Acknowledgments……………………………………………………………………… 5

Foreword…………………………………………………………………………………. 6
Introduction……………………………………………………………………………… 8
Part I: Basics
Chapter 1 Decisions Under Risk: Probability and Expectation……………………… 15
Chapter 2 Predicting the Future: Variance and Sample Outcomes………………….. 24
Chapter 3 Using All the Information: Estimating Parameters and Bayes' Theorem… 34
Part II: Exploitive Play
Chapter 4 Playing the Odds: Pot Odds and Implied Odds…………………………… 47
Chapter 5 Scientific Tarot: Reading Hands and Strategies…………………………… 59
Chapter 6 The Tells are in the Data: Topics in Online Poker……………………….. 70
Chapter 7 Playing Accurately, Part I: Cards Exposed Situations……………………. 74
Chapter 8 Playing Accurately, Part II: Hand vs. Distribution……………………….. 85
Chapter 9 Adaptive Play: Distribution vs. Distribution……………………………… 94
Part III: Optimal Play
Chapter 10 Facing The Nemesis: Game Theory………………………………………. 100
Chapter 11 One Side of the Street: Half-Street Games……………………………….. 110
Chapter 12 Headsup With High Blinds: The Jam-or-Fold Game……………………… 121
Chapter 13 Poker Made Simple: The AKQ Game…………………………………….. 138
Chapter 14 You Don't Have To Guess: No-Limit Bet Sizing…………………………. 146
Chapter 15 Player X Strikes Back: Full-Street Games………………………………... 155
Appendix to Chapter 15 The No-Limit AKQ Game……………………………… 167
Chapter 16 Small Bets, Big Pots: No-Fold [0,1] Games……………………………… 174
Appendix to Chapter 16 Solving the Difference Equations………………………. 191
Chapter 17 Mixing in Bluffs: Finite Pot [0,1] Games………………………………… 194
Chapter 18 Lessons and Values: The [0,1]Game Redux………………………………. 212
Chapter 19 The Road to Poker: Static Multi-Street Games……………………………. 228
Chapter 20 Drawing Out: Non-Static Multi-Street Games…………………………….. 243
Chapter 21 A Case Study: Using Game Theory……………………………………….. 259
Part IV: Risk
Chapter 22 Staying in Action: Risk of Ruin…………………………………………… 274
Chapter 23 Adding Uncertainty: Risk of Ruin with Uncertain Win Rates……………. 288
Chapter 24 Growing Bankrolls: The Kelly Criterion and Rational Game Selection….. 297
Chapter 25 Poker Finance: Portfolio Theory and Backing Agreements………………. 303
Part V: Other topics
Chapter 26 Doubling Up: Tournaments, Part I………………………………………… 312
Chapter 27 Chips Aren't Cash: Tournaments, Part II………………………………….. 324
Chapter 28 Poker's Still Poker: Tournaments, Part III………………………………… 338
Chapter 29 Three's a Crowd: Multiplayer Games……………………………………... 349
Chapter 30 Putting It All Together: Using Math to Improve Play……………………. 359

Recommended Reading………………………………………………………………... 365

About the Authors………………………………………………………………………. 366
About the Publisher…...................................................................................................... 367

_________________________________________________________________________
THE MATHEMATICS OF POKER 4
Acknowledgments

A book like this is rarely the work of simply the authors; many people have assisted us along the
way, both in understanding poker and in the specific task of writing down many of the ideas that
wc have developed over the last few years. A book called The Mathematics of' Poker was
conceived by Bill, Chuck Weinstock, and Andrew Latto several years ago, before Jerrod and Bill
had even met. That book was set to be a fairly formal, textbook-like approach to discussing the
mathematical aspects of the game. A few years later, Jerrod and Bill had begun to collaborate on
solving some poker games and the idea resurfaced as a book that was less like a mathematics
paper and more accessible to readers with a modest mathematical background.
Our deepest thanks to those who read the manuscript and provided valuable feedback. Most
notable among those were Andrew Bloch, Andrew Prock, and Andrew Latto, who scoured
sections in detail, providing criticisms and suggestions for improvement. Andrew Prock's
PokerStove tool (http://www.pokerstove.com) was quite valuable in performing many of the
equity calculations. Others who read the manuscript and provided useful feedback were Paul R.
Pudaite and Michael Maurer.
Jeff Yass at Susquehanna International Group (http://www.sig.com) has been such a generous
employer in allowing Bill to work on poker, play the World Series, and so on. We thank Doug
Costa, Dan Loeb,Jay Siplestien, and Alexei Dvoretskii at SIG for their helpful comments.
We have learned a great deal from a myriad of conversations with various people in the poker
community. Our friend and partner Matt Hawrilenko, and former WSOP champion Chris
Ferguson are two individuals especially worth singling out for their insight and knowledge. Both
of us participate enthusiastically in the *ARG community, and discussions with members of that
community, too, have been enlightening, particularly round table discussions with players
including Sabyl Cohen, Jack Mahalingam, JP Massar, Path Beadles, Steve Landrum. and (1999
Tournament of Champions winner) Spencer Sun.
Sarah Jennings, our editor, improved the book significantly in a short period of rime by being
willing to slog through equations and complain about the altogether too frequent skipped steps.
Chuck Weinstock at Conjelco has been amazingly patient with all the delays and extended
deadlines that come from working with authors for whom writing is a secondary occupation.
We also appreciate the comments and encouragement from Bill's father Dr. An-Ban Chen who
has written his own book and has numerous publications in his field of solid state physics.
Jerrod's mother Judy has been a constant source of support for all endeavors, even crazy-
sounding ones like playing poker for a living.
Throughout the writing of this book, as in the rest of life, Michelle Lancaster has constantly been
wonderful and supportive of Jerrod. This book could not have been completed without her.
Patricia Walters has also provided Bill with support and ample encouragement, and it is of no
coincidence that the book was completed during the three years she has known Bill.

_________________________________________________________________________
THE MATHEMATICS OF POKER 5
Foreword

Don’t believe a word I say.

It’s not that I'm lying when I tell you that this is an important book. I don't even lie at the poker
table -- not much, anyway - so why would I lie about a book I didn't even write?

It’s just that you can't trust me to be objective. I liked this book before I'd even seen a single
page. I liked it when it was just a series of conversations between Bill, myself, and a handful of
other math geeks. And if I hadn't made up my mind before I'd read it, I'm pretty sure they’d have
won me over with the first sentence.
Don’t worry, though. You don't have to trust me. Math doesn't lie. And results don't lie, either. In
the 2006 WSOP, the authors finished in the money seven times, including Jerrod's second place
finish in Limit Holdem, and Bill's two wins in Limit and Short Handed No Limit Hold'em.

Most poker books get people talking. The best books make some people say, “How could anyone
publish our carefully guarded secrets?" Other times, you see stuff that looks fishy enough to
make you wonder if the author wasn't deliberately giving out bad advice. I think this book will
get people talking, too, but it won't be the usual sort of speculation. No one is going to argue that
Bill and Jerrod don't know their math.

The argument will be about whether or not the math is important.

People like to talk about poker as "any man's game." Accountants and lawyers, students and
housewives can all compete at the same level - all you need is a buy-in, some basic math and
good intuition and you, too, can get to the final table of the World Series of Poker. That notion is
especially appealing to lazy people who don't want to have to spend years working at something
to achieve success. It's true in the most literal sense that anyone can win, but with some well-
invested effort, you can tip the scales considerably in your favor.

The math in here isn't easy. You don't need a PhD in game theory to understand the concepts in
this book, but it's not as simple as memorizing starting hands or calculating the likelihood of
making your flush on the river. There's some work involved. The people who want to believe
intuition is enough aren't going to read this book. But the people who make the effort will be
playing with a definite edge. In fact, much of my poker success is the result of using some of the
most basic concepts addressed in this book.

Bill and Jerrod have saved you a lot of time. They've saved me a lot of a time, too. I get asked a
lot of poker questions, and most are pretty easy to answer. But I've never had a good response
when someone asks me to recommend a book for understanding game theory as it relates to
poker. I usually end up explaining that there are good poker books and good game theory books,
but no book addresses the relationship between the two.

Now I have an answer. And if I ever find myself teaching a poker class for the mathematics
department at UCLA, this will be the only book on the syllabus.

Chris “Jesus" Ferguson

Champion, 2000 World Series of Poker
November 2006

_________________________________________________________________________
THE MATHEMATICS OF POKER 6
Introduction
“If you think the math isn't
important, you don't know
the right math.”
Chris "Jesus" Ferguson, 2000 World Series of Poker champion

_________________________________________________________________________
THE MATHEMATICS OF POKER 7
Introduction

In the late 1970s and early 1980s, the bond and option markets were dominated by traders who
had learned their craft by experience. They believed that their experience and intuition for
trading were a renewable edge; that is, that they could make money just as they always had by
continuing to trade as they always had. By the mid-1990s, a revolution in trading had occurred;
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.

If the latest backgammon programs, based on neural net technology and mathematical analysis
had played in a tournament in the late 1970s, their play would have been mocked as
overaggressive and weak by the experts of the time. Today, computer analyses are considered to
be the final word on backgammon play by the world's strongest players - and the game is
fundamentally changed for it.
And for decades, the highest levels of poker have been dominated by players who have learned
the game by playing it, "road gamblers" who have cultivated intuition for the game and are adept
at reading other players' hands from betting patterns and physical tells. Over the last five to ten
years, a whole new breed of player has risen to prominence within the poker community.
Applying the tools of computer science and mathematics to poker and sharing information across
the Internet, these players have challenged many of the assumptions that underlie traditional
approaches to the game. One of the most important features of this new approach to the game is
a reliance on quantitative analysis and the application of mathematics to the game. Our intent in
this book is to provide an introduction to quantitative techniques as applied to poker and to the
application of game theory, a branch of mathematics, to poker.

Any player who plays poker is using some model, no matter what methods he uses to inform it.
Even if a player is not consciously using mathematics, a model of the situation is implicit in his
decisions; that is, when he calls, raises, or folds, he is making a statement about the relative
values of those actions. By preferring one action over another, he articulates his belief that one
action is better than another in a particular situation. Mathematics are a particularly appropriate
tool for making decisions based on information. Rejecting mathematics as a tool for playing
poker puts one's decision-making at the mercy of guesswork.

Common Misconceptions
We frequently encounter players who dismiss a mathematical approach out of hand, often based
on their misconceptions about what this approach is all about. We list a few of these here; these
are ideas that we have heard spoken, even by fairly knowledgeable players. For each of these, we
provide a brief rebuttal here; throughout this book, we will attempt to present additional
refutation through our analysis.

1) By analyzing what has happened in the past - our opponents, their tendencies,
and so on-we can obtain a permanent and recurring edge.
This misconception is insidious because it seems very reasonable; in fact, we can gain an edge
over our opponents by knowing their strategies and exploiting them. But this edge can be only
temporary; our opponents, even some of the ones we think play poorly, adapt and evolve by
reducing the quantity and magnitude of clear errors they make and by attempting to counter-
exploit us. We have christened this first misconception the "PlayStation™ theory of poker" - that
the poker world is full of players who play the same fixed strategy, and the goal of playing poker
is to simply maximize profit against the fixed strategies of our opponents. In fact, our opponents'
strategies are dynamic, and so we must be dynamic; no edge that we have is necessarily
permanent.

_________________________________________________________________________
THE MATHEMATICS OF POKER 8
2) Mathematical play is predictable and lacks creativity.
In some sense this is true; that is, if a player were to play the optimal strategy to a game, his
strategy would be "predictable" - but there would be nothing at all that could be done with this
information. In the latter parts of the book, we will introduce the concept of balance this is the
idea that each action sequence contains a mixture of hands that prevents the opponent from
exploiting the strategy. Optimal play incorporates a precisely calibrated mixture of bluffs, semi-
bluffs, and value bets that make it appear entirely unpredictable. "Predictable" connotes
"exploitable," but this is not necessarily true. If a player has aces every time he raises, this is
predictable and exploitable. However, if a player always raises when he holds aces, this is not
necessarily exploitable as long as he also raises with some other hands. The opponent is not able
to exploit sequences that contain other actions because it is unknown if the player holds aces.

3) Math is not always applicable; sometimes "the numbers go out the window."
This misconception is related to the idea that for any situation, there is only one mathematically
correct play; players assume that even playing exploitively, there is a correct mathematical play -
but that they have a "read" which causes them to prefer a different play. But this is simply a
narrow definition of "mathematical play" - incorporating new information into our understanding
of our opponent's distribution and utilizing that information to play more accurately is the major
subject of Part II. In fact, mathematics contains tools (notably Bayes' theorem) that allow us to
precisely quantify the degree to which new information impacts our thinking; in fact, playing
mathematically is more accurate as far as incorporating "reads" than playing by "feel."

4) Optimal play is an intractable problem for real-life poker games; hence, we

should simply play exploitively.
This is an important idea. It is true that we currently lack the computing power to solve headsup
holdem or other games of similar complexity. (We will discuss what it means to "solve" a game
in Part III). We have methods that are known to find the answer, but they will not run on modern
computers in any reasonable amount of time. "Optimal" play does not even exist for multiplayer
games, as we shall see. But this does not prevent us from doing two things: attempting to create
strategies which share many of the same properties as optimal strategies and thereby play in a
"near-optimal" fashion; and also to evaluate candidate strategies and find out how far away from
optimal they are by maximally exploiting them.

5) When playing [online, in a tournament, in high limit games, in low limit

games...], you have to change your strategy completely to win.
This misconception is part of a broader misunderstanding of the idea of a "strategy" - it is in fact
true that in some of these situations, you must take different actions, particularly exploitively, in
order to have success. But this is not because the games are fundamentally different; it is because
the other players play differently and so your responses to their play take different forms.
Consider for a moment a simple example. Suppose you arc dealt A9s on the button in a full ring
holdem game. In a small-stakes limit holdem game, six players might limp to you, and you
should raise. In a high limit game, it might be raised from middle position, and you would fold.
In a tournament, it might be folded to you, and you would raise. These are entirely different
actions, but the broader strategy is the same in all - choose the most profitable action.

Throughout this book, we will discuss a wide variety of poker topics, but overall, our ideas could
be distilled to one simple piece of play advice: Maximize average profit. This idea is at the heart
of all our strategies, and this is the one thing that doesn't change from game condition to game
condition.

_________________________________________________________________________
THE MATHEMATICS OF POKER 9
Psychological Aspects
Poker authors, when faced with a difficult question, are fond of falling back on the old standby,
"'It depends." - on the opponents, on one's 'read', and so on. And it is surely true that the most
profitable action in many poker situations does in fact depend on one's sense, whether intuitive
or mathematical, of what the opponent holds (or what he can hold). But one thing that is often
missing from the qualitative reasoning that accompanies "It depends," is a real answer or a
methodology for arriving at an action. In reality, the answer does in fact depend on our
assumptions, and the tendencies and tells of our opponents are certainly something about which
reasonable people can disagree. But once we have characterized their play into assumptions, the
methods of mathematics take over and intuition fails as a guide to proper play.

Some may take our assertion that quantitative reasoning surpasses intuition as a guide to play as
a claim that the psychological aspects of poker are without value. But we do not hold this view.
The psychology of poker can be an absolutely invaluable tool for exploitive play, and the
assumptions that drive the answers that our mathematical models can generate are often strongly
psychological in nature. The methods by which we utilize the information that our intuition or
people-reading skills give us is our concern here. In addition, we devote time to the question of
what we ought to do when we are unable to obtain such information, and also in exposing some
of the poor assumptions that often undermine the information-gathering efforts of intuition. With
that said, we will generally, excepting a few specific sections, ignore physical tells and opponent
profiling as being beyond the scope of this book and more adequately covered by other writers,
particularly in the work of Mike Garo.

About This Book

We are practical people - we generally do not study poker for the intellectual challenge, although
it turns out that there is a substantial amount of complexity and interest to the game. We study
poker with mathematics because by doing so, we make more money. As a result, we are very
focused on the practical application of our work, rather than on generating proofs or covering
esoteric, improbable cases. This is not a mathematics textbook, but a primer on the application of
mathematical techniques to poker and in how to turn the insights gained into increased profit at
the table.
Certainly, there are mathematical techniques that can be applied to poker that are difficult and
complex. But we believe that most of the mathematics of poker is really not terribly difficult, and
we have sought to make some topics that may seem difficult accessible to players without a very
strong mathematical background. But on the other hand, it is math, and we fear that if you are
afraid of equations and mathematical terminology, it will be somewhat difficult to follow some
sections. But the vast majority of the book should be understandable to anyone who has
completed high school algebra. We will occasionally refer to results or conclusions from more
advanced math. In these cases, it is not of prime importance that you understand exactly the
mathematical technique that was employed. The important element is the concept - it is very
reasonable to just "take our word for it" in some cases.

To help facilitate this, we have marked off the start and end of some portions of the text so that
our less mathematical readers can skip more complex derivations. Just look for this icon for
guidance, indicating these cases.

In addition,

Solution:
Solutions to example problems are shown in shaded boxes.
As we said, this book is not a mathematical textbook or a mathematical paper to be submitted to

_________________________________________________________________________
THE MATHEMATICS OF POKER 10