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 By the mid-1990s, a revolution in trading had occurred; the old school grizzled traders had been replaced by a new breed of quantitative analysts, applying mathematics to the "art" of trading and making of it a science.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.
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Most Helpful Customer Reviews
99 of 103 people found the following review helpful
Format:Paperback|Amazon Verified Purchase
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! lol
I 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.
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! lol
I 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.
60 of 63 people found the following review helpful
Format:Paperback
Bill Chen and Jerrod Ankenman, The Mathematics of Poker (ConJelCo, 2006)
I should start this review by saying I'm not a math guy. I never was. I failed calculus the first time and had to take it twice (I squeaked by with a C- the second time). Years as a horseplayer, though, made me understand that I was a stats guy, and that the math inherent to the stats was workable even for an English major like me. Then I started playing poker seriously. Probability? Kelly criterion? Game theory? Yeah, I had all that. Then I read The Mathematics of Poker. And there's my old nemesis... calculus.
Chen and Ankenman say in the intro that the book is geared towards laypeople, and that they try to keep the math to a minimum (they separate out the more complex proofs and the like for non-math-guys to skip over). In short, they don't succeed. They can't; in order for you to grasp concepts later in the book, you have to get the math earlier in the book. There's no way to keep it to a minimum, really. There might be a way to make it more palatable, though. I've read probably seven or eight books on horse racing for every poker book I've read. (I was a horseplayer for a decade before I started playing poker with real, honest-to-goodness money.) One thing many of the good ones have in common is that they err on the side of excess when it comes to examples. If there's tricky math involved, the author will take you through it with four or five examples. When you're reading a book on horse racing, sometimes it seems like overkill, and I know I've remarked on that in some reviews of horse books I've written. I am now reformed, and see the light. Had I had that many redundant examples here, I'd probably have gotten it. Theory is great and all, but it's fundamentally useless unless you can put it into practice. Which is the stated goal of the authors here. What's missing is the gateway between theory and practice those examples provide.
One other thing (and this, too, is addressed by the authors towards the end of the book)-- even if you don't get the math, unless you're Daniel Negreanu or someone who plays like he does, you're likely to look at Chen and Ankenman's conclusions and say "whoa, that's some seriously aggressive play." Academically, yes, there will be times when it's right to call a raise with a suited five-deuce. (For that matter, with three-deuce offsuit as well.) There will also be times when it's right to push all-in with it. Would anyone actually do it at the table? The authors say they've been accused of maniacal play, and I have to say that after reading this book, I can see why. So be prepared: if you plan to put the lessons this book teaches you into practice, you're probably going to find yourself well outside your comfort zone for a while. ***
I should start this review by saying I'm not a math guy. I never was. I failed calculus the first time and had to take it twice (I squeaked by with a C- the second time). Years as a horseplayer, though, made me understand that I was a stats guy, and that the math inherent to the stats was workable even for an English major like me. Then I started playing poker seriously. Probability? Kelly criterion? Game theory? Yeah, I had all that. Then I read The Mathematics of Poker. And there's my old nemesis... calculus.
Chen and Ankenman say in the intro that the book is geared towards laypeople, and that they try to keep the math to a minimum (they separate out the more complex proofs and the like for non-math-guys to skip over). In short, they don't succeed. They can't; in order for you to grasp concepts later in the book, you have to get the math earlier in the book. There's no way to keep it to a minimum, really. There might be a way to make it more palatable, though. I've read probably seven or eight books on horse racing for every poker book I've read. (I was a horseplayer for a decade before I started playing poker with real, honest-to-goodness money.) One thing many of the good ones have in common is that they err on the side of excess when it comes to examples. If there's tricky math involved, the author will take you through it with four or five examples. When you're reading a book on horse racing, sometimes it seems like overkill, and I know I've remarked on that in some reviews of horse books I've written. I am now reformed, and see the light. Had I had that many redundant examples here, I'd probably have gotten it. Theory is great and all, but it's fundamentally useless unless you can put it into practice. Which is the stated goal of the authors here. What's missing is the gateway between theory and practice those examples provide.
One other thing (and this, too, is addressed by the authors towards the end of the book)-- even if you don't get the math, unless you're Daniel Negreanu or someone who plays like he does, you're likely to look at Chen and Ankenman's conclusions and say "whoa, that's some seriously aggressive play." Academically, yes, there will be times when it's right to call a raise with a suited five-deuce. (For that matter, with three-deuce offsuit as well.) There will also be times when it's right to push all-in with it. Would anyone actually do it at the table? The authors say they've been accused of maniacal play, and I have to say that after reading this book, I can see why. So be prepared: if you plan to put the lessons this book teaches you into practice, you're probably going to find yourself well outside your comfort zone for a while. ***
43 of 46 people found the following review helpful
Format:Paperback
I just finished my first complete reading of the book. It is absolutely extraordinary.
Those looking for specific advice playing particular forms of poker will not be happy with the book (with one important, and possibly extremely profitable exception). Those who are looking to really understand the depths and complexity of the game, in all its forms, will be rewarded with an absolute masterpiece.
I am a professional poker player, and I've read and studied everything worth reading (and many others not worth reading!) about poker many times. In my opinion, nearly all of the worthwhile stuff is 2+2 books, with a few important exceptions. As stellar as I believe the 2+2 books are, I feel that Mathematics of Poker (MoP) deserves its own category.
Its major departure from most good poker books is to explore the notion of "optimal play" in a great deal of depth. The most powerful tool of this exploration is game theory, and the book contains an extremely rigorous application of game theory to poker using exemplifying "toy" games that illustrate strategic principles of real poker games. Except for what Sklansky has briefly written on the subject (Theory of Poker), this is the only book containing this kind of information that I am aware of.
While the game theory sections seem to be causing the most comments, MoP also contains excellent sections on what the authors call "exploitive play". While optimal play intends to make our own play unexploitable, exploitive play intends to maximally profit from the deficiencies in our opponent's strategies. To do so, we must ourselves deviate from optimal play, which opens us up to be expolited ourselves (what the authors call counter-exploitation). The discussion of identifiying opponent's strategic weaknesses and developing maximally exploitive strategies is fantastic. Related to this whole discussion is the notion of strategic "balance", which is the bridge to the discussion of optimal play -- and the defense against counter-exploitation.
I can't say the book has taught me any new "plays" or given me any one specific thing to improve about my game (I am not a tournament player, the domain of the important exception I mentioned above). Instead, this book has given me something orders of magnitude more valuable: a more sophisticated way of *thinking* about poker. One reading has already prompted me to think about some pretty important aspects of my game -- balanced strategy on the turn in cash NL holdem, in my particular case -- in an entirely different paradigm. This is absolutely NOT just another book showing you how to calculate pot odds and reminding you to consider future action or the chance you'll catch and lose (my opinion of Yao's "Weighing the Odds"). There is some new and very sophisticated stuff here.
The book has introduced me to thinking about poker at the level beyond what's described in the existing literature. As soon as I finished the last page, I started reading it again...
One final comment about the math. I have an extremely strong math background (though not post-graduate level), and I am comfortable reading ideas in a textbook style of writing. However, the math is not difficult in this book, and the most "advanced" math employed is probably finding a minimum by finding the zero of the first derivative. That is calculus, but anyone who's taken basic differential calculus will be able to follow all the math in the book (this includes quite a few high school students). If you're someone who thinks that NL Holdem is a "people game" and so you don't need to know about equity of hands, pot odds, and draw probabilities, skip this book. This book is for people who have that stuff down cold, don't need any clever new ways to think about it (DIPO?!?), and want to go to the next level.
The beginning of the book has a nice introduction to probability and statistics, but I feel that a good understanding of how the authors analyze poker will require some basic training in statistics, particularly a degree of comfort with the idea of distributions. I think that studying the first half of a first-term college statistics book is valuable for gamblers whether they read MoP or not, but it will definitely help you with this book.
Those looking for specific advice playing particular forms of poker will not be happy with the book (with one important, and possibly extremely profitable exception). Those who are looking to really understand the depths and complexity of the game, in all its forms, will be rewarded with an absolute masterpiece.
I am a professional poker player, and I've read and studied everything worth reading (and many others not worth reading!) about poker many times. In my opinion, nearly all of the worthwhile stuff is 2+2 books, with a few important exceptions. As stellar as I believe the 2+2 books are, I feel that Mathematics of Poker (MoP) deserves its own category.
Its major departure from most good poker books is to explore the notion of "optimal play" in a great deal of depth. The most powerful tool of this exploration is game theory, and the book contains an extremely rigorous application of game theory to poker using exemplifying "toy" games that illustrate strategic principles of real poker games. Except for what Sklansky has briefly written on the subject (Theory of Poker), this is the only book containing this kind of information that I am aware of.
While the game theory sections seem to be causing the most comments, MoP also contains excellent sections on what the authors call "exploitive play". While optimal play intends to make our own play unexploitable, exploitive play intends to maximally profit from the deficiencies in our opponent's strategies. To do so, we must ourselves deviate from optimal play, which opens us up to be expolited ourselves (what the authors call counter-exploitation). The discussion of identifiying opponent's strategic weaknesses and developing maximally exploitive strategies is fantastic. Related to this whole discussion is the notion of strategic "balance", which is the bridge to the discussion of optimal play -- and the defense against counter-exploitation.
I can't say the book has taught me any new "plays" or given me any one specific thing to improve about my game (I am not a tournament player, the domain of the important exception I mentioned above). Instead, this book has given me something orders of magnitude more valuable: a more sophisticated way of *thinking* about poker. One reading has already prompted me to think about some pretty important aspects of my game -- balanced strategy on the turn in cash NL holdem, in my particular case -- in an entirely different paradigm. This is absolutely NOT just another book showing you how to calculate pot odds and reminding you to consider future action or the chance you'll catch and lose (my opinion of Yao's "Weighing the Odds"). There is some new and very sophisticated stuff here.
The book has introduced me to thinking about poker at the level beyond what's described in the existing literature. As soon as I finished the last page, I started reading it again...
One final comment about the math. I have an extremely strong math background (though not post-graduate level), and I am comfortable reading ideas in a textbook style of writing. However, the math is not difficult in this book, and the most "advanced" math employed is probably finding a minimum by finding the zero of the first derivative. That is calculus, but anyone who's taken basic differential calculus will be able to follow all the math in the book (this includes quite a few high school students). If you're someone who thinks that NL Holdem is a "people game" and so you don't need to know about equity of hands, pot odds, and draw probabilities, skip this book. This book is for people who have that stuff down cold, don't need any clever new ways to think about it (DIPO?!?), and want to go to the next level.
The beginning of the book has a nice introduction to probability and statistics, but I feel that a good understanding of how the authors analyze poker will require some basic training in statistics, particularly a degree of comfort with the idea of distributions. I think that studying the first half of a first-term college statistics book is valuable for gamblers whether they read MoP or not, but it will definitely help you with this book.
Most Recent Customer Reviews
4.0 out of 5 stars
The Mathematics of Poker
I liked this book very much. It is a very difficult read. But, it is well worth the time spent reading it.
Published 1 month ago by John Herbert
5.0 out of 5 stars
This could be used as an intro to statistical decision theory
Poker is a fascinating game, and when approached from the proper point of view, it can teach you how to make better decisions when faced with uncertainty. Read more
Published 2 months ago by Rob Ryley
5.0 out of 5 stars
Outstanding
When I got this book I was very excited to read it only to put it away because I was puzzled by all the math. A year later I figured why not give this book a try again. Read more
Published on March 23, 2011 by Paolo Francis
5.0 out of 5 stars
Not for the Un-Mathematical
This book is about the mathematics of poker, and it doesn't slouch on the math! Personally, I loved it, but I have a degree in Applied Discrete Mathematics and a strong interest in... Read more
Published on February 14, 2011 by Robert LaPrelle
5.0 out of 5 stars
One of a kind
Such an interesting book, but certainly one that you have to put time into to follow and understand, then use. Read more
Published on December 31, 2010 by Halo
3.0 out of 5 stars
Print too small
I've read the introduction and the beginning of the first chapter and I enjoy the way the authors think, BUT the print on this book is ridiculously small!!!! Read more
Published on June 30, 2010 by J Splinter
5.0 out of 5 stars
Excellent Advanced Poker Book...
I am well-read in poker literature and would recommend this book for advanced players. It will benefit cash game, tourney and SnG players alike. Read more
Published on December 16, 2009 by Gabriel Wechter
1.0 out of 5 stars
If you didn't major in Math, don't get this book.
So I made it through Calculus II in college about 8 years ago and felt like I needed to complete my math major before understanding this book. Read more
Published on October 1, 2009 by Mike
5.0 out of 5 stars
Not for everyone
Firstly, if you have not taken at least one statistics course, you probably won't understand much in this book. Read more
Published on July 23, 2009 by Bookworm
4.0 out of 5 stars
Can you beat poker with numbers?
In the movie 21, which if you will forgive the plug, I have to say is fantastic, a group of MIT students run rampant through Vegas unerringly winning big at the blackjack table. Read more
Published on April 8, 2009 by MHIslander
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