I suspect that many of you got a chance to see The Dark Knight movie this weekend.
Just as an aside, I will say that I thought that the movie was sweet. Definitely the best Batman movie, maybe one of the best superhero movies ever made. Heath Ledger is terrifyingly good throughout. Aaron Eckhart and Christian Bale give excellent performances as well, and Maggie Gyllenhaal is a hell of lot better than Katie Holmes.
Anyway, there is a scene in the movie that got me thinking about game theory, and that is what I want to talk about. Beware, if you haven’t seen the film, this discussion includes some big spoilers.
In the final battle the Joker has rigged two ferries carrying people out of Manhattan Gotham to explode. One ferry carries mostly civilians with a substantial National Guard presence. The other ferry contains large numbers of prison inmates and some guards. The Joker has rigged both to explode, and he has given the crew on each boat the detonators — only they have the detonator for the other boat. He announces the rules of the game to the crew and passengers of each vessel.
- 1) Each of them have the power to blow up the other boat and then their boat will live.
- 2) If they get to midnight with
knowno exploded boats, the Joker will detonate both. - 3) Any attempt to leave or defuse the bombs will result in the destruction of both boats.
Now there is probably some horrible twist that the Joker had in mind that would subvert these rules. Maybe each detonator was actually for their own boat. For the purposes of this discussion, however, we will assume that the rules are actually as the Joker describes them.
Clearly, the Joker’s intention with this scheme was to have one of the boats blow up the other. The boat filled with civilians might feel entirely justified in blowing up a boat filled with felons to save themselves. They have committed no crimes, there are children aboard, etc. The boat filled with criminals could easily overpower the guards and blow up the boat of civilians to save themselves.
The tricky part for the actors on both boats is to try and guess what the other party is likely to do. The longer you wait, the more likely the scheme will be foiled and both will be saved. On the other hand, the longer you wait the more likely the other boat will decide to kill you.
The whole thing strikes me as an interesting application of game theory to decision making. Because the people on the boats have no way of communicating with one another, they have to judge their best course of action based on what they think the other people will do.
Whenever you have a game theory problem, it is best to draw a payoff matrix. A payoff matrix lists each of the actor’s options and the resulting payoff depending on how each actor performs. The interesting part is that the payoffs are contingent on the responses of both players.
To illustrate a point, I want to divide the payoffs matrices into two types: the simple case and the complex case. The simple case doesn’t include any of the complications that are certainly present in the film. It is shown below.
solid windowtext .5pt;mso-padding-alt:0in 5.4pt 0in 5.4pt;mso-border-insideh:
.75pt solid windowtext;mso-border-insidev:.75pt solid windowtext'>
style='mso-bidi-font-weight:normal'>Prisoner Boat
style='font-size:10.0pt;font-family:Arial'>
style='font-size:10.0pt;font-family:Arial;color:#3366FF'>Press button
style='font-size:10.0pt;font-family:Arial;color:#3366FF'>Don’t press button
style='mso-bidi-font-weight:normal'>Civilian
style='mso-bidi-font-weight:normal'>Boat
style='font-size:10.0pt;font-family:Arial;color:red'>Press button
style='font-size:10.0pt;font-family:Arial'>Not Allowed
style='font-size:10.0pt;font-family:Arial;color:red'>– style='font-size:10.0pt;font-family:Arial'>\Dead style='color:red'>
style='font-size:10.0pt;font-family:Arial;color:red'>Don’t press button
style='font-size:10.0pt;font-family:Arial;color:red'>Dead style='font-size:10.0pt;font-family:Arial'>\– style='color:red'>
style='font-size:10.0pt;font-family:Arial;color:red'>Dead
style='font-size:10.0pt;font-family:Arial'>\Dead
The simple payoff matrix assumes that there are no consequences — social or otherwise — to one boat deciding to blow up the other. Further, it assumes that the probability of rescue before midnight is zero. If no one presses a button, they are both dead. The option of both pressing the button is not allowed because presumably the blown up boat would not have that option.
You can figure out pretty quickly how the situation described in the simple matrix will end: both sets of passengers would be running to push the button. That is the best option from the point of view of both boats. (You would be running fast too because a reasonable player would also assume that the other boat knew exactly what you knew.)
However, the situation is much more complicated than the simple payoff matrix suggests because of the two assumptions we had to make. Below is a matrix that includes these complexities.
solid windowtext .5pt;mso-padding-alt:0in 5.4pt 0in 5.4pt;mso-border-insideh:
.75pt solid windowtext;mso-border-insidev:.75pt solid windowtext'>
style='mso-bidi-font-weight:normal'>Prisoner Boat
style='font-size:10.0pt;font-family:Arial'>
style='font-size:10.0pt;font-family:Arial;color:#3366FF'>Press button
style='font-size:10.0pt;font-family:Arial;color:#3366FF'>Don’t press button
style='mso-bidi-font-weight:normal'>Civilian
style='mso-bidi-font-weight:normal'>Boat
style='font-size:10.0pt;font-family:Arial;color:red'>Press button
style='font-size:10.0pt;font-family:Arial'>Not Allowed
style='font-size:10.0pt;font-family:Arial;color:red'>Murderers style='font-size:10.0pt;font-family:Arial'>\Dead style='color:red'>
style='font-size:10.0pt;font-family:Arial;color:red'>Don’t press button
style='font-size:10.0pt;font-family:Arial;color:red'>Dead style='font-size:10.0pt;font-family:Arial'>\Murderers style='color:red'>
style='font-size:10.0pt;font-family:Arial;color:red'>Dead
style='font-size:10.0pt;font-family:Arial'>\Dead
(Possibility of Rescued\
style='color:#3366FF'>Rescued)
The assumptions in the simple payoff matrix are wrong. For one, there would very likely be consequences to blowing up the other boat. For the civilian boat, there is the social stigma of being murderers — even if it is murder of felons. There may well be legal repercussions. For the prisoner boat, the social stigma may be less (they are already in prison), but the legal consequences are probably greater. Thus, in pressing the button, both actors face a non-zero penalty. You could say that the penalty is trivial compared to being dead, but it is still there.
Second, this being a superhero movie there is a non-trivial probability that the people on both boats will be saved before midnight. (You wonder whether the characters know that they are in a movie and account for this in estimating their probability of rescue.) The non-trivial possibility that both boats could live is a strong temptation to wait and not push the button.
The presence of these two complexities explains why this situation doesn’t immediately degenerate into explosion.
It also poses interesting personal questions too: how would you respond? How would you rate the size of the relative payoffs? What is the probability from your point of view of the other ship blowing you up? What is the probability — given your limited knowledge — of being saved? The interesting part of game theory problems (at least for me) is that people vary in their estimation of probabilities of other people’s actions and vary in their weighting of the different payoffs. This variation makes profit-maximizing play even more difficult to engineer.
For my part, I found the actual result in the film to be a little surprising. I mean, come on, a guy on the prisoner ship throws the bomb detonator out the window. Really? Maybe I’m cynical, but that strains credulity. It may have been important to the message of the film — Gothamites are willing to stand up for right and prove that the Joker’s cynicism was unfounded — but it hardly seemed like realistic behavior.
Anyway, I don’t remember the last time I saw such a cunning use of game theory for suspense value in a movie. I am curious to hear what people thought of that situation. (Further, I am hardly an expert in the subject, so if I have committed some substantive error please let me know.)
