The newest Batman flick The Dark Knight absolutely stunned me. The endless mind games between the players are a great chance for someone to learn about several basic tenets of game theory.The newest Batman flick The Dark Knight absolutely stunned me. Not since Dr. Strangelove has a movie contained so much game theory. While many others have noticed the game theory connection, particularly about a scene near the end of the movie, such commentaries miss the big picture: the entire film is a sequence of games and an exploration of strategic thought.

Game theory comes up in many scenes even where it’s not clear what the “game” really is. Strategy is a theme introduced immediately in the opening bank robbery scene. This scene is one of the most powerful movie openings and it foreshadows the chaos and tempo in the story. Today, I’ll analyze the robbery scene using the lens of game theory.

(Fair warning: this article contains spoilers and covers roughly the first five minutes of the movie.)

How can we split up the stash?

The movie starts out with a bang. An aerial attack begins when a window shatters on a skyscraper, allowing two robbers to glide across a zip-line on to a bank’s roof. On the street-level, a car screeches to a stop to pick up the last member of the ground attack.

The first spoken words concern the topic of strategy. These lines introduce the Joker’s character and they foreshadow the punch and counterpunch of the entire movie. The robbers in the car explain the job and how the loot will be divided. It’s apparent they are not happy with the plan:

Driver: “Three of a kind. Let’s do this.”

Passenger side: “That’s it—three guys?”

Driver: “Two guys on the roof. Every guy gets a share. Five shares is plenty.”

Passenger side: “Six shares. Don’t forget the guy who planned the job.”

Driver: “He thinks he can sit it out and still take a slice. I know why they call him the Joker.”

The robbers don’t like that the Joker gets an equal share for doing unequal work. Their complaint raises the issue of fair division, which is central to game theory. In fact, fair division is the first problem that game theory addressed historically. The problem appears in the Babylonian Talmud about how creditors should divide an estate. The text offers a mysterious solution that had baffled scholars for over 2,000 years. It was only very recently that a Nobel Laureate economist deciphered the answer using the tools of coalitional game theory. Let me tell you, the answer is fascinating.

Fair division is about understanding incentives and strategic thought. How can you trust self-interested people? How can you achieve cooperative outcomes with diametrically opposed motives? Such ideas have been applied to important areas such as nuclear disarmament and labor negotiations. But they are even applicable to mundane situations, like dividing up restaurant bills fairly.

The robbers accept an equal division for unequal work, but should they believe things will go as planned? Perhaps they should not, if they really considered the incentives and the possible ways others could tamper with the plan. In game theory you do not trust someone because they are your friend. You trust them because it is in their self-interest to help you. We can learn from a natural example: children should trust their parents on safety rules because parents have a vested interest in seeing their children are safe. They should not, however, trust strangers.

Had the robbers considered these issues, perhaps their fate would have been different. A little bit of thinking ahead and reasoning backwards would have demonstrated flaws in the plan.

The scene is reminiscent of a popular game theory puzzle about pirates and splitting up treasure. Some of you may have even heard this as a technical interview question. The game offers insights to collective voting and the ability of a leader to buy off votes.

The Pirate puzzle

The game

Three pirates (A, B, and C) arrive from a lucrative voyage with 100 pieces of gold. They will split up the money according to an ancient code dependent on their leadership rules. The pirates are organized with a strict leadership structure—pirate A is stronger than pirate B who is stronger than pirate C.

The voting process is a series of proposals with a lethal twist. Here are the rules:

1. The strongest pirate offers a split of the gold. An example would be: “0 to me, 10 to B, and 90 to C.”
2. All of the pirates, including the proposer, vote on whether to accept the split. The proposer holds the casting vote in the case of a tie.
3. If the pirates agree to the split, it happens.
4. Otherwise, the pirate who proposed the plan gets thrown overboard from the ship and perishes.
5. The next strongest pirate takes over and then offers a split of the money. The process is repeated until a proposal is accepted.

Pirates care first and foremost about living, then about getting gold. How does the game play out?

The solution

At first glance it appears that the strongest pirate will have to give most of the loot. But a closer analysis demonstrates the opposite result—the leader holds quite a bit of power.

The game can be solved by thinking ahead and reasoning backwards. All pirates will do this because they are a very smart bunch, a trait necessary for surviving on the high seas.

Looking ahead, let’s consider what would happen if pirate A is thrown overboard. What will happen between pirates B and C? It turns out that pirate B turns into a dictator. Pirate B can vote “yes” to any offer that he proposes, and even if pirate C declines, the situation is a tie and pirate B holds the casting vote. In this situation, pirate C has no voting power at all. Pirate B will take full advantage of his power and give himself all 100 pieces in the split, leaving pirate C with nothing.

But will pirate A ever get thrown overboard? Pirate A will clearly vote on his own proposal, so his entire goal reduces to buying a single vote to gain the majority.

Which pirate is easiest to buy off? Pirate C is a likely candidate because he ends up with nothing if pirate A dies. This means pirate C has a vested interest in keeping pirate A alive. If pirate A gives him any reasonable offer—in theoretical sense, even a single gold coin—pirate C would accept the plan.

And that’s what will happen. Pirate A will offer 1 gold coin to pirate C, nothing to pirate B, and take 99 coins for himself. The plan will be accepted by pirates A and C, and it will pass. Amazingly, pirate A ends up with tremendous power despite having two opponents. Luckily, the opponents dislike each other and one can be bought off.

The game illustrates the spoils can go to the strongest pirate or the one that gets to act first, if the remaining members have conflicting interests. The leader has the means to buy off weak members.

Don’t get caught up in the exact assumptions or outcomes of the game—just remember the basic lesson. In the real world, it might be necessary to buy a vote with 20 gold coins. Nonetheless, the general logic is the same. Here are some of the main insights from the game:

Lessons:

• Players should think ahead and reason backwards
• A leader can win by exploiting conflict among weaker members
• Players derive worth from voting power, and some players can be bought off

The robbery scene in The Dark Knight

The original plan of equal division is flawed. Each robber has incentive to increase his share by killing a fellow team member. Once a member performs his job, he loses his negotiating power and value to the team.

The Joker plays off this conflict by instructing the robbers to take out fellow teammates once their tasks are performed. The game would be different if the robbers were a group and they repeated crimes together—perhaps an even split could be sustainable. But as the movie hints right away with the first backstabbing scene, this robbery will be a one-shot game.

Many of the robbers fail to see they can be victim to the same deceit they pull on others. The second robber on the rooftop is a prime example. After his partner disarms the silent alarm, he quickly kills him and then proceeds to perform his own job. He doesn’t see the same thing could happen to him.

After he disarms the bank vault, he is greeted with a most unpleasant surprise:

Robber: “Where’s the alarm guy?”

Vault guy: “Boss told when the guy was done, I should take him out. One less share, right?” [opens the vault]

Robber: “Funny. He told me something similar.”

Vault guy: “What? No! No!” [gets shot in the back]

By now it’s clear the Joker wants everyone dead, and minutes later we learn the Joker has been present on the job all along. The plan finishes with two more deaths both involving the escape vehicle bus.

The Joker, being the “strongest pirate,” was able to sequentially bribe the weaker robbers one by one. In the end, he puts a twist on the game by taking the whole pie.

Other strategic elements

There are many other mini-strategy elements during the robbery scene. Here are three that came to my mind:

• How can a handful of robbers overtake a bank?

In theory, a mob of unarmed citizens should be able to overwhelm a small group of armed robbers. The problem is there will be casualties, particularly for those that act first. Who is going to step out and be the hero? Robbers make sure that people don’t coordinate to force them to act sequentially. Any individual that attempts to be a hero will be killed as an example, like the angry bank employee. In the movie, the robbers demonstrate they are willing to use lethal violence by shooting up in the air and taking out the bank cop.

• Do you trust your teammate?

This is a subtle point and I loved it. The robbers face a small obstacle when an angry bank employee starts firing his shotgun. The robbers duck for cover, and after a few shots, one robber asks the other if the bank employee is out of bullets.

If you watch closely, you’ll see the robber (really the Joker) ponders the question carefully and then nods his head “yes.” The other robber jumps out and is greeted with a bullet that narrowly misses him. Almost immediately the disguised Joker jumps out and disables the bank employee with a round of bullets. The disguised Joker acted so quickly and without fear, almost as if he was now sure the bank employee was out of bullets. Did the Joker lie on purpose earlier to put the other robber in danger? The other robber, not aware it’s the Joker, is furious that he was almost shot and yells back: “Where did you learn to count?” The disguised Joker looks back in scorn.

• What kind of robber is the Joker?

The cops who arrive on the scene will be stunned. They will see a crime scene with five dead robbers and a bank vault that has been cleaned out. They will likely conclude the Joker is interested in selfish gains, a simple criminal, who wants all the money in Gotham.

As the cops, the mobsters, Batman, and we as the audience experience the complex themes that unfold later in the movie, many of us are left with one thought: if only the Joker were so simple.

The Joker demands money, yes, but is that what he really wants? That’s a question the Joker plays off later in the movie in an explosive fashion.