Hang 'em with probability zero: Why does it not work?

A celebrated result in the economics of crime, which we call the Becker proposition (BP), states that it is optimal to impose the severest possible punishment (to maintain effective deterrence) at the lowest possible probability (to economize on enforcement costs). Several other applications, some unrelated to the economics of crime, arise when an economic agent faces punishments/ rewards with very low probabilities. For instance, insurance against low probability events, principal-agent contracts that impose punitive fines, seat belt usage and the usage of mobile phones among drivers etc. However, the BP, and the other applications mentioned above, are at variance with the evidence. The BP has largely been considered within an expected utility framework (EU). We re-examine the BP under rank dependent expected utility (RDU) and prospect theory (PT). We find that the BP always holds under RDU. However, under plausible scenarios within PT it does not hold, in line with the evidence.