When will this be Over? The Agony of the Drawn Out Mediation Session

After over a decade as a neutral, occasionally I can see where the parties will go and that settlement will be accomplished fairly early in the process.  Last week, I had the pleasure of being "shadowed" by a student of mediation, who asked me quite directly, "Why is it necessary to do this back and forth in these instances?  Why can't the parties just get to their number, since it's clear at the outset there will be an acceptable number in a close range?"
As in everything worthwhile, my response was that if the parties had to work for the resolution, they would also have some ownership in it.  In my hypothetical dispute, the initial demand was $100,000.  The initial offer was $10,000, but confidentially, the defense counsel told me he'd never go beyond $50,000.00.  The case settled at a mid-point of $45,000.00 some 3 hours later, and both sides felt triumphant.  I am confident, as I assured the "mentee" shadowing my hearing that day, that had I made the offer of $45,000 as the first (but final) offer as against a $100,000.00 demand, the Plaintiff and her lawyer would have demanded $90,000 and would not have felt as good about ultimately accepting $45,000. as a compromise.  In fact, they may well have rejected that offer and let slip away a decent compromise.
Although sophisticated parties and their counsel are often impatient and want to begin the negotiating process and conclude it quickly, shortcuts are dangerous when the mediator is attempting to manage expectations, weave in sensible tidbits from the varying versions of the factual history and continually assess the likelihood of success if the matter does not settle on that day, together with the costs attendant to a failure to reach an agreement.
In what ways do you pace your negotiations for maximization of success?