Blue Elephants are patient and polite beings. Therefore they suit perfectly as mediatiators.
You need a mediator whenever you are in a situation to negotiate important deals. When you decide for a certain software that is likely to influence heavily the future landscape of your company, you need to be sure that the offer of the vendor is both honest and reliable.
Getting the best price is certainly the least important goal. If the vendor cannot live happily from the margin he makes from the deal the service will be more than lousy. Bu how can you tell what the true price is? And how can you let your vendor easily now what you are willing to negotiate and where you have no room to move?
A mediator who can moderate your talks but also speak in private with both parties is what will result in what you need. An honest and fair deal that will hold for a long future.
It happens regularialy that a project comes into rough water. When the emotional waves clash high you often mind the necessary distance and objectivity when it comes to review the state of the project. Typically people like to concentrate on blaming past errors. Even worse: when a mistake happened there seems to be an andro-evolutional program that forces people to react on errors with measures to avoid them being repeated. This all leads to emotional stress and sober and souvereign coldbloodedness gives way to emotional stress and arguments. Here you need an external expert advisor who cannot only act as referee in true and reasoned arguments but also show you a way to pragmatically continue your project.