[...] Mr. Nash's contribution was far more important than the somewhat contrived analysis about whether or not to approach the most beautiful woman in the bar. What he discovered was a way to predict the outcome of virtually any kind of strategic interaction. Today, the idea of a ''Nash equilibrium'' is a central concept in game theory.
Modern game theory was developed by the great mathematician John von Neumann in the mid-1940's. His goal was to understand the general logic of strategic interaction, from military battles to price wars. Von Neumann, working with the economist Oscar Morgenstern, established a general way to represent games mathematically and offered a systematic treatment of games in which the players' interests were diametrically opposed. Games of this sort -- zero-sum games -- are common in sporting events and parlor games.
But most games of interest to economists are non-zero sum. When one person engages in voluntary trade with another, both are typically made better off. Although von Neumann and Morgenstern tried to analyze games of this sort, their analysis was not as satisfactory as that of the zero-sum games. Furthermore, the tools they used to analyze these two classes of games were completely different. Mr. Nash came up with a much better way to look at non-zero-sum games. His method also had the advantage that it was equivalent to the von Neumann-Morgenstern analysis if the game happened to be zero sum.
What Mr. Nash recognized was that in any sort of strategic interaction, the best choice for any single player depends critically on his beliefs about what the other players might do. Mr. Nash proposed that we look for outcomes in which each player is making an optimal choice, given the choices the other players are making. This is what is now known as a Nash equilibrium. At a Nash equilibrium, it is reasonable for each player to believe that all other players are playing optimally - since these beliefs are actually confirmed by the choices each player makes.
It's a nice theory. But is it true? Does it describe actual behavior in actual games? Well, no. Game theory is an idealization: it analyzes how ''fully rational'' players should play if they all know they are playing against other fully rational players. That assumption of ''full rationality'' is the problem with game theory. In real life, most people - even economists - are not fully rational.
Consider a simple example: several players are each asked to pick a number ranging from zero to 100. The player who comes closest to the number that is half the average of what everyone else says wins a prize. Before you read further, think about what number you would choose. Now consider the game theorist's analysis. If everyone is equally rational, everyone should pick the same number. But there is only one number that is equal to half of itself: zero. This analysis is logical, but it isn't a good description of how real people behave when they play this game: almost no one chooses zero.
But it's not as if the Nash equilibrium never works. Sometimes it works quite well. Two economists, Jacob Goeree and Charles Holt, recently published a clever article, ''Ten Little Treasures of Game Theory and Ten Intuitive Contradictions,'' that exhibits a number of games in which the Nash theory works well, and then show that what should be an inconsequential change to the payoffs can result in a large change in behavior. In their simplest example, two players, whom we will call Jacob and Charles, independently and simultaneously choose an amount from 180 cents to 300 cents. Both players are paid the lower of the two amounts, and some amount R (greater than 1) is transferred from the player who chooses the larger amount to the player who chooses the smaller one. If they both pick the same number, they both are paid that amount, but no transfer is made. So if Jacob chooses 200 and Charles chooses 220, the payoff to Jacob is 200+R and the payoff to Charles is 200-R. If Jacob thinks Charles will say 200, then Jacob will want to announce 199. But if Charles thinks Jacob will announce 199, then Charles should say 198. And so on. The only consistent pair of beliefs is when each thinks the other will say 180. When Mr. Goeree and Mr. Holt performed this experiment with R=180, nearly 80 percent of the subjects picked 180, which is the Nash prediction. When they set R=5, and reran the experiment (with different subjects), however, the outcomes were completely reversed, with nearly 80 percent choosing 300.
Findings of this sort have stimulated the development of ''behavioral game theory'', which tries to formulate a theory of how to understand games involving real people, rather than those mythical ''fully rational'' people. Consider, for example, the ''guess half the average'' game described earlier. Oscar, a simpleminded player, might think that any number between zero and 100 is equally likely, so he would guess 50. Emmy, who is more sophisticated, might figure that if lots of people were like Oscar and say 50, then she should say 25. Tony, who is yet more sophisticated, figures that if lots of people think like Emmy, then he should say 12 or 13. And so on. An economist named Rosmarie Nagel ran a game like this a few years ago and found that the choices do tend to cluster around 50, 25 and 12. In fact, the winning choice turned out to be close to 13, a number chosen by about 30 percent of the players. In this game the best strategy wasn't the Nash equilibrium, but it wasn't so far away from it either.
Back to picking up women. In the movie, the fictional John Nash described a strategy for his male drinking buddies, but didn't look at the game from the woman's perspective, a mistake no game theorist would ever make. A female economist I know once told me that when men tried to pick her up, the first question she asked was: ''Are you a turkey?'' She usually got one of three answers: ''Yes,'' ''No,'' and ''Gobble-gobble.'' She said the last group was the most interesting by far. Go figure.
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