Fundamental theorem of poker
Encyclopedia
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 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 expected long-term results. This is the mathematical expression of the Fundamental Theorem.
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. (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), 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.
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.
David Sklansky
-Life and career:Sklansky was born and raised in Teaneck, New Jersey, where he graduated from Teaneck High School in 1966. He attended the University of Pennsylvania, but left before graduation. He returned to Teaneck and passed multiple Society of Actuaries exams by the time he was 20, and worked...
that he believes expresses the essential nature of poker
Poker
Poker is a family of card games that share betting rules and usually hand rankings. Poker games differ in how the cards are dealt, how hands may be formed, whether the high or low hand wins the pot in a showdown , limits on bet sizes, and how many rounds of betting are allowed.In most modern poker...
as a game
Game
A game is structured playing, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements...
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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
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 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
Morton's theorem
Morton's theorem is a poker principle articulated by Andy Morton in a Usenet poker newsgroup. It states that in multi-way pots, a player's expectation may be maximized by an opponent making a correct decision....
in multi-way pots.) The less you deviate from these correct decisions, the better your expected long-term results. This is the mathematical expression of the Fundamental Theorem.
An example
Suppose Alice is playing limit Texas hold 'emTexas hold 'em
Texas Hold 'em is a variation of the standard card game of poker. The game consists of two cards being dealt face down to each player and then five community cards being placed face-up by the dealer—a series of three then two additional single cards , with...
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
Draw (poker)
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to...
. She is essentially drawing to 2 outs
Out (poker)
In a poker game with more than one betting round, an out is any unseen card that, if drawn, will improve a player's hand to one that is likely to win. Knowing the number of outs a player has is an important part of poker strategy. For example in draw poker, a hand with four diamonds has nine 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
Pot odds
In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value....
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), 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
Draw (poker)
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to...
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
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
from the other deep stacked opponents. Such a situation is sometimes referred to as implicit collusion
Morton's theorem
Morton's theorem is a poker principle articulated by Andy Morton in a Usenet poker newsgroup. It states that in multi-way pots, a player's expectation may be maximized by an opponent making a correct decision....
.
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.