Revenue in first-price auctions with a buy-out price and risk-averse bidders

We model the "Buy-It-Now or Best Offer" eBay listing option as a first-price auction with a buy-out price and consider seller revenue when bidders are risk-averse. In this model we consider a discrete setting and a continuous setting. We show that a buy-out price in a first-price auction can increase seller revenue in the discrete setting but not in the continuous setting. Furthermore, we prove that in either setting a second-price auction with an optimally chosen buy-out price will never generate more expected revenue than a first-price auction with an optimally chosen buy-out price. These theoretical results support the thesis that a first-price mechanism is superior in expected revenue generation to a second-price mechanism in an online marketplaces where sellers are risk-neutral and buyers are risk-averse and explain the prevalence of "Buy-It-Now or Best Offer" listings on eBay over the traditional eBay auction.