[EM]Definite Majority Choice, AWP, AM

[Jobst Heitzig]

Dear Chris!


First, I'd like to emphasize that DMC, AWP, and AM can be thought of as
being essentially the same method with only different definition of
defeat strength, so it seems quite natural to compare them in detail as
you started.

Recall that the DMC winner is the unique immune candidate when defeat
strength is defined as the approval of the defeating candidate, so with
that definition, Beatpath, RP, and River become equivalent to DMC.

Perhaps it is helpful to look at the defeat strength like this: When A
defeats B, then the defeat strength is composed as a linear combination
of the following three components:
                                          AM   AWP   DMC
  no. of voters approving A but not B     +     +     +
  no. of voters approving A and B         0     0     +
  no. of voters approving B but not A     -     0     0


My second point is this:

Your wrote:
> I used to think that electing the voted approval loser
> was absurd if we assume that the votes are sincere,
> but by that logic we should resolve all top cycles by
> electing the Approval winner. From that point of view,
> sometimes electing the approval loser is only a degree
> "worse" than not always electing the approval winner!

Here you state the obvious problem when looking at both approval and
defeat information. Forest's ingenious argument was that we should at
least not elect a candidate where both kinds of information agree that
the candidate is defeated, leaving us with his set P of candidates which
are not strongly defeated.

But when we take both kinds of information serious, it does not seem
appropriate to me to always elect a candidate from the two extremes of P
like Approval and DMC do. Still, DMC has the obvious advantage of
extreme simplicity.

I would find it much more natural if the winner was somewhere in the
"middle" of P!

The simplest way to achieve this is to use Random Ballot among P, which
adds two nice properties to the method: (i) Randomization for better
long-time fairness and strategy-proofness, and (ii) taking into account
the third major kind of preference information: direct support.

In your second example, this would result in R with 61% probability and
L with 39% probability, so that the R voters would take the risk of
getting a worse outcome than before with 39% probability, which should
suffice to deter them from using that strategy. The results of the first
example were posted by me already.

People who don't like randomization could instead use TAWS (Total
Approval Winner Stays): Process the candidates in order of increasing
approval, always keeping the winner of the pairwise contest between the
next candidate and the candidate at hand. With a large number of
candidates, I guess this will still give a winner more to the less
approved end of P, but in your first example it elects B as DMC does,
and in your second one L as AM and AWP do.

Forest alternatively proposed to choose from P via IRV (not monotonic)
or using Winner Stays as in TAWS but starting from the top candidate on
a Random Ballot.

Yours, Jobst