# Remembering John Nash

The news of his death brought back memories of my first meeting with Nash. In 1989, I was a visiting professor at Princeton. By then Nash’s schizophrenia was in remission. He could be seen strolling in the Princeton lawns, hours on end. For the residents of Princeton this was part of the fixture and so barely noticed. For me and Jorgen Weibull, also a visiting professor, it felt strange. There we were in classrooms analyzing or applying the “Nash equilibrium” and the “Nash bargaining solution” and the man, after whom these concepts were named, was out there, pacing the yard, immersed in his own world.

It was through Jorgen’s painstaking effort that we managed to get Nash to join us for lunch in the university cafeteria. It was exciting to be with a person who was known for his genius, even though he spoke little and, every now and then, seemed to drift off into his own thoughts.

The two vivid memories that stand out for me from that remarkable afternoon was, first, his answer to my question about the origin of his name. He said it came from the Sanskrit nashika, meaning nose. The second was the reaction of Abhijit Banerjee, as he, seeing Jorgen and me, came and joined us at the table, and was introduced to Nash. It was like a young literature student joining friends for lunch and being told that the third person at the table was Shakespeare.

In terms of influence on modern economics and game theory, John Nash has few peers. His work provided foundations for analyzing both cooperative and non-cooperative interactions among rational agents in economic and political settings. In addition, he made contributions to various branches of pure mathematics, including differential geometry, and isometric embeddability of abstract Riemannian manifolds (whatever that may mean), with implications for not just economics, but computer science and evolutionary biology.

This is absolutely remarkable given the brevity of his creative life. His most important papers were written before he was 25; and his creative period was over by the time he was in his late twenties, felled by schizophrenia. By the age of 30 he had to be confined to a mental hospital, and the next thirty years were a battle with paranoia, voices in the head and delusions.

John Nash was born on June 13, 1928, in Bluefield, West Virginia. He was recognized early as a prodigy; he got his PhD in mathematics from Princeton at age 22. His PhD thesis was as short as his productive life—28-pages. One of his most celebrated papers—that lays out the conditions under which we can be sure of the existence of a non-cooperative equilibrium in games—is just one page and a few lines. The entire paper was reprinted on the T-shirt designed by graduate students in the economics department at Cornell in the late 90s.

Of Nash’s many contributions the one that got maximum play is that of the “Nash equilibrium,” which is used to understand the behavior of oligopolistic firms, movements in financial markets, political rivalry and strategizing in conflict, such as during the Cuban missile crisis.

The basic idea of a Nash equilibrium is simple. Consider a group of agents, each of whom has to choose an action or strategy; and the payoff or utility that each person gets depends not just on that person’s choice but on what other agents choose. To take a stark case, you and the oncoming car each can choose which side of the road to drive on. Your well-being depends on your choice and also the other driver’s. A Nash equilibrium is a choice of action on the part of each person, such that no one can do better by unilaterally deviating to some other action.

In economics we know that (under some assumptions) a competitive equilibrium takes society to an optimal outcome. Each individual, acting in his or her self-interest, takes society to an optimum. This was the broad idea that led to the orthodoxy of the invisible hand in economics. Nash equilibrium gained so much currency because it challenged this central idea by illustrating how perfect individual rationality can lead a group to a collectively bad outcome. We can see this in the area of CO2 emissions, the provision of public goods and problems related to the commons.

Here is a simple game theory puzzle for the reader, which illustrates this. Two players are told to each choose an integer from 2 to 100. If both choose the same integer, they are told they will get that number in dollars. But if they choose different numbers, they get the lower of the two numbers with a small adjustment: The person who chose the smaller number gets an additional $2, and the person who chose the higher number has $2 deducted. The question for the reader is this: What choice of numbers by the two players constitutes a Nash equilibrium? This has a unique answer. Try to figure out the answer at leisure. The next time I met Nash was after he had become a worldwide celebrity. He had won the Nobel Prize, sharing it with John Harsanyi and Reinhart Selten, in 1994, and was the subject of a popular Hollywood film, A Beautiful Mind, in which Russell Crowe acted his part.

He was part of a conference in Mumbai in January 2003 that brought several prominent economic theorists to town, including Robert Aumann, Roger Myerson and Amartya Sen. The audience for Nash was large, with some recognizable faces from Bollywood, who may have come expecting to see Russell Crowe. The talk was disappointing, as Nash tried to address some practical policy questions. He was too much of a hedgehog, in the sense of Isaiah Berlin, and so was good at focusing on one thing very deeply, and too unlike the fox to be able to range over many topics, even superficially.

The next day, much to my surprise, as I prepared to give my talk, Nash came in and sat in the front row. I was very thrilled by his presence—the full five minutes that he was awake.

*An edited version of this post first appeared in the Indian Express on June 3, 2015*.

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