There are a few options; obviously all require some sort of pumping
mechanism.

What I think you are talking about is a water powered sump pump. These
are not inexpensive however, and because of that, it's not likely worth
your while to install it yourself (Penny wise pound foolish).

I'd say roughly you are looking at between $1000 - $1500; but it
depends on many things, like if you already have a sump pit, how far
water lines would need to be run to the pump, etc.

As an aside, if you are really going to lay out the jack for a SERIOUS
game room, get a good insurance policy, just to keep all your bases
covered.