John F. Nash Jr died in a car crash on Sunday. He was on his way home from Newark airport, having just returned from Norway, where he received the prestigious Abel Prize for Mathematics. Nash’s work on game theory, for which he got the Economics Nobel in 1994 — he is the only person to have won both honours — is probably his best known. The concept of Nash equilibrium is intuitive, elegant and relatively easy to understand. It is specific enough to generate meaningful results and analysis, yet general enough to be extended and applied to a variety of disciplines — evolutionary biology, economics, defence studies and politics, for instance. But the mathematical community regards his work in geometry and partial differential equations as “his most important and deepest”, according to his Abel Prize citation.
It is incredible that both Nash’s Nobel- and Abel Prize-winning work were completed by the time he was 30. He wrote just one 23-page paper in 1958 on partial differential equations, and as Harold W Kuhn noted during the Nobel seminar in 1994, “the results for which he is being honoured this week were obtained in his first 14 months of graduate study”. Indeed, Nash came to Princeton as a PhD student with a one-line letter of recommendation from R L Duffin of Carnegie Institute of Technology, where he was an undergraduate: “This man is a genius.” A W Tucker, Nash’s thesis advisor in Princeton, wrote years later: “At times I have thought this recommendation was extravagant, but the longer I’ve known Nash the more I am inclined to agree that Duffin was right.”
But in early 1959, Nash began to spin out of control and started exhibiting symptoms of schizophrenia. He became paranoid and delusional, and other than in a few brief periods of clarity, his research came to an end for about four decades. During this same period, Nash’s name, the non-cooperative game that he gave form and definition to, and his concept of equilibrium were becoming part of basic undergraduate game theory training.
So what is a “non-cooperative game”? It is not a game where cooperation is ruled out because of the structure of the payoffs such as in a zero-sum game, where one player’s benefit implies another’s loss. There can be scope for cooperation in the game but it is ruled out because there is no mechanism, like a legally binding contract, to ensure commitment to collusive strategies.
A simple and celebrated non-cooperative game is the Prisoner’s Dilemma (pictured above). Suppose two conspirators are arrested and questioned simultaneously in separate rooms. Each has the option to confess or stay mum and is offered a deal: if she confesses (but the partner-in-crime does not), she can go scot free while the accomplice will go to prison for 10 years. But if both choose to keep quiet, they would go to jail for a year each for petty crime. And if both confess, they’d go to jail for eight years each.
The unique Nash equilibrium of the game is where both players confess. Interestingly, both would be better off if neither confessed. But that isn’t a Nash equilibrium, which is defined as a stable state in which no player can improve the outcome for herself given what the other players are doing. For a moment assume that both players somehow indicate that they will choose not to confess. In such a situation, given that Player B is not confessing, Player A would be better off by “reneging” and choosing to confess instead — no jail time is more appealing than a year behind bars. The same is true for Player B. So both would deviate from their “commitment” to keep quiet and confess instead.
The applications of Nash equilibria and non-cooperative games are infinite. For instance, some in India have noted that today, private capital seems to be waiting for the investment cycle to be kickstarted before putting in their own money. This situation could be modelled as a non-cooperative game between two potential investors, where the benefits of investment are only realised if both sink in their money. In such a game, there are two Nash equilibria: one, where both players invest, and two, where neither invests. We seem to be stuck in the bad equilibrium. And while Nash was able to prove the existence of at least one equilibrium for a non-cooperative game, the theory is silent on why a particular one results and not another. This is where government, society and norms come in — to nudge us from bad equilibria to good.