Yes, there can be bid rigging. You can't know if you don't know who's bidding. Believe me, the media love to investigate these things. There can be bid rigging, and you can pay more than you should. It's not hard to police.
By the way, it's not always recognized, but the government has the power to impose contracts under exceptional circumstances, and also to make certain that all prices are audited so it doesn't have to be gouged. Again, though, I don't know what was in the contract or how the contract was written.