I somehow don't believe the "lowest bidder" thing, otherwise everyone would just be putting out contracts for <= $1. However, IIRC from working with people in the government contracting business, the "punishment" is that you don't very easily get more money if you go over budget. If your contract is $100, you get $100 and if you need another $100 you will spend $400 on lawyers and renegotiation.

As far as I understand, the actual "punishment" is government agencies keep track of how badly particular contractors underbid and will simply add that amount to their bids when considering, so they're going for "lowest bid," but the bid they're considering isn't necessarily the bid the contractor puts forward. So theoretically, that provides an incentive to be more accurate, but it only works if there is sufficient competition in the first place and at least some of the competitors are even capable of being accurate.

I believe when awarding construction contracts, some of the deciders look at past projects, which could discount underbidding. Often, other engineers are deciding on the contracts, so they also have a general idea of how much it should cost.