Poker Strategy  Fundamental Theorem
The Fundamental Theorem is a poker strategy created by David Skalansky, a theorem to express the essential nature of poker and the decision making process when one faces incomplete information. The goal is to visualize yourself on your opponent's shoes and try to figure out what their decision will be, based on the information you have on the table.
"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."  David Skalansky
This theory is based on common language with mathematical reasoning. Each decision made during a game of poker has an expected value. A player is faced with several options, if he makes the correct decision that one will have the highest expected value. If a player could see of the opponent's cards he could take the correct decision, and the better the long term results.
This theorem finds an obstacle when it applies to all players in the same way, and the correct decision for one could actually be beneficial to another player, this is based on Morton's Theorem. In probabilistic this is called the Law of Total Expectation.
Example
Player A is playing Texas Holdem and receives a 9 of clubs and a 9 of spades, before the flop. Everyone on the table folds when the big blind checks with player A remaining and the Player B who stands on the blind. The 2 initial cards of the flop turn out to be A clubs, K diamond and 10 diamond. Player B makes a bet.
Player B finds himself in front of a decision to take based on the incomplete information on the table. The most correct decision to take would be to fold. There are too many outcomes Player B could win this hand, taking into account there are several ways to construct a turn or river. Even if player B does not have pairs for an A or K, there are only 3 cards for a straight and 2 cards for a flush. There is a good chance player B has a diamond card since player A does not and all other players folded. Also, the fact that all other players folded against big cards probably means none of them had matching pairs, player A knows he doesn't have one either so player B has a high probability of having at least one.
On the other hand, player B is counting on 2 outs to get another 9 for a powerful hand that could not even be enough. The other second possibility would be a weak sided river from 9 to K.
However, suppose player A knows exactly what cards player B holds, 8 of diamonds and 7 of diamonds. In this scenario the most correct decision would be to raise even though the odds still favor Player B with a high chance of a flush.
Based on the theorem, player A has played his hand differently from the way player B would if he could see his opponent's cards, and so based on the theorem, his opponent has won. Player A has made a mistake, he played differently from the way he would have played after knowing player B had 8 of diamonds and 7 of diamonds. The "mistake" is the best decision given the incomplete information on the table.
Remember that the goal in poker is to induce the opponent to make mistakes. In this example player B employed selfdeception with a semibluff, he bet a hand hoping player A would fold, but with high chances of winning a hand. Player B induced player A to make a mistake.
This is poker theorem a little hard to get your mind around, involving both mathematics with player psychology. But it helps a player make better decision taking, and inducing the desired decision from an opponent. The perfect place to apply would be online poker at William Hill were most tables are full of knobs.
