Condorcet misinterpretation (was Re: Arithmetic Errors)
Steve Eppley
seppley at alumni.caltech.edu
Tue Apr 9 18:23:31 PDT 1996
Bruce Anderson wrote:
[snip]
>Either I am missing a whole lot here (which is certainly possible),
>or Steve and Mike disagree on which of their clearly explained, well
>understood, and obviously right results is, in fact, the right result
[snip]
I'm the screw-up. <blush>
I've been misinterpreting the meaning of worst defeat. My intuition
caused me to presume this referred to the worst *margin of defeat*
instead of the number of votes against.
So the examples I've been posting, and the notation I've been using
which shows margins of defeat, have been in error.
I'm glad Bruce's questioning brought this to my attention.
>A New Example:
>
>48: A, (B & C tied)
> 6: B, (A & C tied)
>46: C, B, A
>
>Then #(A>B) = 48 and
> #(B>A) = 52, so B beats A by 52-48 = 4.
>Also #(A>C) = 48 and
> #(C>A) = 46, so A beats C by 48-46 = 2.
>Also #(B>C) = 6 and
> #(C>B) = 46, so C beats B by 46- 6 = 40.
>
>A says: "I scored 48 in my only defeat, while C only got 46, and B
>only got 6. So my worst pairwise defeat was the smallest."
>
>B says: "My opponent only scored 46 in my one defeat, while C's
>opponent got 48, and A's opponent got 52. So my worst pairwise
>defeat was the smallest."
>
>C says: "I lost by only 2 in my one defeat, while A lost by 4, and
>B lost by 40. So my worst pairwise defeat was the smallest."
Let's try out a new notation which omits the margins:
Pairing Results For Against
----------------------------- --- -------
A loses to B with 52 against. 48 6+46
B loses to C with 46 against. 6 46
C loses to A with 48 against. 46 48
B wins. (46 is smaller than 48 & 52.)
I hope I got it right this time.
--Steve
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