Wednesday, June 10, 2009

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

(This is the second 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).

We've discussed the decision of whether or not Batman villains should cooperate. Now consider another fun issue: lets suppose some of them do, in fact, work together and somehow manage to capture the Batman. There is now another decision that has to be made by each: whether or not to betray the partner.

Let's talk about Two-Face this time. As discussed in the previous post, the benefits of betrayal are obvious. Two-Face could claim full credit for killing Batman, could win the respect of the Gotham underworld, could elicit fear from the elite, and could potentially acquire some wealth and technology from the Batcave that he would not have to share with, say, Mr. Freeze (I can't think of a conceivable reason for that pair to team up, but I haven't used Mr. Freeze anywhere yet).

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 ↓
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.

If both villains cooperate with one another, they each enjoy a utility of 5 from killing Batman and ruling Gotham (5,5). If Two-Face cooperates but Mr. Freeze betrays, then we will assume that Freeze will eliminate Two-Face, thereby winning all of his utility. In this case, Two-Face would get 0 utility and Mr. Freeze would get a utility of 10 (0,10). If Two-Face betrays but Mr. Freeze cooperates, then the opposite happens: Two-Face gets a utility of 10 and Freeze gets 0 (10,0). If they both betray each other, I'm going to assume that they'll either kill each other (in which case they'd both get 0) or they'll both walk away alive, but each would get less utility than they would have if they had cooperated (obviously they'd prefer not to engage in a near-death battle with one another, so they would lose something). Lets award each a utility of 3 in this situation (3,3).

If this is the scenario, each player would rationally want to betray the other. To see this, we must note that Two-Face and Mr. Freeze are hopelessly self-interested. They only care about their own utilities and not that of the other player. Now assume Mr. Freeze decides to cooperate. So we are restricted to the "cooperate" column of the table above. Looking at Two-Face's utilities in that column, it is clearly a better decision for him to betray, as he would be receiving a utility of 10 as opposed to 5. Let's say Mr. Freeze decides to betray, so we are now in the "betray" column. The best decision for Two-Face is still to betray, as he would receive a utility of 3 instead of 0. Hence regardless of what Mr. Freeze does, Two-Face would still want to betray him. The same is true for Mr. Freeze given Two-Face's actions.

Both players will choose to betray each other and (3,3) is the Nash equilibrium outcome. This means that neither Two-Face nor Mr. Freeze can benefit by deviating from his course of action alone. The dilemma, however, is that there exists an outcome in which both players are strictly better off. If both choose to cooperate, then each receive a utility of 5, which is greater than 3. Thus the outcome (5,5) is the Pareto Optimal point. It is the outcome of the game in which one player deviating would necessarily mean that somebody else is worse off.

So, what we have is a scenario in which Two-Face and Mr. Freeze teamed up and were successful (for some reason). Now it would benefit both of them to continue working together, but neither of them will actually do so. Hence they'll walk away with less than what they could have. As the Prisoner's Dilemma demonstrates, Nash Equilibria are not necessarily Pareto Optima. It's sort of funny to think about, actually. Batman can still claim a small victory even in his death.

E said...

Good analysis. One addition I would make is the possibility of cooperation in the infinitely repeated version of the game (which actually isnt that far fetched, given how long the comic has been going on and how many failed attempts at killing the batman each of these villains has made)

For the infinitely repeated case, we'd be comparing the probability of cooperation in each period vs. a one-stage deviation giving a payout of 10 and then 3 afterwards (as both decide to defect forever).

Doing a bit of math we need 5/(1-d) > 10+ 3d/(1-d) where d is the discount rate. Solving gives d > 5/7.

So if the comic runs long enough and villains dont discount future gains that much, then we should be seeing some more evil team-ups.

Allen Varney said...

Two-Face, of course, wouldn't decide whether or not to betray, but would simply choose randomly by flipping a coin.

Anonymous said...

So, Two-face would be random, which would make Mr. Freeze's best bet be betrayal even in an infinite repetition scenario, right? Since he can't rely on Two-Face thinking about the future.

Unknown said...

Wouldn't the utility of a double betrayal be zero for both villains, since that would inevitably give Batman the opportunity he needs to turn the tables on them... ?

Anonymous said...

Not really Erik, in this scenario they have already defeated Batman.

Peter said...

Great post, but I wonder ...many supervillains (and Batman's villains in particular) seem to be psychotically fixated on the hero's death, beyond any utility from having him out of the way. In any case, Batman's death in itself is a non-rival, non-excludable good, right? It's only the utility of *being the one who killed him* that has to be allocated. So if we imagine a case where this is positive but negligible compared to the utility of having him dead, and betrayal introduces some small risk of failure to kill him, couldn't we get much closer to a point of indifference between betrayal and cooperation?

(if it's not obvious by now I was a C student in economics, so maybe i'm thinking about this wrong...)

pete

Tom said...

Excellent! I move that this post be emailed immediately to DC and incorporated into a story wherein Batman is captured by two villains and, using the Prisoner's Dilemna argument, persuades them to betray each other in order to faciliate his escape.

Anonymous said...

I suppose you can say the probability of Two-Face betraying Freeze would be 50%. His decision is entirely based on the coin...

Anonymous said...

This scenario is a good example of the limits of models. In this case we're taking the generic and generically rational 'prisoners' and replacing them with specific individuals. Specifically, irrationally psychotic individuals.

Mr. Freeze is focused on the long-term goal of either (Depending on which version we're talking about) reviving his cryogenically preserved wife or avenging the horrible slow death of his wife who's frozen corpse he keeps as a momento. To betray or not to betray depends on how the decision serves this goal. Killing Batman is a side-course, not the main dish.

Two-Face is dependent on his coin to make major decisions. Rational utility doesn't even enter the picture.

The Joker... Who is stupid enough to think they can team up with the Joker and not get burned? The man tortures people to death for the lulz.

Models and game theory are useful tools, but all tools have limits. This post is a great example of the limits of the Prisoner's Dilemma.

JLBeck said...

@Anonymous Poster Above Me

While there is room for psychology, you are missing the point of the chart. Each villain, with or without predisposition for betrayal (which I would argue they all have to some degree) and with all else remaining equal (IE: the death of Batman and it's immediate rewards being the only motivation for the team up), then it is logically beneficial to the villains to betray each other, regardless of their personal choice (or impersonal in Two-Face's case). It could also be argued that rational players would choose the 5-5 split because of the guarantee of safety by continual mutual gain (assuming that cooperation implies further criminal activities). Batman villains, as you explained, are irrational and would likely deviate from this rational course of action.

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