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Fundamental theorem of poker
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 concept of expected value. The expected value expresses the average payoff of a decision if the decision is made a large number of times. The correct decision to make in a given situation is the decision that has the largest expected value. (Although sometimes it is correct not to choose this decision for the larger goal of long-term deception.) If you could see all your opponents' cards, you would always be able to calculate the correct decision with mathematical certainty. (This is certainly true heads-up, but is not always true in multi-way pots.) The less you deviate from these correct decisions, the better your long-term results. This is the mathematical expression of the Fundamental Theorem.
Here is an example that illustrates how the Fundamental Theorem is applied. (This example assumes a familiarity with the basic rules and terminology of holdem.) Suppose you are playing limit holdem and are dealt 9♣ 9♠ under the gun before the flop. You call, and everyone folds to the big blind who checks. The flop comes A♣ K♦ 10♦, and the big blind bets.
You now have 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 your 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 she could easily be on a straight or flush draw. You are essentially drawing to 2 outs (another 9), and even if you catch one of these outs, your set may not hold up.
However, suppose you 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. (Calling would be giving the big blind infinite pot odds, and this decision makes less money in the long run than raising.) Therefore, by folding (or even calling), you have played your hand differently from the way you would have played it if you could see your opponent's cards, and so by the Fundamental Theorem of Poker, she has gained. You have made a "mistake", in the sense that you have played differently from the way you would have played if you knew the big blind held 8♦ 7♦, even though this "mistake" is almost certainly the best decision given the incomplete information available to you.
This example also illustrates that one of the most important goals in poker is to induce your opponents to make mistakes. In this particular hand, the big blind has practiced deception by employing a semi-bluff -- she has bet a hand, hoping you will fold, but she still has outs even if you call or raise. She has induced you 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 loose-passive games, when a player has a strong hand, but several opponents are chasing with draws. Sometimes such a situation is referred to as implicit collusion. Experts disagree on the prevalence of implicit collusion in particular games, as well as the extent to which implicit collusion might be unethical.
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.
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