[EM] Another use for Majority Choice style ballots.

[Forest Simmons]

I've been thinking about Richard Moore's Majority Potential idea which is
just Copeland with fictitious candidates distributed uniformly in issue
space.

The method was never intended for public proposal, rather it was intended
as a standard to measure the "Majority Potential" of other methods.

However the following musings have led me from that idea to a way of using
Majority Choice ballots in a new and interesting way.

Remember that Majority Choice ballots distinguish the Favorite category
from the Also Acceptable category.

The standard use of that ballot is to give the win to the candidate who is
acceptable to most voters if no candidate is a favorite of more than fifty
percent of the voters.

Anyway, here are the musings about issue space that led me to another use
of those Majority Choice ballots:

The main reason that Richard's idea is impractical for public elections is
that we have no simple way of locating the candidates' and other voters'
positions in issue space, even if we can agree on what the main issues
are.

What if each voter (including each candidate when possible) were required
to fill out a survey on the current issues like bombing Iraq, global
warming, abortion, gun control, campaign finance reform, affirmative
action, socialized medicine, school vouchers, genetic engineering,
chopping down old growth forests, privatization of aquifers and other
traditional commons, helping the homeless, etc.?

Then similarity in responses would measure the proximity of two
voters (one of which could be a candidate) in issue space, getting close
to Richard's idea.

Nice, but impractical:  who's willing to fill out all of those surveys
(assuming we could ever agree on which issues to include, or on a fair and
impartial wording of the questions)?

Here's the crucial idea that allows us to avoid the survey: a voter's
position in issue space is reflected in the voter's choice of approved
candidates.

The fewer disagreements two voters have about which candidates are
acceptable, the closer they are to each other in issue space (on average).

Suppose we use Majority Choice ballots, and no candidate is a favorite of
more than fifty percent of the voters.  Then (without violating the
secrecy of any voter's ballot) we can ascertain the approximate position
of each candidate by averaging the ballots that list him/her as a
favorite.

Here we are assuming that the most avid supporters of candidate X are
relatively near candidate X in issue space, so their average location is
a reliable estimate of his/her location.

Another reason for estimating the candidate's position rather than taking
the candidate's self estimate, is that the "candidate" might not be a
person; voting methods are used in other contexts as well.

Once we have an estimate of the position of each candidate, the distance
separating each voter from each candidate can be estimated also.

I suggest using the Hamming distance, which is just the L1 norm of the
difference in position vectors.

For example, if there are ten candidates and my (ordered) list of
acceptable candidates is C1, C4, C5, and C9, then my position vector is

[1,0,0,1,1,0,0,0,1,0].

Candidate C7 might have an estimated position of

[.1, .2, .3, .1, .2, .5, 1.0, .6, .4, .8] .

Then my Hamming distance is the sum

.9 + .8 + .7 + .9 + .8 + .5 + 1 + .4 + .6 + .8 .

Note that your Hamming distance is always at least one unit from any
candidate that you do not list as acceptable.

Now how do we figure the winner?

For each candidate X we calculate the sum of the distances from X to each
voter, and then divide by the number of voters.  This number E may be
interpreted as the expected distance from X to a randomly chosen voter.

The candidate with the smallest value of E wins the election.

Any other candidate is more distant (on average) from a randomly chosen
voter.

If there are two seats to be filled, we consider all pairs of candidates,
and average the distances from the voters to the nearest member of the
pair.

In other words, if the pair is {X,Y} and voter Z is closer to X than to Y,
then the distance from X to Z is the number that this ballot contributes
to the sum.

Obviously this method can be extended to multiwinner elections with
several seats to be filled, with no more computational effort than PAV.

In fact, the access to the shape of the issue space that this method
affords us can drastically reduce the number of combinations of candidates
that need to be considered.

For example, if the issue space is essentially one dimensional, and if two
candidates are both to the left of the median voter, then there is no use
in considering them as a candidate pair.

By the way, it is obvious how to adapt this method to CR style ballots,
and, since it is easy to convert rankings to ratings, the method can be
adapted to ranked ballots just as easily.

However, I don't think that the extra refinement of ratings or rankings is
really necessary, or that there is any statistical chance that they would
change the outcome in any election with more than one hundred voters.

The Majority Choice ballots are just the right compromise in simplicity
and expressivity for the day when we can go beyond Approval ballots.

In the mean time, Approval Ballots can be adapted to the method by the
simple device of conflating the two categories of Favorite and Acceptable
with Approval, i.e. approved candidates are considered as both Favorite
and Acceptable.


Forest


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