A query about The Dark Knight

From Brad Allen:

I was watching the Dark Knight on a bus yesterday evening (I’m not sure how familiar you are with the movie) – there was a scene that I thought was pretty interesting to think through, and was curious how you might go about it.

There is a scene where the Joker kills a mob boss, and then gives his 3 subordinates one half broken pool cue – and basically tells them that to live, the other two have to die. You don’t see what happens, but what do you think happens? Is it advantageous to pick up the pool cue, or would that signal the other two to attack you first? Would you try to back out and let the other two fight? Or would that incent them to come after you? OR does everyone do nothing, until a last second dash like bicycle sprints?

Obviously, I’ve had fun thinking about this. Do you have any guesses?


This is a standard variant of a finite sequence game and has a Nash equilibrium point by definition. It's a variant of the Prisoner's Dilemma (as are all such games) with three people not two. Off the top of my head I would stay the equilibrium is to do nothing, kind of like a game of Chicken (which itself is an example of such a game).

No, it's better than the Prisoner's Dilemna because it involves fantasizing about killing people, and there's a big demand for that.

Suppose we run the game with endogenous Joker. If everybody does nothing the Joker probably does something to force the action, up to possibly killing thugs until someone initiates an attack. If it is easier to kill the Joker than two mob thugs, the payoff structure favors doing that obviously. A more complex situation might be where it is easier to kill the Joker, but only if two or three thugs cooperate. Then you might have multiple equilibria. If the thugs can coordinate a cooperative response, maybe none die. If not, your best bet may be to start the killing immediately.

Don't forget that the Joker was brought (ostensibly as a body) by several presumably loyal subordinates. I recall some exhortation that the test should be ended quickly, so I don't think he was going to give them much time to not start killing each other.

One way or another, the Joker wouldn't die. He never does. (I still have my suspicions about Ledger...)

The Joker wants to die, he is just concerned with who kills him. He wants Batman to run him over in the Batpod and wants Harvey to shoot him in the hospital, not to mention laughing when he thinks he is falling to his death at the end.

Read the comics. He never dies.

True, but if I really didn't want to die I'd make some different choices.

Based on his later actions (involving the two boats), The Joker is likely to force action by killing all three thugs simultaneously. The thugs don't necessarily have this information, though.

The Joker is not there by himself. The thugs presumably don't have guns and The Joker's people do. The Joker might lose some of his men but killing The Joker seems impossible in the immediate game. (Becoming a lieutenant who later kills him is still possible.)

So what, no one would have died if the Joker didn't further instigate? Don't we also have to consider how much advantage holding the the stick confers in a fight as well?

Although the Game of Chicken is similar to the Prisoner's Dilemma in some respects, they are two different games with totally different payoff and different equilibrium solutions

The same movie ended with the same game in another context. Two ferries were loaded with explosives. Each ferry had a switch which would blow up the other ferry. The idea is that if you didn't hit the switch first, the other guy would.

The solution in the movie was neither side flipped the switch.

There was signalling. Every minute went by with no explosion was a signal that the other side did not want to flip the switch. The two ferries signalled each other by doing nothing.

It mattered that one ferry was evacuating prisoners. Even the criminals did not want to commit mass murder.

I expect the solution to these kinds of games depends on the characters of the players. Even violent criminals might stand down; it is the sociopaths that you can't trust.

And there I was thinking it was about the greatness of the human spirit...

I was figuring the switches were reversed and the Joker was lying about it...

Woah... I never thought of that.

He already switched Harvey and Rachel earlier, so this is not unlikely.

The earlier switch was such a lead in. I was honestly surprised when this didn't occur. I suppose we never really found out whether or not he'd done it again...

The earlier switch was such a lead in. I was honestly surprised when this didn't occur. I suppose we never really found out whether or not he'd done it again...

I'm worried this will appear twice. MR seems to be flaking out again.

The Joker called it a social experiment, and it would have been fascinating as such with those terms as you supplied.

But The Joker said he would blow both boats if no one took action in 15 minutes. (He only couldn't do this because The Batman interfered, thus totally spoiling the results and costing The Joker his PhD.) Given that The Joker followed through on all previous threats, there was no reason to believe that he wouldn't do it here.

I would say the equilibrium would be to do nothing, and that's not the only Prisoner's Dilemma in the movie. I believe towards the end the people in the ferries have to decide if they have to blow up the other boat.

The entire film is full of this kind of stuff. Batman and Gordon decide to squeeze the mob. The mob decides to let the joker "out of the box." Batman decides to turn himself in but Harvey Dent decides to take the rap. The joker decides to get caught. He tells Batman the wrong locations of Dent and Rachel knowing he'll go for Rachel. Batman believes the Joker ultimately destroying Dent's reputation and dooming Gotham to chaos. It's amazing.

So much stuff is thrown at you (and the characters) without the time to think through the game theory I think that is why some people find the movie to be a mess when that is part of the movie's theme.

It's gotta be one of the best films of the decade.

I'm one of those who found it a mess.

It's a mess because it is Joker's movie. It is the chaos that Joker wants make of Gotham to force the "white knights" to become what they hate because Gotham is corrupt and at least total chaos is fair. It works on every level. It all comes together by about you've watched it for the 30th time.

The Joker keeps presenting the protagonists with choices so rapidly that they keep getting wrong. Even the final decision of the movie made so "the Joker cannot win" while incredibly inspiring we discover turns out to be the wrong decision in The Dark Knight Rises.

That's 29 times too many for me. Actually, looking back, that's 30 times too many.

I was exaggerating, but that does it, I'm going to go watch it again right now.

Well, watched about half. The opening scene is a perfect introduction to the Joker character, but why did the Joker pick a mob bank to rob in the beginning? One reason is he robbed the bank so he could go to the mob meeting the the right amount of respect and craziness to make his proposal to "save" the mob from Batman when his plan from the beginning is to take over the mob for himself.

Why was the Joker able to turn public opinion against Batman? One theme is that the purpose of the symbol and the disguise of Batman is that he could be anyone. This is of course a falsehood as only a billionaire martial artist could be Batman. Secondly, the mob provides some stability to the city where corruption is ingrained. By pushing too fast for 'regime change' Batman and Gordon have destabilized the city and played directly into the Joker's hand.

I'm going to go ahead and call it: best movie ever made.

But you are in the middle of the movie, where you can't count of The Batman saving you. So if all three people do nothing, The Joker will come back in and kill you all. Someone has to break from that equilibrium.

Finally a subject of interest! Not only is it a nearly perfect film, almost an entire film could be made just from what is implied and not shown on-screen. We may be witnessing the director version of Babe Ruth, Daniel Day-Lewis, Micheal Jordan, etc. Clearly you attack the Joker's gang. But, that doesn't work for the movie and the general public doesn't realize how even a contest between a sharpened stick versus a handgun would be at close range. In movies guns are magical talismans. And the problem is that this perception also affects reality. For Gamble's gang under stress maybe the first-mover problem and the psychological anchoring of Joker's proposal would be too much of a cue to think rationally (just like on the 9/11 flights prior to flight 93). Either way, you need to have a stick. But what about the Joker's side? If he understands such calculations might be made, Is such a contest a screen for the craziest of the 3?

There's a competing trope whereby in Hollywood, guns don't work. See, for example, MacGyver, Jurassic Park, and every zombie movie ever made. Tell me how the zombie apocalypse wouldn't be over in an afternoon... It would be like Conquistadors and Incas but even more lopsided.

The gun in film is deus ex machina. It does whatever the director wants it to do. If he wants it omnipotent, it is. If it needs to fail in the face of overwhelming nature, it does.

7 out of 10 people shot by a handgun survive, even with multiple hits.

Handguns are incredibly underpowered. Their bullets have the kinetic energy of tossing up and catching a half pound plate. Bullets tend to kill only when they strike major blood vessels., and those are remarkably well protected.

A rifle bullet - now that's another story.

But yeah, movies exaggerate the usefulness of firearms to the polar extremes. Very astute of you. Reminds me of how Stormtroopers are such poor shots in Star Wars. Cloning must have given them poor hand eye coordination.

The future isn't particularly kind to Stormtrooper lethality... Not only are they poor marksmen, as you say, but their "bullets" have practically no velocity.

When your armour is plastic, your guns are slow, and you can't hit the sides of barn doors, you are doomed to be defeated by guys with medieval technology (shiny sabres).

I believe the correct answer is to flee the city.

You grab the pool cue as fast as possible. As soon as someone doesn't have one, the expected gain from attacking him is greater than the expected gain from attacking a person with a cue, while constantly rotating and backpedaling to never end up between the other two. The tricky part, however, might be to not be the guy to deliver the ultimate disabling blow to the unarmed person, but rather to deliver the penultimate blow, and switching your attention to the 3rd player, in the hopes of initiating the action with the 3rd party. In improvised confrontations, the advantage almost always goes to the aggressor. The order of importance is 1) be more dangerous, 2) use one body as an obstacle to 3rd party. Or whatever.

Even more optimally, say something like "Oh c'mon! I'm not going to do that!!!" just before you reach for the pool cue to smash the first guy over the head.

I say blast off and nuke the entire area from orbit.

It's the only way to be sure.

A broken pool cue is not as significant of advantage in CQC as the first strike. Having to go for the weapon would take focus off of the two targets. There would also likely be an opportunity to eliminate one of the targets while they went for the weapon. If I recall the scene correctly, there were also other improvised weapons accessible.
I think you are attempting to apply ivory tower thought to a problem best solved by animal cunning.

Indeed, the most important strategy is to not waste time thinking about the game-theoretic equilibrium.

The pool queue doesn't matter. There are various levels of dangerous people, not dangerous weapons. Just about anything can be used as a weapon, especially in close quarters.

The most common result is that the second most dangerous will win. The two gang up on the third guy perceived to be most dangerous, then turn on each other when he's out of the picture.

The most common result changes if one of the participants has been deceptive over time about how dangerous he is, but I find that unlikely in the described set of circumstances.

This is exactly my complaint with the C power in the US (the bottom 65%), they agree with the B power (the 1%) to gang up on the A power (the top 1/3).

But that is where the money is.

I do not know the answer, but one factor that must be accounted for is each of the actors are criminal and know the others are criminal as well. This means you cannot count on purely "rational" action - in particular - long term rational action such as Prisoners Dilemma logic.

"The most common result is that the second most dangerous will win. The two gang up on the third guy perceived to be most dangerous, then turn on each other when he’s out of the picture."

You just described World War II and the Cold War.

Ed September 28, 2012 at 2:15 pm
“The most common result is that the second most dangerous will win. The two gang up on the third guy perceived to be most dangerous, then turn on each other when he’s out of the picture.”

You just described World War II and the Cold War."


"You just described World War II and the Cold War."

I described a lot more than that. :)

Just because one hobby is economics doesn't mean you can't have another hobby of military history, including wargaming. The two go together better than some might suppose...

It depends on the disparity between the players. The two weakest players should attack the strongest player who would otherwise be guaranteed to win one on one.

The people in the room are P1, P2, and P3. If P1 > P2 > P3 in terms of dangerousness, then:

If you're P1, you'll likely be ganged up on anyway. Grab the cue and attack P2.
If you're P2, grab the cue and toss it to P3, then attack P1. P3 should join you until P1 goes down.
If you're P3, grab the cue and toss it to P2, and let P1 and P2 duke it out for as long as possible.

If P1 = P2 = P3, grab the cue, break it over your knee, toss half to each of the others, and let them duke it out for as long as possible.

What if dangerousness is not common knowledge?

Between two people being the first to move has obvious advantages in this case.

I don't think this changes between three people. The game won't converge to an equilibrium, as they will only play it once. And i believe that people will have a tendency to hesitate in this game for psychological reasons. So i think you can deal a good first blow and take the weapon without much resistance.

Of course to do this you'd have to be crazy.

I think after some moments of hesitation, people will be ready to react to movements, so moving first might not be so good if everyone hesitated first.

You probably should move between the weapon and the enemies, without taking the weapon, in a non treating way if circumstances allow it, or very quickly and violently, and take the weapon only when the enemies are distracted.

If you can position yourself in a way that every one is aligned to each other and the weapon it would be optimum.Like this:


So the guy next to you would be surrounded and would have to slip attention. They didn't seen to have much room to move, so the guy opposite to the weapon wound't have room to go around.

This way you could take the weapon without much risk of being attacked while not properly focused on your enemy.

I'm quite sure the result of the game would depend heavily on the starting positions of the players.

A situation that comes up in every game of Diplomacy, ever?

Yes, how odd.

Seems to be disagreement over whether it's better to try and eliminate the more dangerous opponent or go after the weaker one with greater expected gain.

I was watching the Dark Knight on a bus yesterday evening

In other news, blah blah blah great stagnation blah blah blah this just replaced a purchase so there was no GDP is unchanged so it doesn't matter blah blah blah ...

Maybe you are the star of the movie. Grab the guy's arm who is first to go for the pool cue, stab the other guy with it. Kick a bad buy. Spin around so the other bad guy shoots the guy you are holding the arm of. Flip the body over knocking over another bad guy. Throw a pool ball hitting another thug. Grab the pool cue and knock the gun out of the thug's hand, pick it up and shoot your way out a window and jump into an open-top trash truck.

LOL - that would be the Lee/Chan/Li subgame perfect equilibrium.

Lee winning and the other two kowtowing, obviously.

As has been pointed out, a broken pool cue isn't a decisive enough weapon to expose yourself to moving first. You should probably do everything you can to avoid fighting for as long as possible. Fighting is exhausting, I can only imagine that fighting to the death is much more so. it's probably not something you want to do twice in a row.

The Wikipedia page on "truel" goes over much of this, though admittedly it's focused more on guns than sharp objects:

My thought also - isn't it a just a truel? The weakest guy should commit to not taking the cue by backing away and let the other two fight and weaken the victor.

Sounds almost like Survivor.

Asking for fighting advice on an economics blog is perhaps not optimal. Could we at least limit comments to those with neck tattoos?

This isn't a prisoner's dilemma, this is a "mexican stand-off."

this mean ''do not trust ''

Okay, here's my take:
Suppose that, you're a thug and that:
a) every time at least two thugs fight, one thug is dead at the end of the fight
b) that when two thugs gang up on one thug, that one thug is automatically considered dead
c) you have a 50% chance of winning against any other thug

On the first move, you have either two choices: attack one of the other thugs, or wait. If you attack one of the thugs, there are three possible outcomes: 1) the other thug joins you in attacking the thug you chose or 2) the other thug stands by or 3) the other thug joins with the thug you attack and they both kill you. For event 1, both of you would kill the first thug and you would have to duke it out with each other, and at the end you either win or die. For event 2, you either kill the first thug, and then move on to the second thug, or die trying to kill the first thug. If you manage to kill the first thug, you would then either win or lose to the second thug. For event 3, you would be dead. So, in total, if you decide to attack one of the other thugs, there are 6 total possible outcomes, and you win in 2 of them.

If you wait, however, there are three other possible events that could result from your decision: 1) one of the thugs attacks the other one 2) one of the thugs attack you, and 3) both of the thugs also wait. For event 1, one of the two thugs would die at the end of the fight and you'd have to end up killing the other one. At the end, you are either killed or you win. For event 2, the third thug that did not attack you either allies with you, allies with your aggressor, or stands by. This event closely resembles the "attack" decision previously discussed, so in total there are 6 possible outcomes, and you win in two of them. For event 3, the three thugs all wait in limbo; I'll list this as a neutral event. So in total, for the "wait" decision, there are nine outcomes, and you win in 3 of them (and you simply get to live in 1 of them, although it's questionable for how long...)

So really, in my view there are no differences in the chances for winning for either of the two scenarios, but waiting may slightly increase your chances of surviving for longer.

"there are three other possible events that could result from your decision": you are missing another possible event - both of thugs attack you.


This is my favorite explication of the opening of The Dark Knight.

The fastest route to victory and surviving the Joker afterwards, is to make a fast, brutal and insane kill. Make the third guy figure his chances are better if he tries to run.

A lot of flat out wrong things said in this thread. First, assuming holding the pool cue is an advantage and no teamwork allowed/possible, you should not attack the other un armed guy. It is clearly better if you are going to attack someone that you fight the guy with the stick since you have to beat him anyway and by fighting him first you get to go into the second fight with the advantage of the que.
So neither cueless guy should attack the other. This means that the guy who grabs the cue has to fight twice while the guys without the cue have to fight on average of 1.5 times so grabbing the cue is only right if it gives you a 33% increased chance of winning a 1 on 1 fight.
Assuming coordination is allowed there are only two options. Either the two cueless guys team up on the cue or one cueless and one cued team up on the other cueless. Because its clearly a zero sum game one player's loss is split between the other two players. This makes teaming up very powerful and also implies the two without cues should team up on the guy with a cue. The reasoning is simple, one the total reward of beating the guy with the stick is higher and two the gains are split evenly while the gains are split unevenly if a cueless guy teams up with the cue.
Grabbing the cue is a losing move unless it confers a huge advantage and if you do grab the cue you should immediately attack one of the other two.

So you agree that the equilibrium is: no one does anything, until the Joker prods them?

Just the equilibrium of grabbing the weapon or not. Making an alliance is priority number 1 and then taking out the mutual enemy, which resets the equilibrium and the enemy should then go for the weapon.

Isn't the answer that you should fire your gun into the air? Oh wait, that's a different one.

It's a Mexican stand off. But for the same with asymmetric information, just watch the last ten minutes of The good, the bad and the ugly where Clint demonstrates the Nash equilibrium. What is there not to like: a gunfight, economics, and the best movie soundtrack ever....

tom, I am totally agree with your thoughts. Keep doing these type of work.

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