Fundamental Theorem of Poker

In David's book The Theory of Poker, he introduces a concept he calls 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.2

The basic idea is that, if you could see your opponent's cards, you'd always choose the "ideal" play, the play that serves you best. You'd never pay off with a second-best hand, and you'd never fail to bet when you should. Every time you make a play other than the "ideal" play, you have made a "mistake," and you've cost yourself some money.

Note that we use the term "mistake" in a specific and somewhat peculiar sense. We don't mean that you played badly, or that a more skillful player would have played differently. We just mean that you played differently than you would have if you could have seen your opponent's hand. For instance, say you have $500 left in a tournament with $100-$200 blinds. You're on the button with pocket kings, and you move in. Your opponent in the big blind calls and shows pocket aces. Raising all-in there with kings is clearly correct. But your raise was a "mistake" in our

The Theory of Poker by David Sklansky, pages 17-18.

terminology because you wouldn't have moved in had the big blind shown you the aces first.

Throughout the book, we will use the term "mistake" in this sense; a mistake is a play other than the play you would make if you knew your opponent's cards, but it's not necessarily a bad play.

The Fundamental Theorem of Poker highlights the value of hand reading and deception. One of your goals when you play no limit hold 'em is to try to deduce your opponent's holding while disguising your own. You try to make few mistakes, while you encourage your opponent to make lots of them. If you do a good job, you will be winning the "battle of mistakes," and over time money will flow from your opponent to you.

Indeed, the format of no limit hold 'em allows the Fundamental Theorem of Poker to blossom fully. In limit poker, many situations arise where you simply cannot entice your opponent to make a mistake no matter what you do. Say you are limited to a $20 bet, and you know that your opponent has a flush draw. If the pot is $200, there's absolutely nothing you can do to encourage your opponent to make a mistake. You can bet $20, and he will call, just as he would do if he saw your cards. The 11-to-l pot odds make the bet and call automatic plays, and neither player has any real opportunity to make a mistake.

In no limit, however, you can choose whatever bet size you want. That ability allows you to deceive your opponents more fully and to encourage them to make mistakes. You could bet $150 into the $200 pot, and the player with the flush draw might no longer be correct to call. If your opponent likes to draw to flushes, and he isn't so concerned about the exact odds he's getting, he may be willing to call your $150 bet even though it's a mistake.

Say you know your opponent well enough to know that he will call a $100 bet correctly, and he will fold to a $200 bet correctly, but he'll mistakenly call bets in between. You can target your opponent's weakness by betting the exact right amount to encourage his mistakes.

No Limit and the Fundamental Theorem of Poker 19

Manipulate your opponents and create situations where they are likely to make mistakes. Don't let them off easy. Place them in situations where their natural tendencies lead them astray.

For instance, some players (and we'll talk about these players more later in the book) are particularly suspicious (especially if you've given them even the slightest reason to be suspicious in the past). They seem to always be worried that every bet is a bluff. Consequently, they tend to call bets (particularly some big ones) that they shouldn't call. These players make for very profitable opponents in no limit hold 'em, and the reason is that they are very likely to pay off with second-best hands when they shouldn't. That is, they systematically tend to make one certain type of mistake.

If you were playing limit hold 'em, there would be only so much you could do to exploit this weakness. You could bet for value somewhat more often against these players, but your bet size would be fixed (and small relative to the pot size). And you'd play many hands exactly the same way, whether your opponent was suspicious or not.

In no limit, however, you can exploit this weakness to its fullest. You can vary your bet size on the river to make it the largest you think your suspicious opponent is likely to call. By betting more against suspicious opponents than against unsuspicious ones, you tailor your play to exploit your opponents' weaknesses and set up situations where their natural tendencies will be their downfall.

And betting more on the river isn't the only thing you can do to exploit this weakness. You can also manipulate the betting and pot size on earlier betting rounds to encourage them to make big river calls even more often than they already do. We'll learn more about this idea in later chapters.

In any event, you should set up pots where your opponents will make mistakes without even thinking about it. Likewise, you

When you are heads-up and last to act on the river with the nuts, your expectation on a bet or raise is given by (ignoring check-raises or bet-reraises)


Pcall is the chance you will be called by a weaker hand, and

S is the size of your bet or raise.

To find the right bet size, you have to estimate the chance of being called by a weaker hand for bets of different sizes. Specifically, let's consider three potential raise sizes for this example: $50 (small), $ 150 (medium), and $450 (large and all-in).

If you make the small $50 raise, you think your opponent will likely call with most of his possible hands. Maybe you expect him to call your minimum-sized raise about 80 percent of the time.

If you make the medium $150 raise, you expect your opponent to fold any hand that doesn't include a seven (making a straight). However, since he bet the river into this scary board, you think he has a relatively good chance of having a seven. Let's say he's got a 40 percent chance to have a seven and call the raise. (Please ignore the chance that he has a ten-seven with you, so your straight will always be bigger than his.)

If you make the large $450 raise, your opponent will again likely fold anything except a seven, and we've already posited that he'll have that hand 40 percent of the time. But say your opponent is a little scared of big bets, and you aren't sure he'll call such a large bet with just a seven (he'll fear you have the hand you have, ten-seven). Say you think there's a 50/50 chance he'll call an all-in raise if he has a seven. Thus, you think he'll call you about 20 percent of the time (half of 40 percent).

To find out which raise size is best, you should calculate the expectation for each size. The expectation for the $50 bet is $40.

Thinking in Terms of Expectation — Playing ... 23

The expectation for the $150 bet is $60.

  • 60 = (0.40)($150) Finally, the expectation for the $450 bet is $90.
  • 90 = (0.20X$450)

The small "don't chase them away" raise works out to be the worst of these three options; moving in makes you the most money on average over the long run.

And while we made up the percentage chances you'd get called for this example to make the mathematical process easy to understand, in practice moving in is likely to be the best play in this scenario.

A one-card straight is possible, but you have that hand beaten because you hold the top card also. Anyone without a straight will be hard-pressed to call a decently-sized raise, and anyone with a straight will be hard-pressed to fold. Your only real decision is whether to make a tiny raise to try to get two pair and trips to call or to forget about those hands and try to get the most out of a trapped straight. Because you have so much money behind, your best play is to move in and hope your opponent has a seven.

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