Bayesian Theorem of Poker (was: Sklansky's vs. Abdul's theorem)
Michael Maurer
I've followed some of this thread and I can't decide which camp I'm in.
People in the Useless camp say "the fundamental theorem of poker is worse
than useless" because it is outcome-based and relies on 20-20 hindsight.
Not only is it impossible to apply hindsight in advance, but a way of
thinking that gives value to hindsight handicaps your thinking when you must
make decisions without it.
People in the Relevant camp say "the fundamental theorem of poker is
relevant because it helps me think about poker situations at an abstract
level". One of Sklansky's examples here is his claim that you should be
happy when a player with a worse hand than you, but with correct odds to
draw, folds to your bet. Another of Sklansky's points is that the theorem
teaches people the importance of reading their opponent's hand and of
disguising their own hands. So people in this camp feel that, although the
statement of the theorem is outcome-based, it helps their situational
thinking even when they don't yet know what the outcome will be.
Finally, people in the Badly Worded camp say that "well, what Slansky really
meant to say was the probabilistic version, but his audience wouldn't have
understood that so he dumbed it down". They argue that it's obvious that
Sklansky didn't mean to say that you should use hindsight to judge your
decisions. Rather he intended that his readers form a conceptual framework
that includes what you think your opponent has and what your opponent thinks
you have, and so on. And ultimately, to act according to the probabilistic
version. People in this camp wish Sklansky had just written the
probabilistic version so we wouldn't be having this pointless discussion.
Before I join a camp, let me first attempt to rename the idea that some are
calling the "Abdul/Maurer Probabilistic Fundamental Theorem of Poker". That
name sucks bigtime. A better name is the Bayesian Theorem of Poker (my
apologies to the Reverend Bayes, who probably would not have approved of
poker at all), and I'll state it as
## Every time you play your hand differently than you would have played had
you known the true probability distribution of your opponent's hand given
all previous actions, you lose; every time you play your hand the same as
you would have played had you known the true probability distribution of
your opponent's hand given all previous actions, you gain. ##
I call it the Bayesian Theorem of Poker because at any given moment in a
hand, whether your decision is good or not is a function of all "prior"
knowledge, that is, everything that you know at that moment, and nothing
that you will learn in the future. Thus, the fact that your flush was
coming on the river does not justify your raise on the turn, but your
estimate of your opponent's cards as of the turn could justify a raise
judged as correct by the theorem. You see that for those players with a
probabilistic bent, when they say they "know your cards", what they really
mean is that they know you have AcAs 14% of the time, AcKc 21% of the time,
and so on.
Now, I leave out the term "Fundamental" because I don't think this theorem
is a complete description of how to play poker well. What it lacks is any
mention of the expected response of your opponent to your actions. And to
know that you must first know your opponent's estimate of your own hand
distribution. In other words, you must know your opponent's strategy which,
when you are in the hand, depends on your opponent's estimate of *your*
strategy! But that's an improvement for another day.
So which camp am I in? On the one hand, I have to agree with the Useless
camp that Sklansky's statement is not prescriptive in a particular
situation. But at the same time I believe that reading the Theory of Poker
was very helpful when I was learning the basics of poker. Although you
don't know what cards your opponent actually has, you always have a
probability distribution. And for each element in that distribution the
FTOP helps you understand how you would like the remainder of the hand to
play out. So, you can form an aggregate view of how you would like to see
the hand play out, based on the information you have at the time and its
effect on the probabilities of your opponent's possible hands. Then you can
act in a way that gives the best result averaged over all the hands.
So I guess that when I talk about Sklansky's version I am actually applying
the Bayesian version. That puts me near the border of the Badly Worded camp
and the Useless camp.
-Michael Maurer
Michael Maurer
> Now, I leave out the term "Fundamental" because I don't think this theorem
> is a complete description of how to play poker well. What it lacks is any
> mention of the expected response of your opponent to your actions. And to
> know that you must first know your opponent's estimate of your own hand
> distribution. In other words, you must know your opponent's strategy
which,
> when you are in the hand, depends on your opponent's estimate of *your*
> strategy! But that's an improvement for another day.
Notice that Sklansky's original wording, which mine is based on, cleverly
avoids this point. He defined the difference between good and bad play as
the difference between your actions when you have a correct estimate or an
incorrect estimate of your opponent's hand. He doesn't try to prescribe the
correct play given a correct estimate, but leaves that up to the player.
Perhaps in that sense the concept can be called fundamental.
Michael Maurer
Michael Maurer
>You see that for those players with a
> probabilistic bent, when they say they "know your cards", what they really
> mean is that they know you have AcAs 14% of the time, AcKc 21% of the
time,
> and so on.
Here is an illustration of how the Badly Worded camp would apply the FTOP as
if it were the BTOP. Also it gives and example of how the concept can
prescribe a call on the river.
Say that you have played an exciting hand with an opponent and after the
betting on the turn completes you have estimated his hand distribution as
Ac Kc - 30%
Qs Qd or Qs Qd or Qd Qh - 70%
The flop was Qc Jc 4d and the turn was 5c. You have Js Jd. Now the river
comes 4s. So you are either ahead or you are behind! The river gives you
no information that modifies your probability distribution, so you think the
chances are 30% that you have the best hand. Your opponent bets.
Of course, at this point the pot is quite large, so folding a 30% winner is
out of the question. And yet, raising is probably not right because your
opponent has a pretty good read on your hand at the moment also. Let's say
that if he has AKc then his read of your hands, after the river card falls,
is {T9c 50%, QQ 20%, JJ 20%, 44 10%} and if he has QQ then his read is {T9c
60%, JJ 30%, 44 10%}. Suppose this read is actually correct. Given his
good read he will reraise with the queens, plus with the flush some of the
time, forcing you to call some of those reraises. Thus, your best play is
probably to just call.
By the way, if anybody is inspired to compute the correct strategy for each
player on the river, under the assumption that the hand distribution
estimates each has for the other is correct, I'd be curious to see how it
turns out. I imagine that because of the nonzero chance that you have QQ
beat with 44, that you would sometimes want to raise without the 44. But I
think those raises would be made with the T9c, since that is a hand that
loses otherwise, and not with the JJ.
Anyway, this example turned out to be less simple than I hoped. My goal was
to find a common real-life situation where the Badly Worded camp could apply
the BTOP. The funny thing is that if you know what your hand distribution
is and what your opponent's hand distribution is, then you can compute the
correct river play as if poker were a game of complete information. In
fact, you should be able to do that even with cards to come, since you know
what cards remain in the deck for each of your and your opponent's holdings.
The way I see it, knowing the probability distribution of your opponent's
hands and knowing his estimate of your distribution is what reading your
opponent means.
-Michael Maurer
Hux Wallace
often confused with the Bayesian TOP, with the result that people are
wrongly crediting FTOP with all sorts of results which do not belong to
it.
The "correct play" determined by using poker principles/Bayesian
theorem and the FTOp can be completely different and they therefore
cannot both be correct. Where the FTOP gives a different result from
the Bayesian theorem, it is always wrong, or useless, and where it
gives the same result it doesn't tell you anything you didn't work out
through normal methods, ie the Bayesian TOP.
Sklansky's FTOP seems to be a sacred cow. With some important
exceptions,(thanks mum, dad, ) my earlier posts raised a diverse group
of respondants who, oddly, while generally (but not unanimously)
agreeing that I must be wrong, at the same often contradicted each
other and SKlansky's own views and statements. Most people found
SOMETHING to like about FTOP, therefore my blanket dismissal of it as
trivial aroused a lot of dissenters, many of whom, as I say, are
actually at odds with Sklansky's own views. For instance, some agree
that it is useless at the table, but useful in other ways, which is
certainly not Sklansky's position.
Sklansky's views are clear: the FTOP stands as written, it's
meaningful, it's important, it's frequently useful at the table. He
also says that it's almost a tautology, which means that it is almost
circular and meaningless. I agree with him on that.
Sklansky's FTOP says clearly that the best way to play is the way you
would play if you could see your opponents cards: this cannot be
reworded into the Bayesian theroem without completely changing it's
meaning, and Sklansky certainly cannot say "I really meant the Bayesian
theorem all along", though some who have defended FTOp have in effect
said exactly that as an argument against my position, and as a defence
of Sklanksy's.
The bayesian theorem in effect says "work out what to do according to
what you know, using probability,etc", and doesn't give any specific
guide for action, it's simply a description of what we do when we play
poker. The FTOP similarly gives no specific (or even general) guide
for action, but it is not in any way a description of what we do when
we play poker because it only applies after the event, never during
play. FTOP therefore has no place in determining how to play poker,
whereas every time we play poker we are constantly "using" the bayesian
theorem.
Claims made for FTOP's "strategic value" are frequent, but rarely
specific: if concrete examples are given they can soon be shown to be
examples of the Bayesian theorem and not the FTOP at all.
To sum up, where the FTOP is true it is trivial, and when a non-trivial
example is put forward it is never an example of the FTOP at all, but
of the bayesian theorem.
Thanks for reading to the end :)
Hux wallace.
In article <ohFU3.178$PR.3...@newsin1.ispchannel.com>,
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Before you buy.