Procurement Auctions with Best and Final Offers

We study sequential procurement auctions where the sellers are provided with a ''best and final offer'' (BAFO) strategy. This strategy allows each seller i to effectively ''freeze'' their price while remaining active in the auction, and it signals to the buyer, as well as all other sellers, that seller i would reject any price lower than that. This is in contrast to prior work, e.g., on descending auctions, where the options provided to each seller are to either accept a price reduction or reject it and drop out. As a result, the auctions that we consider induce different extensive form games and our goal is to study the subgame perfect equilibria of these games. We focus on settings involving multiple sellers who have full information regarding each other's cost (i.e., the minimum price that they can accept) and a single buyer (the auctioneer) who has no information regarding these costs. Our main result shows that the auctions enhanced with the BAFO strategy can guarantee efficiency in every subgame perfect equilibrium, even if the buyer's valuation function is an arbitrary monotone function. This is in contrast to prior work which required that the buyer's valuation satisfies restrictive properties, like gross substitutes, to achieve efficiency. We then also briefly analyze the seller's cost in the subgame perfect equilibria of these auctions and we show that even if the auctions all return the same outcome, the cost that they induce for the buyer can vary significantly.