The great game

Mathematics - Simon Singh on "Nash's equilibrium", the brilliant legacy of an unstable mind

A Beautiful Mind is a film about a mathematician, but it is not a film about mathematics. It concentrates on John Nash's battle with schizophrenia, and barely touches on his great mathematical achievements, which are mentioned only in bar-room scenes or hinted at via arcane equations scrawled on window panes.

To try to complete the picture, it is important to understand Nash's passion - game theory - and how his contribution to that subject has had a huge influence on modern economics. In just a few short years, a man barely out of his teens laid the foundations for a discipline that had an enormous impact on economies all over the world.

Game theory, put simply, is the mathematical study of the strategies used to win games. It began with the study of such games as noughts and crosses and chess, which are relatively easy to analyse because they are games of "complete information" - in other words, each player can see the other's position.

Then mathematicians became interested in games such as poker, which is much more interesting because players cannot see each other's cards. Poker is a game of "incomplete information", so more subtle elements such as bluff come into the analysis.

Eventually, mathematicians attempted to analyse more important games, including economics, warfare and divorce settlement. In each case, you have two parties competing over money or territory; each party develops a strategy based on its own strengths and objectives, and on the perceived mindset and skills of their opponent. Game theory is maths plus a dash of psychology.

And the man who did more than anyone else to apply game theory to the real world was John Nash. Between 1950 and 1953, Nash published four papers that revolutionised game theory. Still in his early twenties, he conducted a deep analysis of a special set of games that were said to be non-zero sum.

In most games, including chess, there is a zero sum, which means that if I win, then you lose, or vice versa. But in a non-zero-sum game, both players can win . . . or both can lose. For example, pay negotiation between management and a trade union can be a non-zero-sum game. The result can be a long strike that hurts both sides, or a fair agreement that benefits both sides.

Nash enshrined his theory in mathematical equations; in particular, he identified a situation, later known as the Nash equilibrium, in which both players have a perfect strategy that results in stability. Players maintain this strategy because anything else will only worsen their own position.

In recent years, Nash's greatest legacy has been in the awarding of third- generation (3G) licences, which assign blocks of radio frequencies to companies so that they can develop video and the internet for mobile phones.

The traditional method for assigning licences involved ministers and civil servants considering proposals from each company and deciding intuitively which was best. This "beauty contest" approach raised little or no money, and often led to the licences being awarded to the wrong companies.

In contrast, economists now encourage the auctioning of licences. Using Nash's equations, economists treat the auction like a game and construct the rules to achieve the seller's goals. The game theorist will optimise the rules of the auction, which involves setting the reserve price, deciding whether to request sealed or open bids, deciding if lots should be sold simultaneously or consecutively, and fixing the penalty for a successful bidder who then defaults on payment.

The UK 3G auction was a great success and raised a phenomenal £22.5bn. Critics have argued that the companies paid too much, but the mathematician who designed the auction argues that the companies paid what they knew they could recoup in future profits. Professor Ken Binmore of University College London argues that "a carefully designed auction achieves this end by creating a competitive environment in which the bidders are forced to put their money where their mouth is".

Binmore's work is a direct consequence of Nash's brilliant mathematics, which enshrined the essence of bargaining, bidding and negotiation within a rigorous framework. But how does the story of his research tie in with the tragedy of his insanity, which is the subject of Ron Howard's film? It cannot be denied that Nash was unable to do research when his schizophrenia took over. However, I still remember the opening page of Sylvia Nasar's biography A Beautiful Mind, the basis for the film, which recounts how a friend visiting Nash in hospital asked how he could believe that aliens were recruiting him to save the world. Nash simply replied: "Because the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously."

Theatre of Science, Simon Singh's show about game theory and risk, is at London's Soho Theatre, W1 (020 7478 0100), in April