Abstract: Recently, many electronic commerce websites have begun to using "pay-per-bid" auctions to sell items, from mp3 to television. In these auctions, bidders pay a fee for placing each bid. Although the winner's final purchase price is much lower than the market price of the item, the auctioneer can still profit from the bid fees.An analysis of descending pay-per-bid auctions is made in this paper. A general, full-information model is proposed and its equilibrium strategies are described. It can be found that the auctioneer's expected revenue is less than the market price. Then, the ending rules are modified to increase the auctioneer's expected revenue. A model is proposed and its equilibrium strategies are then solved. Further, the equilibrium strategies and the auctioneer's expected revenue are compared between in the ascending and descending pay-per-bid auctions. It is concluded that the probability that the descending pay-per-bid auction ends increases with its process, while the auctioneer's expected revenue in the descending pay-per-bid auction is less than that in the ascending pay-per-bid auction. Finally, the impacts of some controllable factors, i.e. the starting price, the bid fee and the bid decrement, on the expected revenue are analyzed. Utilizing numerical examples, results show that the optimal bid fee is 1 yuan and the optimal bid decrement is 0.01 yuan. Moreover, the auctioneer should set the starting price to be 4.5 percent of the market price to maximize his expected revenue. Analysis of descending pay-per-bid auctions represents a well-motivated avenue for further research and practice.

Key words: pay-per-bid auction, online auction, bidding strategy, electronic commerce