Tuesday, June 9, 2009

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).

Reprinted from http://fc07.deviantart.com/fs25/f/2008/035/9/1/Batman__s_Rogues_Gallery_by_Buzz_On.jpgJeph 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.

225 comments:

1 – 200 of 225   Newer›   Newest»
Dan said...

You assumed that working as a group makes the three villains less likely to kill Batman. (If the three of them made separate attempts they would have a total chance of roughly 6%; for the three together you assumed 5.38%.)

Then you made some questionable utility assumptions, hand-waved about the Joker's feelings of personal accomplishment, invoked game theory, simplified your analysis to just one case, did some math, and concluded that the villains shouldn't work together.

I can make your argument more simply: "We assume that the villains are less likely to kill Batman if they work together. Therefore they shouldn't do that."

Cisco said...

Minor point: If x is "other villains", then it should be 10/(x + 1), shouldn't it?

Mikey said...

I think it's a safe bet that the Joker isn't likely to work well with anyone to accomplish his goals. He isn't going to "put his cards on the table" and would no doubt attempt to come out on top not just vs. Batman but anyone he is working with.

How does this same analysis play out if the villain isn't the Joker?

Neotoma said...

Batman's villains can't play well with others long enough to play a game of poker, let alone kill Batman.

Unlike the Flash's villains -- the Rogues work well together and that's why they're a serious threat to the Flash in spite of their goofy Silver Age origins and gimmicks...

Gatsby said...

An EXCELLENT analysis, sir! I tremble to think of how much more complex the formula would be if you figured in Batman bringing his bat friends along. I suspect that Batman's bringing along a Robin or Batgirl would decrease his effectiveness as it would divide his attentions between defeating foe and rescuing friend, plus Joker would have no scruples about killing Batman's buds, thus increasing his foe's utility.

Glen said...

Doesn't the Joker get value from the team-ups beyond the scant possibility of killing Batman? In "Hush" (yes, I know, but bear with me), the Riddler claims that it was easy to get the Joker to go along with the latest Loebfest because he loved the dead Robin/Clayface gag. If we accept Morrison's "supersanity" conceit, then his motivations for joining/not joining the Arkham All-Stars would change every day (not to mention the questionable logic in using algebra to try to predict Mista J's behavior).

Craig said...

This post is amazing

ShadowBanker said...

Dan -- Good point. Except, the decision I was laying out was one villain by himself vs. three combined, rather than three separate vs. three combined. The Joker has to decide whether to work alone or to work with others. Yes the probability of killing Batman if the three of them were working separately would be close to 6%, but the Joker cannot count on other villains simultaneously hatching plots. Even if they did, the issue here is the probability that he does it himself. So it's like flipping a coin three times and asking what the probability is of getting heads on the third flip.

Cisco -- Oops. That was a typo. I meant for x to be the total numbers of villains. Good eye!

Mikey -- Some villains would be more cooperative for sure. The Penguin would likely cooperate well. The Riddler would too (although he is now reformed). But I think most of the A-list rogues (Joker, Two-Face, Scarecrow) are less likely.

Gatsby -- Oh boy, the classic "to save the girl or catch the villain decision." One day I'll have to write this up.

Glen - Nice use of Morrison's supersanity concept. Obviously there are different interpretations of the Joker and some of them might take an odd pleasure in teaming up. I happen to disagree with Loeb's interpretation, but to each his own.

Craig -- Thanks!

Klaus said...

Good stuff. Actually, separate villain attacks would reduce the Joker's chance of success since someone else might kill Batman first--which would prevent Joker from doing so.

The problem with using diminishing returns is that we aren't talking about one Joker vs 3 Jokers, we are talking about 3 villains with a variety of skills, ideas and resources. This can be made up for with Batman's incredible ability to start fights between people who have captured him.

Anonymous said...

Your probability equation is all wrong and your statement that probability is based in economics and diminishing returns is backwards - economics uses probability.

The way to calculate this sort of probability, x = 0.02% and y = number of additives is:
p = 1 - (1 - x)^y

So three villains working together means the probability of killing batman is p = 1 - (0.98)^3 = 5.88%

John Snyder said...

This analysis leads me to wonder about the economic advantage to any one villain, the Joker perhaps, in killing off the other villains, so as to protect the Batman for his own villainous purposes. (Either to be able to kill or unmask him, or preserve the Joker/Bats dichotomy that drives him.) I think the Joker would find that an amusing project, and it might make for a dramatic story.

Follow me? If there is only one Batman-worthy villain left, his odds of eventually killing Batman would seem to rise, at least in contrast with these hypothetical cooperation schemes.

Maybe we introduce a new villain for the purpose. He starts offing B- and C-list enemies of the Bat, the Joker says, "Fantastic, why didn't I think of that?" and then the Joker hunts the hunter to keep the game for himself. Meanwhile, of course, Batman begins investigating the trail of bodies that is leading right to him.

There you go. Top-selling graphic novel rooted in economic theory. All rights reserved.

Flash said...

A good idea, but your analysis is incomplete, as well as contingent on the functional form for the probability you specified and the division of the payoff for killing Batman as part of a coalition.

It is incomplete in that it ignores in the calculation of the expected utilities the value to the villain of *someone else* killing Batman ("Unk" in your terminology). So, for example, if the Joker doesn't join the scheme, a coalition of two other villains might kill Batman. Or, one of the two other villains acting alone might kill Batman. If the Joker does join a coalition, then (a) at least one other player has to join as well -- otherwise, it's not a coalition, and (b) someone else may or may not join, in which he/she will try to kill Batman on his/her own, and might succeed while the coalition fails.

Presumably, the value to a villain of Batman dying, but not by the villain's hand, is between zero and ten, but of course, as you note, this depends heavily on the individual villain's psychological makeup.

With the correctly specified expected utilities in hand (which may, of course, differ by villain, and must be specified for each possible choice by each villain given 0, 1, or 2 other villains wants to join a scheme), you're going to have a 2x2x2 game. (Not, as you analyze it, a 2x1 decision problem). Look for Nash equilibria in pure strategies if they exist; if not, look for mixed strategies (i.e., villains cooperating with some nonzero probability). I think the game as written satisfies Nash's conditions, so at least one Nash equilibrium should exist.

Regarding functional form, we could tweak the (non-sacrosanct) 2*y^0.9 and 10/x formulae and get a completely different answer.

Anonymous said...

Decreasing returns to scale? Please. Didn't your momma teach you anything about cooperation? Two heads are better than one.

Economic assumptions should be carefully considered before being applied. Consider combat, for example. If you can gang up on someone, your chance of beating him is clearly larger than the sum of the individual probabilities. Economics is as its worst when people use math such as expected utility calculations as a substitute to careful thinking rather than as a complement to it.

--annoyed game theorist (who won't waste any further time replying to your rebuttals because he has actual research to do)

CalebGT said...

Dan said...
"If the three of them made separate attempts they would have a total chance of roughly 6%"

Anonymous said...
"So three villains working together means the probability of killing batman is p = 1 - (0.98)^3 = 5.88%"

Dan is laughably wrong, and Anonymous is correct to treat the problem as a binomial distribution with n=3, k>0 http://en.wikipedia.org/wiki/Binomial_distribution

However, "Anonymous," if I may nit pick, that is NOT the probability of success of all three rouges working *together*. It is merely the probability that one or more of the three will have success if they each try to kill the batman in *independent* incidences. Binomial probability can only be applied to independent events.

The probability of success with co-operation is much harder to quantify because of the various ways in which the rouges can aide and/or hinder each others' actions.

Thelvyn said...

I agree with you.

The best example of your analysis would be the Reaper in Batman: Year Two.

I got a question though.

Who is the flying white hooded villain in that picture?

Nefarious Mathematician said...

You're missing a major factor in your calculations - the villains not only get points for actually killing Batman and sharing his loot, they also _lose_ points every day Batman's alive, because he interferes with their nefarious plans. Even though joining the other criminals in ganging up on Batman decreases their share of the loot and glory, it increases the chances that they have a high future income stream from a Batman-free Gotham. It's probably a big win for them, in spite of diminishing returns, though of course they also have some incentive to stab each other in the back to reduce the amount of sharing they'll have to do in the future.

Bryce said...

It sounds like this econoblogger has never been in a fight.

Neither have I. I mean, not a real one. But if you put one fighter against two opponents, each the fighter's equal, the odds of him winning don't drop from 50% to 25%. They drop to more like 10%. Add a third opponent, the odds of him winning are probably 1%.

If, as your model supposes, good cooperation is possible, the utility function of adding more villains should be exponential, not logarithmic.

Now, if you want to complicate the model some, there is some negative utility to killing associated with killing Batman. If you consider that the death of Batman represents the creation of a power vacuum that the dregs of Gotham will be rushing to fill, then the villain who has expended great resources to kill Batman may end up without the resources needed to capitalize on the opportunity.

Anonymous said...

I believe the Inverse Law of Nina should also be taken into account. 1 ninja or villian, badass; a bunch of ninja, lame.

Jonathan Ichikawa said...

I have serious reservations about this argument. I express them here:
http://www.youtube.com/watch?v=Xko-glcr1rI

Anonymous said...

Ugh. This is a sad misapplication of economics. The factors involved in the decisions of these characters (fictional though they may be) are not nearly as simple as some 'utility value' to an objective like killing Batman. It takes a complex human motivation and over-simplifies it to the point that a discussion about the math involved does nothing to actually describe the real (fictional) act of trying to kill Batman.

Econ is about commerce, and does not accurately describe how people really make choices outside of that realm.

Glen said...

Thelvyn: The mystery villain in question is A-Mortal of the Hyperclan. Of "I KNOW YOUR SECRET!" fame from Morrison's first arc on JLA.

Jot said...

I like the gist of this analysis. Of course, it seems like you understand that the variables become much more intricate, as well as hard-to-predict with the use of different specific villians, and different specific scenarios.

The one flaw that I see in the logic, however, is this: The value of teamwork should have a positive modifier, not a negative one.

First, three people attacking one person generally have the advantage of striking while their foe's attention is elsewhere.

Second, is the complement of each Batman foe's derangedness. For all that the foes of Batman have distinct and occassionally crippling neuroses, each of those neurotic compulsions is offset by a complementing form of genius.

Let me say it simply: There are no stupid (or even average-intelligence) characters in Batman.

The Riddler is a stellar example. He views Batman (a noted genius) as the only person worthy of opposing him. He creates games, riddles, and puzzles that require immense IQ to solve.

The benefit to any given villian for working with another villian is that they gain a skillset they do not possess. For example, the Scarecrow, whose genius for hallucinagenic drugs, coupled with the Riddler's skill with the creation of puzzles and traps.
What about the Penguin's fairly vast access to criminal resources (thugs, weapons, plans, illicit information) combined with the unmatched talent for scheming possessed by the Joker?

~Jot

Foilking said...

This is a great starting point on how villains do not work well together. However, there still are many issues that were not covered in this article that greatly affect the way the Batman villains work.
First each has their own way they want to kill or toy with Batman. Villains like the Riddler and the Joker want more to play with his mind while villains like Bane and Clayface want to physically destroy him. This difference in style automatically causes a riff in the teamwork of any villain team.
Secondly, each villain wants the credit of being the one to either kill or unmask Batman and does not want to share the credit of this accomplishment of this with anyone else.

CalebGT said...

We know that the chances of killing batman are zero because the writers would never let him die. The number that matters for this discussion is the Joker's perceived chance of killing batman, as that is what he would need to weigh. Ignoring stories such as Dark Knight in which he has no intention of killing the batman, he usually perceives his chance of killing batman with his latest plan as quite high (higher than 2%, I'm sure). If you're going to get inside of the criminal mind, then try to approximate the numbers inside it instead of making up a number that we all know is zero. You know when you pick up a Batman comic that he's going to survive.

Arkain said...

@CalebGT

*SPOILER*

Apparently you haven't been reading your Batman comics lately (I actually haven't either, but I have my ways), as the good Bat was just killed... Er... Sort of. Darkseid has a messy way of sort of killing people to a semi-permanent degree. I doubt it'll stick (if it hasn't already been reversed somehow).

*/END SPOILER*

@Jot
You are forgetting to take into account (at least to a large enough degree) the weaknesses of each villain, which is also added alongside each strength. While Riddler may be able to cover for a certain degree of Two-Face's coin induced antics, there will be a point where it becomes debilitating to the overall effort (and thus beneficial for him to be cut from the team).

@Original Post
Nice job, I found it very interesting. I also must laugh at the rift caused between mathematicians and psychologists in these comments. While I'm no math wiz (more of a psychologist actually), I think you've pretty much nailed the probabilities minus the infinite array of variables that would be impossible to account for.

-DuLake

Anonymous said...

Jeeze people.

Scrap all the economics. Scrap all the probabilities.

Send Batman on vacation for a month and just let Superman clean up gotham.

Anonymous said...

This example ignores the added probability of being able to multitask, occupy separate places simultaneously, etc., all things Batman cannot do.

Friar said...

"...NOT the probability of success of all three rouges working *together*."

I suspect the eyebrow pencils would be more dangerous to Batman than are the rouges. Mascara applicators could also pose some problems, as might lipsticks. Rouges, it seems, are quite a ways down on the list, unless their colors clash to such a degree as to cause lethal injury.

Anonymous said...

It is not always true that two people attacking one have an advantage greater than simply arithmetic. As in this article, a diminishing probability is also possible. Trained fighters are frequently able to use the poor coordination of fighters not used to working with each other against them.

thenerdwonder said...

I got stuck wondering if Two-Face should be counted as one villain or two.

Elias said...

@Dan

Working as a group is not less likely to kill Batman. It's just that the probability to kill Batman as a group is not proportional to the number of the villains.
For instance, Batman is more likely to be killed by a group of 3 (5,38%) than by one villain (2%)

Fractal said...

Why propose new arbitrary formulas when this one isn't even adequately analyzed? (And yes, any formula describing the effectiveness of comic villains is going to be an arbitrary approximation unless Brainiac provided it himself.)

What I found interesting about the math is that according to the formula provided here, it will take 78 (well, 77.223) cooperating villains to have a 100% chance of killing Batman. That's a certainty that no number of independent villains will ever achieve.

Since having to split the utility pie for working together is apparently exactly the same expected value as randomly choosing which villain gets full credit, the Joker and his buddies should select the method maximizing the probability of Batman's demise. Independent attempts are better for small numbers of villains, but in this case, that means transitioning from independent attempts to cooperation when there are at least 41 (40.317) bad guys.

So, as long as there are are least 41 villains gunning for Batman, they're better off working together. Otherwise, not. How many were in that picture again? ... It looks like only 22. Unless there are 19 more in hiding, the ~36% chance of success they'll have on their own is better than the ~32% chance they'll have together.

Anonymous said...

This feels just a little over my head. But that might be the whole Just getting out of bed feeling.
Now hypothetically, The Joker utilizes two to three others to kill Batman and succeeds. Yay, the Batman is dead, hurrah! now let's celebrate. The Joker's going to be like "Who said anything about WE?" and likely kill his cohorts. He doesn't strike me as someone who would share the glory. Thus, the victory would likely not be so diminished in his delusional mind.

Anonymous said...

Bryce -

Your scenario works only for untrained fighters. This is because at an untrained (or minimally trained) skill level, the chaos of a fight is amplified and an untrained mind is pretty quickly overwhelmed.

However, as the solo fighter's skill increases - regardless of his opponents' skill level(s) - his odds of reaching a favorable outcome decrease only slightly, UNLESS his opposition is specifically trained to fight as a team.

I have been on all sides of this scenario, losing and winning when alone and when outnumbering my opponent. Multiple uncoordinated attackers completely lose the advantage of maneuver, and simply become movable obstacles for their teammates.

Even under your supposition of all having equal levels of training, without the additional team training to delegate zones of control and task handling, one versus 3 is not a 90% loss rate for the individual (not even close, in fact).

My envisioned scenarios, of course, are based on a favorable outcome for the individual, which is to emerge from the confrontation intact and free. I also assume adequate room for maneuver and the existence of an escape route. YMMV, of course.

Mike said...

There are actually 3 stages to a battle, and several levels to an engagement. I'l simplify:

Logistical: Ensuring all necessary personnel and materiel are in the proper places and accessible. This is most important.

Strategic: Holding the proper ground for initiating the battle.

Tactical: Maneuver once the battle is begun.

The "more is better" argument only really applies to the tactical stage, and only assuming they are willing to cooperate. If someone winds up with the garotte around Bats' neck, the others might want to interfere a bit.

I'd guess they'd cooperate in the logistical stage, backstab each other at the strategic level, and infight in the end stage of the tactical engagement.

A clandestine event would follow this progression:

Reconnaissance of possible engagement areas.
Planning.
Placement of resources.
Approach.
ENGAGEMENT.
Reinforcement (if needed).
Conclusion of engagement.
Departure from the area, in this case, with proof of Bats' death/with the captured Bats, OR, strategic withdrawal in event of failure.

You'll need to calculate probabilities at each stage.

cyrebjr said...

@Jot: There are no stupid (or even average-intelligence) characters in Batman.
Wasn't there the Killer Croc, not to mention countless muggers?

cyrebjr said...

@Jot: There are no stupid (or even average-intelligence) characters in Batman.
Wasn't there the Killer Croc, not to mention countless muggers?

kaiki's sketch blog said...

My only comment is your equation for diminishing returns in regards to the probability of villains defeating Batman.

You can set a value for the exponent such that you would continue to get diminishing returns but that the expected utility could either be close enough the expected utility 0.2 that it the difference may not be noticed by the joker.

Example,

If you set your equation to
p = 2*y^0.978
then
p = 2*(3)^0.978 = 5.85672163

plugging that solution into the equation for expected utility you find

EU = 0.0585672163*(3.33) + (0.9462)*0 = 0.195

The expected utility is lower then if the Joker killed Batman by himself, but it is unknown that if the difference in utility would be noticeable.

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