When <= isn't an ordering: Example: teams rated on Offense, Defense, Stamina.

    Sets w/ no natural order:
    The ec-hw had some exercises from the book about partial orderings
     Eg, you can some things are bigger than others 
       (we can say that Gwen Stefani >= Backstreet Boys,
        but we can't compare everything: 
         Gwen Stefani !>= Leonard Bernstein, and Leonard Bernstein !>= Stefani.
        However, we do know that Stefani >= Backstreet Boys and
         Backstreet Boys >= N*Sync is enough to conlude Stefani >= N*Sync.
       That is, "better music than" is not a complete relation, 
         but it is transitive.

    However, sometimes you have a relation that seems like it should
      be transitive, but it's not!
      For example: suppose Ultimate Frisbee teams really have three
      underlying ratings: 
            Offense, Defense, and Stamina.
      And suppose that when two teams play, whoever has a better rating
      in two out of the three categories is the winner

      It seems like the relation "beats(X,Y)" should always be a nice,
      complete ordered relation (especially since we've eliminated all
      randomness and psychology present in real-world sports),
      but it's not!:
      Say team A has offense=3, defense=2, stamina=1.
          team B has offense=2, defense=1, stamina=3.
          team C has offense=1, defense=3, stamina=2.
      We can see that A > B, and B > C, yet C > A.

      The upshot: 
      be careful about trying to project multidimensional data 
         down to a single scalar; 
      you'll induce an ordering which may not reflect the original
      multi-dimensional relation.  
      (And the original relation might not qualify as an ordering -- 
       not even a partial ordering -- even though it sure
       seems like it should be one.)
  
      Alternate upshot, to console Rockets fans:
      It's conceivable that every team in the league is equally good,
      and it's merely the order of the matches which is
      determining the outcome.

=======================
Snippet from Richard Tapia's University Professor talk, 2005.Nov.29:
      ***"Rankings are evil."***
Humans are many-dimensional; you can't meaningfully reduce them to a single number 
(or even, just their academic talent/creativity/ability).
To try to say that every student with a 1425 SAT should be accepted
before students with 1400 SAT is ludicrous &mdash; 
and, ludicrous in a way which is socially damaging.

Instead, set a threshhold or bin (say, 1350-1450), and then view everybody 
within that bin as potentially equal-scoring on the test.
  (Similarly, w/ faculty hiring: schools which hire only the candidate
   from the top-ranked grad schools 
   are missing out on a lot of diversity and talent from students coming 
   from schools ranked #6.
   So set a threshhold of competency, then choose from that entire pool.)