American Scientist

A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. Tom Siegfried. viii + 264 pp. Joseph Henry Press, 2006. $27.95.

Game theorists have won five Nobel Prizes in recent years. The best known of these laureates by far is John Nash, whose remarkable life was the subject of Sylvia Nasar's best-selling 1998 book, A Beautiful Mind. A movie with the same name starring Russell Crowe then made Nash into a modern folk hero.

The highs and lows of Nash's life are out of the range of experience of most human beings. As an undergraduate, he initiated the modern theory of economic bargaining. His graduate thesis formulated the idea of a Nash equilibrium, which is now regarded as the basic building block of the theory of games. He went on to solve major problems in pure mathematics, using methods of such originality that his reputation as a mathematical genius of the first rank became firmly established. But at the age of 30 he fell prey to a serious schizophrenic illness, which persisted for many years, during which time he languished in obscurity. By the early 1990s, he was no longer delusional, although this fact was not widely appreciated. Fortunately, his recovery was brought to the attention of the Nobel committee just as they were deciding who should get a Nobel Prize for game theory, which had by degrees totally transformed the face of economic theory while Nash was out of action.

Tom Siegfried's A Beautiful Math surfs Nash's recent wave of popularity. The book is certainly a lot of fun. One cannot help but be carried away by Siegfried's irrepressible enthusiasm and jaunty style. In a magnificent flight of fancy, he compares John Nash with Hari Seldon, a fictional character in Isaac Asimov's Foundation Series. In the series, Seldon invents the mythical science of psychohistory, a theory capable of predicting the course of humanity's future throughout the galaxy. Siegfried then grandiosely says that the game theorists who follow in Nash's footsteps are engaged in a similar "quest for the code of nature."

It is true that game theorists do believe that their discipline will inevitably be part of any serious science of humanity that may emerge in the future. It is also true that game theory is already a major tool in economics, evolutionary biology and political philosophy. But it doesn't help our project to promote the myth that John Nash is some kind of superman. (Neither is he the evil genius who is responsible for everything that is wrong in the modern world—as is ludicrously argued in a recent television documentary called The Trap.) The truth is that Nash is simply the most talented of a group of talented people whose joint efforts created modern game theory. If Siegfried's concept of a real-life psychohistory is ever realized, Nash will be remembered only as a foot soldier in an army of scientists who will have made discoveries we cannot yet even envisage.

Aside from sanctifying Nash, Siegfried offers a tour of what he thinks are the important new ideas in game theory. However, he doesn't really have the background to distinguish between established lines of research, crackpot notions and speculative ideas that might go somewhere but haven't got there yet. Nor does he know when his informants are taking him for a ride.

It is as though Siegfried is giving us a tour of a city but only offering us a glimpse or two of the downtown skyscrapers, while conducting us around a bunch of new neighborhoods, some of which exist only in the minds of developers who are more interested in parting investors from their money than in expanding the city. It is true that some of the districts we visit are flourishing, but others are still under construction, and some are already in ruins.

Evolutionary biology is an example of a flourishing neighborhood. Why do some species not have the same numbers of males and females? What determines how long a male dung fly waits at a fresh cowpat for a female to come by? How come eusociality has evolved independently at least 12 times in the order Hymenoptera (ants, bees and wasps) but only occasionally elsewhere? Game theory has the beginnings of answers to many questions of this kind. I would have liked Siegfried to tell us more about such bread-and-butter applications of the theory and less about the long-discredited claims that Robert Axelrod makes for the strategy "tit for tat" in his Evolution of Cooperation. It would also have been nice to see William Hamilton credited with having introduced game theory into biology.

The downtown that we observe only from a distance is economics. The book's dust jacket tells us that economists did not get into game theory until 10 years after biologists, but this is quite wrong. Economics is the subject where game theory grew to maturity and has had its biggest successes.

In fact, the theory of games totally transformed economic theory. The most spectacular successes have been in the design of big-money auctions. Billions of dollars have been raised all around the world by specially tailored auctions, in which the right to use segments of the radio spectrum were sold off to telecom companies. The British auction that I helped design made £22.4 billion ($34 billion) all by itself. What we need authors like Siegfried to publicize is that the same tool that was used to engineer these successes can also be used to regulate imperfectly competitive industries. The relevant branch of game theory is called mechanism design—a term that doesn't even appear in Siegfried's index.

All the vested interests are hostile to the use of mechanism design. Lawyers don't want game theorists muscling in on their territory. They are perfectly happy to bank their fees administering laws that implicitly assume that all industries are either monopolies or else perfectly competitive—the two cases that economists could analyze adequately before game theory appeared on the scene. In reality, nearly all industries fall between these two extremes. Current law is therefore based on false premises, which is one of the reasons that legal regulation is notoriously bad most of the time. But the answer isn't to dispense with regulation altogether, as our captains of industry airily advocate. Adam Smith's famous discovery of the advantages of perfect competition is always on their lips, but they know perfectly well that they aren't operating in a perfectly competitive industry. In unregulated industries that aren't perfectly competitive, the producers can (and do) gouge their customers by manipulating the market—as when the electricity-generating companies selling on the old California Power Exchange engineered price hikes that left the hamstrung distributing companies $13 billion in debt. Such naked greed is self-defeating, but the same kind of gouging goes on all the time all over the world on a smaller scale when markets are unregulated or regulation is ineffective or corrupt.

Tolstoy observed that all happy families are the same, but each unhappy family is unhappy in its own way. Similarly, all perfectly competitive markets are essentially the same, but each imperfectly competitive market is uncompetitive in its own way. Game theory allows us to construct economic models that are tailored to each special case and then to predict how different regulatory regimes will work out in practice after firms have figured out how to exploit all possible loopholes. It is a far cry from using psychohistory to reform the galaxy, but game theorists know, in principle, how to do this in a way that will work.

What game theorists need from the public is the kind of pressure that will hold vested interests in check and persuade government officials to spend money on researching the special strategic peculiarities of different markets. What the public needs from talented popular writers like Tom Siegfried is a serious assessment of what game theory really can do for you and me right now—and why we aren't being allowed to enjoy its potential benefits. The froth that we are offered instead is entertaining, but readers who want reliable information on what game theory is all about will need to look elsewhere.