above example, in a first-price sealed-bid auction, there is a SBNE with , i.e., each bidder bids 2/3 of his/her signal. PROOF: The proof takes the point-of-view of Xenia. We assume that she knows that Yakov bids , but does not know . We find the best response of Xenia to Yakov's strategy. Suppose Xenia bids . There are two cases: . Then Xenia wins and enjoys a net gain of . . Then Xenia loses and her net gain is 0. All in all, Xenia's expected gain (given her signal X and her bid Z) is: where is the conditional probability-density of Y given X. Since , the conditional probability-density of Y is: when when Substituting this into the above formula gives that the gain of Xenia is: This has a maximum when . But, since we want a symmetric BNE, we also want to have . These two equalities together imply that . The expected auctioneer's revenue is: Note that here, the revenue equivalence principle does NOT hold—the auctioneer's revenue is lower in a first-price auction than in a second-price auction (revenue-equivalence holds only when the values are independent). Relationship to Bertrand competition Common-value auctions are comparable to Bertrand competition. Here, the firms are the bidders and the consumer is the auctioneer. Firms "bid" prices up to but not exceeding the true value of the item. Competition among firms should drive out profit. The number of firms will influence the success or otherwise of the auction process in driving price towards true value. If the number of firms is small, collusion may be possible. See Monopoly, Oligopoly. References Types of auction