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.

[ QUOTE ]
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.

[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.

[four-flush]

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In response to four-flush's post, I am not here for one moment suggesting that mathematical probability is flawed

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Yes you are because you said, "Probability is to a large extent not a measure of reality."

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What I am questioning is its use as foundation for a prescriptive theory of how to play good poker.

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Just remember that most of the skill used to win at poker is not mathematical, suffice it to say that probabilility has existed for half a millenium, it has been applied successfully to poker in both theory and practice for decades. You are the only person I have ever heard of who contests the method of application of probability to poker. I don't believe the leading authorities on the mathematics of poker (Caro for eg.) would agree with your contentions.

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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

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No, the meaning of probability is this: a measure of how likely it is that some event will occur.



One cannot discount something that is incontestable. I suggest that you write to the author of the `FTOP' and discuss the matter with him, or with a professor of mathematics and statistics. I think I'll leave it at that and let you talk with others in the forum to get their perspective on this. There's not much more I can add, other than what I have already said.
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[four-flush]

[ QUOTE ]
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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[Daisydog]

Most of us would agree that what makes a poker player good is the ability to make good decisions based on available information (which is incomplete). In Texas Hold em, when you are making a decision, information is incomplete for at least two reasons:
1. uncertainty in the board cards to come on later streets
2. uncertainty in your opponent's hole cards.

Both of these can be described with probability distributions.

The FTOP states,". . . every time you play your hand the same way you would have played it if you could see all of their cards, they lose . . ."

Of course, the FTOP doesn't literally mean every time. It is saying "they lose" from an EV perspective. The board cards on a later street may come up in such a way that your opponent sucks out on you. But the board cards follow a probability distribution, and from an EV perspective your opponent loses.

Similarly, your opponent's hole cards also follow a probability distribution. So there are plays that may be bad given an opponents exact hole cards, but may be good from an EV perspective given the opponent's hole card distribution. For example, it might be a good decision to raise UTG with KK, even if a MP player has AA. We don't know the MP player has AA and raising with KK is +EV given the hole card distributions.

I think what lastcardcharlie is trying to say is that the FTOP deals with the probability distributions of future board cards, but not the probability distribution of opponent's hole cards. Therefore, it can tell us that a decision was good even if an opponent sucks out on us. However it will not allow us to say that we made a good decision when we lose because the opponent's hole cards just happen to be on the extreme high end of their distribution.

In poker we make decisions with incomplete information. This incomplete information lends itself to being described by probability distributions and EV analysis. In Texas Hold em the incomplete information stems from uncertainty in future board cards as well as from uncertainty in opponents hole cards. Decisions can be good even if board cards OR opponents hole cards turn out to be unfavorable. The FTOP appears to deal only with one aspect of the incomplete information (board cards) but not the other (opponent's hole cards). Does this mean it is wrong? No. It's correct, it just doesn't deal with uncertainty in opponent's hole cards.

In my mind, the most fundamental thing about good poker playing is making good decisions given all AVAILABLE information. Exact hole cards are not available, but hole card distributions are to those who are observant. Perhaps the FTOP ought to incorporate the idea that decisions can be good even if they turn out to be bad once the hole cards are known.

Thanks, lastcardcharlie, for a very interesting topic. I look forward to future discussion.