With only the first constraint in mind on bid vector

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With only the first constraint in mind, on bid vector v , the auctioneer should never allocate if v 1 < m 1 and v 2 < m 2 , but otherwise, should allocate to the bidder with the higher value of v i - m i . (We can ignore ties since they have zero probability.) It turns out that following this prescription (setting the payment of the winner to be the threshold bid for winning), we obtain a truthful auction, which is therefore BIC. To see this, consider first the case where λ 1 = λ 2 . The auction is then a Vickrey auction with a reserve price of m 1 . If λ 1 6 = λ 2 , then the allocation rule is α i ( b 1 , b 2 ) = ( 1 b i - m i > max(0 , b - i - m - i ) 0 otherwise. (14.23) If bidder 1 wins he pays m 1 + max(0 , b 2 - m 2 ), with a similar formula for bidder 2. Exercise 14.9.8 . Show that this auction is truthful, i.e., it is a dominant strategy for each bidder to bid truthfully. Remark 14.9.9 . Perhaps surprisingly, when λ 1 6 = λ 2 , the item might be allo- cated to the lower bidder. See also Exercise 14.14. 14.9.3. The multi-bidder case. We now consider the general case of n bid- ders. The auctioneer knows that bidder i ’s value V i is drawn from a strictly increas- ing distribution F i on [0 , h ] with density f i . Let A be an auction where truthful bidding ( β i ( v ) = v for all i ) is a Bayes-Nash equilibrium, and suppose that its allocation rule is α : R n 7→ R n . Recall that α [ v ] := ( α 1 [ v ] , . . . , α n [ v ]), where α i [ v ] is the probability 15 that the item is allocated to bidder i on bid 16 vector v = ( v 1 , . . . , v n ), and a i ( v i ) = E [ α i ( v i , V - i )]. The goal of the auctioneer is to choose α [ · ] to maximize E " X i p i ( V i ) # . Fix an allocation rule α [ · ] and a specific bidder with value V that is drawn from the density f ( · ). As usual, let a ( v ), u ( v ) and p ( v ) denote his allocation probability, expected utility and expected payment, respectively, given that V = v and all bidders are bidding truthfully. Using condition (3) from Theorem 14.6.1, we have E [ u ( V )] = Z 0 Z v 0 a ( w ) dwf ( v ) dv. Reversing the order of integration, we get E [ u ( V )] = Z 0 a ( w ) Z w f ( v ) dv dw = Z 0 a ( w )(1 - F ( w )) dw. (14.24) 15 The randomness here is in the auction itself. 16 Note that since we are restricting attention to auctions for which truthful bidding is a Bayes-Nash equilibrium, we are assuming that b = ( b 1 , . . . , b n ) = v .
14.9. MYERSON’S OPTIMAL AUCTION 259 Thus, since u ( v ) = va ( v ) - p ( v ), we obtain E [ p ( V )] = Z 0 va ( v ) f ( v ) dv - Z 0 a ( w )(1 - F ( w )) dw = Z 0 a ( v ) v - 1 - F ( v ) f ( v ) f ( v ) dv. (14.25) Definition 14.9.10 . For agent i with value v i drawn from distribution F i , the virtual value of agent i is ψ i ( v i ) := v i - 1 - F i ( v i ) f i ( v i ) . In the example of § 14.9.2, ψ i ( v i ) = v i - 1 i . Remark 14.9.11 . Contrast the expected payment in (14.25), i.e., the expecta- tion of the buyer’s allocated virtual value , with the expectation of the value allocated to him using a ( · ), that is, Z 0 a ( v ) v f ( v ) dv. The latter would be the revenue of an auctioneer using allocation rule a ( · ) in a scenario where the buyer could be charged his full value. The difference between the value and the virtual value captures the auctioneer’s loss of revenue that can be ascribed to a buyer with value v , due to the buyer’s value being private. We have proved the following proposition: Lemma 14.9.12 . The expected payment of agent i