Seriously: Have you got one ? It's really a good idea to have one around. There's a pretty high risk because a gameroom is generally full of 25 year old electronics. It's not like they catch fire all the time, but it's still a true risk.
The second risk is that a gameroom can be pretty full of people at parties, and I bet a lot of gamerooms only have 1 exit. Think about what you would do in a room full of smoke and games ! Might be a good idea to add a lit exit sign if you have a really big room.
Now, what type of extinguisher to select ? There are various types and each have their advantages/disadvantages.
I wouldn't go with a powder extinguisher because although it may put out the fire, it will also ruin your cabs or whatever you spray it on. It's VERY hard to remove without damage.
Also the foam extinguisher is not very suitable as we could well be faced with an electrical fire.
That leaves the CO2 extinguisher as the one to choose. Those are suitable to safely use on electrical fires while power is still on (max. 1000V at 1 meter distance).
Of course you should have one or more smoke detectors installed too.
I know, it's because I'm in the fire detection business that it's natural for me to think about these things but I hope you think about your situation.