THE FUNDAMENTAL THEOREM OF POKER
There is a Fundamental Theorem of Algebra and a Fundamental Theorem of Calculus. So it’s about time to introduce the Fundamental Theorem of Poker.
Poker, like all card games, is a game of incomplete information, which distinguishes it from board games like chess, backgammon, and checkers, where you can always see what your opponent is doing.
If everybody’s cards were showing at all times, there would always be a precise, mathematically correct play would be reducing his mathematical expectation and increasing the expectation of his opponents. Of course, if all cards were exposed at all times, there wouldn’t be a game of poker.
The art of poker is filling the gaps in the incomplete information provided by your opponent’s betting and the exposed cards in openhanded games, and at the same time preventing your opponents from discovering any more than what you want them to know about your hand.
That leads us to the 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 applies universally when a hand has been reduced to a contest between you and a single opponent. It nearly always applies to multiway pots as well, but there are rare exceptions, which we will discuss at the end of the chapter.
What does the Fundamental Theorem mean? Realize that if somehow your opponent knew your hand, there would be a correct play for him to make.
If,, for instance, in a draw poker game your opponent saw that you had a pat flush before the draw, his correct play would be to throw away a pair of aces when you bet.
Calling would a mistake, but it is a special kind of mistake. We do not mean your opponent played the hand badly by calling with a pair of aces; we mean he played it differently from the way he would play it if he could see your cards.
This flush example is very obvious. In fact, the whole theorem is obvious, which is its beauty; yet its applications are often not so obvious.
Sometimes the amount of money in the pot odds makes it correct to call, even if you could see that your opponent’s hand is better than yours. Let’s look at several examples of the Fundamental Theorem of Poker in action.
Example 1
Suppose your hand is not as good as your opponent’s when you bet. Your opponent calls your bet, and you lose.But in fact you have not lost; you have gained! Why? Because obviously your opponent’s correct play, if he knew what you had, would be to raise.Therefore, you have gained when he doesn’t raise, and if he folds, you have gained a tremendous amount.
This example may also seem too obvious for serious discussion, but it is a general statement of some fairly sophisticated plays. Let’s say in nolimit hold’em you hold the
and your opponent holds an offsuit
The flop comes:
You check, your opponent bets, and you call. Now the ace of diamonds comes on fourth poker street, and you bet, trying to represent aces.
If your opponent knew what you had, his correct play would be to raise you so much it would cost too much to draw to a flush or a straight on the last card, and you would have to fold.
Therefore, if your opponent only calls, you have gained. You have gained not just because you are getting a relatively cheap final card but because your opponent did not make the correct play.
Obviously if your opponent folds, you have gained tremendously since he has thrown away the best hand.
