[EM] Re: Weak INI -- three possibilities

[Richard Moore]

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. If the result is too broad a definition of 
"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