[EM] current version of weighted pairwise proposal

[James Green-Armytage]

Here is a copy of the current version of the weighted pairwise proposal,
which can also be found at
http://fc.antioch.edu/~jarmyta@antioch-college.edu/voting_methods/weighted_pairwise.htm

________________________________________________________

The Weighted Pairwise Comparison Method
            by James Green-Armytage
 
I    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 Bush will win. 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 process of pairwise comparison. That is the
goal of the method which I describe here.


II    The method:
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.

Additional provisions:
1. 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.
2. Once a Schwartz set has been established by the pairwise tally in step
2, it may be a good idea to maximize the voters' ratings differentials in
scale between the candidates in the set. That is, to change each rating
ballot such that the highest-rated Schwartz set candidate is at 100, the
lowest-rated Schwartz set candidate is at 0, and the rating differentials
between the Schwartz set candidate retain their original ratios. (For
example, 50,20,10 would become 100,25,0.)  

III    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
 
Pairwise comparisons (using rankings information)
            Bush     Dean    Kerry
Bush                 52        49
Dean    48                    53
Kerry   51        47
 
Defeats (using rankings information)
Bush --> Dean
Dean --> Kerry
Kerry --> Bush
 
Magnitude of defeats (using ratings information)
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.


IV    A variation
            This variation can be called the approval-weighted pairwise
comparison method.
Ballots: 
Ranked ballots with approval cutoff

Tally:
1. Pairwise tally using the ranked ballots. Elect the Condorcet winner if
one exists.
    If there is a majority rule cycle:
2. Determine the direction of the defeats by using the rankings for a
pairwise comparison tally.
3. Determine the strength of the defeats as follows: For a given defeat A
over B, the magnitude of the defeat is defined by the number of voters who
place A above their approval cutoff and B below their approval cutoff.
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. 

Comments:
            I prefer the primary version of the proposal to the approval
cutoff variation. The only advantage of the latter is that it uses
somewhat simpler ballots. 
 
_________________________
The weighted pairwise comparison method was invented and first proposed by
James Green-Armytage on June 8, 2004. Please cite the author when
discussing this method.