The Mathematics of Poker

[thylacine]

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thylacine said:[ QUOTE ]
What on earth is the quantity that equals 1.61 on page 34? [img]/images/graemlins/mad.gif[/img]

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Oct0puz said:[ QUOTE ]
"As we calculated above, the standard deviation of a sample of 16,900 hands is 1.61 units/100hands" Thats what it is. Or is there something in the quote that is unclear?

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thylacine said:[ QUOTE ]
Yes, we are looking at the same book. The quote is about as clear as pea-soup flavored porridge in a thick fog.

Let me ask again. What on earth is the quantity that equals 1.61 on page 34? What is the precise mathematical meaning of each of the quantities in this calculation?

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thylacine says: Even more to the point: What exactly is being done to a random variable with standard deviation 2.1 to produce another random variable with standard deviation 1.61?

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The SD of 1.61 is a renormalization of the standard deviation of a 16,900 hand sample into BB/100h terms. It's not the standard deviation of a 100 hand sample, of course! It's actually not a random variable, either.

Maybe this would be clearer if we said the following instead -- we won 1.15 BB/100 hands in 16,900 hands, which is 194.35 BB. Our standard deviation for a sample of this size is 273 BB, and change all the confidence interval stuff to read "194BB +/ 273BB" etc.

It's just that it seemed that people are more comfortable with BB/100 as a metric than large raw numbers of big bets over 16,900 hands.

Jerrod

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Okay let me try to reverse engineer what you are saying. Tell me if this is right.

The question is what is the significance of the quantity

S(\sigma,N,K) := K \sigma / \sqrt{N} ---(EQ1)

(Note \sqrt{N} in denominator.) In particular you had \sigma=2.1 N=16900 K=100. This quantity is also given by

\sigma \sqrt{N} / H where H=N/K ---(EQ2)

(Note \sqrt{N} in numerator.) In particular you had H=169.

When K=N, H=1 then the quantity is

\sigma \sqrt{N} ---(EQ3)

which is what is in your equation (2.4) so no problem there. (Note \sqrt{N} in numerator.) But if K=1, H=N then the quantity is

\sigma / \sqrt{N} ---(EQ4)

(Note \sqrt{N} in denominator.) The question is, what is the significance of this quantity and why can we scale by multiplying by K to give a per-K-hand version, which is clearly quite different to your equation (2.4).

So I guess it is this. With K=1 or 100 or 16900 or anything you like, from the statstics gathered (and using "classical method") the "best guess" for the "true underlying mean" is 0.0115K per K hands (which scales as you would expect for the mean of the sum of K samples) and that you could represent your uncertainty about what the "true underlying mean" is as a "standard deviation of the best guess of the "true underlying mean" per K hands" is the abovementioned quantity

K \sigma / \sqrt{N}

(which equals)

\sigma \sqrt{N} / H

The scaling factor K is what you would expect for a mean (or a variance) but not a standard deviation. But since it is actually a "standard deviation of ... mean" it makes sense.

Does this make sense?

Do you understand why the factor K bothered me?

[Jerrod Ankenman]

Yes, this seems right. The key is that *all* the samples we're talking about are 16,900 hands -- since we're not changing the sample size, we can talk about the standard deviation of a sample of that size in whatever units are convenient. But I get why this isn't terribly clear.

jerrod

[Jerrod Ankenman]

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So is this book in Barnes and Noble/Borders yet?

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My wife saw it in our local Barnes and Noble, and both have it on their web sites.

[gdsdiscgolfer]

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So is this book in Barnes and Noble/Borders yet?

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My wife saw it in our local Barnes and Noble, and both have it on their web sites.

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Must be dependent on location. Neither store near me (southwest CT) had it.

[captainwan]

Starting on page 272, for NL hold'em, the authors describe an optimal strategy on the flop OOP and facing a preflop agressor.

What I understand is:

1)on the flop we check 100% of the time even against opponents who don't autobet.
2)as a consequence, betting on the flop is a dominated strategy so we never bet into our opponent.

Am I right?

[Jerrod Ankenman]

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Starting on page 272, for NL hold'em, the authors describe an optimal strategy on the flop OOP and facing a preflop agressor.

What I understand is:

1)on the flop we check 100% of the time even against opponents who don't autobet.
2)as a consequence, betting on the flop is a dominated strategy so we never bet into our opponent.

Am I right?

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The strategy we outline is not optimal! If we had an optimal strategy for NL holdem we sure wouldn't publish it in a book. Instead, it's a case study of applying the principles of optimal play to an actual game. So hopefully the strategy is at least sorta balanced and not that exploitable.

In the case we describe, it's a multiplayer scenario (as there is an "early" raiser) which further prevents there from being an "optimal" strategy.

In any event, supposing that we treat the hand from the flop on as a headsup game where the opponent has some fixed distribution, our contention is that an early raiser in NL playing against the blind should probably autobet as his strategy on the flop (as an equilibrium). If this is the case, then likewise the blind should autocheck to the bettor. This is true regardless of who the player is or what weaker strategy they will employ -- if autobetting and autochecking are the ~optimal strategies, then the raiser can't improve by checking on the flop.

But betting out is not necessarily dominated - in order for that to happen, checking instead of betting would have to have higher or equal expectation against every possible opponent strategy, which is not the case. For example, suppose the opponent played normally if you checked, but automatically folded unless he had the nuts if you bet. Obviously this is a stupid strategy, but betting out would be superior in this case. It's that against the ~optimal strategy, checking has a higher or equal expectation against the opponent's optimal strategy than betting -- hence we check all hands.

That make sense?

Jerrod

[Nsight7]

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My wife saw it in our local Barnes and Noble, and both have it on their web sites.

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Must be dependent on location. Neither store near me (southwest CT) had it.

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Same here. In Austin I haven't found it at ANY bookstores and I frequent 4 or 5 (3 B&N's and a couple of local bookstores). Further, in Arkansas, I went to the local B&N and they didn't have it either.

[uDevil]

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This and other errata can be found at:
http://www.conjelco.com/mathofpoker/...ker-errata.pdf


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p. 41: Typo in heading of last column of first table. Instead of P(B), it should be P(not B).

Wow, I read slowly.

I saw the book at the Barnes & Noble in Cary, NC today.

[captainwan]

Yes, that makes sense.

I missed the point that our opponent hands distribution is constant in the checkraise scenario from the book.