[EM] Head to Head Comparison of Election Methods

[Paul Dumais]

Blake Cretney wrote:
> 
> Paul Dumais wrote:
> 
> > I would like to propose a debate to compare my favorite election method
> > (I call it Dumais) and anyone else's favorite election method (for fun).
> > My goal is to discover which method is the best at fairly and
> > efficiently ranking candidates based a set of voters' choice(s). Though
> > I doubt any method will be universally accepted in every situation; I
> > think such a debate will bring interesting issues to light which may
> > alter people's views. I will take up this challenge with the first
> > serious challenger.
> >       Here is a description of Dumais voting.
> > 1.  Each voter ranks one or more of the available candidates.
> > 2.  Do a standard borda count.
> > 3.  Divide the candidates into two groups. One group (the preferred
> > group) all have a Borda Count (BC) of BC > (C-1)*V/2 where C is the
> > number of candidates and V is the number of voters. The second group
> > (the non-preferred group) has BC < (C-1)*V/2.
> > 4.  Repeat steps 2 and 3 (on the preferred group) until the preferred
> > group is small enough to choose the winner or winners. A complete
> > ranking of all candidates can also be done in this way by repeating
> > steps 2 and 3 on each group.
> >
> > Unranked choices split the remaining borda count (or (U-1)/2 where U is
> > the number of unranked candidates). This method can be interpreted in
> > another way. Each candidate gets one point for every opponent it defeats
> > on a single ranking by a single voter. Each candidate gets 1/2 point for
> > each opponent it ties with (unranked candidates). The borda count is
> > equivalent to asking: "how many times does a candidate (ie A) defeat
> > another candiate when we compare him to every other candidate using all
> > the available information (ie using each voter's ranking of the
> > candidates)?". Elimination is required to reduce the effect of
> > introducing identical candidates (as a stratategy) and to allow us to
> > reduce the effect of "lower" candidates in our choice of the best
> > candidate. A borda count in the only fair way to choose candidates for
> > elimination, since it is the only method that takes into account all of
> > the voting information without predjudice.
> 
> What do you mean by "without prejudice?"

I was probably alluding to the different forms of elimination that have
been proposed. STV is a particular example that comes to mind. In STV we
eliminate the candidate with the lowest number of first place votes.
This seems to "prejudge" that first place votes should be used to
determin the last place candidate.
> 
> Also, I do not agree that Borda uses all the voting information.  For
> example, Borda's method of totalling renders these two blocks of
> voters
> 30 A B C
> 30 B C A
> 30 C A B
> 
> and,
> 
> 30 A C B
> 30 B A C
> 30 C B A
> 
> as equivalent.  That may or may not be a good thing, but it points
> out the fact that Borda ignores certain kinds of information.  Of
> course, this is true of all methods.

As long as no one advocates that the information ignored by borda should
be used in any way, I think the point I made above is still "mostly"
valid.
 
> 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.
 
> 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)
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
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.
 
> This is a serious problem for its own sake.

I disagree that this problem can be serious. You'll have to show me.

> However, I see it as
> particularly unfortunate because this is not a problem in IRV.  If we,
> as Condorcet advocates, propose Dumais, IRV advocates will point this
> failing out.

I will gladly go head to head with a proponent of IRV, but I'll try
Dumais out on Path Voting first if you don't mind.
 
> This is one of the reasons I advocate Path Voting (which seems to
> have been invented by Markus).  If you have a path from one candidate
> to another, like the following from A to D:
> 
> A has a majority over B of 19 votes
> B has a majority over C of 20 votes
> C has a majority over D of 18 votes
> 
> we measure the path's strength by its smallest majority, in this case
> 18.  A will be ranked above D unless D has an even greater path back
> to A (strength measured by smallest majority).
> 
> So, for a simple case
> 40 A B C
> 35 B C A
> 25 C A B
> 
> A>B 65-35=30
> So, A has a path to B of strength 30.
> B's best path to A is
> B>C 75-25=50, C>A 60-40=20
> giving a strength of 20.  So A will be ranked above B.
> 
> A>B [30] B>C [50]
> So, A has a beat path to C of strength 30.
> C's best path to A has strength 20, so A will be ranked above C.
> 
> Looking further we have B...>C [50] and C...>B [20], so B is ranked
> above C.
> 
> So, A B C is the complete ranking.  In this case the same as Dumais.
> 
> Can you give an example where you think the Path Voting result is
> inferior to that given by Dumais?

> Blake Cretney

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


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.

-- 
Paul Dumais