# [EM] Another use for Majority Choice style ballots.

Forest Simmons fsimmons at pcc.edu
Fri Dec 13 14:23:55 PST 2002

```If anybody knows Richard Moore's new email address, please forward him a
copy of this; it bounced back to me.

By the way, the method described below should always give the CW if the
issue space is one dimensional, since in one dimension it is the voter
median candidate that minimizes the sum of the distances in issue space.

[That's because moving the candidate one voter closer to the median voter
reduces more distances than it increases.]

Also, once we have the relative distances from the voters to the
candidates, we can infer their order of preference for the candidates and
generate a ranked ballot for each voter, and find the Condorcet Winner, if
there is one.

This makes the method very easy to test in simulations comparing it with
other methods.

We need a name for the method.  Any ideas?

Forest

On Fri, 13 Dec 2002, Forest Simmons wrote:

> 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
>
>
> ----