[EM] MCA and median

[Gervase Lam]

> Something I've been wondering about... has anyone suggested extending
> the gradation in MCA beyond preferred, approved, and diapproved?  For
> example, why not use MCA with a A,B,C,D,F ballot?  If no candidate has a
> majority of A's, then check for a majority of A's and B's, then check
> for a majority of A's, B's, and C's, and finally just elect the
> candidate with the most A's, B's, C's, and D's.
>
> It seems like an obvious point, but I haven't actually seen any messages
> advocating it.  Call it "extended MCA" or "unconstrained Bucklin" or
> "Approval Bucklin" or "Bucklin done right" or "bubble up approval" or
> whatever.
>
> -Adam

This past weekend, I was thinking over how MCA could be made more
sensitive to the difference between the top two levels (i.e. if no
candidate obtains > 50% of favored, then the favored and acceptable levels
are considered the same).

I had thought of using the upper and lower quartile values many times
before in various voting methods.  But they just seemed too sensitive.
Only 25% of the voters is needed to influence either of the quartile
values.

However, I then realised that if the "top" 50% of the votes for a
candidate are at worst at a certain level, then that certain level must be
the median for the candidate.  It may be easier to see things if I define 
the following "N-level" voting method:

(1) The winning candidate is the candidate with the highest median rank or
score.
(2) If more than one candidate satisfies (1), then break the tie by making
the candidate with the least number of votes below the median the winner.
(3) If more than one candidate satisfies (2), then break the tie by making
the candidate with the most number of votes above the median the winner.

See what happens if you apply the method rules with N being 2, 3 or
possibly 5 (i.e. A,B,C,D,F).  (3) is needed for candidates whose median is
the bottom most level because (2) does nothing in this situation.

I must admit, (2) and (3) aren't good measures of which of two or more
tied medians is the better.  Really [Votes above median - Votes below
median] should be used in step (2).  But I can't picture clearly what
strategic implications this could have.

The Median Rating method has been proposed quite a few times in Election
Methods but dismissed.  See the whole thread starting at
<http://groups.yahoo.com/group/election-methods-list/message/3833> for an
example.  However, what had been discussed previously was a Cardinal
Ratings system with N being very high (e.g. 100).

Thanks,
Gervase.