Game Theory · 2023

Prisoner's Dilemma,
expanded.

Abstract: Real criminal cooperation is neither a single Prisoner's Dilemma nor an infinitely repeated one. It is a game two people keep choosing to play until the investigation reaches a point where one of them stops. We extend the standard model with weights for what holds cooperation together and for the fear that dissolves it, and calibrate the whole thing on a murder trial whose payoffs are a matter of public record.
Case: El Crim de la Guàrdia Urbana. Two Barcelona city police officers, Albert López and Rosa Peral, tried for the murder of a third. Both initially covered for each other; both eventually testified against the other; both were convicted.
Method: Analytical derivation from the standard payoff matrix, with sentences from the actual verdict as the payoffs.

Payoffs from a real verdict

The virtue of this case is that the numbers are not invented. Spanish sentencing gives the whole matrix in years of prison: mutual betrayal cost Albert twenty years and Rosa twenty-five (the difference is an aggravating circumstance, since the victim was her partner). Concealment, for whichever of them stayed silent while the other talked, carries at most three. Had both stayed silent, the evidence supported something in the region of five.

Years of prison. Albert's sentence first, Rosa's second.
		Rosa
		betray	cooperate
Albert	betray	−20 , −25	−3 , −25
Albert	cooperate	−20 , −3	−5 , −5

Notice what happens once one of them talks. The other serves the same term either way: twenty-five years for Rosa whether she stays silent or not, if Albert has already betrayed her, and twenty for Albert in the mirror case. The sentences alone leave that choice tied, so we add Ψ, a perceived injustice, borne only by the player who kept quiet and was betrayed anyway. It is an argument for completeness more than a behavioural claim; the dilemma proper lives on the other diagonal.

Stylised illustration of two prisoners and the ground shifting between them — Fig. 1 · Prisoners in the desert, for some reason.

What holds cooperation together

Two players who care about each other are not playing the textbook game. We let each of them weight the other's outcome by L and their own by Ω, both in [0, 1]:

u(X_{ia} \mid b) = \Omega_i(-\Psi_i - Y_{iab}) + L_i(-\Psi_j - Y_{jba})

The standard Prisoner's Dilemma is the corner where Ω = 1 and L = 0. Set both to one (two people who fully internalise each other's sentence) and cooperation becomes strictly dominant: the equilibrium moves to (C, C) without any repetition, reputation or enforcement. Set Ω = 0 and L = 1 for one player only and you get the version Albert told the court, in which he takes her payoffs as his own and the equilibrium has him sacrificing himself. Rosa's version is the same structure with fear in place of affection, and it produces the mirror-image equilibrium.

That interchangeability is the useful part. L is not really love; it is whatever makes a player internalise someone else's outcome. A hitman weighing what happens to him if he informs on the man who sent him is described by the same parameter, and so is anyone weighing what happens to informants in prison.

Solving the cooperation condition for this case gives Albert cooperating whenever L > 2Ω/25, and Rosa whenever L > Ω/10. Hers is the higher bar, and for a reason worth stating: what restrains each of them is the harm done to the other. Betraying Albert costs him twenty years; betraying Rosa would cost her twenty-five. She therefore has to care slightly more than he does to keep quiet.

What dissolves it

Both versions above are static, and the case was not. Cooperation held for months and then collapsed, without anyone's affections changing on the day it did. What changed was the investigation.

So we add c, the probability of being caught (subjective, a belief rather than a fact, and rising as the police close in), and F, how heavily a player weights their own skin when that becomes likely:

u(X_{ia} \mid b) = \Omega_i(-\Psi_i - Y_{iab}) + L_i(-\Psi_j - Y_{jba}) + c_i F_i \Omega_i(-\Psi_i - Y_{iab})

Below a minimum level of F the term never bites and the player cooperates regardless of the investigation; here that level is 11.5 for Albert and 9 for Rosa. Above it, each player has a threshold belief c* at which cooperation stops, the point where c·F reaches that minimum. Taking F = 18 for both, Albert breaks at c* = 0.64 and Rosa at c* = 0.50.

Rosa breaks first, and she breaks first for a structural reason rather than a psychological one: the same asymmetry between twenty years and twenty-five that set their cooperation thresholds apart also puts her tipping point lower. She was, in fact, the first to talk.

Why it ends where it ends

The moment either player betrays, the game does not simply move to a worse cell; it stops being repeatable. Both statements to the police placed their authors at the scene, which is precisely what the investigation had been unable to establish; each accusation therefore pushed c sharply up for both of them. Cooperation cannot be re-entered from there. (C, C) is unstable under any rising investigation, (B, B) absorbs, and the iterated game collapses into a single decisive round.

Conclusion

Criminal cooperation in the real world is best read as an ultimatum game wearing the clothes of an iterated one. Players keep choosing to cooperate for as long as their belief about being caught stays under their own threshold, and the outcome is decided not by the payoff matrix (which is fixed by the criminal code) but by how fast the investigation moves and how much each player fears its conclusion.