# Auction with one buyer and multiple sellers

In the standard auction model, there are one seller and multiple buyers, the bidders are the buyers.

Consider now an auction with one buyer and multiple sellers, where the bidders are the sellers. Does it fundamentally changes the nature of the auction compared to the standard model? What changes does it make?

Such a setting is called a "procurement auction" or "reverse auction." It does not fundamentally change anything. Instead of buyers with privately known valuations, the auctioneer faces sellers with privately known costs. Hence, bidders underbid each other. An efficient auction is then (typically) one where the bidder with lowest costs wins. An optimal auction is one where the winner has the lowest virtual cost, $$c + \frac{F(c)}{f(c)}$$, the procurement analogue to Myerson's virtual value, $$v-\frac{1-F(v)}{f(v)}$$, where $$F$$ is the cdf of the private type (valuation or cost). The first paper on procurement auctions that comes to my mind is Luton & McAfee, JPubE 1986, but I am sure people were aware of reverse auctions as a "mirror image" of standard forward auctions long before that.

Standard Models for Auction include (mainly) four types:

• First-price sealed-bid auction in which bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying the amount bid.
• Second-price sealed-bid auctions (Vickrey auctions) in which bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying a price equal to the second-highest bid.
• Open ascending-bid auctions (English auctions) in which participants make increasingly higher bids, each stopping bidding when they are not prepared to pay more than the current highest bid. This continues until no participant is prepared to make a higher bid; the highest bidder wins the auction at the final amount bid. Sometimes the lot is only actually sold if the bidding reaches a reserve price set by the seller.
• Open descending-bid auctions (Dutch auctions) in which the price is set by the auctioneer at a level sufficiently high to deter all bidders, and is progressively lowered until a bidder is prepared to buy at the current price, winning the auction.

If you looking at the First-Price or Open-Descending Model as one of the models for the many-buyers/single-seller auction, where the buyers are the bidders, and each bidder has a private valuation for the item independently, the results such auctions will result in the the MAX(buyers_max_value), where buyer_max_value is a list of each bidders maximum valuation of the item.
Where we are considering single-buyers/many-seller auction, where the sellers are the bidders, and each bidder has a private valuation for the item independently, the results to such auctions will result in the the MIN(sellers_min_value)

Note: I am assuming that in this case the each of the bidders valuation is a given. There are likely to be many significant changes as to how the bidders themselves evaluate their valuation for the item.

In the spirit of the recent Nobel prize winners, who noted that an auction can be a learning experience which reveals the true value of a good:
It seems conceivable that, with many sellers, the price could go to zero or even negative. This assumes the sellers learn the true price of their goods during the bidding process, and determine that the goods are worthless or even garbage.

This seems highly unlikely to me. But note that oil prices did briefly go negative a few months ago (because oil storage capacity was maxed out).