Thursday, June 11, 2009

Follow-Up to Batman Villains and Cooperation Post

Thank you all for your comments on our post analyzing the Joker's decision of whether or not to cooperate with other villains. It seems as though many of you have taken issue with a few assumptions that I have made. I want to address those issues here.

1) The probability of killing Batman given that three villains attack him separately should be equivalent to adding the probabilities--or 6%.

I never actually said this--in fact, I did not mention the probability of them working separately at all. Suppose the Joker attacks Batman on Monday, Two-Face on Tuesday, and the Riddler on Wednesday. In effect, this means the three villains would be attacking Batman separately. The probability of killing Batman in this case would actually be 5.88%. It would be a geometric series of probabilities. In other words, the probability of killing Batman on the third day (by the third villain) would be equal to:

sum(1-p)^(k-1)*p, where p=probability of killing Batman that day and k=the day #
--> (1-0.02)^(1-1)*0.02 + (1-0.02)^(2-1)*0.02 + (1-0.02)^(3-1)*0.02
==0.0588 --> 5.88%

2) Diminished returns does not apply in this situation.

Upon reevaluation, I concede that the probability equation I offered for cooperation does not work for this scenario. The probability of killing Batman given cooperation among Batman villains should exceed the probability of killing Batman if they were to attack him separately. So if instead of attacking on separate days, the Joker, Two-Face and the Riddler were to plan a coordinated, simultaneous attack, the probability should exceed 5.88%, as dictated by the concept of synergy (the whole is greater than the sum of its parts).

However, this should only be the case up to a point. I am surprised to see that so many commentators seemed to ardently deny the theory that as you add more villains, the marginal effectiveness would diminish. If adding more villains to the plot increased the probability exponentially (or even linearly), then this means that eventually there is a number such that the probability of killing Batman is 100%, or that death is certain. This cannot be the case for Batman--who survived an attack by OMACS, who lived through an attack by the Black Glove, and who survived Darkseid's Omega Sanction. Further, this means that if the battle hits this point of absolute insurmountable odds, then adding more villains could not possibly increase the probability of death (being that it is already 100%).

Instead, the graph should be convex (increasing returns) up to a certain point and then switch to being a concave graph (diminishing returns). That is, cooperating up to a certain number of villains should increase the marginal probability of killing Batman, but after that point the marginal probability should start decreasing. This would be an "S" curve, similar to a learning curve or a logistic function. It should look like the following shape:

As an example, suppose that Batman is fighting the Joker and Two-Face. If the Scarecrow suddenly joined the party then Batman would have a significantly harder time fighting the three of them simultaneously. But now imagine Batman fighting 100 villains. If one more villain joins the party (making it 101), does this last villain induce the same marginal probability increase as the Scarecrow did? I certainly don't think so. In a battle with 100 villains, there are two outcomes. The first is that Batman withstands the 100 villains by himself, in which case adding one more would increase the probability of killing him, albeit not by much. The second is that Batman loses the fight against 100 villains, meaning that the 101st villain would have been ineffective.

Finally, this is, after all, the Batman universe we are discussing here. Cooperation means not only that the villains have to forgo their already significant hostility towards one another (which would involve a cost), but hatch a plan predicated on compromise. And as many commentators pointed out, compromise is not particularly easy for these villains. These are the sort of people who each want to play a prominent role in the demise of Batman. Yet they all have different talents and different means of achieving that goal, all of which cannot be fulfilled in a cooperative plan. The Scarecrow, who prefers psychological means of destruction, would not be able to poison Batman with fear gas and let him destroy himself, while letting Deadshot shoot him in the head from a distance. The group would have to sustain the interest of each individual member (who have short attention spans) and keep close monitor of these villains as their numbers increase. As the group surpasses a certain point, there becomes a huge potential for villains to become contentious, get in each others ways, foil the plan, or weigh the group down. It is not unlike working on a school project with a group who, though having equally effective means of achieving a goal, cannot agree on the particular method.

All in all, eventually we should be seeing some diminishing marginal probability increases with respect to the probability of killing Batman.

3) This sort of analysis should not be applied to Batman villains since it assumes they are rational actors, when they are in fact, irrational.

First: I already mentioned this in the previous post.

One of the distinguishing features about most of the notable Batman villains is that they all have distinct neuroses and pathologies that render most of them utterly incapable of working together. It is not a rational decision, rather that most of these rogues have deep-rooted psychological afflictions, many of which mirror some aspect of the Batman. As such, they have different motivations and goals, different means to achieve those goals, and different reasons to kill Batman.

As such, the analysis is purely academic. We know that the Joker is not actually making utility calculations in his head when he is deciding. The post was designed for fun and to engage the readers in debate. In no way am I actually prescribing that writers start figuring these calculations into the books or start having the characters engage in mathematical debates.

Secondly, by extension, arguing that this sort of analysis should not be applied to Batman given the nature of their villains' irrationality also implies that economics should not be applied to real-world, human decisions. Human beings are also irrational. Our preferences do not always make sense and our decisions are not always exercised with rational caution. If every human being acted rationally, nobody would have ever won a tic-tac-toe game in the history of human civilization. Yet, we still apply economic theory, as we do political theory, social theory, psychological theory, etc. as a guidance in an attempt to explain the world with the means and evidence available to us.


Ken said...

I like where you're going with the blog, especially using comic-book examples to illustrate some basic concepts. But I also think that you're going to keep running into the problem you do here, which is that some aspects of the comics world just aren't susceptible to this kind of analysis.

In this particular case, Batman has the "hero's death exemption". The probability of any villain killing him is zero[*], so there can't be any considerations of joint probabilities, payoff matrices, diminishing returns, and so forth. You need to pick a character that can be killed for the analysis; or perhaps some situation where a team-up of villains would make sense, like a plot to kidnap the United Nations using a dehydration ray[**].

[*] All right, essentially zero; but if he dies, the writers will eventually retcon it away, especially with all the parallel-worlds stuff that DC does.

[**] Sorry.

Anonymous said...

Regarding point #3:

I commented that your post was a good example of the limits of modeling, but that doesn't mean I think modeling is useless or that it shouldn't have been applied here. Any model should be tested to find where it breaks. You can then use that breaking point to build a better model, find out where not to use this current model, or both. Behavioral modeling and game theory are useful, but it's dangerous not to know the limits of your tools.

Your post was a good example of how the Prisoners Dilemma works. It was also a good example of how to break the model.

robsalk said...

First, wow. What a cool blog.

Second, your analysis indicates to me that the most perilous situation in terms of villain team-ups is when the nemesis of one hero gets together with the nemesis of another hero to take on both (preferably, for them, one at a time). That way, you get the synergies of collaboration without the complicating competitiveness and issues over who gets to deal the fatal blow.

If Joker teams with Two-Face to kill Batman, it is the maximum negative utility to each one if the other gets the glory of the kill, for the game theory-related reasons you describe. This negative incentive diminishes the benefits of collaboration for sure.

But if Joker teams with, say, Shadow Thief to take on Batman and Hawkman, the two can help each other with some assurance that the other partner has no special interest in dispatching the other's primary adversary. Plus the villains get the element of surprise by mixing it up. I'd rate even Batman's theoretical chances of survival much lower in those circumstances than if faced only with his own rogues, unless the two heroes are themselves able to collaborate.

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