In a recent New York Times article, Hal Varian — a respected mainstream economist and textbook author — describes the contributions of Nobel Laureate John Nash:
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.
Mr. 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.
Before proceeding, I must endorse Varian's critique of the bar scene from the movie. For those who have seen it, be assured that the (implausible) strategizing — in which Russell Crowe instructs his friends that the only way to success is for them all to ignore the pretty girl and focus instead on her plainer friends — does not constitute a true Nash equilibrium. The situation is comparable to that faced by OPEC countries bargaining on restrictions in output: Even if all the boys would be better off if all ignored the pretty blonde, there would still be an incentive for each one to deviate from the pact and approach her. (In any event, she seemed rather stuck-up to this viewer.)
Having said this, I must disagree with the rest of Varian's analysis. Nash equilibrium is defined as a situation in which each player has chosen an optimal strategy, given the choices of the others. Such a situation constitutes an equilibrium because, if reached, there would be no reason for anyone to change strategy; it would be a stable resting point.
On the face of it, there is nothing objectionable with this definition, assuming one wants to model strategic interactions in the formal manner of game theorists. Yet even so, there is no reason that in any game the economist should predict that players will pick strategies to form a Nash equilibrium. The players are out to maximize utility; they care nothing for the stability of the outcome. This distinction underscores the flaws in Varian's subsequent commentary.
Varian states that game theory assumes full rationality among all players, and admits that this assumption is far from reality. With this I am in complete agreement. However, Varian seems to think that this explains the failure of human players to reach Nash equilibria in experimental settings. As I shall argue, however, it is not the assumption of rationality, but the obsession with equilibrium analysis, that has baffled mainstream economists in these games. This confusion is epitomized in Varian's illustrative example:
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 everyone1 … 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.
Varian — like most game theorists — here confuses rationality with omniscience. It is one thing to say players should perform correct calculations of probabilities and payoffs; it is quite another to say they should be able to know the moves of their opponents beforehand. Moreover, Varian simply asserts that if everyone is equally rational, everyone should pick the same number; I personally do not see why this should be true.
But the fundamental weakness in Varian's analysis is his criticism of real-life players who fail to conform to Nash equilibrium. Varian believes this failure demonstrates their irrationality, when in fact it only proves the limitations of his equilibrium concept.
To see this, imagine that there are only two players, John and Jane, playing the game Varian describes. Suppose that John picks the number 3 while Jane picks the number 1. One-half the average of their picks is 1, and therefore Jane wins the prize; she has played perfectly.
According to Varian's logic,2 however, Jane's play would be irrational; rather than playing the perfect move of 1, she should have opted for the "rational" choice of zero.
It is clear then that the word "rational" means much more to the typical game theorist than simply "flawless calculation." Ultimately, "rational" describes a player who analyzes games in the way that typical game theorists do. We have seen that a player may choose a number different from zero in this simple game and still win, and so it is difficult to argue that a group of perfectly rational players should all assume everyone else is going to play zero.
The other game that Varian describes (equivalent to a game known as the traveler's dilemma), as well as other such "puzzling" games (including the centipede game, finitely repeated prisoner's dilemma, and ultimatum game) studied in the experimental literature, all have this flavor. They are games with a unique3 equilibrium in which the Nash strategy is disastrous if the other players choose a non-Nash strategy. In other words, the only reason players would ever choose the Nash strategy in these games is if they were absolutely certain their opponents would do the same. Real-life players know that there is no such thing as certainty, and consequently make much better moves than those recommended by the ostensibly rational game theorist.
Varian concludes his article with this amusing anecdote:
Back to picking up girls. 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.
At the risk of priggishness, I must point out that Russell Crowe did look at the game from the woman's perspective; that's why hitting on the pretty blonde first, and then settling for her friends, was a losing strategy.
Finally, I must also disagree with Varian's colleague (who is no doubt a typical game theorist): Never once has my reply of "Oink-oink!" to a female's inquiry led to success.
- 1In Varian's article, he says the winner picks one-half the average of what everyone else says, but I have changed it to the more conventional game. The logic remains the same.
- 2Varian would probably say that not only is he assuming full rationality among players, but also that all players know they are all fully rational. As I argue above, however, I still believe his conclusion does not follow.
- 3For the ultimatum game, it is the stricter subgame perfect Nash equilibrium that is unique.