[EM] Some FWD letters from Paul Dumais
Donald E Davison
donald at mich.com
Mon Apr 19 21:11:58 PDT 1999
Greetings List,
Paul Dumais is now posting to this list.
There are three letters below that he send before he started posting
to the EM list.
Donald
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-----------Forwarded Letters ------------
Date: Mon, 19 Apr 1999 09:53:27 -0600
From: Paul Dumais <paul at amc.ab.ca>
Organization: AMC
Subject: Re: is Salva better than Borda?
In the original Borda count (my favorite), I think the point assignment
is quite logical. You get one point for each opponent you defeat. So
with 5 candidates you get 4 points as the winner (per voter) and 0 if
last. The total number of points can be thought of as the number of
candidated you defeated/vote times the number of voters. Divide this
total by the number of voters and you get a really meaningful number
which represents the average number of candidates defeated for a single
vote.
The original borda count cannot result in an unfair result, while Salva
can. As I said earlier, Salva requires votes to be transferred. To do
this you must choose "the lowest/lower candidate(s)" to transfer votes
from. Basing such a choice from the first choices of voters can result
in an unfair transferring of votes from one or more candidates. This
makes all the difference between somone being elected or not. There have
been many examples of such things happening for STV. The same can occur
with the Salva method. Perhaps some examples will help. Here's one:
ABC 34
ACB 0
BAC 16
BCA 16
CAB 0
CBA 33
If you go through the various methods:
STV: B is eliminated right away, then C to give A the win.
Condorcet: B wins its pairings, A wins one pairing
Salva: Voters who choose B or C as their first choice get their votes
transferred. The winner is A.
It is my opinion that B is the clear winner. Borda count confirms this.
Condorcet does too, but I dissaprove of the votes wasted in each paring
and of course the circular tie issue. The result for Borda count (in
average points / voter is: A: 0.848484848 B:1.323232323 C:0.828282828.
A's "borda number" could represent the the number of candidates that A
"defeated". This number would be meaningful no matter how many
candidates or voters there were.
I think we should look at which voting method produces the least chance
of getting such a clearly unfair voting result. Borda count measures up
quite nicely here. Any thoughts?
mayoral at redestb.es wrote:
>
> What Salva Voting is also good at is to completely eliminate tactical
> voting. When people has only a single opportunity to vote they split up in
> two: those who tactical vote (ie vote for a candidate that is not his-her
> more preferred) and those who are willing to waste their vote (vote for
> his-her most preferred although they know it will not obtain
> representation). With Salva Voting you can safely vote for your most
> preferred candidate, because you know you will have a vote in any case,
> cause it is a one-man-one-vote method instead of methods in which
> only-some-men-one-vote.
> However, Borda count has the same advantages than Salva Voting (no waste of
> ballots, no tactical voting, no circular ties, majority can not be defeated,
> etc). Can somebody tell some advantage Salva has over Borda or some
> disadvantage Borda has and Salva doesn't?
> I think Borda count has an important component of arbitrariety in allocating
> points to choices. Why should the difference between choices be one point
> instead of two points or half a point or 0.1 points or an exponential
> progression? I think that depending on how many points are allocated on the
> different choices it can arise some sort of tactical voting (people may
> alter their wished ranking in order to help lower candidates over
> for-sure-will-have-seats candidates, for instance).
> Salva Voting has no arbitrary decission involved, And there is no way I can
> think of, than can lead somebody using Salva to tactical vote.
> SO SALVA IS BETTER!!!!!!!!!!!!!!!!!!!!
--
Paul Dumais
- - - - - - - - - - - second letter - - - - - - - - - - - - -
Date: Mon, 19 Apr 1999 11:23:29 -0600
From: Paul Dumais <paul at amc.ab.ca>
Organization: AMC
To: Michael Catchpole <catchpol at dolphin.net.au>, cdd-l at cdd.bc.ca
Subject: Re: Fw: A further elucidation of my arguments
Hi,
My thoughts:
> -----Original Message-----
> From: Michael Catchpole <catchpol at dolphin.net.au>
> To: d_saari at math.nwu.edu <d_saari at math.nwu.edu>
> Cc: David Catchpole <s349436 at student.uq.edu.au>
> Date: Monday, March 29, 1999 10:33 PM
> Subject: A further elucidation of my arguments
>
> Dear Professor Saari,
>
> I apologise that this is the second e-mail I have sent, but I hope it
> will shed further light on the arguments I make about your comparisons
> of Borda and Condorcet voting methods.
>
> (i) I believe some of your assumptions about the transitivity or more
> accurately the rationality of social choice, especially in "Explaining
> all 3-alternative voting outcomes," are false. A way to demonstrate
> how this might be is to imagine that a family have entered a
> greengrocer's to decide what to purchase for lunch. Available are- a
> bag of apples, a bunch of bananas, and a cabbage. They can only afford
> one of these choices and the grocer is adamant that they must be
> bought as whole units.
>
> Both parents would prefer bananas best, then the cabbage, then apples
> last. All three children would prefer apples, then bananas, then the
> cabbage. The voting profile then looks like this-
>
> A>B>C
> A>B>C
> A>B>C
> B>C>A
> B>C>A
>
> Borda happens to be walking by and uses his pet method. "I say your
> family prefers bananas," he pronounces. Condorcet overhears, briskly
> dashes in, and yells "I say you prefer apples!"
>
> At the same time, a social choice theorist wishing to stir the pot a
> little strolls in and in the confusion buys the cabbage. Borda and
> Condorcet look confused. "Your family prefers apples to bananas!" says
> Borda. Condorcet nods his head, and wonders whether Borda believes
> what he has just said...
>
> Borda has said two things which are contradictory for a rational
> society- how can it be that a society would prefer B over both A and C
> AND still prefer A over B? It can be proved that as long as we expect
> that majority rule should hold for two-candidate elections this type
> of irrationality always exists for any set of candidates and any
> voting profile except where the society returns a Condorcet winner
> (/ordering) for every subset of candidates.
Probablility theory will clarify this apparent paradox. According to
accepted theory if you have three cards face down (only one of which is
an ace) and you must choose one, the probaility of choosing and ace is
33%. If after you have chosen a card (not revealed yet), a card that you
have not chosen is shown not to be the ace. What is the probability that
the card you chose is the ace? The answer is still 33%. Why? This is
because the probability of choosing an ace is set when you make your
choice. Information gained after the choice will not affect this
probabilty. Your grocery paradox is similar to this one. Borda count
gives a number which is equal to the number of pairings that each
candidate wins. In your example, there was three choices which allows 3
pairings. A vote of ABC means 2 pairings for A (AB and AC) and 1 for B
(BC). The totals in the borda count are: A:6 B:7 C:2. Divide each result
by the number of voters and you get the average number of pairings won
per voter A:1.2 B:1.4 C:0.4. Divide this by the number by the number of
opponents that each candidate has (pairings/ candidate) and you get
A:0.4 B:0.47 C:0.13. This should be (I havn't done my proper math yet)
the probability that a given candidate will win against a random
opponent. When the number of opponents is reduced from 2 to one, the
result is A:60% and B:40%.
So what does this mean? It means precisely what probabilty theory says
and nothing more: if we choose an opponent at random for a given
candidate, bananas are most likely to win. If the number of candidates
are reduced to two, apples are most likely to win. It is my oppinion
that borda count does not produce a parodox as long as voters understand
what the vote means. It seems reasonable that the family may want to
choose apples because they are preferred over bannanas. Therefore, I
propose a borda count with the following modification.
Paul's borda count:
Do a standard borda count, then eliminate the lowest candidates. The
number of candidates kept is equal to the number of choices (seats, one
in this example) + 1. Do a second borda count. Eliminate the last
choice.
This method should greatly reduce the possibility that the choice made
differs when comparing it to the first eliminated candidate only. The
"perfect" method would probably be to eliminate the lowest candidate one
at a time. I'm not sure what would be gained by this however.
> It may well be that there is some kind of "implication" of individual
> irrationality for a Condorcet winner (and I have voiced my concerns
> about the "implication" in my earlier e-mail)- this does not detract
> from the fact that where the Borda result differs from a Condorcet
> method's results it is either true that the Borda result makes a
> direct assault on social rationality where it would otherwise exist
> (and does in a Condorcet method's results) or no such rationality
> exists.
Condorcet and Paul's Borda count should always agree and Paul's borda
count will be considered rational by anyone whene the results of the
vote are properly quatified - for example, candidate B has the highest
probabilty (47%) of being chosen among the three candidates, or
candidate A is preferred 60% of the time when compared to bannanas
alone.
> (ii) Your theorem 11 in "Explaining..." is useful, but not quite so
> biased to Borda Count as the paper seems to imply. The exclusion of
> all voting profiles which cause Condorcet paradoxes is to some extent
> nihilistic, but this is not my problem- you seem to have failed to
> acknowledge what a Condorcet paradox implies, namely- that a rational
> ordering (as described above) cannot exist for any set and profile for
> which a Condorcet paradox occurs and that IIA cannot exist for any set
> and profile for which a Condorcet paradox occurs. In fact after
> proving the strict correspondence between R, IIA and the election of a
> Condorcet winner, the theorem simply becomes one where anonymity, weak
> Pareto, and binary independence for a set always may occur where the
> possibility for Rationality/IIA exists for that set, and may exist
> where the possibility for R/IIA exists but R/IIA is not satisfied. In
> other words, for any set of candidates and any voting profile, R/IIA
> (the existence and election of a Condorcet winner for all subsets of
> candidates) implies all the rest.
Sorry, I havn't read the previous posts re R/IIA etc. I would like to
say however, that condorcet is equivalent to borda except, pairings in
condorcet produce votes which are wasted (you can waste them by winning
a pairing strongly and by losing a close pairing). Condorcet treats each
pairing equally. Paul's Borda count would consider only the pairings of
the top candidates without wasting any votes. It is the low precision of
condorcet which produces the frequent circular ties.
--
Paul Dumais, B.Eng.
Project Engineer
- - - - - - - - - - - - - - third letter - - - - - - - - - - - - - -
Date: Mon, 19 Apr 1999 12:28:42 -0600
From: Paul Dumais <paul at amc.ab.ca>
Organization: AMC
To: Donald E Davison <donald at mich.com>
Subject: Re: [EM] FWD: Borda Count by Paul Dumais
Donald E Davison wrote: [correction: Bart wrote]
>
> ----------- Forwarded Letter ------------
> Date: Sat, 17 Apr 1999 23:48:16 -0700
> From: Bart Ingles <bartman at netgate.net>
> MIME-Version: 1.0
> To: election-methods-list at eskimo.com
> Subject: Re: [EM] FWD: Borda Count by Paul Dumais
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>
> Here is a reply to Mr. Dumais's request.
>
> --Bart
>
> 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.
>
> -----------------------------------------------------
>
> 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.
>
> Note that under Approval Voting, the voters could vote as follows:
>
> 45 A
> 15 BC
> 40 C
>
> C would probably win, with around 55 votes, versus 45 for A and 15 for
> B. The A voters could prevent C from winning by voting for both A and
> B, but are unlikely to for two reasons First, A has enough votes that
> they may think that A has a chance of winning. Second, since they think
> that B is a poor candidate, it will be unlikely that many of them will
> be willing to vote for B the same as for their favorite.
Under my proposed borda count (Paul's borda count) unranked candidates
split the available points (this is equivalent to giving 0.5 points for
all tied pairings. It is similar to approval voting in that a candidate
would vote A or BC or C. It is implied that A means A gets 2 points and
B and C get 0.5 each. BC implies 2 for B and 1 for C. You will see that
this will give a fair result in the example above. A: 110 B:72.5
C:117.5. To ensure extra fairness, I've included a second step, which
compares the top candidates only. For this example A and C are compared
and C stil wins: 55:45.
>
> ----------------------------------------------------
>
> 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
> of the A and C supporters to engage in the same tactic. The sensible
> strategy would be to try to defeat the major opponent, while being
> careful not to give the election away to a "dummy" opponent. The result
> would be a tendency to equalize the scores of all three candidates,
> resulting in a nearly random outcome.
>
> Note again that under Approval voting, voters could vote:
>
> 45 A
> 15 BC
> 40 C
>
> Here, the A and C voters have no incentive at all to vote for a second
> candidate. The B voters know that they have no chance of winning, but
> can get a moderate improvement in outcome by including a vote for C
> along with B.
Paul's Borda Count gives:
A: 110 B:72.5 C:117.5
Or if no strategic voting was done:
45 AB
15 B
40 CA
This gives: A:137.5 B:75 C:87.5
after eliminating b: A:52.5 C:47.5
This means Borda is not immune to strategic voting, but so is approval
voting. In any case approval voting (AV) is similar to borda count
except AV gives one point to it's highest choices and 0 to all others.
Borda count is the simplest and more accurate way to match sincere
ratings.
>
> -----------------------------------------------------------
>
> While a couple of examples alone don't prove that Approval is a better
> method than Borda (or other ranked methods), they do seem to poke holes
> in some of the arguments to the contrary. How good a method is at
> distinguishing between to nearly equal candidates is immaterial when the
> method is also capable of letting a very bad candidate defeat both of
> them.
My modified borda count should rarely produce unfair results. Thank-you
for the great examples!
--
Paul Dumais, B.Eng.
- - - - - - - - - - - - - Paul's letter to EM - - - - - - - - - - -
Date: Mon, 19 Apr 1999 13:44:56 -0600
From: Paul Dumais <paul at amc.ab.ca>
Reply-To: paul at amc.ab.ca
Organization: AMC
MIME-Version: 1.0
To: election-methods-list at eskimo.com, cdd <cdd-l at cdd.bc.ca>
Subject: Re: Reply #2: [EM] FWD: Borda Count by Paul Dumais
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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.
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Dear Paul,
Blake Cretney wrote the letter which you are replying to in this letter.
I merely forwarded it to you.
Donald
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