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

[Kristofer Munsterhjelm]

On 01/12/2017 06:49 AM, Rob Lanphier wrote:
> On Tue, Jan 10, 2017 at 3:56 PM, Kristofer Munsterhjelm
> <km_elmet at t-online.de> wrote:
>> 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).
>
> This is an intriguing corner case.  Let's define a goal, and see if we
> can find a system that meets it.
>
> Possible constraints:
> *  An open primary that allows for two viable candidates to emerge
> *  A general election where the largest percentage possible has
> expressed approval of at least one candidate in the primary
>
> Approve-and-removal seems to result in the outcome you suggest.
> Worse, the lack of summability makes the method uncomfortably complex.
>
> It seems that it's worth exploring the full range:
> #1:
> 51: A B
> 49: C
>
> #2
> 70: A B
> 30: C
>
> #3:
> 99: A B
> 1: C
>
> Scenario #1 seems to clearly call for some form of
> approval-and-removal to ensure proportionality.  A vs C would be
> respectable choices for the general.
> Scenario #3 is an extreme case where a crackpot is able to gain the
> approval of a tiny slice of the electorate, and the general election
> seems a waste of time.  A vs B seems the best choice for the general.
>
> However, what about scenario #2?  Allowing A & B to move the general
> seems to marginalize the views of C in a way that seems like a
> disadvantage when considered against old-school, closed-primary FPTP.
>
> The percentage chosen in my scenario #2 was somewhat arbitrary, and
> was one that involved working within the two candidate general
> election constraint.  Perhaps that number is worth playing with.  Or,
> perhaps "number of candidates in the general election" is instead a
> number worth playing with.

Suppose we use Droop quota logic. In each of these, the Droop quota is 
33 + 1/3, say 33 for simplicity's sake. So in scenario 1, the surplus is 
18; in scenario 2, it is 37, and in scenario 3, it's 66. Thus the 
outcome would be {A, C} in #1 and {A, B} in #2 and #3.

>> 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.)
>
> This deserves more thought than I can give it right now.  It seems
> worth examining further.

I haven't had time to look at it thoroughly, but it seems to be 
summable, which ordinary approval-and-removal isn't (as you said).

To calculate X's adjusted Approval score, you only need:
	- The number of voters who approved both W and X
	- The number of voters who approved W but not X
	- The number of voters in total (for the 1/3 calc)

If you have the number of voters who approved W but not X, and you have 
the unadjusted Approval counts for every candidate, you can get the 
number of voters who approved both W and X. So it seems all you need is:

- The number of voters in total: O(1)
- For each candidate X: the number of voters approving X: O(n) in total
- For each pair of candidates W, X: the number of voters who approved W 
but not X: O(n^2) in total

which gives summability with order n^2.

So we have an interesting contrast: One method is simple to explain but 
not summable. The other one is more complex, but is summable.

(It's vaguely Condorcetian if you read "approves W but not X" as W>X.)

The 1/3 threshold would probably mean it fails IIB. It should be 
possible to fix this (at the cost of more complexity still), with the 
general idea being to eliminate candidates with the fewest Approvals 
until the winner has at least 1/3 of the remaining votes.