Elected officials unfortunately don't have that much incentive to hire the "cheapest" company as the debt will be incurred over the next 100 years while they will be long gone.
They probably hire the company that they feel will give them the least amount of trouble, which is the easiest to navigate or that will do something for them in exchange. It's the "not my money" issue at play.
The reality of most construction projects? You don't need them. You're not going to get better with them and you're not going to get worse without them.
A stretch of road is past its shelf life and scheduled to be repaved. The city does some analysis, and estimates the cost for that mile of road to be 1M^1. Then they put out a Request for Bids for contractors to do the work, lowest bidder wins. Corrupting the auction is how things fail.
The simplest corruption is to leak sealed bids so that you know how much to bid. Remember, this isn't a vickery auction, bidders are paid what they bid and no more. If you want to maximize your profits, all else being equal you want to bid one dollar less than the 2nd best bid. If you were going to bid 800k, someone who informs you the second highest bid is 900k is worth 100k. Sending them a bribe for that tip will still leave you ahead, and the city behind. And if you repeat this, eventually that 2nd bidder is discouraged enough to leave the market, or at least stop participating in Corrupt City IFBs. Which raises the cost per mile even higher.
The other method is to disqualify competitors and proposals. Contracts often give a measure of leeway to officials disqualify bids. Or, to advise a bidder to change their proposal before they get disqualified. Or you can just write the work to be done in such a way that only one bidder could feasibly win.
^1 I really don't know the costs here, just examples.
0: https://en.wikipedia.org/wiki/MacArthur_Maze#2007_I-580_East...