[EM] How would you fix California's top two primary?

[Kristofer Munsterhjelm]

On 01/10/2017 09:13 PM, Monkey Puzzle wrote:
> If Top Two is required, I would prefer Approval voting, then include the
> Approval winner (AW1), plus the approval winner after all AW1-approving
> ballots are removed.  This would be clone independent and would
> generally tend to include candidates from two different parties.  It is
> basically a 2-person multiwinner election using Approval reweighted voting.

That goes a bit too far in the other direction. Consider a profile like
this:

99: A B
 1: C

It seems pretty clear that the candidates to go to the second round
should be A and B, but Approval-and-removal will pick either {A, C} or
{B, C}. Of course, the real world probably won't have this kind of
pathological election situation, but the bias is still there to a lesser
degree: it disproportionately picks "extremists" for the second seat
(i.e. candidates whose voters wouldn't vote for the first winner).

If your complexity budget is so that you can't do anything more complex
than approval-and-remove, go with approval-and-remove because it's
better than just picking two Approval winners right out. But if you can,
the following might be better:

A little bit more complex: First pick the Approval winner. Then randomly
remove ballots that approved of this winner until you've either removed
every ballot that approves of the winner, or 1/3 of the total number of
ballots. Then pick the Approval winner by the remaining ballots (ignore
the first winner if he's still number one).

This is closer to Droop-proportional, but has a vote management
incentive. The following mitigates the vote management incentive, but is
more complex still:

First pick the Approval winner W.
For each other candidate X:
Until you have removed 1/3 of the total number of ballots or every
ballot that approves W, first remove ballots that approve W but not X,
then ballots that approve both W and X. Count X's approval according to
the remaining ballots after removal, then put the ballots you removed
back in the pile so you can repeat for the next candidate.

The candidate with the greatest thus counted approval score gets the
second seat in the runoff. (This is essentially the constraint method
with two seats and Approval.)