Dent and Gordon's Uneasy Alliance: A Game Theoretic Analysis

Watching the film, The Dark Knight, got me thinking more about trust and cooperation. In fact, trust seems to be a big theme in the movie (as it was in Batman Begins). For example, though we know that future Batman and future Commissioner Gordon form a close bond predicated on absolute trust and respect for one another, the relationship was more uneasy at the dawn of Batman's career. Gordon knew that in order to aid a masked vigilante, he had to bend the rules of the law and risk his career. Batman knew that in order to gain help from the inside, he needed to accept someone into his operation who could at any moment turn on him and compromise his mission. In Frank Miller Batman: Year One, we are actually given access into Gordon's mind as he struggles with the implications of partnering with Batman.

There are other examples of cooperation and trust that are presented in the Christopher Nolan films as well. Batman's gradual but cautious partnership with Harvey Dent, Lucius Fox's faith in Bruce's use of company technology, even mob loyalties towards one another. And the movie certainly does pose a few interesting questions. Namely, "Are these partnerships a good idea" and "What are the potential consequences of such relationships?" We also are allowed to see the short-term effects of some of these, particularly Batman's relationship with Gordon. SPOILER ALERT. Towards the end of the film, we in fact see their ties severed and the good that they had accomplished slowly wither away.

But let's take it a step back and think about a specific example in game theoretic terms to see whether cooperation makes sense. Earlier, we had considered the case of villains betraying each other after neutralizing Batman's threat, which we likened to the famous example of the Prisoner's Dilemma. Now, we consider a different scenario--one in which cooperation, rather than defecting, might be an optimal choice for each party involved.

Let's take the example of Gordon and Harvey Dent in the Dark Knight. Recall the scene in which Gordon meets with Dent at the DA's office following an unsuccessful trial in which the latter attempted to convict Boss Maroni for his connections to the mob. In this scene, Dent proclaimed that despite all of his best efforts, he was at best only able to put some mobsters away and was ultimately incapable of rooting out their money laundering schemes and halting the transfer of illegal money. Conversely, Gordon, with Batman's help, was able to craft a scheme to seize the mob's finances from five notable banks. However, he needed Dent to issue warrants to back the search and seizures on the banks. Without Dent's help, Gordon and Batman could only continue stopping pockets of crime here and there.

It is clear that the benefits of cooperation in this case would be a significant reduction in mob power and influence. However, both Gordon and Dent have some reservations about partnership. Gordon was skeptical about allowing a third party to meet Batman, fearing both that it would make the operation too large and that it would increase the chances of either sensitive information being leaked or the operation being compromised. Dent feared that he could not trust Gordon's Major Crimes Unit after having investigated several of its members in Internal Affairs.

Though they had met and ultimately agreed to cooperate, let us suppose that their mistrust was considerable enough to cause serious doubts. That is, neither Gordon nor Dent were 100% certain of the actions of the other player. What would their optimal choices be in this situation?

We need to assign some utilities here. Let's say if both players cooperate then each would receive a utility of 4 for the success of having rooted out the mob's finances. If neither chooses to cooperate, but rather continue with their separate means of crime-fighting, then each would receive a utility of 3. They would continue catching mobsters and putting some in jail, but of course this would not be as beneficial to either as "hitting them where it hurts: their wallets." Suppose that Gordon decides to cooperate, setting up the scheme to seize assets from the five banks and involve Batman, but Dent backs down at the last minute. In this scenario, Dent would still receive a utility of 3 for catching criminals on his own, but Gordon's utility would be reduced, say, to a utility of 1 for having planned the operation and expending all the resources of his unit. Conversely, if Dent decides to cooperate and drafts up the warrants, but Gordon decides not to include him after all, then Gordon would receive the utility of 3, while Dent would receive the utility of 1 for having wasted the effort.

This scenario is similar to the popular game known as the Stag Hunt. In it, two hunters decide whether to cooperate to catch a stag (from which they each derive greater utility) or go separately and each catch a hare. The following normal-form matrix represents these utilities.

Harvey Dent -->> James Gordon ↓	Cooperate	Don’t Cooperate
Cooperate	(4,4)	(1,3)
Don’t Cooperate	(3,1)	(3,3)

Let's find the pure strategy Nash Equilibria of this game. Suppose that Dent decides to cooperate. Then it would be in Gordon's best interest to cooperate as well, for he would receive a utility of 4 instead of a utility of 3. However, if Dent decides not to cooperate, Gordon would be better off by also not cooperating, for he would only receive a utility of 1 for cooperating, rather than a utility of 3 for going on his own.

Similarly, if Gordon decides to cooperate, Dent should choose to cooperate as well, as he would also receive a utility of 4 as opposed to 3. Should Gordon choose not to cooperate, Dent would also be best served by not cooperating, as he would receive a utility of 3 rather than 1.

Notice that the results of this game are quite different than that of the Prisoner's Dilemma. Namely, this game actually has two pure strategy Nash equilibria (strategies in which neither player can benefit by deviating alone). Either both players will cooperate or both players will work alone. Recall that in the Prisoner's Dilemma (Batman villains betraying each other), there was only one pure strategy Nash Equilibrium: both villains betray each other. Although both players would benefit by cooperating, it did not make sense for either player to do so individually. Rather, each should have always chosen to betray each other. In this case, however, Gordon and Dent choosing to work together could make sense.

Note that there is a difference between the two Nash Equilibria in this game. If both players cooperate, this is known as a payoff dominant equilibrium. This means that it is Pareto superior to all other outcomes in the game, i.e. should players choose to cooperate, each would receive more benefits than any other outcome. However, the other equilibrium, in which both players work on their own, is risk dominant. This means that as players become more uncertain of the actions of the other player, they would be more likely to choose the strategy that leads to this outcome. The reason is that choosing not to cooperate guarantees a utility of 3, whereas an individual choosing to cooperate bears the risk of receiving a utility of 1.

In fact, one major implication of this game (aside from the fact that there are situations in which cooperation makes sense) is that an individual's view of another one's actions matter. Suppose Gordon knew the probability that Dent would cooperate. How would this affect his actions? This is where mixed strategies and randomization come into play. As it turns out that this "Stag Hunt" game contains one mixed strategy Nash equilibrium. We shall consider these mixed strategies in an upcoming post.