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With only the first constraint in mind, on bid vectorv, the auctioneer shouldnever allocate ifv1< m1andv2< m2, but otherwise, should allocate to thebidder with the higher value ofvi-mi. (We can ignore ties since they have zeroprobability.)It turns out that following this prescription (setting the payment of the winnerto be the threshold bid for winning), we obtain a truthful auction, which is thereforeBIC. To see this, consider first the case whereλ1=λ2.The auction is then aVickrey auction with a reserve price ofm1. Ifλ16=λ2, then the allocation rule isαi(b1, b2) =(1bi-mi>max(0, b-i-m-i)0otherwise.(14.23)If bidder 1 wins he paysm1+ max(0, b2-m2), with a similar formula for bidder 2.Exercise14.9.8.Show that this auction is truthful, i.e., it is a dominantstrategy for each bidder to bid truthfully.Remark14.9.9.Perhaps surprisingly, whenλ16=λ2, the item might be allo-cated to the lower bidder. See also Exercise 126.96.36.199.3. The multi-bidder case.We now consider the general case ofnbid-ders. The auctioneer knows that bidderi’s valueViis drawn from a strictly increas-ing distributionFion [0, h] with densityfi. LetAbe an auction where truthfulbidding (βi(v) =vfor alli) is a Bayes-Nash equilibrium, and suppose that itsallocation rule isα:Rn7→Rn.Recall thatα[v] := (α1[v], . . . , αn[v]), whereαi[v] is the probability15that the item is allocated to bidderion bid16vectorv= (v1, . . . , vn), andai(vi) =E[αi(vi, V-i)].The goal of the auctioneer is to chooseα[·] to maximizeE"Xipi(Vi)#.Fix an allocation ruleα[·] and a specific bidder with valueVthat is drawn fromthe densityf(·). As usual, leta(v),u(v) andp(v) denote his allocation probability,expected utility and expected payment, respectively, given thatV=vand allbidders are bidding truthfully. Using condition (3) from Theorem 14.6.1, we haveE[u(V)] =Z∞0Zv0a(w)dwf(v)dv.Reversing the order of integration, we getE[u(V)] =Z∞0a(w)Z∞wf(v)dv dw=Z∞0a(w)(1-F(w))dw.(14.24)15The randomness here is in the auction itself.16Note that since we are restricting attention to auctions for which truthful bidding is aBayes-Nash equilibrium, we areassumingthatb= (b1, . . . , bn) =v.
14.9. MYERSON’S OPTIMAL AUCTION259Thus, sinceu(v) =va(v)-p(v), we obtainE[p(V)] =Z∞0va(v)f(v)dv-Z∞0a(w)(1-F(w))dw=Z∞0a(v)v-1-F(v)f(v)f(v)dv.(14.25)Definition14.9.10.For agentiwith valuevidrawn from distributionFi, thevirtual valueof agentiisψi(vi) :=vi-1-Fi(vi)fi(vi).In the example of§14.9.2,ψi(vi) =vi-1/λi.Remark14.9.11.Contrast the expected payment in (14.25), i.e., the expecta-tion of the buyer’s allocatedvirtual value, with the expectation of thevalueallocatedto him usinga(·), that is,Z∞0a(v)v f(v)dv.The latter would be the revenue of an auctioneer using allocation rulea(·) in ascenario where the buyer could be charged his full value. The difference betweenthe value and the virtual value captures the auctioneer’s loss of revenue that canbe ascribed to a buyer with valuev, due to the buyer’s value being private.We have proved the following proposition:Lemma14.9.12.The expected payment of agenti