Probablility and The Fundamental Theorem of Poker

[lastcardcharlie]

I'd like to make the following observation on the FTOP.

Probability is to a large extent not a measure of reality as it is but of what one knows about reality. To see this, consider for example if you flip a coin and you can see the outcome and I can't. For me, the probability that the outcome is Heads is 0.5, whereas for you it is either 0 (you know the outcome is Tails) or 1 (you know the outcome is Heads). You and I are talking about the same reality (the outcome of the coin flip) but the respective probabilities of what that outcome is are very different (as you know more than I).

The FTOP seems to be a theorem based on reality as it is (i.e. on how one would act with perfect knowledge of what all the hole cards are) rather than on one's knowledge of reality. But one's knowledge of others' hole cards is usually limited, and the correctness of a decision is surely relative to this limited knowledge rather than to perfect knowledge?

To make a probably-very-unfair comparison, some people have great difficulty understanding that, say, the correctness of a turn bet has absolutely nothing to do with what card actually falls on the river. That e.g. I happen to make my gutshot in a small pot does not in any way mean that my call on the turn was the right thing to do. Thinking it was the right thing to do seems to be a reality-based view (the reality is that the river was favourable and I won the pot and am glad I called the turn) rather than a limited-knowledge-of-reality-based view (the contingency of what the river actually is has nothing to do with the correctness of the turn call).

[four-flush]

I understand what you are saying although I'm not sure I agree.

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you flip a coin and you can see the outcome and I can't. For me, the probability that the outcome is Heads is 0.5, whereas for you it is either 0 (you know the outcome is Tails) or 1 (you know the outcome is Heads).

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The 0.5 heads/tails can be used as a prediction of what will happen, so your example doesn't apply because it assumes one of us knows the outcome to start off with! (Probabilities imho are about seeing the chance that something will happen, not whether you or I know the answer beforehand. That's not probability).

Now, if you say that the body of knowledge that human beings have amassaed is certainly limited, can improve and is by no means "perfect" than I am on your side! Then, I can say, "the way we do things now, (theories of poker) is subject to our knowledge of mathematics and in particular probability and, as we improve this knowledge, we shall advance poker theory." However, to say that our "theorum" of poker or probability is flawed because it is not a measure of reality, is not true. It may limited - because we can improve theory - but it is certainly not flawed.

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[lastcardcharlie]

Although probability theory is used to predict future events the vast majority of the time, it's not necessarily restricted to that. Probability = 1 is just a mathematical term for "certainty", whether about the past, present or future - I don't believe there's anything controversial about that.

As for the second part of your post, my point wasn't that human knowledge can't be improved but rather that I'd like to see a "fundamental theorem of poker" as measured against subjective human knowledge (i.e. more in accordance with the meaning of probability) rather than against a "God's eye" view of the world.

[numeri]

I'm not sure what your point is here. You stated:

"But one's knowledge of others' hole cards is usually limited, and the correctness of a decision is surely relative to this limited knowledge rather than to perfect knowledge?"

The FTOP is saying that we can make a "perfect" decision based on all the information we know, but that decision will not be "perfect", as we do not know our opponent's cards. This doesn't mean that our decision is necessarily wrong, because we made it with limited information. In theory, it may have been wrong since we may have assumed we were ahead when we were in fact behind. This does not change the fact that we made what we thought was a correct decision at the time, but our opponents benefited from that decision - thus satisfying the FTOP.

In fact, your final paragraph in the OP is saying that the turn decision is correct or incorrect based on the theoretical probability that the river card will fill the gutshot. In this context, you are exactly following the FTOP. You are assuming you are behind, (And therefore, you do know your opponent's cards, to some extent.) so the call is unprofitable. If you call, the FTOP states then that your opponent has gained, as you should have used that knowledge to fold.

One other concern of mine. You stated in your second post that, "I'd like to see a "fundamental theorem of poker" as measured against subjective human knowledge (i.e. more in accordance with the meaning of probability)". What meaning of probability is this? When I play poker, I assume that the deck is fairly shuffled and each card is equally likely. If I happen to flop a set two hands in a row with my pocket pair, that doesn't change the probability that the next flop will give me a set with my pocket pair.

[four-flush]

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Although probability theory is used to predict future events the vast majority of the time, it's not necessarily restricted to that. Probability = 1 is just a mathematical term for "certainty", whether about the past, present or future - I don't believe there's anything controversial about that.

As for the second part of your post, my point wasn't that human knowledge can't be improved but rather that I'd like to see a "fundamental theorem of poker" as measured against subjective human knowledge (i.e. more in accordance with the meaning of probability) rather than against a "God's eye" view of the world.

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If someone came up with a `fundamental theorum of poker', which in your words is "as measured against subjective human knowledge," I wonder how many professors of mathematics/probability/statistics in the U.S. will agree with the new theorum and will also admit that their use of probability is flawed. Probability has been around since the 1600s, don't you think since that time mathematicians would have found out whether their approach/definition/methods of probability are flawed?

If probability theory is not useful to poker, then why can people make mathematically correct decisions based on such things as `pot odds' and the chance of improving their hand? etc. Are you suggesting such knowledge is incorrect? That if I withdraw an Ace of Clubs [img]/images/graemlins/club.gif[/img] from the deck there is still a chance that four more Aces will come out?
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[lastcardcharlie]

In response to four-flush's post, I am not here for one moment suggesting that mathematical probability is flawed or that it is not applicable to poker. I don't see how my posts could be interpreted in this way.

The FTOP states (roughly, from memory; I don't have the text to hand) that:

Every time you act as you would if you could see the others' hole cards you gain, and every time you don't you lose.

Now this statement is not prescriptive in that it refers explicitly to correct or incorrect play, but Sklansky does go on to build a prescriptive theory of how to play good poker out of it. One of my points is, what is the essential difference between this statement and, say:

Every time you act as you would if you could see the next card to be dealt you lose, and every time you don't you gain.

To return to the meaning of probability, I repeat that it is a measure of what one knows about reality rather than a measure of reality itself. So I would like to see a theorem of poker in which the criterion for correct play is not reality itself (in this case, what the others' hole cards actually are) but what one knows about reality.

But, to be clear, I am NOT saying that the FTOP as it stands is a flawed or false statement. As a matter of fact it is probably a true statement. What I am questioning is its use as foundation for a prescriptive theory of how to play good poker.

[zoobird]

I think we'd get a little bit closer to what you're looking for if it said something like "We lose every time we play our hand differently than we would if we knew exactly what range of hands our opponent would play this way and what the probability of each is".

[numeri]

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Every time you act as you would if you could see the next card to be dealt you lose, and every time you don't you gain.

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This statement is not helpful in poker. It is not possible to gain any information about the next card, so your "theorem" does not help us decide whether to call, fold, or raise.

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Every time you act as you would if you could see the others' hole cards you gain, and every time you don't you lose.

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This statement, on the other hand, is applicable. We can get information about the other players' hole cards, and we can then use that information.

Your "theorem" is not helpful because it is not possible to know that information - even partially. Therefore, it is of no use to poker players.

[lastcardcharlie]

Yes, zoobird, that's exactly the point I'm making: the criteria for good play should be measured against what one knows rather than what actually is.

With regard to knowledge of opponents' cards vs knowledge of next card, I suspect there probably is an essential difference but I'd like to understand more what it is. It is not entirely true to say we can have no knowledge of what the next card is as the cards already seen give an indication. Obviously this is more of a factor in stud than it is in holdem.

[numeri]

You're going to have to figure this out yourself. You're either not reading my posts, misreading them, or just not understanding them. I'm not sure how else to explain it.

The FTOP is not a "criteria for good play", as you state. It is a theoretical reference to the "perfect" decision. In practice, we will not often be making the "perfect" decision, as we are working with imperfect information.

Read the strategy forums. The replies there do not reference the FTOP. They do not know what the opponenents' cards are. Instead, they place the opponent on a range of hands, and make the best decision based on that information. There is often disagreement about that decision, but sometimes there actually is one that is accepted as the best given the limited information.

And don't be obtuse. When I wrote that we have no knowledge of what the next card will be, of course I am implying that of the remaining cards in the deck, we have no knowledge. On the other hand, even though I only know the two cards in my hand and the cards on the table, I can get information about my opponents' cards.

Again, the FTOP is talking about the theoretical best decision. It is not talking about the best decision based on reality.

If you would like another theorem that talks about the best decision in reality, go for it.