Optimal crime sentencing

I have been watching some true crime videos recently, and a phenomenon that seems relatively common is that the criminal sets out to commit one crime, which ends up turning into a much more serious one. For example, in one video I saw the other day a couple kidnapped a woman hoping to take a lot of money from her, but after she saw pill bottles with their names on them decided to kill her because they knew she would be able to identify them. It would be nice if the criminal justice system could prevent this type of situation from occurring (of course it would be even better if it could prevent all crimes from occurring). In particular, we would like the sentences for the kidnapping and murder convictions in this case to be such that the criminals reason as follows: “Since she knows our identities we will definitely get caught for kidnapping if we don’t kill her. If we kill her there will be a larger sentence, but there is some probability that we will actually get away with it and not get caught. However, taking this all into account, we are still better off not killing her and accepting the kidnapping punishment.”

To make the model more formal, suppose there are two crimes, a lesser crime A (e.g., kidnapping), and a more serious crime B (e.g., murder). Suppose that A has already been committed, and the perpetrator is considering whether to also commit B. Suppose they believe that they will be caught and convicted of A with probability p, and that if they also commit B they will be convicted with probability q. Assume that the utility from their perspective of being convicted of A is C, and the utility of being convicted of B is D. Finally, assume that their utility of not getting caught is F. Without loss of generality assume that F = 0, so that D < C < 0.

The expected utility of not committing crime B is: p * C + (1-p)*F

The expected utility of committing B is: q * D + (1-q) * F

So it is rational to not commit the crime if: p * C + (1-p)*F ≥ q * D + (1-q) * F

Doing some algebra, this is equivalent to the condition:

C ≥(qD+pF-qF)/p

Setting F = 0, this gives us C ≥ qD/p

As an example, suppose that p = 1 (which was the case in the motivating example above), and assume q = 0.75. Then in order to make it rational for them to not commit B, we must have C ≥ 0.75 D.

Now it is impossible to come up with a system that chooses values for C and D such that this condition holds for all values of p and q (obviously the trivial solution of C = D = 0 is nonsensical). To see this, suppose that p = 1 and that q < C/D. Then qD/p < (C/D)D/1 = C, which violates the condition.

So the only way to achieve the condition would be if the values for C and D are not fixed in advance, and can be modified during sentencing given the specifics of the situation. On the one hand, kidnapping is kidnapping. It seems kind of crazy to “reward” criminals who committed kidnapping and strongly contemplated also committing a murder but didn’t by giving them a lesser sentence than criminals who simply committed kidnapping without contemplating the murder. But as we showed, if we want to choose values of C and D that truly disincentive committing B, we need to make C and D dependent on p and q (specifically on q/p).

In order for this system to be practically implemented, at the sentencing hearing the criminals should be trying to make their best argument as to why q/p is as small as possible given the specific circumstances of the case (and the prosecution should argue why q/p is as large as possible). The smaller q/p is, the greater the minimum value of C is that satisfies the condition C ≥ qD/p, since D < 0 and qD/p decreases as q/p increases.

As far as I know, this type of analysis is not typically done during sentencing procedures.