[EM] Head to Head Comparison of Election Methods
Paul Dumais
paul at amc.ab.ca
Wed Jun 2 09:47:21 PDT 1999
> > > > 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
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.
>
> --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.
>
> > 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
>
> Yet you yourself are willing to accept a candidate who does very
> poorly by this standard, as in the following example.
>
> 51 A B C D E F G H I J K L
> 49 B C D E F G H I J K L A
>
> A total=561/1100
> B total=1049/1100
> If this is "clearly relevant," why don't you pick B? If you would
> instead pick A, then how can clearly relevant information go so far
> wrong?
The clearly relevant information is very important here. A is the winner
(by Dumais). The clearly relevant information has not gone wrong at all.
This information makes sure that A is never gets eliminated. It also
makes sure that B is not eleiminated until the last step. In my above
examples, some relevant information is ignored by path voting which
leads to a poor result (ie a clearly inferior candidate is elected). I
don't pick B because that is what borda count would do. Dumais is a
compromise which removes "lower" alternatives in a monotomic fashion by
a borda-like process. This difference from borda produces a method which
is disproportionately better than borda itself.
> > 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?
>
> This information must be tabulated in order to find the Path Vote
> winner. To go back to my Country, Rock, Folk example, the voters
> preferences are such that Folk songs will always lose to Country
> songs. PV therefore ignores whether a Folk song loses to 1 or 1000
> Country songs, since this is more a feature of how many were nominated
> than what the voters actually want. In a sense this is ignoring
> information, but information about how many songs were nominated from
> each genre, not about peoples actual preferences.
>
> ---
> 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
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). Dumais ensures that duplicates are eliminated such that the
final comparison(s) will be between Rock and Country anyway. We'll have
to get some more concrete examples to shed light on this; since
duplicates appears to be your major arguing point, perhaps you should
start. I'll be away on business until Sunday, so if I don't reply right
away, rest assured I'll be back at it next week. Thanks for the great
discussion!
--
Paul Dumais
More information about the Election-Methods
mailing list