Two problems:
1) There's no guarantee the second lowest bidder has low balled any less egregiously based on their understanding of the project. In fact, there's no statistical evidence to even suggest this is correlated in any way.
2) Often times there are only two qualified bidders. So your suggestion then really becomes to take the highest bidder
A better strategy may be to give the city discretion over which bidder to choose as long as the lowest bids are all within a small range. But for all I know that is actually already the case.