I also have your intuition but after thinking about a couple of the police's jobs I do understand how it could easily go the other way. You have the somewhat o

I also have your intuition but after thinking about a couple of the police's jobs I do understand how it could easily go the other way. You have the somewhat obvious want of a visible police officer ranging an area to deter criminal activity such as shoplifting/vandalism/mugging. But you also want the police in an area to deal with domestic dispute type crimes which you'd expect to increase at least linearly with density, possibly more so as the closer you are to your neighbors the more you may see them and the more their behavior may impact you. Which may mean that you need several more cops to respond to concurrent incidents even if the average rate of incidents is lower than what a single cop could deal with. That's without getting in to any other issues such as bureaucracy needs/wants of larger departments. Or the odds of more dense areas being more culturally diverse/allowing higher differentials in income against distance. So after that I think it would be quite hard to judge what would be required.