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Review  A Beautiful Math  Tom Siegfried

No, you didn't misread what I very nearly typed as "A Beautiful Mind"  it is "A Beautiful Math". This confusion isn't accidental  this book is strongly and misleadingly packed to ride on the tail of the (now rather dated) Beautiful Mind bandwagon. Even the subtitle, "John Nash, Game Theory and the Modern Quest for a Code of Nature" doesn't really give the game away. Because although Nash does appear in the book, he's not a major character. It would be much more honest if the title read something like "The Real Psychohistory: Hari Seldon, Game Theory and...".
The driving force here  and none the less interesting for that  is Isaac Asimov's imagined mathematical concept psychohistory used in his 1950s Foundation trilogy of books and later sequels. The idea is to treat masses of human beings in the same way as statistical mechanics treats masses of gas molecules  individually unfathomable, but with the right maths, predictable on the large scale. Asimov's description of psychohistory is beguiling. I have to confess it was probably one of the reasons I went into Operational Research after university. OR is a way of applying mathematical methods to solve all sorts of real world problems, from queuing theory and route planning to the simulation of complex processes. Operational Research seemed to me to be the closest thing to Asimov's imagined concept.
Tom Siegfried never even mentions OR, however. For him the real world parallel of Hari Seldon (Asimov's fictional mathematician) and his psychohistory is game theory. This concept, originated largely by John von Neumann and developed by Nash of Beautiful Mind fame is not how to win at chess or Monopoly, but starts as a much more abstract attempt to show what the best strategy is to take in very simple two person choices, where the choices made influence the outcome for both individuals.
There's no doubt that game theory has proved of great interest in fields from economics to biology, but as you read this book it's hard not to get the feel for someone with a theory that he is desperately trying to make the facts fit. Time after time, game theory is squeezed into a situation where its value is hard to spot. One of the problems is that traditional game theory doesn't in any sense model real world behaviour, but rather the most logical behaviour if the players had full information and perfect ability to calculate the outcome. Although this can be modified with probabilities of different approaches, it still has trouble with human beings' almost limitless ability to fail to engage with probability (visit any casino if you doubt this). At one point, Siegfried shows us how different cultures will play games involving consideration of other people in totally different ways. This shows so well how game theory is better suited as an analytical tool for what's known than the predictor of human behaviour Siegfried wants it to be.
Perhaps the strangest omission here is the lack of mention of chaos theory. We now know that a system like the weather, for instance, will never be susceptible to long term prediction, because it is a chaotic system (in the mathematical sense). It would not be at all surprising if making predictions of large scale human behaviour suffered similar chaosderived problems, and Siegfried's optimistic assumption that you only need enough data and good enough development of game theory to get effective predictions is likely to prove as illusory as early meteorologists' assumptions that once we had enough data and computing power we would see pretty well perfect weather forecasts.
The book provides interesting insights into the modern applications and developments of game theory, but it suffers a lot from this dubious underlying theme, and a poor structure that makes it sometimes seem rather repetitious and uninspiring reading, even though the text itself is well written and enjoyable. If you are interested in game theory, I'd strongly recommend it, but as a general popular maths book it doesn't quite make the grade.
Only in hardback.
Reviewed by Brian Clegg
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Last update 05 June 2007