there are some privately owned cities in India, one is great and one is mixed
Jamshedpur (1.7M people) is wholly owned (at least, the middle half) by Tata Steel and has really good services and utilities (competitive for most reliable electricity, and at one point was the only city with 100% treated wastewater. no municipal government at it it seems
Gurgaon is bunch of privately owned office parks and apartment buildings, each of which are nice and have good services internally, but anything shared is a disaster (roads, etc). reliable power because everyone does it themselves, inefficiently. ship sewage out on trucks to dump somewhere