(see first-price sealed-bid auction). Independent signals, second-price auction The following example is based on. There are two bidders participating in a second-price sealed-bid auction for an object. Each bidder receives signal ; the signals are independent and have continuous uniform distribution on [0,1]. The valuations are: where are constants ( means private values; means common values). Here, there is a unique SBNE in which each player bids: This result is in contrast to the private-value case, where in SBNE each player truthfully bids her value (see second-price sealed-bid auction). Dependent signals, second-price auction This example is suggested as an explanation to jump bidding in English auctions. Two bidders, Xenia and Yakov, participate in an auction for a single item. The valuations depend on A B and C -- three independent random variables drawn from a continuous uniform distribution on the interval [0,36]: Xenia sees ; Yakov sees ; The common value of the item is . Below we consider several auction formats and find a SBNE in each of them. For simplicity we look for SBNE in which each bidder bids times his/her signal: Xenia bids and Yakov bids . We try to find the value of in each case. In a sealed-bid second-price auction, there is a SBNE with , i.e., each bidder bids exactly his/her signal. PROOF: The proof takes the point-of-view of Xenia. We assume that she knows that Yakov bids , but she 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) is: where is the conditional probability-density of Y given X. By the Fundamental theorem of calculus, the derivative of this expression as a function of Z is just . This is zero when . So, the best response of Xenia is to bid . In a symmetric BNE, Xenia bids . Comparing the latter two expressions implies that . The expected auctioneer's revenue is: In a Japanese auction, the outcome is the same as in the second-price auction, since information is revealed only when one of the bidders exits, but in this case the auction is over. So each bidder exits at his observation. Dependent signals, first-price auction In the