Prisoner's Dilemma

Invented by Merrill Flood and Melvin Dresher in 1950 and later formalized by Albert Tucker, one of the most famous dilemmas in Game Theory is the Prisoner's Dilemma.

Unless the names, the story is as follows:

In Chicago district a crime was committed and the district attorney, although it has no sufficient evidence, know that it was committed by two criminals who, for simplicity we will call Bob and Adam. However, the attorney may convict only if at least one of them confess to the crime. For this reason, he orders arrest the criminals, put them in separate rooms and offers each of them the following agreement:

In Chicago district a crime was committed and the district attorney, although it has no sufficient evidence, know that it was committed by two criminals who, for simplicity we will call Ivo and Bruno. However, the attorney may order only if at least one of them confess to the crime. For this reason, he orders arrest the criminals, put them in separate rooms and offers each of them the following agreement:

• If you confess and your accomplice does not confess, you go in freedom and to your accomplice will be sentenced to maximum penalty of 10 years in prison. But if your accomplice confess and you do not confess, he goes free and you will be sentenced to 10 years in prison.
• If both confess, both will be arrested but will not be sentenced to maximum sentence for collaborating with justice, that is, will be sentenced to seven years.
• If no confession, the evidence already gathered is sufficient to give a sentence of one year in prison each.

The question can be summarized as the following table, where in each cell the first number corresponds to the prison sentence to be awarded to Ivo, and the second corresponds to the prison sentence to be given to Bruno according to the possible options that each of the criminals can take.

Note: The use of negative numbers is only because the prison sentence is a negative consequence.

In this case, criminals are faced with a dilemma: confess or not to confess?

Let us analyze the situation in Bob's perspective having only in mind the effect of the attorneys's agreement:

If Adam does not confess, Bob will have a penalty of one year in prison if not confess to the crime and leave freely if he confess;

If Adam confess, Bob will have a 10-year prison sentence if not confess to the crime and a 7-year prison sentence if he confess.

The same analysis can be made for the case of Bruno. Which means that, overall, taking into account what the accomplice can do to either of them, confess to the crime is always the best option, even at the risk of being sentenced to a prison sentence of seven years.

And what this dilemma is interesting is precisely that. Although there is the possibility of having both a prison sentence of a year, which would be possible if the criminals were able to coordinate the best strategy to respond to the agreement of the prosecutor and not confess to the crime, this not being possible, the best strategy for both is to confess the crime and sentenced to a prison term of seven years.

As this, there are many situations in life where we could take advantage if it were possible to coordinate our action with the response of other actors, but as this is not always possible, we ended up taking the option that puts us in a position to minimize losses, whatever the choices of other players. Even if this option can bring us to very negative consequences.

But, it is clear that this is only a theoretical formulation, in real life, each person is an individual and there are more averse to the risk then others, more cooperative and less cooperative. In addition, in real life often an option of this nature can be taken considering that a person has knowledge of the other. But this is nonetheless a model who turns out to illustrate many aspects of life.