[EM] PR methods and Quotas

[Andy Jennings]

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!

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.

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.

Thoughts?

Andy
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