The Dark Knight — Game Theory
Of Hope and Dilemma
Breaking down the Finale of The Dark Knight via Game Theory
The Dark Knight will forever be hailed as a masterpiece, and the primary reason behind that — hands-down — is The Joker. Through him, Heath Ledger and Christopher Nolan successfully brought to life one of the most iconic antagonists ever. He was ruthless, determined, and had a pretty good sense of strategy; that’s one word game theorists need to hear, and voilà, here comes another article.
In the finale of the movie, The Joker puts the people in the two ferries into a paradoxical situation, which Game Theorists call The Prisoner’s Dilemma. It depicts a situation where acting selfishly leads to an outcome that is worse for each player involved than the one achieved if all those involved cooperate.
The Joker was not the one with the guns, but the one with the brains, and that is precisely what made him so difficult to defeat! He crafted such an adverse situation that the players involved had no option other than dying — how crazy is that! Well, that’s the genius of The Joker, and now it’s time for us to break it down!
The Social Experiment
The scene introduces two ferries filled with explosives triggered by a detonator. One detonator was there with The Joker, but a couple were there with the ferries as well. This is where the situation gets tricky. Let’s call the ferries ‘Ferry A’ and ‘Ferry B’. Ferry A had the detonator to Ferry B, while Ferry B had the detonator to Ferry A. The Joker gave both the ferries time till midnight to decide whether to blow up the other ferry or not and calmly added that if neither ferry could decide, The Joker would blow both of them up! However, if one of them blew the other up, The Joker would let that ferry live. This was The Joker’s ‘Social Experiment’ with the participants being Gotham’s most wanted criminals in one ferry (say, Ferry A) and innocent civilians in the other (Ferry B).
Those hardcore criminals would have simply decided to kill the civilians; after all, they were felons, right? While the civilians reasoned that the criminals were the ones who must die, they had already had their chances. Both of them thought that killing the other was better rather than everyone dying, and that’s where the Prisoner’s Dilemma comes into play. The Joker must not have scribbled a payoff table, but I’ll do to make things clear.
Taking the payoffs to be the time that the people in the ferries have left to live (in minutes), we have the following table:
The payoffs for the fourth outcome can be variable as time was ticking, but that won’t affect the nature of the Dilemma. So, to keep things simple, we have it as 30 minutes, considering the table to be drawn when The Joker announced the experiment. To get a brief on the Prisoner’s Dilemma, check this article.
The Prisoner’s Dilemma
In the classic example of the Prisoner’s Dilemma, we have a strict inequality between the payoffs. Consider the table below:
While in the payoffs for the ferries, we have ‘a’ and ‘c’ as equal. If a slack inequality such as b>d>a ≥ c exists, we still have a Prisoner’s Dilemma, only this time it’s a weak one. Although, the Dilemma’s nature still remains the same, with selfish decisions leading to unfavorable outcomes. The only minor difference is that ‘Confessing’ in the classic Prisoner’s Dilemma is a ‘Strictly Dominant’ strategy in the sense that regardless of what the other prisoner does, ‘Confessing’ always provides a better payoff. While, in our case, if Ferry B chooses to ‘Detonate’, Ferry A can choose either to ‘Detonate’ or to ‘Not Detonate’ because that doesn’t make a difference as both strategies result in the same payoffs. But, if Ferry B chooses to ‘Not Detonate’, Ferry A receives a higher payoff of L than 3 if it ‘Detonates’. Thus, to ‘Detonate’ is still a dominant strategy but, in the weaker sense, i.e. it does at least as good as ‘Not Detonate’ and sometimes does better. So to draw parallels, we have ‘Confessing’ as the ‘Strictly Dominant’ strategy in the classic example, making ‘Cooperating’ as the ‘Strictly Dominated’ strategy. While in our model, to ‘Detonate’ is the ‘Weakly Dominant’ strategy, making, to ‘Not Detonate’ a ‘Weakly Dominated’ one, and that’s the only difference!
The Strategy
Being rational players and acting selfishly, to ‘Detonate’ was the best possible option for each ferry regardless of what the other ferry did. They couldn’t just standstill. Time was ticking, and none of them knew when the other ferry would blow them up. There was no option for the civilians in deciding against killing the criminals because if they did so, the criminals might not have been so noble. This same reasoning was there in the criminals’ minds, which is precisely what The Joker envisioned. Wanting to live would fuel the darkness deep inside all of them. They would thus end up killing each other, exhibiting the Prisoner’s Dilemma. The ‘Dilemma’ here is that by acting selfishly (blowing up the other ferry) in hopes of the best payoff, the players would reach an outcome detrimental to all of them (both ferries blowing each other up), while on the other hand, mutual cooperation would, to some extent, result in a better outcome.
This ‘detrimental to all’ outcome is the one on which the Prisoner’s Dilemma usually settles. We call this the ‘Nash Equilibrium’. The classic example has the mutual confession outcome as the only strict Nash Equilibrium, where, by switching strategies, a player would only worsen their outcomes given that the other player sticks to their original strategy. While, in our model, the mutual detonation outcome is the only Nash Equilibrium, but in a weaker sense, where no player can improve their payoff by switching strategies given that the other player sticks with their original strategy.
In contrast, the outcome resulting from mutual cooperation is called the ‘Socially Optimal’ outcome, where none of those involved would gain or lose, and the result would be in the best interest of everyone. If both the ferries would cooperate and decide to ‘Not Detonate’, it would result in the fourth ‘Socially Optimal’ outcome, which is not as good as ‘L’ but not as bad as ‘0’ either. Thus, mutual cooperation results in a better payoff of 30 for everyone than mutual detonation with a payoff of 0. Clearly, the ferries would be mutually better off cooperating, resulting in the ‘Socially Optimal’ outcome rather than acting selfishly and resulting in the ‘Nash Equilibrium’. Thus, simply enough, this is the strategy they needed to go for! But as detonation was the dominant strategy, clearly, the ‘Nash Equilibrium’ and the ‘Socially Optimal’ solution didn’t coincide. So what really happened?
The Solution
Being a bearer of logic, my work here is done. The Joker expected the ferries to blow each other up, and so does the Prisoner’s Dilemma. But what really happened is where natural human behaviour comes into play. This wasn’t an ordinary situation; it was one of Life and Death! Put simply, the ferries would either blow each other up or end up being blown by The Joker. The situation would never settle on the blue outcomes because no ferry would want to stay put and let the other one decide their fate. When death is near, even years seem less, let alone 30 minutes. Thus the expected outcome was obvious, humanity dying by its own hands! But this was a superhero movie, and we can’t begin to lose faith in humanity just yet!
The civilians resorted to democratic trials, but even they weren’t able to help them decide. At the same time, even the prisoners were unable to pressurise the officials enough to kill all the civilians on the other boat. The people had one thing that The Joker didn’t have (hint: it wasn’t Batman). It was their compassion and their ability to hope and to believe in good. The situation was complicated. Both the ferries had the power to kill almost 500 people, which is too much power in anyone’s hand. While the civilians were overcome by indecision, the most fierce-looking prisoner confidently walked up to an official, took the detonator, and threw it out! (On a philosophical note, this shows that even the worst of criminals may still have some good left in them.)
In the end, the clock struck the dawn of a new day, and neither of the ferries had used the detonator. Hope had triumphed over darkness and death. That was the only time in the whole movie that The Joker had seemed so disappointed and lost. The Joker’s goal was not to kill the people, but he wanted to prove that, deep down, everyone was as ugly as him. His battle for the ‘Soul of Gotham’ had been lost. The ferries successfully avoided being intimidated by death through their ability to hope and believe. Batman finally caught The Joker and handed him over to the S.W.A.T team saving everyone from doom.
The Conclusion
Situations in their entirety are thus never ideal. Choices may seem binary and logical, but with homosapiens, there is always a human side to solving all the problems faced. This example clarifies that a simple straight-up strategy is not the one-stop solution to solving real-world issues. Several other factors often differentiate the actual situation from an ideal one, and they need to be considered. Sometimes you can afford to cooperate owing to group rationality, while other times, you may decide to play safe and be rational, individually. Figuring out the motives behind an action is the key to solving any Prisoner’s Dilemma.
References
That’s all for today! Stay tuned for more interesting insights on how Game Theory affects all the decisions we make.
Author — Game Theory Enthusiast Mrigank Sondhi, The Indian Game Theory Society — Delhi Technological University
