It really isn't so simple. RR Trent 3-shaft design gives a lower moving parts count, lower spares requirements and less degradation in performance over engine life. (Fewer variable stators are required because the fan and the IP shafts rotate at optimal speeds)
At the time of the GE-90 vs Trent competition on the early 777's, the Trent was 3 tons lighter and the GE-90 had a year of failed bird strike problems. The P&W engine was heavier than RR and lighter than GE. On the A330 the RR and P&W engines offer significantly better performance than GE.
Clearly the GE-90 was rubbish and after the initial orders it struggled. So GE paid Boeing huge money for the privilege of being exclusive GE-115 supplier on the -200LR and -300ER. The EA GE/P&W hybrid for the A380 is also heavy, but since it woud have been a financial disaster without the Emirates order, they gave Emirates a hell of a deal. The GenX seems much better, but its weight saving has partly been achieved at the expense of changing the blade-off test to favour GE, allowing them to use a lightweight fan shroud. Meanwhile RR had been complacent in its assumed engineering advantage over GE and has had to redesign the IP section of the 787 Trent to get close to EIS consumption targets. And P&W has responded to losing out big time on big engines to advance the GTF. Competition is a wonderful thing!