The Mathematics of Poker by Bill Chen (PDF)

Description

In the late 1970s and early 1980s, the bond an option markets were dominated by traders who had learned their craft by experience. They believed that there experience and intuition for trading were a renewable edge; this 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 By the mid-1990s, a revolution in trading had occurred; the old school grizzled traders had been replaced by a new breed of quantitative analysts, applying mathematics to the “art” of trading and making of it a science. Similarly in poker, for decades, the highest level of pokers 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 has risen to prominence within the poker community. Applying the tools of computer science and mathematics to poker and sharing the 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 is a reliance on quantitative analysis and the application of mathematics to the game. The intent of this book is to provide an introduction to quantitative techniques as applied to poker and to a branch of mathematics that is particularly applicable to poker, game theory. There are mathematical techniques that can be applied for poker that are difficult and complex. But most of the mathematics of poker is really not terribly difficult, and the authors have sought to make seemingly difficult topics accessible to players without a very strong mathematical background.

User’s Reviews

Editorial Reviews: Review “For those who think poker math is only about probability, pot odds, and straightforward, rote play, think again. Chen and Ankenman do a terrific job explaining how math can, among other things, show you exactly how to mix up your play in such a way that even champion players cannot get the best of you. Especially those who don?t read this book.” — David Sklansky “Rear Cover” From the Inside Flap “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, 2000 World Series of Poker Champion. “In the same way that quants and mathematicians took over Wall Street in the late 80’s, mathematical methods will dominate poker in years to come. Chen and Ankenman have written the book that every serious poker player must read.” –Jeffrey Yass, Founding Partner, Susquehanna International Group. “For those who think poker math is only about probability, pot odds, and straightforward, rote play, think again. Bill Chen and Jerrod Ankenman do a terrific job explaining how math can, among other things, show you exactly how to mix up your play in such a way that even champion players cannot get the best of you. Especially those champions who don’t read this book.” –David Sklansky, Author, The Theory of Poker. About the Author Bill Chen was born in Williamsburg, Virginia in 1970 and grew up in Alabama. He received a PhD in Mathematics from the University of California at Berkeley in 1999. He started thinking about poker while a grad student when he augmented his National Science Foundation grant with poker winnings from local clubs. Bill represented PokerStars from 2002 to 2011 and has participated in High Stakes Poker and Poker After Dark. Bill currently lives near Philadelphia. Bill won two World Series of Poker bracelets in 2006.Jerrod Ankenman grew up in Orange County, California. He received a BS in Business Management from Pepperdine in 2004 and an MS in Applied Mathematics from Columbia in 2010. Jerrod picked up poker as a hobby from reading rec.gambling.poker, and as a career from collaborating on poker with Bill (and others). Jerrod won a World Series of Poker bracelet in 2009. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Wow, I’m very impressed with the book. I think it’s touched ground that isn’t available anywhere else. I’m sure that many programmers (myself included) have attempted to solve this game, and have discovered how burdensome the simple odds calculations are, nevermind the strategy and decision trees. Poker will not soon be solved by computers, like chess is. However, Bill Chen’s ideas of “Toy games” help humans get insight into the character of the solution.Anyone picking up this text should be warned of several things: 1) It is not for beginners. Strong poker takes judgement and experience, and basic hand/situational values cen be best learned from Dan Harringtons books or Sklansky’s No-Limit book. I’ve read over 20 poker books, and Harrington and Sklansky stand out as the best. Harrington’s books are very practical, with detailed analysis of situations. 2) It is not for the timid, foggy headed, or undisciplined. The new concepts in his books require for you to stop and think. If your instinct is “gee, this sounds complicated”, then give up now. Some people will have the same backlash that regular people have with math. If you’re from the “Math is hard” philosophy, this is not for you. 3) This book does not read fast. You should read it 3 times slower than a normal book to really appreciate it. The math shold not just be understood, it should be questioned. 4) The book highlights theory behind game strategy, but does not connect the dots with real hands or real situations. It would be good to connect the check-call, check-raise, check-fold, bet-raise, bet-call, bet-fold, bluff, check-raise bluff, etc… thresholds with actual cards. What would be most cool is for software to perform this analysis, although I imagine only one-street analysis could be performed, but it would still be insightful. 5) Personally, I cannot recommend the first 40 pages of this book. They really didn’t dig into the meat of the game and I found it quite mundane.That said, here are the good things I can say about it: 1) It is nothing like you’ve ever read in any other poker book before! Many poker books overlap eachother, reminding pot odds, hand values, tournament phases, etc. This book dives into the fundamental theory. The interesting math of poker is not related with mundane matters of probabilities, pot odds, etc. The interesting math is the math behind bluffing, calling, and value-playing. BTW, there is a math essay by Chris Ferguson about game theory and poker. 2) It will remind you about why you bluff. One of the most practical lesson I learned from this math is that if you are bluffing optimally, YOU SHOULD BREAK EVEN ON YOUR BLUFFS! That was revolutionary for me. If you’re winning on your bluffs, you’re not bluffing enough. If you’re losing, you’re bluffing too much. If you break even, you get paid most on your value. This is not exclusively true, but becomes more true the more solid your opponent is. If your opponent is weak tight, then you should probably profit on your bluffs. Exploit appropriately. 3) Optimal play gives you your “center game”, which you use before you know your opponents. When you adapt to exploit your opponents, be aware that you are opening holes in your own game to perform the exploit. 4) The material covered in this book is shore of an undiscovered land. It is only the beginning. Since the game appears unsolveable, there are riddles and puzzles at every corner. New insights can drive a stronger game. Who knows? You may have some clever insight beyond what the author discovered.I hope he writes a sequel to this book. Material I would love for him to research for the sequel:1) Preflop single-full-street play, but with real holdem. For a given bet-size some actual card thresholds would be given for bluff, check-fold, checkraise, bet-fold, bet-call, etc… Translate this basic game concept to card thresholds. Include the fact that hands only have equity, not some automatic ranking (like 0-1 game).2) Actual single-street post-flop play for some example flops. Again, card thresholds would be great. Ideally, if some representation could be shown for card thresholds as a function of bet & raise sizes. Maybe a few pages of tables are required. There should be at least 10 distinct flop examples and this should probably consume more than 30% of the book.3) Optimal exploit as a function of opponent’s deviation from optimal play. Again, make it practical with card thresholds.4) The math of Caution vs. Aggression. I know that the deeper the stacks are, the more that play should steer towards caution. At 30 blinds, top pair is a push-push-push hand. At stack=pot middle pair is an allin hand. At 200 blinds, suddenly top pair seems like it should be sometimes checked, because it’s tough to fold later. My question is, how does caution show up in the math? And how does it balance with the common notion that Aggressive play is best? I know it’s often better to bet-fold a medium hand, but definately sometimes it’s smartest to check-call it, to make your opponent indifferent to bluffing.5) The math suggests that you should be check-calling and bet-calling with some expected losers to make your opponent indifferent to bluffing. What is the real threshold for these check-calls? Are check-calls with 2nd pair smart? bottom pair? What is really the right threshold? How does this change with multiple “bullets”?6) The math suggests you should only bluff your trash. But then in multi-street poker with draws, we put many of our bluffs on medium drawing hands. How do the partially made hands with draws fit in?7) More analysis about mult-way pots. Try to solve the full street 3-way 0-1 game. In a multiway pot, which player will take the burden to bluff-call and make the opponent indifferent to bluffing?8) Any deeper material which cannot be described absolutely with math can probably be backed only by simulation. The readers are pragmatic people (just trying to improve their game) and do not need a systematic analysis for everything.9) Figure out every secret that Chris Ferguson knows and squeeze it in here! lolI very much believe there needs to be a sequel to this book. A foundation was layed, but the dots were not completely connected together. It’s kinda like a movie where you’re left in the middle, waiting for the sequel. The theory needs to be grounded to some practice.

⭐This is the fifth poker book I have read so far, and I found no useful information for me in this one. The statistics are very basic and essential for those who is major in math/statistics, and you cannot find real implementation tips based on the knowledge you have. Everyone knows you need to adjust your action based on the other’s behavior and statistical conclusions /probabilities asap, but how? and how to make sure your corresponding modifications on your actions are sufficient and efficient is the real problem. I feel this book is teaching statistics not real life poker game. I admit that when you take it to learn STAT, it is not bad. I like the print quality and composition.

⭐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.

⭐The modern poker player is a different breed than the poker player of the previous decade. The skills that make a poker player valueble today are skills in cutting edge mathematics, data science, and game theory. Players of the previous era had to learn to follow strategies but the players of today have to frequently invent entirely new ones and they need the mathematical tools to analyze their play within the context of optimal frequencies and moves.This is not a book that will give you a simple strategy that you can follow and then use to beat small stakes games. This is a book that will equip you with the mathematical reasoning and tools to analyze massive solver files and make sense of that data to build an unbeatable strategy.

⭐I can sympathise with readers who had a hard time with this book. When this book came out I was in my first year of a maths-based undergraduate degree, but even I found the maths so hard going that I had to put the book to one side until I had started my second year. However, if you are a serious poker player you really should persevere, as this book’s message has totally revolutionised poker at the highest level — it has become, in every way, the poker geeks’ bible.The first three chapters are a statistics primer for poker. This material is all good to know, and the treatment is much easier than many statistics textbooks, but I have a feeling that these chapters put off a lot of readers. If you don’t have an interest in mathematics, then you can actually get by in poker with very little knowledge of statistics, beyond knowing a few probabilities specific to your form of poker. The vast majority of the material in these chapters — manipulating the normal distribution, Bayesian vs. classical statistics, and Bayes’ theorem — can be skipped without much loss to your poker game or your understanding of the rest of the book.Chapters 4-9 cover exploitive play. Not only is this material essential, but it is also intuitive, non-mathematically demanding, and very similar to most readers’ conception of poker. Just about every beginning- and intermediate-players’ thought processes are almost entirely exploitive. They form an idea of how their opponent is playing or thinking, and then they find an exploitive counter-strategy to use. Understanding these chapters will leave you well-placed to follow the rest of the book.Chapters 10-21 cover optimal play, and this is where the book’s key strategic insights are located. Optimal play has some other names: “unexploitable play”, “game theoretic strategies”, “GTO poker”. Whatever name used, this is the way many top players approach the game. The core idea is, that instead of attempting to exploit your opponents, you should primarily adopt a safety-first approach by preventing players from exploiting you.Chen and Ankenman take unexploitable poker to new levels. To keep the mathematics tractable, the authors primarily study “toy games” — simple representations of poker — instead of analysing actual situations that might arise at the table. In one of their two main toy games the 52 card deck is replaced by a three card deck; in the other, hands are replaced by a random number between zero and one for each player. These models capture the essence of poker — the ranking of hands — while removing the many complicating factors of specific games and situations.This analytical method — creating a “model” of a situation — is very common in science. It might seem far removed from actual poker, but applying these methods to your game can lead to bountiful improvements in your results. Probably the biggest idea you can apply is their systematic solution to river play: they show exactly how the hands you bluff with, the hands you value bet with, and the hands you bluff-catch with are related to each other. In two chapters — “Chapter 21: A Case Study”, and “Chapter 30: Putting It All Together” — the authors become a little more accessible by showing the reader how their models can be related to real poker games.I really can’t recommend this book highly enough; I think it offers the best roadmap for the truly dedicated player. Their “top-down” approach, starting off on the level of overall strategies and games, and then finally moving down to the level of playing individual hands, is in my opinion the best way of approaching poker. This is because ideas can easily be transplanted from one game or situation to another; given the variety of situations and differing forms of poker on offer, this is probably the most efficient way to become an expert player.

⭐the book is quite amazing but you can easily good the mathematics of poker pdf and get the book for free and also the fact that they dont offer a kindle version is annoying save yourself the amount it costs by getting it for free

⭐Very detailed and enlightening. I am not a mathematician so I can not follow all of it but some of the conclusions have made me think again about some assumptions that I had taken for granted.

⭐Oh well.

⭐good product

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