The Mathematics of Poker

[JaredL]

I picked it up yesterday. While I haven't read much (I read the intro and then skipped to the "optimal" (ugh) play section and read through a couple chapters there) here are my first impressions.

FWIW I was a math/econ double major as an undergrad. I started in Math, taking the pure track (topology, analysis etc.) and added Econ as a major when I got interested in game theory problems that were discussed in an intro class I took to fulfill general education requirements. I am currently in my fourth year of grad school in economics doing research in micro theory and experimental economics. Thus, much of my research is in game theory.

Firstly, based just on a glance through some sections and the reading I've done I think this is much more a math book about poker than a poker book with a lot of math. The book is written very much like a math book. This I really liked, though people that haven't read a lot of math texts, or haven't for a while may not find it too appealing. I'm not sure where it will end up in the store or how relevant this is, but when I was looking for it on the computer at Borders it said it didn't have it but it was categorized in the Mathematics section not the gambling section. The target audience seems to be people who have a deeper interest in why things go the way they do in gambling, what's behind the EV calculations, etc. As sputum said, people who like to think about poker as much as playing it.

The good:
It is very well written. The concepts are explained quite well. The format is easy to follow if you're accustomed to math books. There is something for pretty much everyone, no matter how expert. The section introducing game theory I didn't really need, nor the basics of probability (though I haven't read that section so I can't guarantee that I won't get more out of that than a refresher). However, just on what little I've read I've already come across things I haven't seen before, most notbaly some of the toy games, and looking in the table of contents there are a number of other things of interest. What I'm trying to say here is that this isn't a book that explains very basic EV equations that anybody with knowledge of probability would get nothing out of. I haven't read much of the toy game stuff, but it seems very interesting. It seems they've solved some more interesting versions than just the standard von Neumann poker and they suggest something a bit deeper with them than "bet your good and bad hands, check your mediocre hands." The chapter on bet sizing in NL was quite good for example, I plan on rereading it in the next couple days.

Another plus (sorry I'm thinking of them as I'm writing stuff below so this post is a bit disorganized), is that the way the book is designed, it's easy to read just stuff in which you're interested.

The bad:
I've ranted about this on numerous occasions, but I don't understand at all their nonstandard use of words. They use optimal when they should use equilibrium and use exploitive when they should use optimal or best-response. It takes some getting used to for people who have read other texts in game theory and those who haven't will have to readjust should they decide to take a course in game theory. The latter is a pretty likely outcome - those who haven't studied GT or microeconomics and enjoyed the book would do well to do so. Their (mis)use of these terms is not only not standard, but it is dare I say suboptimal. While theorists use optimal in a way that approximates the standard real-world definition of optimal (optimal given what other people are doing), they do not. They say a strategy is optimal if it's a best-response to the equilibrium strategy of the opponent, which means it's likely not optimal given what the opponent is actually doing. This doesn't really make sense, and they really would have been better off using equilibrium or non-exploitable (whichever makes the most sense each time).

To see a clear example of the problem look at rock-paper-scissors. In equilibrium a player is mixing 1/3,1/3,1/3. If I always play good old rock, then it is clearly not optimal for you to mix but to always play paper. Saying that mixing is the optimal strategy here is confusing and perhaps more importantly doesn't make sense.

Other downsides to the book you can get straight from the upsides - if you haven't read a math book in a while the way it's written might not appeal to you. At least the stuff I read shouldn't be particularly challenging to the reader, but the style may make it more of a challenge.

Also, I'm not sure how indicative it is of the book, but after reading not much of the book, I have found either an error. On the bottom of page 102 it says "Only games with either hidden information or sequential play can contain optimal mixed strategies." It's possible that they meant simultaneous instead of sequential, either way it's wrong. Matching pennies, battle of the sexes, and rock-paper-scissors all have no hidden information nor are they sequential move games. All three have a mixed-strategy equilibrium. In case they meant simultaneous, one can construct examples of sequential games with no hidden information that have equilibria where at least one player is mixing.

I don't think the above error is serious, but if these types of errors are present throughout it would indicate that the book wasn't proofread very carefully.

Overall I have really enjoyed the parts I have read thus far. I suspect that it will be among my favorite poker books. I will give a more full review later when I've read the whole thing.

Jared

[Jerrod Ankenman]

[ QUOTE ]
To the authors of this book,

kudos on bringing up some great topics....HOWEVER, it appears that this book has some errors, specifically w.r.t how you try to lay out the Central Limit Theorem in Chapter 2. It appears that you guys are claiming that if you collect enough data, then it starts to follow a normal distribution and you attempt to use this throughout the book. The problem with this claim is that this is not what the CLT says at all. The data is the data, no matter how much of it you collect, and it is never guaranteed to follow any type of distribution at all. What the CLT says, in fact, is that sums/means will follow the normal distr, not the actual data. Its very confusing and makes the book very hard to follow in places.

Please address,

SB

[/ QUOTE ]

The discussion of the Central Limit Theorem (p24) says:

The distribution of outcomes of a sample is itself a distribution, and is called the sampling distribution. An important result from statistics, the Central Limit Theorem%

[Jerrod Ankenman]

[ QUOTE ]
To the authors of this book,

kudos on bringing up some great topics....HOWEVER, it appears that this book has some errors, specifically w.r.t how you try to lay out the Central Limit Theorem in Chapter 2. It appears that you guys are claiming that if you collect enough data, then it starts to follow a normal distribution and you attempt to use this throughout the book. The problem with this claim is that this is not what the CLT says at all. The data is the data, no matter how much of it you collect, and it is never guaranteed to follow any type of distribution at all. What the CLT says, in fact, is that sums/means will follow the normal distr, not the actual data. Its very confusing and makes the book very hard to follow in places.

Please address,

SB

[/ QUOTE ]

The discussion of the Central Limit Theorem (p24) says:

The distribution of outcomes of a sample is itself a distribution, and is called the sampling distribution. An important result from statistics, the Central Limit Theorem, describes the relationship between the sampling distribution and the underlying distribution. What the Central Limit Theorem says is that as the size of the sample increases, the distribution of the values of the samples converges on a special distribution called the normal distribution.

The CLT doesn't say anything about the distribution of individual data points, but only about aggregated samples. If there's an example of a place where we've conflated data with sample sizes incorrectly, I'd be happy to review it and make errata, but I'm pretty confident that throughout the text, we only deal with aggregated samples, which are generally subject to the CLT.

jerrod

[Python]

On pg. 22 you define a sample as a set of observed outcomes to a particular probability distribution, e.g. for a coin flip [ 1 0 0 0 1 1 0 1 ].

On pg. 24 you define a sample as the summation of these outcomes; in my example 4.

That's cleary contradictory, isn't it?

The last definition should be for an aggregated sample and it's the distribution of outcomes of an aggregated sample, that follows the CLT.

[Megenoita]

What mathematics background do I need to understand this book? I need statistics, right? Some calc?

[Ace-Ex]

I'm considering buying this book or Ace on the River by Barry Greenstein as my next poker book purchase. I'm an engineer, so the mathematics of this sounds fascinating. I'm not convinced this will immediately impact my game, however. Mathematicians love to present theories but real-world application can be lacking at times...

[Jerrod Ankenman]

[ QUOTE ]
What mathematics background do I need to understand this book? I need statistics, right? Some calc?

[/ QUOTE ]

A little statistics will be helpful, a little calculus will be helpful. If you have netiher of these, there's a little review and harder things are marked off so you can skim or skip through the derivations. Some readers have reported that they think the book is just too hard if you don't have a math background - others have said that they were able to follow most things with a little review.

jerrod

[Jerrod Ankenman]

[ QUOTE ]
On pg. 22 you define a sample as a set of observed outcomes to a particular probability distribution, e.g. for a coin flip [ 1 0 0 0 1 1 0 1 ].

On pg. 24 you define a sample as the summation of these outcomes; in my example 4.

That's cleary contradictory, isn't it?

The last definition should be for an aggregated sample and it's the distribution of outcomes of an aggregated sample, that follows the CLT.

[/ QUOTE ]

Yeah, I guess this isn't exactly clear. The "value of a sample" (as opposed to the value of the outcomes in a sample) should be the total value of the measured outcomes added together. THen all this stuff follows. When I get back from my trip, I'll put this in the errata.

jerrod

[BillChen]

Actually the use of the term "optimal" is fairly common when describing equilibiria in zero-sum two player games. In fact the equilibrium is the same as the minimax solution and the term optimal for the strategy pair is used for example by Gilpin and Sandholm from CMU in their "Optimal Rhode Island Holdem" paper. This is also the term used by the Manitoba group and Daphne Koller at Stanford. It seems it's a term adopted ny the Computer Science people working on algorithms for these problems.

It's a fairly common if not standard usage in this field. Even in the field of cooperative game theory, there is the term Pareto-optimal, of course all zero-sum 2-player equilibria are also Pareto optimal.

Now, this is a general problem across fields (and sub-sub-fields). For example mamy of the problems in finance, gambling, and operations research and control theory turn out to be very similar, and while surveying the literature about a particular problem, you realize many of the results in one field are ignoired by researchers in another because of notation. I mean if we had used "equilibria" instead of "optimal" I am sure there would also be complaints. Our goal is not to unify notation across fields, but to clearly define our terms so people across different fields can read and understand what we are talking about.

Bill

[JaredL]

[ QUOTE ]
Actually the use of the term "optimal" is fairly common when describing equilibiria in zero-sum two player games. In fact the equilibrium is the same as the minimax solution and the term optimal for the strategy pair is used for example by Gilpin and Sandholm from CMU in their "Optimal Rhode Island Holdem" paper. This is also the term used by the Manitoba group and Daphne Koller at Stanford. It seems it's a term adopted ny the Computer Science people working on algorithms for these problems.


[/ QUOTE ]

This is fair enough. I would argue that it would be better to use the terminology that is standard for the field at large and not just a very small sliver of it, but in your defense your book, at least as far as I've gotten, only covers two-player zero-sum games. As I wrote in my post above, I think it's better for the readers if you use the terminology that they would have seen or will see in coursework or reading other texts. While you could argue that the texts your readers are most likely to see/have seen are those by these computer scientists, I'm inclined to disagree. I would also argue that these people should not be using optimal in this way either because as soon as player A deviates from her minimax strategy, it is (most of the time) not optimal for B to play his.

I suspected while reading that you chose to use optimal in this way because you are putting forth the idea that people should play equilibrium strategies as a default. This is apparently not the case based on what you've said here and a search of your past work.

[ QUOTE ]

It's a fairly common if not standard usage in this field. Even in the field of cooperative game theory, there is the term Pareto-optimal, of course all zero-sum 2-player equilibria are also Pareto optimal.


[/ QUOTE ]

Pareto optimality is something completely different. An allocation (in this case payoff n-tuple) is pareto optimal (also often called pareto efficient) if you cannot make one player better off without making some other player worse off. In fixed-sum games, every strategy combination will lead to a set of payoffs that is pareto-optimal by definition.

Note that a pareto-optimal allocation is optimal in the sense that nothing is being wasted - there are no $20 bills on the sidewalk if you will.

[ QUOTE ]

Now, this is a general problem across fields (and sub-sub-fields). For example mamy of the problems in finance, gambling, and operations research and control theory turn out to be very similar, and while surveying the literature about a particular problem, you realize many of the results in one field are ignoired by researchers in another because of notation. I mean if we had used "equilibria" instead of "optimal" I am sure there would also be complaints. Our goal is not to unify notation across fields, but to clearly define our terms so people across different fields can read and understand what we are talking about.


[/ QUOTE ]

This too, is fair enough. As I said, the book is well written and reads like a math book. As such, you certainly did define the term and hence can use whatever words you like (perhaps calling an equilibrium strategy an Ankenman-Chen strategy and a best-response strategy a Sklansky would have worked [img]/images/graemlins/wink.gif[/img] ).



Once again, I want to clarify that I am very much enjoying the book. The further I've gotten into it the more I have liked it. If the biggest complaint of a reader is the admittedly nit-picky argument over the use of certain terms, then you guys have done a great job. I am nearly finished with the toy-games section and am looking forward to reading the other sections as well.

Thank you for taking the time to respond to my post.

Jared