The best replacement for bidding is negotiation.

The best alternatives to bidding are rated as following:
★★★★★  negotiation
★★★★★  auction
★★★  competitive proposals
★★  buy now
★★  sniping
★  hotwire
★  buy it now
★  sole source
★  competitive negotiation
★  job posting
★  two step sealed bidding
★  negotiated
  tendering
  biding
  negotiating

Your alternative for bidding is not listed? Suggest!
Speaking about bidding, one probably shouldn't forget about priceline.

Other search terms related to bidding and their ratings are:
★★★★★ priceline
★★★★ ebay
★ hotwire
You have in mind a companion for bidding and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of bidding in all years. Click on graph for more.
bidding
negotiation
auction