head to head & Condorcet winner (was RE: Eye Ball Mathematics)

[Steve Eppley]

Donald asked:
>Is not head to head pairing the same as Condorcet? I am confused.

Try reading the report on EM's sw poll.  It has a useful alphabetized
glossary of methods at the end.  Read the definitions of both "Condorcet" 
and "pairwise methods".  (I suggest you read the definition of "pairwise
methods" before you read the definition of "Condorcet.")

You'll find that pairwise methods elect the candidate who beats all
others head to head *IF THERE IS SUCH A CANDIDATE*.  You'll find
that if there *isn't* such a "beats all others" candidate, different
pairwise methods resolve this occasional lack in different ways, by
identifying the candidate with the minimum "beatenness" using different 
notions of how to measure "beatenness".

You'll find that Condorcet's method is a pairwise method.  
You'll find that Condorcet's method resolves the occasional lack of 
a "beats all others" candidate by selecting the one with the smallest
largest pairloss.  It measures "beatenness" by the number of voters
who prefer the pair-opponent in a candidate's worst head to head
loss, and elects the candidate which minimizes this.  This is the
same as electing the candidate which minimizes the number of voters
who would prefer someone else.

You won't find in the report an explanation of why the "beatenness" 
notion measured in Condorcet's method makes it better than other 
pairwise methods.  That's been the subject of the EM discussion for 
9 months (majority rule, LOE, spoilers, deterrence of misrepresentation, 
etc.) and it's my hopeful plan that we'll distill the important 
points into the commentary message(s).

One thing which may contribute to your confusion is that the term
"Condorcet winner" unfortunately means, for academics, the same thing
as the "beats all others" candidate.  This term was coined by
academics years ago to pay homage to the Marquis de Condorcet. 
Electing this candidate, if s/he exists, is the gold standard which
methods need to satisfy if they're to be taken seriously--and is why 
Instant Runoff isn't taken seriously by serious people.  The serious 
debate revolves around what to do if there isn't such a candidate,
and around removing incentives for voters to misrepresent their true
preferences.

We've pointed out this source of confusion several times in the EM
list.  I avoid using the term "Condorcet winner" entirely, and I
recommend everyone else stop using it too in non-academic settings. 
It's less confusing to use unambiguous phrases like: 
   "the one who beats all others head to head, if there is one"
   (or its short but non-introductory form:  "the BeatsAll winner")
and
   "the one whose largest pairloss is the smallest, if there isn't 
    one who beats all others head to head"

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