your electrician probably never passed grade school.
standard house hook ups are 200A split phase 240VAC. Since games use 120V you effectively have 400A 120Vac service.
Most games use about 2VA of power with a few exceptions being pole position at 3VA and games after about 1993 being around 3-4VA.
I have a 60A service out to my barn and have it split up in 6 20A lines. I have put 8 games per 20A line and have never had any issues. You could probably run 12 but you might want to invest in a $20 killawatt meter to make sure you aren't going to blow your breakers.
In short, use 20A breakers, 12 gauge solid wire and ideally nema 5-20R receptacles. I used 5-15P's because they are dirt cheap and they are just fine even though its a 20A line. Only downside is, its technically not code but meh, breaker will go before that plug will. Don't put more than 8 games per 20A line and you will be fine.
The only thing you need to be careful with is if you have an electric hot water on demand, those things can draw like 125A and if you loaded the rest of your service up with games you might blow the input breaker.
Well said and great info!
If your panel is completely full. You can add a sub panel. Might get a little costly at that point.