Fundamental Theorem of Poker

[bradha]

I am trying to figure out whether this theorem would say I have gained in the following case:

I am the big blind in holdem (limit), and have Qx. There is one limper, and the small blind folds. Flop is A-K-K.
I decide to bet out, folding to a raise, and checking/folding the rest of the hand if I am called. The limper folds.

Suppose the limper had a hand that is even worse than my Q. FTOP says that I gain if I forced my opponent to play differently than they would have if they could see my cards.
1. If my opponent can see that he/she is behind my hand, and doesn't have pot odds to catch up, folding makes sense and FTOP would indicate that I have lost by not having them call or raise me.
2. On the other hand, if my opponent could see how weak I am, they might realize that I could be bluffed out, so then FTOP indicates that they have lost my not raising me.
3. If I could see my opponents hand, I certainly wouldn't want to give a free card, and I would keep betting as long as I was ahead.

For most poker hands there are many different lines depending on knowledge assumptions.
1. The way play would go if all hands were face up - we each know the others hand, and know that the opponent knows our hand.
2. The way play would go if each knew the others cards, but didn't think that our cards were known.
3. The way play would go if I know my opponents cards, they don't know mine, but they know that I know their hand.
4. The way play would go if I don't know my opponents hand, but know that they know mine.
5. I know my opponents cards, but they don't know mine or realize that I know theirs.
6. Opponent knows my cards, but I don't know it, and don't know their cards.
7. Noone knows the others cards.

It seems to me that FTOP talks about two of the cases, but I am a little unclear on whether it is 5 and 6 or 3 and 4. Then again it might be referring to 1 or 2 in comparison with 7. I just got through this chapter in Theory of Poker, and I am still unclear on which knowledge assumptions FTOP is referring to.

I think you had the best possible outcome according to the Fundamental Theorem of Poker.

If you could see his cards, you still would have bet, knowing that you were ahead. The situation was such that you would not want to give him a free card, and he would likely have checked on his turn. If he had bet, you would have had to fold.

So, with a check, your best hope is that he checks and you hit your queen on the turn - if you don't, you are back at square one - plus he may hit one of his cards, in which case you will likely lose the hand.

Now, assuming your bet, if he had seen your hand, the best move for him would have been to raise since there is no way you could continue playing with your hand, and you would have had to fold.

[R_Ellender]

1&2. If your opponent saw your hand, his optimal play would have been to raise, because he gains nothing by folding. This is especially true since you indicated in the beginning of your post that you planned on folding to a raise, so his raise had +EV.

Psychology is an extension of the Fundamental Theorem, as you'll learn later in TOP. If you only focus on the cards, then he played correctly by folding since your Q-high was probably best. If you factor in psychology, if your opponent knows how you play your hand(folding to a raise), raising becomes the correct play. His raise induces a mistake by you.

3. Yes, it would be correct to keep betting with the best hand, even if it was something as weak as Q-high. However, with a hand that weak, you're not really hoping for a call, since your bet is pretty much a semibluff. You should be satisfied with his fold, because a call on the flop may have caused you to play incorrectly on the turn and the river(if your opponent has nothing). You may be afraid to bet, and if you have the best hand, its clearly a mistake to give a free card.

As for the rest of your post, the summary of the chapter says it best. You should play the way that you would if you could see your opponents cards, simple as that. It's a general statement that takes into account everything from knowing what cards your opponent holds to knowing how he will play those cards(tendencies). When you analyze a poker play, you look at it from every angle to try and decide whether or not it was the optimal play in terms of +EV. You must take advantage of your opponents tendencies(folding too often, calling too much with bad odds) while not allowing him to take advantage of yours(folding if raised when you held Q-high earlier).

To clarify about the FTOP . . .

"Every time you play a hand differently from the way you would have played it if you could see all your opponents' cards, they gain; and every time you play your hand the same way you would have played it if you could see all thier cards, they lose. CONVERSELY, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose."
THE WORD CONVERSELY IS WHAT MAKES IT TRUE THAT YOU BOTH KNOW EACH OTHERS CARDS

I take it like this in your case:

Assuming your opponent has a hand like 910s (worth limping in) he would have to fold otherwise you gain being the big favorite that you are. Even IF he raises your bet, you would then lose by folding. Obviously you would have to fold in this situation, and there lies the game of poker. He then made you make the wrong play.

It is a very touchy situation u have there, b/c you're right u can't continue in the face of a raise, but according the FTOP (seeing his cards) you would then put him all in and being the 2 1/3 - 1 favorite that you are. The mere fact that you would rarely do that is what seperates poker from math.

JPMurph, the word conversely in the theory could (and probably does) mean that the plays of both players should be looked at individually, not as a whole. So on each hand, there would be a case where player A knew player B's hand, and a case where player B knew player A's hand. But the two would be seperate cases. Therefore, if the OP knew his Q high was best, or that his opponent was likely to fold for a bet, the bet was the best play. (A side note to the OP: If your opponent folded, how would you propose to make more on the hand? Them not calling is not a failure according to FTOP, because there was no way to induce a call. A check-raise might work, but relies on knowing your opponent would bluff in this situation). Then, after examining his play, we look at his opponents play. His opponent, if he could have seen Q high, would have known one of two things:
1) My hand is the best, or
2) He will fold to a bet
In this problem (completely independant of the problem the OP faced), we can see that folding is the wrong play.