One reason we punish is to deter the offender and others from doing naughty things. But how does deterrence work?
The simple idea is that if the punishment (pain) is worse than the gain achieved by doing a naughty thing (pleasure), than we, as rational people, won’t do that bad thing. I won’t steal a loaf of bread if my hand would get chopped off. Deterrence works where Pain > Pleasure.
Add a complication. What if we don’t expect to be punished? Fines for smoking inside in public places are relatively high to mitigate against the fact that it might not be applied in all cases. This is the maths of probability. A fine of £50 that is applied in (say) 25% of cases creates a probable pain of £12.50. Deterrence works where Probable Pain > Pleasure.
Add another complication. What if I don’t know smoking in a museum is wrong, or if I think the punishment for stealing bread is community service, or if I wrongly guess that only 2% of offenders get caught? My calculation is skewed by my misinformation. So, Deterrence works where Expectation of Probable Pain > Pleasure.
My last complication. Humans aren’t rational. I’ve been reading a book by Daniel Kahneman, the psychologist who has made behavioural economics a thing. He writes about (among other things) how humans assign weights to probabilities incorrectly (overplaying small chances, or negating them entirely), about how they are risk averse when it comes to losses, and how people have optimistic views about how their own plans will work out. Apply that to our formula, and we can see that our irrationality mucks up our calculation. A 5% of apprehension might loom larger, and feel like 10%. I might be really worried about the cost of the fine. I might think that I am such a good loaf stealer that I won’t get caught. Therefore, Deterrence works where (Expectation of Probable Pain > Pleasure) (as affected by Human Irrationality).
And this is all unapplied theory. Criminologists don’t know how well punishments deter, because it’s very hard to do controlled experiments. Further, Pain and Pleasure might be felt differently by different people. Finally (for now), a lack of other options (the hungry loaf-stealer) must have effects. With calculations this hard, who’s to say whether punishment X will deter Mr Droog or Fyodor Plc from offending or reoffending?