Colloquial rendering: If you are drawing to the nuts, then no matter how long the odds are, you should sometimes play your hand.
Precise formulation: Assume that you and and your opponents have sufficiently large stacks. Let N* be the condition that you hold the nuts on the river. Let S = {s} be the set of all possible sequences of actions by you and your opponents prior to the river. Let X characterize the sequences in which you wind up contesting the river heads up with the possibility of N*, i.e., if you make it to the river with one opponent and it is logically possible that you hold a hand that your opponent cannot beat, then X(s) = 1 ; otherwise X(s) = 0.
Then if you play with a strategy such that there exists s*, a sequence of actions such that Prob[s*] > 0 and X(s*) = 1, but Prob[N*|s*] = 0, your strategy is sub-optimal. In words, if your strategy is responsible for eliminating the possibility that you hold the nuts heads up on the river, then you are not employing an optimal strategy.
Illustration.
Suppose you hold 3c2c in the big blind. Everyone folds to the small blind, who raises. You call. The flop comes AdTd9c. You have the lowest ranking hand in this situation, but you can make the nuts if a 4 and 5 hit without making a flush possible. According to the Fundamental Theorem of Chasing, if your opponent bets, you shouldn't always fold.
In fact, the Fundamental Theorem of Chasing says more than this. If in this situation there are some hands with which you'd raise, and other hands with which you'd call, then you should play a mixed strategy with 3c2c -- sometimes call, sometimes raise. Also, if your opponent checks and there are hands you would bet, you should sometimes bet 3c2c. If you get check-raised, and there are hands with which you would then re-raise, you should sometimes reraise with 3c2c. Etc.
Now suppose a 5h falls on the turn. You may have only 3 or 4 outs and the pot contains just 3 big bets. Yet if your opponent bets, then according to the Fundamental Theorem of Chasing, you should sometimes call and sometimes raise with 3c2c, since there are other hands you'd play that way in this situation.
Motivation for the Fundamental Theorem of Chasing.
In order to play optimally, you must maintain the threat of holding the nuts on the river. You need this threat in order to make your bluffs, raise-bluffs, re-raise bluffs, etc., credible.
Proof of the Fundamental Theorem of Chasing.
If your opponent doesn't find these threats credible, you will be able to achieve an aggregate expected profit from playing the potential nuts because on those rare occasions you hit your hand, your opponent won't stop raising with what he believes must be the best hand.
Here we see the necessity for the assumptions of sufficiently large stacks and heads up play on the river: with a short stack, your opponent will stop raising when he runs out of chips; with multi-way play, there is a limit to the number of raises. In either case, you may not get paid off enough to make a long shot draw to the nuts profitable.
What's sloppy about the "colloquial rendering" of the Fundamental Theorem of Chasing?
The proof demonstrates that optimal play never eliminates the probability that you will hold the nuts on the river. This is not the same thing as saying that any time you have a draw to the nuts, you should (at least sometimes) play the hand. This is because there can be more than one hand you could hold that can draw to the same nuts. For example, if we substitute 3s2s for 3c2c in the illustration above, it may be correct to always fold 3s2s, since it's dominated in expectation by 3c2c (which also has a back door flush draw).
Also, the colloquial rendering doesn't mention the assumptions of heads up play and sufficiently large stacks. And it lacks the full scope of the precise formulation -- as noted in the illustration, not only should you play the hand, you should mix your play with the hand so that it appears in every possible sequence of actions arising from optimal play on your part.
Finally, consider five card stud. Because the river is dealt face up, it's possible no matter what your hole card is, you can't beat your opponent! But the precise formulation of the Fundamental Theorem of Chasing is careful to exclude this type of situation.
Implication for different poker games.
The Fundamental Theorem of Chasing reveals some essential differences between different poker games.
In Hold'em, it means that before the flop, you should play any suited hand that can make a nut straight, no matter how expensive it might be, with positive (but perhaps exceedingly small) probability. For example, suppose under the gun raises, the next player re-raises and you hold 52s. Since you wouldn't fold AA or AKs in this situation, you shouldn't always fold 52s. If, for example, you call with AKs and re-re-raise with AA, then you should sometimes call and sometimes re-re-raise with 52s. Thus it is correct to make some very peculiar plays in Hold'em, at least occasionally.
Contrast this with Omaha. In Omaha, there's no need to play any really bad starting hands, because no matter what the nuts might be on the river, you can have those cards plus AA, giving you a hand with plenty of pre-flop equity.
In both Hold'em and Omaha, it's always possible that you might be drawing to the nuts. So, as a corollary to the Fundamental Theorem of Chasing, it's always possible for you to have sufficient implied odds to continue playing no matter how ugly your draw. Unless your opponent holds the nuts, if you play optimally, he has to fear even the most apparently insane bad beats. Restating the corollary: there are no true bad beats in flop games.
But in stud games, there are situations before the river where no hand you might hold could outdraw the best hand your opponent might hold. (This demonstrates the importance of paying attention to up cards.) Thus there are situations where your opponent can reasonably expect you to fold a hand that could draw out on his strong but not best possible hand. And thus there are situations where you can truly put a bad beat on your opponent, in the sense that without question, you had to be playing incorrectly in order to get there.
