Nash Equilibria

Whenever a great man dies it’s a tragedy in the abstract; but like so much in life, this death means more to me because of the history I’ve laid upon it, the meaning his life has had for me, personally.

I read A Beautiful Mind – the book on which the movie was based – soon after it came out.  I was working at a hospital gift shop in my hometown of Detroit at the time; and at lunch and after work I’d sit in my car (which also occasionally doubled as my residence at the time) and read about this man and his life.  He became my image of learning, of erudite scholarship, as I headed towards college.  The core of a Nash Equilibrium, as I inexpertly understand it, is a situation where your best strategy is affected by the strategy others take; where best responses are mutually dependent and stable.  Information, or examples, are not strategy, and thus his influence on me is no Nash Equilibrium – but Nash affected my strategy through life all the same.  While life’s choices are complicated and multi-causal, it is not implausible that I would not be where I am today without Sylvia Nasar’s book and John Nash’s life.

One day a decade later I was living in Princeton, working on a Masters.  I was on my way to NYC, a trip that starts with the one-stop train (the “dinky”, as it’s called) that connects Princeton to the main train line.  I’d already sat down on the train when an odd man walked in, speaking to himself.  He sat down next to me, though many rows were unoccupied.  I smiled at him; he seemed to acknowledge me; the dinky moved.  The ride was all of 5 minutes, and we passed it in silence.  As we exited, and after the man got up, a Princeton undergrad came up to me. “You know who that was?  That’s John Nash.”

A few months later, it happened again; the only difference was that this time, I knew who he was.  I was sitting in a different row (I think); Nash got on; and again, he sat down next to me when other rows were entirely free.  Maybe it was random chance; but maybe he remembered me, and sat next to me again intentionally.  I like to think it was the latter; that in a very small way I had affected his strategy.  That in some small way I had changed his play of life’s game in a tiny echo of the way he’d changed mine.

When I heard Nash died this morning it was from my wife.  She remarked that there was in some ways something beautiful about a couple seemingly still in love in their eighties, dying together without pain.  It was a much more empathetic reaction than mine; which was there are very few ways of dying that could plausibly be framed as involving Nash equilibria.  My driving on the right side of the road while you do likewise is, I believe, a Nash equilibrium; our best response strategies are mutually dependent on one anothers, and give us no incentives to deviate.  Nash’s car crossed over the line, the equilibrium disturbed likely not by intent (as there is nothing to be gained) but rather an icy patch, or a faulty break, or some other external perturbation.  As the car that hit his taxi sped towards them on that dark New Jersey road, I doubt that other car had any idea that the taxi had deviated from the (expected) equilibrium, that his best response needed to change to accommodate these altered circumstances.  Seen one way, Nash died because his taxi deviated from Nash’s equilibrium not by design but because the taxi driver’s intent to maintain equilibrium behavior could not make it so, the driver’s scope for action limited by technology, physics, and – more broadly – the world outside himself.  I wonder if it occurred to Nash, in his last moments, that this was all a possible instantiation of his work. If so, I hope it gave him comfort, and that – were he able to look down and see this note – it would please him.  I like to think it might.

Nash changed my world, as he did so many of us.  May his passing remind us all of his Beautiful Mind, as well as the limitations of his or any mind’s in the world we live in.  That we must work to make our lives as full as we can, to let our talents flourish, and to realize the edge that delimits what we can control from what we cannot.  As the saying goes, may G-d grant us all the strength to change the things we can, accept the things we cannot, and – particularly – the wisdom to know the difference.

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