John Nash changed economics with his work on game theory. It won him the Nobel Prize. Photo / AP
John Nash changed economics with his work on game theory. It won him the Nobel Prize. Photo / AP
The world will be a poorer place without John Nash, one of the pioneers of game theory, who died on Saturday at 86 in a car crash along with his wife.
Most Nash retrospectives (and the movie about him) focus considerable attention on his mental illness, but I think thisis overdone. We should celebrate people's ideas, not obsess over their personal struggles. And Nash's ideas were in a class of their own.
We think of Nash as an economist, because he won the Nobel Prize in economics (in 1994). But, like people at the very top of many quantitative fields, he thought of himself as a mathematician. Nash's fatal car crash came just as he was returning from receiving the Abel Prize, a prestigious math award, for solving a thorny geometry problem back in the 1950s. Economics was just one more target for Nash's mathematical brilliance.
Nash actually invented several things in economics. His first major foray was the theory of Nash bargaining, which is really just a method of bargaining that satisfies some intuitive conditions of fairness. The basic result is that a "fair" bargain is one where both parties benefit equally from accepting the deal.
Nash figured out the math of this result in a paper for a college class, and his solution is still used in many applications to this day. For example, Nash bargaining is the method that labour economists often assume is used to split the "surplus" that results when an employer and an employee find each other.
Nash with his wife Alicia at the Academy Awards in 2002. Both were killed in the taxi crash. Photo / AP
But as Rotman School of Management economist Kevin Bryan points out on his blog, Nash bargaining isn't very realistic. In the real world, bargains aren't fair - they depend on things such as outside options and threats. To find out what really happens in bargaining and many other competitive situations, you need to use Nash's truly monumental contribution: game theory.
When two or more people play a game, what will they do? Game theory is meant to answer that question. It actually existed before Nash, but Nash developed a concept to select from among all the different possible game outcomes. This is known as the Nash equilibrium.
It's actually pretty simple to explain - it means a situation in which everyone is making their best choice, given the choices of the other players in the game. That choice could include randomisation, e.g. if I decide to randomly rock, paper and scissors one-third of the time in a game of rock-paper-scissors. When there's a Nash equilibrium, it means that everyone is making their best response to what the other people in the game are doing.
That concept turned most of economics on its head. Before Nash, the dominant idea of "equilibrium" in economics was the Walrasian equilibrium. It was sort of a Panglossian idea - everything that people want to sell gets bought, and markets are efficient - no money gets left on the table, so to speak.
But in Nash equilibrium, you can get outcomes that are far darker - for example, in the Prisoners' Dilemma, where two rational prisoners will always betray each other, leading to a worse outcome for both than if they had both kept silent.
Russell Crowe stars as John Nash, a mathematic genius fighting his own personal demons in the Oscar-winning film, A Beautiful Mind. Photo / AP
When you go from Walrasian equilibrium to Nash equilibrium, you get all kinds of non-optimal economic outcomes. You can get markets that break down completely, because no one can trust their counterparties. You can get companies hurting the economy by competing with each other too much.
Suddenly, you go from a perfect free-market paradise to an uncertain, confusing thicket of potentially destructive competition. Economists before Nash had created a few isolated models of strategic interaction, but Nash created a system.
The game theory concepts that Nash's math brought to the field were a true paradigm shift in economics. Macroeconomists, who continue to use the old Walrasian notion of equilibrium typically engage in hand-waving about how the macroeconomy is too big for strategic interactions to matter.
But most of the economics profession has gradually shifted toward Nash equilibrium. The 2014 Nobel winner, Jean Tirole, is emblematic of the new economics. And many of the biggest successes in applied economics, like the auctions that power Google's advertising, rely on Nash's technique.
Of course, Nash's concept of game theory is probably not the end of the story. For Nash equilibrium to give reliably good predictions, you probably need common knowledge of rationality - that's when I know you're rational, and you know I know, and I know you know I know, and so on. That is a pretty strict condition. That is probably a good approximation for companies bidding over Google ad space, but in some situations people may suspect that others are irrational. For those situations, it may help to go beyond Nash equilibrium, and economic theorists are now working on these extensions.
But in the end, economics will be John Nash's field. We owe him as much as we owe Adam Smith. Not bad for a guy who probably wouldn't even have called himself an economist!
