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