The Mathematics of Poker

by Bill Chen, Jerrod Ankenman

Thirty years ago the bond and option markets were dominated by traders who had learned their craft by experience. By the mid-1990s the old school grizzled traders had been replaced by a new breed of quantitative analysts, applying mathematics to the "art" of trading and making of it a science. A similar phenomenon is happening in poker. The grizzled "road gamblers" are being replaced by a new generation of players who have challenged many of the assumptions that underlie traditional approaches to the game. One of the most important features of this new approach is a reliance on quantitative analysis and the application of mathematics to the game. This book provides an introduction to quantitative techniques as applied to poker and to a branch of mathematics that is particularly applicable to poker, game theory, in a manner that makes seemingly difficult topics accessible to players without a strong mathematical background.

Genres: Mathematics, Nonfiction, Games, Science, Finance, Sports, Reference

392 pages, Paperback

First published November 30, 2006

Reviews

[Caroline · Rating 5 out of 5]

When I first picked up this book, I wasn’t sure if I would enjoy it. The introduction hit me with a couple sentences that gave me pause. First: “If you have never played poker before, 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.” I have never played poker, and nor am I about to start. I wondered if I wouldn’t be able to follow the book. It was rough going for a while, since the book did not include even a basic overview of the rules of poker, nor was their a glossary for the substantial amount of poker jargon used. Poker tutorials online were also surprisingly bad. For example, one hold’em tutorial for beginners only said that rounds consist of checking, betting, and raising, without explaining how those actions may be used. I also learned that for any conceivable rules question in poker, the answer is “it depends on the variant.” In the end, I managed via a mix of googling and asking my husband questions, since he has played poker.

The second concerning sentence: “The primary goal of our work here is not to solve game theory problems for the pure joy of doing so; it is to enhance our ability to win money at poker.” Since my interests are the exact opposite, I thought maybe the book would not be for me. However, the approach was very careful and precise, exploring simplifications where optimal strategy could be derived completely. Some of the chapters got a bit too detailed for me to be interested in following, given that I'm not planning to apply them, but overall I found an incredible amount of joy in this book.

The biggest insight I got from this book is how fundamentally defensive Nash equilibria are. For example, consider a game of Rock Paper Scissors where the utility of winning with scissors is doubled. That is, if you win with scissors, you gain 2 points and your opponent loses 2 points. If you win with rock or paper, you gain 1 point and your opponent loses 1 point. If there is a tie, no one gains any points. I didn't pause to predict the result before reading on, but I might have expected the result would be that scissors is played more often. Instead, the Nash equilibrium is playing rock with probability 1/2, scissors with 1/4, and paper with 1/4. Why did rock increase? Because rock is the counter to scissors. The Nash equilibrium strategy usually makes your opponent indifferent between their options. To make the opponent indifferent to playing scissors when its utility increases, you have to play rock more often. At the same time, if the opponent is also playing the Nash equilibrium, you cannot gain utility by playing scissors more often. Your opponent has made you indifferent.

This concept applies to poker as well. For example, suppose you are playing limit poker, meaning there is one fixed bet size that is allowed. As the size of the pot increases, how should bluffing and calling behavior change? The answer is that you should bluff less and call more. This example was discussed in the book before the Rock Paper Scissors example, and I had a lot of trouble seeing why. After all, the utility of bluffing and calling both increase as the pot increases, so why should you do one less and the other more? The answer is that this is correct defensive play. The counter to bluffing is calling, so as bluffing gets better you call more. The counter to calling is bluffing less, so as calling gets better you bluff less.

Some other highlights for me: Chapter 7 discussed games with a made hand (who has the best hand currently on the board, e.g. a pair of aces) and a draw (who is trying to draw into a better hand, e.g. they currently have 4 card of the same suit and would like a flush). The draw often prefers to get all-in before more cards are drawn, going against my intuition that the draw would like to see more cards as cheaply as possible. This is because the made hand wants to preserve the option to make large bets in cases where the draw misses. Chapters 22-25 had some interesting analysis of how to play poker professionally without running too great a risk of losing all your money at some point. And Chapter 29 had some strange scenarios that can occur in 3-player games.

Finally, one big takeaway from this book for me was just how high a variance there is to poker. Whenever the authors put in numbers they thought were realistic assumptions as to how much of an edge a player might have, I was shocked by how easy it was to lose money despite being the best player. They gave one example of an overall winning player who nonetheless had a 45% chance of being down money after 300 hands! If I had any inclination to play poker before reading this, it completely vanished. It just doesn't sound fun to play a game where you need statistical analysis to know if you're doing well.

[Bryce Johnson · Rating 5 out of 5]

If you happen to be a person that

1) does not cry when you see a bunch of equations (and in fact do not mind writing a few more equations to find the answers you really want)
2) can overlook some ugly formatting errors
3) plays poker and would like to know how to approach some situations in a game theoretically optimal fashion

then this will be an extremely enlightening book for you. It was for me. If you just want charts of which hands to play in hold 'em, this will be less useful.

It's not like you need to know what a Radon-Nikodym derivative is (though it couldn't hurt), but this is a math book first.

[Ben Phillips · Rating 3 out of 5]

Practically speaking, poker) is too complex of a game for humans to analyze completely using math (it will eventually be done using computers). While this book provides detailed mathematically analyses of many poker "mini-games" the onus is on the reader to figure out how to apply that to his/her strategy in "real" games. Unfortunately, it is far from obvious as to how to go about doing this which probably explains why this issue is hardly addressed in the book.

[Gabriel Osborne · Rating 3 out of 5]

The math is solid, but the writing is fairly hard to decipher. A classic for poker theory, but maybe not the best read for most players.

[Kelsey · Rating 3 out of 5]

I stopped reading after chapter 4 because as a beginner poker player, the first few chapters were sufficient to give me a basic understanding of how to calculate expected value for hands and how to evaluate pot odds, implied odds, and the like. This book is not the easiest to comprehend, though I'll admit I'm also not used to reading texts about math.

I might return to this book after I've developed an informal understanding of basic strategies and therefore feel more ready to delve into the more heavy-duty mathematical analysis in later sections.

[Alexandra Chauran · Rating 5 out of 5]

This book was a treasure found in a thrift store. Don't be fooled, though, when it says on the cover that you don't need a mathematical background in order to understand this book, it lies. I have an education that includes advanced statistics and calculus, and I still had gaping holes in my knowledge of game parametrization. If you're not that much of a math wiz, I'd recommend starting with Practical Poker Math: Basic Odds & Probabilities for Hold'em & Omaha for background, and if you're pretty math savvy move on to Poker Math That Matters Simplifying The Secrets Of No Limit Hold'em for the essentials. Having read those two books I totally enjoyed diving into this one. I would describe it as dry, but not too dense. The nuggets of GTO wisdom were hidden in between plenty of proofs that finally explained to me why poker players do a lot of things that they do. Ever wondered why no limit players often bet a third or half pot instead of some random arbitrary number every time? This book will explain it to you.

[Corwin · Rating 4 out of 5]

Poker bug, super involved analysis on the math behind poker. Optimal strategies, balance, aggression, odds, EV all calculated out to a tee. Enjoyed working through the toy games to find the Nash equilibria and reasoning through strategically parameterizing certain hands. Some parts had too much math though, for example bankroll management and tournament chip mapping took up too much space. I appreciated the thorough analysis of preflop, flop, turn, and river play though(merit to many options and strategies)

[John · Rating 5 out of 5]

This isn't really a strategy book, but rather sets a standard for those interested in the math behind poker.

[A. Gazes · Rating 4 out of 5]

Forget relying on "feel" or reading tells; The Mathematics of Poker by Chen and Ankenman is where the real advantage comes from. This book is a deep, rigorous dive into the fundamentals of poker theory, teaching you how to build robust, mathematically sound strategies based on game theory optimal (GTO) principles. It’s dense, yes, but it is the definitive guide to understanding variance, pot odds, implied odds, and expected value (EV) on a level your opponents simply won't reach.

The best part? Once you internalize these concepts, the game changes forever. I can walk into my poker club and beat the table without even having to be in the mood. My game is so fundamentally sound thanks to this text that I can practically autopilot to victory.

Watch me—this book is the reason why. If you're serious about mastering poker, not just playing it, this is mandatory reading.

[Mark Albrecht · Rating 5 out of 5]

Hard stuff. Revolutionized the game and introduced game theory to a broader audience

[Shokhjakhon Giyosov · Rating 5 out of 5]

Good book

[Nathan Mannes]

DNF

[Jon · Rating 4 out of 5]

Classic introduction to analysing the game.

[42 · Rating 5 out of 5]

"Oh My God This Surely Is The Work Of Satan"
For math geeks who love poker and first-principles sort of thinking.

[CV Rick · Rating 5 out of 5]

This is a fantastic poker book, it's a great math book, and it's a fabulous game theory book. I wish this were a textbook for a graduate course because that's the way it reads. I'm going to be going over the work in this volume again and again in order to understand the concepts. By far the best examination of poker I've encountered.

[Christoph Weber · Rating 5 out of 5]

http://www.weberseite.at/buecher/math...

[Richard Herbert · Rating 4 out of 5]

SERIOUS math poker read.

[Robin Spano]

Interesting math. Made me think. More complex than you need at the table, but I like that they back up their results with logical proof. A good read for a learning player.

[Lindsay · Rating 2 out of 5]

Very dry and very technical. Just boring, frankly, though I suspect I'm not the target audience.