We've had something like this happen to us, where a company won bids over us, usually because of politics (and connections in the local governments). Eventually, a number of contractors started realizing they wanted "quality work", vs. cheap bids fed to them via politics. So then we started winning a lot the bids, even though we came in with somewhat higher numbers. Contractors started realizing that when we give them a number, that's what they'll pay. When going with the "low number" from their political connections, they experience either lots of errors (which greatly increase costs) or they get nickle-and-dimed for "restakes" that the OC says were not included in the contract. So we've had a number of GCs who have learned to trust us, vs. the "other bid", knowing that that "other bid" ends up costing them more, by the time the contract is complete, even though it may seem lower to start with.
We've actually seen several local companies go out-of-business due to this. They consistently deliver sub-par performance, so even when they win contracts by giving low-balled bids, they end up losing in the long-run, because they deliver poor results.