The fair way to do it is to give everyone numbered tags for their bags. The rule then is that if someone wishes to place their #1 bag in the overhead bin and there isn't room, bags with higher numbers on them are removed to make room. If you can fit bag #1 overhead and bag #2 under the seat then no problem.
Implement this on flights taken regularly by EU bureaucrats so they get the inconvenience of the resulting delays as baggage is reshuffled.
