Proof of Fundamental Theorem of Poker?

[de Moivre]

Sklansky's Fundamental Theorem of Poker says essentially that it is optimal to play your hand in the way you would play it if you could see your opponents' cards. It is said to apply always in heads up play and usually with more than two players.

Can anyone supply a proof or refer me to one in the poker or game theory literature? (There is no proof given in my 1994 edition of The Theory of Poker.)

Actually, I suspect it is a general result in game theory, which is not limited to poker. If so, I would like a precise formulation and a proof. Thanks in advance for any leads.

[icheckcallu]

x(yz/p)+1= butcho

[RobNottsUk]

If there was a proof it wouldn't be a theory would it?

[RustyBrooks]

I think you're perhaps misunderstanding the classical use of the word "theorem". For example, the Pythagorean Theorem is not only provable, it's well proven. Mathematicians just don't call these things "The Pythagorean Fact"

[RustyBrooks]

Or, I got leveled?

[de Moivre]

Thinking about it some more, I may have a proof.

One formulation of the theorem is: If you play your hand the way you would play it if you could see your opponent's cards, you gain. I regard "you gain" as meaning "your expected gain increases."

Assume a heads-up game. If the game matrix is A, player 1 has a mixed optimal strategy by the minimax theorem. Any departure from this will reduce the expected payoff for player 1 if player 2 plays optimally, which seems to contradict the theorem.

So maybe the meaning is "your expected gain, conditioned on your opponent's cards, increases." For if we condition on our opponent's cards, our payoff matrix changes and is now B, say. Here it would be optimal for player 1 to use his minimax strategy for matrix B, while using that for matrix A would be suboptimal.

So I think I've found an interpretation for the FTP that makes it correct and provable, but rather simple. But Sklansky did say the theorem is obvious, so maybe I'm on the right track.

[indianaV8]

Proof is obvious. If you know more than your opponent you have an edge from GT PoV. And if you manage to play LIKE you have an edge from GT PoV then you have an edge.

But if you are interested in theory, drop the sklansky FTOP nonsense and study game theory (start with mathematics of poker).

[Mark1808]

Knowing your opponents cards will not mean you will play optimally. To do that you must also know how the opponent will play his various hands. In other words if you have the best hand it is not optimal to bet if you know he will bluff if you check.

[RustyBrooks]

But you could not play better by not knowing his cards. Knowing his cards will always allow you to perform better than not.

[Mark1808]

[ QUOTE ]
But you could not play better by not knowing his cards. Knowing his cards will always allow you to perform better than not.

[/ QUOTE ]

Not always. If I know his cards but don't know how he plays his cards I may play more optimally not seeing them in certain situations. The pefect example is check calling a bluff. If I could see his cards and see he has bottom pair to my top pair I might bet for value. However it may be more optimal to check if I knew he would fold to a bet and bet when checked to.

Even when bet in to on the river when you have the best hand knowing how your opponent plays his cards can add more value then just seeing them. How much of a raise will he call?

Obviously given the choice I could make more money seeing the hole cards than knowing how he plays his cards, but you need to have both to fully optimize play.