The theoretical results mentioned above are based on the assumption of a fixed and known number of bidders, who incur no cost of entry. When the number of bidders is endogenous (i.e. bidders can choose whether or not to participate) and bidders have some cost of entry, it may be advantageous for the seller to set the reserve price no higher than her valuation in order to encourage efficient levels of entry (Samuelson 1985, Engelbrecht-Wiggans 1987, McAfee and McMillan 1987, Levin and Smith 1994). The theoretical effects of secret reserve prices are also rather mixed. The obvious market- design question is whether the use of a secret reserve price is more beneficial than a public reserve price (minimum bid). Tatsuya Nagareda (2003) models a second-price, sealed-bid auction where the seller can either set a public or a secret reserve price. He finds that no symmetric equilibrium exists in which secret reserve prices increase the expected revenue of the seller. Other researchers, such as Elyakime et al. (1994) analyze an independent-value, first-price
17 auction and conclude that a seller is strictly worse off using a secret reserve price versus a minimum bid. Not all theoretical models predict a disadvantage to secret reserve pricing. Li and Tan (2000) focus on risk-averse bidders rather than risk-neutral bidders. The authors find that with risk-averse bidders, a secret reserve may increase the seller’s revenue in an independent, private- value, first-price auction. On the other hand, in second-price and English auctions, risk preference does not play a role and the seller should be indifferent between a private or public reserve price. The work of Vincent (1995) provides an example where setting a secret reserve price in an English or second-price auction can increase a seller’s revenue in an affiliated-values setting. He argues that since a nonzero minimum bid can cause some bidders to avoid the auction entirely, the attending bidders will have less information than in an auction with a secret reserve price, but no minimum bid. As usual, less information on other bidders’ signals in an affiliated- values auction leads to more cautious equilibrium bidding and hence lower prices. With shill bidding, the informational situation is rather different than in the case of secret reserve prices. For one thing, the bidders in such auctions receive no explicit notice that seller is effectively imposing a reserve. Of course, in an institution in which shill bidding is possible, buyers may expect it to happen. In fact, Izmalkov (2004) shows that in an English auction with asymmetric independent private values, there exists an equilibrium with shill bidding that has an equivalent outcome to that of Myerson’s (1981) optimal mechanism. The intuition for this result is best described by Graham et al. (1990 and 1996), who show that setting a reserve price dynamically, that is after having observed some bidding, can increase the seller’s revenue. The effect is due to the fact that the longer the seller can observe bidding in the auction, the more precise becomes the seller’s information buyers’ values. In the absence of penalties for shilling,
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