You know the way you'll sometimes see someone with a rag tied over a jar? I tried to make this effect with a jar and a ring (torus) where the cloth falls over the top of the jar (this part works perfectly) and then the ring cinches around the cloth. The jar and ring are both set to collision. When the ring reaches the cloth it seems to cut right through it and distorts it beyond recognition. I'm certainly not getting the nice cord around a cloth effect I'm going for. I've also tried using a force field on the ring with similar outcome.
For starters, I'd say model the thing manually if its not necessary to use the cloth sim. Cloth sim is for interactivity, i.e. if you want the cloth to interact with something later in an animation.
However, if you want to use the cloth sim, try using Cloth Sewing (new since 2.70), here is a general tutorial: https://www.youtube.com/watch?v=ySyQsLD_7GM
In your case, i would use a similar approach as in the video with the cloak, wrap the cloth mesh around the jar lid, connect all open vertices with edges and mark them as seams to pull together.
I would model the string holding it together entirely manually.