[EM] How to convert a set of CR ballots to a set of Approval ballots

[Forest Simmons]

Kevin, I think that the refined method can be slightly improved as
outlined below.

First some terminology to simplify the description.

At any stage, after the CR score has been calculated for all of the
candidates, let's say that the "viability" of a slot on a ballot is the
highest CR score achieved by any candidate occupying that slot on that
ballot.

Now we can talk about one slot being more or less viable than another.

In practical terms, at any stage on any ballot, the more viable of two
slots has a greater chance of containing the eventual winner.

In our refined method we amalgamated adjacent slots with minimal combined
viability.

A slightly simpler and (I believe) improved rule would be to amalgamate a
slot of minimal viability with the less viable of its two neighbors (or
with its only neighbor if it is one of the two extreme slots).

I would like to call this method Max Power Cardinal Ratings (MPCR) since
its hueristic is to convert the CR ballot into an Approval ballot with the
maximum likelihood of being positively pivotal, i.e. maximizing the voting
power of the ballot.

I was tempted to name it Max Power Approval, but as Bart has reminded me
we have already taken too many liberties with the name "Approval."  Since
the method uses CR ballots as input, and gives approval ballots only as
output, I'll resist the temptation to raise Bart's hackles again :-)

Note that all of the various versions of MPCR from the crude to the more
refined agree in their conversion of a resolution three ballot to an
Approval ballot.

With that in mind, I wonder how three slot MPCR compares with three slot
Condorcet, especially in the case of only three candidates.

In the following example three slot MPCR agrees with Ranked Pairs and SSD
rather than Black:

40 A>B>C
35 B>C>A
25 C>A>B

Details:  The CR (on a scale of zero to three) is the same as the Borda
Count score:  A, B, and C get 105, 110, and 85, respectively. So B is the
Borda (hence also Black) winner.

The MPCR amalgamations yield ...

40 A>B=C
35 B>C=A
25 C=A>B

So the final approval scores are 65, 35, and 25 for A, B, and C,
respectively.

It's easy to check that this is the Ranked Pairs order, since the weak
link in the A beats B beats C beats A cycle is the C beats A link.

Thanks for your continued valuable contributions.

See further (interlinear) response below:

On Thu, 31 Jul 2003, [iso-8859-1] Kevin Venzke wrote:

> Forest,
>
>  --- Forest Simmons <fsimmons at pcc.edu> a écrit :
> > Kevin, thanks for the feed back.  I cannot prove relative immunity to
> > manipulabilty, but see my second post for some thoughts in that regard.
> >
>
> I implemented the method this morning.  (I'm not sure how likely it is that
> I'll be able to do the refined version, though.)  While I was at it I did
> that last method, where a revised cutoff for each ballot was calculated as
> ((Sum of the two front-runners' scores) + (midrange value)) / 3.  The new
> method appears to be much better at picking the CW.
>
> Actually, the question occurred to me of whether you consciously decided that
> only the leading front-runner should be considered.

That evolved gradually from a desire to simplify along with the
realization that the creative procrastination strategy allowed one to
forget about the second placer except in cases where a symmetry breaker is
needed (when the front runner is in the center slot of an odd resolution
stage).

Even the front runner can be ignored (except as a tie breaker) in the
refined methods since the attention is now focussed on amalgamating
neighboring slots with minimum viability.

Note that IRV judges viability by first place votes, and then eliminates
the minimum viability candidate.  Perhaps it would do better to amalgamate
rather than eliminate.  That might save the CW from early elimination.

Forest

> (I had an idea of
> factoring in ALL the candidates' initial scores, to get a better idea of what
> "expectation" is, but I doubt that could be made clone-proof.)
>
> When the new method doesn't pick the CW, it seems to be because the CW
> never managed to become the leader in any round, and so the ballot
> restructuring wasn't relevant to electing the CW.  A result, for instance,
> might be that factions who liked the CW ended up approving too many
> candidates.
>
> If the leader changes frequently or abruptly, it can give garbage results (like
> electing someone with a very low Borda score).
>
> I see the refined method tries to address some of this.  I'll go back and see
> if I can manage it.
>
>
> Kevin Venzke
> stepjak at yahoo.fr
>
>
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