on the floor—any floor within the United States. (The map represents the crumpled up piece of paper.) No matter where you place the map, there will be one point that is directly above the corresponding actual location in the United States. Applying the same principle to the players in a game, Nash showed that there was always at least one “stable” point where competing players’ strategies would be at equilibrium.
“An equilibrium point,” he wrote in his Ph.D. thesis, “is … such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed.”11 In other words, if you’re playing such a game, there is at least one combination of strategies such that if you change yours (and nobody else changes theirs) you’ll do worse. To put it more colloquially, says economist Robert Weber, you could say that “the Nash equilibrium tells us what we might expect to see in a world where no one does anything wrong.”12 Or as Samuel Bowles described it to me, the Nash equilibrium “is a situation in which everybody is doing the best they can, given what everybody else is doing.”13
Von Neumann was dismissive of Nash’s result, as it did turn game theory in a different direction. But eventually many others recognized its brilliance and usefulness. “The concept of the Nash equilibrium is probably the single most fundamental concept in game theory,” declared Bowles. “It’s absolutely fundamental.”14
Nash published his equilibrium idea quickly. A brief (two-page) version appeared in 1950 in the Proceedings of the National Academy of Sciences. Titled “Equilibrium Points in n-Person Games,” the paper established concisely (although not particularly clearly for nonmathematicians) the existence of “solutions” to many-player games (a solution being a set of strategies such that no single player could expect to do any better by unilaterally trying a different strategy). He expanded the original paper into his Ph.D. thesis, and a longer version was published in 1951 in Annals of Mathematics, titled “Non-cooperative Games.”