Book Review: The Mathematics of Poker

The game of poker is a fascinating mix of mathematics and psychology. The combinatorial and probabilistic nature of the shuffling and dealing of cards -- as well as the uncertainty associated with competing against people with differing personalities and backgrounds -- suggests that a good poker players should be, at least on an intuitive level, good mathematicians. The need to make major decisions under pressure also suggests that poker is a case study in applied psychology.

As someone fascinated by both econophysics and poker, I drew the conclusion that mathematics could be fruitfully applied to both the more obviously 'mathematical' aspects of poker as well as the psychological/decision-making aspects of poker (via game theory). With the ongoing popularity of poker on television, on the internet, and in card rooms, as well as a growing body of poker 'literature,' I naturally expected there would be at least a few books available that took on poker mathematics. Sadly, the vast majority of poker books only deal with 'math' on a purely computational or numerical basis -- i.e., they explain or offer up various types of odds associated with poker and not much beyond that -- rather than applying the analytical tools provided by mathematics in its more purer sense.

The Mathematics of Poker by Bill Chen -- a mathematician, part time pro poker player, and full time financial quant at Susquehanna International Group -- and poker pro, Jerrod Ankenman, finally addressed many of my musings on the intricate relationship between poker and mathematics. Although this book is not for everyone, especially those who are faint of heart when it comes to equations and formulae, it should be highly useful to quantitatively minded poker players and fans as well as to finance types that may (or may not) be surprised to find so many commonalities between poker and quantitative finance.

One of the interesting attributes of this book is that, despite a plethora of equations and semi-formal mathematical expressions that very few poker players will actually work through at the table, the authors take the position that the mathematical analysis and reasoning they use in this book is designed to make the reader more profitable poker players. This practical stance actually enhances the intellectual credibility of this book: While the more self-consciously 'intellectual' works that tackle 'poker' deal merely in abstract scenarios that differ dramatically from real-world poker, Chen & Ankenman's book offers up analysis (even when using 'toy games') that is tied to what poker actually looks like.

Some of the high points of this book are: the book's explanation of how mathematical statistics and probability -- especially Bayesian approaches -- might apply to poker decisionmaking (Parts I & IV), the application of mathematical game theory to the game of poker (Parts II, III, V), the concept of "effective tournament size" (where tournament payout structure alters the number of 'double ups' needed to make money in a poker tournament) (Part V), a quantitative approach to poker backing agreements (Part IV), the important role that exponentials and logarithms play in the mathematical analysis of poker (Parts III and IV), risk management for poker players (Part IV), and the scientific approach to the 'art' of hand reading (i.e., making an educated guess of the distribution of cards your opponent may have) (Part II).

An especially fascinating aspect of this book is how the authors make some interesting analogies that connect poker to quantitative finance. The idea of maximizing logarithmic utility, which is at the heart of a lot of quantitative finance, is discussed in detail in connection with profitability in poker play (I should note that the authors call this concept the 'Kelly Criterion' or 'Kelly betting,' but for reasons beyond the scope of this blog post, I don't quite agree with this characterization because the Kelly Criterion is a much deeper concept, IMHO, than maximizing log utility). The book also bring in other concepts from quantitative finance and financial economics into poker analysis, including the Sharpe ratio, financial options (real options), and modern portfolio theory.
The best part of this book was its explanation of the 'risk of ruin' and how it relates to poker play over time. I have read many books on probability, statistics, general mathematics, and gambling over the years and I have always felt frustrated by the lack of a clear explanation of the rather useful but basic concept of the risk of ruin. The Mathematics of Poker gives, by far, the best explanation of the risk of ruin I have ever read.

Although this book makes many very excellent points and should be a valuable addition to anyone intersted in the subject matter, this book does have some flaws. One of the most obvious flaws is that it has a number of typos and errors. I wished the authors or the publishers had invested in LaTEX typsetting (it doesn't appear that way to me). Having pointed this out, however, I should, in the book's defense, also note that: (a) most of the errors are minor and of a typographical nature rather than a substantive nature, and (b) the authors and publishers are putting out an errata and have been making corrections to new printings of the book (details on this can be found on the book's website) -- which are the responsible things to do (that many others neglect to do in the technical publishing world). Another criticism along the same lines is that some of the notation is confusing (e.g., the Greek symbol for 'alpha' means radically different things in different parts of the book) and should have been better thought out.

The only other major criticism I have the book is 'Part III: Optimal Play.' Although game theory is utilized throughout the book, Part III is where the bulk of the application of game theory to poker takes place. So it frustrated me to find this part to be tedious even to someone who is an avid reader of highly mathematical and technical material. Having said that, however, I should note that there are definitely interesting and worthwhile points of wisdom made in Part III. Furthermore, the other parts of the book are so interesting in and of themselves that Part III can be safely skimmed in order to 'enjoy' (to the extent one can 'enjoy' a mathematics book) the book as a whole.

The final point that needs to be made about this book are the mathematical prerequisites needed to read this book. The authors state that they have kept the prerequisites to a minimum and that someone with a very solid high school college prep mathematics knowledge base can understand this book. Although I think the authors are sincere in their claims, I think this book would be a challenging read for those who are limited to that criteria. I think a more realistic mathematical prerequisite for reading this book is either someone who has had at least some education in calculus (which the authors occassionally throw in) or someone who regularly reads math books -- they could be 'pop' math books -- that have equations and algebraic manipulations in them. If you have that level of mathematical sophistication -- which is still a fairly low standard -- you should do alright with going through the type of reasoning used in this book.

In summary, I believe that this book does an excellent job of finally addressing what had been a glaring omission in the poker literature: the application of mathematics (as opposed to just numbers and computations) to poker. As Chris Ferguson, a World Series of Poker main event champion and a holder of a PhD in computer science from UCLA, said in his endorsement of the book, "If I ever find myself teaching a poker class for the mathematics department at UCLA, this will be the only book on the syllabus."

Bill Chen & Jerrod Ankenman's book may do more than offer up a route to intellectual exploration, however. As Jeffrey Yass, Bill Chen's boss at Susquehanna, states "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."