Reply #2: [EM] FWD: Borda Count by Paul Dumais

[Paul Dumais]

Donald E Davison wrote:

> 
> Just because I don't choose to rely on ratings doesn't mean I pretend
> they don't exist.  However, I don't view the goal of the election to
> be to find the highest average rated candidate.  Instead, I see the
> goal as finding the most likely best candidate based on the ballots.
> >From this perspective, average ratings would still make sense if you
> believe that a voter's rating is a good measure of the likelihood of
> accuracy.  That is, if the voter says that A is MUCH better than B, we
> would think that this increases the probability that A is in fact
> better than B.  I, however, view allowing people to self-rate the
> certainty of their opinions as misguided.  Often the strangest
> opinions are the most strongly held.  The same errors in reasoning
> that result in a faulty preference can also cause the strength of the
> preference to be exaggerated.
> 
> In particular, if 10 people say that A is better than B, do you
> really think that should be equaled by one person saying that B is
> better than A, even if that person feels 10 times as strongly about
> it?
> 
> Another related point is that Ratings is often advocated as an
> attempt to maximize utility.  Actually, this is an error because
> people will be affected by the outcome of an election by different
> amounts.  By having each persons lowest candidate rated 0 and highest
> 100, we ensure that true utility is not being measured.  Of course,
> Path Voting isn't based on maximizing utility either.
> 
> > In the first example, 45% of the voters prefer A over all other
> > candidates.  15% prefer B, and 40% prefer C.  The full ratings are as
> > follows:
> >
> >
> > EXAMPLE 1:  Voters' private suitability ratings
> >
> >       Rating:
> >       100    80    60    40    20    0
> >       ----------------------------------
> > 45     A                          B  C
> > 15     B  C                          A
> > 40     C                          B  A
> > ---
> > 100 votes total
> >
> >
> > Average sincere ratings:
> >
> > Candidate A = (45% x 100)              = 45.0 points
> > Candidate B = (15% x 100) + (85% x 10) = 23.5 points
> > Candidate C = (15% x 90) + (40% x 100) = 53.5 points
> >
> >
> > Borda Count:
> >
> > Candidate A = 90 points
> > Candidate B = 115 points
> > Candidate C = 95 points
> >
> >
> > Borda picks B as the winner based on rankings, although B has only half
> > the rating of the other two candidates.
> 
> This first example assumes that highest average rating is the best
> possible goal.  Since I disagree with this, I am unconcerned with the
> result of this example (which also apply to Condorcet and PV, if
> anyone is wondering why I'm defending Borda).
> 
> > As bad as the Borda results were in the last example, it gets even
> > worse.  In the following example, B is considered totally unsuitable by
> > all of the A and C supporters.
> >
> >
> > EXAMPLE 2:
> >
> >       Rating:
> >       100    80    60    40    20    0
> >       ----------------------------------
> > 45     A              C              B
> > 15     B                        C    A
> > 40     C              A              B
> > ---
> > 100 votes total
> >
> >
> > Average sincere ratings:
> >
> > Candidate A = (45% x 100) + (40% x 50)              = 65.0 points
> > Candidate B = (15% x 100)                           = 15.0 points
> > Candidate C = (45% x 50) + (15% x 20) + (40% x 100) = 65.5 points
> >
> >
> > Under Borda C should get 140, A should get 130, and B should get 30,
> > right?  The only problem is, since B is perceived as a weak candidate,
> > the A supporters are more concerned with defeating their major opponent,
> > C.  The A voters rank insincerely as:
> >
> >
> > 45   A B C
> > 15   B C A
> > 40   C A B
> > ---
> > 100 votes total
> >
> >
> > Now A has 130, C has only 95, and B has 75.  What should the C voters
> > do?  If they adopt the same strategy by voting CBA, they can knock out
> > A, but will be worse off with last choice B winning (as in the Borda
> > results for Example 1).  They may have to do so, though, in order to
> > discourage the other side from attempting the strategy again in the
> > future.
> >
> > Granted that such severe Mutual Assured Destruction tactics involving
> > every voter are unlikely.  However, it is possible for smaller numbers
> 
> That would be my argument.  I think, however, that Borda has a
> somewhat worse, but similar problem in that if there are a large
> number of fringe candidates, as there often are, then a voter can
> greatly magnify his vote buy burying strong contenders below them.
> Consider the following example
> 
> Sincere preference.  Here, C through I are considered fringe
> candidates:
> 45 A B C D E F G H I
> 55 B A C D E F G H I
> Borda and Condorcet winner is B
> 
> 43 A B C D E F G H I
> 2 A C D E F G H I B - burying tactic
> 55 B A C D E F G H I
> 
> A 45*8 + 55*7 = 745
> B 55*8 + 43*7 = 741
> 
> So, a small number of clever voters can greatly magnify their power,
> at very low risk.
> 
> > > Date:   Fri, 16 Apr 1999 10:22:44 -0600
> > > From: Paul Dumais <paul at amc.ab.ca>
> > > Reply-To: paul at amc.ab.ca
> > > Organization: AMC
> > > MIME-Version: 1.0
> > > To: Donald E Davison <donald at mich.com>
> > > Subject: Re: Salva Voting - multi-seat example
> > >
> --snip--
> > >         I can construct examples where borda count is superior to other
> > > methods. I have not found any examples where other methods are superior
> > > to borda count. Perhaps someone can help.
> >
> 
> Here is the example that I think shows that Borda is unusable.
> Imagine that there are two parties, the D's and the R's.  Now, assume
> there are more R than D voters as in the following example.
> 
> 56 R D
> 44 D R
> 
> Obviously, the R candidate wins, as one might expect.  But what
> happens if the D party runs two candidates, D1 and D2.  The following
> is a possible outcome.
> 
> 56 R D1 D2
> 44 D1 D2 R
> 
> R 112
> D1 144
> D2 44
> 
> One of the D party candidates wins.  Note that there are no more D
> voter's in the second example.  Running more candidates in Borda is a
> very good strategy.  If a legislative body is choosing between
> multiple proposals using Borda, it makes sense to submit as many
> nearly identical copies of your proposal as possible.  This is both
> absurd, and a real practical problem.

Paul's Borda Count reveals: 56 R, 44 D1D2 or 56RD1 44 D1D2
R: 112     or   R:112  
D1: 116         D1:144
D2: 72          D2:44
after first count
eliminate d2 gives
R:56 D1:44

Using borda count to eliminate minor candidates reveals a fair result.
The burying tactic is just another form of voting voting for minor
candidates over a percieved favorite. It will always be a problem in
voting methods. I feel however, that strategic voting will either be
rare or self-moderating. When rare, it's effect is limited. When not
rare, voting results will rebeal a closer race which will make it more
difficult to chose who is favorite, second etc. If we use one or more
steps with borda count (Paul's Borda count) to eliminate the minor
candidates, the vote will always come down to "how many voters chose X
over Y" in a single seat race. As you can see Borda Count converges with
condorcet however, borda count has a higher degree of success due to no
wasted votes.
> 
> My view is that changed results should be the result of new
> information.  Since you can predict the way an identical proposal will
> be ranked on peoples ballots, it should not be considered new
> information, and should not affect the result.
> 
> ---
> 
> Blake Cretney
> My Election Method Resource is at
> http://www.fortunecity.com/meltingpot/harrow/124/



-- 
Paul Dumais