[EM] Interesting use of Borda count

Bart Ingles bartman at netgate.net
Wed Jan 2 21:02:21 PST 2002

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.


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
> http://www.math.nwu.edu/~d_saari/vote/triple.pdf
> > Date: Tue, 01 Jan 2002 15:17:21 -0800
> > From: Bart Ingles <bartman at netgate.net>
> > To: election-methods-list at eskimo.com
> > Subject: Re: [EM] Interesting use of Borda count
> >
> > I wonder if Bennett's ballot was counted per Borda rules -- i.e.
> > Bennetts's first choice receiving 10 points, the remaining nine
> > receiving 5 points each.
> >
> > If this were a public election held in Florida, Bennett's candidate
> > would have contested the election, claiming that either the election
> > method or the voter instructions were at fault, causing Bennett to cast
> > non-optimal ballot.  The result being, of course, that Bennett was
> > robbed of her voting rights by being allowed to vote incorrectly.
> >
> >
> > Anthony Simmons wrote:
> > >
> > > Interesting use of Borda count.  Note that one voter insisted
> . . .
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