Royal Caribbean does not have a booking category equivalent to Carnival's Early Saver. Although their official policy says you have 48 hours after making a booking to tell them you found the same cruise and cabin category for a lower price - which will result in an OBC equal to 110% of the price difference - if the cruise price drops before final payment my agent has always been able to get me the new fare.
However, once final payment has been made they won't change your price. They are quite firm about that, even for upper tier loyalty members. However, they might upgrade you to a better cabin, as recently happened to my friend Bill when he learned the price had dropped for his cruise after final payment had been made.
A key thing with Royal Caribbean is that an agent is often more able to get you the new fare as opposed to doing your own booking with RCI online or on the phone with their version of the Carnival PVPs.
It is no secret that Royal Caribbean is my preferred line. They simply 'suit' me best.