Game Theory and “Confession”

Among the Italian Supreme Court’s reasons for overturning Amanda Knox’s acquittal was their belief that her statement to the police, in which she placed herself at the murder scene and named bar owner Patrick Lumumba as the killer was powerful evidence of her guilt. (Never mind that the Supreme Court had, in earlier proceedings, found that the interrogation–which wasn’t recorded and for which Knox’s lawyers hadn’t been present–violated Knox’s rights. They threw it out as evidence in the original criminal trial; but, thanks to the oddities of Italian jurisprudence, it was allowed to be used as evidence in the simultaneously running civil proceeding.)

That Knox did eventually name Patrick Lumumba, an innocent man, to authorities is of course quite true. The question that must be addressed is why. For Knox’s prosecutors and their supporters, the answer is that she named Lumumba to deflect blame from herself and avoid punishment for Meredith Kercher’s death. Knox’s defenders point out that it was the police, not Knox, who insisted that Lumumba was the killer, and that Knox named him only after a lengthy interrogation that included sleep and food deprivation and physical pressure.

So how should we read this?

Unlike the physical evidence at the murder scene, which leaves the disinterested observer with little doubt as to what happened, the evidence regarding Knox’s interrogation is murky. We don’t have any videos or recordings of the police interrogation sessions. We’re instead asked to weigh narratives and, based on the few facts we can gather and guided by our own experiences, logical inferences, and prejudices, decide whom to believe.

To weigh these narratives appropriately, I suggest we try to answer two questions, each including a fact inconvenient for advocates of either side.

1. Assuming Amanda Knox is innocent, why would she make a statement placing herself at the scene of the crime and falsely accusing her boss of murder?

2. Assuming Amanda Knox is guilty, why would she accuse a man whom she knew wasn’t there and whose alibi could be easily discovered by her interrogators?

I’ll defer to this page for the answer to question #1. I have little to add to it, and I think it does a good job of linking the Knox interrogation to the wider phenomenon of forced false confessions. Open it, read, then come back.

So on to question #2. To explore it, we’re going to do a little exercise. For the purposes of this exercise, we’re going to assume that the prosecution was, in the working essentials, right. Amanda Knox and Raffele Sollecito participated in Meredith Kercher’s murder. They’re in police custody, held in separate rooms, and can’t communicate with each other. We’ll also assume that the police are not acting coercively and are instead calmly apprising Knox of her situation and her options.

Why should we assume these things? Because we’re setting up a game, a classic game, known as the Prisoner’s Dilemma. This game is used to analyze and predict behavior in business, macroeconomics, biological evolution, and foreign policy, but here we’ll use it to analyze the behavior of two actual prisoners.*

The original version was laid out as follows:

Tanya and Cinque have been arrested for robbing the Hibernia Savings Bank and placed in separate isolation cells. Both care much more about their personal freedom than about the welfare of their accomplice. A clever prosecutor makes the following offer to each. “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 does serious time. Likewise, if your accomplice confesses while you remain silent, they will go free while you do the time. If you both confess I get two convictions, but I’ll see to it that you both get early parole. 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.”

What tends to happen in this game? The smart player betrays her comrade, even though cooperation between the two would result in the best outcome for both. Betrayal is the most rational move if the individual player’s prime motive is avoidance of punishment.

So, let’s try to set up a game in which Guilty Amanda and Guilty Raffele are the players. Here are the rules, with C representing cooperation between the players, and D representing the defection of one of the players:

If Amanda and Raffele cooperate with each other (Condition CC), they’ll each spend two years in jail on marijuana charges.

If Amanda betrays Raffele or Raffele betrays Amanda (CD or DC), the betrayer will spend 1 year in prison for marijuana while the other goes to prison for 15 years for murder.

If Amanda and Raffele betray each other (DD), each goes to prison for 5 years.

So far, we’re in line with the classic game, but there are some additional conditions we need to account for. The police believe there’s another accomplice, not yet in custody. They want that person’s name too. We’ll denote defecting against the third party by D*.

If Amanda and Raffele cooperate with each other but betray a third party, (CD*CD*) they each get a year knocked off their marijuana sentences.

If Amanda or Raffele cooperates and betrays a third party, while the other player merely cooperates (CCD*), the cooperator who also betrays gets 1 year in prison while the other gets 2.

If Amanda or Raffele betrays the other plus a third party, while the other person cooperates (CDD*), then the betrayer goes free while the cooperator gets 20 years in prison.

If Amanda and Raffele betray both each other and the third party (DD*DD*), each gets 4 years in prison.

If Amanda and Raffele betray each other, but only one of them betrays a third party (DDD*), the one who betrays the third party gets 3 years in prison, the other gets 6.

If one player cooperates but betrays the third party while the other player defects without betraying a 3rd party (CD*D), the defector gets 1 year while the cooperator gets 4 years.

If each player betrays the third party but only one defects on the other (CD*DD*), the cooperator gets 4 years while the defector goes free.

But there is one more thing: what if one of the players falsely accuses a third party and the police find out about it? We need to add a penalty for that in our game. Let’s make it that the proven liar gets the maximum possible sentence on the board, 20 years, plus 5 years.

Now let’s play. Here’s what our game matrix looks like:



You’re playing Amanda, and since we’re playing under the assumption of Amanda’s guilt, assume that you’re a cold-blooded killer with no feeling for other people but with an intense desire to avoid doing hard time in an underfunded foreign prison. Also, you have no idea what Raffele, in the other interrogation room, is going to do. Can you trust him not to betray you? You’ve known him all of a week. You know he’s probably getting a shot at the same deal, and if you stand up for him and he betrays you, you’re looking at, at least, fifteen years in prison. Maybe even twenty. So your best move where he’s concerned is betrayal, which costs you no more than five years on the down side if he makes the same move, and could mean a single year in prison for you if Raffele sticks.

Where the third party is concerned, what incentive do you have to hide Rudy’s identity? He’s nothing to you. If you name him and that besotted sap Raffele strings along, you could walk out of this interrogation room right now, free and clear. At worst, you’re out in four, right? Defecting, and turning in Rudy, has to be the best move for you, regardless of what Raffele does.

But in the real world, Amanda didn’t name Rudy. She chose CD*, but the D* in this case was Patrick Lumumba. But here’s the thing. If Amanda’s guilty, she knows Patrick wasn’t at the crime scene. She also should have a pretty good idea where Patrick was: at work at the bar, the place from which he texted her to tell her she didn’t have to go to work that night. Remember the liar’s penalty? The police could easily figure out that Patrick was at the bar the night of the murder. (That the police failed to divert an officer or two from Knox’s interrogation to check up on Lumumba’s alibi is, to put it mildly, a severe indictment of their professionalism.) If Amanda’s guilty, and she knows Patrick has an alibi, by what logic would she choose to take the liar’s penalty by naming him and do 25 years in the slam? Betraying Rudy, whom the police didn’t yet know about (or at least weren’t asking about), would have been the infinitely smarter play.

The liar’s penalty explains why most killers who name an alternate suspect accuse someone whose identity is impossible to verify. They know getting caught in a lie makes them look guilty. When Susan Smith, the woman who drowned her children in a car, claimed that a black man had carjacked her vehicle with her kids inside, she provided a description that was just good enough to produce a sketch, but vague enough that it couldn’t be traced to anyone specific. Rudy Guede, in this very case, for months refused to name Amanda or Raffele as accomplices, choosing instead to proffer a vague description of a bushy-haired white man. (I take it he figured the police would soon eliminate Amanda and Raffele as suspects, so if he committed himself to their presence, he’d have been left alone, with a liar’s penalty of his own to deal with.) Guede would eventually transform the bushy-haired man into Raffele and later add Amanda to his story once it became clear doing so would improve his position with the court. (A rational response to incentives, in game theory terms.)

It should be said here that game theory has its limits. It does assume that players are rational, in the sense of being able to accept the reality of a given situation and understand the applicable rules. (Note: to be rational is not to be moral. The mere fact that someone is a murderer does not preclude their acting rationally.) Also, the prisoner’s dilemma assumes that the players care less for each other than for their own liberty. It could be argued that Guilty Amanda and Guilty Raffele, who’d had a romantic liaison in the days prior to the Kercher killing, might have had a sentimental attachment that would make them less likely to betray each other; but the cooperation options embedded in our version of their prisoner’s dilemma would still make cooperation plus betrayal of Rudy an attractive proposition for both players.

Game theory can’t give us a definitive answer about either Knox’s guilt in the murder or whether a guilty Knox named Patrick Lumumba in hope of avoiding punishment, but it can tell us that a guilty Amanda Knox in full possession of her faculties probably wouldn’t have made the needlessly self-destructive choice of accusing Lumumba. It would have been a disastrous lapse in logic, a total misread of the situation and its opportunities. I can picture a three year-old failing this badly, or someone with a severe brain disorder, but not an ordinarily calm, educated, rational person capable of functioning in the world and making successful long term plans, capacities Amanda Knox had amply demonstrated before Kercher murder and has continued to exhibit since.

To believe that a guilty Amanda Knox would have been irrational enough to decide to name Lumumba to avoid punishment for her actions is, in my view, irrational. As Judge Hellman wrote in his motivation document regarding Knox’s naming Lumumba: “If Amanda Knox had found herself in the house…at the time of the murder, her easiest way of defending herself was to give the real name of the murderer, who was present in the house, because this would have made her credible, rather than give the name of someone who was totally extraneous.” That the Italian Supreme Court thinks otherwise suggests that they haven’t analyzed this situation with sufficient care. If they were to do so, they’d realize that the more likely scenario is that an innocent Knox named Lumumba because she was placed under enough physical and psychological pressure that she’d accuse him in hope that giving her interrogators the name they wanted would end her torment. If avoiding prison time was her game, a guilty Knox would have probably betrayed both accomplices, in far less than 40 hours of questioning. In the end, though it’s not definitive on its own, Knox’s “confession” provides better evidence of her innocence than it does of her guilt.

*Interestingly, a German study tried out the prisoners dilemma on both prisoners and college students. It turned out convicts were more likely to choose cooperative game strategies than college students were. The reasons for this are interesting and merit further study. Read about it here.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.