[EM] Tideman and GMC

[Blake Cretney]

I'm going to do some rethinking of an argument that occurred some time
ago on this list.  It will likely mainly be of interest to Markus
Schulze, but others familiar with the original argument may be
interested.  

Back at one time, on this list, one popular criterion was GMC.  This
said,

If any candidate has a pairwise absolute majority against it, that
candidate should not win, unless all candidates have pairwise absolute
majorities against them.

By pairwise absolute majority, I mean that a majority of the voters
have ranked one candidate over another.  Note that I am speaking of a
majority of ALL the voters, not just those with a preference between
the two candidates.

I will also define an absolute Condorcet winner to be a candidate
with an absolute pairwise majority against each other candidate, and a
sincere absolute Condorcet winner (SACW) to be the absolute Condorcet
winner among voters actual, as opposed to expressed preferences.

Now, the idea was that a method that met GMC would not allow a group
of voters to defeat the SACW in favour of a candidate they like
better, if they use only truncation.  Truncation is insincerely
leaving candidates unranked.

If there is a SACW, this means that the candidate has an absolute
pairwise majority against each other candidate.  Truncation cannot
create an absolute majority against the SACW, as it does not actually
add votes.  As well, since the truncator already has the preferred
candidate ranked over the SACW, truncation will not reduce its loss
against the SACW.  So, even after truncation, the truncator's
candidate still is excluded by GMC, and the SACW isn't.  Therefore,
the truncator's candidate cannot win.

However, GMC was found to be in conflict with the Smith criterion
(and GITC depending on how you define it).  Certainly the methods that
passed GMC failed both these criteria.  As a result, GMC fell out of
favour, and was superseded by a new criterion, beat path GMC.

This new GMC relied on the idea of a path of absolute pairwise
majorities from one candidate to another.  So, if X has an absolute
majority over Y, and Y has an absolute majority over Z, there is a
path of these majorities from X to Z.

The new GMC said,

For any candidates X and Z, if X has a path of absolute majorities to
Y, then Y must not win, unless Y has a path of absolute majorities to
X.

Markus Schulze showed that this new criterion was consistent with
GITC and Smith, by proposing a method that satisfied all 3.  Another
method, Tideman, was shown to fail beat path GMC, and largely for this
reason was considered defective.  

I should point out that I never agreed with GMC, or with the argument
about truncation.  However, the purpose of this posting is to suggest
that Tideman was unfairly criticized, even assuming the argument about
truncation.  I am going to suggest that beat path GMC is much more
restrictive than necessary to be a successor of the original GMC. 
Also, although I think that majority strengths should be measured in
margins, I will be considering the version of Tideman which only uses
the number of votes on the winning side.

Beat path GMC seemed a natural revision of GMC, but it was not the
only one possible.  Consider instead the following definition, GMC2

For any candidates X and Y, if X has an absolute majority over Y,
then Y must not win, unless Y has a path of absolute majorities to X.

Now, it is clear that the winning-votes version of Tideman has this
property.  The only way for the X vs. Y victory to not get locked, is
if there is already a path of locked victories from Y to X.  This can
only happen if they are considered greater victories by the method.

GMC2 implies the same kind of truncation resistance as did GMC.  The
truncator's candidate has an absolute majority against it from the
SACW.  Because no absolute majority exists against the SACW, it
follows that there is no path from truncators candidate to SACW, and
the truncators candidate is excluded by GMC2.

So, my conclusion is that Tideman (winning-votes) was rejected on
faulty grounds.
---

Blake Cretney