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Examples Of The Fundamental Theorem Of Poker

 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 no-limit hold 'em you hold and your opponent hold an off suit

You check, your opponent bets, and you call. Now the ace of diamonds comes on fourth 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.

Example 2

Suppose there is $80 in the pot, and you have two pair. You are playing draw poker, and you bet $10, which we will assume is all you can bet. Your single opponent has a four flush - that is, four cards to a flush. The question is - are you rooting for him to call or fold? Naturally you want him to do what is most profitable for you. The Fundamental Theorem of Poker states that what is most profitable for you is for your opponent to make the incorrect play based on complete information about both hands. Since your opponent is getting 9-to-1 odds (his $10 call might win him $90) and is only about a 5-to-1 underdog to make a flush, it is correct for him to call because a call has positive expectation. Since it is correct for him to call, following the Fundamental Theorem, you are therefore rooting for him to fold.

This sort of situation comes up frequently. You have the best hand, but your opponent is getting odds good enough to make it correct to call if he knew what you had. Therefore, you want your opponent to fold. By the same token, it is correct for you to chase when you are getting sufficient pot odds. If you don't chase, you are costing yourself money and, therefore, making money for your opponent.

Example 3

Since it is correct for your opponent to call when he is getting sufficient pot odds; you can sometimes make an opponent fold incorrectly by showing more strength than you actually have on an early betting round.

You are fairly sure he has kings. You now proceed to make a pair of 6s on board, and you bet. Your opponent will almost certainly fold a pair of kings since he is afraid you have made aces up.

Some people might say, "Well, wait a second. Why don't you want your opponent to call as long as his pair of kings is worse than your two small pair?" The answer is that if there are cards to come and your opponent is getting proper odds, you do better to win the pot right there. A pair of kings versus two smaller pair needs very short odds to justify a call. Since your opponent would have been correct to call, you gain when you make him fold.


Example 4

In razz, a seven-card stud lowball games in which the lowest hand wins, we can see another example of showing more strength than you have to make an opponent fold incorrectly.

If you think your opponent has a four-card 8 - and you have a pair and only a four-card 8-7 - it is important to bet, even though you know you will be called. The bet gains you some extra equity, should you happen to catch a little card on sixth street, giving you an 8-7 low. If your opponent catches a big card or a pair, still having a draw to a better 8 than yours, he will fold, since your previous bet indicated you had an 8 made already. The little card you've now caught suggests you have made a 71ow, which makes your opponent think he is drawing dead - that is, drawing with no chance of winning.

Notice that once again you want your opponent to fold even though you have the best hand. You have an 8, 7 low and are drawing to a 7, while all your opponent has is a draw to a better 8. However, you gain by his folding because, had he known you had only an 8, 7, he would be getting proper odds to call in the hope of drawing out on you. By not calling he made a mistake, and you have gained. (You gain even more when that sixth street card makes you two pair, and your opponent folds the best hand.)

Example 5

Just as you are rooting for an opponent to fold when he is getting sufficient pot odds, you are rooting for him to call when he is getting insufficient pot odds. Thus, it is frequently correct to play a strong hand weakly on an early round - the converse of your plays in the previous two examples - so that your opponent will make a bad call when you do improve.

A good play against some people with this hand would be to check and just call if your opponent bets. Many players would now put you on a pair or a bad card in the hole. If you do catch a 4, 5 or 7 on board, giving you a 6 or 7 low, your opponent will probably still call, even if he is drawing dead, because your earlier play along with his pot odds make him think it's worth a call. This is exactly what you are hoping for. Your deceptive play early has caused your opponent to make an incorrect play on a later round.

Example 6

Any time an opponent is not getting close to proper odds against you, you are rooting for him to call, even if by calling he has a chance of drawing out on you. If in the flush example at the beginning of this page, the pot were $20 instead of $80, you would be rooting for your opponent with the four flush to call your $10 bet because he is a 5-to-1 underdog getting only 3-to-1 for his money. If he calls and makes a flush, those are the breaks. Nevertheless, his play is incorrect because it has negative expectation, and you gain any time he makes it.

When you have a hand that is rooting for a call, you should not try to make your opponent fold by betting an exorbitant amount in a no-limit or pot-limit games. Such a situation came up one day when you were playing no-limit hold 'em. There was one card to come, and you had a straight which, at that point, was the nuts - that is, the best possible hand. You bet something like $50, the player to your left called, and the player behind him called the $50 and raised the rest of his money, which was about $200.

Since you had the best possible hand, the question was, should you raise or just call? There was something like $500 in the pot. Because the third man was all-in, you only had to think about the man behind me. You knew if you re-raised, say, $400, making it $600 to him, he definitely would fold; in fact, if you raised almost any amount he would fold. But if you just called the $200, he would probably call.

What did you want him to do? You were pretty sure he had two pair. You called the $200 there would be about $700 in the pot, which would give him 7-to-2 odds to call $200 with his two pair. However, the odds against his making a full house with two pair were 10-to-1 (there were 40 cards in the deck that didn't help him and 4 that did). Therefore, if he knew you had a straight, it would be incorrect for him to take 7-to-2 odds on a 10-to-1 shot. So you just called the $200, and as you expected and wanted, he did too.

The sad conclusion to this story is that he made a full house and bet a very small amount, which you paid off. Many people argued you had been wrong to let him in rather than raise him out, but in fact they are wrong. You had to give him a chance to make a mistake, which he did, because whenever your opponent makes a mistake, you gain in the long run.

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