The general term economists use for a system like this, where sellers compete to offer the lowest price, is a reverse auction ("Reverse" with respect to a traditional auction, in which buyers compete to offer the highest price).

In his book "Gomorrah", Roberto Saviano describers something similar as being used in the Italian fashion industry: the design house convenes its suppliers and says "we need 1,000 purses like this one", and the suppliers try to out-bid each other to make the items fastest, and at the lowest price.

(While Gomorrah is a fictionalized work, Saviano is one of the most respected Italian journalists covering the mafia- and Camorra- ridden areas of southern Italy. I would consider the fact that the clans put a price on his head, requiring him to go into hiding for many years, a strong indication that it's a reliable source.)

It can also be used in government or corporate procurement - I remember in my business classes people talking a lot about its potential, but I can't find any specific examples (that wiki article is way too full of [citation needed]'s for my taste). The idea is to run a procurement contract like an Ebay auction, so all the bidders can see the current best price and make a lower offer. This is different from a "traditional" procurement process, in which the bids are secret, but the contract still goes to the lowest bidder.

While not entirely like your scenario, it was common during the 19th century for the parish or muncipality to auction off orphans in Sweden. The lowest bidder (ie the one who would charge the parish or muncipality the least to care for the child) would win and take the child in. There were occasions where children were taken in as labour and badly abused or maltreated, but also cases of people taking in orphans and adopting them as their own kids.

It was not uncommon for the kids to be auctioned off several times when their hosts tired of them or did not think them suitable for the free labour they wanted anymore.

The system was banned in 1918.