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 situation 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
Here is a version of the story, adapted from Steven Kuhn’s entry on the prisoner’s dilemma in the Stanford Encyclopedia of Philosophy. Ben and Alice have been arrested for robbing Fort Knox and placed in separate cells. A prosecutor makes 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 the rows; Alice’s moves are the columns. 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).
| Alice: Confess | Alice: Silent | |
|---|---|---|
| Ben: Confess | Alice: 1, Ben: 1 | Alice: 0, Ben: 5 |
| Ben: Silent | Alice: 5, Ben: 0 | Alice: 3, Ben: 3 |
Payoffs: 5 = best (go free), 3 = good (token sentence), 1 = poor (light sentence), 0 = worst (heavy sentence). Green cell (lower right) = cooperative outcome; gray cell (upper left) = dominant-strategy equilibrium.
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 token 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). In the economist’s vocabulary, the outcome in which both confess is Pareto inferior to the outcome in which both stay silent: moving from (1, 1) to (3, 3) would make both players better off. 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.
In the terms of Table One, the Herreras landed in the upper-left cell: both talked. Notice, too, that Geraldo’s retaliation came only after Cynthia had made her choice, when it could no longer affect it. A threat of retaliation that is credible before the choice is made is another matter, as the next section explains.
Communication and Bargains
How can we overcome a prisoner’s dilemma? You have probably noticed that the prisoner’s dilemma assumed that the two prisoners were isolated from each other. This was not an accident. If the two prisoners 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. And if Alice does keep the bargain, I do even better by breaking it—I go free.” The agreement has not changed the payoffs, and confessing is still the dominant move. 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 iterated (repeated) prisoner’s dilemma.
Iterated Game
As described above, the prisoner’s dilemma is a one-shot game. But in the real world, many prisoner’s dilemmas involve repeated plays. You can imagine a series of rounds, 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—Alice 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 iterated prisoner’s dilemma, go to this website, where you can play against an opponent who uses tit for tat, or try Nicky Case’s The Evolution of Trust, which lets you pit a whole range of strategies against one another.
One more twist. Suppose that this game is finite, i.e., it has a fixed number of rounds, e.g., ten. How will Ben and Alice play in the “end game”? Ben might reason as follows: “If I defect and confess in the tenth round, Alice cannot retaliate in the eleventh round (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.
This pattern of reasoning has a name: backward induction. You start at the end of the game and reason your way back to the beginning. But notice what the argument assumes—that Ben and Alice both know which round is the last. Suppose instead that the game will be repeated indefinitely, or that neither player knows when it will end. Then there is no final round from which the unraveling can begin. So long as the players care enough about the future, the prospect of retaliation in later rounds—what Robert Axelrod called “the shadow of the future”—gives each of them a reason to cooperate today. That is why strategies like tit for tat can sustain cooperation in the real world, where most relationships come without a fixed expiration date.
The Prisoner’s Dilemma and the Law
So far we have had two players, but the same structure shows up when there are many. Think of a public good like clean air. Everyone is better off if everyone pays for pollution control, but each individual does better still by letting the others pay and breathing the clean air for free—so, unless something changes the incentives, no one pays. That is the free-rider problem, discussed in the Lexicon entry on public and private goods. The “tragedy of the commons” has the same shape: each fisher has a reason to take more from the shared fishery no matter what the others do, and the result is a depleted stock that leaves everyone worse off (see the entry on property theory). Legal theorists call these collective action problems, and a good deal of law—taxation, environmental regulation, the creation of property rights—can be understood as a response to them.
The prisoner’s dilemma also helps explain what law is for. Recall why Ben and Alice’s agreement failed: neither could count on the other to keep it. Now imagine that a promise is backed by a legally enforceable contract, with damages for breach. If the damages are large enough, breaking the promise no longer pays; the law has changed the payoffs, so that cooperation becomes the rational move for both parties. On this view, one central function of contract law is to make promises credible, allowing the parties to escape prisoner’s dilemmas that would otherwise block mutually beneficial exchange. In the vocabulary of the Coase Theorem, the inability to make a binding agreement is a transaction cost. (Ben and Alice’s agreement to stonewall the prosecutor is, of course, not one that any court would enforce—and that is no accident.)
Here is the twist. Sometimes the law wants a prisoner’s dilemma. The prosecutor in our story is not a bystander to the game; the prosecutor designed it. Offering leniency to the first codefendant who talks is a familiar move in criminal cases, and antitrust enforcers run leniency programs that reward the first member of a price-fixing cartel to confess. In each case, the law places wrongdoers in a prisoner’s dilemma on purpose, so that the cooperation that would benefit them—and harm the rest of us—breaks down. The Herreras (discussed in Judge Easterbrook’s opinion) were on the receiving end of exactly this strategy. Whether a prisoner’s dilemma is a problem to be solved or a tool to be used depends on whose cooperation is at stake.
One caution. The prisoner’s dilemma is only one game among many, and not every social problem has its structure. Sometimes the parties’ main difficulty is not the temptation to defect but the need to coordinate—to settle, for example, on which side of the road everyone will drive. Once a convention is in place, no one is tempted to depart from it. Richard McAdams has argued that legal scholars reach for the prisoner’s dilemma too often and neglect these coordination games. So before you say “that’s a prisoner’s dilemma,” check the payoffs: is defection really each player’s best move, no matter what the other player does?
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
Legal Theory Lexicon 002: The Coase Theorem
Legal Theory Lexicon 029: Public and Private Goods
Legal Theory Lexicon 045: The Attitudinal Model and the New Institutionalism
Legal Theory Lexicon 058: Contractarianism, Contractualism, and the Social Contract
Legal Theory Lexicon 060: Efficiency, Pareto, and Kaldor-Hicks
Legal Theory Lexicon 064: Possibility and Necessity
Legal Theory Lexicon 113: Property Theory
Legal Theory Lexicon 114: Contract Theory
Online Resources
Nicky Case, The Evolution of Trust (2017).
Steven Kuhn, Prisoner’s Dilemma, Stanford Encyclopedia of Philosophy (2025).
Don Ross, Game Theory, Stanford Encyclopedia of Philosophy (2023).
Serendip Studio, Prisoners’ Dilemma (interactive game).
Bibliography
Robert Axelrod, The Evolution of Cooperation (Basic Books 1984; rev. ed. 2006).
Douglas G. Baird, Robert H. Gertner & Randal C. Picker, Game Theory and the Law (Harvard University Press 1994).
Richard H. McAdams, Beyond the Prisoners’ Dilemma: Coordination, Game Theory, and Law, 82 S. Cal. L. Rev. 209 (2009).
William Poundstone, Prisoner’s Dilemma (Anchor Books 1993).
Anatol Rapoport & Albert M. Chammah, Prisoner’s Dilemma (University of Michigan Press 1965).
Link to the Most Recent Version of this Lexicon Entry
Legal Theory Lexicon 007: The Prisoners’ Dilemma
This entry was last revised on September 20, 2026.
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