[EM] Head to Head Comparison of Election Methods
Paul Dumais
paul at amc.ab.ca
Tue Jun 1 08:21:47 PDT 1999
Blake Cretney wrote:
>
> Paul Dumais wrote:
>
> > > You seem to believe that the fact your method uses Borda is a selling
> > > point. I view the use of Borda to be a fatal flaw in your method.
> > > The problem with Borda is that it allows the same information to be
> > > re-used.
> > >
> > > For example, if candidate X loses to candidate Y, the method uses
> > > this as evidence against candidate X. Fair enough. But if two
> > > candidates Y1, and Y2 are both in the running, and are identical in
> > > the mind of voters, it will follow that X will lose to both by the
> > > same margin. So the information, that a certain majority of voters
> > > prefers a candidate like Y to one like X, is essentially counted
> > > twice.
> >
> > The result would be Y>X or Y1>Y2>X which is quite valid. We'll have to
> > look at some concrete examples to campare.
>
> 1 0 1
> 40 X Z Y
> 25 Y X Z
> 35 Z Y X
>
> X wins
>
> 3 1 1 3
> 40 X Z Y1 Y2
> 25 Y1 Y2 X Z
> 35 Z Y1 Y2 X
>
> Z wins
>
> This example shows the situation I was describing. The first example
> shows that each candidate has some evidence against it being the best
> candidate. Z loses to X, Y loses to Z, and X loses to Y. What
> decides the winner is the margins of these pair-wise choices.
>
> The second example shows what happens when someone suggests a second
> Y alternative that is almost (or exactly) the same as the first. Now
> the fact that a policy like Y is preferred by a majority to a policy
> like X is counted twice, and so X loses to Z. If you were to propose
> more policies like Z, then X would probably win again.
>
> > > As a result, Dumais suffers from vote-splitting (as demonstrated in
> > > my previous email on the subject), although not to the same extent as
> > > plurality. This means that the more candidates there are representing
> > > an ideology, the more likely it is to lose.
> >
> > This is not true for Dumais. In rare cases (involving circular ties)
>
> It is true that it only happens when there is no Condorcet winner.
> Of course these are the only times when Path Voting and Dumais are
> different.
>
> Please avoid the term "circular tie," as it causes certain people on
> this list a lot of confusion. They believe that "circular tie" refers
> to a state of the method rather than the ballots, and will claim that
> this means that Dumais cannot handle these situations. I am not
> joking.
>
> > clones may cause a different candidate to be eliminated. If that
> > candidate was most often beaten by the clone, then it could affect its
> > standing. Using your example and Dumais:
> > > 35 A B C
> > > 33 B C A
> > > 32 C A B
> >
> > We get
> > No clones: A>B>C
> > clone A: C>A1>A2>B
>
> This is the problem. You clone A, and as a result C wins.
>
> > clone B: A>B1>B2>C
> > clone C: B>C1>C2>A
> >
> > As you can see, cloning an ideology can hurt you, may make you
> > indifferent or
> > help you depending on how you measured up in the vote.
>
> For many purposes, only the top ranked candidate matters. Therefore,
> the "helping" example of cloning C is not going to make any
> difference, since this just moves C from third to second place.
> Presumably if you were picking the top two candidates, then a single
> clone could only help.
>
> >
> > > This is a serious problem for its own sake.
> >
> > I disagree that this problem can be serious. You'll have to show me.
>
> For example, in Canada, a plurality-based elimination procedure is
> used to elect party leaders. Probably a major reason that this method
> is used instead of simple plurality is that it does not give the same
> incentive to form smaller sub-parties in order to prevent vote
> splitting. It seems unlikely to me that parties could be convinced to
> use Dumais instead, since Dumais would also suffer from vote
> splitting, although to a lesser degree.
>
> My problem with this goes beyond the issues of strategy, however. It
> seems to me that a method should as much as possible base the result
> on the voting process rather than the nominating process.
>
> Obviously, the nominating process can affect the result if an
> alternative does not get nominated that would otherwise win. That
> cannot be avoided. As well, an alternative may be suggested which
> allows the voters to give more information about the other
> alternatives. For example, if we have
>
> 60 A B
> 40 B A
>
> It looks like A is better than B. But if a third candidate (C) is
> introduced
>
> 35 A B C -- was A B
> 25 C A B /
> 40 B C A
>
> Now we know that there is a candidate who is preferred by a majority
> to A, but not B. Such a candidate need not have existed. The fact it
> does can reasonably be said to be new information. This extra
> evidence suggests a different outcome.
>
> However, the fact that if you run 20 candidates equivalent to B,
> voters will make the same decisions about them that they did for B is
> not new information. It should therefore not affect the result.
>
> --snip--
>
> >
> > I can create any number of examples (using 4 or more candidates in a
> > circular tie) that have the winner (via path voting) being defeated
> > pairwise by all other candidates except one. Here's a small example:
> > a b c d tot
> > a 53 1 47 101
> > b 47 54 52 153
> > c 99 46 52 197
> > d 53 48 48 149
> > 600
>
> Are you saying that you would abandon Dumais if it could be shown
> that for any number of candidates an example can be constructed in
> which the Dumais winner beats only one other candidate? If not, why
> hold this against Path Voting.
Perhaps not quite. I'm saying we should abondon any method which picks a
winner which pair-wise loses against all but one candidate while not
picking a candiadate that pair-wise wins against all candidates but one.
This is obviously far more grosly unfair than any shortcoming that you
might point out in Dumais voting.
>
> >
> >
> > Dumais gives us B>C>D>A while Path gives us D>B>C>A. Notice that D is
> > more often defeated than it wins when compared to every other candidate;
> > it gets 149/300. This effect gets even more pronounced when we use more
> > candidates. I could give you an example where there is 100 candidates in
> > a circular tie. D could defeat only one of his opponents (head to head)
> > yet still be declared the winner (under path voting) while C could
> > defeat 98/99 opponents (head to head) and still lose to D under Path
> > voting. The borda count in this example could also show C to be vastly
> > superior to D when compared to all candidates. The problem with path
> > voting is that it doesn't use a lot of information. If you used all
> > relevant information, you would converge on a method similar to Borda or
> > Dumais.
>
> For a complicated example like the one you propose, it is very hard
> to conclusively say which candidate should win, or which information
> should be used.
I think that it's not that complicated and it can be quite clear that
the winner by path voting should not win. Here's an example with 5
candidates:
a b c d e tot
a 54 1 1 46 102
b 46 54 1 53 154
c 99 46 54 53 252
d 99 99 46 53 297
e 54 47 47 47 195
The path winner is E even though it loses pair-wise against all
candidates but one. It seems clear to me that D should at least be
considered since it defeats all other candidates (including E) except C.
I think this is grossly unfair. Dumais could never produce a result this
unfair. Candidate C would rightfully complain (under path voting) that
it lost because the path voting system picks the winner using an
arbitrary subset of the available information. Path voting ignores
informatiuon about B>E C>E D>A D>B. Certainly this information is useful
and important. Any method which ignores this much information cannot be
fair. If we examine the ballots E is preferred only 195/400 times. D is
preferred 297/400 times. Clearly this is relevant information. When
picking the winner don't you want to know how it faired against all the
candidates - especially if these candidates are part of a cycle or
circular tie?
> However, since in simple examples Borda and Dumais
> seem to be strongly affected by irrelevant and duplicated information,
To very different degrees. See my simple example above.
> I see it as most likely that this is the cause of the difference
> between these methods and PV in this example as well.
>
> ---
> Blake Cretney
> See the EM Resource: http://www.fortunecity.com/meltingpot/harrow/124
--
Paul Dumais
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