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PostSubject: Game Theory   Thu 03 Dec 2009, 6:30 pm

Game Theory in The Dark Knight: the opening scene



The newest Batman movie The Dark Knight absolutely stunned me. Not since Dr. Strangelove has a movie contained so much game theory. A lot of people have focused on a scene near the end of the movie. But there is so much more to see. In fact, I’ll be so bold as to suggest the following: the entire film is a sequence of games and an illustration of strategic thought.

There is a lot of creativity going on. Game theory is illustrated even at times when it’s not clear what the “game” really is, like the opening bank robbery scene. This scene is one of the most powerful movie openings and it foreshadows the chaos and tempo that will follow. Let’s take a moment to analyze it in detail.

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 when you’re eating out. I hate splitting the bill when I ordered a less expensive item (don’t you?). Understanding game theory can bring us to a better answer (finally, a return to splitting the bill fairly).

Ultimately, the robbers accepted an equal division for unequal work. Given that, should they have believed things would have gone according to plan? Perhaps they should not have, if they would have considered the incentives and possible landmines about trust. You see, in game theory you do not trust someone because they agree with you on a plan. You trust them because it is in their self-interest to help you.

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 (see next post in thread) 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 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 by forcing 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.
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PostSubject: Re: Game Theory   Thu 03 Dec 2009, 6:37 pm

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
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PostSubject: Re: Game Theory   Thu 03 Dec 2009, 8:48 pm

Very nice post. I have never studied game theory and the correlation to Dark Knight fascinated me. Definitely a subject worth studying.
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 1:55 am

The movie looks fascinating. I'm definitely going to see it, and thanks for posting this intriguing advance look.

From what I can glean, game theory seems to rely on the theory that, at bottom, all human beings are thoroughly, ruthlessly selfish. Is this pretty accurate?
Then, and IF so, given ONE benevolent, or Christ-like, 'player', the theory would be crippled and kaput, would it not? (VBG)
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 2:57 pm

ScoutsHonor wrote:
The movie looks fascinating. I'm definitely going to see it, and thanks for posting this intriguing advance look.

From what I can glean, game theory seems to rely on the theory that, at bottom, all human beings are thoroughly, ruthlessly selfish. Is this pretty accurate?
Then, and IF so, given ONE benevolent, or Christ-like, 'player', the theory would be crippled and kaput, would it not? (VBG)
Did you see the documentary called "The Trap"? If not, I recommend viewing it, for they discuss Game Theory and how John Nash (from the movie The Beautiful Mind) developed game theory while working at RAND Corporation. The problem is Nash has severe mental problems and they ultimately determined that much of his theories broke down, becuase, as you say SH, it assumes that humanity is comprised of selfish machines that act 100% of the time in a self interested manner.

If I have time, I'll post The Trap video in another thread in the Library subforum. If someone else beats me to it, then great, because its good material to reference back to from time to time.
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 3:03 pm

So, I've just been reading through the comments at the original blog post for this article and found some interesting comments by readers that I think are worth sharing here....


#
isn’t this how we’ve been to behave? Our policians bring home the bacon and we dutififully keep electing them to office. It matters not that as a country we’re broke. Add to this the class warfare ( make the wealth pay their share of the taxes – even though nobody talks about the freebies which has led us into this quagmire). The truely wealthy have no reason to fear the socialist (Democrats) and wimps (the Republicans) because their money is shielded from such higher taxes – ask Buffet or Ted Kennedy anbout such things as trusts and foundations – the higher taxes will ultimately fall on the middle class. Yes game theory, and we willingly suspend our disbelief and follow in line.

By Erwin Rysz on Aug 25, 2008






#
I think you hit the nail on the head with one of the earliest assessments (and an above commenter did too). The way these scenes throughout the movie develop appear to be classic Game Theory problems perverted by the Joker to never exactly be fair, to always either allow him to come out ahead or to not let the participants arrive at an equilibrium by any means.

Most Game Theory involves complete information, this is more poker than chess – the participants can’t see all the pieces, or aren’t able to communicate. More, they’re being told they’re playing chess going in and only later is it revealed that not all the pieces are visible.

By John on Aug 26, 2008






#
The Joker used various game theory approaches throughout the Dark Knight — it’s not over-analysis. There’s the bank scene and the ferry scene, obviously, the latter of which is almost a classic “game”. There’s also the issue about the choice between saving the girl or the lawyer. And the whole question about killing the guy vs. the hospital. This particular Joker, unlike the Nicholson Joker, is all about creating extremely tough choices. And what is game theory other than the systemization of tough choices?

The real beauty of the Joker, of course, is that he sometimes breaks the rules—as Batman says, at one point, “with the Joker, it’s never that simple.” Even when he tells you that you are in a game you can’t trust that the rules of the game are going to be enforced. It took me two viewings to realize that when the Joker tells them that the girl is in one warehouse and the lawyer is in the other, that he’s lying—they’re switched. So factoring in the veracity of the apparent game is yet another wonderful wrinkle.

By Alex W. on Aug 26, 2008



#
“what is game theory other than the systemization of tough choices?”

game theory is about making choices, but it’s specifially about making choices where the benefits that you get from choosing are completely dependent on the choices that others make [like 'the prisoner's dilemma', 'battle of the sexes', and so on].

in the ‘choose to save the girl or the lawyer’ example the joker had already set both the bombs ticking, & implied that he wouldn’t intervene further whatever batman did, meaning that batman had a straight decision to make make based on his own preferences, independent of any further actions made by others.

so it was no more an example of game theory at work than someone in a burger king deciding whether to buy a whopper or a chicken royale.

to my mind the scene was intended to remind the viewer of the one in the first [recent] spiderman film where spiderman was given a choice to save his girlfriend or a cable car full of schoolchildren… but in a feelgood film of that sort spiderman’s ingenuity was such that he was able to save both… but this was a much darker film, where the choice offered to batman was a serious one, he genuinely did have to choose to consign at least one person to being blown up.

the ferry scene can be analysed quite easily using game theory [as done on the links provided by the author], but it was a complicated one that is hard to analyse with simple game theory. in the simplest prisoner’s dilemma each prisoner cares only about the number of years’ sentence that he personally receives, & is indifferent to everything else. if, in the ferry scene, players had cared only about their chances of survival, the nash equilibrium would have been a race to the buzzer on both boats, with the fastest boat surviving. as it was, it seemed that, whilst the game itself was very simple, the payoffs of the passengers on each boat were potentially complex, with issues of morality coming into it.

By SA on Aug 27, 2008



#
Awareness:
Good question–there is no reason to kill the pirate who gives away all the gold, except pirates are considered to be ruthless. Perhaps they keep their group small so each gets a larger share for the next job.

I think people get lost in game theory since the mathematical models can lead to some strange predictions of reality. This is unfortunate and causes much problems since they should be seen more as guides, not as full-scale predictions.

To me, the pirate game makes sense for a handful of people. When you get to 200+ pirates, almost all intuition is gone and it’s more of a math problem. Still interesting, but nothing we should take so seriously.

By Presh Talwalkar on Aug 28, 2008




#
I re-watched the movie with game theory in mind and the whole thing is full of it.

I think it’s most apparent when Joker says he destroyed Harvey as part of his master plan. This is after he told Harvey he is only a “dog chasing cars” and while it’s true to some extent, it’s a lie when he later reveals to Batman that he was changed as part of the plan to create chaos.
Even the end result where they choose to chase Batman is based on the desired outcomes of possible reactions to Harvey’s killings. They are playing on public reactions but also playing there own game with the public as pawns in order to prevent the chaos the Joker planned for.

By Dj.Jk on Sep 21, 2008
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 4:15 pm

IP wrote:
ScoutsHonor wrote:
The movie looks fascinating. I'm definitely going to see it, and thanks for posting this intriguing advance look.

From what I can glean, game theory seems to rely on the theory that, at bottom, all human beings are thoroughly, ruthlessly selfish. Is this pretty accurate?
Then, and IF so, given ONE benevolent, or Christ-like, 'player', the theory would be crippled and kaput, would it not? (VBG)
Did you see the documentary called "The Trap"? If not, I recommend viewing it, for they discuss Game Theory and how John Nash (from the movie The Beautiful Mind) developed game theory while working at RAND Corporation. The problem is Nash has severe mental problems and they ultimately determined that much of his theories broke down, becuase, as you say SH, it assumes that humanity is comprised of selfish machines that act 100% of the time in a self interested manner.

If I have time, I'll post The Trap video in another thread in the Library subforum. If someone else beats me to it, then great, because its good material to reference back to from time to time.

The Pathocrats Achilles' Heel, maybe. That they're unable to know or understand good will or benevolence. Wish I could figure out (was smart enough) how this basic deficiency of theirs could be used to defeat them. I suspect it might be used against them, except that this isn't a novel and we (normal mankind) don't seem to have the time, for thinking...

Just thinking out loud... Smile

Thanks for all the good info, IP. I'm going to watch The Trap a little later and will also try to
post it to the forum.
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 9:04 pm

ScoutsHonor wrote:
The Pathocrats Achilles' Heel, maybe. That they're unable to know or understand good will or benevolence. Wish I could figure out (was smart enough) how this basic deficiency of theirs could be used to defeat them. I suspect it might be used against them, except that this isn't a novel and we (normal mankind) don't seem to have the time, for thinking...

Just thinking out loud... Smile

Thanks for all the good info, IP. I'm going to watch The Trap a little later and will also try to
post it to the forum.
First, right now I think all of us are going through really steep learning curves, so I don't think the issue of whether we're smart enough or not is relevant. I think just by the fact that you are here demonstrates that you are, because you're engaging in the learning process. It is those who refuse to engage in the learning process that we must be most concerned with.

As far as their theories, I do think that have badly misjudged humanity, so I do think that their calculation are misguided. When you watch "Batman: the Dark Knight," you'll see some references by the Joker regarding humanity's breakdown as chaos is increased. I think the Joker represents their plans, but on the contrary, I think humanity will revert to it's most basic tendancies when they breakdown their system. I think instead of turning on each other, we will naturally help and support each other. I turn to my direct experience in hurricane zones as evidence of this, no matter what the media portrays, which were all lies.

Of course, they will insert their teams of nihlists in order to try and destablize everything, and their media will report horrific events, of course, which they instigate and control, but the reality will be that communities will form and support systems will grow, and we will protect our souls from them and their actions.

They will never ever be able to tap into "the Ghost in the Machine," which is what they refer to as the human soul which they have never ever been able to uncover or figure out. God gave us a failsafe mechanism, to protect ourselves from ourselves.
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PostSubject: Re: Game Theory   Fri 04 Dec 2009, 11:16 pm

IP wrote:
ScoutsHonor wrote:
The Pathocrats Achilles' Heel, maybe. That they're unable to know or understand good will or benevolence. Wish I could figure out (was smart enough) how this basic deficiency of theirs could be used to defeat them. I suspect it might be used against them, except that this isn't a novel and we (normal mankind) don't seem to have the time, for thinking...

Just thinking out loud... Smile

Thanks for all the good info, IP. I'm going to watch The Trap a little later and will also try to
post it to the forum.
First, right now I think all of us are going through really steep learning curves, so I don't think the issue of whether we're smart enough or not is relevant. I think just by the fact that you are here demonstrates that you are, because you're engaging in the learning process. It is those who refuse to engage in the learning process that we must be most concerned with.

As far as their theories, I do think that have badly misjudged humanity, so I do think that their calculation are misguided. When you watch "Batman: the Dark Knight," you'll see some references by the Joker regarding humanity's breakdown as chaos is increased. I think the Joker represents their plans, but on the contrary, I think humanity will revert to it's most basic tendancies when they breakdown their system. I think instead of turning on each other, we will naturally help and support each other. I turn to my direct experience in hurricane zones as evidence of this, no matter what the media portrays, which were all lies.

Of course, they will insert their teams of nihlists in order to try and destablize everything, and their media will report horrific events, of course, which they instigate and control, but the reality will be that communities will form and support systems will grow, and we will protect our souls from them and their actions.

They will never ever be able to tap into "the Ghost in the Machine," which is what they refer to as the human soul which they have never ever been able to uncover or figure out. God gave us a failsafe mechanism, to protect ourselves from ourselves.

Ahh. That is a message worth its weight in gold.
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PostSubject: Re: Game Theory   Sat 05 Dec 2009, 2:34 am

Well no matter what the Oligarchs think or do they are bound to the same rules as we are "Probability". There are mathematic formulas and game theories to minimize their risk but yet they cannot make guarantees. They could and probably do have a quantum computer but as quantum physics states it is just statistical probabilities otherwise our electronic devices would work much better and last longer. If they did have a quantum computer they could have countless number of outcome models but with 6 billion people making choices no quantum computer can compute that high and if it could no person could analyze that many outcomes to choose which he or she felt was the best to go with.

I really like the Vendetta movie in which to me the most important concept is you must lose your fear and then you will be free. I dont know if they meant all fear, I certainly dont believe in foolishness but certainly not in fearing most of what they try to scare you with. I still have fears but not many that they create in their world.
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