Cinema Signals: ~ An addendum to the film, "A Beautiful Mind" MOBILE: |

Strategies and Games
A discourse on game theory
applied to economics and business

. The real Nash Equilibrium

If you liked "A Beautiful Mind" because you cared for American mathematician John Forbes Nash Jr.'s achievements and character, you might want to know how his theory of equilibrium really works and what it means. If so, you're correct in coming here because the movie did what movies do, commercialized it, sexed it up and got it wrong.

That doesn't mean they shouldn't have simplified it. Simplifying a theory is not only good, it's ability to be simplified is a mark of elegance and, often, applicability to the real world (though not always). With those thoughts in mind, let's see what the theory posits.

The Nash Equilibrium shows how selfish competitors should act in relation to those whom they compete against. It deals with individual needs and desires in a group context and must take into account likely human behavior.

Jean-Jacques Rousseau, a Swiss political philosopher, suggested a situation in his 1755 "Discourse on Inequality" that has led to something called the Deer Game. In it, four starving people are in the forest where they might catch a rabbit or a deer. A deer is the better catch but one may be caught only if the four work together.

If three go out and catch rabbits, the chance for a deer to be caught is lost. The fourth must forget venison and content himself with rabbit. This 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, order arises from competitive struggle.

In a slightly altered scenario, three of the group want to catch a deer. The fourth must decide if he wants to just catch his rabbit and spoil the dreams of venison for the others or join them in the hunt for the deer and participate in the feast that will follow. It's in the fourth's selfish interest, then, to join the others in the hunt for the deer. This is also a Nash Equilibrium because each person pursues his own interest and does the best he can.

Now let's see how the movie gets it backward. In this poorly contrived scene there are four men in a bar, including Nash. A fairly flashy blond walks in with four brunet companions, an event immediately noticed by the horny male quartet. Russell Crowe suggests at this point (after some wild and meaningless scribbling) that it would be in their combined interest for the four of them to go after the brunets on the supposition that they'd all at least get a woman. But it's not simply "a woman" that each man wants. They all want the blond and, because she can't be shared in the way that the deer can be, they would each want to trade their brunet for the blond. This is a complete misstatement of Nash's Equilibrium. No group equilibrium is attainable without resolution or satisfaction, clearly missing in the situation.

The film's logic would say that children will pick up only the pennies on a sidewalk and not the hundred dollar bill lying next to them because they can't all have the bill. All sorts of motivations suggest otherwise, like normal human greed, acquisitiveness, need, security, etc.

If the century note, on the other hand were at a height roughly equal to the combined height of the children if they formed a human ladder, they would then be acting in their own selfish interest, with the prior agreement, of course to split the $100. Each child then walks away with $25 -- a far better outcome than a penny, and an equilibrium attained. Another victory for Nash's theory.

Why would director Ron Howard and his screenwriter, Akiva Goldsman, get it so wrong? Could it be that they didn't understand it and, if so, why wouldn't they have studied it closer? Could this be their belief that most people wouldn't care or notice a misrepresentation for the sake of a little spice to boost the commercial prospects? Was it marketing potentials that lead them to choose a blond over a deer? They're not alone. Other directors/writers/studios have done as much and it's always an insult to the intelligence of the audience.

I hope this little run-through of so fascinating a subject will serve to correct the error of their ways, but let's be real and not expect too much from an expos‚ of commercial motivations. Still, let's urge filmmakers to be more careful with their material and more respectful of their audiences.

Of further interest for scholars and followers, Nash's autobio for the Nobel site covers his work in game theory (but it doesn't even mention the Equilibrium).

                                      ~~  Jules Brenner  

To further corroborate what we've written here, an interesting letter to the editor of the L.A. Times appeared on March 20, 2005 from the science consultant on the film, William Zame.

Click this button for the text of Mr. Zame's letter (if it's still carried on their server).

Note: this is not the review of the movie. If you found this page directly from outside our site, you can read the full review by using the link below.

Back to our review of "A Beautiful Mind"

The Soundtrack album


Academy Award Nominations, 2002 (with some commentary)

Opinion Section
Comments from readers:
Well written
Rating: 7

This review was one of the best explanations for laypersons I could find. The ease with which the concept could be absorbed from this review, and the reviewer's passion, keeps interest in the subject aflame. Does this mean there is an equilibrium between the movie maker's aims and the reviewer's aims relative to the contribution of John Nash?

                                                      ~~ Karen 
[Editor's reply: Excellent question, Karen. I believe you'll get an A this semester. ~~ FC]

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