be the expectation of the value to a bidder when the signal he receives is and the highest signal among the other bidders, is . We assume that is nondecreasing in and strictly increasing in and that . For each standard auction format , suppose that the auction has a symmetric and increasing equilibrium , which is a mapping from a bidder's observed signal to its bid. Let denote the expected payment by a bidder if he is the winning bidder when he receives a signal but bids as if their signal were , i.e., he bids . Let denote the derivative of with respect to its first argument and the derivative with respect to its second argument, evaluated at . For specific examples, in a first-price sealed-bid auction, labeled , where the high bidder wins and pays the amount of their bid, we have and in a second-price auction, labeled , where the high bidder wins and pays the amount of the second-highest bid, we have Now we may state: Linkage Principle. (Krishna, 2002, Proposition 7.1) Let and be two standard auctions, each having a symmetric and increasing equilibrium such that (i) for all , (ii) Then the expected revenue in is at least as large as the expected revenue in . Proof: The expected payoff of a bidder with signal who bids is . In equilibrium, it is optimal to choose and the resulting first-order conditions imply that which we can rewrite as Letting we conclude that . By hypothesis (i), the second term is positive, and by hypothesis (ii), which implies , it follows that and cannot be of different signs, implying that for all , . Q.E.D. To use this proposition to rank, for example, the second-price and first-price auctions, we need to assume that the bidders signals are affiliated (see Milgrom and Weber, 1982, Appendix on Affiliation, pp. 1118–1121), which implies that is decreasing and that . Note that . Thus, under the assumption of affiliation, . In addition, so the Linkage Principle implies that expected revenue from a second-price auction is at least as great as that from a first-price auction. To use this proposition to show that expected revenue is greater when public information is made available, consider the first-price auction. Let be a random variable denoting the information available to the seller and suppose a symmetric equilibrium strategy that is increasing in