This is classic case of free rider problem. For providing the police service the city government will have to raise taxes. Once the tax has been imposed then everyone will have to pay for it. Thus the tax is levied on everyone.
Now there are people who will evade the tax. So some people will not pay and get the benefit in terms of increased security. So, people know this and thus they are arguing against the extra increase in police force.
In the case when they are hiring private forces only those who truly need it will pay and there will not be any problem of free rider in this scenario.
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