All-Pay Auctions with a Buy-Price Option

This paper studies all-pay auctions for a product in which there is a buy-price option for bidders to guarantee purchases at a seller-specified price. We analyze symmetric pure-strategy equilibria in the first- and second-price all-pay auctions with the buy-price option. Under these equilibria, the buy-price option will affect bidders' bidding behavior: while it has no impact on low-value bidders, it makes middle-value bidders bid more aggressively and high-value bidders less aggressively. Under the optimal buy-price, the first- and second-price all-pay auctions maintain the same expected profit while the buy-price in the second-price is higher than those in the first-price all-pay auction. Compared to the uniform posted-price selling mechanism, all-pay auctions with the buy-price cannot attain more expected profit under the fixed number of consumers. However, if there is an endogenous entry process, they can attract more consumers and then reach more profit in expectation.