Thanks to a mistake in one of my brackets this year¹ I had the following she’eilah:

Question: Are you allowed to have North Carolina winning a Texas vs. UConn final?²

Putting aside the merits of the specific teams for a moment, is this a legitimate bracket? Do we count the winners of the games or simply who advances to the next round?
The answer I believe depends on how your bracket is scored. Most brackets are weighted such that victories in the second round are worth more “points” than the first round games, third round more than second, and so forth. The reason behind this system is obvious – the odds of a given team winning in the third round are significantly decreased when you consider that that team may not make it out of the second round. There are far too many variables and possibilities such that correctly picking the tournament champion ought to be worth more than correctly picking the 1-16 game.
Since weighted brackets are predicated on the logic of a formalized tournament, you cannot count a victory which would be impossible in the actual tournament. If you have a team eliminated in the sweet sixteen, that team cannot be counted in your final four. This error could be grounds for disqualification, but I’d be content to treat the errant pick as a loss even if that team does in fact advance in the appropriate round of the tournament itself.
However, in the unlikely event you’re involved in a pool which only scores the total number of wins – possible for a secondary prize – then the placement of these victories is no longer dependent on the actual tournament. As such, logic may be safely ignored and you’re free to pick whomever at any given stage even if you have that team losing in the first round.
UPDATE: Apparently, it’s not just me as even the famed sports guy made a similar mistake.

1. In a non-gambling pool, so no cracks about me being pasul l’eidut. Not for this anyway.
2. By “winning” I mean the actual game, not in some after-the-fact economic or recruiting benefits or the hana’ah (benefit) of mocking Duke for losing earlier in the tournament.