Sure, if an area is low crime, I'd say your policing strategy doesn't matter a ton, assuming your cops don't engage in police brutality or corruption. Probably in a low crime neighborhood, competing departments would focus on friendly interactions with the public and making really sure that none of their officers look bad in viral videos.
I know I mentioned randomization, but I don't think uneven distribution is necessarily a problem. Ultimately, a voter who looks at 3 different logos on the ballot is going to think back to interactions they had with the cops, and what their friends tell them about the cops, in order to figure out which logo to vote for. Different voters are going to see different distributions of police presence, but that's probably fine. In the same way people tend to vote for the incumbent candidate, I would guess that voters will tend to vote for the logo they see the most in their neighborhood, assuming that logo seems to be doing an OK job.
Another way to approach it is that each competing department could have a different service area, and the service areas grow or shrink every election based on election results. This could be done algorithmically by splitting the city in to precincts and finding an assignment of precincts to departments that approximately conforms to the election results, while also keeping service areas for any given logo contiguous through time and space as much as possible (minimize travel and switching costs).
All this stuff could work for schools too btw. It achieves the "skin in the game" aspect of charter schools while reducing the ability/incentive to shuttle problem students elsewhere in order to juice your school's numbers (potentially, depending on implementation).
I know I mentioned randomization, but I don't think uneven distribution is necessarily a problem. Ultimately, a voter who looks at 3 different logos on the ballot is going to think back to interactions they had with the cops, and what their friends tell them about the cops, in order to figure out which logo to vote for. Different voters are going to see different distributions of police presence, but that's probably fine. In the same way people tend to vote for the incumbent candidate, I would guess that voters will tend to vote for the logo they see the most in their neighborhood, assuming that logo seems to be doing an OK job.
Another way to approach it is that each competing department could have a different service area, and the service areas grow or shrink every election based on election results. This could be done algorithmically by splitting the city in to precincts and finding an assignment of precincts to departments that approximately conforms to the election results, while also keeping service areas for any given logo contiguous through time and space as much as possible (minimize travel and switching costs).
All this stuff could work for schools too btw. It achieves the "skin in the game" aspect of charter schools while reducing the ability/incentive to shuttle problem students elsewhere in order to juice your school's numbers (potentially, depending on implementation).