I can only say that you get what you pay for, private vs. public non-withstanding. I don't put it past any private company to put in an unrealistically low bid, and hope for one of three exit strategies (based on what I've seen in my short lifetime):

1) Bill the difference later, resulting in huge overruns. 2) Be out of the picture when the hits the fan. 3) Declare bankruptcy, all stakeholders walk away with their gains