing holds for median ratings in general. If 
you want to find someone who represents the whole population, median 
ratings can pick someone who is loved by 51% and hated by 49%, rather 
than someone that 80% think are okay (and I think Warren have made 
arguments to the effect that this makes Range better than median).

The question then is: what makes that logic okay when you're electing a 
single representative for the whole population, but not okay when you're 
electing one of ten representatives for 10% of the population? Is it the 
fluid nature of the clustering - that the optimizer could try to 
artificially inflate the scores by packing "hate A" voters into the A-group?

>     Then the last candidate is only the one with the best worst votes in
>     the sense that there are only ten voters left.
> 
>     How about using the midpoint? That is, you find the 5th voter down,
>     not the 10th. Then when you're down to the last 10 voters, the 5th
>     voter down is the median. Doing so would seem to reduce it to median
>     ratings in the single-winner case, since 100/1 = 100, so you'd pick
>     the midpoint, i.e. at the 50th voter, which is the median.
> 
> 
> True, but in filling the first seat, I don't think we should take a 
> candidate loved by 5 and hated by 95 as the first choice to represent 
> one-tenth of the population.

I guess you could be more gentle by placing the point at 50% (1/2) for 
one winner, 1/3 for two, 1/4 for three ... 1/11 for ten. That would be 
more Droop-like and less Hare-like. But then you can't simply eliminate 
those who contributed to the voting, I think.

> With any finite number of voters, the median is still the score of one 
> voter, who can change the median by changing his vote.  But you are 
> right that if the scores follow a normal distribution, then he probably 
> can't change the median very much before he crosses another voter's 
> score and is not the median vote anymore.   But that's not true for a 
> bimodal distribution.

He can't alter the median to an arbitrary extent, however. An outlier at 
the mean can do so by setting his score arbitrarily high (or low), and 
the max or min voter can do so, but to a limited extent, by raising his 
score (if he's max) or lowering his score (if he's min). If the median 
voter alters his score by too much, he's no longer the median voter. 
That may change the median result by some amount (unless the new median 
voter expresses the same score as the old one used to), but it's limited.

Ah, there is a term for this reasoning. 
https://secure.wikimedia.org/wikipedia/en/wiki/Breakdown_point#Breakdown_point

I haven't investigated it in detail though :-)