John Nash’s many equilibria
4 min read . Updated: 27 May 2015, 12:45 AM IST
PremiumOne thing that marked John Nash clearly from his rivals, including the talented but far less generous John von Neumann, was humility
In the death of John Forbes Nash Jr, the world has become poorer by a great intellectual. Along with John von Neumann, Nash will be remembered as a founder of game theory. The subject was invented in the 1940s after von Neumann and Oskar Morgenstern authored Theory of Games and Economic Behavior (1944). In the years that followed, Nash wrote three papers that revolutionized the subject and continue to provide the bread and butter concepts of game theory.
One definition of game theory describes it as “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers". This is a definition that is certain to put off anyone. However, if one can overcome one’s phobia of mathematics, game theory is an elegant way to examine almost all kinds of strategic interactions in the world.
The original formulation of the subject was in terms of a zero sum or pure conflict games where one player’s gain is matched exactly by the other’s loss. Nash vastly extended this formulation by von Neumann. Real life examples are complex: most situations have a mix of conflict as well as cooperation. Nash’s equilibrium idea was more versatile and applicable in a far more diverse range of games. Nash’s idea is simple even if the formal proof of his theory requires sophisticated mathematics known as fixed point theorems.
Nash’s 1950 paper (Equlibrium Points in n-Person Games) changed not only the intellectual climate of economics but of almost all social sciences. He wrote his last game theory paper of that period in 1953 and returned to the subject after a gap of almost four decades. In that interval economics gained and then lost, to an extent, its charm with game theory.
Twelve years after the Theory of Games and Economic Behavior was published, Daniel Ellsberg—then a graduate student—wrote a devastating review of the book. Ellsberg showed that far from being a theory of “complete principles which define rational behaviour", what von Neumann and Morgenstern had developed did not lead to certainty even in a simple class of games. Some of this scepticism spilled over to Nash’s theory as well.
The truth is that in the end a Nash equilibrium requires two players to have correct beliefs about each other’s actions. As an example, consider two players 1 and 2. Both are planning to meet each other and have two choices where to meet. Either they can meet at a café both visit usually or they can meet at a metro station from where they go to work together. Suppose on a given day, they are not able to speak with each other (say 1 forgets to bring a mobile phone along). How will they meet each other? Both have to guess—accurately—where the other is going. Unless both do that, they can’t meet. An accurate guess—for example both meet at the café—is a Nash equilibrium.
The real world, it is safe to say, is far more complicated than two people trying to meet each other. One of the assumptions behind such games is that players know each other’s actions, payoffs to those actions and knowledge that both know that they know each other’s actions regressed endlessly (Yes! That wonderful assumption is known as “common knowledge"). The fraction of cases where all this is known is, well, negligible in the real world.
Generations of exceptionally talented practitioners of the subject have tried hard to overcome these foundational weaknesses. For most part the results have been discouraging. Writing on the 60th anniversary of the Theory of Games and Economic Behavior in 2004 one such theorist Ariel Rubinstein dismissed any hope of game theory improving performance in real-life strategic interactions. That has not ended the temptation of using the subject to explain everything. From history to evolutionary biology, anthropology to political science, few subjects have retained immunity from the charm of game theory. The genie Nash unleashed cannot be bottled again.
In the interlude when these adventures were taking place, their founder moved on. In the mid- and late-1950s Nash wrote papers on algebraic manifolds, ventured into a formidable, and still unsolved, mathematical problem known as the Riemann Hypothesis and partial differential equations. These papers, written in very different domains of mathematics, demonstrated his versatility. His personal hardship and mental tumult—borne with great fortitude for over three decades—were a measure of his great strength.
One thing that marked Nash clearly from his rivals—including the talented but far less generous von Neumann—was humility.
In his autobiographical sketch issued when he received the Nobel Prize, Nash said, “Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts, to add much to his or her previous achievements. However, I am still making the effort and it is conceivable that with the gap period of about 25 years of partially deluded thinking providing a sort of vacation my situation may be atypical." It is safe to say that barring game theory, there has not been any other mathematical idea that has influenced 20th century economics so widely.
Siddharth Singh is Editor (Views) at Mint. Reluctant Duelist takes stock of matters economic, political and strategic—in India and elsewhere—every fortnight.
Comment at siddharth.s@livemint.com. To read Siddharth Singh’s previous columns, go to
www.livemint.com/reluctantduelist
Follow Mint Opinion on Twitter at https://twitter.com/Mint_Opinion
