Let's explore that theory some. There's not much driving the Ball Index relay:
Space Odyssey Ball Index relay (resized).jpg
If you block the three switches in red with pieces of paper (and check for shorts, bent solder tabs, etc.), do things work better? If so, remove the paper bits one at a time to see which switch allows the relay to fire when it shouldn't.

/Mark

The way I think of it is that a relay has a bunch of switches, each switch has two or three leaves, and each leaf has a contact. To block a switch I usually fold a small piece of paper in half, then put the fold over one switch leaf and contact to prevent it from touching the other leaves and contacts. Blocking an entire relay affects all of its switches and changes too many things at once.

If you block the three switches in red, the Ball Index relay should never fire. If it does, we have more digging to do.

If you press the start button and things work better with all three switches blocked, try again with with one switch unblocked. Repeat that test for all three switches, unblocking just one at a time. Try to identify which of the three switches lets the Ball Index relay fire when it shouldn't.

Stick with the Start button/ball count problem when you have the switches blocked. If you try to play a game with the switches blocked other things will probably not work right.

I think the wire colors should be the same, but here's the color code from Space Odyssey:
Space Odyssey wire color (resized).jpg
Wires usually have one, two or three colors which could be specified on the schematic spelled out (e.g. grey-blu) or abbreviated (e.g. br-g). An optional number at the end indicates multiple uses of the same color. So br-g-4 would be in a different circuit than, and electrically isolated from br-g-3, or br-g.

Ok, so we can eliminate the two lower parts of the circuit. The problem(s) must be in the red box:
Space Odyssey Ball Index relay 2 (resized).jpg
Somehow current is running from the yellow wire on the left through the Extra Ball relay switch to fire the Ball Index relay. You need to figure out how that's happening.

One of the switches on the 10 point, 100 point or 1000 point relays may be stuck closed or shorted somehow. Shut off the power, set your multimeter to the lowest resistance setting and clip the probes to the brown-green and grey-red wires on either side of one of those three switches. I suspect you'll measure just an ohm or so between those two wires which means that they're shorted together. You can use the folded paper trick on those three switches to see if you can get the resistance to be higher, maybe 10-15 ohms or more which would mean the wires are no longer shorted together.

Once you've sorted out and fixed one of the three switches, you need to figure out where the connection is between those switches and the yellow wire on the left. It could be that there's a short to a yellow wire on one of the 10, 100 or 1000 point relays, or it could be that your Bonus Unit is stuck in one of the positions (1 or 2) that lets current pass from the yellow wire to the grey-red wire.

Did you use a meter or just check visually?

I don't think the Ball Index relay should fire until the 10, 100 or 1000 point relay fires. None of those should fire during reset.

Online videos show that the bonus steps down to zero, then back up to 1000 at about the same time the ball is kicked into the shooter lane.

So given that the Bonus Unit steps to the 1000 point (or #1) position at the start of a ball, that provides a path from the yellow wire on the left of the schematic to the grey-red wire on the other side of the Bonus Unit. There should be no electrical path from that grey-red wire to the brown-green wire until you launch the ball and score 10, 100 or 1000 points. I'm pretty sure you have a maladjusted switch or short between the grey-red and brown-green wires.

Does the Ball Index relay relax after the ball is kicked to the shooter lane? If not, maybe you could poke around and see if you can make it relax by slipping paper between switches, etc. Only do this if you're comfortable and know what you're doing poking around a game with the power on. If not, don't try.

Earlier you said that if you use paper to block the switch on the Extra Ball relay that the Ball Index relay doesn't fire, things work better and you can add 2 players. What happens if instead you block all three switches on the 10, 100, & 1000 point relays? Do you get the same behavior? You should.

So if you block the red Tilt relay switch from the schematic in reply #2 things work better? I don't think the Tilt relay should fire nor should that switch close. If the Tilt relay is firing we'll need to sort out why.

There may be a couple of things wrong. Your photo of the Tilt relay shows that you've blocked the normally open switch with a yellow wire on one tab. That's the Tilt relay switch circled in red in the Ball Index circuit below:
Space Odyssey Ball Index relay 3 (resized).jpg
Either that switch isn't opening, or the Tilt relay is firing. Either way that would cause your Ball Index relay to fire.

The other blocked switch in your Ball Index relay photo is harder to identify but it looks like the normally closed switch in the red box that connects the white-black wire to the yellow-red wire. If removing you blocking paper from that switch causes your Tilt relay to fire, have a look at the Spinner Lane Rollover switch in the first red box that connects the yellow and white-black wires. That could be stuck closed or shorted.

Resurrecting this old thread to give a big shoutout to @MarkG!

I recently got my first machine, a Space Odyssey. Game wasn’t resetting when I got it, managed to troubleshoot it and get the machine to do basic one player mode.

The next problem to tackle was the exact problem described in this thread. Using a schematic and the excellent suggestions by MarkG, I was able to find that my problem had a badly misadjusted Spinner Lane Rollover switch. Fixed it and now the game plays well in both single and two player mode.

Still some minor issues to diagnose, but the game is now fun for two people to play.

Thanks to MarkG and everyone who is willing to spend their time and share their expertise with us new guys!