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 for each player. Any player who deviated from his 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 open-handed 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 multi-way 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 pal flush before the draw, his correct play would be to throw away a pair of aces when you bet. Calling would be 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 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.