# [EM] Fw: IBCM, Tideman, Schulze

MIKE OSSIPOFF nkklrp at hotmail.com
Mon Jul 24 23:50:38 PDT 2000

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Dear Markus--

You wrote:

But as far as I have understood the definition of "innermost
unbeaten sets" properly, then "innermost unbeaten set" is
only a different word for "Schwartz set."

No, because there can be more than one innermost unbeaten set,
and the Schwartz set is then the union of all the innermost
unbeaten sets. As I usually word it, "The Schwartz set is the
set of candidates who are in innermost unbeaten sets."

Of course, when defining SSD, there's no need to talk about
the possibility of more than one innermost unbeaten set. That
unnecessarily complicates the method definition.

You wrote:

If I haven't understood the definition of "innermost unbeaten
sets" properly, then I want to ask you to define "innermost
unbeaten sets" without using Schwartz sets or beat paths.

Certainly:

An unbeaten set is a set of candidates none of whom is beaten
by anyone outside that set.

An innermost unbeaten set is a an unbeaten set that doesn't
contain a smaller unbeaten set.

***

The way there could be 2 innermost unbeaten sets would be if
the candidates in {A,B,C} are each tied with each of the
candidates in {D,E,F}, and the members of both sets are
unbeaten by the candidates outside those sets. Then we've
got 2 innermost unbeaten sets. {A,B,C,D,E,F} isn't an
innermost unbeaten set, because it contains 2 smaller innermost
unbeaten sets.

But that situation is so rare & improbable that there's no need
to talk about it to people or word the SSD definition so as to
talk about the possibility of having more than 1 innermost
unbeaten set.

Mike Ossipoff

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