# 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*

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

```