How much would it cost for a large airline to switch engine manufacturer for their planes?
Many (especially larger) airlines have a history and infrastructure for engines from a specific manufacturer and are unlikely to change that, the cost would extend far beyond just the cost of changing the contract. E.g. KLM has a massive investment in maintenance facilities for GE engines, workshops, spare parts warehouses, highly trained specialised technicians. Retooling to RR would be prohibitive for them. 

Of course a smaller operation without maintenance facilities of their own might be more flexible in that and just buy whatever is cheapest at the time, contracting out maintenance to various companies as needed. But even then, it's often more economical to have a single maintenance provider doing all your work for you, which might well mean a provider who's specialised in a single series of engines, for example if you're based at Schiphol there's a good chance you'll be contracting out your maintenance to KLM and their GE specialised shops.

So there are many good reasons for airlines to stick with a single engine supplier (and to a degree with a single aircraft manufacturer), even if the cost of buying those engines is higher than selecting competing engines from another manufacturer.
Say you're going to save 0.1% in your fuel bills for the 2 new aircraft you're buying to increase the fleet of 20 with other engines you have already (and that's a big if, those engines are pretty close between manufacturers for a specific aircraft). You'd need to fly a LOT for it to be worth building an entire duplicate maintenance network just for those few new aircraft.
Of course if it's part of a long term plan to replace the entire fleet with aircraft using that other manufacturer's engines, that's a different story. But mixed engine fleets tend not to be the most economical way to go, even if on one type manufacturer A is marginally more fuel efficient and on type B manufacturer B is marginally more fuel efficient (and that's what you're going to see, marginal differences only).
If you're talking an operation like Delta or United, with hundreds of aircraft of each type, it might be a different story. But then too you're talking about more aircraft with each manufacturer's engines than most other airlines have in total, and they'll have multiple maintenance shops themselves, making it easier to have a facility for each manufacturer (and of course in case of the big US airlines, they tend to have ended up with that in the course of gobbling up other airlines anyway).