Condorcet(x( ))

[Steve Eppley]

DEMOREP1 at aol.com wrote:
[snip]
>In other words, the whole issue of x being 0, 0.5, 1 or some other
>number is a waste of time. 

Since it can make a difference in the outcome, I wouldn't say this is
a waste of time.  Perhaps I misunderstood your point, though; it
looked like you were making a technical point that x couldn't affect
the outcome, but maybe you meant that the effects are unimportant?

If I understood you right, you've apparently overlooked that an A=B
in someone's ballot might (if x>0) or might not (if x=0) contribute
to one candidate's *worst* pairwise defeat and not the other's,
thereby changing the outcome.  If the voting method used margin of
defeat (instead of votes for opponent), then x would make no
difference to the outcome because the equals would cancel out.  
But Condorcet doesn't use margin of defeat.  Back when I thought
Condorcet used margin of defeat, I too thought that x doesn't make 
a difference.

Here's an example showing how x can affect the outcome.
Suppose with two ballots remaining to be tallied, the results 
so far are:
  A losing to B (45 to 51)
  B losing to C (40 to 52)
  C losing to A (43 to 53)

If the two remaining ballots are both {A=B > C}, who wins?  

It depends on x.
With x=0:
  A lost to B (45 to 51)
  B lost to C (42 to 52)
  C lost to A (43 to 55)
  A wins with smallest worst defeat = 51.
With x=.5:
  A lost to B (46 to 52)
  B lost to C (42 to 52)
  C lost to A (43 to 55)
  This is a tie between A and B, with smallest worst defeat = 52,
    so some other tie-breaking method would pick one of them.
With x=1:
  A lost to B (47 to 53)
  B lost to C (42 to 52)
  C lost to A (43 to 55)
  B wins with smallest worst defeat = 52.

Since the choice of x can affect the outcome, I'm not convinced yet
that discussing it is a waste of time. 

--Steve