The Mathematics of Poker

[Patrick Sileo]

[ QUOTE ]
P.48:
<B,call> = p(A has nuts)(-1) + p(A has a bluff)(+5)
<B,call> = (0.2)(-1) + (5)x

Shouldn't it be:
<B,call> = p(A has nuts)(-1) + p(A has a bluff)(+5)
<B,call> = (1-x)(-1) + (5)x

[/ QUOTE ]

There is a problem in the original text. The correct formulation follows from conditioning on the right information set. I previously posted the correction here:

http://forumserver.twoplustwo.com/sh...age=0&vc=1

The authors' general conclusions are unchanged by the correction even though the value function (and graph thereof) becomes non-linear.

[Jim C]

I just finished my first complete reading of the book. It is absolutely extraordinary.

Those looking for specific advice playing particular forms of poker will not be happy with the book (with one important, and possibly extremely profitable exception). Those who are looking to really understand the depths and complexity of the game, in all its forms, will be rewarded with an absolute masterpiece.

I've read and studied everything worth reading (and many others not worth reading!) about poker many times. In my opinion, nearly all of the worthwhile stuff is 2+2 books, with a few exceptions. As stellar as I believe the 2+2 books are, I feel that MoP deserves its own category.

Its major departure from most good poker books is to explore the notion of "optimal play" in a great deal of depth. The most powerful tool of this exploration is game theory, and the book contains an extremely rigorous application of game theory to poker using exemplifying "toy" games that illustrate strategic principles of real poker games. Except for what Sklansky has briefly written on the subject (ToP), this is the only book containing this kind of information that I am aware of.

While the game theory sections seem to be causing the most comments, MoP also contains excellent sections on what the authors call "exploitive play". While optimal play intends to make our own play unexploitable, exploitive play intends to maximally profit from the deficiencies in our opponent's strategies. To do so, we must ourselves deviate from optimal play, which opens us up to be expolited ourselves (what the authors call counter-exploitation). The discussion of identifiying opponent's strategic weaknesses and developing maximally exploitive strategies is fantastic. Related to this whole discussion is the notion of strategic "balance", which is the bridge to the discussion of optimal play -- and the defense against counter-exploitation.

I can't say the book has taught me any new "plays" or given me any one specific thing to improve about my game (I am not a tournament player, the domain of the important exception I mentioned above). Instead, this book has given me something orders of magnitude more valuable: a more sophisticated way of *thinking* about poker. One reading has already prompted me to think about some pretty important aspects of my game -- balanced strategy on the turn in cash NL holdem, in my particular case -- in an entirely different paradigm. This is absolutely NOT just another book showing you how to calculate pot odds and reminding you to consider future action or the chance you'll catch and lose (my opinion of Yao's "Weighing the Odds"). There is some new and very sophisticated stuff here.

The book has introduced me to thinking about poker at the level beyond what's described in the existing literature. As soon as I finished the last page, I started reading it again...

One final comment about the math. I have an extremely strong math background (though not post-graduate level), and I am comfortable reading ideas in a textbook style of writing. However, the math is not difficult in this book, and the most "advanced" math employed is probably finding a minimum by finding the zero of the first derivative. That is calculus, but anyone who's taken basic differential calculus will be able to follow all the math in the book (this includes quite a few high school students). If you're someone who thinks that NL Holdem is a "people game" and so you don't need to know about equity of hands, pot odds, and draw probabilities, skip this book. This book is for people who have that stuff down cold, don't need any clever new ways to think about it (DIPO?!?), and want to go to the next level.

The beginning of the book has a nice introduction to probability and statistics, but I feel that a good understanding of how the authors analyze poker will require some basic training in statistics, particularly a degree of comfort with the idea of distributions. I think that studying the first half of a first-term college statistics book is valuable for gamblers whether they read MoP or not, but it will definitely help you with this book.

Jim

[thylacine]

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

[Oct0puz]

"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?

[thylacine]

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

[/ QUOTE ]


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?

[/ 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?

[thylacine]

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

[/ QUOTE ]


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?

[/ QUOTE ]

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?

[/ QUOTE ]

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?

[uDevil]

[ QUOTE ]

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?


[/ QUOTE ]

The 2.1 is the SD/hand, 1.61 is the SD/100 hands. They want to compare the WR (in BB/100) to the SD so they need to adjust the units of SD from /hand to /100 hands.

Edit: Uh, yeah. So shouldn't the converted value be 27.3, not 1.61?

[Jerrod Ankenman]

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

[/ QUOTE ]


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?

[/ QUOTE ]

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?

[/ QUOTE ]

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?

[/ QUOTE ]

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

[Jerrod Ankenman]

[ QUOTE ]
A question to Bill Chen. In "Jam or Fold Tables" for attacker (i.e. small blind) with KT offsuit strategy is "JAM" meaning SB should push with stack/BB ratio 50 or less. I've made some calculations using linear programming method and got another results:

stack/BB KTo % jam
50 41.08%
49 0.00%
48 0.00%
47 8.73%
46 100.00%
45 83.50%
44 100.00%
43 6.25%
42 21.70%
41 19.34%
40 20.33%
39 43.36%
38 38.07%
37 32.53%
36- 100.00%

Comments?

Andrzej Nironen

[/ QUOTE ]

Checking these against our results for specific stack sizes, your results match ours. In formulating the jam or fold tables (which are sort of a simplification of this type of thing) we had to choose which oddities to report on and which to sort of ignore, so I think that here we simply decided to mark the hands as JAM as it is jammed above 50 at least sometimes and it can't be terribly costly to jam it when mixed.

Jerrod

[johnnybeef]

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