Skip to main content



Can the Joker Beat Game Theory?

In 2008, one cinematic adaptation of classic comic book series, Batman, became an international sensation after breaking multiple box records in the opening weekend. The Dark Knight directed by Christopher Nolan is not only deemed to be a masterpiece, but also created the trend among Hollywood movie scenes, followed by numerous cinematic adaptations of classic superhero series. Although this movie is frequently used as a subject of filmmaking study or cultural behavior, it also discusses an interesting topic of Economics: game theory. (This article contains spoilers.)

During the third act of The Dark Knight, the main villain, the Joker, challenges Batman to prove the validity of his nihilism, by taking 2 groups of people of Gotham hostage: low abiding citizens and prisoners of Gotham. 2 populations are separately trapped in 2 ferries which carry enough explosives to kill each group of passengers. The rule of the Joker’s game is as follows. Each ship is given a detonator for the “opponent” ship. When one player uses the detonator, they win the game, meaning the survival of the passengers, and the opposing ship will be destroyed. If either ship fails to use the detonator after 30 minutes, the Joker will destroy both ships and both passengers will die. For the sake of argument, we assume that if 2 detonators are used simultaneously, the passengers of both ferries will die. The simplest analysis of this game is shown in the Figure 1.

screen-shot-2016-09-15-at-8-49-32-pm

The Joker’s game has some similarities with Prisoner’s Dilemma. For both ships, using the detonator is a dominant strategy, which means, theoretically both ships will explode (as 2 prisoners will eventually confess in Prisoner’s Dilemma). This was the Joker’s original intention; however, this does not happen in the movie. Decision making in real life is far more complex especially if it involves the extreme situation like this. For example, one passenger on the ship of normal citizens is convinced to use the detonator, but his sense of morality hinders him from doing so at the very last moment.

screen-shot-2016-09-15-at-9-04-36-pm

In the second diagram, we assume that people value their morality unless their survival is jeopardized. In this case, 2 players will no longer have a dominant strategy. This game has the pure strategy Nash equilibrium at (Detonate, Cooperate) and (Cooperate, Detonate) with a mixed Nash equilibrium of playing each moves with a probability of 0.5. (This probability varies based on the distribution of the passengers who value survival over morality) In other words, we only have 25% chance of cooperate-cooperate happening. For other 75%, the detonator will be used and the Joker wins, proving his theory that anyone can become a criminal like him.

screen-shot-2016-09-15-at-9-16-31-pm

Finally, let’s consider the case where each passengers always values morality over his own survival. In this situation, the dominant strategy of 2 players is to cooperate. Only in this case can all passengers survive (if batman appears and beats the Joker before the time limit).

In this case study, I focused on the balance of morality and willingness to survive; however, other factors can also have an important role. The calculation is extremely simplified by denoting death as 0 and survival as 1 while those values can value by individuals. To some people, death could go down to negative infinity or could even become positive for some cases. In the movie, both passengers give up using the detonators and batman saves the day; however, in real life, the Joker could have proven his nihilism by killing one of 2 populations. Was the Joker simply a bad economist? Did the people of Gotham prove themselves to be selfless? The answer is better remain secret.

The Dark Knight and Game Theory

 

Comments

Leave a Reply

Blogging Calendar

September 2016
M T W T F S S
 1234
567891011
12131415161718
19202122232425
2627282930  

Archives