John Nash's 'beautiful mind' opened whole new vista for mathematics
Many mathematicians, including fellow Nobel laureates, have delved further into the game theory he pioneered at the age of only 22
When I read the tragic news of John Nash's death at age 86 in a car crash last week, my first thought was not of his Nobel Prize in economics, nor of his redemptive life story dramatised in the film . The memory that came to mind was of a summer night in 2002 and an event in the faculty club at Princeton University.
The room was abuzz with anticipation: Nash was guest of honour and scheduled to make some remarks - rare for the eccentric mathematician.
A hush fell. My host whispered: "Nash is here." I saw a tall, stooping, white-haired figure enter the room and shake a few hands. Then he turned around and left without a word.
But that was part of his lingering legend as "the phantom of Fine Hall". He was often spotted wandering the corridors of Fine Hall, where Princeton's mathematics department is located, during his decades of mental illness. By the time I saw him he had recovered from his illness, received the 1994 Nobel Prize, and lived as normal a life in Princeton as fame from the 2001 Oscar-winning film allowed.
It was at Princeton in 1950 that Nash submitted a 27-page, double-spaced doctoral dissertation that pioneered modern game theory and its application to economics. He was 22 years old, only two years into his PhD programme.
At the time, Princeton was the centre of the mathematics universe: Albert Einstein, the polymath John von Neumann, the logician Kurt Gödel and other intellectual titans were working there. Yet Nash developed his theory alone.
He was not the first to study game theory, which is the mathematics of conflict and cooperation between rational decision-makers in which each person's behaviour will affect the other's welfare.
In the 1920s, von Neumann began to examine how decision-makers behave in competitive situations and to analyse the mathematical structure of that behaviour. In 1944 he and Austrian economist Oskar Morgenstern published their "Theory of Games and Economic Behaviour", the first attempt to derive logical and mathematical rules about rivalries.
But their focus was the zero-sum game in which one player's gain is another's loss. Zero-sum games, such as chess or tennis, have a winner and a loser.
Real-world interactions, however, are seldom zero-sum. Interaction between parties usually provides the opportunity for mutual gain.
For example, Iran and the United States both have their own expectations as they negotiate limits on Iran's nuclear capability. Each deploys a negotiating strategy to affect the other's expectations while realising its own as much as possible.
On the level of individuals, an example might be a couple who want to go out for the evening. They decide to go either to a ballet or to a boxing match. Both prefer to go together rather than going alone.
While the man prefers to go to the boxing match, he would prefer to go with his wife to the ballet than to the fight alone. Similarly, the wife would prefer to go to the ballet, but she too would rather go to the fight with her husband than go to the ballet alone.
This situation represents a non-zero-sum, non-competitive conflict. The common interest between the husband and wife is that they would both prefer to be together than go to the events separately. The opposing interest is that the wife prefers to go to the ballet while her husband prefers to go to the boxing match.
Nash resolved the non-zero-sum and non-cooperative game where players make decisions independently with the concept of equilibrium. He provided the mathematics to analyse what will happen if two or more parties are making decisions and the outcome depends on the decisions of the other(s).
One must ask what each player would do, taking into account the decision-making of the others. The strategy of each party will lead to an equilibrium - the Nash equilibrium - if no party can benefit from changing this strategy if others do not change theirs.
Applying this approach to the couple's ballet-or-boxing dilemma, three Nash equilibria are possible: One is a mixed strategy where each person attends either event for a certain proportion of time, and the other two are pure strategies where both parties either go to the ballet or boxing.
Another powerful concept that Nash described in his dissertation is that every game that can be characterised by his mathematical model has an equilibrium point. So, theoretically, every conflict that can be modelled as a game involving multiple parties under various circumstances has conditions where each party is getting as much as it can, and there is no incentive to continue the conflict by changing their positions.
An arms race between two countries need not escalate indefinitely. It can stop when certain conditions are met. It is up to the countries involved to adopt strategies that create this condition, the Nash equilibrium.
Nash's work opened up a new vista of research. Problems analysed through the lens of game theory now include market structure, entry and exit, bargaining, auctions, insurance, public policy, arms control and even biology. Eleven game theorists have received the Nobel Prize in economics, reflecting its central role.
Nash's contribution to the science of decision-making is far from a state of equilibrium.
