I am sure that there have already been a number of game theorists who have commented on Joker’s ship game in the Dark Knight. For example, the following link analyzes the game while factoring in human desires to survive and to obey morality. Also, it introduces a probability that batman will intervene and save the ship.
However, one shortcoming of this model is that the ship game is treated as a one-shot game, which does not take into account the changes that may occur in human thought and behavior over time. For example, suppose that Joker started the game at 6pm, and the ships are set to detonate at midnight. You are aboard one ship, and you know that the other ship has no intention to blow you up. However, you feel morally obligated to keep the other ship alive as long as possible, in case batman does intervene successfully. Would you end up blowing the other ship up? Well, if you ultimately want to live, you would probably want to blow the other ship up a few seconds before midnight, granted that batman has not yet succeeded. However, at 8pm, you may want to hold onto your remote and wait to see the outcome.
In this entry, I will use a simple mathematical model to predict the change in human behavior over time as a function of their sense of urgency.
Partition the game into T discrete time periods (not necessarily equal) during which Batman has a fixed, independent probability Z of intervening successfully. Call the value of life X, and the value of obeying ones conscience Y. (In our game matrix, we will use 0 for clear conscience, and -Y for a guilty conscience.) If batman successfully intervenes with Joker’s plan first, then the game ends, and both ships survive with clear consciences. The game also ends if one ship destroys the other ship, or vice versa, but the surviving ship is left with a guilty conscience. Finally, we will also assume that if both ships flip the switch during the same time interval, then both ships are annihilated, but no moral consequences are felt.
Some Trivial Cases:
Case X > Y, Z = 0. Suppose that batman does not exist, and people do not have a conscience, or a conscience that is worth less than life. Then Joker’s repeated game has a dominant strategy: blow the other ship up as soon as you get the trigger.
Case Y > X, Z = 0. Suppose that batman does not exist, and people have a collective conscience greater than life. If the penalty of violating the conscience exceeds that of dying, then the dominant strategy is to wait until midnight, and let Joker blow them up.
A more interesting case, the game of hope:
Case X > Y, Z > 0. Here batman has a chance Z during each time interval of saving both ships. Hence, at the beginning of the game, the value of the game is a geometric sum from t = 0 to T-1 of Z(1-Z)^t. Provided that batman defeats joker, the resulting utility derived by both ships is X, since both conscience and life are satisfied. Hence, at each time interval t between 0 and T-1, the matrix is given by:
| blow ship | not blow ship | |
| blow ship | 0, 0 | X -Y, 0 |
| not blow ship | 0, X -Y | (1-(1-Z)^(T-t))*X, (1-(1-Z)^(T-t))*X |
where the bottom right entry is the expected value derived if both ships decide never to pull the trigger.
When t becomes large enough, the value of not blowing the other ship may become less than X – Y, which leads to a weakly dominant strategy of blowing up the other ship. Note that in the very last interval T-1 (right before midnight), the game reduces to:
| blow ship | not blow ship | |
| blow ship | 0, 0 | X -Y, 0 |
| not blow ship | 0, X -Y | Z*X, Z*X |
Hence, if the chance of batman’s intervention is greater than (X-Y)/X, the dominant strategy throughout the entire game is to not blow the other ship. However, if the chance of batman is smaller than (X-Y)/X, somewhere along the way, maybe at 10pm, or 11pm, or 11:30pm, the expected value of not pulling the trigger will become less than pulling the trigger. The best strategy then becomes to blow the other ship up, because it is time to give up hope on Batman.
The game of paranoia
However, this game can be extended once more! Suppose that you also know that the other ship is also reasoning using the same selfish game theoretic principles as you are. You know that at 10pm, both of you are going to give up hope on Batman, and hence blow each other up. In this case, your real deadline is not midnight, but rather 10pm! What happens then? Should you try to blow the other ship up at 9:59pm, knowing that if you do not, then you will die?
However, the other ship also knows that you are thinking this, and hence decides that it would be better to blow you up at 9:58pm. I mean, 1 minute is not enough for Batman to intervene right?
Using this argument recursively, we see that this game becomes one of paranoia, and a dismal game this is! Each ship comes to the conclusion that it is best to press the trigger upon getting it, to minimize the chance of the other ship pressing it first!
The game of mutual trust or cooperation
I do not have a nice rigorous argument for this, but: If there were such a thing as a game of cooperation, it seems likely that the two ships will talk to one another at t=0 and decide to throw out the triggers, even if each ship values life over morality. (Of course, the movie suggests that they did so for moral reasons.) By tossing out the triggers on each ship at time 0, granted that Batman has a sufficiently high probability of intervening, they play a one-shot game based on the expectation that Batman will successfully intervene before midnight. Furthermore, by preventing an extended game, they avoid the terrible equilibrium induced by paranoia.
Of course, because the ships are unable to talk to each other, this decision can only be reached based on some form of “mutual trust” or “faith in humanity”.
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Other notes:
In reality, each human has a different weight attached to life and morality, but here I simplified the problem into a symmetric game assuming that the law of large numbers reduces the variance of each ship’s collective value system, such that the utilities are nearly identical.
Also, Joker could have been lying, just for the heck of it. But such a game would not be meaningful to play, would it?
Finally, does anyone really know the probability that Batman will pull through?
Posted by Foo 
Theory World 