A Simple Example

The School Prank:

Jack and Susan both work together on planting stink bombs throughout the school as a prank. The prank is a great success resulting in the school having to be evacuated and all the children having to be sent home!

Their teacher Mrs Smith suspects the pair but does not have any evidence to prove it. Being a fan of game theory she separates the two of them and offers them both the same deal on their own:

"This is a serious matter. I know you were both involved in this but I can't prove it. If you confess now you can go home and your partner in crime will get 3 detentions. If you both confess you will get 2 detentions. If you both say nothing you will still get 1 detention."

What should Jack or Susan do?

1. If neither of them confess then they both only suffer the single detention.
2. If Jack confesses and Susan stays quiet, then he goes home and she gets 3 detentions.
3. If Susan confesses and Jack stays quiet then she goes home and he gets 3 extra detentions.
4. If they both confess then they both get 2 detentions.

We can express this in a table like so:

• The first value to the left of the comma expresses the time in detention that Jack gets.
• The second value to the right of each comma expresses the time that Susan gets.

This is the basic convention used in these matrices.

Sometimes the options for each "player" may also be slightly raised or lowered to highlight the distinction.

I have also seen some tables that make them different colours for the same reason but I can't do that using Markdown!

So for example if we look at the top right situation where Jack is quiet but Susan confesses.

The result 3,0 indicates that Jack gets 3 detentions whereas Susan gets zero.

This is how to read these tables/matrices.

This can help us to break down the choices in a more simple visual way.

Jack's Options:

It can sometimes be easier to break things down to the individual viewpoints.

If we consider Jack's choices if he thinks that Susan will stay quiet we can simplify the matrix/table as follows:

If he stays quiet he gets one detention. If he confesses he goes home without any further detention.

Therefore the optimal choice for him is to confess.

What if he thinks Susan will confess?

Let's isolate what happens to him then:

In this case if Jack remains quiet he gets 3 detentions, if he confesses he gets 2. Again the best choice is confession.

This tells us that the best strategy for Jack regardless of what Susan does is to confess.

What about Susan?

If she knows Jack will be quiet:

The situation is exactly the same for Susan as it was for Jack.

The best option is for her to confess as she goes home with 0 detentions.

If she thinks Jack will confess:

Again as with the situation with Jack, the optimal solution for her is to confess, because it only gives her 2 detentions vs 3.

The best strategy for Susan, as for Jack is to confess.

Confession is the Dominant Strategy

Since the strategy of confession is better for both parties than staying quiet, confession is said to dominate keeping quiet.

It might seem counterintuitive, since if both kept quiet they would only get 1 detention each.

The point is this is not the optimal strategy from either individual standpoint.

The way I conceptualise it is that dual confession is the equilibrium point if both parties are trying to minimise their punishment.

This becomes all the more important if you don't know for certain how your partner or opponent for that matter may behave.

In most real world situations I think that is actually the case.

Hierarchy of Options is More Important than Absolute Values

It should also be noted that in the Prisoner's dilemma the actual relative values of the punishments are not important.

All that matters is that there is a hierarchy of punishments or losses.

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