This reminds me of the airplane on the conveyor belt, in that the only confusion comes from the question being insufficiently specified.
- Friction of the pulleys
- Mass of the pulleys
- Moment of inertia of the pulleys
- Mass of the rope
- Unit of mass of the weights
- Is there a surface that the weights are resting on?
- What's the local gravity like?
- Others
If we assume the things we're likely supposed to (rope mass, pulley friction, pulley mass and moment of inertia all insignificant, gravity tending down, resting on a surface), it's clear that the lightest weight will rise first. If, on the other hand, we make ridiculous assumptions (weights mass in AMU, in a no-gravity environment, high moment of inertia pulleys), then the "heavy" weight will lift first (because it's easier to lift the weight than to spin the pulleys).
The airplane on the conveyor belt is more interesting in this respect, in that different people have different ideas about how the question should be interpreted.
For this question, I think everyone will agree on the likely expected assumptions. Incidentally, the mass of the rope doesn't affect the answer so long as it's uniform, and the mass of the pulleys doesn't matter so long as the pulleys directly attached to the weights all have the same mass. And once you assume the pulleys are frictionless, their moment of inertia doesn't affect the answer either.
