# [EM] Re: Weak INI -- three possibilities

Richard Moore moore3t1 at cox.net
Wed Apr 14 20:59:16 PDT 2004

```Jobst Heitzig wrote:
> Since only X beats Y directly, there is no beatpath from X to Y through
> Z. (Only beatpaths without loops can be meant, otherwise (at least when
> winners belong to the Smith set) there would in general be a beatpath
> from X to Z and one from X to Y which together with Z>X would combine to
> a loopy beatpath from X to Y through Z...)

I thought the definition of beatpath (as given by Markus) excludes
paths with loops. If my memory is incorrect on that matter, then your
qualification is appropriate.

> However, although beatpaths are a wonderful construct from a
> mathematical point of view, I'm not sure whether the existence of a
> beatpath without a direct beat has that much significance to become part
> of a meaningful condition...

In the case of Weak INI, it seems proper to consider a beatpath to be
"supporting evidence" for the candidate at the top of the beatpath to
defeat the candidate at the bottom of the beatpath. This broadens the
definition of "supporting" -- i.e., narrows the definition of
"nonsupporting" -- which is a good thing given that the original
definition of "nonsupporting" was quite broad . When Y has better
margins to Z than X has, a beatpath from X to Y might offset this
nonsupport with a different form of support. Note that the "direct
beatpath" concept (Y>Z>X) is already covered by the margins-only
definition of "nonsupporting", but I consider it a narrow special case.

In the case of Weak ICI, leaving out the beatpath provision leaves
only the Y>Z>X direct beatpath cases to be considered "contradictory",
which would make "contradictory" merely a special case of the
margins-only definition of "nonsupporting". This narrow definition of
"contradictory" would make for a very weak ICI, in my opinion.

Strong ICI considers many more possible configurations to be
"contradictory", then feel free to consider Strong ICI as too strong
to be of great use, just as Strong INI is too strong to be of great use.

Here is another way to define "supporting" or "contradictory" using
only beatpaths: Compare the strength of the strongest X-->Y beatpath
through Z with the strength of the strongest Y-->X beatpath through Z;
if the first is stronger than the second then Z supports X over Y; if
the second is stronger than the first then Z contradictts X over Y. I
haven't had time to consider the ramifications of these definitions,
but clearly there are many ways these terms could be defined, and
correspondingly many versions of ICI and INI that could be stated.

-- Richard

```