A Beautiful Theory: But It Is Only a Game

Listener 13 July, 2002.

Keywords History of Ideas, Methodology & Philosophy

The film A Beautiful Mind was another version of Love Story. Instead of the lovers having to overcome health, race, other obligations, or location, this time the obstacle was mental illness. Ultimately John and Alicia Nash triumph, with a Nobel Prize in economics to boot. The true story is far more complicated, and in some key places different. More disappointing, the film made only the feeblest attempt to explain what Nash actually did. Hollywood must have thought the concept too difficult for the average film-goer. Let me accept the challenge.

‘Game theory’, the topic for which Nash was awarded the Nobel, might better be called ‘interactive decision theory’. It is not really about games like netball and rugby, but bridge, chess, Diplomacy, draughts, go, Monopoly and poker (which can all be played against a computer). There are a well defined set of rules. There are two or more players each of whom has to choose a strategy. The outcome may be ‘zero-sum’ (the winners’ gains are offset by the losers’ losses) or ‘non-zero-sum’ (where there is a net gain in total). The players may be allowed to cooperate, or not. They may have complete or incomplete information.

Game theory explores whether there is an optimum strategy, which will generate the best possible outcome for each player. In some games (e.g. noughts-and-crosses) the strategy is known. In others (e.g. chess) the best strategy is not known. Nash’s greatest contribution was to prove that in a very wide range of games (including all those mentioned above) there is an optimum strategy. (Even for chess – it is just that we dont know it.) The outcome of optimum strategies generates a ‘Nash equilibrium’, a term which will give Nash immortality. (His later work involved fundamental mathematics. It seems possible that but for his mental illness he would have won a ‘Field Medal’, the Nobel Prize in mathematics. Noone has won the mathematics Field and the economics Nobel.)

New Zealand economist John McMillan writes in Games, Strategies and Managers (a book I recommend to the general reader) that ‘game theory cannot, by its nature, give all the answers to how any particular strategic situation works.’ Its importance is the conceptual framework it offers.

Initially economics was not greatly interested in game theory when it focussed on zero-sum games, for the discipline is inherently about both sides benefiting. Many people see all or a lot of economic activity as zero sum, as when they think that if someone made a profit it must be at the expense of someone else. Economists think there are often gains to both in a trade.

When game theory was extended to non-zero-sum outcomes, economics hit a different problem, captured in the well known game ‘the prisoners’ dilemma’. It concludes that if two prisoners compete against one another the outcome is worse than if they cooperate. So it challenges the assumption that competition always gives the best results. Again you can see economists wrestling within the underlying concepts of game theory.

Game theory may have also led to a whole new branch of ‘experimental economics’. When the prisoners’ dilemma game was first identified, they played it to see whether people actually followed the logical behaviour. The overall conclusion is ‘sort of; sort of not’. (Among the developments is the ‘Ultimatum Game’ of my column of 23 December, 2000.) A practical application of game theory has been the design of efficient auctions.

Nash’s biographer, Sylvia Nasser, says that game theory was important in the early 1950s to breaking the cold warrior’s obsession with zero-sum nuclear war. It could not tell them what the best strategy was, but non-zero-sum games pointing to the potentialities of cooperation led to the development of limited responses rather than all-out nuclear war.

So it was absolutely right to award its instigators a Nobel Prize in economics. Nash was accompanied by Hungarian John Harsanyi and German Reinhard Selten, who extended Game Theory in different ways to Nash. In some ways the awards were really to Hungarian (and polymath) John von Neumann and Austrian Oscar Morgenstern, whose seminal Theory of Games and Economic Behaviour triggered the development of game theory. The book was published in 1944, but they both died before Nobel economic prizes were instigated. Exactly fifty years later, the Nobel Prize committee awarded the prize to game theorists to recognize the earlier achievement too.

So Nash’s story is empowering to those who have had to struggle with mental illness personally or in the family, and his work has enriched our understanding of how humans can behave. He offers one further lesson. Asked how his paranoia led to a belief in aliens, he said ‘the ideas I had about supernatural beings came to me the same way that my mathematical ideas did.’ But mathematics is about the deep elegant world of logical systems. Aliens are, or are not, about the real world. The criteria for truth in the two are quite different. Mathematics is essential for modelling , but there has also to be a test of reality. Sometimes non-paranoiac economists forget the distinction.