I think most people don't realize a microwave oven is just a radio transmitter. The cavity you put your food into is part of a tuned area for this transmitter. Instead of feeding the output of this transmitter into an antenna, the power is contained within the cavity and, thus, warming the food. Note: extremely simplified statements.
Mistuned transmitters can cause a mismatch with their antenna. The same thing happens when there is little to no food inside the microwave cavity. Such issues can cause the magnetron, the microwave's "tube", to overheat and cause damage to itself. However, by not being a solid state device, and by being more of a mechanical device, it can withstand mismatches and overheating for a more extended period of time whereas, a semiconductor device (transistors and integrated circuits), could fail within seconds.
This is why manufacturers warn not to run the microwave without anything in the oven. But you did not do that. For most of the 12 minutes, you had potatoes in there. The quality of this "load" on the microwave element deteriorated over time but it wasn't zero. If you had left it like that for half an hour or more, I could claim there would be a deterioration of the magnetron tubes "quality" but it is not anything I would concern myself with and I don't think it would even be measurable.
Source: me. Electronic engineer but haven't dealt with transmitters in a long time.