For this alternate universe, we have an eccentric billionaire in a modern day setting.
He's a resident of a fairly large city - several million in the metro area - and he is very fond of it. However, the city's administration is known for being inefficient/corrupt, and no other apartment is more plagued by this than FictionalTown's fire department.
The fire department has a lot of free riders in its employ, and those that are actually trying are woefully stretched thin and low on manpower and resources. Emergency calls are responded to at what can only be described as a stately pace, with many more simply going unanswered.
The billionaire simply tolerated this state of affairs for some time, lobbying for change, until FictionalTown had itself an enormous fire that wiped out a considerable percentage of the town before it was extinguished. It was at this point he became so furious that he decided to take matters into his own hands.
He quietly acquired the engines, facilities, equipment, and professionals, necessary to start his own private department.
Then when the time came, he unleashed them on the city, tapping into the city's calls to know what emergencies were not being responded to, as well as having their own private emergency number. Normally his contribution wouldn't have been as noticeable, but due to the fact that many emergencies were literally left unmitigated by the local department, that every call they responded to either had a positive change or turned out the same.
What would be the ramifications of him starting this department? Whose toes would be stepped on, and would he be allowed to continue operating?