Econoblogger explains why Batman villains shouldn't cooperate

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.

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

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.
Batman Villains and Cooperation: A Utility Analysis

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

(via io9)


  1. Sheesh – what an amateur. He didn’t allow for the “worthless minion” factor which greatly complicates the picture.

  2. Econoblogger illustrates why we should treat economists (other than behavioral economists, the only true scientists in the field) with intense skepticism. Human beings are not rational in the sense that economists use that word, that is, they do not have a utility function in mind that they seek to maximize. And “Prisoner’s Dilemma” is a poor model for the real world, because people have more than one encounter with each other; the payoffs aren’t bounded and the choice a player makes in one “game” will affect how future players choose to act.

    Besides, let’s face it: the real reason the bad guys don’t collaborate to kill Batman is that then the comic series would be over, or they’d have to start up a new alternate timeline or something like that. And that would cost DC Comics money.

    Aha, we’re back to actual money, in the real world. Perhaps the econobloggers should focus on why DC Comics would find certain plot lines profitable and others not. But I guess that wouldn’t be as much fun.

  3. This is exactly what kills usually superhero movie franchises too; for some reason studio execs always want to pack sequels with an ever-growing number of adversaries for the hero to face off against.

    Not only does this leave less time for character development, it often means putting together villains that have incompatible goals. I don’t think a plant fanatic would voluntarily team up with a guy known for turning landscapes into frozen wastelands, for example.

    Also: you cast a bodybuilder as a Batman villain and you decide he should be Mr. Freeze instead of, I dunno, BANE? What the fuck?

  4. The whole equation is faulty: The Joker would NEVER intentionally kill Batman. He may threaten and plot, but The Bat is just too much fun to have around.

  5. Why is the utility of capturing batman splitted between the involved villains? That is a big supposition that obviously concludes that the villains should not cooperate.

    What’s the difference for Two-Faces if he lets the Joker kill batman alone? He doesn’t get his shares of bat-gadgets?

  6. Presumes a rational actor- especially inappropriate in the case of Batman villains.

  7. “I don’t think a plant fanatic would voluntarily team up with a guy known for turning landscapes into frozen wastelands, for example.”

    I take it you haven’t seen Batman and Robin then. Consider yourself blessed.

  8. Flawed. Ignores tactical advantage of simultaneous, multi-pronged attack. The whole *is* greater than the sum of its parts.

  9. This theory is based on the presumption that the principal motivation of all of the villains is the capture and death of Batman.

    While that may be the central motive for the Joker, other Batman villains are driven by financial interests.

    The fact that the villains don’t have a common objective can not be discounted in cases of antagonist team-ups.

    To the Dork-Cave, Robin!

    — MrJM

  10. If you start with the wrong premise, you will end up with the wrong conclusion.

    If the Joker kills Batman himself, the Joker gets +10 utility. But if Two-face and Mr Face collaborates, they each only get +5 utility? And the Joker gets nothing because he simply stood by and watched?

    I think our economist have been playing too much RPGs. Do the villains benefit from actually killing Batman by their own hand, or simply from having him dead?

    How does he explain the fact that crime bosses frequently do not kill people themselves, but tell their underlings to do it, or hire independent contractors?

  11. Assuming rational actors doesn’t even work in the real world, much less Batman’s world of lunatics and insanity. That kills this analysis as a prediction, but it doesn’t necessarily kill it as a recommendation.

    Diminishing returns doesn’t entirely work here either. Diminishing returns pertains to ongoing production of goods, not combat. Anybody who’s played a strategy game knows that a a large group of enemies attacking simultaneously is exponentially more difficult to deal with than a steady stream of individual enemies. It’s more costly to organize, but the chances of success increase EXPONENTIALLY, not logarithmically.

    Also, his definition of “utility” is extremely vague. What is utility to a Batman villain? Being free of Batman, who thwarts their plans routinely, so that they can pursue their other goals? In that case, the utility applies equally to every villain at full value. Getting the satisfaction of revenge? In that case, the utility decreases exponentially as you add more villains. Having another party to you revenge is less than half as satisfying as doing it yourself.

    In short, don’t fuck with us comic nerds. We’re way nit-pickier than you can imagine. We’ve already run dozens of (admittedly informal) analyzes in greater detail in our minds and our conversations with friends than you could possibly fit in to a half-assed blog post. Do your goddamn research, and you better nail every little detail, or it’s going to bite you in the ass.

  12. Considering how territorial the Joker gets at times over Batman, I think it’s fair to say that he’d much prefer to catch Batman all by his lonesome, rather than in partnership with another villain. And I think he’d be particularly bent out of shape if someone took the B out without his input at all.

    The other villain may not feel that the “value” of Batman’s capture is diluted by the Joker helping take him down, but working with the Joker at ALL can’t be a pleasant (or safe!) experience. That would reduce their perceived value of the team-up – don’t worry about being thwarted, worry about being fatally poisoned by a plastic flower, or blown to smithereens by a prank cigar. Or just shot.

    And of course wherever you get two or more villains working together, you increase the odds of thwarting in general – the more villains, the more likely this is a team-up event comic and the scheme attracts a GANG of heroes to ruin everything.

  13. Whut.

    If anything the utility of splitting the kill is greater because it takes less effort on Joker’s part to get Batman out of the way.

  14. @joe: No, econoblogger only shows that we should treat him with intense skepticism. Any professional economic theorist knows that cooperation can very easily be sustained in a repeated prisoner’s dilemma. For example, Google the “folk theorem.”

    Also, assuming diminishing returns to scale is ridiculous here. Remember, just because you use math in your analysis doesn’t mean that your analysis makes any sense.

    –annoyed game theorist.

  15. @vnend
    No, synergy (which we call increasing returns to scale) is typically not assumed because it makes the analysis much more difficult (it’s harder to find equilibria and us mathematical optimization techniques). Any good economist knows that there are limitations to this approach, but uses it anyway because it usually doesn’t ruin the analysis as it did for the original blogger. The difference between a good economist and a bad one is knowing when you can get away with assumptions like that and when you can’t.

    –annoyed game theorist

  16. The two main flaws (which have been mentioned in various ways above) are:

    1) the supposition that the utility is 10/x

    2) the supposition that the law of diminishing returns applies to capturing batman

    Point 1 has been pretty well covered above: The utility for two villains capturing batman is only half if they have to equally split the “reward” of capturing him. However, this fails to account for (a) the villains having differing goals: e.g. if Joker wants chaos in the world and Two Face wants revenge (say), they both get 100% reward. (b) Even they both have the exact same goal (which in unlikely), the end result of THAT, obviously, is that one kills the other, so for at least one villain, the reward would be 100%.

    Point 2 fails because the law of diminishing returns does not apply to all situations. For instance, with just me, the chance of being able to grab a balloon hovering eight feet of the ground is exactly zero. With someone on my shoulders, the probability is almost 1.0, Likewise, while it may be virtually impossible for one villain to catch Batman, two or more working strategically have a much better chance. A good example is lions hunting in packs — the only way this strategy could have evolved is if the expected utility of each lion is greater if they hunt in packs than if they hunt alone.

  17. As an addendum to my statement above, the expected utility for both lions hunting in packs and super villains probably follows and sigmoid curve as the number of participants increases.

    Going from one lion (or villain) in the hunt to three may increase the total expected utility by more than 3X, while going from 100 lions to 300 probably won’t.

  18. Batman’s death isn’t a bag of loot and there’s law of conservation of utility.

    There’s nothing stopping two villains from both getting 100% utility out of it, or one getting 80% and the other 60%.

  19. Ridiculous. And not just for using something as contrived as the Batman series of comics for the purposes of illustrating a real-world economic problem. Obviously, if the Batman were to be killed all criminals in Gotham would benefit whether they were involved in his demise or not. In that part of the fantasy universe, crimes would suddenly become easier to get away with. I would even argue that crime would go up in the other locations in the DC universe as well because characters like Superman and the Flash would be spending at least some of their time in Gotham, thus increasing crime in places like Metropolis and Central City.

  20. Actually I remember comics where the Joker captures Batman but deliberately refuses to unmask him because it would have ended the cat and mouse game that they keep playing. There are also scenes where he refuses to kill Batman because he’s “too much fun”.

  21. All this talk of “utility” and not one mention of the utility belt? Meh.

    Anyway, I think ShadowBanker simply misinterpeted this classic scene:

  22. Classic theoretician stacking the deck to support pet theories.

    Even more damning when there is plenty of empirical data available: I was hoping for an analysis of the success and failure of solo vs team-up results over the canon.

    There’s no real need to involve Batman in the Econoblogger post at all.

  23. Lots of people have already pointed out many of the flawed assumptions contained within this “analysis”. But here’s a flat-out mathematical error. Consider this part:

    “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.”

    No you wouldn’t, and the reason has nothing to do with diminishing marginal returns. It’s all about probability. Probabilities are not addtive. If I have a 2% chance to hit the target and you have a 2% chance to hit the target, then then probability at least one of us will hit the target is 1-(0.98^2), i.e., 100% minus the probability that we BOTH miss. This equals 3.96%. The formula for n villains is 1-(0.98^n).

    Moreover, if each villain is INDEPENDENTLY trying to kill Batman, there is no reason to assume diminishing marginal returns from an increasingly complex scheme with multiple participants. (There might be diminishing returns if the villains get in each other’s way, though.)

  24. Aside from everything, I’m sorry but the Joker’s nose is waaaayyy too pointy and is bothering me.

    I guess I can look at it as exaggerated for emotional purposes,though.

  25. I’m nominating Cory for the prestigious Golden Pocket Protector Award for turning a silly comic into a math equation. Geeesh! I had to break my glasses and tape them back together just to read this!

  26. Utility derived to villain from death of batman is non-rivalrous. EPIC FAIL for assuming Joker gets less utility from dead batman, (ie freedom to wreak havoc) if someone helps him.
    Betrayal is really weak in this example too.

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