Ah, this is what I do! As you can see in my signature line, Construction Services. I am a PM for hire and this is how most of my clients pay me.
I have one client that pays me a flat weekly fee + expenses + a per diem for travel if necessary. he negotiates all contracts then gives me his schedule, I make revisions and then we discuss. When we are done, we have negotiated on the time frame and if I go over the agreed schedule I work for free, however, if I complete the project early then I get a bonus, usually the remaining weeks + $5-10k depending on size of project.
The rest of my clients pay 15% of construction costs + expenses + per diem if necessary. They have me do the bidding with their drawings, I present them with the bid results and they select the subs. Their architect sets the schedule and before I sign a contract I review the schedule to see if it's reasonable, if it is, I sign and commence construction. If it's not, I meet with the client and their architect and discuss the schedule. If I bring the project in early and within 1% of the budget (over or under), I get a 5% bonus. For each percent over 1% over budget, I lose a percent (i.e.: 3% over = 3% bonus) Note: Most of my projects are commercial and in the $250k to $600k range so my bonuses are substantial.
Quality in materials usually isn't a problem because most of my projects have national suppliers and the subs for the most part are just bidding their labor.