ShadowBanker, a comics-oriented econoblogger examines the economic rationality of the Batman villains depicted in Jeph Loeb comics like The Long Halloween and Dark Victory, which show the colorful villains acting in unlikely concert. From Batman Villains and Cooperation: A Utility Analysis:
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…
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.
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 –>>
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.