Hi there tdub,
Sound's a funny thing. Most rooms are horrific for containing sound, as the design never includes this as a feature and most building practices actually go against what you would do to isolate sound.
People get paid a lot of money to sound proof rooms, but I can tell you what you can do to take care of the majority of your problem.
First, sound is dampened by absorbing the waves, so that's what you want to do.
1) First, place
LOOSE layers of open faced insulation in between the floor joists. This of course means you have to pull the sub floor in the game room up, but if you don't do this, you won't even come close to getting rid of the noise.
Do not pack the insulation into the bay. Tightly packed insulation is completely useless, as it will just pass the sound on. It's best if you can actually place two thin runs of insulation with an air gap in between them in each bay. This may sound impossible, but you can either use wood blocking, or simply use the wire tension rods that are used to keep the insulation in the bays overhead. This would be much more effective than just a single layer. If there are any fixture boxes in the ceiling below that you see with the floor off, use a thick rubber (like the shower pan liner) and glue it to the drywall around it, leaving a slit for the wires penetrating the box to get through. Use a rubberized caulk to seal around any holes either in your rubber diaphrams or any other holes that happen to penetrate a top of a wall that you happen to be able to see.
2) Going to sound crazy, but now you want to apply a rubber membrane along the floor(again, think shower liner, or if you're pinching a penny, pond liner, which is thinner and won't work as well). Install it onto the floor with an adhesive and roofing nails, using just enough to keep it from going anywhere. MARK THE FLOOR JOISTS! :) Seal it at all edges with rubberized caulk, running up the wall just enough to ensure you block off the bays in the floor joists.
3) Once you've done that, you can install your floor. Use at least two sheets of 5/8"(19/32" seems to be the popular size). Do not use a single layer of subfloor. Run a
thick bead of construction adhesive
Along the edge of the floor joists and install your first layer of subfloor, using screws every 8-10" on the joists. Once you have installed you're first layer, install the second keeping the following in mind:
1) Always straddle joints
2) The new layer's joints must straddle the lower layer's.
3) the second layers screws must also run into the floor joists, and not just the lower layer of floor.
4) Both layers' long side must run parallel to the joists for strength.
Apply a thick bead of construction adhesive again where each new piece will go, first around the perimeter of the new board then in a zigzag pattern throughought.
What you've done here is you have used the floor as a damper, by utilizing a dampening device in between the floor layers.
If you have anything that breaches the floor, use a rubberized caulk where it penetrates the floor.
Install the finished floor.
You're not done yet.
There are two more things to take into account. First is that a square room is the worst thing in the world for controlling sound waves. If you are dealing with a square room, you should make it not-square :). Do this either by a folding divider, a hanging blanket, anything that will cut a corner off the wall, or break up the shape somewhat.
Next is air duct. Air ducts are terrible for passing sound. You fight this by making the sound change direction. This is what fatigues a sound wave. Build a baffle box to route the air to your room. I can provide a diagram, but to do so, I'd need to know where your duct exits(floor, wall or ceiling).
Finally, you need to design the room to absorb the waves before they get a chance to leave the room. You can do this tons of ways, but here are some ideas.
Think of a movie theater. The fabric panels you see are sound dampeners. You can do the same with acoustical board and burlap or canvas. Rough, loosely woven fabric does a better job than finely woven fabrics. Place some panels along some areas. Use the canvas to dampen sound, but dye it to work with the look of the room.
If you don't feel like getting crafty enough to make the panels, consider
these, instead. Working in a game room, you have some leeway in what works visually and can even use some of the funkier colors. You don't have to do a whole room or even a whole wall for these to be effective. You can use them in a checkerboard fashion along a wall and still get a great reduction in errant waves.
Finally, anything that generates vibrations(bass tubes, jukeboxes, media stand, etc) can be helped with vibration isolation blocks. You can make them yourself with some thick rubber and cork, or you can buy washer/dryer isolation blocks at lowe's or HD. This will help keep those vibrations from making it to your floor.
There's always more you can do, but the price progressively increases in proportion to the reduction :). This should dramatically reduce your problem, though. If it doesn't, there's an aspect of the room that we don't know about.
I know this sounds like a lot, but if you were already considering pulling the floor, then you're not dealing with too much more here :).
Good luck on your project!
Thanks,
Json