Game Theory, Signaling and Comic Book Crime

by David G. Klein from the New York Times (2009)

Can Batman devise a scientific way to fight/reduce crime?

A recent article by Robert H. Frank in the New York Times described a more cost-effective method of reducing crime in the United States that borrows from game theoretic principles and the effect of signaling. The theory, proposed by Professor Mark Kleiman, basically says that rather than attempting to deter criminals by offering more severe punishments, law enforcement officials should instead attempt to increase the odds by which they are apprehended.

This is based on the premise (and apparently some empirical data) that most criminals are not rational actors in the sense that they would be deterred if the expected punishment was greater than the expected benefit of committing the crime. Instead they are "more like impulsive children, blinded by the temptation of immediate reward and largely untroubled by the possibility of delayed or uncertain punishment."

However, Kleiman argues that if a criminal were to think that the probability of being caught were high enough, then even more modest punishments would likely be enough to make him or her think twice about committing the crime in the first place.

If we were to accept this premise, what then would be an effective method of increasing the probability of criminals being apprehended? Kleiman argues that this could be done by using the effect of signaling--that is making the criminals think that by engaging in a crime, they would likely be caught. One method he suggests is publicizing a criminal priorities list. The game would work as follows:

Suppose that all drug violence in a city is committed by members of one of six hypothetical gangs — the Reds, Whites, Blues, Browns, Blacks and Greens — and that the authorities have enough staffing to arrest and prosecute offenders in only one gang at any one time. Mr. Kleiman proposes that the police publicly announce that their first priority henceforth will be offenders in one specific gang — say, the Reds (perhaps because its members committed the most serious crimes in the past).

This simple step quickly persuades members of that gang that further offenses will result in swift and sure punishment. And that is enough to deter them.

With the Reds out of action, the police can shift their focus to the Whites. They, too, quickly learn that violent offenses result in swift and certain punishment. So they quiet down as well, freeing the police to focus on the Blues, and so on.

But why don’t the Reds, seeing that the police have moved on, start committing violent offenses again? The reason is that they always remain atop the enforcement priority list. If they start offending again, police attention will again quickly focus entirely on them.

It is an interesting proposition to be sure, but would this method work against comic book supervillains? Suppose Batman and Commissioner Gordon read Professor Kleiman's new book and decided to employ these tactics on the Gotham villains. Let's say they take a group of notable supervillains--The Joker, Two-Face, The Penguin, Mr. Zsasz and The Riddler--and make a priorities list, which they intend to signal to them. By severity of past crimes, lets assume the ordering is (you can debate this if you want; I imagine people will argue the fact that I put the Riddler as more severe then Mr. Zsasz, but there should be no arguments that The Penguin is dead last).:

1) The Joker
2) Two-Face
3) The Riddler
4) Mr. Zsasz
5) The Penguin

Here's the problem with this method. Batman is already a signal. Everything he does, from the batsignal to the bat-tracking devices, is meant to deter criminals. He uses fear to thwart them. After all, this is the importance of dressing up like a bat. Otherwise, he would just be some goofball with a fetish.



Yet, this signaling unfortunately had the opposite effect of what Batman had originally intended. He may stand as a symbol of fear and deterrence for ordinary street thugs (which is incidentally what he started out fighting), but his actions have produced an unintended consequence of breeding more insane villains and "freaks." Indeed, this has been explored in numerous books including The Dark Knight Returns, The Killing Joke, Batman: Ego, and others.

The idea is that many of these villains have motivations other than the immediate reward of the crime. They like having Batman chase them. They try time and time again to outsmart him. For many of these villains, it has become a game to try and thwart the Batman. His very presence in the city, rather than deterring them, has actually and somewhat counter-intuitively inspired them to commit more clever crimes. And without that game, these villains tend to revert back into catatonia. In one story called "Going Sane," The Joker, after thinking he had killed Batman, actually ends up retiring from crime, changing his name to "Joe Kerr," getting reconstructive plastic surgery, and moves into suburbia to live a quiet and peaceful life. Once he finds out Batman hadn't been killed, he mutilates himself and gears up into action once again.

Given these motivations, it is unlikely that Kleiman's strategy would work against the Batman villains should he order his list as he did above. If Batman publicizes pursuing the Joker above all else, he would not be deterred in the slightest. In fact, I'd bet that other villains, in an attempt to grab Batman's attention, would start committing more crimes.

However, this does not necessarily mean the signaling strategy is hopeless. It might be able to achieve some benefits for Batman if used appropriately. For instance, although many villains see Batman's presence as motivation, there are still supervillains who unequivocally want him gone. Consider our list above. The Penguin and Mr. Zsasz two such villains.

Let's look at The Penguin. During Batman's peak, The Penguin operated primarily from behind the scenes. Rather than being out on the street, he owned and operated a "legitimate" business--a restaurant/night club. Sure the club was a hot spot for villains and sure he laundered lots of money through the business, but The Penguin did not want to be caught. That's why he was Batman's primary source of information regarding the other villains. Yet, when Batman went missing after the events of Batman R.I.P., guess who formed a major crime gang and tried to take control of all of Gotham City? That's right, the Penguin is now one of the major problems that Dick Grayson and the crew have to deal with.

So what would happen if Batman decided to order his list another way? Instead of having the most dangerous villains first, suppose he decided to put The Penguin and Mr. Zsasz as top priorities. Then, those two would probably stop committing as many crimes. With them out of the way, as the argument goes, Batman would now be able to focus his attention on The Joker, Two-Face and the Riddler without expending anything on additional resources or sidekicks. It's a cost-effective way of getting rid of the guys on the sidelines--those that really want Batman dead--and focusing on the truly insane ones. The problem, of course, is unlike the gangs mentioned in the NYTimes example, that these latter villains will not be deterred. However, even a small amount of sideline villains out of the way is a bonus.

The other major problem is that, as mentioned above, we don't know how these latter villains will react. It is entirely possible that the signaling efforts of Batman and Gordon will cause them to commit more crimes that are more clever or dangerous in an attempt to win Batman's attention. It is also possible that with Batman's attention elsewhere, they will simply get bored (until such time that the Penguin and Mr. Zsasz are completely out of commission). We don't really have a great way of knowing.

Given Batman's circumstances, I can't think of a great scientific means of fighting crime effectively. He's a bit trapped--if he continues his war on crime, villains like The Joker are born and remain in play. If he retires, then the city becomes weak and vulnerable to villains that, despite not being The Joker, are just strong enough to resist capture by standard law enforcement.

Can anyone devise a way that Batman should/could be fighting/reducing crime effectively?

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Should Batman Villains Betray Each Other? (Analysis using the Prisoner's Dilemma)

(This is the second part of a post that will include some very light and simple algebra and game theoretic concepts. Not to worry--it is pretty crude and easy to follow along with. Also, please note that the assumptions made are rudimentary and based off of my own view of Batman and his villains. I welcome everyone to debate them with me).


We've discussed the decision of whether or not Batman villains should cooperate. Now consider another fun issue: lets suppose some of them do, in fact, work together and somehow manage to capture the Batman. There is now another decision that has to be made by each: whether or not to betray the partner.

Let's talk about Two-Face this time. As discussed in the previous post, the benefits of betrayal are obvious. Two-Face could claim full credit for killing Batman, could win the respect of the Gotham underworld, could elicit fear from the elite, and could potentially acquire some wealth and technology from the Batcave that he would not have to share with, say, Mr. Freeze (I can't think of a conceivable reason for that pair to team up, but I haven't used Mr. Freeze anywhere yet).

This situation is a nice example of the Prisoner's Dilemma. So, let's do a really quick summation of this two-player (Two-Face, Mr. Freeze), two-choice (Cooperate, Betray) game in Batman terms to show that it would actually make sense for the two of them to continue to cooperate, even though neither will. We must again assign some utilities for each player. I have done so, as the following normal-form game matrix represents:

Mr. Freeze -->> Two-Face ↓	Cooperate	Betray
Cooperate	(5,5)	(0,10)
Betray	(10,0)	(3,3)

In this matrix, Two-Face is the player on the left and Mr. Freeze is the player on the top. Each has the choice of either cooperating after capturing Batman or of betraying the other. In each cell, the numbers represent the utilities awarded to the respective players given their choice of action.

If both villains cooperate with one another, they each enjoy a utility of 5 from killing Batman and ruling Gotham (5,5). If Two-Face cooperates but Mr. Freeze betrays, then we will assume that Freeze will eliminate Two-Face, thereby winning all of his utility. In this case, Two-Face would get 0 utility and Mr. Freeze would get a utility of 10 (0,10). If Two-Face betrays but Mr. Freeze cooperates, then the opposite happens: Two-Face gets a utility of 10 and Freeze gets 0 (10,0). If they both betray each other, I'm going to assume that they'll either kill each other (in which case they'd both get 0) or they'll both walk away alive, but each would get less utility than they would have if they had cooperated (obviously they'd prefer not to engage in a near-death battle with one another, so they would lose something). Lets award each a utility of 3 in this situation (3,3).

If this is the scenario, each player would rationally want to betray the other. To see this, we must note that Two-Face and Mr. Freeze are hopelessly self-interested. They only care about their own utilities and not that of the other player. Now assume Mr. Freeze decides to cooperate. So we are restricted to the "cooperate" column of the table above. Looking at Two-Face's utilities in that column, it is clearly a better decision for him to betray, as he would be receiving a utility of 10 as opposed to 5. Let's say Mr. Freeze decides to betray, so we are now in the "betray" column. The best decision for Two-Face is still to betray, as he would receive a utility of 3 instead of 0. Hence regardless of what Mr. Freeze does, Two-Face would still want to betray him. The same is true for Mr. Freeze given Two-Face's actions.

Both players will choose to betray each other and (3,3) is the Nash equilibrium outcome. This means that neither Two-Face nor Mr. Freeze can benefit by deviating from his course of action alone. The dilemma, however, is that there exists an outcome in which both players are strictly better off. If both choose to cooperate, then each receive a utility of 5, which is greater than 3. Thus the outcome (5,5) is the Pareto Optimal point. It is the outcome of the game in which one player deviating would necessarily mean that somebody else is worse off.

So, what we have is a scenario in which Two-Face and Mr. Freeze teamed up and were successful (for some reason). Now it would benefit both of them to continue working together, but neither of them will actually do so. Hence they'll walk away with less than what they could have. As the Prisoner's Dilemma demonstrates, Nash Equilibria are not necessarily Pareto Optima. It's sort of funny to think about, actually. Batman can still claim a small victory even in his death.

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