[EM] A simpler approval based way of replacing the CA jungle primary

[Kristofer Munsterhjelm]

On 2018-07-19 00:26, Rob Lanphier wrote:
> Hi folks,
> 
> During the California jungle primary, it's become clear to many of my
> fellow activists that our top-two jungle primary doesn't make sense.
> A while back, I posted "Party-based top two with approval", which
> resulted in a great on-list conversation with Kevin Venzke.  As I've
> been talking to folks, I've been afraid of bringing up my complicated
> rules or pushing my system via my blog/wherever.
> 
> After mulling it over, I think this simple version could get traction.
> The rules, in a nutshell:
> 
> a. All candidates who receive over 50% approval advance to the general election
> b. If less than two candidates get 50% approval, then advance the two
> candidates approved by the most number of voters
> 
> There's some intentional ambiguity in that second bullet point.  The
> goal would be to find two candidates for whom the largest portion of
> the electorate approves at least one of.  So, let's say we have three
> candidates: A-left, B-center, and C-right.  A-left and C-right are
> popular with their respective base voters, and B-center is a weak
> centrist.  Let's the voters vote like this:
> 
> 25% - approve only A-left
> 20% - approve A-left and B-center
> 10% - approve only B-center
> 20% - approve B-center and C-right
> 25% - approve C-right
> 
> This would result in the following approval scores for the individual
> candidates:
> 
> A-left: 45%
> B-center: 50%
> C-right: 45%
> 
> However, in this version of the rules, we look for the pair of
> candidates where at least one candidate meets with approval.
> 
> AB (A-left and B-center): 75%
> BC (B-center and C-right): 75%
> AC (A-left and C-right): 90%
> 
> Using the rules in this proposal, A-left and C-right would advance to
> the general election.  It would be possible to layer some more
> slightly more complicated rules on top of this system to avoid this
> flavor of center squeeze.  However, these rules dissuade candidates
> from relying on a "mushy middle" lesser-of-either-evil campaign, but
> instead, push candidates to earn the approval of either left or right
> base voters.  Moreover, in this scenario, both A-left and C-right
> would be worried about the possibility of their ideological opposite
> getting 50% approval thus making it so that two candidates have the
> required 50%.  Trying to divide the electorate rather than achieving
> 50% approval would an extremely risky strategy.

This sounds like a combination of majoritarian and minmax Approval. I 
think that a two-round system should contain the winner of the first 
round so that if the first round is correct, the winner is preserved.

So how about making that more explicit? Say something to the effect of:

The Approval winner advances, and
two other winners advance so as to maximize the number of ballots that 
approve of at least one of them.

By having three instead of two, a center candidate won't bias the runoff 
towards either the left or right in an LCR situation. On the other hand, 
it's harder to justify once opinion space becomes multidimensional; and 
having three instead of two does lead to the Approval dilemma of whether 
to approve your favorite only or also approve your second best.

You could run the second round with a ranked method, but that would make 
it a lot more complex. Letting two candidates pass instead of three 
would solve the problem, as you could use Plurality for the second 
stage, but any LCR situation would be biased: either it would be 
left-leaning (LC), right-leaning (RC), or miss the centrist (LR).

 From a design perspective, the best would probably be to have the 
Approval winner advance, and as many additional candidates as are needed 
to achieve proportional representation up to some set level (which would 
act as an effective threshold). That suggests using a house-monotone PR 
method to pick the candidates for the second round; but that would be 
anything but simple.

> There are many ways of dealing with this:
> 
> 1.  Only allow 2 candidates to advance, keeping with the spirit of
> "top two", and use simple plurality in the general election.
> 2.  Have higher limit (e.g. 5 candidates) and only allow the top 5
> approval getters to advance.  Tally the general election using
> approval voting > 3.  Choose the 5 candidates for whom at least one is approved.
> Calculating this seems complicated, but the goal would be the 5
> finalists would be A1-left, A2-left, A3-left, B-center and C-right.
> Once again, tally the general election using approval voting.

If I recall correctly, choosing the candidates to maximize the number of 
voters who approve of at least one of them is NP-hard for general n. See 
https://en.wikipedia.org/wiki/Maximum_coverage_problem.

The best you can do without P = NP is a very obvious greedy algorithm 
(successively pick the candidate that maximizes the number of additional 
voters covered by the set of candidates so far). The greedy algorithm 
makes it relatively easy to start with one or more candidates picked by 
other means (e.g. the Approval winner) and then filling in the rest.