🔴 CAUTION 🔴 There will be a few spoilers ahead.. But, I guess it's a little too late for that now :D
There are multiple scenarios in this movie that contain different dilemmas that can be analyzed with game theory; however, I will be focusing on the bank robbery scene as well as the ferry scene.
And Here. We. Go!
https://giphy.com/gifs/quote-batman-joker-2GjgvS5vA6y08
The Bank Robbery 💰
Let's watch the opening scene, where the Joker robs a bank with some accomplices:
Opening Dark Night Scene!
As you just saw, the Joker seems to have gotten away with murder (lol) pretty well. Let's dissect what happened here:
Definition 1. The Pirate Game¹ is a game containing the following rules:
1. Three or more pirates need to split 100 gold coins
2. The pirates vote on the distribution plan made up by the most dominant pirate, if the dominant pirate looses the vote, then he walks the plank
3. Pirate A > Pirate B > Pirate C
Solution-- The Joker bribes pirate C to vote for his distribution plan of pirate A gets 99 coins, B gets 0 coins and pirate C gets 1 coin. This works because if pirate A dies off, then B can just overrule C, so C gets nothing (which C does not want).
Tying this back to The Dark Night, the Joker is pirate A. He tells pirate C that if he kills the next guy that he will have a bigger piece of the pie. In this way, the Joker has bribed each of his men to make the plan go through.
https://giphy.com/gifs/cartoon-batman-comics-a0NtFqyoHWjhS
The Ferry Scene ²🛥
In this scene, there are two ferries loaded with hundreds of people. One ferry has civilians and the other ferry is holding prisoners. The Joker has rigged each ship with explosives and gave the detonators to the people on board. Except it's the detonator for the other ship! Either someone pushes the button, or everyone dies at midnight. To keep it simple, let's assume there are only 50 people in total.
There are two types of hostages on each boat:
a. Selfish: Want to live no matter what. (Payoff is -1 from dying, 0 for living)
b. Sacrificial: Doesn't want to bad guy to get what he wants (Payoff is -9 from Joker winning, -1 from dying)
Let's assign probabilities to (a) and (b) hostages. Since we want generally make it hard for any sort of villain to get what they want, let us say that P(a)= 0.05 P(b)= .95
Before moving forward, answer this TRAIN of thought: What is the percentage that the bad guy will get what he wants in this type of situation?
--100% (We will see why in a bit)
Case 1:
The selfish person always would rather be a rat (payoff= 0) than die (payoff = -1). So, this case is easy and the Joker will get what he wants.
Case 2:
The sacrificial person is a more complicated person. We need to look at the expected value of "talking" or "keeping quiet".
Probability nobody talks = 0.95⁴⁹= 0.08.
Probability someone talks = 1-0.08 = 0.92.
EV of keeping quiet = (-1)(0.08) + (-10)(0.92)= -9.28
EV of talking = -9
What's going on here?
It looks like the expected value for the sacrificial person is less if they talk!! Crazy. And so theoretically, this is why the Joker should get what he wants 100% of the time. It is more valuable for the hostages to snitch every time.
How does the movie end?
This approach to solving this game doesn't line up with the character of Gotham City. At the end of the scene, a dominant prisoner throws the detonator out of the port hole. Big moral win for him! Also, the civilians voted to blow up the other ship! However, no one had the guts to do it because they felt it was wrong. So, the Joker never got what he wanted because no one blew anyone else up and the Batman stopped him. One thought is the people of Gotham value morality rather than life. This is shown in the payoff matrix below:
http://quantitativepeace.com/blog/2008/07/the-dark-knight.html
ConclusionI believe that the take away here is that the pure strategy Nash equilibrium is to not detonate the other ship. As we can see from Figure 3, nobody gains anything from changing their strategy from both cooperating.
Work Cited
1. Presh Talwalkar, "Dark Knight Game Theory: The Robbery Scene And The Pirate Game," YouTube, October 25, 2013, , accessed April 12, 2017, https://www.youtube.com/watch?v=ZKvEFZbmp7M.
2.Spaniel, William . "Game Theory 101: The Dark Knight Rises (Hostage's Dilemma)." YouTube. July 31, 2012. Accessed April 12, 2017. https://www.youtube.com/watch?v=g2Oxg1oAKpk.
3. "The Dark Knight and Game Theory." The Quantitative Peace. May 13, 2016. Accessed April 12, 2017. http://quantitativepeace.com/blog/2008/07/the-dark-knight.html.
It's interesting that my favorite movies are relating back to game theory...maybe that's why I like them in the first place (subconsciously)! Can we go over both of the cases you mentioned in the blog in class again tomorrow? I didn't fully understand the second case specifically.
ReplyDeleteYes, certainly!
DeleteAre there any ways to prevent Pirate A from getting what he wants in the Pirate's Dilemma? For example Pirate B offering C a better bribe, then the two of them teaming up against Pirate A so he walks the plank?
ReplyDeleteI remember a scene in the movie where the Joker says that he will kill a civilian everyday until Batman reveals himself. I won if either of the situations you talked about can relate to this.
ReplyDeleteI will try and incorporate that scene within the ferry scene. Great question!
DeleteJoker plays a lot of games! I Liked both examples, but I thought the hostage example would have involved a handful of hostages like the ones that were in the bank. Di you read anything on how negotiators try to talk down bank robbers if they have hostages? Would be interesting to see if any of these theories apply.
ReplyDelete