**Introduction**

One of the most useful tools in analyzing legal rules and the policy problems to which they apply is *game theory*. The basic idea of game theory is simple. Many human interactions can be modeled as games. To use game theory, we build a simple model of a real world situations as a game. Thus, we might model civil litigation as a game played by plaintiffs against defendants. Or we might model the confirmation of federal judges by the Senate as a game played by Democrats and Republicans. This week's installment of the *Legal Theory Lexicon* discusses one important example of game theory, the prisoner's dilemma. This introduction is very basic--aimed at a first year law student with an interest in legal theory.

**An Example**

Ben and Alice have been arrested for robbing Fort Knox and placed in separate cells. The police make the following offer to each of them. "You may choose to confess or remain silent. If you confess and your accomplice remains silent I will drop all charges against you and use your testimony to ensure that your accomplice gets a heavy sentence. Likewise, if your accomplice confesses while you remain silent, he or she will go free while you get the heavy sentence. If you both confess I get two convictions, but I'll see to it that you both get light sentences. If you both remain silent, I'll have to settle for token sentences on firearms possession charges. If you wish to confess, you must leave a note with the jailer before my return tomorrow morning." This is illustrated by Table One. Ben's moves are read horizontally; Alice's moves read vertically. Each numbered pair (e.g. 5, 0) represents the payoffs for the two players. Alice's payoff is the first number in the pair, and Ben's payoff is the second number. Larger numbers represent more utility (a better payoff); so 5 is best, then 3, then 1, then 0 (the worst).

`Table One: Example of the Prisoner's Dilemma.`

Suppose that you are Ben. You might reason as follows. If Alice confesses, then I have two choices. If I confess, I get a light sentence (to which we assign a numerical value of 1). If Alice confesses and I do not confess, then I get the heavy sentence and a payoff of 0. So if Alice confesses, I should confess (1 is better than 0). If Alice does not confess, I again have two choices. If I confess, then I get off completely and a payoff of 5. If I do not confess, we both get light sentences and a payoff of 3. So if Alice does not confess, I should confess (because 5 is better than 3). So, no matter what Alice does, I should confess.

Alice will reason the same way, and so both Ben and Alice will confess. In other words, one move in the game (confess) dominates the other move (do not confess) for both players. But both Ben and Alice would be better off if neither confessed. That is, the dominant move (confess) will yield a lower payoff to Ben and Alice (1, 1) than would the alternative move (do not confess), which yields (3, 3). By acting rationally and confessing, both Ben and Alice are worse off than they would be if they both had acted irrationally.

**The Real World**

The prisoner's dilemma is not just a theoretical model. Here is an example from Judge Frank Easterbrook's opinion in *United States v. Herrera*, 70 F.3d 444 (7th Cir. 1995):

Cynthia LaBoy Herrera survived a nightmare. She and her husband Geraldo Herrera were arrested after a drug transaction. The couple, separated by the agents, then played and lost a game of Prisoner's Dilemma. See Page v. United States, 884 F.2d 300 (7th Cir.1989); Douglas G. Baird, Robert H. Gertner & Randal C. Picker, Game Theory and the Law 312-13 (1994). Cynthia told agents who their suppliers were. Learning of this, Geraldo talked too. When both were out on bond, Geraldo decided that Cynthia should pay for initiating the revelations. Geraldo clobbered Cynthia on the back of her head with a hammer; while she tried to defend herself, Geraldo declared that she talked too much to the DEA. As Cynthia grappled with the hand holding the hammer, Geraldo used his free hand to punch her in the face. Geraldo got the other hand free and hit Cynthia repeatedly with the hammer; she lapsed into unconsciousness.

**Communication and Bargains**

How can we overcome a prisoner's dilemma? You have probably noticed that the prisoner's dilemma assumed that the two prisoner's were isolated from each other. This was not an accident. If the two prisoner's can communicate with each other, then they might reach an agreement. Alice might say to Ben, "I won't confess if you won't," and Ben might say, "I agree." Of course, this might not solve the prisoner's dilemma. *Why not?* Suppose they do agree not to confess, but each is then taken to a separate room and given a confession to sign. Ben might reason as follows, "If I keep the bargain, and Alice does not, then she will get off while I get a heavy sentence." So Ben may be tempted to defect from their agreement. And Alice may reason in exactly the same way. On the other hand, it may be that Ben and Alice have a reason to trust one another. For example, they may have had prior dealings in which each proved trustworthy to the other. Of course, trust can be established in another way. If each party can make a credible threat of retaliation against the other, then those threats may change the payoff structure in such a way as to make the cooperative strategy dominant. One situation in which the threat of retaliation is built into the model is the iterative (repeated) prisoner's dilemma.

**Iterated Game**

As described above, the prisoner's dilemma is a one-shot game. But in the real world, may prisoner's dilemmas involve repeated plays. You can imagine a series of moves, for example:

Round One--Alice Confesses, Ben Does Not Confess

Round Two--Alice Confesses, Ben Confesses

Round Three--Alice Does Not Confess, Ben Does Not Confess

We can imagine various strategies of play for Ben and Alice. One of the most important strategies is called tit for tat. Alice might say to herself, "If Ben Confesses, then I will retaliate and confess, but if Ben does not confess, then neither will I." Add one more element to this strategy. Suppose both Ben and Alice say to themselves, on the first round of play, I will cooperate and not confess. Then we would get the following pattern:

Round One--Alice Does Not Confess, Ben Does Not Confess

Round Two--Does Not Confess, Ben Does Not Confess Round Three--Alice Does Not Confess, Ben Does Not Confess

Thus, if both Ben and Alice play tit for tat, the result might be a stable pattern of cooperation, which benefits both Ben and Alice.

If you want to get a really good feel for the iterative prisoner's dilemma, go to this website, where you can actually try out various strategies.

One more twist. Suppose that this game is finite, i.e. it has a fixed number of moves, e.g. ten. How will Ben and Alex play in the "end game." Ben might reason as follows. If I defect and confess on the tenth move, Alice cannot retaliate on the eleventh move (because there is no eleventh round of play). And Alice might reason the same way, leading both Ben and Alice to confess in the final round of play. But now Ben might think, since it is rational for both of us to defect in the tenth round, I need to rethink my strategy in the ninth round. Since I know that Alice will confess anyway in the tenth round, I might as well confess in the ninth round. But once again, Alice might reason in exactly this same way. Before we know it, both Alice and Ben have decided to defect in the very first round.

**Conclusion**

This has been a very basic introduction to the prisoner's dilemma, but I hope that it has been sufficient to get the basic concept across. As a first year law student, you are likely to run into the prisoner's dilemma sooner or later. If you have an interest in this kind of approach to legal theory, I've provided some references to much more sophisticated accounts. Happy modeling!

**Related Lexicon Entries**

**Bibliography**

- William Poundstone, Prisoner's Dilemma (Anchor Press, 1993).
- Anatol Rapoport and Albert M. Chammah, Prisoner's Dilemma (University of Michigan Press 1965).

**Resources on the Web**

- Steve Kuhn, Prisoner's Dilemma, Stanford Internet Encyclopedia of Philosophy (2010).
- Don Ross, Game Theory, The Stanford Encyclopedia of Philosophy (2010).
- "Comprehsive Repository" for information about the iterated PD (compiled by group at Laboratoire d'Informatique Fondamentale de Lille).
- Axelrod and D'Ambrosio's annotated bibliography (1988-1994).
- Spatial IPDs by Norman Siebrasse
- Spatial IPDs by Jason Alexander.
- Spatial IPDs by William Harms.
- Site designed to accompany Axelrod's
*Complexity of Cooperation.* - Miscellaneous PD Resources (compiled by Constitution Society).
- Interactive Prisoner's Dilemma (at the Serendip pages at Bryn Mawr).

(Last revised on April 7, 2013.)