If it's a dedicated gameroom, black would certainly work, and I'd expect it to drop the glare and reflections quite a bit.
But getting it a consistent deep black (and them putting it back to white when you move) is going to be one +bear+ of a project. It'll likely take 3-4 coats for good coverage.
It'll also make the room feel MUCH smaller. If it's a decent sized room already, probably not a big deal, but if it's small now, it'll feel like a coffin with the ceiling black.
That said, some friend of mine painted their +living room+ ceiling black and it worked amazing well. But their house was an older home with 11' ceilings and a wall of windows on both ends.
Just a few things to consider.