Fundamental theorem of poker
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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.
|“||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.||”|
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 Alice is playing limit Texas hold 'em and is dealt 9♣ 9♠ under the gun before the flop. She calls, and everyone else folds to the big blind who checks. The flop comes A♣ K♦ 10♦, and the big blind bets.
She 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 her hand. Even if the big blind 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 he could easily be on a straight or flush draw. She is essentially drawing to 2 outs (another 9), and even if she catches one of these outs, her set may not hold up.
However, suppose she knew (with 100% certainty) the big blind held 8♦ 7♦. In this case, it would be correct to raise. Even though the big blind would still be getting the correct pot odds to call, the best decision is to raise. Therefore, by folding (or even calling), she has played her hand differently from the way she would have played it if she could see her opponent's cards, and so by the Fundamental Theorem of Poker, her opponent has gained. She has made a "mistake", in the sense that she has played differently from the way she would have played if she knew the big blind held 8♦ 7♦, even though this "mistake" is almost certainly the best decision given the incomplete information available to her.
This example also illustrates that one of the most important goals in poker is to induce the opponents to make mistakes. In this particular hand, the big blind has practised deception by employing a semi-bluff — he has bet a hand, hoping she will fold, but he still has outs even if she calls or raises. He has induced her to make a mistake.
Multi-way pots and implicit collusion
The Fundamental Theorem of Poker applies to all heads-up decisions, but it does not apply to all multi-way decisions. This is because each opponent of a player can make an incorrect decision, but the "collective decision" of all the opponents works against the player.
This type of situation occurs mostly in games with multi-way pots, when a player has a strong hand, but several opponents are chasing with draws or other weaker hands. Also, a good example is a player with a deep stack making a play that favors a short-stacked opponent because he can extract more expected value from the other deep-stacked opponents. Such a situation is sometimes referred to as implicit collusion.
The Fundamental Theorem of Poker is simply expressed and appears axiomatic, yet its proper application to the countless varieties of circumstances that a poker player may face requires a great deal of knowledge, skill, and experience.