Customer Reviews

The Mathematics of Poker

5 star:		(20)
4 star:		(4)
3 star:		(8)
2 star:		(5)
1 star:		(6)

 

 

Wow, I'm very impressed with the book. I think it's touched ground that isn't available anywhere else. I'm sure that many programmers (myself included) have attempted to solve this game, and have discovered how burdensome the simple odds calculations are, nevermind the strategy and decision trees. Poker will not soon be solved by computers, like chess is. However, Bill Chen's ideas of "Toy games" help humans get insight into the character of the solution.

Anyone picking up this text should be warned of several things:

1) It is not for beginners. Strong poker takes judgement and experience, and basic hand/situational values cen be best learned from Dan Harringtons books or Sklansky's No-Limit book. I've read over 20 poker books, and Harrington and Sklansky stand out as the best. Harrington's books are very practical, with detailed analysis of situations.
2) It is not for the timid, foggy headed, or undisciplined. The new concepts in his books require for you to stop and think. If your instinct is "gee, this sounds complicated", then give up now. Some people will have the same backlash that regular people have with math. If you're from the "Math is hard" philosophy, this is not for you.
3) This book does not read fast. You should read it 3 times slower than a normal book to really appreciate it. The math shold not just be understood, it should be questioned.
4) The book highlights theory behind game strategy, but does not connect the dots with real hands or real situations. It would be good to connect the check-call, check-raise, check-fold, bet-raise, bet-call, bet-fold, bluff, check-raise bluff, etc... thresholds with actual cards. What would be most cool is for software to perform this analysis, although I imagine only one-street analysis could be performed, but it would still be insightful.
5) Personally, I cannot recommend the first 40 pages of this book. They really didn't dig into the meat of the game and I found it quite mundane.

That said, here are the good things I can say about it:

1) It is nothing like you've ever read in any other poker book before! Many poker books overlap eachother, reminding pot odds, hand values, tournament phases, etc. This book dives into the fundamental theory. The interesting math of poker is not related with mundane matters of probabilities, pot odds, etc. The interesting math is the math behind bluffing, calling, and value-playing. BTW, there is a math essay by Chris Ferguson about game theory and poker.
2) It will remind you about why you bluff. One of the most practical lesson I learned from this math is that if you are bluffing optimally, YOU SHOULD BREAK EVEN ON YOUR BLUFFS! That was revolutionary for me. If you're winning on your bluffs, you're not bluffing enough. If you're losing, you're bluffing too much. If you break even, you get paid most on your value. This is not exclusively true, but becomes more true the more solid your opponent is. If your opponent is weak tight, then you should probably profit on your bluffs. Exploit appropriately.
3) Optimal play gives you your "center game", which you use before you know your opponents. When you adapt to exploit your opponents, be aware that you are opening holes in your own game to perform the exploit.
4) The material covered in this book is shore of an undiscovered land. It is only the beginning. Since the game appears unsolveable, there are riddles and puzzles at every corner. New insights can drive a stronger game. Who knows? You may have some clever insight beyond what the author discovered.


I hope he writes a sequel to this book. Material I would love for him to research for the sequel:

1) Preflop single-full-street play, but with real holdem. For a given bet-size some actual card thresholds would be given for bluff, check-fold, checkraise, bet-fold, bet-call, etc... Translate this basic game concept to card thresholds. Include the fact that hands only have equity, not some automatic ranking (like 0-1 game).
2) Actual single-street post-flop play for some example flops. Again, card thresholds would be great. Ideally, if some representation could be shown for card thresholds as a function of bet & raise sizes. Maybe a few pages of tables are required. There should be at least 10 distinct flop examples and this should probably consume more than 30% of the book.
3) Optimal exploit as a function of opponent's deviation from optimal play. Again, make it practical with card thresholds.
4) The math of Caution vs. Aggression. I know that the deeper the stacks are, the more that play should steer towards caution. At 30 blinds, top pair is a push-push-push hand. At stack=pot middle pair is an allin hand. At 200 blinds, suddenly top pair seems like it should be sometimes checked, because it's tough to fold later. My question is, how does caution show up in the math? And how does it balance with the common notion that Aggressive play is best? I know it's often better to bet-fold a medium hand, but definately sometimes it's smartest to check-call it, to make your opponent indifferent to bluffing.
5) The math suggests that you should be check-calling and bet-calling with some expected losers to make your opponent indifferent to bluffing. What is the real threshold for these check-calls? Are check-calls with 2nd pair smart? bottom pair? What is really the right threshold? How does this change with multiple "bullets"?
6) The math suggests you should only bluff your trash. But then in multi-street poker with draws, we put many of our bluffs on medium drawing hands. How do the partially made hands with draws fit in?
7) More analysis about mult-way pots. Try to solve the full street 3-way 0-1 game. In a multiway pot, which player will take the burden to bluff-call and make the opponent indifferent to bluffing?
8) Any deeper material which cannot be described absolutely with math can probably be backed only by simulation. The readers are pragmatic people (just trying to improve their game) and do not need a systematic analysis for everything.
9) Figure out every secret that Chris Ferguson knows and squeeze it in here! lol


I very much believe there needs to be a sequel to this book. A foundation was layed, but the dots were not completely connected together. It's kinda like a movie where you're left in the middle, waiting for the sequel. The theory needs to be grounded to some practice.

Bill Chen and Jerrod Ankenman, The Mathematics of Poker (ConJelCo, 2006)

I should start this review by saying I'm not a math guy. I never was. I failed calculus the first time and had to take it twice (I squeaked by with a C- the second time). Years as a horseplayer, though, made me understand that I was a stats guy, and that the math inherent to the stats was workable even for an English major like me. Then I started playing poker seriously. Probability? Kelly criterion? Game theory? Yeah, I had all that. Then I read The Mathematics of Poker. And there's my old nemesis... calculus.

Chen and Ankenman say in the intro that the book is geared towards laypeople, and that they try to keep the math to a minimum (they separate out the more complex proofs and the like for non-math-guys to skip over). In short, they don't succeed. They can't; in order for you to grasp concepts later in the book, you have to get the math earlier in the book. There's no way to keep it to a minimum, really. There might be a way to make it more palatable, though. I've read probably seven or eight books on horse racing for every poker book I've read. (I was a horseplayer for a decade before I started playing poker with real, honest-to-goodness money.) One thing many of the good ones have in common is that they err on the side of excess when it comes to examples. If there's tricky math involved, the author will take you through it with four or five examples. When you're reading a book on horse racing, sometimes it seems like overkill, and I know I've remarked on that in some reviews of horse books I've written. I am now reformed, and see the light. Had I had that many redundant examples here, I'd probably have gotten it. Theory is great and all, but it's fundamentally useless unless you can put it into practice. Which is the stated goal of the authors here. What's missing is the gateway between theory and practice those examples provide.

One other thing (and this, too, is addressed by the authors towards the end of the book)-- even if you don't get the math, unless you're Daniel Negreanu or someone who plays like he does, you're likely to look at Chen and Ankenman's conclusions and say "whoa, that's some seriously aggressive play." Academically, yes, there will be times when it's right to call a raise with a suited five-deuce. (For that matter, with three-deuce offsuit as well.) There will also be times when it's right to push all-in with it. Would anyone actually do it at the table? The authors say they've been accused of maniacal play, and I have to say that after reading this book, I can see why. So be prepared: if you plan to put the lessons this book teaches you into practice, you're probably going to find yourself well outside your comfort zone for a while. ***

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 am a professional poker player, and 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 important exceptions. As stellar as I believe the 2+2 books are, I feel that Mathematics of Poker (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 (Theory of Poker), 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.

I really didn't expect to like this book. Honestly, the only reason I bought it was because I like to read everything that my opponents possibly have read. I don't want them to have an edge simply because I slept on something. So I thought: "Fine, I'll read this book. But I'm not going to like it." Boy, was I wrong. This might be the most thoughtful and intelligent poker book that I have ever read.

I learned poker by using my instincts, not math. So I expected to disagree with a lot of what this book would present. On the contrary, the opposite took place. Not only was I agreeing with the authors, it also reinforced what I had figured out intuitively on my own. Instead of contradicting my play, it reinforced that what I was doing was usually correct.

I haven't taken a math class since high school pre-calc in 1998, and I did alright. I didn't understand ALL the math in these pages, but one doesn't need to. As long as you get the gist of what they're saying and can apply it to poker, which an intelligent person should be able to do, you'll be fine. If you find yourself lost without a map (which you probably will at some point in the book), take a break and come back to it.

Flipping through this at your local book store, you might be intimidated by all the charts, graphs and equations. Don't be. Just buy it, read it with an open mind, and watch your poker game soar.

I finished this book last week and was pretty amazed. I think, at least for non-mathematic experts like this reviewer, going through it a couple more times is the best way to make use of the author's endeavor. This book is not huge but its pages are swelled with information. It is broken down into five major parts; each of these support the central theme of maximizing average profit. By the second page of the Introduction--in which the common misconceptions of play are examined--readers will discern that there is no fluff in these 350+ pages. Parts II and III embody its intellectual core as they outline the mechanics of both exploitative and optimal play. Exploitative play is defined as maximizing expectation in lieu of your opponent's strategy; whereas, optimal play makes use of fundamentally sound strategies which are independent from your opponent's actions. While most players strive to be exploitative with their play, the better ones compete at a "near-optimal" level which is an evolutionary advancement over taking advantage of mistakes. Other than Roshambo [rock, paper, scissors] and the The Jam or Fold Game for no limit, many examples will not be familiar to the average person. A lack of familiarity is not a problem, however, because studying games like Clairvoyance, AKQ, Cops and Robbers, and Auction strengthen the mind and provide valuable perspective. Of course, novices should be forewarned to put off this purchase until they become fully grounded in the elementary facets of poker. This text does not address the majority of the decisions one makes at the table. In this way, Chen and Ankenman are more Plutarch than Sklansky by treating the mind as "a fire to be kindled, not a vessel to be filled."

Poker fans may be worried about the difficulty of the math presented, and whether or not the possession of serious quantitative skills mandatory for getting something out of it. Not surprisingly, the answer is, "It depends." Assuredly, most members of the book consuming poker public meet the author's criteria in this area, which is the completion of eighth grade algebra. Although, what Chen and Ankenman may forget is that many of us no longer remember most of what we learned during those dark days of middle school. Understanding the proofs so prevalent hinges on the retention of information that might have been long deleted from our memory banks. Furthermore, a rudimentary background in statistics is also necessary for apprehending the meaning behind the equations. Those with no knowledge of statistics and algebra will be slightly stunned by the extent of the quantitative detail on display. The math impaired might become slightly demoralized, but the good news is that some amazing ideas are presented above and below the ubiquitous expressions. The sections concerning bankrolls, backing agreements, and tournaments will be of value to everyone as will the chapters devoted to the Risk of Ruin model, the use of math to improve play, and a no limit hold `em case study used as the basis for justifying the precepts of game theory.

Yes, this book is quite challenging, but self-improvement is rarely accomplished via easy endeavor. It is important to recall that this text is not an end point. Mountain ranges worth of mathematical information remain in need of interpretation. The Mathematics of Poker is a thorough introduction, and there is little doubt that future works will build upon its foundation. Chen and Ankenman offer something here that is totally unique due to its avoidance of felt level tactics and its emphasis on strategy--which is its essential virtue.

Chris Ferguson states that if he ever teaches a poker related math class he'll use this book. That observation really fits what this book accomplishes. The book is very much like a text book. A fair amount of study is required to fully understand what's in this book. Most of the information is quite terse and, like other reviewers have mentioned, you get out of it what you put in. If you're looking for specific advice for specific poker situations then this isn't for you. If you do want to gain a deeper understanding of the maths behind poker, with some effort, this book will help with that.

This book is difficult for people without an advanced math background. It is a theoretical book that will evolve your thinking about the game. The content however is ground breaking as it is the only book that brings game theory into the game of poker.

I am a 2-4nl online regular (6-tabling) and have every poker book worth mentioning. This is a must have!!! Having a master in theoretical physics obviously helped me understand the notation but I don't think you need to understand all math to get something from reading this book. It has lots of good stuff that can be applied like, how fast does my win rate (BB/100) converge to 95% probability.

If you are interesting in game theory but not so familiar with the math, then I could recommend some of the less technical books on the subject (just search on game theory) and get a feeling for the subject. If you haven't already reached an advanced level in poker (or have a master in mathematics) I recommend that you first make sure you have read: Harrington, Sklansky, Supersystem II and Maybe Phil Gordon's green book.

I purchased this book based primarily on Chen's reputation. My first thought on opening it was that "this is math, not poker."

I had expected something on the order of King Yao's "Weighing the Odds in Hold'em Poker" on steroids. Instead I got a rigorous exploration of the mathematics of this fascinating game.

The authors point out quite rightly that one may get more insight into poker through exploration of what they call "toy games" than by tackling the monster head-on. This is much like the dilemma posed in performing quantum mechanical calculations - either solve an exact equation approximately or solve an approximate equation (toy game) exactly.

Some reviewers express disappointment in that they apparently wanted a book that would show them directly how to win at the poker table. Instead they got a book that will give them insight that will increase their probability of winning, provided they are willing to work through the math.

If you want the "smart pill" that produces instant understanding this book probably isn't for you. However, if you want to rigorously explore the mathematical underpinnings of the game I strongly recommend it.

I am by no means finished with this book yet, and I'm having to dust off a lot of mathematics I haven't used since college (a while ago), but I have already learned a great deal from this book.

I would encourage anyone who seriously wants a better understanding of how to play this complex game to take a look at this book, but be forewarned, it isn't a quick read. You will get out of this book in proportion to the amount of work you're willing to put into it.

Here is a quote from the book, "This is the general recursion that describes the relationship between successive values of Rp. To find specific values, we can use the fact that we know that Rp converges on r as P goes to infinity (since this is an approximation of the no-fold game as shown in Example 16.4). Hence, we can choose an arbitrarily large value of n, set it equal to square root of 2 - 1, and work backwards to the desired P. As it happens, this recursive relation converges quite rapidly. Of course, we can see that for the no-fold case, Rp simply is r for all thresholds, and this game simplifies to its no-fold analogue."

Of course! Duh, who didn't know that! If you enjoyed reading that, there are 375 more pages of this stuff. Oh, and in case you get bored with such elementary level stuff, there are sections throughout the book, where the authors have "marked off the start and end of some portions of the text so that our less mathematical readers can skip more complex derivitions." I can't even give you a quote from one of these sections because I don't have those keys on my keyboard...

I have been buying poker books for a while now and I finally found my limit. While I did slug my way through this book, I doubt it will make any difference in my game. I don't see how knowing that, quote, "skew instead sums the cubes of the distances, which cause the values to have sign," will improve my game. Whatever the heck that even means!

My advice, save your money and time, unless you want to put this book on your shelf and use it to intimidate your opponents. If one of them asks for advice you could give them this book, tell that it is a "must read" if they ever want to be a good poker player, and say that you LOVED IT! If they read it, they'll never look at you the same way again, I guarantee it. And they might just quit poker all-together. Like I said above, if THIS is poker, I might quit myself!
UPDATE Sep 4, 2008: I recently found a computer program that anyone who buys this book really should also seriously consider investing in. No serious poker player should be without a retro encabulator. For more info see this informational video: http://video.google.com/videoplay?docid=5125780462773187994&hl=en