This hand from the 2+2 high-stakes multi-table tournament forum
got me thinking about what it means to be "at the top" of a polarized
range. Here's a quick summary, for those who can't/won't follow the
link:

It's a tournament, and blinds are 100/200. Hero raises to 475 UTG
with KK, and good loose aggressive regular calls out of the SB. The
flop comes Qd 4s 9d. SB checks, Hero bets 625, Villain calls.

The turn is the 4d, pairing the board and putting three diamonds
out.  Both players check. (I don't think it matters much, because the
turn decision isn't the important thing here, but Hero has the Kd.)

The river is the 4h, giving Hero Kings full of Fours. Villain
checks, Hero bets 1400 into a 2300 pot, and Villain shoves for 11080.
Hero has him covered.

Most of the forum seemed to think this was a call, and in many cases not a particularly close one. To me, it's a clear fold.

Not everyone articulated it this way, but the general sense seemed
to be that a pair of K's is at the top of Hero's range, and that
folding hands at the top of your range isn't what you do against a LAG.

The problem here is that Hero actually has two very separate ranges:
a bluff range and a value range. KK is actually only around the middle
of Hero's value range, which I would set at roughly TT+. Unless Villain
is floating the flop from out of position, a possibility made even less
likely since Hero can account for two of the K's, meaning he won't see
stuff like AK/KJ/KT very often, Villain probably has showdown value. He
called a bet on the flop, and the most obvious draw came in on the
river.

This means that Villain will primarily combat Hero's bluff range by
calling rather than by check-raise bluffing. When Villain check-raises,
it no longer matters much where KK fits in Hero's full range, because
Villain isn't really playing against the bluff portion of that range
any longer. Villain's shove is rarely a bluff, so it matters only where
KK fits relative to Hero's value range, and more importantly relative
to Villain's value range.

If you're thinking that this is just a very fancy way of saying that
Hero has a bluff catcher in a spot where Villain is rarely bluffing,
you're correct. The thing is that many people tend to use this flawed
"top of my range" or "hand under-represented" logic (which are really
the same argument, in my opinion) to justify some calls that I would
say are clearly bad. Yeah, your hand can be a bluff quite often, but
Villain can usually combat that by calling. When he shoves, KK is in
trouble.