[EM] Interesting use of Borda count
Bart Ingles
bartman at netgate.net
Thu Jan 3 20:48:58 PST 2002
Steve:
I agree with your Saari results, if the two voters are ignorant enough
to actually bullet vote (even though this may accurately represent their
preferences).
One of the ways to defeat Saari's variation is for the two voters to
collaborate: One voter agrees to rank A, B, and C in order, while the
other ranks A, C, B. So the individual ballots are worth (2, 1, 0) and
(2, 0, 1). The combined total is (4, 1, 1), hence the per-ballot
average (2, .5, .5) I claimed below.
Thus the suggestion that Saari's variation could function as a sort of
voter intelligence screen, since a potential bullet voter who doesn't
understand the above strategy has his voting power reduced by one-third.
Samuel Merrill III (Making Multicandidate Elections More Democratic,
1988) includes the following citation:
Black, D. (1958) *The Theory of Committees and Elections*, Cambridge
University Press.
I haven't read Black's work. But the issue seems moot to me, since with
the voter strategy above (or the equivalent coin-toss strategy), the
three variations we have discussed (Borda, Black, and Saari) are all
equivalent.
Bart
Steve Barney wrote:
>
> 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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