The Dark Knight movie has a lot og game thoery. Here I chose one scene to describe:
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.
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.
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. 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. 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.
Wednesday, December 10, 2008
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment