John Nash's Beautiful Mind and You

John Nash and the logical dilemma that bedevils our relationships

Posted Jan 01, 2010

Today marks the 60th anniversary of John Nash's discovery of a major stumbling block in human relationships. It is something that we should all know about - a Catch-22 logical trap that can damage relationships, foul up society and even block our efforts to resolve important problems like global warming. The discovery won him a Nobel Prize, and fame as the subject of the Oscar-winning film A Beautiful Mind. Yet few people know what Nash's discovery actually was, and fewer still realize just how often we encounter it in our daily lives.

Nash made his discovery when he was just twenty-one years old, and not yet suffering from the schizophrenia that was to blight much of his life, and which formed the focus of the film. The idea is called a "Nash equilibrium." It is a point of balance in a social situation from which neither party can escape independently without landing up in a worse situation.

A simple example is when two people are walking along a narrow sidewalk towards each other, and there is no room to pass without one of them stepping into a muddy gutter. They are in a Nash equilibrium, because neither of them can independently get out of the situation without ending up in a worse position.

Nash's great discovery was his rigorous mathematical proof that there is always a Nash equilibrium waiting to trap us in any situation of competition or conflict where the parties cannot or will not communicate. This suggests a solution - communicate to agree on a coordinated strategy, and act together to escape from the trap. With the two people confronting each other on a narrow sidewalk, for example, one could hold on to the other to help them step round without having to get muddy feet.

Coordinated action doesn't always work, though, because there are many situations where people may agree to cooperate, but later think better of it. This possibility of cheating after agreeing to cooperate is what gives the Nash equilibrium its bite, as illustrated in the famous story called The Prisoner's Dilemma. It is a made-up story, invented by Princeton University mathematician Albert Tucker to illustrate the problem to a group of psychologists. It has strong resonances with real life, however, especially with the practice of plea bargaining, which is now outlawed in some countries precisely because of the impact of this story, which has appeared in various incarnations.

In one version of the story, two thieves (let's call them Bernard and Frank, after two of the conspirators in the Watergate scandal) have been caught by the police, but the prosecutor has only enough evidence to put them behind bars for two years, on a charge of carrying a concealed weapon, rather than the maximum penalty of ten years that they would get for burglary. The thieves know this, and have agreed with each other to plead "not guilty" so that they will both get only two years. The prosecutor, however, has a persuasive argument to get them to change their pleas.

He first approaches Bernard in his cell, and points out that if Frank changes his mind and pleads guilty, but Bernard doesn't, Frank will receive a reduced sentence of four years for pleading guilty, but Bernard will get the maximum ten years. So Bernard's best bet, if he believes that Frank will plead guilty, is to plead guilty as well, so as to receive four years rather than ten. "Furthermore" says the prosecutor "I can offer you a deal that if you plead guilty and Frank doesn't, you can go free for turning state's evidence!"

No matter what Frank does, it seems that Bernard will always do better for himself by pleading guilty. The logic seems irrefutable - and it is. The trouble is that the prosecutor has made the same offer to Frank, who has come to the same conclusion. So they both plead guilty - and they both end up in jail for four years, rather than the two years that they would have received if they had both kept their mouths shut. There are two Nash equilibria here (both getting four years, or both getting two years), but the logic of self-interest has landed both thieves in the wrong one!

The insidious logic of The Prisoner's Dilemma affects us in many situations, from divorce to war - so many, in fact, that it has been proposed as the basic problem of sociology, since our efforts to live together in a cooperative and harmonious fashion are so often undermined by it.

Once you catch on to the basic logical paradox of The Prisoner's Dilemma, you will start to see examples all around you. The underlying scenario is always the same. It goes like this: The logic of self-interest tells us that we will do well by cooperating - in a relationship, in a social setting, or in the wider global context. When two or more individuals cooperate, though, the same logic of self-interest can often suggest to each individual that they could do better by breaking the cooperation - cheating on a partner, pushing in to a traffic queue, or plundering a resource, for example. The logic is impeccable, but if one individual can use it, so can the other(s). When they do, the cooperation collapses, chaos ensues, and the paradoxical outcome is that the logic of self-interest has led to a situation where self-interest is the last thing that is being served.

This is not to say that social problems are always that simple, but very often The Prisoner's Dilemma lies at the heart of breakdowns in cooperation. This is especially so when more than two people are involved in an effort to cooperate. In this case the situation is called The Tragedy of the Commons - a scenario that was brought to public attention by the Californian ecologist Garrett Hardin in 1968, although philosophers have been worrying about it since the time of Aristotle.

Hardin illustrated it with the parable of a group of herders each grazing an agreed quota of animals on common land. Then one herder thinks about cheating on the agreement by adding an extra animal to his herd. An extra animal will yield a tidy profit, and the overall grazing capacity of the land will only be slightly diminished, so it seems perfectly logical for the herder to add the extra animal. The tragedy comes when all the other herders think the same way. They all add extra animals, the land becomes overgrazed, and soon there is no pasture left.

The Tragedy of the Commons makes its unwelcome presence felt in the over-exploitation of resources (from minerals to fish stocks), rainforest clearances, territorial disputes and host of other circumstances where greed leads to the breakdown of cooperation. The logic that underlies it is very similar to that which underlies the Prisoner's Dilemma - in fact, game theorists have proved that the Tragedy of the Commons is really a series of Prisoner's Dilemmas enacted between the different parties. We saw it acted out in just this way when the stories about negotiations at the Copenhagen climate change conference began to emerge. The tragedy unfolded in full as each nation remained determined not to commit to the economic sacrifice that reducing carbon emissions would entail, until in the end there was no firmly enforceable agreement at all.

In a later blog I will discuss what measures we may be able to adopt to escape from the problems posed by The Prisoner's Dilemma and The Tragedy of the Commons. In the meantime, let us celebrate John Nash's great achievement - a remarkable insight into the underlying causes of breakdowns in cooperation, and one that everyone has a right to know about if we are to have any hope of resolving such problems in the future.
Len Fisher

John Nash's original paper was published in the Proceedings of the National Academy of Science of the U.S.A. (Vol. 36), January 1st, 1950, pp.48 - 49. Be warned - although the conclusions are simple and transparent, the supporting mathematical proof is only within the reach of experts.