Re: Fuzzy preferences

RWTodd (rwtodd@aol.com)
Sun, 15 Mar 1998 22:53:09 +0100 (MET)

>I call this theorem the Collapsing Principle. It says that if the
>degree to which you prefer x to y is greater than the degree to which
>you prefer y to x, then it is plain true that you prefer x to y. So the
>vagueness disappears and you are left with a complete preference
>ordering.

I'm new to fuzzy sets and am just learning also, but I think the answer is:
Being able to make a judgement like that doesn't break fuzzy logic.
With crisp logic, you either prefer x (x=1) or you don't (x=0). If you
prefer x more than you do y then x=1 is your crisp answer. But this doesn't
tell you how close the race was, so to speak. Like you said, there are degrees
to which you prefer one over the other.

Simple case. Say y=.5. Maybe x just wins and gets a confidence value of
around .6 or so. This is different (from a fuzzy standpoint) than having x=.9.
Why? Because the engine may be(should be?) set up to recognize that a
difference of around .1 means that x is slightly preffered, but strongly
suggests that there is no real preferrence. A difference of around .4 might
mean that x is strongly preferred, with a very slight degree of confidence that
there is no real preferrence. Fuzzy systems have overlapping boundaries for
classification that make them act this way. If necessary a fuzzy system may
also be set up to decide that if both numbers fall within a certain (fuzzy)
range then there is no real preferrence no matter who wins (like if both x and
y are considered 'very unsatisfying'). This doesn't work with crisp cutoff
points (consider the SHORT 5' guy and the TALL 5'1" guy. Fuzzy logic doesn't
draw strict lines like that).

So x may numerically win, but the fuzzy system may deem them equally preferred
given other considerations. It's a lot more vague than if (x>y)

Once again I'm no expert and could be off....

Richard Todd