A key scene in the movie "A Beautiful Mind" purports to show how mathematician John Nash came up with the equilibrium theorem that won him the 1994 Nobel Prize. Attempting to pick up women at a Princeton bar in 1948, the young Nash realizes that he and his four friends will have a better chance of scoring should none of them approach the most beautiful woman in the room. Instead, he suggests, they should turn to less sought-after dates, boosting their chances of success. "If none of us goes for the blonde, we all win," muses Nash. What's so revolutionary about that? To find out, NEWSWEEK's Anna Kuchment spoke with Stanford game theorist Paul Milgrom, as "A Beautiful Mind" prepared to compete for the best-picture Oscar. Excerpts:
KUCHMENT: How accurately did "A Beautiful Mind" portray Nash's ideas?
MILGROM: I thought the movie did a horrible job.
First, let's go back before Nash. [Until the early 1900s] economists had abstract solutions for markets, but no role for people and strategy. They would say, "Supply equals demand." Well, what happens if the price is not one where supply equals demand?
[In the 1920s, mathematicians John] von Neumann and [Oskar] Morgenstern were the first to study two-player games like chess, but most economic transactions are not zero-sum games. They're situations, like bargaining, where all the players hope to end up better off.
What was Nash's breakthrough?
He found a solution that worked for any game. His theory is very simply this: a set of strategies, one for each player, is a Nash equilibrium if each player's strategy is the best he can do, taking as fixed the other players' strategies.
But isn't that what happens at the bar?
Each player at a Nash equilibrium takes account of what he expects the others to do and, using that, makes the best choice for himself. In the movie, Nash was saying they could all be better off if they all behaved differently. A calculation like that is irrelevant for Nash equilibrium analysis.
One reason this distinction is important is that it explains how bad things can happen. For example, overfishing, overgrazing or overdevelopment. None of us has an individual incentive to change our behavior, but our collective behavior is damaging. We could all be better off if we could coordinate our behavior, but acting as individuals, we can't.
You're a pioneer in modern auction design. How does the Nash equilibrium work there?
Bidders deciding how much to bid in an auction often start by forecasting how much others are likely to bid. For example, in a sealed-bid auction to buy a spectrum license, a bidder who expects that the highest opposing bid will be 500 million euro might plan to bid just a bit more than that. When all bidders base their bidding decisions on intelligent forecasts, the result is a Nash equilibrium.
Where else is it applied?
One other very important area is matching. College-football bowl games in the United States, for instance, are examples of something that's quite common in matching markets. There was a problem in organizing the [championship] bowl games in that the bowl offers started coming in earlier and earlier. The Cotton Bowl might approach Notre Dame after the sixth week of the season, when Notre Dame had a very good record, and try to lock them in to playing in their bowl game. They would make them an offer, saying: we'll guarantee you a spot if you'll agree to play in the Cotton Bowl. And it would be better, of course, if those offers were made at the end of the season, when we knew who the good teams were. But there was an incentive for the bowls to make their offers early in order to try to get good teams.
That problem arises all over in matching markets. For example, there are children in the developing world who are betrothed before they're born. Parents will make a deal that says: my wife is pregnant; if our next child is a daughter, she will be betrothed to your 2-year-old son. For the same sorts of reasons: people want to make good matches, and so they had to go earlier and earlier.
What do all these bad outcomes say about the equilibrium?
This is the way a lot of academic applications go. Is there a Nash equilibrium in the matching game in which everyone waits until the kids have matured before we make marriage matches? In these cases, the good outcomes aren't Nash equilibria. We know that some Nash equilibrium exists--Nash proved that--so it must be that the only Nash equilibria are the ones involving bad outcomes.