Game Theory in The Dark Knight!

I personally love when I find places that Math relates to everyday life. And while The Dark Knight might be an extreme relation to real life, it is still a movie, so why not talk about it!

At this point, I hope that everyone who has an interest in seeing the movie already has, I don’t want to ruin the plot for anyone. So if you really want to see the movie, I guess I am assigning you homework to watch it tonight, or at least read about it. The Link below to the blog has a good overview of the movie before getting into the Game Theory discussion.

Game Theory of the Dark Knight 

So without further delay, here is the game theory in The Dark Knight, or at least some of it (there is a lot).

For those who have seen it, I am sure that the first thing that you think about is when the Joker fills the two ferries with people. One ferry carries prisoners from Gotham while the other carries innocent civilians. But here is the twist that gives us the game theory! The Joker has rigged each of the ferries with explosives; he then precedes to hand each of the ferries the detonator to the other ferry. If neither ferry blows up the other ferry before midnight, than the Joker will blow up both of the ferries and everyone will die.

So lets take a look at the payoff matrix for this situation:

So we see that the highest payoff is not dying, for both the prisoners and the civilians.

The civilians recognize this and take a vote on whether or not to blow up the ferry carrying the prisoners. When the vote is tallied, the result is to detonate the ferry. Seems simple in practice, press a button, kill prisoners that have “had their chance,” and you get to live. But it is not as easy as that. There are other factors that go into play. We are going to try and understand these factors by taking a look at a different type of hostage dilemma.

Hostage Dilemma

Lets paint the scene:

-50 people locked in a room by a bad guy

-Bad guy needs a password from at least one of them

-Bad guy will ask hostages one by one what the password is

            -Hostage spills the information, “game” ends

            -Hostage stays quiet, he dies

Now for the payoffs:

Sacrificial and Selfish types

Sacrificial receives -1 from dying and -9 for giving away the password

Selfish receives -1 from dying and does not care if the password is given up

For this case we will make the hostage sacrificial 95% of the time, and selfish the other 5%.

So how often will the bad guy get the password? Turns out that it is 100% of the time, because the first person will always give up the answer.

 Proof

How:

Case 1, the first hostage is selfish.

They will receive a -1 payoff if they die, and a 0 payoff if they give up the password. And thus they will always talk, and so that wraps up case 1.

Case 2, the first hostage is sacrificial.

No one talks, then they die and receive a -1 payoff, if they talk they get a -9 payoff.

Seems simple, but you need to take a look at the future, aka the people to go afterwards. Selfish type will always talk, and so the best-case scenario is that all sacrificial types remain silent. The probability that no one will talk is .95^49=.08.

And thus the probability that someone talks 1-.08=.92, or 92%.

So now this allows us to take a look at the expected utility.

EU(keep quiet)=(-1)(.08)+(-1+ -9)(.92)=-9.28  

EU(talk)=-9

And so the hostage will receive more of a payoff if he/she decides to talk, even though it is –9. And sadly the bad guy will always get the password.

So in the Dark Knight, the hostages, the civilians and the prisoners, have two decisions, detonate or be detonated.

So while the connection between the two is not exactly clear and straightforward, it still helps us to understand that there is more behind the decision.

The civilians who are selfish want to detonate, and the sacrificial ones will not detonate. Detonating will not receive any negative payoff, but sacrificial ones will, and also sacrifice the rest of the ferry.

So I hope this gives you an overview of the hostage dilemma and an intro to the Dark Knight. I hope I didn’t spoil anything yet, but warning there will be a spoiler in class tomorrow.

Thanks guys,

Please post any questions that you have, and I will do my best to answer them in the comments, or in the presentation tomorrow.

Sam Horan