## The Fundamental Theorem of Poker

### Poker Concepts, Strategy

September 9, 2007

Before we get started on the fundamental theorem of poker, I want to introduce a concept that I hope is very obvious nay axiomatic – conservation of money. Simply put, playing poker neither creates nor destroys money. Or put another way, if your results are that you won X everyone else at the table’s collective result is that they lost X and vice versa.   Stated in mathematical terms, everyone’s results sum to zero. Note that the house, if collecting rake, is one of the participants in this equation. Hopefully we can agree this is pretty obvious stuff.  I mention it because it’s the starting point for the fundamental theorem of poker.

David Sklansky states the fundamental theorem in Theory 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.

Now, this idea is very close to the concept of conservation of money, but it’s really what I like to call conservation of accurate expectation. Extending the concept of conservation of money to conservation of expectation isn’t that hard. If I expect to make \$X on some play, the rest of the table expects collectively lose \$X on that play. This is just saying that you believe conservation of money will continue to hold in the future. As always, these expectations are average results over a lot of trials. Now, the key here is that these expectations have to be accurate for the conservation to apply. If the people holding the expectations are a) misinformed or b) irrational or c) incapable of the reasoning needed to form correct expectations, then there’s no reason to believe that everyone’s inaccurate expectations add up to zero on a given play.

I’m not a big fan of the way Sklansky phrased the fundamental theorem. In particular, I think he’s incorrectly adopted seeing your opponents cards as a proxy for accurate expectations. It should be fairly obvious that he’s slightly off base here, as an example will show.

Suppose you’re playing limit holdem, are in early position, and have called your way to the river with a flush draw which misses leaving you with no pair. The only other person remaining in the hand has been betting into you the whole way, and after the river card is dealt accidentally exposes his hand, which is pocket 3s giving him an underpair to the board. Despite being able to see your opponent’s hand here, it’s really not clear what course of action yields the best expectation. Depending on what line he intends to adopt, you best expectation might come from checking with the intention of folding, checking with the intention of raising on the bluff, or betting out on the bluff. So in fact, the accuracy of your expectation is not a function of knowing the cards, but rather a function of knowing the cards and how your opponent intends to play them.

In other words, holding accurate expectations is much more difficult than Sklansky makes it out to be. However, putting that aside, the fundamental theorem is still an incredibly important concept. If you have accurate expectations taking all factors into account, and take the course of action that maximizes those expectations, you come out ahead. If you fail to do this, you lose. And the same, vice versa, goes for you opponent. So the goal is to, as often as possible, adopt the same strategy that you would if you could see the cards and know how your opponent would play them.

There is one other minor criticism of the fundamental theorem worth knowing about: it only applies to play once it becomes heads up. In multi-way play, you can get a horse race scenario where any individual opponent would be making a mistake to play against you (hence you would gain) but two or more would be correct to do so. This little caveat technically negates the fundamental theorem when there are more than two people in the hand and at least two left to act, but most of the time in practice the fundamental theorem applies still applies even in those situations because true horse race paradoxes are somewhat rare.

So remember: accurate expectation is conserved (ie. everyone’s accurate expectations sum up to zero)