The Mathematics of Poker
by Bill Chen, Jerrod Ankenman
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Book Reviews of The Mathematics of Poker
Book Review: A masterwork introduction to real world-class poker thought
Summary: 5 Stars
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.
Book Review: Challenging but Superb.
Summary: 5 Stars
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.
Book Review: Excellent Advanced Poker Book...
Summary: 5 Stars
I am well-read in poker literature and would recommend this book for advanced players. It will benefit cash game, tourney and SnG players alike. [If you are an improving intermediate player, but have a math or economics background, I think you would also find it helpful. If you have read less than 5 poker books or need to learn more fundamentals, you would likely get more value from reading other books first.] Many of the concepts presented apply broadly across different types of poker games.
This book exposes the true (and sometimes counter-intuitive) nature of common situations that arise in games of confrontation. The authors do an excellent job making game theory accessible to those without a math or econ/finance background, while not being afraid to ply into the underlying math for those interested. They successfully keep the topics focused on their relevance to poker, trimming out discussion that would be more mathematically complete, but bog down the poker learning.
They also do an excellent job of building up from fundamental concepts and examples (variable-constrained "toy" games) to more complex and applicable scenarios similar to those found in the mathematically complex world of poker. They gradually introduce new concepts and variables into the toy games as they come to resemble actual poker scenarios, with insights gained along the way. It helps solidify a meta-understanding of the game, and I suspect that even most advanced players will at least refine aspects of their play after reading. Note that it is, however, written in a heavy, take-your-time-digesting-it style, so it's not a quick breezy airport read. Some highlights:
+ The best poker book by far that I have encountered for game theory topics.
+ There are plenty of new ideas presented even for well-read players.
+ Delves elagantly into the nature of optimal play, exploitation and counter-exploitation. Learn to dominate others to extract value and how to adjust to prevent yourself from being dominated.
+ Presents concepts of risk, bankroll management, etc that I have yet to see elsewhere in poker literature.
+ Challenges some common conventions of most poker players and some authors.
+ Many concepts are digestable for most without a heavy math backgroud, though those gifted in math will likely take away a deeper understanding after reading.
- One aspect I would have liked to see more about is the Jam-or-fold scnearios. They do an excellent job of showing solid HEADS-UP strategy, but I would have liked to see this expanded to include 3+ player ICM-type scenarios. I would have also liked to see them quantify some of the weaknesses in ICM modeling, such as accounting for impending blinds, etc. In other words, a position- / equity- / blind-dependent jam-or-fold framework for a several player games would have been very helpful (even if it accepts ICM as-is, but especially if it refined it).
Book Review: Warning: a lot more advanced than the authors think it is.
Summary: 3 Stars
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. ***
Book Review: Mistargeted
Summary: 2 Stars
The two stars that I give the book is not, by any means, a way of rating the quality of the material within. It is meant to illustrate a different problem with the book - lack of any real target audience for it.
I am a rather mediocre poker player (who has read all of the basic poker theory books out there, but who has little practical experience and almost no time to improve on that) and a reasonably competent math user (having graduated in Computer Sciences). However, this book is of no interest to me in either aspect. For the poker player part of me, it is too detached, too much time is spent analyzing the theoretical aspects, too little intuitive data is given on how to actually improve your game (you might say that some of the game-theoretic results given within are something that cannot be inferred from the other poker books out there, but those pearls of wisdom could be easily compiled in a 10-page article). Maybe for a really world-class poker player, one who has mastered all the basics, can count all the relevant odds and take into account all the factors, this book might provide the pieces of intuition that might give him the competitive edge, but how many players like this are there?
For the math-user part of me, this book is too specific. I do not specialize in probability or game theory and even if I did, this text has actually little scientific value. The results contained within, from a scientific point of view, are not very interesting (which is why it's a 'popular' book and not a scientific publication). Most of this describes calculations that actually lead to nowhere because, as the authors themselves admit, poker is a too complicated game to solve by game-theoretical means with the current computing power. The few special cases which are actually broken down in the book are too specific to be actually used without modifications in real-game situations, which can be seen in the case of poker-playing bots, who cannot beat even a fair poker player in 1-1 situations in limit games. In what is currently probably the predominant game type on the internet, 10-seat no limit hold'em games, they are practically helpless. Human intuition quickly closes the gaps left by the increasing number of factors a computer program cannot cope with.
If you're the specific case of a math person who's recently done a course in probability, this subject interests you and you play poker competitively, this book is for you. If you're not, it's probably not. Hence the two stars: it represents the rough average of the usability of this book for the standard reader out there.