The Coin Unit keeps track of how many players there are. When you start a new game it resets to position 1. Then, each time you press the replay button the Coin Unit takes another step up, up to 4 players. Once the game starts the Coin Unit stays put until the end of the game. In the schematic above imagine the red double ended arrow moving one position to the right for each player added. The arrow represents a wiper on the Coin Unit that determines which of four paths through the Coin Unit the circuit to the Ball Count S.U. (step up) solenoid has to use. So in a three player game for example that arrow will be over the 3 instead of over the 1 as shown and positions 1, 2 and 4 are not used.
The Player Up Unit keeps track of which player is currently playing. It starts the game in position 1 as shown. If it's just a one player game the Player Up Unit isn't used because each time the Outhole relay closes (when the ball drains) the circuit to the Ball Count step up solenoid is completed and the Ball Count advances to the next ball. That's the path in red.
In a multi player game the Player Up Unit takes a step up each time the ball drains. In a 3 player game for example when the 1st player drains, the Player Up Unit steps from position 1 to position 2. When the 2nd player drains, the Player Up Unit steps from position 2 to position 3. Now the Player Up Unit and Coin Unit are both in position 3 so when the 3rd player drains, the Outhole relay fires which completes the circuit to the Ball Count Unit step up solenoid through position 3 in both the Coin Unit and Player Up Unit.
Not shown above is the Player Reset relay which fires when the Ball Count Unit steps up. The Player Reset relay is what causes the Player Unit to reset back to Player 1.
So in your game, only the 1st path through the Coin Unit is working. When it is in positions 2-4 somehow the circuit to the Ball Count Unit never completes so you end up playing Player 4, Ball 1 over and over.