Auctions versus private negotiations in buyer-seller networks

Buyer-seller networks where price is determined by an ascending-bid auction are important in many economic examples such as certain real estate markets, radio spectrum sharing, and buyer-supplier networks. However, it may be that some sellers are better off not participating in the auction. We consider what happens if sellers can make a take it or leave it offer to one of their linked buyers before the auction takes place and thus such a seller can choose not to participate in the auction. We give conditions on the graph and buyers valuations under which the buyer and seller will both agree to such a take it or leave it offer. Specifically, the buyer-seller pair will choose private negotiation over the auction if the seller acts as a network bridge with power over the buyer and if there are enough buyers with low valuations so that the seller does not expect to receive a high price in the auction.