[EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)

[Abd ul-Rahman Lomax]

At 11:29 AM 3/28/2010, Chris Benham wrote:
>Jameson Quinn wrote (26 March 2010):
>
><snip>
>"Right now, I think MCV - that is, two-rank, equality-allowed Bucklin, with
>top-two runoffs if no candidate receives a majority of approvals in those
>two ranks - is my favorite proposal for practical implementation."
><snip>

I agree with him, except that I'd make it three-rank, unless there 
were fewer than four candidates. (i.e., four explicit on the ballot 
plus write-in). This allows full ranking of four candidates, which 
may be completely adequate, given the equal ranking allowed, even for 
very large candidate sets. (Three-rank Bucklin allows four actual 
ranks, when "no rank" is considered the bottom rank.)



>Jameson,
>
>What does "MCV" stand for?
>
>Does "top-two runoffs" mean a second trip to the polls?

Yes, I'm sure. That's a "runoff election."

>How are the candidates scored to determine the top two? Is it based on the
>candidates' scores after the second Bucklin round?

Probably. I'm not sure that the limitation to two candidates in the 
runoff is good. Bucklin could surely handle three, so the runoff 
could be two-rank Bucklin, with top three, but using three ranks is 
harmless and allows full ranking with a write-in in the runoff. Yes. 
Most-approved top three would be one choice; another would be top-two 
plus a Condorcet winner, if one exists that is not in the top two.

However, I'd probably prefer this algorithm: Condorcet winner, if 
apparent from ballots, will either win or be included in a runoff. If 
no candidate gains a majority, considering all Bucklin ranks, then a 
runoff would include a Condorcet winner, plus one or two of the 
most-approved candidates, as necessary to show two. So there are two 
selected candidates for the runoff. Write-ins would be allowed in the 
runoff, so the Bucklin runoff would then be two-rank Bucklin.

The remaining question: what if there are two candidates gaining a 
majority, one candidate is leading in approvals, but the other is the 
Condorcet winner? In theory, a runoff should be held, but it might 
not be considered to be worth the cost. I'd want to study this case 
more closely, and I don't propose Condorcet analysis as part of the 
first implementations, but study of ballots later from real elections 
will ultimately reveal how significant this might be. In that event, 
yes, the runoff would be between the most-approved candidates 
(considering any Bucklin vote, at any rank, as an approval.)