Friday, January 18, 2008

Econ 101: The Prisoner's Dilemma

[This post was motivated by a question in the comments to my post on self-fulfilling expectations]

One of the most canonical examples in all of modern economics is the game theory story of the prisoner’s dilemma. The reason this has become so well-known and almost always used as the very first example of game theory in economics classes is because of the power of the result. It very simply and elegantly shows how, in strategic situations, the basic efficiency of free markets fails.

A little background: in economics perhaps the most powerful result of all is what is known as the first welfare theorem. The first welfare theorem states that when markets are free and complete (which is a technical term but actually implies some fairly strong conditions), the resulting equilibrium is efficient. This is also related to the invisible hand which states that economic agents, by solely pursuing their own self-interest, will maximize social welfare. Now, there are lots of things that make this not true in reality, the most common are the existence of externalities, public goods and asymmetric information. But, the prisoner’s dilemma shows us that it is also not true in strategic situations. Strategic situations are where the actions of one player determine the ‘payoffs’ of the other players. (NB: you can think of this as a type of externality, but a in a specific and complicated way – more complicated than the standard externality stories) To give a simple example if we are roommates and you are considering purchasing a new iPhone, the joy you get from that purchase may depend on you being the first to own one and show it off to our friends. If I buy one before you, then the enjoyment you get from yours diminishes. So my actions have affected your payoffs from the purchase of an iPhone. We are now no longer in the pure free market case which assumes that others decisions have no affect on your costs or benefits from market transactions.

The Game: there are infinite variations on the same story but they all follow similar lines. Suppose that two people are arrested of a crime. The police put them in two separate interrogation rooms where they cannot see or talk to each other. The police have enough evidence to get convictions only for a lesser charge which would lead to 1 year jail terms, but if one confesses to the crime they can convict the other of the more serious crime and in this case the confessor gets probation only (o years in jail) and the other person gets a 5 year jail sentence. If they both confess then they both get convicted and sentenced to 3 years in jail. It is assumed that they both know all of this and they know the other person knows as well (and technically each knows that the other knows that the other knows, and on, and on, and on…). They also have to make their final decision without knowing what the other has done.

So now consider the choices of each person. If they decide to not confess they get: 1 year in jail if the other does not confess and 5 years if the other confesses. If they decide to confess then they get 0 years in jail if the other does not confess and 3 years if the other confesses. So no matter what one person thinks the other person will do, confess is always the better choice. So they both confess and the result is that they both spend 3 years in jail.

Note, however, that there is a better out come for both, the one where both don’t confess and get only 1 year in jail. So the result of the prisoner’s dilemma game is sub-optimal because both could be made better off by not confessing. But acting in their own self-interest leads to this poor outcome – contrary to the invisible hand in free and complete markets.

So you see the power of the story: in strategic situations, all the efficiencies of free markets are no longer guaranteed. You could easily change the story to my iPhone story or any other more standard market stories and motivate this more obviously, but the essence is always the same.

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