(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.