John Nash Equilibrium Game Theory

As Chariots Of Fire did for Eric Liddell and Braveheart did for William Wallace, the 2002 film A Beautiful Mind made mathematician John Forbes Nash a household name - without necessarily rendering his life, or his work, much better-understood. Audiences and critics welcomed the movie - it won a 2004 Academy Award - but enthusiasts of Nash's work insist that even bigger rewards await those who study Nash's real-life work, and the esoteric discipline, game theory, in which he made his name.

Born in Bluefield, West Virginia, in 1928, Nash was already carrying out bedroom scientific experiments at the age of twelve. He didn't excel in sports or other stereotypically youthful pursuits, instead fixing on E.T. Bell's book Men of Mathematics with the same intensity that a young would-be guitarist might bring to, say, Led Zeppelin IV. While still in high school, he took college-level math classes, and a Westinghouse scholarship to the Carnegie Institute of Technology (a school known, and revered, today as Carnegie Mellon) seemed to confirm his vocation as a mathematician - a vocation only confirmed when Princeton aggressively recruited him to its Ph.D. program in mathematics. He finished his doctorate in 1950.

Much of his important early work - including the three scholarly articles that defined and explained the tendency that came to be known as "Nash equilibrium" and which (many years later) helped secure him a 1994 Nobel Prize - had to do with game theory, a branch of mathematics that analyzes the ways people interact. Game theorists construct equations that reflect peoples' assumed motives in entering a situation, and then analyze the range of possible actions they may take. They use mathematical modeling to determine what the actual outcomes of the situation, then, will be.

Publisher's note:

John Nash famous theory is the basis for many heads up confrontational decisions made in poker. By committing the outcome of a game to a certain push mentality, you are essentially making it difficult for your opponent to take advantage of you, especially when the blinds get high. That is becuase it is almost always wrong to call all ins with weak hands, yet it is valid to be pushing with them.

A logical puzzler known as the Prisoner's Dilemma offers a good quick example of how basic game theory works. Imagine two prisoners caught near the scene of a burglary and hauled in by the police. The cops know that they've found their suspects, but they can't get either person to admit guilt, so they offer each man a deal. As Michael A.M. Lerner, writing in Good Magazine, describes it: "If they both confess and cooperate, they'll both get a minor sentence of five years. If neither man confesses, they'll both only get one year - But, and here's where it gets interesting, if one confesses and the other doesn't, the one who confesses walks out scott-free while the other will do 10 years. What will they do? Will they trust each other and do what's obviously in their best interest, which is not confess?" Game theorists assume that each person in this dilemma is out for themselves; assigning values accordingly, they come up with equations that predict the two burglars will betray each other - even though it makes more sense to cooperate.

It may sound crazy - how on earth can something that seems as cut-and-dry as math make successful, predictive models of how humans will behave in a real-world situation? But mathematicians, economists and political scientists have used game theory to yield some startlingly accurate predictions. Game theorist Benito de Mesquita used his own equations to predict the Ayatollah Khomeini's successor, in 1984; when his answer proved, several years later, to be correct, it launched a career that now includes a rich consulting firm and several Pentagon collaborations. Game theory may not be uncontroversial, but it does look to be here to stay.

Nash's own most famous work has to do with the way we can assume people will behave in certain "non-cooperative" games, i.e. situations in which people compete against each other. He showed, in general, that there are limits on the degree of success that can be achieved by people in competition against each other - that, contra Adam Smith (the father of modern economics), some kinds of competition tend to reduce the amount of good stuff available for everyone (rather than making the total size of the pot bigger, as Smith is usually assumed to have taught). This is the insight for which - decades later, after his protracted struggle with schizophrenia, and along with Reinhard Selten and John Harsanyi - he won the Nobel Prize. It may not be as photogenic as Russell Crowe (who played Nash in the movie), but it's - who knows? - probably more relevant to your life.

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