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In the final battle in the movie, the Dark Knight (2008), 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 know no exploded boats, the Joker will detonate both.
  3. Any attempt to leave or defuse the bombs will result in the destruction of both boats.

Is this a known variant of the prisoner's dilemma? How to model this game and what is the actual solution? Why is the equilibrium as shown in the game attained?

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    $\begingroup$ (+1) This is more complex than it appears. It has to do with how each group ranks the various outcomes in terms of utility, but also with the beliefs that each group holds about the outcome-ranking of the other group. Finally, this is a repeated game -and every second that passes with both ferries intact, provides information and so beliefs are updated. $\endgroup$ – Alecos Papadopoulos Aug 2 '15 at 3:51
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    $\begingroup$ @Batman what do you mean by "Why is the equilibrium as shown in the game attained?" Do you mean shown in the movie? If so, I would not read too much into it - it's a movie. $\endgroup$ – Giskard Aug 2 '15 at 4:50
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    $\begingroup$ I guess any feasible outcome can be attained for specific preferences and beliefs. $\endgroup$ – FooBar Aug 2 '15 at 8:23
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    $\begingroup$ It seems fairly straightforward that a set of altruistic preferences could generate delay as outlined by Olive below. More interesting to me is the question of whether one can find an empirically plausible set of preferences and type-space such that there is an interior solution in which each type detonates after some amount of time $t$ has elapsed. Such an equilibrium would require continuous Bayesian updating intil beliefs about something become so pessimistic that the group chooses to detonate. $\endgroup$ – Ubiquitous Aug 2 '15 at 13:47
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Suppose first that the groups are not altruistic and care only about their own survival. It is not exactly a prisoner's dilemma since the outcome obtained by mutual cooperation (if both groups wait) is not Pareto-improving: everyone dies in that case. The only equilibrium is that one of the groups destroys the other boat as soon as possible; and any action played initially by the other group is possible in equilibrium, since this team is indifferent between waiting and triggering other the bomb (they will die one second later anyway).

As @Alecos_Papadopoulos wrote, the game becomes more interesting if the groups have pro-social preferences. For instance, they might be reluctant to sacrifice the other group and prefer that everyone dies (including themselves). If there is no uncertainty, the outcome is trivial: the only equilibrium is that both groups wait until the Joker triggers the bombs.

The most interesting scenario is the one in which the types of the groups are uncertain: each boat can be either selfish or altruistic. In that case, it seems reasonable (but other specifications are possible) to assume that cooperation is desirable only if the other group is also altruistic, but if the other group is selfish the individuals prefer to kill them first and survive. The equilibrium strategies are the following:

  • If the group is selfish, it triggers the bomb of the other group as soon as possible (it is a dominant strategy).
  • If the group is altruistic, its initial action depends on its beliefs about the other boat. If it is sufficiently optimistic (i.e it believes that its opponent is altruistic with a sufficiently high prior), the group waits. If it has not exploded one second later, the group understands that it faces an altruistic opponent and therefore waits until the Joker kills everyone. Notice that the explosions happen either at $t=0$ (as soon as the game starts) or at midnight (when the game ends) but never in between.
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  • $\begingroup$ If selfish types do not prefer mutual destruction to their individual destruction then one group can also blow the other boat at any time, not just $t = 0$, because blowing the boat at $t = 0$ is only weakly dominant. $\endgroup$ – Giskard Aug 2 '15 at 10:14
  • $\begingroup$ @denesp I might be mistaken, but selfish types playing a mixed strategy would be possible in equilibrium only if they were sure they faced an altruistic type, isn't it? If selfish types mix between destroying and not destroying, it is strictly dominant for a selfish type to destroy at the initial stage. $\endgroup$ – Oliv Aug 2 '15 at 10:41
  • $\begingroup$ Yes, only boat can have a $t \neq 0$ strategy for its selfish type, the other has to have $t = 0$. The boat that has $t \neq 0$ can also mix. It might be worthwhile making your solution more technical to see if we have missed other equilibria. $\endgroup$ – Giskard Aug 2 '15 at 10:50
  • $\begingroup$ @denesp That is true, I didn't mention that I was looking for symmetric equilibria, but the type of equilibrium that you mention also exists. $\endgroup$ – Oliv Aug 2 '15 at 10:53
  • $\begingroup$ Besides altruism, I think the actions are also affected by the fact that the Joker's actions are uncertain: it is possible that the Joker is neutralized, or regrets, before midnight, or that he is just lying, etc. $\endgroup$ – Erel Segal-Halevi Aug 24 '15 at 8:21
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I recently watched again the Dark Night movie and refreshed the game in question. First it is clear in the dialogues that beliefs on the other group's ranking of outcomes and intentions are updated as time passes (and the more time passes the more each group believes that the other group won't push the button). Second, what I believe would be interesting here is to determine what kind of preferences and beliefs must be in place in order to experience the outcome seen in the movie: neither group pushes the button, meaning that they accept to die rather than kill the other group, with perhaps some hope (some strictly positive probability) that the Joker's assertion of blowing both up may be a bluff.

I think an important aspect is what happens to the boat with the civilians: they take a vote in order to decide (i.e. a collective action), and decide to push the button. But then, the burden is down to a single person to do it, and we see that, even though the vote has taken the burden of decision of the individual back, no one can do it -perhaps because he feels that he will kill many people to save just himself (not the group). The individualistic experience of performing the final dreadful act appears to be a strong deterrent, and this act is not seen as just the procedural step of a collective decision but something that bears a special burden, a burden that even a person that voted "yes" to push the button cannot bear. The group voted to kill in order to survive - but there is no single person in that group that can execute this decision.

So it is not even clear that we can model only group-preferences here, but maybe one would need to start at the individual level and aggregate.

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