If you have not already seen The Dark Knight, then you should read this post at a later time as to avoid spoilers after the jump. Otherwise, follow the jump for a game theoretic discussion derived from the movie that has already broken two box office records for both best midnight opening and top first day revenue. I personally enjoyed the movie immensely, but that discussion does not follow.

So, officially, consider this a spoiler alert for what follows.

I am sure there were plenty of other individuals who immediately conceptualized parts of the movie in game theoretic terms as one’s mind is prone to do after sufficient indoctrination; however, after discussing the scenario with Julie for a bit, it seemed to be appropriate blog material and worth formalizing. Without further introduction (as I am assuming you have already seen the movie), we can begin:

The Story

The Joker’s final act as criminal mastermind and agent of nihilism (or, seemingly, to show Gotham city that we are all Homo Economicus when the structure of the game forces us to be) involves two ferries filled with people. The first ferry is filled with normal, law abiding citizens while the second ferry is filled with the population of Gotham Prison. The Joker, doing so without prior knowledge of the passengers and city officials, wired the ships with powerful explosives such that their explosion would destroy the entire ship and everyone aboard. No single individual is allowed to escape. Each ship is given a detonator for the other ferry. The use of the detonator saves the ship while killing everyone aboard the opposing ship. Thus, if any member of Ship A pushes the detonator, then Ship B is destroyed and all of Ship A is saved. Additionally, if either ship fails to use the detonator to destroy its opponent, then both ships will be destroyed by the Joker. Assuming that the actors must make their decision simultaneously, this would lead to the following game:

Solving this game is pretty straightforward and (detonation, detonation) becomes the dominant strategy as, at best, cooperation is weakly dominated. Thus, homo economicus and politicus have a very clear strategy to, without fail, destroy their opponents, and in so doing, both will ships will be destroyed.

However, in Gotham City, this does not happen – nor would it necessarily happen in a laboratory experiment either as a few more complications are introduced into the game. First, the game appears to be pseudo-sequential or, perhaps, a series of simultaneous game with a finite end. The Joker gives both ferries 30 minutes in which they can detonate the other side. Even with this complication, the outcome should be the same and both actors ought to choose detonation at the first node (that is, t =1 or when the first move is available). Using backwards deduction, both players recognize that their opponent will choose to detonate in the final iteration even if there are some gains to short-term cooperation. This effect cascades backwards to the initial decision node and mutual detonation occurs to prevent receiving the sucker’s payout of cooperating while the opponent defects.

Decision Rules

Beyond this, it appears that the decision making process for both ships is different. In the ship containing prisoners, the decision to detonate becomes decentralized and any one actor willing to grab the detonator could do so. While the armed guards gives the opposite impression of clear authoritarianism, Decentralization becomes apparent as the time moves on. Thus, decentralized decision making should lead to the optimal play as any single individual among the 500 or more sub-actors should have a preference for survival.

On the civilian ship, the decision mechanism becomes a simple majority vote. When the votes are aggregated, the decision to detonate the other ferry is chosen at a rate of almost 3 to 1. Yet, there is no executive to carry out the decision and the majority will does not prevail as no single sub-actor is willing to push the button. This psychological separation between deference of active responsibility (voting to kill others) versus carrying out the task, has been illuminated in some experimental settings as shown in the video Julie blogged about previously.

The civilians act rationally as long as they, individually, are not too involved in carrying out a potentially morally reprehensible act.

Morality?

Perhaps social norms mattered for the actors in the game? In the second game, we can assume that there are some social benefits from being a moral agent; however, being moral is not as beneficial as being alive. Since survival trumps morality, we get the second game:

As Yev Kirpichevsky notes in his comment, the pure strategy equilibria are {cooperate-detonate} and {detonate-cooperate} with a Mixed Nash Equilibrium of playing cooperation and detonation with a probability of .5 for both players with this specification (this would change depending on how we parametrize the value for being moral and the value for surviving). Perhaps, then, we saw the cooperate, cooperate cell (with probability .25) in the movie and if we watched the movie infinite more times, we would see a nice distribution where cooperate-cooperate occurs 25% of the time, both ships defect 25% of the time, and only one ship explodes 50% of the time. I plan on seeing the movie again, and I will let you know if the outcome of this scene changes.

However, this is not the full specification of the game, let’s assume that the parameters allow for morality to trump survival:

The pure strategy Nash equilibrium is to never detonate as cooperation is always more beneficial than destroying the other ship. While both specifications may be able to explain the movie, I think there might be a fourth game that explains the game the best.

The Joker Misspecified his Game?

While not the moral of the story that Batman wants the Joker or Gotham City to learn (he tells the audience and the Joker that people are not all evil – there is some good in Gotham), adding an additional parameter to the game can easily induce a cooperation solution. If the people aboard both ships believe that the cost of dying is probabilistic, then they would cooperate as long as the benefit to cooperating trumped both the probabilistic chance of punishment and the likelihood that the other ship decides to detonate. Thus, each ship has to calculate the value for morality (m) minus the cost of punishment P(c) and the cost of the other ship defecting P(dx)(d) where x is the other actor. The cost of opponent defection and punishment are the same (the value of survival), so the equation reduces to:

U(Ship A) = M – (Pc + P(dB))*(S) U(Ship B) = M – (Pc + P(dA))*(S)

Each equation must still be greater than the value of Survival Minus the Cost of morality such that for Ship A (or B with appropriate substitution):

S – M < M – (Pc + P(dB))*S

If M = S, then the joint probability of punishment and opponent detonation must be less than 1 for cooperation to be preferred, else defection dominates. As M and S vary in relation to each then, the threshold for the joint probabilities to make a preference in strategy changes. Note, if the value for M is zero for the observed actor and survival is infinitesimal greater than 0, detonation dominates. Likewise, if we build in opponent morality into the probability of opponent detonation, and opponent morality is zero, then the cost of morality becomes too high. Obviously, the game becomes increasingly complicated as parameters are made endogenous.

The actors must determine a subjective probability of the game being false: that is, the probability of punishment is not 1. The probability of punishment could go down from 100% if the passengers believe that the Joker is lying about the deadline, if they think there is a chance of technological failure, that their detonator may indeed destroy their own ship (adding a hidden cost to detonation), or if they live in a world with Batman. I would venture forth the assumption that the probability of punishment is inversely proportional to the probability that Batman exists with some modification based on his likelihood of failing to save people — but that is just me.

There are a few other situations in this movie that are prone to game theoretic applications and I have not exhausted the ways in which this game could be modeled. However, I think this calls for a new villian in the third movie of the trilogy: The Game Theorist. Much like the riddler, but deadlier and requiring Batman to use mathematics to fight crime.

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