Maybe some pictures and a little more information?
I'm assuming your harnesses are all properly grounded...i.e. someone didn't go in and rip out the ground wire from every harness? And to that end, you are simply inquiring about the green equipment ground wire? Assuming that's true, it needs to terminate at any metallic piece of equipment that has the potential to become energized. To the best of my knowledge, this is the coin door, the speaker, each joystick, the monitor frame, and the power supply is the return path. At the power supply, the connection is made with an eyelet terminal at one of the screws securing it to the vertical board, between nut and power supply frame. Wherever the continuous wire is making connection at each, you want to be sure it's making good metal-to-metal contact...if it's making contact at a painted area of the coin door, use a star washer between eyelet terminal and coin door, and make sure the star washer pierces the paint.
In more simple terms, take a 18 ga. wire (should be more than sufficient), and cut a length that will route cleanly from each of those piece of equipment to a central location at the back of the cabinet. Terminate an eyelet terminal on one end of each wire. Then, use a butt connector to join the opposite end of all the wires. Find a nut and bolt at each piece of equipment to secure each eyelet in direct contact with metal. At the coin door, use a star washer to penetrate the paint and make contact with metal, if needed.
If there's any metal parts I missed, just cut another length of wire for it.
