Personally I just use 10k as a starting place, and move up or down accordingly.

But yea, after having a closer look at that DS, I would say 10k is too low.

And I also suggested using a cap to bypass any RFI...

 

How is your Arduino attached to the device? Could be you have a ground loop from case ground

The Arduino is only connected to the device via the optos. I have the Arduino plugged-in to my laptop's USB.

 

I finally managed to figure out what was happening. When I first turn the machine on it goes into "Tilt" mode and for some reason, all the switches get bypassed and there actually is current flowing through all the lines so the optos were reporting accurately. Once the game is started, it goes back to "normal" mode where the switches are in the loop again. Thanks for the help!