More than 70 years ago mathematicians started realizing that analyzing simple parlor games could illuminate many situations in which people compete with one another and have to decide what strategy to adopt. The principles they discovered have shed light on subjects from how nations interact in a nuclear arms race to why some organisms cooperate with one another. And in one of its most striking successes, game theory has led to a revolution in the way economists understand auctions.
The renowned Hungarian mathematician John von Neumann, a lecturer at the University of Berlin at the time, launched the field in 1928. He was curious about how game players should choose their strategies: When, for instance, should a poker player bluff? He studied two-player “zero-sum” games, such as chess and tic-tac-toe, in which the players’ interests are entirely at odds: in the simplest manifestation, one player’s gain is the other player’s loss. As any child knows, in tic-tac-toe both players can avoid losing; if they each follow their best strategies, they force the game to end in a draw. Von Neumann proved that in any two-player zero-sum game, not just in tic-tac-toe, there is a certain “right” outcome, in the sense that neither player can reasonably expect any better outcome unless the other player makes a mistake. This implies, for example, that if two chess players follow their best strategies, the game will always have the same outcome. Luckily for the excitement of the game, however, no one has ever figured out what that outcome is—a win for white, a win for black, or a draw?
Von Neumann and economist Oskar Morgenstern of Princeton University became convinced game theory would illuminate economic questions, and in 1944 they published a book, The Theory of Games and Economic Behavior, arguing that point. At the time, the prevailing approach to economics was to look at how each individual responds to the market as a whole, not how individuals interact with each other. Game theory, von Neumann and Morgenstern argued, would give economists a way to investigate how each player’s actions influence those of the others.
Von Neumann and Morgenstern’s book analyzed zero-sum games and cooperative games, in which players can form coalitions before the game starts. But many economic interactions don’t fall into either of those categories; for instance, von Neumann and Morgenstern’s cooperative framework doesn’t apply to situations in which the players have valuable secrets to preserve. For that reason, although cooperative game theory was useful for studying certain economic questions, such as problems of supply and demand, it was less useful for such subjects as auctions.
In the late 1940s mathematician John Nash, then a young graduate student at Princeton, realized that in any finite game—not just a zero-sum game—there is always a way for players to choose their strategies so that none will wish they had done something else. In 1949 he wrote a two-page paper whose ideas would change forever how economics research is pursued. Nash came up with the notion of a “strategic equilibrium”: a collection of strategies, one for each player, such that if all the players follow these strategies, no individual player has an incentive to switch to a different strategy. In the setting of two-player zero-sum games, Nash’s equilibrium gives exactly the same solution as von Neumann’s analysis. But Nash’s concept goes far beyond this scenario: He proved that even non-zero-sum games and games with more than two players must have at least one equilibrium.
Consider, for example, a three-person “duel” in which Alex, Barbara, and Chris will fire simultaneous gunshots at each other once every minute. Alex and Barbara are sharp-shooters who hit their target 99 out of 100 times. Chris, however, only makes his shot 30 percent of the time. Surprisingly, if all the players follow their equilibrium strategies, Chris is the most likely to survive! Alex and Barbara’s equilibrium strategy is to fire first at each other, since it is in their best interest to kill their most dangerous opponent first. The most likely outcome is that Alex and Barbara will kill each other on the first shot, and Chris will escape unharmed.
In some games the Nash equilibrium predicts an even more counterintuitive outcome. Imagine, for example, that you belong to a criminal gang, and you and one of your accomplices have been caught. The police don’t have enough evidence to convict you, and if you both stay silent then the best they can do is convict you on a lesser charge with a one-year prison sentence. The police offer you a deal: If you squeal on your accomplice, they'll let you off with a half-year sentence, while your hapless accomplice will get 10 years. But you know that in the next cell over, the police are making the same offer to your accomplice, and if you both rat on each other then you’ll each spend seven years inside.
In this famous “Prisoner’s Dilemma” game you’re better off if both of you stay silent than if both of you squeal. But that's not what will happen: Staying faithful to each other is not a Nash equilibrium, since you can improve your lot by squealing. The only Nash equilibrium is for both of you to squeal. In fact, squealing is what is known as a dominant strategy: It is the best thing for each of you to do, no matter what the other player does. Assuming you are both motivated by pure self-interest, you are inexorably driven toward seven-year sentences, while by cooperating you could have gotten one-year sentences.
Nash’s equilibrium concept gives economists a precise mathematical approach to analyzing how people will behave in competitive situations. But, perhaps because of its very simplicity, for a couple of decades after Nash wrote about the equilibrium, most economists didn't realize just what a powerful tool he had handed them. Even Nash’s dissertation advisor thought Nash's theorem was an elegant result, but not a particularly useful one.
Part of the reason many economists didn’t immediately see the value of Nash’s equilibrium concept was that in Nash’s formulation, each player knows ahead of time what payoffs the other players will earn from the different possible outcomes. But in many economic interactions this is not the case. In an auction, for instance, a bidder generally doesn’t know how much the other bidders value the item being sold, making it harder to guess their strategies.
In 1967 game theorist John Harsanyi of the University of California, Berkeley, developed a method to do Nash equilibrium analyses even when players have incomplete information about each other's values. Twenty-seven years later Nash and Harsanyi shared the Nobel Memorial Prize in Economics with another game theorist Reinhard Selten, of the University of Bonn in Germany.
With these ideas in hand, more and more economists started feeling that game theory might have some important things to say about their field. Auctions, whose precise rules make them akin to games, seemed like a natural testing ground for the theory. Researchers interested in auctions began to roll up their sleeves.
