Nash equilibrium

The mathematician John Nash, who died on Sunday, provided a foundation for economics that is rich, not dismal

Tony Curzon Price
Tony Curzon Price
24 May 2015

John Nash's great contribution to economics - the one for which he was awarded the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1994 - is his proof, in his 1950 Princeton PhD thesis, of the Fundamental Theorem of Game Theory which states, roughly, that every game has at least one solution.

A little more needs to be said: a game in this context is a set of possible actions and interactions between players where each outcome has some value to each player. And a solution is that the game has a Nash equilibrium, where this means that it is true of each player that they are doing the best they can given what the others are doing. The key breakthrough in Nash's PhD is in this concept of the equilibrium. I am doing the best I can do given what you're doing; and you're doing the best given what I'm doing. It is natural to call such a situation an equilibrium because, if you're in it, there's no good reason to deviate from it. Since everyone's doing best, no one has a reason to deviate.

To me, there are two very striking things about the result:

  1. It is absolutely not obvious that all games would have such stable points. It is quite plausible that many human interactions can be represented as games. But it could quite easily have turned out that optimal behaviour could not be determined in the way that Nash's Fundamental Theorem suggests.
  2. It presents a view of stability and optimality that is very relative: a course of action is best for me only because another course of action is best for you, which is itself only best for you because a course of action is best for me ... which is only best for me because that one is best for you ... and so on. When Marx, in the Communist Manifesto, speaks of a future in which "The free realisation of each becomes the condition for the free realisation of all," I take him to be describing a Nash Equilibrium: a state in which I'm doing what leads to my self-realisation because it is a best response to what you're doing for your self-realisation and vice versa.

There are many important questions on which the Fundamental Theorem is completely silent. Probably the most important in terms of social outcomes is the question of how players rank different outcomes. There is nothing intrinsic to game theory that says that preferences should be self-regarding or that players should not care about the pay-offs to others. That is a layer of psychology and sociology on top of Nash's mathematics and utterly separable from it. Nash's result will apply as much (or, perhaps, as little) in a den of thieves as in a paradise of saints.

Once preferences are given, Nash points the way to some powerful shaping forces for social outcomes. These are shaping forces rather than hard and fast predictions, because the result requires some stringent conditions to be met, especially about the nature of rationality and the form of common knowledge - conditions which are too strong to be realistic and are rightly questioned in many disciplines - psychology, sociology and anthropology.

Nash Equilibrium gave economics foundations that have two features we don't usually associate with the dismal science: first, it emphasises the stability of optimal outcomes; and second it addresses the question of our collective interdependence. However dismal the uses to which the theory be put, these two features make it capable, I think, of formulating important practical truths for our common existence. The second is particularly striking and is worth contrasting with the usual telling of Adam Smith's invisible hand. Under the invisible hand, I can do what is best for me regardless of what you are doing because the hand will guide society to the best outcomes. Under Nash, there's a very visible minds (not hand), that, in deciding what I should do, needs to think about what you should do. The mythology of the invisible hand is a socially isolating one - one in which I can be a social atom - whereas the world as represented by Nash equilibrium is one in which we must always consider interdependence. 

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