What is the benefits of using the game center?

I have one Apple ID and multiple iphones and ipads being used by individuals in my family, the single apple id used to purchase apps to be used across the devices, I am having problems with the game center when trying to save the game progress on the cloud. If I could have multiple players setup in the game center this would eliminate losses of game progress. It seems to me the game center is useless if I can't setup more than one player. Is there a way to setup multiple players in the game center using the same Apple id?