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.

[RustyBrooks]

I don't know of a "proof" of the theory but I think it is obvious to pretty much any poker player that you could not do better against an opponent than if you could see his cards, and he could not see yours (well, you could gain an improvement if you knew which cards were coming). Therefore, any time you take the same action you would have taken if you knew his cards, you took the best option possible.

[rufus]

"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 their 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."

Assuming (1) rational knowledgable players, and (2) using full-knowledge value as a baseline, this should be obviously true for heads-up play - in multi-player pots, implicit collusion limits it.

Sklansky's theorem is based on complete game-state information, which means that it's applications to actual play decision making (where there isn't nearly so much knowledge) are very limited.

[The 13th 4postle]

AP superuser account is your proof

[Vetgirig]

Poker is a Zero sum game

http://en.wikipedia.org/wiki/Zero-sum

The wins of one player comes from the loss of another player. So to win one must get the opponent to make mistakes.