Nash's theories are no longer confined to texbooks
By J.P. | São Paulo
JOHN NASH may be best known for being portrayed by Russell Crowe in the biopic “A Beautiful Mind”, but his essential and lasting contribution is in the world of economics. In 1994 he shared the Nobel prize in the field with two other scholars for work in “game theory”, which helps individuals and firms understand the way their own decisions affect the decisions of others (see article).
Yet until his untimely death on May 23rd, aged 86, Mr Nash was always first and foremost a mathematician. When he and his wife were killed in a car crash in New Jersey, they were on their way home from the airport after a trip to Norway, where days earlier he had picked up the Abel prize, one of the field’s most illustrious, for advances in the theory of “non-linear partial differential equations” (he shared it with Louis Nirenberg of New York University).
In pure maths, where Mr Nash turned his attention after his game-theory work in the early 1950s, such equations are used to analyse abstract geometric objects, such as “submanifolds of Euclidean space”. They may make non-mathematicians cringe, but the equations themselves are indispensable for understanding how fluids flow, chemicals react or gravity pulls. These days, engineers cannot imagine life without them.
Game theory, too, is a branch of mathematics—and no less complicated than abstract geometry. It was first formalised in “Theory of Games and Economic Behaviour”, a doorstop of a book published in 1944 by John von Neumann, a mathematician, and Oskar Morgenstern, an economist. But they concentrated narrowly on “zero-sum” games, where one player’s gain is another’s loss, and “co-operative” ones, where binding commitments can be made and enforced. Most interesting real-world situations do not fit into either category.
Only Mr Nash lifted these constraints, and unleashed game theory’s potential. Remarkably, he did it in four concise papers, none longer than a dozen or so pages. What was arguably his most important result—a proof that in any game with any number of players, each player has a strategy available to him which he has no interest in changing, so long as the other players’ stick with theirs—took up merely a page in the Proceedings of the National Academy of Sciences (and did not include a single formula).
Nowadays “Nash equilibria”, as such unimprovable sets of strategies came to be known, are no longer limited to economics and business textbooks. Plenty of phenomena are now thought of in terms of games.
But in the late 1950s an inability to square Mr Nash’s theories with human psychology led researchers to lose interest. Critics quibbled that the Nash equilibrium theorem presumes that players would have to choose a random mix of pure strategies for behaviour (such as “always back down when provoked” or “always attack”), yet people rarely resort to randomness when making decisions. Instead of flipping coins or rolling dice to decide what to do next, most people rely on deterministic reason, or equally deterministic instinct. As economists turned away from Mr Nash’s game theory, the man himself turned towards abstract geometry.
Then in the 1970s John Maynard Smith, a British biologist, suggested that Mr Nash’s probabilistic strategies can be thought of as characterising populations, not individuals. For instance, some male mule deer will, in ritual combat for mates, be hard-wired to back down when faced with a threat of real injury (call them “doves”), whereas others (“hawks”) will always attack, risking injury themselves. The mixture of hawks and doves then creates the effect of randomness whenever one male faces off against another (drawn randomly from the population). Natural selection will, the late Maynard Smith noted, move a population to a strategy that, if adopted by most members (eg, the majority of male mule deer being either “hawks” or “doves”), will make it impossible for some other strategy, introduced through genetic mutation, to outcompete the incumbents.
Such “evolutionarily stable strategies” pervade biological systems. They are also a special case of Nash equilibria. Although Maynard Smith’s seminal paper in the journal Nature, which he co-authored with George Price in 1973, does not cite Mr Nash explicitly, the influence is clear, says Jorgen Weibull of the Stockholm School of Economics (and a former chum of Maynard Smith’s).
Maynard Smith may have arrived at this collective interpretation of equilibrium independently, but Mr Nash hinted at it 20 years earlier. Tucked away near the end of his unpublished doctoral thesis from Princeton—a parsimonious 30 pages, generously double-spaced, including contents and acknowledgements—was the notion that an equilibrium of a game could be understood as the average behaviour of a population of individuals. Just as with Maynard Smith’s animals, whose choices were more random than reasoned, humans often behave in ways that defy the rational archetype of Homo economicus.
By the time Mr Nash's full dissertation was dug up in the early 1990s, just before he won his Nobel, Maynard Smith’s work was common currency. The retrieved thesis nevertheless helped renew intellectual ferment among game-theorists, recalls Peyton Young, now at Oxford University. Mr Young still uses Mr Nash's tools to develop, among other things, strategies for such demonstrably non-rational “players” as wind turbines on a wind farm, each of which causes turbulence that affects its and others’ power output. In only a few elegant pages and some equations, Mr Nash managed to capture many facets of a complex, ever-whirring world, and inspire the imagination of generations of scholars.
