[EM] Head to Head Comparison of Election Methods

[Blake Cretney]

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.

> 
> 
> 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.  However, since in simple examples Borda and Dumais
seem to be strongly affected by irrelevant and duplicated information,
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