Gabriele Gratton

John Forbes Nash died last Monday. Economist and mathematician, winner of the 1994 Nobel Memorial Prize in Economics, Nash made substantial contributions during the 1950s to algebraic geometry, non-linear differential equations, and some fundamental aspects of economic theory.

He also invented a beautiful board game now known as Hex, but whose early name at Princeton was simply “Nash”.

But economics students all over the world know him for the equilibrium that bears his name and that economists seem to use to predict what people will do in almost any situation.

The game-theoretic concept we know as the Nash equilibrium is at the core of the economic analysis of strategic interactions: each person (player) can take one of many actions, the outcome of which depends also on the actions taken by other persons.

Game theorists believe that when a person is in a strategic interaction she should – so to speak – put herself in the shoes of her opponent and wonder what he is going to do before choosing what to do herself.

In doing so, she will realise that her opponent is also doing the same: he is wondering what she is going to do before choosing what to do himself.

The implication is that she should do what is best for her, given that her opponent is doing what is best for him, given that she is doing what is best for her, and so on forever. This mind-blowing infinite loop finds a solution in a Nash equilibrium.

In essence, a Nash equilibrium indicates to each player to follow a certain plan – perhaps different for each player – and, if a player expects her opponent to follow his plan, then she would have no incentive to do anything different than following her own plan.

Thus, a Nash equilibrium is a good description of how persons would behave in a strategic interaction, as they would find it optimal to do so given what they think all other persons are actually doing.

This basic idea pre-dates the work of Nash. Early examples of applications of what we know today as Nash equilibrium are already present in the 1838 work of French economist and mathematician Antoine Augustin Cournot.

The first systematic study of games and equilibria is probably the 1944 book, Theory of Games and Economic Behavior, by mathematician John von Neumann and economist Oskar Morgenstern.

But it was Nash, in 1950, who gave us the fundamental tools to study games. In as little as 32 pages of PhD dissertation, Nash proposed his equilibrium as the correct solution of any game and, most importantly, he showed that a Nash equilibrium indeed existed for all (finite) games: the mind-blowing infinite loop has a solution. Always.

After Nash’s work, his equilibrium has been applied to almost anything. It helped us understand the military and diplomatic strategies of the Cold War, how to promote efficiency in markets and deter crime.

It is extensively used by economists, political scientists and evolutionary biologists alike. And it recently proved very useful in designing policies to help patients in need of a kidney to find a suitable donor, saving perhaps hundreds of thousands of lives.

The Nash equilibrium continues to be a cornerstone of economic analysis and researchers all around the world keep investigating and perfecting it.

UNSW Business School is no exception, where researchers in the school of economics have in recent years employed the Nash equilibrium to study electoral systems, terrorist activity and how government should react to it, how rules develop and how information spreads in organisations.

Some even keep up with Nash’s quest to find an ever more compelling equilibrium concept.

Nash’s fundamental contribution is taught at all levels, from undergraduates studying game theory to PhD course work in economic theory. Some of our honours students have also learned to play Hex.

Every economics student has heard of him and, most importantly, seen the power of his ideas at work.

Gabriele Gratton is a lecturer in the school of economics at UNSW Business School.