We typically deter crime via a chance of punishment. Someone who commits a crime might get found out, prosecuted, convicted, and sentenced. So the amount of deterrence should increase with the chance of that sentence give committing a crime, with the size of the punishment implied by the sentence, and with the delay in that punishment relative to when the crime was committed.
We currently let people pay more to increase the chance of punishment for crimes where they are the victim. For example, they can hire a private investigator, pay snitches to talk, or install cameras to give more info to investigators.
We could in principle also let individuals pay to increase punishment levels. For example, if the federal sentencing guidelines said that a particular crime should be punished by 41-51 months in prison, we might let each person pay so much per month to increase those levels. Such people might might announce such new levels in an attempt to especially deter crime against them personally. Just as they now try to publicly show that they have paid for cameras, etc.
But note, many would be outraged by such a policy, and we don’t actually allow this. That is, we let people pay to influence the chance but not the level of punishment, even though the chance matters at least as much as level for deterrence. We could forbid many of the ways that people increase punishment chances, but we don’t. Why?
My guess: we often like the appearance more than the substance of equality. So we want to have some things we can point to and say “see, we treat everyone equal”, as long as there are other less noticeable parameters where we can more freely pay to make things unequal.
Note that we allow city mayors to do something similar. A city mayor can typically instruct the police to invest different levels of police resources into different neighborhoods, in effect making the chances of getting caught vary greatly by neighborhood. But we wouldn’t let mayors set levels of punishments given a conviction differently for different neighborhoods; that would be unacceptably overt inequality.
As another example, consider that in legal trials we don’t let you pay the judge to lean their decision toward you, or pay for better trial rules of evidence, or for more time to present your case. Or to stand closer to the jury when talking to them. But we do let you pay for a better lawyer, which most people see as making a big difference to getting more favorable rulings. We could assign trial participants to get random lawyers, but we don’t. Again, we are proud to highlight some dimensions on which we insist on equality, refusing to let people pay for advantages, but on other similarly influential dimensions we do let you pay.
There are probably many more related examples; please do share them in the comments.
Isn't the difference that the cases we let people pay more for are things that are only constrained by resources (so naturally we let people use their resources to increase them) while we forbid things that are already the "the right level"?
Ideally, the chance of catching any criminal would be 100% and everyone would have a brilliant lawyer, but we would want every crime to have infinitely long punishment.
I’m unsure whether this matters for your overall point, but do you have references for the claim that chance and level of punishment "both contribute the same to deterrence”? I’ve not looked at this literature, but I remember being told that the chance of punishment matters more than level. My first result on Google for ‘punishment severity frequency’ is this paper, which suggests that chance is more important than level. https://www.pnas.org/doi/10.1073/pnas.2108507118
On your 'pay for increased punishments' proposal: people might be willing to tolerate inequality in cases where the inequality sufficiently improves the chance of justice, because they believe (e.g.) that allowing people to hire private investigators or install cameras is more likely to lead to correct pronouncements of innocence/guilt. If you allow people pay to increase the severity of punishment, there isn't a comparable increase in the probability of a correct outcome. Admittedly, 'allowing people to pay for better lawyers' doesn't obviously increase the probability of a correct decision either, but I expect that most people who'd denounce the 'pay for punishment' proposal would also object to the status quo where rich people can hire better lawyers.