Interpreting the Fundamental Theorem

[Siegmund]

Quoting as closely as I can recall it without the book in front of me, it says something like: "You make a mistake anytime you play differently than you would have if you could see your opponents' cards. Your opponent makes a mistake anytime he plays differently than if he could see your cards [and his other opponents'.]"

Consider the alternative statement "A player makes a mistake when he plays differently than he would if everyone knew everyone else's cards."

Also consider the usual notion of 'regretting your choice' used to assess whether Nash equilibrium has been achieved: if you chose a different strategy than you would have if your opponents' strategies had been fully disclosed before you made your choice. (That is, knowing the exact range of hands your opponent might hold given his betting, but not knowing until the end which one of them he actually did hold.)

Two quick questions.

1) Is the alternative statement simply a rephrasing of the FT as it was originally intended, or was the FT deliberately worded as it was to ask each player to imagine what he would do if he could see his opponents' hands but they could not see his?

2) Neither the FT nor the alternative statement is logically equivalent to 'regretting your choice' in the Nash sense. Which do you regard as "closer" to it?

[bobman0330]

Very interesting post.

I don't think either of your 2 statements of the FT are very close to the NE statement. In order to think in those terms, you need to imagine a complete mixed strategy for every situation. For example, your mixed strategy would probably call for always raising KK UTG. If you look at your opponent's strategy and see that he will usually 3-bet with AA, you won't feel like you made a mistake with your strategy. However, the FT says that you made a mistake if one of your opponents has AA.

I think in general, the FT sweeps wider than the NE formula. Every Nash mistake leads to FT mistakes, but not every FT mistake is indicative of a Nash mistake.

I'm not sure about your first question. I'd be fascinated to hear what other people have to say though

[AaronBrown]

I think the big difference is Nash equilibrium considers non-constant sum games. The Fundamental Theorem of Poker takes advantage of the fact that every time your opponent does something bad for him, it has to help you.

In the usual game theory formulation, whether Von Neumann or Nash or other thinkers, you worry about your strategy assuming the other players know it and react optimally. You don't start out trying to induce mistakes by the other players, although that sometime occurs naturally in the analysis.

Sklansky starts right in with inducing mistakes. You don't worry about whether the mistakes are folding when he should call or raising when he should fold or calling when he should raise. Any mistake lowers his expected value, so it increases yours. Of course, you have to know how to take advantage, otherwise you might induce a mistake and then make an offsetting mistake that leaves you worse off than if both of you played perfectly.

The FT is not a mathematical truth, it's a useful way of conceptualizing poker. It emphasizes that perfect play by you won't win, you need your opponents to err. It also defines mistake relative to what you would do if you knew all the hands. So losing on the river when your opponent hits one of his three outs is not a mistake, but going all in with K K when your opponent had A A is; even if you had no way of knowing and you draw it out on the board.

[ianlippert]

I always thought the FTOP was kind of useless. It doesnt seem to have any practical application. Sklansky made the comment in TOP that there should a FTOP like there is in calculus. But unlike poker the fundamental theory of calculus is useful and used when differentiating/integrating.

I also think that sklansky's FTOP breaks down in certain situations. For example when you are about to be blinded out of a tournament, and you need to double up. Your choice to go all in with AK vs someones JJ may be wrong according to the FTOP, but a coin flip is probably the best chance you are going to get to get some more chips before you are blinded out.

I mean obviously if I knew what my opponents had I wouldn't make 'mistakes', but poker is a game of incomplete information. There isnt a single player that doesnt break the FTOP on a regular basis.

[J. Sawyer]

The fundamental theorem is very simple. It says exactly what it says, there is no interpreting.

[Siegmund]

[ QUOTE ]

bobman:

I think in general, the FT sweeps wider than the NE formula. Every Nash mistake leads to FT mistakes, but not every FT mistake is indicative of a Nash mistake.


[/ QUOTE ]

Yes.

[ QUOTE ]

ianlippert:


I mean obviously if I knew what my opponents had I wouldn't make 'mistakes', but poker is a game of incomplete information. There isnt a single player that doesnt break the FTOP on a regular basis.


[/ QUOTE ]

Yes. An FTOP-mistake is not always a technically wrong play. But it is still, perhaps, interesting to examine a hand and see which player made more FTOP-mistakes; playing to minimize one's number of FTOP-mistakes amounts to the same thing as playing to make the correct play as often as you can.

I asked about the original vs. alternative statements because it's fairly easy to analyze what would have happened had everyone been able to see everyone else's cards. (If you have a complete hand record, that is.) But ToP, in the section on bluffing, talks a great deal about inducing your opponent to make mistakes -- yet avoids defining a bluff as "deliberately making a FTOP-mistake in hopes of inducing a larger one from your opponent." Under the alternative statement, all bluffs are indeed mistakes; under the original statement, the door seems to be left open for some bluffs to not be FTOP-mistakes (if you think your opponent is likely to fold because of the range of hands you have represented, you might bluff even if you could see your opponent had you beaten.)

My mind actually likes the idea of analyzing all bluffs as deliberate FTOP-mistakes in hopes of inducing larger ones in return. My main reason in asking about the phrasing, was to see whether this idea seemed to others to be intended in the original FTOP.

[BluffTHIS!]

[ QUOTE ]
I always thought the FTOP was kind of useless. It doesnt seem to have any practical application.

[/ QUOTE ]

Perhaps you just have a shallow understanding of the TOP and poker itself. I suspect that is the case. If you actually read TOP all the way through, then you would see that the entire book, and all the plays described therein, flow from that fundamental theorem.

[Guruman]

In response to the OP's question:

I think that the FTOP is very different from the second statement made "A player makes a mistake when he plays differently than he would if everyone knew everyone else's cards." The difference lies in the element of deceptive play that is an integral part of fundamentally sound poker. If everyone knew everyone's cards, there could be no deception. I think that the FTOP allows for deceptive play by specifically not including the viewpoint of the opponent.

Accordingly, I disagree with the idea that a bluff is a FTOP mistake - here's why:

Most well-placed bluffs are made with the purpose of folding a better hand. In order to make a play that will actually accomplish this, two conditions have to be met: A)your opponent has to actually hold a better hand, and B)your opponent must be willing to fold that hand to a properly sized bet. If you can see that your opponent holds 33 on a AJQ board where you have raised pre-flop, then a bluff may be appropriate here no matter what two cards you have.

If I've made the read of a small pp, he checks to me, and I bluff here, I've made the same play that I would have made even if I couldn't see my opponents cards. Therefore this is not a FTOP mistake.

Now if I bluff too much, forget to discount outs, or place my opponent an a totally wrong range of hands, then the FTOP is going to send knuckles to my house to kick my shins in. [img]/images/graemlins/grin.gif[/img]

[ianlippert]

[ QUOTE ]

Perhaps you just have a shallow understanding of the TOP and poker itself. I suspect that is the case. If you actually read TOP all the way through, then you would see that the entire book, and all the plays described therein, flow from that fundamental theorem.

[/ QUOTE ]

I guess I just wonder how the FTOP is supposed to help me when I am at the table and cannot see the other players cards. The strategy of poker is to try and estimate as closely as possible to 'perfect play' (ie if we played with our cards face up). But we make many FTOP 'mistakes' because we can never be 100% sure of our hand. let me give you an example.

Two players limp in you raise on the button with AK, blinds fold. Flop comes A82, two spades. It gets checked to you, and you bet out. One caller, one fold. Turn is a blank, you bet and get called. River brings another spade and the caller bets into you, what is your play? Obviously we call here, and if we are beat we are beat. But according to the FTOP a call is always the wrong play. Either we still have the best hand and we should reraise, or we are beat and we should fold.

I just dont understand how the FTOP is supposed to help me at the table when it assumes the cards are being played face up. I'm sorry but the game that I play is played with the cards face down, and there are other strategic concepts that need to be taken into consideration when playing this game that the FTOP doesnt address. I think sklansky took the idea of a 'theory of poker' a little too far when he penned the FTOP.

[AaronBrown]

[ QUOTE ]
The strategy of poker is to try and estimate as closely as possible to 'perfect play' (ie if we played with our cards face up). But we make many FTOP 'mistakes' because we can never be 100% sure of our hand.

[/ QUOTE ]
This is a reasonable theory, but it's not Sklansky's theory. The FOTP doesn't say you make money by playing perfectly, it says you make money by inducing your opponents to play less than perfectly. Since poker is zero sum, what matters is the difference in performance between you and your opponents.

In your example at the river, the FTOP doesn't help much. Your opponent doesn't care what you have, he either has a flush or he doesn't. You're not going to convince him to fold his flush with a raise, you're not going to fool him into calling with a bust. That doesn't make the theorem wrong, it just says you can't get anything extra out of this situation. Just compute the pot odds and call. As you say, you get what you get.

But go back to the flop. Your opponents should fold any hand not composed exclusively of 8's and 2's, which they are unlikely to have and less likely to have called pre-flop with. So you want to get them to call or raise.

I'm not saying that means you check, you have to weigh the risk of giving them a free card; and you also have to consider this in relation to all the other hands you're going to play tonight. But the FTOP says to think about how your opponents perceive your hand, and how they are likely to react to your actions; rather than trying to guess what they have to make your play more perfect.

I think the FTOP helps a lot of good cardplayers make the leap to poker. It emphasizes what makes poker different from other games. But it's not a magic formula that makes bad players into winners. And you can certainly play good poker in many other ways.