Attachment COND-FLA

[Steve Eppley]

Bruce A wrote:
>> Bruce, do you have any examples showing flaws of Smith//Condorcet?
>> ---Steve
>
>See Examples 4 & 5 of that paper.

Quoting from his paper (Attachment COND-FLA), about 4 & 5:
>These examples show that Condorcet's method can elect pretty strange
>winners even when every alternative would beat at least one of the
>other alternatives in the head-to-head pairing between them.  In
>particular, note that, in Example 5, the winner according to
>Condorcet's voting method receives the least number of first-place
>votes of all of the candidates, has a lower average ranking than
>any other candidate, wins fewer of its head-to-head pairings than
>any other candidate, and loses more of its head-to-head pairings
>than any other candidate. 

Ok, I've looked at them, and I conclude that I disagree with Bruce's
diagnosis of "strangeness."

It's a fallacy to consider first-place rankings privileged.  It's
pretty obvious that when there are a lot of candidates (and maybe
even when there are only three) a second place ranking (or 3rd, etc.)
may actually be an *excellent* ranking by the voter.  

Here's a simple example showing the fallacy of criteria (or methods
like Hare aka MPV aka Instant Runoff) based on the candidates'
first-place rankings.  Consider what ballots might look like if
voters could sincerely *rate* the candidates:
   Voters    Ratings:   A    B    C                Ranked Ballots
   ------    ----------------------                --------------
     v1               100    0   99           -->      A>C>B
     v2                 0  100   99           -->      B>C>A
Here candidate C gets fewer first-place rankings than any other
candidate, but C is clearly (to my intuition, at least) the best
choice for the group.

Another point to remember: in pairwise methods the vote {A>B>C>D etc.} 
is just a *shorthand* for the separate pairvotes {A>B}, {A>C}, {A>D}, 
{B>C}, {B>D}, {C>D}, etc.  Pairwise methods rely on the reasonable
assumption that voters' preferences are transitive (i.e., a vote 
of A>B>C implies A>C).  When viewed separately--and properly--this
way, there's no such thing as a first place ranking; the shorthand
format misleads the intuition.

* *

The average ranking is also meaningless.  Consider again what voters'
*ratings* might look like:
   {A=100  B=99  C=98  D=97  E=0}
Here D has a below-average ranking.  But it's obviously fallacious
to hold this against D, since this voter really likes D.  

If Bruce thinks this isn't fallacious, I think there's something
wrong with his intuition.  Any criterion based on average rank
position or on number of first-place rankings should be rejected,
since it would be based on inferring info from the ballots that
is not really implied by them.

* *

His other "strange" attribute, losing many head-to-head pairings, is
also unimportant: there is no rigorous correlation between how many
pairings a candidate wins or loses and how much more or less popular
the candidate is compared to the others, so that's a bad criterion on
which to judge methods.  See my recent post entitled "The rich party
problem" which clearly demonstrates this.  In that post, I identified
a criterion which I called "Independence from Twins" and I think it
will be easy to rigorously define this criterion and explain why
it's a good criterion.  Condorcet passes this criterion, but Copeland
and Regular-Champion fail it.

Here's a definition of this criterion which I believe is sufficiently 
rigorous.  (If anyone has suggestions for improving it, they're welcome.)

First a brief explanation, then the definition:
T1 and T2 are politically identical twins, in the eyes of all the
voters.  If Twin1 loses the election when Twin2 doesn't compete,
then Twin1 should also lose the election when Twin2 does compete.

Definition of "Independence from Twins" criterion (ITC):
The winner of an election held without candidate T2 must not change
if T2 is introduced and no voter ranks T2 above or below some other
candidate T1 (i.e., on every ballot, either T1 and T2 are both
unranked or are both equally ranked).  Exception: if T1 was the
winner before T2 is introduced, then it's not considered a changed
result if T2 ties T1 for the win.

It's a particularly egregious failure of ITC if including T2 turns a
losing T1 into a winner, the way Copeland can.  It's not quite as
bad if T2 would turn a winning T1 into a loser by fratricide, since
T1 and T2 have mutual incentives to avoid this fate (the same way 
candidates today avoid running if they believe they'll spoil the 
election for their lesser of evils compatriot).  The example in
my message about Copeland and Regular-Champion's "rich party
problem" demonstrated these methods' egregious failure of ITC.

* *

I believe Bruce's examples 4 and 5 illustrate not real flaws with
Condorcet, but flaws with Bruce's intuition and his notions of
"strangeness."  Unfortunately, there will be many other people who
share these flaws.  Hopefully, criteria like ITC will enhance
people's intuitions. 

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)