A Beautiful Equilibrium

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Beautiful Mind is the first film that deals with a profound social issue that touches everyone and yet that few people mention or even understand – a Nash equilibrium. The film also focuses on the mental illness of mathematician John Nash but only because Nash won the 1994 Nobel Prize in economics for a two-page paper called “Equilibrium Points in N-Person Games” that he published in the Proceedings of the National Academy of Sciences in 1950 when he was just 20 years old. There would have been no film without the Nobel Prize.

Yet the filmmakers failed to. correctly explain a Nash equilibrium. And a mere flash of text at the film’s start only hints at why a review article in the September 1999 issue of the Journal of Economic Literature said that the impact of Nash equilibrium in the social sciences “is comparable to that of the discovery of the DNA double helix in the biological sciences.”

So what is a Nash equilibrium?

Nash equilibrium shows how selfish competitors should act given how their competitors act. A Nash equilibrium has a simple mathematical definition. Here is how Nash described it in words in his 1950 paper called “Non- Cooperative Games”: “Each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Thus each player’s strategy is optimal against those of the others.” So each player does the best he selfishly can given the competitive context of his competitors doing the best they can. The competitors constrain one another’s selfishness. And no competitor has an incentive to change his strategy once all the competitors are in a Nash equilibrium.

Jean-Jacques Rousseau suggested a hunting example in his 1755 Discourse on the Origin of Social Inequality that has led to something called the deer game. Suppose you are one of four starving people in the forest. You have just two options: You can try to catch a rabbit or a deer. But you can catch a deer if and only if all four of you try to catch one. How each person behaves depends on how others behave.

Suppose the other three persons catch rabbits. Then the best you can do is catch a rabbit too because you have no chance to catch a deer. This rabbit hunt is a Nash equilibrium because each person does his selfish best given what the others do and because no one has an incentive to switch hunting strategies.

Thus does order arise from competitive struggle.

Now suppose the other three persons try to catch a deer. You still eat if you catch a rabbit but then they can’t catch a deer. They will catch a deer if you help them and then you can all have a feast. So it is in your selfish interest to try to catch a deer. The deer hunt is also a Nash equilibrium because each person does the best he can and has no reason to change his hunting strategy. The latter point is essential. A player cannot have any incentive to switch strategies.

The movie gets this backward when it concocts a “blonde game” in a bar. Each young man in the bar wants to pick up a blond woman rather than a brunette. Then a blond beauty walks in with several brunettes. The Nash character (Russell Crowe) conceives the Nash equilibrium in this fictitious scene. He claims that no man should pursue the blonde because they can’t all have her (and this will insult the brunettes). So he claims that the optimal strategy is to pursue only the brunettes.

But each man will want to switch from his brunette to the blonde if all the other men have brunettes. So this is not a Nash equilibrium. The film’s logic says that children will pick up only the pennies on a sidewalk and not the hundred dollar bill lying next to the pennies because they can’t all have the bill. Our own selfishness says otherwise.

Studies of ultimatums have shown that we can be so self- ish that we become envious and we don’t achieve Nash equilibrium. Suppose I have a hundred dollars and I offer you a share of it. The rules let us keep our shares if you accept my offer. But neither of us get anything if you reject my offer. Then I should offer you as little as possible and you should accept anything I offer. But more than half of players reject an offer less than twenty dollars even though accepting even one dollar is better than nothing.

Nash equilibrium can also explain the darker side of behavior. I published a paper I wrote as a student about outlaws who grow and steal marijuana plants (you can download it from my USC web page). It pays to steal if there are many more growers than thieves because growing pot is so risky. But it pays to grow if there are too many thieves because a grower has some chance of harvesting something while thieves find little to steal and other thieves will steal from them. Players adjust their strategy mixes of growing and stealing until they reach Nash equilibrium (whereas legalization lets growers organize and use the police).

Yale economist Stephen Morris applied Nash equilibrium to political correctness in a 2001 issue of the Journal of Political Economy. Political correctness deals with not telling the truth because of fears to one’s reputation. An advisor may lie to her boss if she fears some words or opinions will harm her reputation. The extreme case leads to a “babbling equilibrium” where the advisor’s advice is no better than flipping a coin and so her boss ignores her. This can apply to advisors from stockbrokers to astrologers to political consultants.

John Nash deserved his Nobel Prize – and a more accurate movie.

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