From: Alan Dechert <alan_at_openvotingconsortium_dot_org>

Date: Thu Oct 28 2004 - 10:48:48 CDT

Date: Thu Oct 28 2004 - 10:48:48 CDT

----- Original Message -----

From: "Hubert Bray" <bray_at_math_dot_duke_dot_edu>

To: "Alan Dechert" <alan@openvotingconsortium.org>

Sent: Thursday, October 28, 2004 8:46 AM

Subject: Re: counting preferential ballots

*>
*

*> Dear Alan,
*

*>
*

*> Sure, I would like to join your email list. Please add me. Also, if you
*

*> could forward this email to the list that would be great. I have spent
*

much

*> time studying preferential ballot vote counting methods and would be happy
*

to

*> contribute in this area.
*

*>
*

*> I am a very enthusiastic proponent of preferential ballot elections, but
*

only

*> if the votes get counted in a reasonable manner. However, it is not
*

generally

*> understood by the general population how one determines a winner with
*

*> preferential ballots. Even worse, there are dozens, if not hundreds, of
*

*> different preferential vote counting methods which, unfortunately, produce
*

*> -different- results typically. I have focused on determining criteria for
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*> deciding which are the best vote counting methods.
*

*>
*

*> Bad vote counting methods would include those that can be manipulated by
*

*> politicians too easily. For example, point counting methods (5 points for
*

*> each 1st choice, 4 points for each 2nd choice, etc.) can be manipulated by
*

*> political parties by flooding the election with numerous similar
*

candidates.

*> If one thinks about it, it is possible to come up with scenarios in which
*

the

*> most popular candidate does not win (even when a candidate is the first
*

choice

*> of 60% of the people).
*

*>
*

*> The two most important properties that I think that preferential vote
*

counting

*> methods should have are
*

*>
*

*> 1) If Candidate A can beat any one of the other candidates in a
*

head-to-head

*> vote, then Candidate A should be declared the winner.
*

*>
*

*> 2) "Cloning" candidates (that is, having your 5 brothers running on the
*

*> ballot representing exactly the same viewpoint) should not make a
*

difference

*> in the outcome of the election.
*

*>
*

*> Candidates who can beat any other candidate in a head-to-head election
*

(which

*> is possible to simulate based on people's listed preferences) are called
*

*> "universal winners" (also called Condorcet winners). A universal winner
*

does

*> not always exist, but there can be at most one universal winner. I think
*

that

*> property (1) is important because this property gives a clear goal to each
*

of

*> the candidates: If you can become more popular than any other single
*

*> candidate, then you win the election. Also, it seems unfair to me to say
*

that

*> Candidate B wins the election even though Candidate A would beat him (and
*

*> every other candidate for that matter) in a head-to-head contest.
*

*>
*

*> Property (2) is important for two reasons. Clearly you don't want
*

candidates

*> to purposely clone themselves for a political advantage, so cloning
*

oneself

*> shouldn't help that candidate. On the other hand, you don't want to
*

penalize

*> candidates who happen to all want to represent a similar, popular
*

position.

*> Since cloning should therefore neither help nor hinder a candidate,
*

cloning

*> shouldn't make any difference at all.
*

*>
*

*> Philosophically speaking, one might want to add other "Ideal Properties"
*

for a

*> vote counting system to have, but it turns out that it is easy to come up
*

with

*> so many "good" properties than no vote counting system can satisfy all of
*

*> them at the same time.
*

*>
*

*> My two favorite vote counting methods are the "Least Worst Defeat" method
*

*> (LWD) and the "Ranked Pairs" method (RP). RP is arguably the best method,
*

but

*> LWD is almost as good and is easier to understand. Both methods determine
*

a

*> winner using the "Margin of Victory Matrix," which we will call M.
*

*>
*

*> If there are n candidates in the election, M is an n x n matrix. The
*

number

*> in row i, column j of M (which we will call M_ij) is the "margin of
*

victory"

*> of candidate i over candidate j, defined as follows: Going through all of
*

the

*> preferential ballots, define M_ij to be the number of voters who prefer
*

*> candidate i to candidate j MINUS the number of voters who prefer candidate
*

j

*> to candidate i. In other words, if these voters had been just voting
*

between

*> these two candidates, M_ij is the number of votes candidate i would have
*

beat

*> candidate j by (and is negative is the case that candidate j would win).
*

All

*> unranked candidates are assumed to be tied for least favorite for that
*

voter,

*> and if both candidates are unranked by the voter, that vote does not
*

*> contribute to M_ij. Also, since every candidate ties with himself, the
*

*> diagonal entries of M are all zero.
*

*>
*

*> Since M_ji = - M_ij, the n x n matrix M is called an "anti-symmetric
*

matrix."

*> If the election is being held in multiple precincts, then each precinct
*

should

*> determine the Margin of Victory Matrix for their precinct and then all of
*

*> these matrices should be added together to get the Total Margin of Victory
*

*> Matrix for the election.
*

*>
*

*> In other words, at some point, I would like to suggest that your software
*

*> should compute the Margin of Victory Matrix M defined above in each
*

*> precinct. This is the most important statistic to keep track of.
*

*>
*

*> Given M, we now know how each candidate would do against any other single
*

*> candidate. In particular, each candidate has a "worst defeat" to some
*

other

*> candidate by some number of votes. Whichever candidate has the least
*

"worst

*> defeat" is the winner as determined by the LWD method. Notice that LWD
*

*> satisfies both (1) and (2). Property (1) is satisfied since a universal
*

*> winner doesn't have any defeats (so one could say his worst defeat equals
*

zero

*> since he ties with himself). But since a universal winner beats all other
*

*> candidates, every other candidate will have at least one defeat by some
*

*> positive number of votes. Property (2) is satisfied since cloning
*

candidates

*> does not change the value of a candidate's worst defeat to any of the
*

other

*> candidates.
*

*>
*

*> The Least Worst Defeat (LWD) vote counting method is great because it is
*

easy

*> to understand and has good properties. I can describe the Ranked Pairs
*

(RP)

*> method at some future date if you like. Philosophically the RP method is
*

*> slightly better (and not too hard to understand). However, as long as
*

your

*> software computes the margin of victory matrix M, the RP method can still
*

be

*> used to determine a winner.
*

*>
*

*> Bottom line, let me strongly suggest that the margin of victory matrix M
*

is

*> the most important statistic to keep track of when counting votes. Then
*

the

*> people conducting the election will have a number of very good
*

preferential

*> vote counting methods by which to determine a winner.
*

*>
*

*> Best,
*

*> Hubert Bray
*

*>
*

*>
*

*>
*

*>
*

*> On Oct 27,2004 16:00:54 -0700, Alan Dechert
*

<alan@openvotingconsortium.org> wrote :

*> >Greetings Hubert!
*

*> >
*

*> >We usually refer to these as "ranked preference contests." We plan to
*

*> >support whatever scoring methods are in use. Our prototype doesn't, at
*

this

*> >point, go beyond the precinct level. Here we only tally "so many 1st
*

place

*> >votes for candidate x, so many 2nd place votes for candidate x, etc.
*

*> >
*

*> >If you want to join our email list to discuss your ideas, I would be
*

happy

*> >to add you. Right now, we're focused on raising some money. With luck,
*

*> >we'll have this worked out within the next month or so.
*

*> >
*

*> >Alan Dechert
*

*> >
*

*> >
*

*> >>
*

*> >> Question: Does your software count votes as well? If so, how do you
*

*> >> count the preferential ballots? If not, do you have plans in this
*

*> >> direction? I have studied the different vote counting methods for
*

*> >> preferential ballots extensively and would be happy to help in this
*

*> >regard.
*

*> >>
*

*> >> Best Regards,
*

*> >>
*

*> >> Hubert Bray
*

*> >> Professor of Mathematics,
*

*> >> Duke University (on leave from Columbia and MIT)
*

*> >>
*

*> >
*

*> >
*

*>
*

*>
*

*> Hubert Bray
*

*> Mathematics Department
*

*> Duke University
*

*>
*

*>
*

*>
*

*>
*

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Received on Mon Nov 1 15:28:52 2004

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