If the punishment is equal to the crime such as eye for an eye,or apply for an apple, then given the <1 probability of punishment, it is always rational to com

If the punishment is equal to the crime such as eye for an eye,or apply for an apple, then given the <1 probability of punishment, it is always rational to commit the crime because the expected value is profit. This is true regardless of whether there is police. When you add police then the utility of everyone goes down, i.e the police may randomly shoot you for doing nothing whatsoever. In the middle ages ( > 100 years ago) the english tried making the punishments more severe than the crime, e.g hanging or transportation for stealing a loaf, but that didn't result in decrease in crime, even if it made the crimes unprofitable. If the punishment is random and unfair, as it will be, Then the deterrent effect is negligible because the probability that you will get away with it increases more than the severity of punishment. Humans mostly avoid crime not because it is unprofitable, but because humans are mostly nice. Adding police to society decreases the probability that they will be nice. SO in most police states police and police penalties on society are heavy yet nobody wants to live in a police state. tldr: police do not work as a deterrent to crime.