I lived this quandry with a previous startup. I made the mistake of giving my partner an equity position from the start, even though it was my concept and my sweat equity to get the company up and running. I offered him the equity because he had good credentials and seemed to eager to contribute and grow the business. Like the stories above, he didn't follow through and his involvement did not increase as the company grew- he basically sat on the equity.
My two biggest mistakes were:
1) giving him a significant amount of equity without a vesting schedule and/ or specific milestones
2) not buying him out sooner.
I'll echo the comment that in general, my biggest mistakes weren't making decisions that didn't work- it was sticking with those decision too long.
Here are two applicable and somewhat humorous quotes:
"When you sit down to a ham and eggs breakfast, the chicken was a contributor, the pig was committed"
(I was the pig, my partner was the chicken)
and from Willie Nelson:
"Why is divorce so expensive? Because it's worth it!" (it was worth the money to buy out a partner who was a drag on the company and a distracting aggravation)
Good luck to all!