N-Person Prisoner's Dilemma

The n-person prisoner's dilemma (NPD) is basically the Prisoner's Dilemma with more than two players. The NPD emerged during the early 1970's and quickly became popular among social theorists and economists. The sudden interest in NPD occurred mainly because of the economic and social developments during the late 60s and early 70s. At this time, problems such as inflation, voluntary wage restraint, the energy crisis, and environmental pollution were pressing issues. This era of history, however, is better known for the increasing international tension between the U.S. and the Soviet Union. Both superpowers were engaged in mass production of nuclear weapons, creating a very real threat to the existence of the entire world. With the proliferation of nuclear weapons came the issue of multilateral disarmament. The various social, political, and economic tensions of the 70's can all be modeled by the NPD, indicating the remarkable range of real-world problems that NPDs can simulate.

Many real-world problems, be they social, political, or economic, can be modeled as an NPD. In economics, an interesting example concerns the "invisible hand theory" and how it applies to the labor market.

In 1776, economist Adam Smith introduced the theory of the "invisible hand" which still remains the cornerstone of traditional economics. The "invisible hand", in short, is what dictates the motion of the economy. It is not a single individual or government that controls its motion, but is instead motivated by every person who participates in the economy.

In the labor market, companies hiring workers are consumers and those looking for jobs are the producers. That is, job seekers have a product to sell, namely their skills and the companies want to buy their labor. The "invisible hand" which dictates the labor market decides the wages that companies will pay.

An example of how and NPD can be used to model the labor market is as follows: Every trade union's individual self-interest is to negotiate wages that exceed the rate of inflation in the economy as a whole. However, if all trade unions negotiated wages to benefit their own self-interest, the prices of goods and services go up and everyone is worse off than if they had all exercised restraint.

In order to solve this problem, the British Labour Party issued a Manifesto (1974) which contained an outline of a "social contract" whose aim was to encourage trade unions to exercise voluntary wage restraint in order to decrease the rate of inflation. The "social contract" was designed to encourage collective rationality in wage bargaining over individual rationality. However, this solution was unsuccessful because it did not change the underlying strategic structure of the wage bargaining game.

Another type of NPD that is readily evident in the real world is that which simulates situations where resources are scarce. For example, when there is a shortage of any resource, such as water or energy, there is usually a call for conservation. However, and individual only benefits from restraint if everyone else restrains as well. However, restraint of an individual is unnecessary. That is, if everyone else restrains then it would make much of a difference if you didn't restrain. On the other-hand, if you restrain and no one else does, then your attempt at conservation is futile. Therefore, it is everyone's individual self-interest to NOT conserve. However, if everyone acts individualistically, all are worse off.

One last interesting example of an NPD is called the tragedy of the commons. Suppose there are six farmers who each owns one cow that weighs 1000 lbs. These six farmers share one plot of grazing land, a plot that can maximally sustain six cows without deterioration from overgrazing. For every additional cow that is added, the weight of every animal decreases by 100 lbs. Suppose every farmer had the opportunity to add one cow. If one farmer decides to add one cow, then his wealth increases since he will now have two cows that weigh 900 lbs each instead of just one cow that weighs 1000 lbs. Each of the six farmers, if they act in their own self-interest, will also add another cow. However, if all six farmers do add another cow, then each farmer ends up worse off. That is, each farmer will have two cows that weighs 400 lbs each instead of one cow that weighs 1000 lbs.

Small farmers in England during the period of the enclosures in the eighteenth century became impoverished because of this NPD situation.

All multi-person prisoner's dilemmas share a common underlying strategic structure. Therefore, any game that satisfies the following criteria is an NPD by definition:

  1. each player has two options: cooperate or defect
  2. defecting is the dominant strategy for each player (i.e. each player is better off choosing to defect than to cooperate no matter how many other players choose to cooperate)
  3. the dominant strategies (to defect) intersect at a deficient equilibrium point (if all players choose to defect, the outcome is worse than if each player had chosen non-dominant strategies (to cooperate))

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