So what did John Nash actually do? Viewers of the Oscar-winning film "A Beautiful Mind" might come away thinking he devised a new strategy to pick up girls.
Nash's contribution was far more important than the somewhat contrived analysis about whether or not to approach the most beautiful girl 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-1940s. 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 classes of games were completely different.
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 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. Nash proposed that we look for outcomes where 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 is 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 is 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.
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