[EM] Head to Head Comparison of Election Methods

[Blake Cretney]

Paul Dumais wrote:

> Blake Cretney wrote:
> > 
> > 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?

--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.

> 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?

> 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