[EM] proposal: weighted pairwise comparison

[James Green-Armytage]

Dear election methods fans,

	I've finally come up with an idea that seems workable for combining
Condorcet with cardinal ratings ballots. I’ve been working on the project
on and off since March. I spent quite a lot of time working on ideas which
seemed promising for awhile, some of them quite complex, but in the end
each method seemed to be overrun by problems, bringing me back to square
one again and again. However, last night, after having set it aside for
some time, I sat down to the problem again. I was in a state of total
despair as to actually making any progress after all the fruitless hours
that I had spent before, but I came upon a relatively simple idea that
seems like it might work. Please let me know what you think.


The problem that this method addresses:
	Consider the following set of sincere preference rankings and ratings
26: Bush 100 > Dean 10 > Kerry 0
22: Bush 100 > Kerry 10 > Dean 0
26: Dean 100 > Kerry 90 > Bush 0
1: Dean 100 > Bush 50 > Kerry 0
21: Kerry 100 > Dean 90 > Bush 0
4: Kerry 100 > Bush 50 > Dean 0
	If you use a standard Condorcet method (such as minimax, ranked pairs,
beatpath), with winning votes or margins, and everyone votes sincerely,
then there will be a majority rule cycle which Bush will win. (Bush
--52-->Dean, Dean--53-->Kerry, Kerry--51-->Bush.) The problem with this
isn’t just that Kerry has a (very slightly) higher utility score. The
problem is that the 26 Dean > Kerry > Bush voters consider the Kerry -->
Bush defeat to be much more valuable than the Dean --> Kerry defeat, and
many of them would probably be willing to change their votes to Dean =
Kerry > Bush, if given an opportunity after learning the results. This
actually may create a subtle barrier against the entry of additional
candidates in some situations.
Again, there is nothing strategically underhanded about this result; all
of the voters were being good sports and voting out all of their sincere
preferences. The problem is that some preferences are more important to
voters than others, and a straight ranking ballot does not give voters an
opportunity to express the relative strength of their preferences.
Hence, it seems that an ideal voting method would incorporate cardinal
ratings information. But what is the best way to integrate this
information into a Condorcet-efficient method? Simply running an ordinary
pairwise tally and then falling back on the sum of cardinal ratings scores
in the event of a majority rule cycle seems to be unsatisfactory. In the
event of a cycle too much of the ranking information would be lost, and in
many ways it would be equivalent to starting over from scratch with
cardinal ratings instead of Condorcet. Cardinal ratings is problematic for
known reasons, such as the strong strategic incentive for compressing
preferences. 
I think that a better method would be one that would integrate the ratings
data more carefully into the pairwise comparisons. That is the goal of the
method which I describe here.

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. Using the ranked ballots only, construct a map of which candidates beat
or tie which other candidates. At the point we are only concerned with the
direction of defeats, not the magnitude.
3. For each defeat, fill in 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, 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 only candidates who rank A over B contribute to the sum, so the
sum is never negative.)
4. The strength of each defeat is defined by its weighted magnitude. Now
that you know the magnitude of the defeats, you can choose from a variety
of Condorcet completion methods to determine the winner. Beatpath and
ranked pairs are my preferred choices.


Example:
26: Bush 100 > Dean 0 > Kerry 0
22: Bush 100 > Kerry 0 > Dean 0
26: Dean 100 > Kerry 100 > Bush 0
1: Dean 100 > Bush 50 > Kerry 0
21: Kerry 100 > Dean 100 > Bush 0
4: Kerry 100 > Bush 50 > Dean 0

Ranked ballot pairwise comparisons
	Bush	Dean	Kerry
Bush		52	49
Dean	48		53
Kerry	51	47

Defeats
Bush --> Dean
Dean --> Kerry
Kerry --> Bush

Magnitude of defeats
Bush --> Dean
	(26x(100-0)) + (22x(100-0)) + (4x(50-0)) = 5000
Dean --> Kerry
	(26x(100-100)) + (26x(100-100)) + (1x(100-0)) = 100
Kerry --> Bush
	(26x(100-0)) + (21x(100-0)) + (4x(100-50)) = 4900

Completion by minimax
1. No unbeaten candidates
2. Drop defeat of least magnitude, Dean --100--> Kerry 
3. Kerry is unbeaten

Completion by beatpath
beatpath Bush-->Dean:	Bush--5000-->Dean: 5000
beatpath Dean-->Bush:	Dean--100-->Kerry--4900-->Bush: 100
	Bush has a beatpath victory over Dean.
beatpath Bush --> Kerry:	Bush --5000--> Dean --100--> Kerry: 100
beatpath Kerry --> Bush:	Kerry --4900--> Bush: 4900
	Kerry has a beatpath victory over Bush.
beatpath Dean --> Kerry:	Dean --100--> Kerry: 100
beatpath Kerry --> Dean:	Kerry --4900--> Bush --5000--> Dean: 4900
	Kerry has a beatpath victory over Dean.
	Kerry is a beatpath winner. Complete ordering is Kerry-->Bush-->Dean.

Completion by ranked pairs
5000: Bush-->Dean	keep
4900: Kerry-->Bush	keep
[100: Dean-->Kerry]	skip (would cause a cycle, Bush-->Dean-->Kerry-->Bush)
	Kept defeats produce ordering Kerry-->Bush-->Dean.

Possible names for the method:
	"Weighted pairwise", "Condorcet completed by winning marginal utilities"
(CCWMU), "Condorcet with integrated ratings" (CIR), or (if it turns out to
be worth using ; ) the Green-Armytage method.