[EM] FWD: Borda Count by Paul Dumais

Blake Cretney bcretney at postmark.net
Sun Apr 18 14:28:21 PDT 1999


Now that we have the approval perspective on Borda, I'm going to argue
what I see as the Condorcet view.  Just so it's clear, I advocate a
particular kind of Condorcet completion method called Path Voting. 
This is defined as follows:

For any two candidates X and Y, if more people rank X over Y than
vice versa,then X has a majority over Y.  The number of votes by which
the X over Y exceeds the Y over X is the margin. If, starting with a
candidate X and moving only from the winner to the loser of these
majority decisions you can reach a candidate Y, we say there is a path
from X to Y.

If there is a path from X to Y, made up only of majorities above a
certain margin, and no path back made up of majorities of the same
margin, then X is declared as ranked over Y by the method.

If the method ranks one candidate over all others, that candidate is
the winner.
I describe this method in detail at:

http://www.fortunecity.com/meltingpot/harrow/124/path

Bart Ingles wrote:

> The following are examples of failures in the Borda Count voting
> method.  The first example shows a failure mode common (in some form) to
> most if not all popular election methods that use ranked ballots, and
> the second shows a failure that appears to be peculiar to Borda.
> 
> The examples show voters' suitability ratings of the candidates on a
> scale of 0-100.  I am assuming that each voter has the right to give a
> maximum score (100) to his/her first choice, and minimum score (0) to
> his last choice, and that the voter has the right to place middle
> candidates at any position along that scale.  I use average ratings to
> score the overall suitability of a candidate.
> 
> While voters' sincere ratings are probably not measurable, I believe it
> is a mistake to pretend they do not exist.  Even though we may not be
> able to use ratings directly in an actual election, we can set up
> examples based on sets of voter ratings and see how various election
> methods behave.

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.

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/



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