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 -->>|
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.