Published in 2006, The Mathematics of Poker, by Bill Chen and Jerrod Ankenman, is a monumental poker classic.
And now I need to write more about it for this to qualify as a "book review". A genre I have hardly touched before. Is it just greed that propels me into this contest, with thousands of dollars to be won? Well, greed and poker seem like a natural couple — although one could argue that poker has a certain educational value as well. The game teaches decision-making under uncertainty (among other things). The young hotshots that dominate it these days are clearly highly intelligent; but even were one to travel back to the 1980s or 1990s and find a table of degenerate gamblers playing poker in some casino at four in the morning, the regulars among them, even those with little talent, would know that you can make a good decision and lose because of it. Had you made a worse decision, you would have won. Strange, but it can happen; it’s just that the probability for it to happen is low. These gambling types constitute a cognitive elite of sorts, understanding something important that many normal, reputable people fail to appreciate.
Defending poker as worthy is not the point of the book, though. What is the point? For one thing, as the title indicates, there is certainly mathematics there. The book looks in large parts almost like a maths textbook. If you hate equations, it is not for you — except perhaps to justify an interest in poker to someone else who thinks of the game as a pure gambling pastime like roulette, where there is no such thing as a good or bad decision in terms of expected value. Seeing a book like this might alert people that there must be more to poker.
So showing off is a notable use case of the book, but as long as you don’t hate equations and are into poker, actually reading it should be as well. (For those not into poker but ACX-style intellectually curious, I hope that the “philosophical” discussion further below could still be interesting.) More forbidding sections, think of integral signs taking over from plus and minus signs, are marked and can be skipped. Probability and expected value are crucial, naturally, but hardly foreign to the ACX readership and in any case introduced at the beginning. Now, let’s hone in further on the point. Consider two different reader profiles:
One might think that here we have a book for the practitioners, who will find the formulae they are looking for. But in fact it’s the other way round. True, Chen & Ankenman, clearly skilled applied mathematicians and “very focused on the practical application” (page 5), throw quantitative techniques at whatever seems amenable, like economists do in their papers. Some of these topics I will skip completely here, such as win-rate estimates, bankroll-management issues, and also everything related to tournaments. But ultimately the authors have written “not a book about how to play poker. It is a book about how to think about poker” (page 8). This is really about insights more than about formulae. Thus if you hate equations and are not interested in showing off, what you could still actually do if you somehow stumble on the book is read just the “Key Concepts” at the end of each chapter, and gain something that way — in fact, starting with them and then following up on what seems interesting could be an efficient approach quite generally to the book. Moreover, a few sections, such as the excellent final chapter (out of 30), read like normal advanced poker-strategy texts, without all those equations.
Meanwhile, the chapter preceding it is devoted to the case of three players, rather than only two, and illustrates how the phenomenon of shifting implicit alliances causes the core methods from before in the book to no longer apply. But in real poker confrontations, there are often three or more players involved. This just underlines that pure practitioners are not the ideal audience. And in that regard, here is an additional consideration. The book had better not be centered around recipes for the practitioner simply because, considering the rapid advance in poker knowledge, it would be outdated! By contrast, many of its philosophical lessons, often derived from toy models, are timeless. The authors do try to apply their insights to real poker, of course; and their advice has broadly aged better than I might have expected given the year 2006; and their focus on game-theoretic equilibria as opposed to second-guessing the opposition’s strategy seems to have since become dominant among strong young players. Still, you can see the danger in the chapter on the "Jam-or-Fold Game", a type of scenario which actually does occur at real poker tables: the assumptions on page 137 (that rule out "limping" as a viable middle ground between jamming and folding) are superseded as far as I know.
Can we have an example of what I mean by “philosophical lessons”? In this review I will develop one particular aspect of the game a bit, position, and also add my own spin (so don’t rely on the following takes). Good poker players know that it is advantageous to “have position” on one’s opponents, i.e. to act after them. But why exactly? It’s obvious, many will answer. Most importantly, you know the opponents’ moves and can use that information when choosing yours. However, is it really so clear? I might imagine there to be an expensive buffet, so that I wouldn’t know any of the delicacies on offer. What to choose? Fortunately, I have position on the field. I watch what others choose, learning what is (especially) desirable — except that if this is a small buffet and there are lots of hungry people, having position is not at all what one wants!
Not the strongest of analogies, perhaps. But the point is that, generally in games, going first can well be an advantage. You can present the other players with a fait accompli. Take example 16.1 from the book, a toy poker game of two players and one betting round where folding is not allowed. The latter assumption models a limiting case where the given pot of contested chips in the middle is so huge that even with terrible cards you would want to stay involved, on the off-chance that your opponent’s are worse still. If in that no-fold game the player first to act just always goes all-in, regardless of cards, then the player with position obviously has no advantage.
To gain insight into real poker, we need to allow folding. Once we do that, the essence of poker is in place (as long as cards are hidden). Let’s say you have good cards, me bad ones, so you bet and I fold, so far so good. But if you have even worse cards, you can also bet and thus “smuggle” them past my merely standard-bad ones, thereby stealing the pot, using as cover the good cards that you could have as far as I am concerned. On page 121, Chen & Ankenman write that John von Neumann and Oskar Morgenstern solved this basic game and others in 1944 (for a standardized infinite deck of cards). So perhaps von Neumann was the first to explicitly write about, and calculate the game-theoretic value of, what I shall call here the “smuggling thing”.
From a basic scenario of one player doing the smuggling thing and the other defending, one can gradually move towards a fuller betting round, by letting both players have a go at betting and thus smuggling, then allowing raising, then allowing check-raising (i.e. not betting but then actually raising if the opponent bets), and more. Chen & Ankenman calculate the values of these games, for fixed bet sizes at least, and thus of these options — those are rather fundamental results if you are interested in poker as an abstract platonic concept.
They also examine Mike Caro’s “AKQ game” as the simplest model for the three levels of card strength I noted above. Ace is “good”, King is “bad”, Queen is “even worse”. The players get one card each, the third card remains unused. Notice that the problem card in a sense is not the queen, which can just be folded, without regret. Rather, it is the king. If you call with that, you risk paying off the ace. If you fold, you risk losing the pot, which “should” have been yours, to the queen. Meanwhile, if you are the one to bet, you execute the smuggling thing by betting when you have the ace but sometimes also when you have the queen. Or, in poker parlance, you bet “all your aces and some of your queens”. How many queens exactly can be calculated for given parameters, but we’re not concerned with numbers here. Of course, betting a queen is commonly known as a bluff.
We’re in a position now to look at position again. For simplicity, assume there is just one bet to be made, always all-in. In the book, this is toy game 15.1. Let’s imagine you have position on me. That means I am first to act and hence actually first to execute the smuggling thing. After which the game may already be over. Sounds good . . . but alas, whenever I get dealt the problem card, the king, betting would make no sense for me, so you still get your turn doing the thing. Worse, I cannot now have the ace with which I could happily catch your bluffs! For I would have bet with the ace! So your smuggling thing is more lucrative than mine was! Depending on the size of the pot, it may even make sense for me to forgo mine and use my aces to catch bluffs from you instead, so at least yours does not become extra-lucrative. Either way, I’m at a disadvantage. Well, I might be overcomplicating things, and the straightforward idea of position as informational advantage is basically just confirmed; but still, I feel like the issue has become clearer.
And then, once we introduce variable bet sizes and raising, there is a twist. Possibly the highlight of the book. Namely, the game-theoretic equilibrium includes me betting a king. Any readers who made it this far, please take a moment to appreciate how utterly crazy this is, on the face of it. Why would I ever bet a king? You either have the ace and happily collect my stupid bet; or you have the queen and can fold without regret! Indeed, Chen & Ankenman write they were quite convinced at first of having made a mistake, see page 170, when they came across this in their calculations.
Actually, can we not just formally prove that betting a king is bad. Is it not never worse, but sometimes better, to check over to you and call in case you bet? In other words, is betting a king not game-theoretically dominated by check-calling? But there’s a catch. I could be faced with a bigger bet from you than the one I would have made myself. So this proof does not work. Still, if I bet a king, then instead of betting big with your aces and some of your queens, you could just raise big with a similar card range, so how could my bet not be foolish?
I mentioned already that forgoing one’s smuggling thing can be the best strategy against the player with position. If I keep the threat of having an ace alive, my kings are less bluffable by your queens. But if my aces do want to be bet, in order to serve as cover for smuggled queens, then there is one reason to bet kings after all: to put them under the aces’ protective umbrella. Furthermore, these strategies, in games 15.2 and 15.3 in the book, bet fewer kings than aces, so the kings that are bet are protected by a bigger number of aces. Such majority protection cannot be had by passively checking everything. Bluffraise big against these kings at your peril. The other kings, those left behind, are even more vulnerable now, but somehow with the right bet size and frequencies it works out overall to reduce the positional disadvantage. The size won’t be too big, as we can gather from our failed proof attempt.
So much for position (and the AKQ game). What is notably still missing, compared to real poker, is to have more than just one round of betting. Further rounds are added in chapter 19, named “The Road to Poker”. It includes a classic and important result about how a betting line of “geometric growth of pot” can be the best strategy. That applies to a certain static scenario, however, “static” meaning that card holdings cannot improve between rounds. The next chapter looks for insight into dynamic scenarios. I found its beginnings very instructive, but its end less so, and from 20.3 onwards the examples are riddled with errors (on errors in general, more below). In any case, the gap between the toy examples and recipes for practitioners remains large. The complexity of real poker is just too big. Recent “solver” software tools are better at coping with it, but I lack the knowledge to expand on that.
In concluding this survey of some topics from the book, I note that the issue of bluffs/smuggling, which I called the “essence” of poker, does not exhaust the game. No-fold games still offer room for analysis, see chapter 16. Also, even if all the cards are exposed, some room remains. Not for a single betting round, obviously; and with two rounds, it amounts just to calculating whether the player who is behind has enough “outs” to improve for an engagement to be justifiable (if you fold because it’s not and then one of your rare outs does hit after all, that is a salient case of a good decision punished, mentioned at the beginning here). But with three rounds, things can already get surprisingly tricky, see chapter 7.
Hopefully I managed to convey an impression of what the book is about, and I recommend the book to the poker philosopher. A drawback should be mentioned, however, that does affect its value a little.
Start with the fact that quite a few of the intended subscripts in the formulae appear in the normal line, which lowers readability at best and creates confusion at worst. But there are also just so many inconsistencies, inaccuracies and mistakes. An errata compendium on the publisher’s website lists 22 errors that are already corrected in my copy, mostly successfully. Yet in my estimate there could be even ten times that number left, not counting harmless typos. I cannot help but wonder, what were the authors thinking? They must have known they have produced a classic in the field that will be in the spotlight. Were they not afraid of embarrassment? If your book is one for the ages, then people might enter book-review contests, for instance, and take shots at you for sloppiness, if perhaps secretly only in order to brag about how thoroughly they studied your book.
The authors may have found themselves simply too exhausted to polish the text, after years of work. But can that be the whole story? I’m asking since occasionally in calculations mistakes appear and then somehow disappear again. Almost as if, whenever faced with an implausible result, the authors just seized on any amendment that looked like it would restore plausibility. This kind of phenomenon reaches a peak on page 203, where they derive a value that is known in advance (from a more general formula) and get the result (b = 8/15) only after three independent mistakes of different types magically cancel out overall.
So that I have a backup for this strange grumpy end to my review, let me close by quoting “StMisbehavin”, from the poker community website Two Plus Two, who worked through the game-theoretic core part of the book, clearly with a deep understanding of the matter, and recorded a number of errors (including some I missed). Here is how StMisbehavin’s thread opens:
This book at once thrills and infuriates me. It thrills me in that it is the book I wish I had written myself. It infuriates me in that the little niggling errors — typographic and logical — drive me to distraction as I read it.