fied with the strategy they’ve adopted, in the sense that no other strategy would do better (as long as nobody else changes strategies, either). Similarly, in social situations, stability means that everybody is pretty much content with the status quo. It may not be that you like things the way they are, but changing them will only make things worse. There’s no impetus for change, so like a rock in a valley, the situation is at an equilibrium point.

In a two-person zero-sum game, you can determine the equilibrium point using von Neumann’s minimax solution. Whether using a pure strategy or a mixed strategy, neither player has anything to gain by deviating from the optimum strategy that game theory prescribes. But von Neumann did not prove that similarly stable solutions could be found when you moved from the Robinson Crusoe–Friday economy to the Gilligan’s Island economy or Manhattan Island economy. And as you’ll recall, von Neumann thought the way to analyze large economies (or games) was by considering coalitions among the players.

Nash, however, took a different approach—deviating from the “party line” in game theory, as he described it decades later. Suppose there are no coalitions, no cooperation among the players. Every player wants the best deal he or she can get. Is there always a set of strategies that makes the game stable, giving each player the best possible personal payoff (assuming everybody chooses the best available strategy)? Nash answered yes. Borrowing a clever piece of mathematical trickery known as a “fixed-point theorem,” he proved that every game of many players (as long as you didn’t have an infinite number of players) had an equilibrium point.

Nash derived his proof in different ways, using either of two fixed-point theorems—one by Luitzen Brouwer, the other by Shizuo Kakutani. A detailed explanation of fixed-point theorems requires some dense mathematics, but the essential idea can be illustrated rather simply. Take two identical sheets of paper, crumple one up, and place it on top of the other. Somewhere in the crumpled sheet will be a point lying directly above the corresponding point on the uncrumpled sheet. That’s the fixed point. If you don’t believe it, take a map of the United States and place it

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement