# [EM] Interesting use of Borda count

Steve Barney barnes992001 at yahoo.com
Thu Jan 3 13:54:14 PST 2002

```Bart:

You lost me with your bullet voting example under Saari's modified Borda Count.
Under Saari's modified BC method, two bullet votes for A (1,0,0) add up to
(2,0,0); that is A gets a total of 2 points, while B and C get none. Can you
give me a citation for Merrill's and Black's "adjusted" Borda Count method?
Saari's comment about the relationship between pairwise voting and the BC,
which I tried to include in my last message got cut short due to my time
restrictions. Rather than trying to resend it, you may read it online in PDF
format - see table 2.2 in the bottom half of page 4:

"EXPLAINING ALL THREE-ALTERNATIVE VOTING OUTCOMES," DONALD G. SAARI
http://www.math.nwu.edu/~d_saari/vote/triple.pdf

Steve Barney

> Date: Wed, 02 Jan 2002 21:02:21 -0800
> From: Bart Ingles <bartman at netgate.net>
> To: election-methods-list at eskimo.com
> Subject: Re: [EM] Interesting use of Borda count
>
> Merrill calls this "adjusted Borda", and attributes it to Black in the
> late 50's.  Evidently strict ranking is required in plain Borda.  In any
> event, a voter can always accomplish the same thing either by voting
> randomly or by cooperating with another voter.
>
> So in a three-candidate election with strict ranking, the point
> assignments would be 2, 1, 0 (or equivalently 3, 2, 1 using the counting
> method I used in my 10-candidate example).
>
> If the voter is indifferent between two candidates, each receives the
> average of what the they would have received under strict ranking.  So
> if the voter bullet votes, the candidates receive 2, .5, .5 (or
> equivalently 3, 1.5, 1.5).
>
> The first I heard of Saari's proposal was from Saari himself in an
> e-mail a couple of years ago.  He basically acknowledged the equivalent
> as "the correct values" but went on to state his preference for 1, 0, 0.
>
> But Saari's method would be impossible to enforce, since the voter can
> always defeat it through randomization or through cooperation.  Two
> voters who wish to bullet vote for A can get together and vote ABC on
> one ballot and ACB on another, so that the two ballots each average 2,
> .5, .5.
>
> Or a single voter can toss a coin to decide whether to rank ABC or ACB.
> Assuming there are other voters who do likewise, these ballots should
> also average out to 2, .5, .5.
>
> So adjusted Borda merely does what the voters could do in any event.
> Although I suppose one could argue for Saari's variation as a sort of
> "voter intelligence test", in that it rewards voters who are
> sophisticated enough to get around the restriction.
>
> Bart
>
>
>
>
> Steve Barney wrote:
> >
> > Bart:
> >
> > Where are these Borda rules? I know they are not in the article by Jean
> Charles
> > de Borda, which I referred to in my previous message. I also know that
> Donald
> > Saari, probably the worlds leading exponent of the BC, says otherwise.
> > According to Saari it is essential to treat a bullet vote as an indication
> of a
> > weak preference for one candidate and indifference between all of the
> unmarked
> > candidates. He recommends giving 1 point to the preferred candidate, and 0
> > points to all the rest, in such a case. The essential thing is to always
> give
> > the same difference to each successive ranking. In _Chaotic Elections_,
> Saari
> > suggests that if voters are permitted to give (2,0,0) points to the 3
> > candidates by bullet voting, they will have:
> >
> > "an incentive to vote for only one candidate to give the candidate a boost.
> A
> > simple way to minimize this strategic action is to interpret the BC as
> giving a
> > point differential to each candidate. Thus a truncated ballot assigns only
> one
> > point to the candidate."
> > --Donald Saari, _Chaotic Elections_, pg 151.
> >
> > This interpretation preserves the nature of the BC as a method based on
> > pairwise compairisons. Here is excerpt (from an online article) about the
> way
> > in which the BC can be interpreted as based on pairwise comparisons:
> >
> > An important relationship (probably due to Borda and known by Nanson [17])
> > between the pairwise and the BC tallies can be described by computing how a
> > voter with preferences A > B > C votes in pairwise elections.
> >   Thus the sum of points this voter provides a candidate over all pairwise
> > elections equals what he assigns her in a BC election. This means (along
> with
> > neutrality 2 and the fact that each pair is tallied with the same voting
> > vector) that a candidate's BC election tally is the sum of her pairwise
> > tallies. (See Saari, _Basic Geometry of Voting, Springer-Verlag, 1995
> > "EXPLAINING ALL THREE-ALTERNATIVE VOTING OUTCOMES," DONALD G. SAARI
> > http://www.math.nwu.edu/~d_saari/vote/triple.pdf

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"Democracy"?: