As the title suggests, Bill Chen and Jerrod Ankenman’s The Mathematics of Poker is by far the most mathematically rigorous poker book on the market. The mathematics to which the title refers is not the simple stuff, like calculating pot odds and counting outs; it’s the complex game theory that underlies even the most basic poker situations. Although the authors repeatedly insist that their objective is not “to solve game theory problems for the pure joy of doing so” but “to enhance our ability to win money at poker,” the bulk of the book consists of an academic examination of highly simplified “toy games” that isolate a particular aspect of poker, such as semi-bluffing or river bet sizing. The application of the solutions to actual poker situations is largely left to the reader, though a few chapters at the beginning and end offer some guidance.

Diligent readers who invest the necessary effort to follow Chen and Ankenman’s arguments and consider their implications, however, will not be disappointed. Careful study of this text will reveal a world of insight into poker concepts such as value betting, range balancing, and optimal strategy. Although more could have been done to elucidate its practical applications, The Mathematics of Poker is nevertheless an extremely valuable text for any poker player willing to give it the thoughtfulness it deserves.

It is not an easy read, but it should not be beyond the grasp of anyone with a high school education. Game theory is serious mathematics, and nearly every page of this book is packed with equations, charts, and graphs. This looks intimidating, but in fact, the authors do all the heavy lifting and help a diverse audience follow along in a variety of ways. For the real mathematicians, they show their work and occasionally suggest follow-up problems that readers might consider attacking on their own. However, they bracket these sections so that the mathematically challenged can skip past them to the (relatively) plain-language explanations of the process and results that follow. Chen and Ankenman do a remarkably good job of elucidating the conceptual meaning of equations and solutions, and every chapter concludes with a summary of the “Key Concepts.” Still, a passing acquaintance with statistical notation, graphical representation, and high school algebra is all but required to make sense of the text.

After some opening chapters that cover concepts like variance, sample sizes, hand reading, and pot odds, Chen and Ankenman introduce the concept of optimal strategy, a style of play that cannot be exploited even if your opponents knew ahead of time exactly how you would be playing. In other words, they are interested in finding solutions such as the exact ratio of bluffs to value bets that would make your opponent indifferent to calling or folding with a weak made hand on the river.

All commonly played versions of poker are far too complex to solve with the tools of game theory, however. Instead, poker must be attacked indirectly, through a series of “toy games” that represent highly simplified poker situations. One oft-revisited example involves two players each dealt one card from a three-card deck containing exactly one A, one K, and one Q. Throughout the book, the authors consider situations where the second player to act knows what card his opponent holds, situations where he does not have this information, situations where the first player is forced to check dark, and finally a full-street game where neither player knows the other’s card but may bet, check, or raise as he sees fit.

The text is very helpful in explaining the optimal strategy for each player in each game and how the addition of new strategic options affects these results. Still, the ultimate solution is nothing but the optimal strategy for a game that no one will ever play. Undoubtedly, understanding what optimal play entails and how it is derived can be enormously valuable at the poker table. But these games are accompanied by a few sentences, at best, explaining their relevance to actual poker situations.

Part of my frustration stems from the fact that the tidbits of practical advice that Chen and Ankenman do include are tantalizingly thought-provoking: “Bluffing in optimal poker play is often not a profitable play in and of itself. Instead, the combination of bluffing and value betting is designed to ensure the optimal strategy gains value no matter how the opponent responds”; “Betting preemptively is a perfect example of a play that has negative expectation, but the expectation of the play is less negative than the alternative of checking”; “it’s frankly terrible to find oneself in a situation with a marked open draw.” Any of these insights is worthy of several pages of extrapolation, but this work is largely left to the reader.

This is more than an error of omission. When Chen and Ankenman do directly address the question of how their material translates into poker strategy, they concern themselves overly much with pursuing optimal versus exploitive strategies, which is to say strategies that deviate from unexploitable, or optimal, play in order to capitalize on perceived weaknesses in an opponent’s strategy.

Their stated reasons for this preference are that it is difficult to determine another player’s strategy with certainty and that opponents will eventually change their play to counter your exploitation. While these are both reasonable concerns, neither is prohibitive. Although certainty is impossible, the ability to make quick and reasonably accurate assessments of a player’s strengths and weaknesses and the ability to adapt and re-adapt to him more quickly than he can do the same are skills from which a successful poker player derives his edge.

Whereas Chen and Ankenman advocate playing optimally against unknown opponents, I would argue that you often can and should assume and default to exploiting certain weaknesses until you see some evidence to the contrary. Balancing one’s river betting range in order to avoid exploitation by a check-raise bluff, for instance, is a poor default strategy because very few poker players are capable of such a tactic.

The real strength of The Mathematics of Poker, in my opinion, is not that it will help you to play a near-optimal strategy. Rather, it will help you to understand optimal strategy so that you can better recognize and exploit your opponents’ inevitable deviations from it. For example, one toy game illustrates how a player in position must value bet and bluff fewer hands when his opponent is allowed to check-raise than in a game where he is not. The lesson I take from this is that against an opponent who rarely check-raises the river, I should value bet and bluff more often than optimal strategy would suggest.

But ultimately, these are shortcomings, not flaws. The mathematics are there to illustrate the game theory that underlies poker. Even with the supplemental explanations and synopses, The Mathematics of Poker is a demanding read. It asks a lot of the reader both in following the arguments and in making the jump from toy games to real life poker. Those who invest the requisite time and energy, however, will be rewarded with a deeper understanding of how to exploit their opponents and how to avoid such exploitation themselves.

Great review

I think i need the Cliff Notes version of this book.

Thank you for writing all of that out. It sounds like a book to read when midterms are over... Or hell, a required text for a class that the math dept should hold and give midterms of its own.

this is pretty funny.

Wow This is what i needed. 2000 FPPs well spent. Book is on its way

This concept sounds awfully familiar, where have I seen this explored recently?

Thanks for the review.

Have to say, though, I agree with Beavis. I think you're seriously understating the difficulty of the maths. IMO there are plenty of people with a "high school education" who will find this book tough going.

Is this sarcasm? At the risk of looking foolish, Sklansky had an article tangentially about this in last month's magazine, and I started a thread about it in the Mag forum.

Right, but the point is that you don't have to follow most of the math. Of course you'll get more from the book if you can, but to follow along with the general drift of ideas requires only the ability to make sense of sentences that include mathematical notation and to interpret some simple graphs.

Yeah, it was semi-sarcastic. Just surprised you didn't tie the two together since you had commented on DS's article.

The full value of this book is understanding the math and deriving important concepts from it. If the point was to only get the drift, there would be no reason to include the math. It is like TOP, it is a valuable read without understanding the math, but the derivative concepts are more important than the face value.

First off, this was an excellent review. Thank you for the thought and time it took to put this review together!! I do have a bone to pick though. With everything your review said, I agree with almost everything, however, you made a serious error when saying this:

No, this book requires more than a general high school education to have an understanding of all the theories that presented within this book. I have a high school degree and couple years of college, too. I am also very mathematically and logically minded, by nature. I understand that you want to make this book a reachable goal, but saying that a high school degree is all that is required is a terrible understatement. Please correct what you have said, as it is not accurate at all.

The rest of your review is excellent. I have learned tons from this book. This book has served me as something that has made me take the time to learn mathematics even when out of school. Great book but high school math is only going to work if you were advanced math in certain schools. The majority is going to need college Probability and Statistics courses.

Anyways, this was one of the best reviews I have seen. I hope you continue to post your thoughts, because you seem to have put a great deal of thought into your review, and overall I was way, way more pleased than displeased.

Fair enough, cato, you're not the first to take me to task for this. I certainly didn't mean to imply that a high school education would be sufficient to follow all of the math. But it seems I'm underestimating how worthwhile the book would be if you didn't follow any of the math.

I also overstated the value of a high school education in the US. I really ought to know better, having worked in the Chicago Public Schools, but there are plenty of high school graduates who couldn't follow much simpler poker books. Not sure what I was thinking, there.

Foucault, I can't speak about the US, but you sure as heck overstated the value of a 'high school' education here in the UK. (And I speak as someone who, like you, has worked in schools).

But hey, that's what a forum is for - exchanging ideas and opinions.

Still a very good review IMO.

Nice review, but as people have already stated there's quite a lot of higher level math in this, certainly enough that a high school level of education won't really suffice in places. Heck, it's been a few years since I finished my BSc in it but there's stuff that has me confused