[EM] Proportional Representation from Ratings Ballots

[Raph Frank]

On Thu, Nov 19, 2009 at 4:51 AM, Brian Olson <bql at bolson.org> wrote:
> Oh, that is a problem. It gets the right answer if I use L1 norm instead of
> L2. I think L2 norm is going to work better for single-seat IRNR but L1 norm
> better for multi-seat. L2 inflates the amount of vote that winds up getting
> applied to multiple choices.

The L1 norm mean that each voter always gets to cast exactly 1 vote
(ratings add to 1).  Thus the total number of votes cast is always
constant.  This means that a quota can be easily determined.

You could use a different rule for eliminating than you use for
electing (and I think that is a good idea anyway).

For example, for electing, each ballot is scaled so that

w(a)*r(a) + w(b)*r(b) + ..... = 1

All eliminated candidates have a w(x) = 0 and all non-elected
candidates have a weighting of 1.
Elected candidates have weighting so that they have exactly a quota of
the votes.

If any candidate meets the Droop quota, that candidate is declared
elected and the next round is started.

If no candidate is elected, a different rule is used, each ballot is
scaled so that

[w(a)*r(a)]^2 + [w(b)*r(b)]^2 + ... = 1

The running candidate the the lowest score is then eliminated.

(The weights are based on the L1 calculation)

This process has the nice feature that a group of voters equal to a
Droop quota will decide their candidate using the L2 single seat (L2)
version of the process.  (This assumes that they rate all non-party
candidates at zero and all voters outside the group rate their
candidates at zero).

Also, there is also a question if the weights assigned in step 1 will
always yield a unique set of weights.  Hopefully there is a Meek's
method like proof.