Note: The GE engined aircraft has the best range as indicated on the Airbus graphs.
I have read and re-read this thread with some interest, but you know it hasn't really addressed the original question.
UT has made a good case for saying that the GE engine has a better fuel consumption than the RR design, but the question was not does GE do better than RR, but what are the advantages of two shaft engines against three shaft designs.
The debate about relative fuel burn at take-off, climb and approach doesn't really address the question since by far the biggest amount of fuel is consumed in cruise and so far as I am aware there is no simple relationship between TO and cruise sfc performance, it depending amongst other things on the BPR (and cruise thrust matters as well). The only truly valid comparison I have seen mentioned in this debate is that given by UT as quoted above. These are estimates by a single manufacturer, for the same airframe and for the same MZFW and TOW, i.e. the same amount of fuel on board and as such should give a coherent assessment of relative performance on real missions.
Sure, the GE aircraft with new engines goes farther, but it isn't all that much of a difference. The graphs are not all that big, but my reading for range taken from the AI graphs is, with 175t MZFW and 233t TOW:
@ Max structural P/L/MTOW PW4000 - 3700 n.mls, RR - 3700 n.mls, GE 90 - 3750 n.mls
@MTOW/Max tankage PW4000 - 5500 n.mls, RR - 5500 n.mls, GE90 - 5550 n.mls
@ Max tankage/zero P/L PW4000 - 6500 n.mls, RR - 6500 n.mls, GE90 - 6600 n.mls
In other words the GE is about 1% better than either the PW4000 or the RR engines when new, but that is a difference that could be easily lost by deterioration.
Some airlines ask for, and are given, performance guarantees to be met after X years in service, so the retention of performance is also very important. Without, as put earlier, wanting to get into a pissing contest, I have to say that when I was in work the RR engines had a good reputation in this respect - certainly there was less heart searching about giving 'in service' guarantees if the airline had chosen RR engines.
What might one deduce from this?
I would say since the PW engines are 2 shaft and identical to the 3 shaft design in aircraft performance terms one cannot conclude superiority of 2 shafts over 3 so far as fuel consumption is concerned, just that one company has better technology than the other two.
Other issues of cost, maintainability and reliability I leave to others who are better informed