Any game room with more than about 5-10 cabinets, especially dedicated ones with lots of extra gee-gaws such as lights, force feedback, huge monitors, mega audio, etc. will probably need at least 2 15A circuits. If you're installing stuff special, go with 3x 20A. My basement gameroom has 3x 20A + 2x 15A (the 15A are both shared with some lights), and I've got the materials to install a 20A 240V circuit in case I ever need it e.g. for a giant air compressor for a deluxe driving game or foreign games w/o transformers, but haven't had the need yet.
For just a couple plain-jane video cabinets or pinballs, a standard household 15A circuit with some lights and other minimal loading will probably be fine.
The Kill-A-Watt people have mentioned can be handy, but be aware that it tends to read improperly on non-linear loads like switch mode power supplies commonly found in modern arcade games (late 80s and beyond), PCs, and monitors (including those without a SMPS as they still have the input rectifier). SMPSs with PFC (power factor correction) may read a little more accurately, but the load is still usually non-linear, which is what really messes the thing up. It's meant for measuring appliances like refrigerators with motors.