Book Review: The Mathematics of Natural Catastrophes

I haven't written a pure book review blog post since coming to the UK, so I thought I would write one reviewing a book by a British author. Gordon Woo -- the author of The Mathematics of Natural Catastrophes -- was a senior wrangler in the Maths tripos as an undergraduate at Cambridge University. He went onto a PhD in theoretical physics at Cambridge as well (while taking a detour through MIT and Harvard). He is currently on the staff of RMS (a risk management consultancy) doing research on catastrophic risk and the risks of terrorism.

The title of Dr. Woo's book is slightly deceptive (but in a good way). Yes, it is literally about the mathematics of natural catastrophes, but it's a lot more than that. It is an insightful yet accessible guide to the philosophy of risk and chance.

Although the book has its share of mathematical formulae and equations, the book -- despite the title and the academic background of its author -- is less about maths (British English) in the colloquial sense (i.e., complicated symbolics) and more about maths in its true sense, a logical way of tackling problems. In that way, the book is highly readable and should be accessible to people with even rudimentary mathematical backgrounds, yet remaining sophisticated enough for people with more formal training in maths.

The types of natural disasters discussed in the book focuses on examples from geology and meteorology, but the discussion can easily be extended to other types of events (including terrorism). Not surprisingly, given Dr. Woo's background, the author applies the tools of mathematics, statistics, probability theory, physics, geology, meteorology, engineering, and actuarial methods, in order to get a grip on these thorny issues.

But he doesn't stop there; he also demonstrates -- in the framework of Isaiah Berlin's The Hedgehog and the Fox -- a certain intellectual 'foxiness.' Dr. Woo incorporates history, philosophy, and even literature into his analysis of catastrophic risk. For those interested in the increasingly important links between natural catastrophes and financial instruments, Dr. Woo devotes a chapter to financial issues (including Cat bonds) and other chapters to issues relevant to the insurance industry and the 'management' of extreme risk.

The most fascinating aspect of this book, beyond the purely practical (and there are definitely practical aspects to this book), are the philosophical aspects of the book. As a mathematician philosophizing about the nature of probability, Dr. Woo reminds me of another Cantabrigian mathematician (unfortunately, known more as an economist), John Maynard Keynes (especially in his magisterial, A Treatise on Probability). Dr. Woo's book has not received as much attention as Nassim Nicholas Taleb's Black Swan (my book review can be found here), but I strongly believe that fans of the Black Swan will enjoy reading The Mathematics of Natural Catastrophes.

Book Review: Chaos Theory Tamed

I'm presuming that most of you have either read my musings (The Black Swan ... "Well That's Life!") on Nassim Nicholas Taleb's new book, The Black Swan, and/or read the book on your own accord. It occurred to me that a large segment of the reading public may have difficulty following the mathematics presented in Part Three and in the Notes of Nassim Taleb's book. If your mathematical skills or training is limited or rusty, then you will certainly have a hard time following the logic of the Black Swan, fractal randomness, power law, etc. Even if you have had a fair amount of technical education, many of the topics covered in The Black Swan aren't things that are normally covered in the standard curriculum.

Fortunately, there is a (partial) solution to this problem. Chaos Theory Tamed, by Garnett P. Williams, clearly explains much (but not all) of the mathematics invoked in The Black Swan: the mathematics of complexity theory. (Note: There are epistemological distinctions between complexity theory, chaos theory, fractals, etc. However, for the purposes of this book review, I will mostly ignore those distinctions. At any rate, Garnett Williams' book covers math that are cross-disciplinary and would be useful for any of those aforementioned categories.) Uniquely, Garnett Williams' book manages to explain the mathematics of complexity, chaos, and (to some extent) fractals in a way that is both accessible yet sophisticated.

Most books on chaos theory, complexity theory, fractals, etc., fall into two categories. The first category are books that are of a 'pop science' / 'pop math' variety; relatively easy to understand but whatever knowledge one can glean from these books are given with a lot of hand-waving and not a lot of ways for more sophisticated readers to get beyond generalities and the 'gee whiz' factor. The second category of books are for the more technically minded. The more technical books are (hopefully) good for the initiated but are over the heads of the uninitiated. Frankly, even for those comfortable with the mathematics and scientific jargon invoked by these technical works, the more formal papers and books are usually unpleasant to read and may deny people the sense of epiphany that one should get from good science.

There are a handful of books that attempt to bridge this gap between pop sci/math and more formal literature, but most of these books, frankly, fail. Garnett William's book is the one bright exception ... a positive Black Swan.

The mathematical prerequisites for reading Chaos Theory Tamed are light. The reader only needs to have the equivalent of a good American high school math education (basic algebra, and hopefully, some exposure to pre-calculus), and the patience to go through the logic (and equations) presented throughout the book. Unlike the few other gap bridging books that are in the marketplace, Chaos Theory Tamed's mathematical prerequisites are very minimal.

This low level of mathematical pre-knowledge should not be mistaken with a low level of mathematical sophistication. On the contrary, Garnett Williams' book does a great job of covering math that is actually invoked and used by professional researchers in the fields of chaos and complexity theories (as opposed to hand-waving toy models presented in other pop sci/math books and, even, in the gap bridging books on chaos theory). Without assuming an active knolwedge of calculus, Garnett Williams explains the workings of difference equations and differential equations of the type that Edward Lorenz and Mitchell Feigenbaum used to 'discover' chaos theory. Like other books on chaos theory, Chaos Theory Tamed discusses topics like 'strange attractors','bifurcation,' etc.; unlike those other books, Garnett Williams actually explains what those terms mean and the logic (math) behind them.

More relevant to Nassim Taleb's The Black Swan, Chaos Theory Tamed does an excellent job of explaining probability theory and the mathematics of the power law.

Garnett Williams does an excellent job of explaining the nuts and bolts of probability theory and mathematical statistics to the uninitiated. In fact, if that is all he did in his book (and he did much more than that), it would be a worthy book in it of itself since he provides the sort of explanations that would be useful to anyone wanting to learn probability and statistics beyond an elementary level. He also explains information theory along with the idea of entropy (in both the thermodynamics and Shannon information theory senses) and ties those ideas in with probability theory. Again, Garnett Williams' explanation of these complex topics -- topics that are so riddled with difficulties that more technical books try to avoid it -- are both so ambitious and helpful to the uninitiated that I could recommend the book on these grounds alone.

But Chaos Theory Tamed doesn't stop with explanations of 'strange attractors,' 'entropy,' and 'autocorrelation in time series.' Garnett Williams' book gives the best accessible explanation of power laws that I've encountered. Chaos Theory Tamed explains power law and scalability (scale free, scale invariant, etc.) in terms of 'dimensions.' Dimensions are essentially the kind of dimensions that we are all familiar with ... three dimensions of space (four dimensions if you include time), two dimensions of a sheet of paper or a fictional 'flatland,' and the single dimension of a Platonic straight line. With power laws, the dimension of what is being measured is the exponent and is relatively invariant (within some range). When graphed on log-log axes, a power law is a straight line and the power law exponent (represented as 'alpha' in The Black Swan) is the slope (usually, negative) of this line.

Garnett Williams uses the concepts of dimensions and scales in order to give a very clear-headed explanation of fractals. Chaos Theory Tamed doesn't dwell on fractals as much as other books that are more specifically focused on fractals, but, when the book does deal with fractals, the explanations of what fractals are and how they relate to chaos theory are brilliant for its clarity. Basically, fractals are defined in this book as being "fractional dimensions" -- i.e., instead of 1, 2, or 3 dimensions (integer dimensions), fractals are fractional (non-integer) dimensions (1.2, 1.4, 2.8, etc.). I actually find this sort of definition to be much more useful and interesting then the thousands of pretty pictures of fractals I have seen in other books, magazines, and on the web.

One can tie this fractal dimension idea in with power laws in the following way: The power law exponent (the dimension) is usually not a whole number. For example, 1.1 is the exponent (as provided by Nassim Taleb in The Black Swan, p. 264) for the net worth of Americans; that power law exponent translates to the top 20% of the wealthiest Americans having 86% of the wealth (from p. 265 of The Black Swan). Again, this way of thinking about fractals is at least as valuable as seeing a pretty picture of fractals.

Another distinguishing feature of Chaos Theory Tamed is that, unlike other books on chaos theory, it goes into how one might empirically detect and measure chaos and/or complexity; i.e., it goes beyond mere scientific speculation or even theorizing. The concept of dimensions (i.e., the power law exponent; the 'alpha' in The Black Swan) is important to the empirical study of chaos and complexity. There are different standards/measures of dimensionality including Hausdorff dimension, information dimension, and correlation dimension; they are distinct but similar ways of measuring dimensions. Information theory, in the form of Kolmogorov-Sinai entropy and mutual information, is also important to the quantitative study of chaos, and Garnett Williams does a commendable job of explicating these topics.

Garnett Williams is also quite frank about the limitations and difficulties inherent in trying to empirically detect and measure complexity. This note of caution fits in well with Nassim Taleb's warnings about not placing too much weight on 'precise' (frankly, there aren't any) measures of 'alpha' when thinking about randomness from a Mandelbrotian perspective.

One final point to praise in Chaos Theory Tamed is its glossary. Its glossary -- clearly defining terms in chaos theory, probability, mathematical statistics, information theory, etc. -- alone is worth the price of the book!

Garnett Williams' professional background is worth noting. He was a geologist for the U.S. Geological Survey. A geologist might not be the first person to come to mind in writing a book about the mathematical backbone of chaos theory, but in many ways he is the ideal person to write the book. The distribution of earthquakes and other natural geological phenomena follow power laws and have been studied by complexity theorists as being examples of complexity in nature. In fact, many natural -- as opposed to social -- phenomena seem to be consistent with Black Swan theory. Thus, Nassim Taleb's ideas should not be thought of as being confined to social sciences only but as being applicable to natural sciences as well.

Bottom-line: If you're looking for a clear-headed explanation, that is both accessible and sophisticated, of the mathematics behind The Black Swan (and complexity/chaos/fractal theory, in general), then Chaos Theory Tamed is a great place to start ... in fact, it's the best place to start. Frankly, I wish there were more books like Chaos Theory Tamed and more authors like Garnett Williams; books that aren't afraid to cater to the intellectually ambitious and science/math writers who aren't afraid to lay bear the equations that tend to mystify science and math to the uninitiated.

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."