A Beautiful Mind

Thanks to Sylvia Nasar's book and the Hollywood movie, John Nash is a household name. His battle with schizophrenia is now well known, but what did he do to earn his Nobel Prize? Why was he a mathematical genius?

New Statesman, March 2002

A Beautiful Mind is a film about a mathematician, but it is not a film about mathematics. Consequently, since the film’s release, I have regularly encountered people who are moved by the extraordinary story of Nash’s battle with paranoid schizophrenia, but who are curious about why he was considered a genius and why he won a Nobel Prize.

So, if you are one of those people who people who want to know more about Nash’s genius and less about Russell Crowe’s temper tantrums, then here is a crash course in Nash equilibria, game theory and billion dollar auctions.

Game theory is simply the mathematical study of the strategies used to win games. It began with the study of games like 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 like 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, and each party develops a strategy based on their 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 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, and in particular he identified the Nash equilibrium, a situation 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) licenses, 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 licenses 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 awarded the licenses to the wrong companies.

In contrast, economists now encourage the auctioning of licenses, which earns money for the government and awards licenses to the companies than can make best use of them. Using Nash’s equations, economists treat the auction like a game and construct the rules to achieve the seller’s goals, which might be maximising income or encouraging new companies or a mix of the two. Moreover, the game theorist will pick the rules so that the Nash equilibrium puts everybody in a win-win situation.

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.

You might think that running an auction is common sense, but a recent Turkish one went horribly wrong, because the bidding rules were poorly constructed. Two licenses were auctioned sequentially, with the rule that the second one would have a reserve price equal to the sale price of the first one. One company appeared to ridiculously overbid for the first license, but they knew exactly what they were doing. The reserve price for the second license was now so high that nobody bought it, therefore the first company had a monopoly, which then made the price it paid for the one and only license seem more than reasonable. A monopoly was an unforeseen and undesirable outcome.

In contrast, the UK 3G auction was a great success and raised a phenomenal £22.5 billion. 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 via future profits. 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."

But doesn’t this mean that the phone companies will pass on the cost of the license to consumers and charge us more? Binmore feels that this is a naive view. He believes that companies will always charge as much as they can, rather than as much as is necessary to make a moderate profit. Hence, the license will reduce the profits of the shareholders rather than increase the price paid by phone users.

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 tragedy of his insanity, which is the subject of Ron Howard’s film? It is undeniable 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."

You can find out more about John Nash  and the film 'A Beautiful Mind' by clicking on the links below:

The Man:
Read an American Mathematical Society article about John Nash, his work and his life. 

The Book:
See what Sylvia Nassar has to say about her book 'A Beautiful Mind', and find out what others thought of it.

The Film:
You can find out more about the film by visiting its official website, or reading the Hollywood.com guide.