# The Prisoner's Dilemma

Posted by Jason Polak on 08. March 2018 · 2 comments · Categories: math · Tags:

Suppose the police suspect you and your friend of robbing one billion dollars in bitcoin. The police can only charge you and your friend with the small crime of possession of a sawed-off shotgun, however, and want both of you to confess to the robbery. So, you and your friend are put in separate rooms and given two decisions:

1. Stay silent, in which case you'll only be charged with weapons possession
2. Confess

If both of you stay silent, then both of you will get a minor prison term. If you confess and your partner stays silent, then the police will let you go in exchange for prosecuting your partner, and vice-verson. Since your partner stayed silent, they will get the maximum sentence. If both of you confess, both of you get serious prison time but a little less than the maximum because both of you cooperated.

Since each of you is in separate rooms, you have to make the decision without knowing what your partner decides. What is the best strategy?

Game theorists like to put all the possibilities in a 'payoff matrix', which describes the possible outcome for each combination of decisions. Here is the payoff matrix in this case:

 Confess Stay Silent Confess (-3,-3) (0,-4) Stay Silent (4,0) (-1,-1)

The choices in the column represent your choices, and the choices in the row are those of your partner in crime. In each tuple, the first number is your payoff, and the second number is that of your friend's. Negative numbers are bad.

The specific numbers don't matter, but it is their ordering that is important. It is the ordering that corresponds to the verbal description.

## The Strategy

In this game, you consider your strategy for each choice of your friend's. If your friend confesses, then the best choice for you is to confess. But if your friend stays silent, the best choice for you is still to confess. Therefore, you conclude that no matter what your friend does, you should confess.

Unfortunately, your friend reasons this way also and also confesses. Therefore, both of you get a pretty severe prison sentence.

This type of situation is called the prisoner's dilemma. It is a dilemma because using the above rational reasoning, both of you end up confessing. However, had you both stayed silent, you would have ended up with a light prison sentence and possibly gotten out young enough to thoroughly enjoy all those bitcoins.

If you could only cooperate with your friend and somehow ensure this lesser outcome, that would be better for both of you.

The prisoner's dilemma is one of the most studied games (in the sense of game theory). That is because it comes up in all sorts of real-life situations. In fact, a good exercise would be to think of some situations that are closely modelled by the prisoner's dilemma game, taken from your own experiences.

## The prisoner's dilemma in scientific publishing

Here is one example that I find particularly amusing. Imagine two scientists, Charles and Francine. Both want to understand the world. That's why they became scientists after all.

It turns out that in addition for their desire to understand the world, both of them also want grant money. And as it so happens, one of the main criterion used to evaluate the merit of their research is the number of papers they publish.

Now it comes time for the research. They have two choices: publish good papers or publish bad papers. If they choose to write bad papers, they can write more papers and increase their chances for grant money.

If Charles and Francine both decide to publish good papers, then they will contribute to the understanding of the world with quality results (+2 points) and since they have roughly the same number of papers, each has a fifty percent chance of getting the grant money (1/2*10 points = 5 points). Therefore, they both get 7 points.

If Charles publishes bad papers and Francine good papers, Francine will contribute to the knowledge of the world (+2 points) but won't get the grant money, and Charles is guaranteed the grant money (+10 points) but won't get much of an intelletual reward.

If both of them publish bad papers, then both of them have an equal chance at the grant money (1/2*10 = 5 points) but neither gets the intellectual reward. Here is the payoff matrix:

 Bad Papers Good Papers Bad Papers (5,5) (10,2) Good Papers (2,10) (7,7)

This payoff matrix is another version of the prisoner's dilemma. Charles wonders what he should do. Francine publishes good papers, he should start publishing bad papers because then instead he will guarantee himself the grant money.

If Francine publishes bad papers, Charles realises he again has to publish bad papers otherwise he is guaranteed not to get the grant money. Therefore, Charles chooses to publish bad papers. Using the same reasoning, Francine chooses to publish bad papers as well.

That nicely illustrates the dilemma. If Francine and Charles cooperated and only published good papers, each would be able to do good science and contribute great knowledge to the world, while still having a chance at getting the grant money. However, because Charles and Francine play rationally, they choose to publish bad papers, and forgo the benefits of doing good research.

Both of them lose, and as a source of grant money, the public loses. Who wins? Journal publishers!