Fundamental theorem of poker

The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information.

The fundamental theorem is stated in common language, but its formulation is based on mathematical reasoning. Each decision that is made in poker can be analyzed in terms of the expected value of the payoff of a decision. The correct decision to make in a given situation is the decision that has the largest expected value. If a player could see all of their opponents' cards, they would always be able to calculate the correct decision with mathematical certainty, and the less they deviate from these correct decisions, the better their expected long-term results. This is certainly true heads-up, but Morton's theorem, in which an opponent's correct decision can benefit a player, may apply in multi-way pots.

In probabilistic terms, this is an application of the law of total expectation.

Suppose Bob is playing limit Texas hold 'em and is dealt 9♣ 9♠ under the gun before the flop. He calls, and everyone else folds to Carol in the big blind who checks. The flop comes A♣ K♦ 10♦, and Carol bets.

Bob now has a decision to make based upon incomplete information. In this particular circumstance, the correct decision is almost certainly to fold. There are too many turn and river cards that could kill his hand. Even if Carol does not have an A or a K, there are 3 cards to a straight and 2 cards to a flush on the flop, and she could easily be on a straight or flush draw. Bob is essentially drawing to 2 outs (another 9), and even if he catches one of these outs, his set may not hold up.

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