The Region

Nash's Nobel idea

Published September 1, 1998  |  September 1998 issue

When one billiard ball hits smack into an identical ball, the first stops dead in its tracks and the second starts moving in the same direction, and at the same speed, as the first was going. This fact has been known since the beginning of time or at least since—practically as long ago—the game of billiards was invented. Surely, you would think, only a charlatan or a fool would take the trouble to express this obvious fact as a mathematical formula and then have the nerve to proclaim it as a fundamental scientific advance. Yet that is essentially what Isaac Newton did when he formulated the conservation of momentum as one of his three laws of physics. In fact, rather than being an uninterpretable tangle of math hieroglyphics and Greek letters, a fundamental scientific law is more often a suggestion that something obvious but not previously recognized as worthy of attention actually has far-reaching implications.

So it is with the idea of "Nash equilibrium" of a game, for which Nash won the Nobel Prize in 1994. Economists use the word "game" to refer to any situation in which people have to worry about what one another are going to do. By an equilibrium, economists mean a specification of what those people do and why they choose to do it. Let's consider two brothers, Click and Clack, who live five miles apart, connected only by a two-lane road with a footpath beside it. Click and Clack each own a jalopy, and their lives revolve around travelling up and down the road and waving at one another as they pass. Click and Clack each have three choices: to drive on the right, to drive on the left or to walk. Each would like to take a spin in his jalopy if the other is going to make a compatible choice, but would prefer to walk rather than to have a collision. Without being connected by a telephone, they cannot coordinate their choices. This is a game. What is going to happen, and why? That is, what will be an equilibrium?

Back in the dark ages before Nash, economists used to analyze this situation by supposing that Click and Clack would each make a choice that offers the best insurance against the worst-case choice by the other. If Click drives on the right, then the worst-case choice by Clack would be to drive on the left when going in the opposite direction. A collision would result. Similarly, Click would suffer a collision with Clack if he chose to drive on the left and Click made the worst-case choice of driving on the right. No matter which choice Clack makes, though, all that happens to Click if he walks to Clack's place is that the trip takes longer than he wishes—his beloved jalopy is not wrecked. So, among the worst-case outcomes of the choices that Click can make, his choice to walk will ensure the best (or, at least, the least bad) of the lot. That is the prediction that the dark-ages economist would make regarding Click's choice, and by parallel reasoning the economist would also predict that Clack would walk.

In reality, all over the world, there are drivers who do not explicitly communicate with one another but who nevertheless enjoy their jalopies because they expect not to get into collisions most of the time. Americans may expect to keep safe by driving on the right while British drivers expect to keep safe by driving on the left, but the fact that either convention can work does not contradict the point: that people make choices by basing their forecasts of benefits and costs on rational expectations of what other people are going to do—not on pessimistic conjectures about worst-case outcomes—and that the various people's expectations are rational because the resulting choices make them mutually self-confirming. This, in a nutshell, is Nash equilibrium. Simple, isn't it? But it does explain all those cars being on the road, and much else besides, as the deliberate reference to "rational expectations" ought to suggest.

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