Question

How are the games used to maintain control over the districts? conversely, how are the games used to keep the capitol happy and content?

Answer 1

I assume that this is referring to The Hunger Games.

The Capitol uses the games to maintain control over the districts by showing the citizens that the capitol is in power. By making the people fight each other it causes a divide between them which prevents a potential rebellion against the Capitol.

The capitol uses the games as entertainment which keeps them happy and content.