A game-theoretic reading of Batman: The Dark Knight
(Some spoilers… but you have seen the movie, right?)
One recent discussion with one of my advisors brought up the subject of game theory crossing over to movies. It prompted me to think about which movies featured a typical game-theoretic situation, where players act competitively (or cooperatively) and receive a payoff from their actions. If this sounds abstract, let me introduce the idea with a game theory flavoured reading of The Dark Knight.
Knowing the other
Before we start, let’s set up the stage for the coming discussion. A situation can be written in terms of game theory if two (or more) players take decisions that affect the welfare of the other players. Think of the classical example of the Battle of the Sexes, where in the (dated) version of the problem, a man wishes to attend a Boxing match, while his female partner would rather go to the Opera. Assume that they have no means of communication before the events and that their enjoyment of the evening is worse if they are separated (i.e one goes to the Boxing match while the other goes to the Opera). Our two players are then both cooperating (because they want to spend the night together) and competing (because they have different preferences over the activity): this is where game theory starts.
A central question in game theory is thus, how do I know what the other party will decide to do? This seems like a reasonable question to ask if I want to maximise my enjoyment of the evening. If I know my partner is definitely going to the Opera, well I must resign to join too, even though I would rather see a Boxing match. But here is the catch: my partner is (assuming she is rational to a point, more on this later) doing the same thing! We are both building a model of what the other is thinking. And inevitably, this model must contain the fact that the other is trying to anticipate my actions. Which itself contains the fact that I am anticipating her actions. Etc. Etc. I think that you think that I think that you think… and on we go.
The Dark Knight, however, does not start with such an abstract exposition, though close. In a beautifully edited bank heist led by a (for now) mysterious leader, a team of gangsters proceeds to empty the safe. It is understood that they are going to be sharing the loot, but they soon start turning on each other to reduce the number of parties involved in the deal. Gangster D received instructions to shoot Gangster E. But Gangster C received the same instruction, only his target his Gangster D. And this climbs up to the top dog, Gangster A, revealed to be the elusive Joker, leaving the scene with the whole treasure.
His minions may be fine gangsters, but they clearly make a terrible team of game theorists. Putting myself in the shoes of Gangster C, about to shoot Gangster D after witnessing his own murder of Gangster E, how do I not anticipate that Gangster B has received similar instructions? If I had let my reasoning go one step further, I might have avoided death through a cognitive model of Gangster B (what I believe he will do). The Joker, on the other hand, is an exceptional game theorist, because he obviously did not believe his men would go the extra step. Indeed, the best game theorist knows that these assumptions of rationality (assuming that I will think about what my opponent thinks of me etc.) fail very quickly in reality, especially when large monetary gains blur your thinking. Which leads us to the Dictator Game.
Humanity: Too nice!
The Dictator Game is not exactly a game per se, but analysing it with the lens of game theory is interesting. There are two players, although player two does not get to act in any way. Player one is the sole decision-maker, and is tasked with dividing $100 between the two of them. An easy problem when looked at it with game theory: take the money! The whole of it! It is a Nash Equilibrium: in this case, nobody wants to change his action. Player one would get less by giving more to player two, and player two anyway does not get to negotiate.
Well, that is the theory. In practice, observed in real experiments made with real people and real money, the money is split, however not equally. Player one would usually offer herself more than her fair share, and player two would go home with a strictly positive amount of money. Huh? If we solely trust the Nash equilibrium to give us the solution of the game, well, these players have failed. Irrational beings! This of course does not account for the weight of societal norms, where individuals possess some notion of fairness. It is all relative of course and there are many ways to define it.
In the greater number of cases, game theorists tend to use the Social Welfare as a measure of how well the system performs. We add up the payoffs of our players and see if that sum can be made greater with a different choice of actions. In the Dictator Game, this does not lead to any good notion of fairness, as the game is zero-sum (everything player one gets is what player two does not get). So the Social Welfare will invariably be $100. A different measure of the social success of our choices may be more appropriate here: the makespan. Here we define as Social Welfare the minimum amount received by all players. In the “player one keeps everything” scenario, the makespan Social Welfare is literally at its worst: $0, which is what player two gets. To maximise our Social Welfare we better find a way for the players to agree on a fifty-fifty split.
But Joker’s philosophy is different. Through his twisted games, during the whole movie, he will try to prove with scientifical precision that most people are not the nicely-behaved fairness-endowed individuals they think they are. In the bank heist, a dictator game played with many players, Joker goes straight for the Nash: he is at the top, commands his own profit, and chooses it to be the whole of the social cake. For an economist, the irony is particularly acute: Joker is in fact the prototypical homo economicus, the rational beast that always acts in his own self-interest. So much for an insane clown with green hair. This analogy may be useful to deflate the weight that the rational agent imposes on the microeconomic theory…
Chicken on boats
The Joker does not only boast remarkable game theory intuition, he is also a mighty fine experimental economist. That is, if the ethics committee of his institution allows for life-or-death experiments involving people on boats waiting to be blown up. This is a scene that looks remarkably like the Chicken game, already pictured in the classic Rebel Without a Cause, only much deadlier.
Two ferries are navigating in the Gotham bay. One of them is boarded by “regular” people (men, women, children) while the other boat’s passengers are prisoners in the orange jumpsuit. One thing in common between the boats: both have enough explosives to be blown up with everyone on board. Each ferry is given the detonator that activates the explosives on the other vehicle. An additional rule: if no one detonates within the imparted time, both will explode.
The homo economicus would rush to the button in the hope of blowing up the other boat first. This is what Joker would like to prove, his hypothesis. But our unfortunate players ponder over the question, in what becomes a strange voting game of what shall be done by whom. In what seems like an argument defending Joker’s thesis, the non-prisoners boat starts declaring that maybe they should consider the prisoners worthier of being blown up. The balance of justice seriously weighs on their side, so that sounds like a reasonable option. Meanwhile, on the other boat, in a typical Hollywood moment, one of the prisoners takes the remote and throws it out to the water. If the non-prisoners decide to push the button, they would get to live, according to the rules. If they don’t, nobody will. But their odds have never been better.
But oh! the irrationality. Again, again defying all logic and theory and infuriating a Joker who probably thought his masterful proof of his hypothesis would get him at least a publication in Science, the non-prisoners find themselves unable to push the button. Even though it’s a dominant strategy! So frustrating.
Some concluding words
The Dark Knight is often seen along the lines of the Good vs. the Evil, and how these lines blur all too easily. Here we offer an alternative explanation, that of the Irrational (or the human) and the Rational. Interestingly enough, another notion of game theory dubbed Price of Anarchy seeks to quantify the gap between a situation where every player is solely concerned with her own well-being, and a situation where the Social Welfare as a whole is maximised. The Joker would very much like to give literal meaning to this Price of Anarchy, letting the selfish actions of his unwilling subjects develop to their (theoretical) conclusion: chaos. But you can always count on our decidedly irrational penchants to perturb the show…