In the Dark Knight one of the Joker’s schemes to bring anarchy to Gotham makes use of a classic game theory puzzle called the Prisoner’s Dilemma. However it isn’t clear whether the film’s writing team didn’t understand game theory or didn’t care. One thing is for sure, they got it wrong…

Most of this is reprinted from my blog.

The Joker has rigged two boats filled with people with explosives. One boat has ordinary law-abiding citizens. The other is filled with prisoners. He has given each boat the detonator to the other boat’s bomb. He tells them that he will blow both boats up in one hour unless one boat blows the other boat up first.

Not really that tough of a prisoner’s dilemma on the surface. Let’s look at how it breaks down.

I’ll assign the following values…

Life Spared: 1
Life Taken: 0

There is no third value as there would be in other dilemmas or even games of chicken.

Cooperate: Don’t detonate
Defect: Detonate the other boat

So the matrix looks like this…
B
C | D
A C|0,0 | 0,1
D|1,0 | 0,0

There is no 1,1 outcome in this dilemma, and there’s only one round (one shot dilemma). Naturally, defection is the rational choice, and defection quickly before the other side could defect first.

This leads to an interesting problem not presented in the film. If both boats are rational actors, then both boats would recognize immediately that a single minute wasted leaves you vulnerable to the other boat’s defection, so the rational action is to push the button right away. But if both boats reached this conclusion simultaneously, then that leads to the 0,0 outcome above. Of course neither party can control the other’s actions in any way so this is a moot point and you would push the button immediately anyway.

Of course in the film they both cooperate, and only after a ridiculous scene where the members of the law-abiding boat decide to put the whole thing to a vote (an incredibly irrational thing to do given the circumstances, and for me anyway an illustration of the limits of majority-rule democracy). But they vote to defect yet nobody on the boat has the balls to push the button so they end up cooperating against their own collective will, another totally irrational outcome, but one that serves the plot. Batman of course prevents the Joker from blowing both boats so all are saved and rewarded with a 1,1 outcome.

I’d have like to have seen some discussion amongst the prisoners of “Batman will save us” or at the very least “let’s hope the police can stop him! We don’t want this on our hands!” That would have changed the matrix somehow for me and would have made the people on the boat’s actions seem more rational to me.
There was an interesting line where one guy looks at a clock and remarks that the other boat hasn’t blown them up yet. The assumption being that the other boat intends to cooperate. In a matrix where mutual cooperation rewards the parties, this would change everything, because every minute that passes by, rather than leaving you vulnerable to the other side’s defection, confirms the other side’s cooperation and would inspire mutual cooperation.

There is one other possible matrix that I didnt consider but was suggested at this blog. He basically argues that because cooperating is the morally correct thing to do, that there is some value that should be assigned to it. So his matrix has defection and dying 0, cooperating and dying 1, defection and surviving 2.

I don’t disagree with this setup at all. And to be clear, I don’t disagree with the outcome in the film, either. I think it is absolutely probable that both sides would cooperate even in the face of death, even though to do so would be irrational. However I do think the actions of the “good citizens” boat, namely taking a vote, were completely irrational and improbable. If anything, the film proved some of the inconsistencies with political “games” and sound game theory, even if it didn’t mean to.