[EM] Re: Definite Majority Choice

[Araucaria Araucana]

On  5 Apr 2005 at 23:51 UTC-0700, Russ Paielli wrote:
> Araucaria Araucana araucaria.araucana-at-gmail.com |EMlist| wrote:
>
>> I happen to think that DMC is the simplest-to-grasp version of all
>> three methods.  Here is one way to find the winner:
>>       Eliminate any candidate defeated by another candidate with
>>       higher total approval.
>>       Among the remaining candidates, the candidate with the lowest
>>       approval defeats all others and is the DMC winner.
>
> I was just thinking about this procedure some more, and I came up
> with a simple way to visualize the procedure (for simple-minded
> folks like me).  Order the pairwise matrix with Approval scores
> decreasing (or non-increasing) on the diagonal, as usual. Then color
> the winning cells of the pairwise matrix black and the losing cells
> white. The winner is then the candidate who has a solid black row
> all the way from the left column to the diagonal.
> 
> If I am not mistaken, no more than one candidate can have that,
> barring ties.

Sorry, you are mistaken -- that is not a unique characteristic.

> If no candidate has it, then the Approval winner is also the CW and
> takes the enchilada.

Color the diagonal as a winning cell and you don't have to have a
special case rule.

>
> The RAV procedure can be visualized exactly the same way, thus
> demonstrating that DMC and RAV are equivalent, if I am not mistaken.
>

That's what I said!  They are equivalent since they find the same
winner.  But the CW concept is a big leap.  The procedure can be
automatic without mentioning the Smith set or Condorcet winner.

If you will allow to modify the visualization slightly:

 - Reorder the pairwise array as you specify above.

 - Instead of black and white, I'd suggest highlighting winning (and
   approval!) scores, rather than blacking them out and obscuring
   their values!  With a yellow highlighter pen, you look for a solid
   yellow row up to (and including) the diagonal.

Here is the crucial difference:

 - You need to start checking left-side to diagonal cells starting
   with the last (least-approved) candidate, and work up the diagonal
   until you find the first candidate with a solid row of wins to the
   left of the diagonal.

For DMC, I would first travel down the diagonal from the upper left,
looking for defeats to the right of the diagonal.  Then I would draw
lines (strike out) through the rows and columns of those
correspondingly defeated candidates to indicate that they have been
eliminated, and move to the next diagonal cell (even if it has been
eliminated).  You can stop once there are no more non-eliminated
candidates with lower approval.  Once all lower-approved candidates
have been eliminated, move back up the diagonal again until you find
the lowest-approved non-eliminated candidate.

The higher-approved remaining candidates are the other members of the
definite majority set.  Each of them will also have a solid row of
wins from the diagonal to the left side.

Re your other message about the name: Ranked Approval Voting is fairly
descriptive and probably as good as any other choice, but it is just
as fuzzy as IRV's "Ranked Choice Ballot" -- it describes the ballot
method and only hints at how they're tallied.  It also implies that
Approval Voting is the primary characteristic of the method and that
the ranking is a slight modification, when what we're doing is
actually the opposite.

Ted
-- 
araucaria dot araucana at gmail dot com