When one of the pieces is the "Flag" and the other piece is something that can capture the flag, it plays a song to indicate that the game is over. So it has to have more than just logic.
There's a flowchart showing the behaviour in the patent:
There's nothing there that you couldn't do with state machines, so it could very well just be logic. The question is whether it would have been cost-effective to produce a dedicated logic chip given the volume the game shipped in. It may have been cheaper to order a mask-programmed MCU. In any case, we probably can't work it out without a decap.