Who would win a piece down?

In my chess club, we had a discussion who would have a higher chance of winning: the 1800 in a 1200 vs 1800 match with the 1800 having to give up a minor piece (knight or bishop) in the starting position, or ,with the same composition in mind, a 2600 in a 2000 vs 2600 match. It was my believe that the 1800 would have an easier time, while others said that the 2000 would have no chance against a GM even a piece up.
Any thoughts? 

To say that a 2000 would have no chance a piece up against a GM is an understatement. Do note that the Chess Rating System is not linear, that is, the difference between a 1800 and 1200 is not at all same as the difference between a 2000 and a 2600. 

Generally, a 2000 playing a 2600 is stronger compared to the strength of a 1200 playing a 1800. 

Can you say that Boris Gefland (around 2700) would have no chance against Houdini (around 3200)? A slight change in point of view changes the whole meaning of the question itself.

To be honest I do not understand what you are trying to say. Are you saying that the GM has a better chance or the 1800? 
To further explain my train of thought: I think the 1800 has a better chance since a 1200 simply does not yet have a sound understanding of chess and is more likely to be the victim of simple tactics like winning a piece etc... Put this in contrast with the 2000 who does have a sound base understanding and will not simply hang the piece without a reason. 

I clearly mentioned my point of view, though I agree it was not direct. I said,"Generally, a 2000 playing a 2600 is stronger compared to the strength of a 1200 playing a 1800. " Therefore, it is obviously clear that the 1800 would have more chances than the 2600. I also opened up the discussion by saying that a 2000 would have better chances to win than a 1200. 

I cannot see where did you go about not understanding my point.

I can see how he didn't understand what you were trying to say. It was not al all clear.