# The Mathematics of Poker

### The Mathematics of Poker

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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 By the mid-1990s, a revolution in trading had occurred; the old school grizzled traders had been replaced by a new breed of quantitative analysts, applying mathematics to the "art" of trading and making of it a science.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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- ISBN13: 9781886070257
- Condition: NEW
- Notes: Brand New from Publisher. No Remainder Mark.

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RatingThis book is difficult for people without an advanced math background. It is a theoretical book that will evolve your thinking about the game. The content however is ground breaking as it is the only book that brings game theory into the game of poker.

I am a 2-4nl online regular (6-tabling) and have every poker book worth mentioning. This is a must have!!! Having a master in theoretical physics obviously helped me understand the notation but I don’t think you need to understand all math to get something from reading this book. It has lots of good stuff that can be applied like, how fast does my win rate (BB/100) converge to 95% probability.

If you are interesting in game theory but not so familiar with the math, then I could recommend some of the less technical books on the subject (just search on game theory) and get a feeling for the subject. If you haven’t already reached an advanced level in poker (or have a master in mathematics) I recommend that you first make sure you have read: Harrington, Sklansky, Supersystem II and Maybe Phil Gordon’s green book.

RatingMake no bones about this, this book contains some pretty complicated and esoteric stuff, but then again, poker can be a pretty complicated game!

RatingThere is nothing more to say other than this book is the complete math book of poker. Nothing is left out and some of the material is fairly advanced. So, if you’ve not taken a college-level math course, you may be lost.

Soooooo much material but so very, very good. Chris Ferguson said it right … if there were ever a university-level class on poker, THIS would be the textbook.

RatingI am well-read in poker literature and would recommend this book for advanced players. It will benefit cash game, tourney and SnG players alike. [If you are an improving intermediate player, but have a math or economics background, I think you would also find it helpful. If you have read less than 5 poker books or need to learn more fundamentals, you would likely get more value from reading other books first.] Many of the concepts presented apply broadly across different types of poker games.

This book exposes the true (and sometimes counter-intuitive) nature of common situations that arise in games of confrontation. The authors do an excellent job making game theory accessible to those without a math or econ/finance background, while not being afraid to ply into the underlying math for those interested. They successfully keep the topics focused on their relevance to poker, trimming out discussion that would be more mathematically complete, but bog down the poker learning.

They also do an excellent job of building up from fundamental concepts and examples (variable-constrained “toy” games) to more complex and applicable scenarios similar to those found in the mathematically complex world of poker. They gradually introduce new concepts and variables into the toy games as they come to resemble actual poker scenarios, with insights gained along the way. It helps solidify a meta-understanding of the game, and I suspect that even most advanced players will at least refine aspects of their play after reading. Note that it is, however, written in a heavy, take-your-time-digesting-it style, so it’s not a quick breezy airport read. Some highlights:

+ The best poker book by far that I have encountered for game theory topics.

+ There are plenty of new ideas presented even for well-read players.

+ Delves elagantly into the nature of optimal play, exploitation and counter-exploitation. Learn to dominate others to extract value and how to adjust to prevent yourself from being dominated.

+ Presents concepts of risk, bankroll management, etc that I have yet to see elsewhere in poker literature.

+ Challenges some common conventions of most poker players and some authors.

+ Many concepts are digestable for most without a heavy math backgroud, though those gifted in math will likely take away a deeper understanding after reading.

- One aspect I would have liked to see more about is the Jam-or-fold scnearios. They do an excellent job of showing solid HEADS-UP strategy, but I would have liked to see this expanded to include 3+ player ICM-type scenarios. I would have also liked to see them quantify some of the weaknesses in ICM modeling, such as accounting for impending blinds, etc. In other words, a position- / equity- / blind-dependent jam-or-fold framework for a several player games would have been very helpful (even if it accepts ICM as-is, but especially if it refined it).

RatingWow, 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?

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.

RatingBought this book to understand the finer aspects of the math behind poker. Knowing that Bill has won a couple of WSOP bracelets led me to be confident that the book was worthwhile.

My statistics knowledge is quite rusty, so the reading is slow-going for me to this point. My eyes have honestly glazed over quite a few times in the early going…

Vanessa Russo, an up-and-coming poker pro, basically endorsed this book on one or more occassions on TV, citing the value of reading the latter chapters of this book. Admittedly, she teaches a poker course on Game Theory, so the material probably comes easy to her

RatingTwo brilliant minds who also happen to be brilliant poker players have written a challenging but accessible explanation of what truly lies beneath the surface of the game of poker – the numbers. They begin with the basics of probability, then move to exploitative play (taking advantage of your opponents’ errors), then to optimal play (playing so that no one can take advantage of you). The book also includes an important discussion of bankroll size, bet size, and risk of ruin, concepts rarely seen in the poker literature. Finally, they wrap up with an “Other Topics” section that provides critical insight into proper poker tournament play.

If you want to get beyond what you can accomplish by rote memorization of starting hand charts, give this book a try. It should not be the first poker book you read, but it may be one of the most important.

P.S. Unlike many poker books, this one is beautifully put together, without the typos and bad sentences that make me cringe as I read.

P.P.S I received my copy today. Why does Amazon still say “it’s not available yet and we’ll e-mail you when it does become available”???

RatingI 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.

RatingIntriguing and thought-provoking analysis of bluffing, mostly in no-limit hold’em, although some analysis of limit hold’em. Focuses on toy examples that are nonetheless educational. Writing at times a bit choppy and elliptical. Nevertheless, full of fun and entertaining examples that will at a minimum give much scope for future research.

RatingChris Ferguson states that if he ever teaches a poker related math class he’ll use this book. That observation really fits what this book accomplishes. The book is very much like a text book. A fair amount of study is required to fully understand what’s in this book. Most of the information is quite terse and, like other reviewers have mentioned, you get out of it what you put in. If you’re looking for specific advice for specific poker situations then this isn’t for you. If you do want to gain a deeper understanding of the maths behind poker, with some effort, this book will help with that.

RatingFor the first time, poker strategy and analysis advances past the simple mathematical concept of pot odds to incorporating more subtle and sophisticated mathematical tools to refine your game. As with the best poker books, this is a book that will require repeated readings to fully absorb. The book covers basic game theory, applications to poker strategy, optimal play in poker, managing risk to one’s bankroll, and misc. topics including tournament play concepts.

That these concepts work is evident by the fact that the authors cashed in 7 WSOP 2006 events and won 2.

RatingThis book is a must have. Ive read many books on poker and this book alone is worth more than the rest of them. Of course if you are going to read it you have to be rather smart and have some poker experience. If you have those characteristics then my advice is: Throw the rest of the books away and read a master book on poker

RatingFirstly, if you have not taken at least one statistics course, you probably won’t understand much in this book. There are also some topics using calculus, but are explained in a way to where you may not actually need to understand the calculus in order to understand the point they are trying to make. I am currently in college and have only taken Statistics I, and Calculus I & II so far and I really had to focus in some parts of the book in order to follow the math.

I learned A LOT from this book and was very happy that I read it early in my “poker career.” It goes extremely deep into seemingly-simple topics and shows how you can eek out a slight percentage advantage in certain situations. I benefited the most from the parts covering “optimal play” where some interesting points were made.

I own ~10 no-limit hold’em books (most of the popular ones) and this book was by far the most advanced book I have read. I would highly recommend this book to any intermediate/advanced players looking to improve their game, but not if you can’t handle a lot of math; this book isn’t for everyone.

RatingI 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.

RatingIn 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. While I’m always skeptical about the way that Hollywood simplifies everything and this was no different. Just because you can count cards, doesn’t make you infallible. But still, the general point holds true: if you know the math and can create a math-based system of play around blackjack, you can dramatically improve your chances to win.

I’m sorry for the long lead in, but this is what I thought about when I started reading The Mathematics of Poker. The book doesn’t propose that a knowledge of great math will allow you to beat the game, and it really is less immediately practicable than Sklansky’s Theory of Poker. But it does give a detailed mathematical analysis of the game that goes beyond the basic pot odds, which have been discussed to death in any number of books by this point. I wouldn’t call this calculus, but I also wouldn’t call it basic math, by any stretch of the imagination.

The place where this book gets crazy interesting is in the math surrounding less predictable poker concepts like bluffing, and how it should be considered through the percentages of hands won. If you’ve won too many of your bluffs, it means you’re not doing it enough. Too few, and you should tone it down. Bluffing is a break even strategy. It is in ideas like this where the book extends beyond Sklansky’s idea that there is a proper play for each situation, assuming you have total awareness of all the cards. MoP tries to take this thinking to a different level, to where there is a mathematical benefit to near optimal play, and that the human dynamic of poker can be understood, if not fully accounted for through math.

So to get back to my original point. Can you beat poker with numbers? Of course not. The game has far too many variables to allow for a purely mathematical approach to be as successful as it would be in blackjack. That being said, when you’re facing a tough river call, I would rather have Chen, Ankenman, and their math by my side, than to just rely on gut instinct.

RatingIn the beginning, God created Supersystem.

The final “Revelation” in poker books is ‘The Mathematics Of Poker”.

With the release of this book now underway, the collective body of poker authorship is now complete. There needn’t be any more.

With the boom in pokers popularity, a lot of books have recently come out that do nothing more than repeat what other authors have already said. At best, many of these books are simply pages upon pages that reflect the individual game-play philosophy of that particular author. Are such books useful? Sure. The more perspectives you understand as a player, the better. You really can’t blame someone like Annie Duke for getting her piece of the poker boom pie by writing a book, but it isn’t like it’s going to make any difference in your game if you bother to read it or not…

Such is *not* the case with this book.

In my opinion, “The Mathematics Of Poker” is the first poker book in a long time that conveys information about the game which is truly essential. Kinda shocking that such a book hasn’t been written before, but it was worth the wait.

This is not the first poker book you should read, but in my opinion, it is definitely the second or third.

Five Stars.

RatingComplex, yes, but nothing beyond high school algebra is necessary to grasp the essentials. Teaches you a way to think about poker that is unlike any other book. The best poker book in my library.

RatingI really didn’t expect to like this book. Honestly, the only reason I bought it was because I like to read everything that my opponents possibly have read. I don’t want them to have an edge simply because I slept on something. So I thought: “Fine, I’ll read this book. But I’m not going to like it.” Boy, was I wrong. This might be the most thoughtful and intelligent poker book that I have ever read.

I learned poker by using my instincts, not math. So I expected to disagree with a lot of what this book would present. On the contrary, the opposite took place. Not only was I agreeing with the authors, it also reinforced what I had figured out intuitively on my own. Instead of contradicting my play, it reinforced that what I was doing was usually correct.

I haven’t taken a math class since high school pre-calc in 1998, and I did alright. I didn’t understand ALL the math in these pages, but one doesn’t need to. As long as you get the gist of what they’re saying and can apply it to poker, which an intelligent person should be able to do, you’ll be fine. If you find yourself lost without a map (which you probably will at some point in the book), take a break and come back to it.

Flipping through this at your local book store, you might be intimidated by all the charts, graphs and equations. Don’t be. Just buy it, read it with an open mind, and watch your poker game soar.

RatingThis book takes around 15 very narrow specific “toy games” (ie. i can only make one bet, and you can only call or fold), and runs thru some very dense, mathematical calculations to draw some conclusions and principles for real poker play. I found the principles and math to be very valuable, and I learned alot about how to play drawing hands, and how to think about what actions to take with a range of hands.

But it took alot of thinking and re-reading to get there. and it is not presented in an easy-to-digest way. However, if you are pretty smart, mathematically-oriented, like to study things to get better at them, then this is a very good book because it is the only one i’ve found that applies games theory to poker and gives you advice on how to holistically play across many hands (as opposed to advice on how to play one specific hand). but this book is only for the serious advanced player.

RatingI purchased this book based primarily on Chen’s reputation. My first thought on opening it was that “this is math, not poker.”

I had expected something on the order of King Yao’s “Weighing the Odds in Hold’em Poker” on steroids. Instead I got a rigorous exploration of the mathematics of this fascinating game.

The authors point out quite rightly that one may get more insight into poker through exploration of what they call “toy games” than by tackling the monster head-on. This is much like the dilemma posed in performing quantum mechanical calculations – either solve an exact equation approximately or solve an approximate equation (toy game) exactly.

Some reviewers express disappointment in that they apparently wanted a book that would show them directly how to win at the poker table. Instead they got a book that will give them insight that will increase their probability of winning, provided they are willing to work through the math.

If you want the “smart pill” that produces instant understanding this book probably isn’t for you. However, if you want to rigorously explore the mathematical underpinnings of the game I strongly recommend it.

RatingI am by no means finished with this book yet, and I’m having to dust off a lot of mathematics I haven’t used since college (a while ago), but I have already learned a great deal from this book.

I would encourage anyone who seriously wants a better understanding of how to play this complex game to take a look at this book, but be forewarned, it isn’t a quick read. You will get out of this book in proportion to the amount of work you’re willing to put into it.