Negotiating with the mob – Game theory in the Dark Knight part 2

In my original post about Game theory in The Dark Knight, I discussed the opening bank robbery scene and compared it to the pirate game. I mentioned the film was full of strategic thought and a great way to illustrate game theory concepts.

In light of recent Batman news, like Facebook offering rentals for The Dark Knight, and the debut trailer for the new Batman game Arkham City, I thought now would be a good time for a follow-up article on the topic.

I have seen The Dark Knight more times than I would like to admit, but on each viewing I notice something different. In a recent viewing, I was struck by the scene where Joker crashes a meeting of the mob videoconferencing with Lau to discuss how to preserve their money. The Joker establishes his presence with the gruesome “pencil trick” and the conversation changes into more of a negotiation. I will analyze this scene in detail to show the connection to game theory.

Here is a clip of the relevant scene:

Youtube video: Joker negotiating with the mob

Make me a fair offer

The scene is a meeting where mob leaders are deciding how to preserve their bankroll. The Gotham police and its energetic District Attorney Harvey Dent are cracking down on crime. Already they had a major sting operation that would have seized the mob’s funds, except Lau was tipped off and hid the money, fleeing to Taiwan with apparent immunity.

As the gang leaders debate their next step, the Joker sneaks in and offers his services. While not everyone is receptive to take him seriously, two mobsters–the Chechan and Maroni–realize how desperate they are and reluctantly hear his offer. The dialogue introduces the topic of negotiation and sets up the Joker’s plan to squash law and order in Gotham.

The Chechen: What do you propose?

Joker: It’s simple, we, uh, kill the Batman.

[The mobsters laugh]

Maroni: If it’s so simple, why haven’t you done it already?

Joker: If you’re good at something, never do it for free.

Now comes the really interesting part. The mob is somewhat interested in the plan to get rid of Batman, but they need to know how much this is going to cost.

The Chechen: How much you want?

Joker: Uh, half.

[the mobsters laugh again]

Gambol: You’re crazy.

Joker: I’m not. No, I’m not. If we don’t deal with this now, soon, little uh, Gambol here, won’t be able to get a nickel for his grandma.

Although the scene is fictional and in a fantasy world, the dialogue reveals the negotiation shares features common to many bargaining situations.

At the heart of the discussion is the concept of fair division. The Joker and the mob are trying to figure out how to split up the mob’s empire, and at what price it would be fair. They start out in disagreement.

The other interesting aspect is how the negotiations change over time. In the beginning, there is some amount of money at stake, and each side has an opinion on how to split up that pie. If they do not agree, then they can continue the discussions another day. But waiting has its disadvantages–every day the cops are closer to shutting down the mob, and the pool of mob profits shrinks.

This feature–that waiting is costly–is extremely important to negotiations. In the current NFL labor dispute, the waiting has already been costly to players and coaches. NFL players will have to cough up for their own health insurance, and owners have slashed pay to coaches by as much as 25 to 50 percent. If the lockout persists to the regular season, then the roughly 9 billion dollar revenue stream of the NFL will be in jeopardy, and owners will take a bigger hit too.

Because time is money, there is often an incentive to agree to a deal immediately with less than ideal circumstances. The Joker’s offer to take half of the profits now–while the mob still has a lot of money–has some appeal because waiting too long may leave the mob with nothing.

There is an interesting game theory problem about bargaining over a shrinking amount of money, and it indicates why the Joker’s offers is not so crazy after all.

The shrinking pie game

This is a game described in the fun game theory book Thinking Strategically.

Consider a simple bargaining problem between two children, Alice and Bob. They are arguing over how to split an ice cream pie, which melts over time. For simplicity, assume the pie melts in equal amounts at every offer or counter-offer in the game. Let’s see how the division changes with different time intervals.

One time period: only Alice can make an offer

To begin the analysis, suppose there is just one offer. The older sister Alice can make an offer to Bob. If Bob agrees to the split of the ice cream pie, then both receive those amounts. But if he does not, then the pie melts and is ruined, so they both end up with nothing. This is a bit extreme case, but let’s see what happens for the sake of illustration.

How will this negotiation turn out?

Alice has a very strong negotiating position. If Bob refuses what she offers, then they both will end up with nothing. Thus, Alice knows she can offer Bob little if anything, and he will have to accept it. So what happens is Alice offers Bob almost nothing, and he accepts.

This one-stage bargaining problem is so interesting that it has its own name: it is called the ultimatum game.

While the theory says Bob should take a small offer–for if he refuses he gets nothing–experiments have shown this is not always how people play the game. When the ultimatum game is played by children, for instance, offers that stray too far from 50/50 are routinely rejected. People may act out of anger or because they do want to establish a reputation for repeated play.

Still, if one were playing the game strictly by the rules, then there is a case to be made why the theory is right. Let us assume that Alice can get away with offering Bob basically nothing, and Bob will accept in the one-stage game.

So how will the game change if Bob can make a counter-offer?

Two time periods: Alice offers, then Bob counter-offers

The game completely changes when Bob has a chance to make an offer. Now Bob does not have to simply accept any small offer that Alice gives. He can wait until his turn and then give his division of the ice cream pie.

The only hitch is there is a cost to waiting. In the two-period scenario, the ice cream pie melts in two turns. If Bob rejects Alice’s offer, then there is only half of the pie leftover. If his offer to Alice is rejected, then that remaining pie melts as well and there is nothing.

How will this game play out?

Alice makes the first offer, but she has to think ahead. If she offers Bob too little, then he can reject and wait for his turn to offer. Half of the pie will melt in the process, but Bob will be the one making the offer, and so he will have negotiating power. If the game goes until the second round, Bob can basically offer Alice nothing and take half the pie to himself.

Alice can see she has limited power. If she does not give Bob a reasonable offer, he will reject and she will end up with nothing in the second round.

So Alice will go ahead and make an offer of half to Bob in the first round, which is just enough for Bob to accept.

Both end up with a 50/50 share of the pie. It is nice how that works out, as nothing is wasted and both parties end up with something.

3 periods: Alice, then Bob, then Alice

We can continue to extend the game into more time periods. In this case, the pie melts in three periods, so it loses one-third of its size in each period.

How will this game play out? We can look ahead and reason backwards.

If the game goes until the last period, then one-third of the pie remains, and Alice is able to take it all.

Bob would like to avoid getting nothing. So he needs to think carefully in his turn, during the middle period. At this point two-thirds of the pie remains. He has to offer one-third share (half of what is left) so that Alice can accept. This means Bob can at best get a one-third share of the original pie.

Knowing this, Alice can offer Bob one-third in the first period, and Bob will have to accept.

Notice that Alice ends up with two-thirds of the pie, which is more than half. But this makes sense because she has two possible turns to offer versus Bob only having one.

In fact, Alice has this first-mover advantage any time the game has an odd number of periods. Now we can generalize.

n periods: Alice and Bob alternate

The division changes depending on whether the game has an even or odd number of periods. I will spare the gory details and just say that it can be shown that:

–If n is even: then the division is 50/50. This makes sense as Alice’s first-mover advantage gets balanced by the fact that Bob gets to offer last, a nice checks and balances system.

–If n is odd, then the division is favored for Alice. Specifically, Alice ends up with a share of (n + 1)/(2n) whereas Bob gets only (n – 1)/(2n).

For example, if the game has 11 periods, then the division would be 12/22 = 54.5% for Alice and 10/22 = 45.5% for Bob. So even though Alice makes the first and last offer, her advantage is only about 5 percentage points over the fair division.

Back to the Joker’s offer

Now we can see that the Joker’s offer to take half was not entirely unreasonable.

Every day the mob does not act, their share is lost as the city gets cleaned up. If they do not do something soon, then there may be nothing left for them at all.

In their desperation, they accept the Joker’s offer of half of the money.

The Joker eventually delivers on his promise to get the money. But in spite of all his masterful negotiating skills, he eventually demonstrates that he is as crazy as they all suspected.

When he reclaims the money, he organizes it in one giant pile, douses it with gasoline and sets on fire. When the Chechan protests, the Joker sarcastically remarks that he’s only burning his half of the money. So as it turned out, the mob leaders were doomed either way–there’s just no way to negotiate with the Joker.