Nash Rambler

Ron Howard's new film, A Beautiful Mind, opens with scenes of John Nash in his student days at Princeton wrestling with abstract ideas that no one else understands or appreciates.

Nash diagrams on his window pane the choreography of pigeons chasing after crumbs, and the pursuit-flight diagram of a man trying to steal a woman's pocketbook. He is obsessed with the deeply hidden mystery of competitive struggles, which he calls Governing Dynamics.

Nash sees structure and patterns everywhere. He has the mind of mathematician, hauntingly gifted with Pattern Recognition Intelligence. Where others see meaningless noise, Nash sees the hidden pattern, the secret message, the encrypted code.

John Nash is the quintessential Absent-Minded Professor, lost in the reverie of thought, oblivious of the social niceties that occupy most people's day. He forgets to eat, loses track of time, doesn't bother to learn anyone's name. He doesn't much like people and people don't much like him. He alienates people with ease and without a clue as to why they shun him.

His closest friend at Princeton is an amiable if puckish roomate who distracts Nash from his studies long enough to go out for pizza and beer. The astonishing part is that the roomate, like Hobbes to Calvin, is entirely an invented character, and not the only one in the story.

Nash, it turns out, suffered from Paranoid Schizophrenia. Ron Howard's movie portrays Nash's delusions as so real that we are shocked to discover that three of the story's characters are entirely imaginary. One of them is Marcie, his roommate's niece, who calls him Uncle John.

Nash's breakthrough in recognizing his delusions comes when he realizes that Marcie never ages. Like a cartoon character, she remains forever young, never maturing into a young woman.

The third delusional figure is the sinister G-Man, Parcher, who enlists Nash in a top-secret spy operation, to decode a Russian plan to smuggle the bomb into the US. Parcher's story is that the Russians are planting instructions for their agents in newspapers and magazines, concealed within ordinary articles and advertisements. Nash, with his extraordinary pattern recognition skills, fills his time looking for hidden messages in Life Magazine.

"Math," says one of Nash's classmates, "is boring." Nash's imaginary forays into spycraft were anything but boring. They were frightening, both to him and to those around him.

Fear of boredom is a good excuse to invent a fantasy life. But fear of boredom wasn't what drove Nash. His deepest fear was fear of humiliation, fear of failure. More than anything, he craved a stunning achievement, an accomplishment that would propel him to the winner's circle.

I first learned of the Nash Equilibrium Solution when I studied Game Theory in graduate school at Stanford. In a competitive game, as the players try out different strategies, they try to take advantage of any weakness in their opponent's play, to gain the upper hand. The Nash Equilibrium Solution corresponds to the detente, where no player can gain further advantage, and where any deviation would hurt their cause. Nash showed that such solutions existed, and he gave criteria for recognizing them. That is, if you found the Nash Equilibrium Solution, you could tell you were there. But finding it was another matter. Many problems did not admit of a systematic search strategy to arrive at the Equilibrium Solution efficiently.

As competitive games go, there are some which are so embedded in our culture and our psyche that we don't recognize them as games. As I watched A Beautiful Mind, I recognized a recurring competitive game that is now the subject of ongoing research. In this game, there are perhaps 5 or 6 scoring axes.

The first is to gain respect and to avoid contempt. The club of mathematicians to which Nash belonged played this game fiercely. Nash was quick to dispense contempt for inferior work. When Alicia brings him a solution to a problem he has posted in class, he dismisses it scornfully, saying, "I never said it was a rational vector field." The poetry of that remark is probably lost on most viewers.

The second dimension in this game is to win approval and avoid disapproval. Alicia was not so wrapped up in the game that she gave up in despair upon receiving his callous disapproval of her solution. Instead she proposed they have dinner.

The third axis is antagonism vs. cooperation. If the competitive stance puts people into a state of mutual antagonism, the inverse would be cooperation. Moving from competition to cooperation remains an unsolved problem in game theory. But Alicia did it effortlessly. She merely had to conquer her pride.

The fourth axis is trust vs. mistrust. Learning to trust means learning to let go of fear. Alicia demonstrates trust, even where it is not rational to do so. But as John had said, it's not a rational vector field.

The fifth axis is autonomy vs. constraint. Most people prefer power and freedom to being held back. There is irony here. Autonomy often means secrecy, since no one can object, interfere, or restrain someone if they don't know what they're up to. Parcher, the imaginary G-Man, embodied this irony, imprisoning Nash in his code-breaking reverie, which Alicia twice uncovered and arrested.

The above model goes by the technical name of Facework Theory, but it could just as easily be seen as an instance of Nash's Governing Dynamics -- the drama and choreography of contestants competing for respect, approval, cooperation, trust, and autonomy.

Now that's a hard problem. How do you compete for cooperation?

Alicia demonstrated that in her first scene. She asked politely. And she won easily.

Marcie had charm, too. And like Marcie, charm never ages.