The Mathematics of Poker

by Bill Chen and Jerrod Ankenman

Like many poker players, old and new, I was attracted to poker primarily by the mythos of the steely eyed staredown and the battle of wits and will that make for a game anyone can play, but only the toughest can win. This is the way poker has been played by everyone from Doc Holliday to Doyle Brunson, and for some it is the way poker should always be played. Bill Chen and Jerrod Ankenman would beg to differ. In The Mathematics of Poker the authors present a vision of math based reasoning that is not only elegant and accessible to the average reader, but is the wave of the future for the poker world.

A word of warning: The Mathematics of Poker reads like a math textbook. If you flip through it you will see pages filled with formulas, diagrams, graphs, charts and more formulas. There will be plenty of Greek letters, x’s, y’s, and p’s, not to mention a few I don’t even know how to type. If this sort of thing makes you run screaming, do not buy this book. If you think you have the stomach for it, or at least the patience to give it an honest try I can’t recommend it enough.

For some, The Mathematics of Poker will not be a groundbreaking book, since they have been onboard with Math driven analysis for a while now. Many of the best and brightest in online poker are deeply grounded in Game Theory and mathematical reasoning. For the rest of us, we may have been vaguely aware of the many of the concepts but the math either scares us, or we are too lazy to devote the time to learning what is going on in the Left Brain of the poker mind. It is long past time to try to catch up to these largely self-taught math wizards and tap into some of that magic for ourselves. Thankfully Ankenman and Chen have provided a volume that makes that possible.

In the past several years, it is fair to say that the market for Poker literature has experienced a glut of supply. Nearly every TV poker personality has his or her own Play Poker Like the Pros, or Making the Final table. Some of these are good, some of these are great, and many of them are indifferent or worse. Unlike the majority of the books on the market today, Chen and Ankenman have specifically geared this book towards the intermediate and up player. They do not waste your time with basic concepts, and specifically say to the beginner, “the best course of action is to put this book down, read some of the other books in print aimed at beginners, play some poker, learn some more, and then return after gaining additional experience.”

Aside from that proviso, the authors do intend this book for a wide audience. The math is accessible, but not easy or dumbed-down. You will have to exercise some brain power to get through this book, and then, if you’re smart you’ll turn around and read it again. I have to admit on my first time through the book I feel dumber, but not for having read it, more so that I haven’t spent more time thinking along these lines before. I have a decent math background including Calculus in high school, but very little math education or study since then and little to no statistics background, so the wheels are a bit rusty when it comes to this stuff.

The Mathematics of Poker is organized into five main parts. After a well written introduction, Part I introduces the basic concepts that the rest of the book will be built on. If you do not have a background in statistics I highly recommend you spend some time in this chapter really absorbing the terms and coming to understand how Chen and Ankenman talk about math, because it will really pay off as you proceed through the book.

Part II is about exploitive play which takes the conversation from general math concepts and applies them to the game of poker. This section focuses on the many different situations of playing your hand versus a range of opponents’ hands and making profitable plays. It then goes a step further and encourages you to think outside the box of your two cards and play a range of possible hands against your opponents’ range, which, in the opinion of the authors, “leads to truly strong play.”

Part III is the heart and soul of the book. Focused on Optimal Play, chapter three can be likened to the search for a Unified Theory in physics. Ankenman has a very strong interest in Backgammon, which with computer modeling is coming closer and closer to being “cracked”, establishing an optimal way to play the game. With the greatly increased human element in poker, no to mention the multi-player aspect this is simply not possible given modern computing power. This is a win for the Luddites and poker traditionalists out there who have nightmares of a game dominated by robotic math geniuses.

Ankenman and Chen have no such fear, in fact they stress that the level of creativity possible in poker play increases with mathematical analysis. They are not out to rob poker of its artistry, but they are out to remove some of the subjectivity. “It depends,” while very often the only reasonable answer to a poker question, is the enemy of reasoned analysis and comparative discussions. The greatest contribution mathematical concepts lend to the discussion of poker is a reliable and objective way to discuss hands. You no longer have to rely on a “better player” to tell you the “right way to play” a certain hand. Instead you can create a mathematical model to fully analyze the hand and compare notes with another player based not on opinion, but some measure of science.

Part IV provides an in depth discussion of bankroll management from a mathematical perspective. The authors discuss the affects of variance and the need to protect against the “risk of ruin”, which is of course the chance of going busto for a player with an otherwise positive expectation. Part V is the catch all, Other Topics, which may be of the most interest to the tournament specialist as it focuses on calculating equity and making accurate decisions in tournament play.

For those who are afraid of math, or are just not “mathematically inclined” I cannot recommend this book to you, because it will very likely be a waste of money. Not because you won’t or can’t get it, but because you will most likely let the book gather dust on your shelf, unread. If you are ready to take the next step in poker and add a whole host of tools to your poker toolbox, then The Mathematics of Poker is for you. Even if you already have a an advanced understanding of the math behind decisions at the poker table this book can be a great resource to compare notes, refresh concepts or go a little further in depth in areas you haven’t studied. For myself, as soon as I post this article, I’m going to take the book, go outside and return to the sections that made me particularly dizzy.