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.

Batman Villains and Cooperation: A Utility Analysis

(This is the first 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).

Jeph Loeb has a tendency to depict Batman villains in a strange way. In The Long Halloween and Dark Victory, we see the bulk of his rogues gallery actually working together to achieve a common goal. In fact, the latter has Two-Face actually conducting a sort of mock trail in his lair with all of the villains (including the Joker) in attendance, watching and participating as Two-Face prosecutes witnesses as part of his deranged scheme.

Of course, this is ludicrous. 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. In fact, it's been argued ad nauseum that the Joker may not even want to kill Batman, for this act would extinguish his very nature of being. The one truth is that, barring certain less-insane villains like the Penguin, most Batman villains have a burning desire to find, identify and kill the Batman on their own.

Most people would intuitively argue that it would make more sense for these villains to pool their skills and cooperate in order to finally rid the world of Batman. However, the decision to work alone is not entirely irrational. In fact, we can use very basic tools of utility and game theory in order to work out such a decision for a Batman villain (let's say the Joker, being the most insane and distinguished) and show that cooperation is not necessarily the optimal choice.

First, we have to make certain assumptions. Specifically, we need to assign probabilities of capturing Batman and figure out how much these probabilities increase due to the addition of a new cooperating villain. We also need to assign utility values for the Joker for each scenario. Let's start with utilities.

For not killing Batman, we can obviously assign the Joker a utility of 0.
For capturing Batman on his own, let's assign the Joker a utility of 10.
For capturing Batman with the help of x other villains, the utility would be 10/x.

The last one is sort of tricky. This means that if the Joker cooperates with one other villain (say Two-Face) and together they manage to kill Batman, then the utility for each would be 5. In effect, this means that the villains "split" the utility of 10.

Many of you might be wondering why it is that the more villains there are, the less utility one of them receives from killing Batman. Well, consider the pathology argument above. Obviously, if we factor in the Joker pathology, his utility for killing Batman with cooperation would be less than that of capturing Batman on his own, but greater than 0 as Batman would still be out of the picture. The thing is, as more and more villains enter the party, the Joker will feel less and less accomplished if they wind up killing Batman. Again, it is his very essence of being. He wants nothing more than to kill the Batman on his own, so it should make sense that the satisfaction he derives diminishes as more rogues are brought on board for the mission.

Aside from pathology, there are other reasons why the Joker's utility would diminish as such. One is that the villain who finally achieves victory would be held in the highest esteem among the criminal underworld and feared the most by the Gotham elite. Victory over Batman is largely a symbolic projection of status to the rest of Gotham City--and this is something the villains all desire. Therefore, if the Joker were to finally kill Batman, he would effectively "rule" Gotham.

Consider an even more tangible reason. The villain who kills Batman would gain access to his identity. He could therefore do with this identity anything he pleases. Assuming Nightwing and Robin won't be a problem to neutralize, the Joker could gain access to the batcave: a goldmine of wealth and technology. He could further auction off Batman's identity to the highest bidder. Even though he would be dead, I have no doubt that most of the villains would want to purchase this information to exact revenge on Alfred, Dick, Tim, etc.

For all the reasons then, it makes sense for the Joker's utility to diminish as more villains are added. The more rogues that take party to the death of Batman, the less the Joker will feel satisfied, the less influence he will have over Gotham City, and the less actual benefits he will reap after his death. For simplicity's sake, I assume that the villains "split" the utility of 10.

Now, let's assign the probabilities. I'm going to assume that each Batman rogue has a 2% chance of killing Batman alone (and this is being very, very generous and neglecting the individual skills of each rogue for simplicity). You would then think that adding villains to the scheme would increase the probability of killing Batman by 2% with each new rogue. Except, this ignores the economics law of diminishing returns, which states that as you increase the factors of production, the marginal benefit of those factors decreases. Usually, this applies to outcomes which are continuous (such as production of goods) rather than binary (to kill or not to kill Batman), but we can apply diminishing returns in this case to the probabilities. The theory is that as you add villains, working together will prove more difficult and planning more arduous. Therefore, the probability of getting Batman will increase, but by a marginally smaller amount with each villain added.

Thinking of probability as output, let's assume that in each state,
p = 2*y^0.9, where
p = probability of killing batman and
y = number of villains involved in the scheme.

Hence, we have a diminishing returns function. If there is only one villain involved in the scheme, the probability of killing Batman is 2%.

If there are 2 villains involved in the scheme, the probability becomes:
p = 2*(2)^0.9 = 3.73% (the probability increased by 1.73 percentage points)

If there are three villains involved, then:
p = 2*(3)^0.9 = 5.38% (probability increased by 1.65 percentage points)

And so on and so forth. Now armed with the knowledge of probabilities and utilities, let's conduct an analysis of whether it makes sense for the Joker to team up with Two-Face and the Scarecrow. We must analyze the expected utility of each scenario (teaming up and working alone).

First let's calculate the expected utility of working alone for the Joker. The equation is:
EU = p * (Uk) + (1-p)*(Unk) where
EU = expected utility
p = probability of killing Batman
Uk = utility of killing Batman.
Unk = utility of not killing Batman

We know that for the Joker, the utility of killing Batman alone is 10 and the probability of killing Batman by himself is 0.02. Hence:
EU = 0.02*(10) + (0.98)*0 = 0.2
Hence the expected utility of the Joker killing Batman on his own is 0.2.

Now, we analyze the expected utility of the team-up. We know that the probability of the Joker, Two-Face and the Scarecrow killing Batman is 0.0538. The utility would be 3.33 each. Hence:
EU = 0.0538*(3.33) + (0.9462)*0 = 0.179
Hence the expected utility for the Joker of the trio killing Batman is 0.179.

Since the expected utility of the trio killing Batman is less than the expected utility of the Joker doing it by himself, the Joker should prefer to work alone. Hence using simple economics, we have shown that it makes perfect sense for the Joker not to cooperate with other villains. Of course, this is incredibly simple and there are many other issues to consider. One of these issues is whether it would make sense for the Joker to cooperate, but then backstab the other villains. This issue will be considered in a subsequent post.