[EM] PR methods and Quotas

[Kristofer Munsterhjelm]

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.

Actually determining where to move each voter to optimize this might be 
quite hard, though. But if you could make it work, then that would seem 
to do what you wanted: it gives one candidate to represent the first 
n/k, one candidate to represent the next n/k, etc, and picks the council 
that makes these people most happy.

> Say there are 100 voters and you're going to elect ten representatives. 
>  Each representative should represent 10 people, so why not choose the 
> first one by choosing the candidate who makes 10 people the happiest? 
>  (The one whose tenth highest grade is the highest.)  Then, take the 10 
> voters who helped elect this candidate and eliminate their ballots. 
>  (There might be more than ten and you'd have to choose ten or use 
> fractional voters.  I have ideas for that, but lets gloss over that 
> issue for now.)  You can even tell those 10 voters who "their" 
> representative is.

I imagine you could eliminate the voters directly, though that would 
have some path dependence problems (which was why I suggested the 
above). Say you make use of highest tenth grade. Then you know which 
voters voted the candidate in question that high. Eliminate these. Find 
the highest tenth with those voters elminated, among uneliminated 
candidates. Again, you know the 10 voters who voted the next winner at 
that level or higher. Eliminate *them*. And so on down.

Is that what you're suggesting? 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.

> Electing the next seat should be the same way.  Choose someone who is 
> the best representative for 10 people.  Repeat.
> 
> The only problem is when you get down to the last representative.  If 
> you follow this pattern, the last candidate is the one whose LOWEST 
> grade among the remaining ballots is the highest, which is rather 
> unorthodox.  You could change the rules and just use the median on the 
> last seat, but using the highest minimum grade does have a certain 
> attraction to it.  You're going to force those last ten voters to have 
> some representative.  It makes some sense to choose the one who 
> maximizes the happiness of the least happy voter.  (Though ties at a 
> grade of 0 may be common.)
> 
> But this system doesn't reduce to median voting.  Which got me 
> thinking...  Is there anything that special about the 50th percentile in 
> the single-winner case anyways?  I can imagine lots of single-winner 
> situations where it's more egalitarian to choose a lower percentile.  In 
> a small and friendly group, even choosing the winner with the highest 
> minimum grade is a good social choice method.  It's like giving each 
> person veto power and still hoping you can find something everyone can 
> live with.  This is the method we tend to use (informally) when I'm in a 
> group choosing where to go to lunch together.

I think the median is used because it's robust. If you assume unlimited 
ratings, the maximum and minimum could be altered by a single voter 
(whoever's at the min or max), as could the mean (by any outlier). 
However, the median is robust to distorted values - quite a number of 
voters would have to change their votes to alter the median.

In one way, then, the median is a way of robustly estimating a property 
related to the shape of the function given by the voters' ratings, even 
in the presence of noise (or strategy). To keep this reasoning for 
multiwinner, one should find out what properties one want to know about 
for multiwinner elections, then find a way of robustly estimating these.