How can you predict what people will do when each one's best move depends on the others?

You look for a Nash equilibrium: a combination of choices where nobody can do better by changing their own move while everyone else keeps theirs. The trick, John Nash realized, is that you can't predict people one at a time. You have to ask what each person would do given what they expect the others to do, and look for the point where all those expectations hold up at once.

Take driving. If everyone drives on the right, nobody gains by switching to the left alone; the same is true if everyone drives on the left. So that game has two equilibria, and everyone simply has to coordinate on one. In the prisoner's dilemma, by contrast, there is only one: both betray, even though both would be better off cooperating. An equilibrium is stable, not necessarily good.

Some games have no stable pure choice at all. In matching pennies, one player wins if the coins match and the other if they don't, so any predictable choice can be exploited. The only equilibrium is to pick heads or tails at random, 50/50, which leaves the opponent nothing to exploit. Game theorists use the same game to study penalty kicks in football.

Nash's big result, in a 28-page doctoral thesis in 1950, was that every finite game has at least one equilibrium, if you allow such random mixes. The idea earned him a share of the 1994 Nobel Prize in Economics, decades after illness had interrupted his career.