[EM] Head to Head Comparison of Election Methods

Blake Cretney bcretney at postmark.net
Sun Jun 6 12:23:03 PDT 1999


Paul Dumais wrote:

> > > > > 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.
> > >
> > 
> > This isn't obvious to me.  It sounds like you have taken a reasonable
> > sounding, but not necessarily good standard that the number of
> > pair-wise wins should be taken into consideration, and hypothesized
> > that this should not be too dramatically violated.  The result is a
> > plausible sounding criterion that may not have much value.
> > 
> > Imagine a vote held to find the best song of the year.  Imagine, for
> > simplicity, that all songs are either Country, Rock, or Folk.  Now
> > imagine that there are three groups of voters:
> > I   C>F>R
> > II  F>R>C
> > III R>C>F
> > 
> > The more Country songs are suggested, the more victories the Rock
> > songs will have, and the more losses the Folk songs will have.  If we
> > use this as a standard, we are likely to decide the issue as much on
> > how many of each genre are nominated as the actual preferences of
> > voters.
> > 
> > > 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.
> > 
> > Dumais fails this criterion.
> > 
> >   3 1 1 3
> > 
> > 3 A B C F
> > 4 A C B F
> > 1 B C F A
> > 1 C B F A
> > 4 F A B C
> > 4 F A C B
> > 
> > F loses to everyone but A by 9 to 8
> > A wins against everyone but F by 15 to 2
> > 
> > I do the tabulation a little differently from you, but the answer is
> > the same.
> > A 21-6+8 >0
> > B 3-4+3+1-4-12 <0
> > C -3+4+1+3-12-4 <0
> > F 24-2-21>0
> > 
> > So, we get A vs. F, F wins.  Since you say that this is obviously
> > grossly unfair, and that we should abandon any method that does this,
> > will you abandon Dumais?
> 
> I guess I should (good example!). I still have a misgiving about how
> path voting can
> choose a winner which is pair-wise beaten by all candidates but one
> while
> a losing candidate wins pair wise against all candidates but one
> including the path voting winner. Such a result is a much more unfair
> result which path voting produces. I don't think Dumais could produce
> such a result (but I've been wrong before). Here is a more severe
> example:
> 
> 	a	b	c	d	e	tot
> a		100	0	0	0	100
> b	0		100	0	99	199
> c	100	0		100	99	299
> d	100	100	0		99	299
> e	100	1	1	1		103
> 

If everyone votes e ahead of a, and everyone votes a ahead of b, then
everyone must vote e ahead of b.  The above, where 99 out of 100 vote
b above e, is not possible.  This kind of problem is why I tend to
give my examples starting with ballots instead of tables.  

Your previous example is also not possible, since if only one ballot
gives a above c, it is not possible to have a beat b 53-47 and c lose
to b 46-54.  This would imply at least 7 ballots where a is above c.

> e is the path winner though it is not preferred 297 out of 400 times (in
> the first count). Both C and D should be ranked higher than E since it
> is preferred 299 out of 400 times (in the first count). Can you create
> an example where the Dumais winner pair-wise loses against all
> candidates but one while there exists another candidate who pair-wise
> wins against all candidates but one, including the Dumais winner?
> Perhaps this is how I should word the criterion.

Here is an example where Dumais violates your revised criterion.
   2 1 0-1-2
49 A B C D E
14 B C D E A
37 C D E A B

C beats every candidate except B.  A loses to every candidate except
B, including C.  A is the Dumais winner.

A 98-28-37>0
B 49+28-74>0
C 14+74>0
D -49+37<0
E -98-14<0

   1 0-1
49 A B C
14 B C A
37 C A B

A 49-14>0
B 14-37<0
C 37-49<0

I should point out that I think Dumais is actually coming to the
correct decision in this example.

> > --snip--
> > > >
> > > > 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
> > 
> > E wins.  If the B>E contest gave 55 to 45, instead of 53 to 47 then E
> > would not win.  So, clearly the result of this contest is not being
> > altogether ignored.  It is very important that B doesn't do better
> > against E.
> 
> That is true of path voting. However, I have extreme issue with the way
> path voting allows information such as E>A far outweigh B>E, C>E. The
> stipulation that the margins of B>E and C>E must be less than E>A is
> better than nothing, but it doesn't take into account the cumulative
> effects of having a possibly infinite number of candidates beat E by a
> margin less than E>A. I really believe that path voting doesn't produce
> good reslts in these situations.

The Rock-Country-Folk example shows why I don't want a method that
uses cumulative effects.

--snip--
> 
> I'm having difficulty believing that this is a significant problem with
> Dumais. Can you show me an example where this is a very significant
> problem? It seems that a single clone can change slightly who is
> eliminated, but it seems to be only such that the candidate close to the
> elimination line is the only one affected. In your country, rock, folk
> example if we have 1000 country songs, about half would be eliminated in
> each round under Dumais (the single folk song would of coarse be
> eliminated). 

That's the problem.  Let's assume that the genre preferences are as
follows
I   35 C>F>R
II  40 F>R>C
III 25 R>C>F

Under Dumais and Path Voting, Folk is the favourite genre, and if one
song from each is nominated, Folk will win.  This is because although
a majority prefer Country to Folk, it is the weakest majority.

However, if three country songs are nominated, the preferences might
look like this:
       2  1  0 -1 -2
I   35 C1 C2 C3 F  R
II  40 F  R  C1 C2 C3
III 25 R  C1 C2 C3 F

F  -35+80-50<0
C1 70+25>0
C2 35-40<0
C3 <0
R  -70+40+50>0
So, R v. C1, R wins.

Remember your criticism of path voting that it

> doesn't take into account the cumulative
> effects of having a possibly infinite number of candidates beat E by a
> margin less than E>A.

What is defeating the folk song is those cumulative margins from all
the country songs in the running.  If there were 1000 country songs in
the running, then, based on cumulative margins, it would look like the
folk song was a terrible choice.  But this would just be the result of
voters' genre preference combined with the number of each being
nominated, not an actual change of attitude on anyone's part towards
Folk in general or the song in particular.

Let's imagine that the nominating committee dislikes folk music, they
can ensure folk is eliminated just by nominating more country songs,
while still appearing to give this genre a fair chance.

To take this back to the world of politics, if a legislative body is
holding a vote between several proposals using Dumais, then it is
possible for people to cleverly nominate similar proposals with the
hope of having the above effect.  

If you are unaware of voter preferences, and just want to torpedo a
particular proposal, there is a more direct way.  Just nominate other
similar proposals.  I'll show this vote splitting effect for the music
example.
       2  1  0 -1 -2
I   35 C  F1 F2 F3 R
II  40 F1 F2 F3 R  C
III 25 R  C  F1 F2 F3

C 70-80+25>0
R -70-40+50<0
F1 35+80>0
F2 40-25>0
F3 -35-50<0

More folk songs have made the rock song seem like a worse choice. 
The effect is that the rock song is eliminated.  This weakens the
standing of folk, since it won against rock; in general, it is bad to
have alternatives that you defeat eliminated.  This effect is also
what causes the violation of monotonicity.

I   35 C  F1 F2
II  40 F1 F2 C
III 25 C  F1 F2

It is obvious that the country song now wins, since it is the first
choice of a majority among the non-eliminated alternatives.

---
Blake Cretney

See the EM Resource:
     http://www.fortunecity.com/meltingpot/harrow/124
My Path voting Site:
     http://www.fortunecity.com/meltingpot/harrow/124/path



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