The activation of the 2nd player in a 2 player game would be controlled by the 2nd coin relay. Do you have a schematic for the machine? It sounds like power isn't reaching the 2nd coin relay coil when you press the replay button again after the initial reset of the machine. Check the switches on the game relay to make sure they are clean and adjusted properly.

All relays on the relay bank need to be in the reset position after the machine is reset for a 1 player game. When this happens, the game relay switch will be able to send power to the 2nd coin relay for a 2 player game. As soon as the 1st player ball scores a point on the playfield, the game relay will trip.

Referring to your photo of the relay bank, if this was the case after the machine was reset for a 1 player game, one of the ten relays, the game relay, did not reset. This would prevent power from being sent to the 2nd coin relay if the replay button was pressed to set up a 2 player game.

If the game relay is not resetting like the other nine relays when the machine is reset for a new game, then this would be a mechanical issue with the game relay. It might be a weak reset spring or wear on a portion of the reset bar that contacts the game relay reset plate.

The game relay coil should never be constantly energized. Can you tell what switches are closed when the game relay is in the "trip" position? Is the switch closed so power can go to the 2nd coin relay coil? Or is the other side of the switch closed so power can go to the start trip relay coil?

I would confirm there are no shorts at the game relay. You can block the game relay switch using a piece of paper or tape between the two switch contacts.

Game relay switch - black/white wire and jumper wire (might be a black wire or a bare wire)

With this game relay switch blocked, the game relay coil should never energize. Also, you should be able to reset the machine to start a new game, and then press the replay button again to add a second player.

If you block this switch, and the relay coil still energizes, then there would be a short somewhere in the circuit...

There are two circuits that can send power to the game relay coil. And, yes, one of those circuits that include the Tilt Latch Relay, Start Trip Relay and Game Over Relay, all have closed switches in the circuit. This would mean the game relay coil would be always energized at the start of the game. Perhaps there is a three position switch in the circuit or there is a mistake on the schematic???

The other circuit that includes the label "to 36-2" might be a source. This circuit goes to the score relays, which would send power to the game relay coil when a point was scored on the playfield. I would check the following switches on the score relays and match unit:

1 pt score relay switch - yellow/brown wire and green/white wire
10 pt score relay switch - yellow/brown wire and grey/white wire
100 pt score relay switch - yellow/brown wire and black/white wire
10 pt score relay switch - yellow/brown wire and yellow/black wire
100 pt score relay switch - yellow/brown wire and yellow/black wire
Match step up EOS switch - yellow/brown wire and green/yellow wire

Make sure these switches are clean and adjusted properly. Look for shorting wires at the switch stacks and switch stack lugs.

Another related "gateway" switch in this circuit is the outhole index relay switch. Check if there is a short at this switch - shorting wires at the switch stacks and switch stack lugs. You can measure voltage at each switch blade - yellow/brown wire and black/white wire.

The schematic should show the state of the machine when it has reset and the ball is in the shooter lane waiting to be shot into the playfield. Looking at the "logic" of the schematic, if you start at the game relay coil and follow back through the circuit, the first switch after the coil is the game relay switch. This switch is in the closed position. So, this would mean that the switch was armed and ready to send power to the game relay coil. When power travels through the circuit, the electricity passes through the game relay switch and then energizes the coil. As soon as the game relay coil is energized (and the relay would be tripped to the down position), it would move all of the switches to a different position. Thus, the game relay switch would be moved to the open position. When the game relay switch was in the open position, it would block power from getting to the game relay coil.

So, the first question would be why is the game relay coil constantly energized when the energized coil would open the game relay switch that would "block" power to the coil. When the game relay coil is energized, is the game relay switch (black/white wire) open or does it remain closed? If the switch is closed while the coil is energized, then the switch is not adjusted properly and it needs to be adjusted to the open position when the coil is energized. If the switch is open while the coil is energized, then there would be a short in the circuit - examples: a short in the game relay switch stack, wires touching at the switch stack lugs, a stray jumper wire touching the coil lug, etc.

To confirm, the machine will reset the relay bank two times when setting up a 2 player game - once during the initial machine reset, and a 2nd time when pressing the replay button to add a 2nd player?

It's great you found the problem! The confusing part about this issue was how the schematic showed the tilt latch relay switch, with the black/white wire and green wire, in the normally closed (NC) position. So, this switch should be open before the replay button is pressed and after the initial reset of the machine. Still learning something new everyday! Enjoy the machine!