If you've reached an impasse in a dispute with someone, why bother bringing in yet another person to complicate things? How could someone else, even someone who professes to be "neutral," make a difference in an already bad situation?
Think about it this way: let's go back to the playground, where we all probably learned the most important lessons in our lives. You and your rival are fighting over that yellow dump truck. Push pull push pull. Neither of you lets go, and maybe one of you even gets a little bloodied in the process. And you're right where you started, no one's won, and you've both still got your claws firmly planted on the truck.
Along comes the playground monitor. "What's all the fuss about?" she asks. First you're annoyed that someone's interfering in this battle royale just when you think you're about to overpower your 'mate.
Then you realize you're still stuck in this sandbox and haven't gotten any closer to really taking that truck away from your nemesis. Its almost snack time and you don't want to miss out on cookies and juice just because of this darn dump truck, which isn't even yours, really.
So you take a breath. Just the presence of the playground monitor has forced your brain to pause. Maybe even long enough to think, "I've been at this for so long and haven't gotten anywhere."
That pause is where mediation begins its work.
"Everyone put down the yellow dump truck and take a step backwards, and no one will get hurt," says the monitor.
The reality of your situation sinks in, to both of you. You haven't been able to get anywhere here, maybe the monitor has a point. You just want a little time with the dump truck anyway, right? This is not a forever thing for you.
There's the incentive. Nothing I've done so far has gotten me what I want. Maybe if I step away a bit, I can use the space to develop a real winning strategy, and get my cookies and juice too!
When both parties take that step back, the mediation has begun. Of course if your nemesis grabs the truck and runs, then you have a whole different kind of fight on your hands.
But if you're both motivated to let something new and different happen, like say taking turns (what a concept) or some bartering with that whistle in your pocket, you'll both get time with the dump truck, and everyone will get their snack today too.