The first circuit I would check is the normally open (NO) switch on the 1st-Player 100,000-pt relay. That relay is a latch-trip design and the switch that closes to energize the lamp should have a green-black wire and possibly a yellow wire soldered to the switch tabs.
If that switch is gapped too close, it may be resulting in the lamp coming on when it shouldn't.
If that does not turn out to be the issue, then may have to determine if the 1st player 100,000-pt relay is latching prematurely.