This happens all the time with public procurements in Sweden. It's not uncommon to see negative bids as well. After you've got the contract you can bill them for service hours etc so it's a win anyway.
Do you think bids would come in negatively for Firm Fixed-Price work. In this case (micro-purchase), you are not allowed to bill again for the same service so the $1 is pretty much where you stop, unless further contract work is pursued which can take months. Genuinely curious.
Sure, that could be a solution. The problem I see is that any future work is more or less guaranteed to go to the winning company (because of system lock-in, knowledge etc). So if you're a big company who can afford the upfront loss you can still cash in in the future. Having the government as a permanent customer is more or less the dream. I'm no expert though.