[EM] possible amendment to weighted pairwise proposal

[James Green-Armytage]

Dear election methods fans,

	Here I am suggesting a new version of the weighted pairwise method which
I proposed. The change I'm proposing is that defeats where the winning
side constitutes a majority (of the valid vote) should always be
considered to be stronger than a defeat where the winning side does not
constitute an actual majority, regardless of the winning marginal
utilities involved.

To sum up:
	If there are two pairwise defeats that are both by a majority, the one
with the higher marginal utility is stronger.
	If there are two pairwise defeats such that neither is by a majority, the
one with the higher marginal utility is stronger.
	If one pairwise defeat is by a majority and another isn't, then the
defeat by a majority is necessarily stronger.


<beginning of new definition>

Ballots:
1. Ranked ballot. Equal rankings are allowed.

2. Ratings ballot. e.g. 0-100, whole numbers only. Equal ratings allowed.
Note: You can give two candidates equal ratings while still giving them
unequal rankings. However, if you give one candidate a higher rating than
another, then you must also give the higher-rated candidate a higher
ranking.

Tally:
1. Pairwise tally, using the ranked ballots only. Elect the Condorcet
winner if one exists.
If no Condorcet winner exists:

2. Determine the direction of the defeats by using the ranked ballots for
a pairwise comparison tally.

3. Determine the strength of the defeats by finding the weighted magnitude
as follows. We’ll say that the particular defeat we’re considering is
candidate A beating candidate B. For each voter who ranks A over B, and
*only* for voters who rank A over B, subtract their rating of B from their
rating of A, to get the marginal utility. The sum of these winning
marginal utilities is the total weighted magnitude of the defeat. (Note
that voters who rank B over A, or rank them equally, do not contribute to
the weighted magnitude; hence it is never negative.)

4. Now that the directions of the pairwise defeats have been determined
(in step 2) and the strength of the defeats have been determined (in step
3), you can choose from a variety of Condorcet completion methods to
determine the winner. Beatpath and ranked pairs are my preferred choices.

5. There is one situation in which a defeat with lesser weighted magnitude
is considered to be stronger than a defeat of greater weighted magnitude:
If the winning side of one defeat constitutes a majority (of the valid
vote), and the winning side of another defeat does not constitute a
majority, then the majority defeat is necessarily considered to be
stronger. Otherwise, the weighted magnitude is always the determining
factor in relative defeat strength.

<end of new definition>


	I'm not yet entirely convinced that this amended proposal is superior to
the original one, but I thought that I would at least make the amended
proposal, and go on considering both versions separately. 
	Here is why I'm suggesting the amendment.

a slightly modified version of David's example
sincere votes:
45: A 100 > B 40 > C 0
10: B 100 > A 90 > C 0
 5: B 100 > C 90 > A 0
40: C 100 > B 40 > A 0

	If everyone votes sincerely, then B is a Condorcet winner, and wins in
weighted pairwise. However, using the original weighted pairwise rules,
the A voters can steal the election for A by truncating.

altered votes:
45: A 100 > B 0 = C 0
10: B 100 > A 90 > C 0
 5: B 100 > C 90 > A 0
40: C 100 > B 40 > A 0

	A now wins with my original proposal. (I double-checked on Brian's
calculator : )
	The defeats in terms of votes are:
B>A : 55>45
A>C: 55>45
C>B: 40>15
	And the weighted defeats are
B>A: 2200
A>C: 5400
C>B: 2400
	The least heavy defeat is B>A, so A wins. So the problem is that my first
proposal is rewarding strategic truncation here, the A voters burying B.
	However, that the C>B defeat is the only one that isn't supported by a
majority. It is not majority-supported specifically because it was only
made possible by truncation. Hence, my current proposal would consider the
C>B defeat to be the weakest.
	I should note here that even in my original proposal, the C voters can
protect B without altering their rankings. All they have to do is give B a
very high rating, strengthening the B-->A defeat. So the strategic problem
in the original proposal isn't as bad as all that. 
	Also, note that, of course, if the A voters decide to strategically
order-reverse instead of truncating, voting A>C>B, the majority/minority
distinction won't come into play. But this is as much a problem with any
Condorcet method as it is with this one. Arguably this method offers an
additional level of protection in that the C voters can protect B by
giving B an insincerely high rating and leaving their rankings sincere.
	Still, I guess it would be nice to let the B voters not have to worry
about doing this, and to be able to give sincere-ish ratings even when A
voters were likely to truncate. If it became known that the A voters were
likely to order-reverse, then the C voters would do well to adjust their
ratings.
	The only question is whether this additional rule has any drawbacks. If
so, I'm hoping that one of you will be able to point them out. In general,
I think that the fewer rules a method has, the better, so I'm always wary
about adding a new one. I guess that a method that is more "continuous" is
likely to offer less complex strategy incentives.
	So, at present, I don't know which version of the proposal is better. I
feel that the original one is pretty good already. If this new rule
doesn't cause any awkward strategic incentives, then I suppose that it is
marginally better.
	I hope that I will get some feedback on this question.

best,
James Green-Armytage