QUARTERLY JOURNAL OF ECONOMICS

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1 THE QUARTERLY JOURNAL OF ECONOMICS Vol. CXV May 2000 Issue 2 EFFICIENT AUCTIONS* PARTHA DASGUPTA AND ERIC MASKIN We exhibit an efficient auction (an auction that maximizes suplus conditional on all available infomation). Fo pivate values, the Vickey auction (fo one good) o its Goves-Clake extension (fo multiple goods) is efficient. We show that the Vickey and Goves-Clake auctions can be genealized to attain efficiency when thee ae common values, if each buye s infomation can be epesented as a one-dimensional signal. When a buye s infomation is multidimensional, no auction is geneally efficient. Nevetheless, in a boad class of cases, ou auction is constained-efficient in the sense of being efficient subject to incentive constaints. I. INTRODUCTION We study efficient auctions auctions that put goods into the hands of the buyes who value them the most. 1 Ou inteest in them is not puely theoetical but also pactical: a leading ationale fo the widespead pivatization of state-owned assets in * We ae gateful to the National Science Foundation and the Beije Intenational Institute of Ecological Economics, Stockholm, fo eseach suppot. A vesion of this pape was pesented as pat of the second autho s Aow Lectues at Stanfod Univesity, Apil Fo impotant emaks, we thank Pete Eso, Philippe Jehiel, Paul Milgom, Benny Moldovanu, and Philip Reny. Fo helpful suggestions, we thank Pete Camton, Lawence Ausubel, Motty Pey, Eddie Dekel, Paul Klempee, Robet Wilson, Aiel Rubinstein, Geoffey Tate, and audience paticipants in talks at Nothwesten Univesity, the Mayland Auction Confeence, the Univesity of Pittsbugh, New Yok Univesity, Texas A&M Univesity, the Univesity of Noth Caolina, Chapel Hill, the Univesity of Floida, Summe in Tel Aviv 1997, Pinceton Univesity, the 1997 Decentalization Confeence at Pennsylvania State Univesity, and the 1998 Intenational Game Theoy Confeence at the State Univesity of New Yok, Stony Book. Fo detailed comments we thank one of the editos and the efeees of this Jounal. 1. In this espect, we depat fom most of the theoetical liteatue on auctions, which pimaily concentates on evenue-maximization (see fo example, Myeson [1981], Riley and Samuelson [1981], and Milgom and Webe [1982]). Ausubel and Camton [1998b] daw a fomal connection between the two objectives by the Pesident and Fellows of Havad College and the Massachusetts Institute of Technology. The Quately Jounal of Economics, May

2 342 QUARTERLY JOURNAL OF ECONOMICS ecent yeas is to enhance efficiency. 2 Of couse, if, when assets ae pivatized, thee ae a sufficiently lage numbe of potential buyes, the question of which auction is most efficient may not matte vey much, since competition will ende vitually any kind of auction appoximately efficient (see Swinkels [1997]). But, in pactice, the numbe of seious biddes is often seveely limited, 3 and so the choice of auction fom may well be impotant fo efficiency. When the value that a given buye attaches to the good (o goods) being sold is independent of infomation that othe buyes may have (the case of pivate values), thee is a well-known and simple answe to the question of how to achieve efficiency: the Vickey (second-pice) auction does so (this is the auction in which the high bidde wins but pays only the second highest bid, see Vickey [1961]). Futhemoe, one of the attactive featues of the Vickey auction is that it extends via the Goves [1973]-Clake [1971] mechanism to the sale of any numbe of goods. 4 Unfotunately, the Vickey auction is no longe efficient once we leave the pivate-values setting (see Example 3). Yet, fo pactical applications, it is the case of common (o intedependent) values whee one buye s valuation can depend on the pivate infomation of anothe buye that is usually petinent. Suppose, fo example, that seveal wildcattes ae bidding fo the ight to dill fo oil on a given tact of land. If the amount of oil unde the gound is unknown, then as long as at least one bidde has some pivate infomation about this quantity (say, fom pefoming a geological test), we ae aleady in the ealm of common values Fo example, the U. S. Congess explicitly mandated the Fedeal Communications Commission to pomote efficiency in its auctions of fequency bands fo telecommunications. 3. Fo many popeties sold in the FCC spectum auctions, the numbe of biddes submitting ealistic bids was as low as two o thee. 4. Engelbecht-Wiggans [1988] and Kishna and Pey [1997] show that the Vickey-Clake-Goves auction is essentially the unique efficient auction, whee uniqueness means that any othe efficient auction would induce each buye to make the same expected payments (see also Williams [1994]). 5. We shall use the tem common values to efe to any situation of such intedependency; that is, we shall invoke the boad denotation of the tem. The naow denotation o the expession pue common values applies to the case whee all buyes shae the same valuation. Much of the liteatue on common values, including seminal papes by Wilson [1977] and Milgom [1979] and the ecent majo contibution by Pesendofe and Swinkels [1997], is limited to pue common values. That is, these papes assume that all buyes have the same valuation, in which case the issue of efficiency is tivial: allocating the good to any buye is equally efficient. Finally, those (elatively few) papes, such as Milgom and Webe [1982] and Ausubel [1997], that accommodate heteogeneous valua-

3 EFFICIENT AUCTIONS 343 The pincipal contibution of this pape is to show that, unde standad conditions, the Vickey auction can be genealized so as to attain efficiency even when thee ae common values (Popositions 1 and 2). It is also shown that ou auction emains efficient egadless of the numbe of goods being sold 6 and of the natue of those goods, e.g., whethe they ae substitutes o complements (Poposition 5). Howeve, ou genealized Vickey auction will be fully (i.e., fist best) efficient only if each buye s pivate infomation can be summaized by a one-dimensional signal. We show (Poposition 3) that, if buyes signals ae multidimensional, full efficiency is, in geneal, unattainable by any auction (Maskin [1992] establishes a vesion of this poposition; an even moe geneal impossibility esult is developed in Jehiel and Moldovanu [1998]). This impossibility esult suggests, howeve, that the elevant optimality citeion in the multidimensional case is constained efficiency (i.e., efficiency subject to the buyes incentive constaints what Holmstom and Myeson [1983] call incentive efficiency ). We show that, in a boad class of cases, ou genealized Vickey auction is efficient in this sense (Popositions 4 6). We poceed as follows. In Section II we conside the sale of a single good. Maskin [1992] shows that the English auction is often efficient in this setting (povided that signals ae one-dimensional). Howeve, thee is no known geneal extension of the English auction to moe than one good. 7 The Vickey auction, by contast, does extend to any numbe of goods. Thus, in this pape we concentate on Vickey. In subsection II.A we show that thee exists a genealization of the Vickey auction that attains full efficiency in the allocation of a single good, povided that buyes signals ae one-dimensional. In subsection II.B we give conditions unde which this same auction achieves constained efficiency (efficiency subject to incentive compatibility constaints) in the multidimensional case. Then in Section III we extend ou efficient tions in a common values setting almost invaiably suppose that the distibution of signals is symmetic acoss buyes. This implies, in standad mechanisms like the high-bid o second-pice auction, that the winne in equilibium will be the buye with the highest valuation. That is, given symmety, the standad auctions tun out to be efficient, even though they fail to emain so once symmety is elaxed. 6. In ecent wok, Pey and Reny [1998] have obtained a beautiful extension of the Vickey auction fo the case in which all goods ae identical. 7. Ausubel s [1997] elegant extension equies that all goods be identical and that maginal valuations be deceasing. Moeove, it does not geneally attain efficiency except in the case of pivate values.

4 344 QUARTERLY JOURNAL OF ECONOMICS auction to multiple goods. We conclude, in Section IV, with ou view of the most impotant emaining issue. II. SINGLE-GOOD AUCTIONS II.A. One-Dimensional Signals A.1. Fomulation Suppose that thee is a single (indivisible) unit of a good available fo auction. Thee ae n isk-neutal buyes. Buye i obseves a pivate eal-valued signal s i S i [s i,s i ]. Fom an ex ante standpoint, the signals s 1,...,s n can be thought of as andom vaiables. Let F(s 1,...,s n ) be thei joint distibution. Let v i (s 1,...,s n ) be buye i s expected valuation fo the good, conditional on all the signals (s 1,...,s n ). 8 Because we pemit v i ( ) to depend on s j, with j i, we ae allowing fo the possibility of common values (i.e., intedependence of valuations). If buye i is awaded the good and pays pice p, his net payoff is v i (s 1,...,s n ) p conditional on the vecto of signals (s 1,...,s n ). Assume that, fo all i, v i ( ) is continuously diffeentiable in its aguments and that a highe signal value s i coesponds to a highe valuation: v i (1) 0. s i We shall assume that a buye who is not awaded the good (and pays nothing) has zeo utility. Fo now, we will assume that, although signals ae pivate infomation, the functional foms v 1 ( ),...,v n ( ) ae common knowledge (this assumption can be elaxed; see Remak 1 following Poposition 1). We give two examples of this fomulation. Example 1. If (2) v i (s 1,...,s n ) s i, then we ae in the ealm of pivate values. Example 2. Suppose that buye i s tue valuation is given by the andom vaiable y i, which he does not obseve. Assume, 8. In fact, if each buye i has no esidual uncetainty about his valuation conditional on all the signal values, ou analysis extends immediately to the case of isk-avese buyes.

5 EFFICIENT AUCTIONS 345 moeove, that (3) y i z z i i, whee z is a nomal andom vaiable common to all buyes, z i is a nomal andom vaiable idiosyncatic to buye i, and i is a constant (the constant is intoduced to ceate some possible ex ante asymmety acoss buyes). Finally, suppose that buye i obseves signal s i, whee (4) s i y i i, whee i is a nomal andom vaiable; z, all the z i s, and all the i s ae jointly independent; all the z i s ae identically distibuted, and all the i s ae identically distibuted. In this case, (5) v i (s 1,...,s n ) E[ y i 0s 1,...,s n ]. We now tun to auctions, which ae selling pocedues in which the good is awaded to (at most) one buye, and tansfes ae made between the buyes and the selle, all on the basis of the buyes bidding behavio. Familia examples include the high-bid auction, in which buyes submit sealed bids, the winne is the high bidde (ties may be boken by some stochastic device such as flipping a coin), he pays his bid, and all loses pay nothing. The second-pice (o Vickey) auction has the same ules as the high-bid, except that the winne, instead of paying his own bid pays only the second-highest bid. Finally, in the open o English auction, buyes call out bids publicly, with the stipulation that each successive bid should be highe than its immediate pedecesso. The winne is the last buye to bid, and he pays his bid. We seek auctions fo which, in Bayesian equilibium, the good is allocated to the buye who, conditional on all available infomation, values it the most. We call an auction efficient if, fo all signal values (s 1,...,s n ), the winne in equilibium is buye i such that v i (s 1,...,s n ) v j (s 1,...,s n ) fo all j. It is eadily seen that, even with pivate values, the high-bid auction is not, in geneal, efficient. Actually, if the distibution F is affiliated (see Milgom and Webe [1982]) and symmetic, then, with pivate values, thee exists a symmetic equilibium in which all buyes use the same bidding function b(s i ), which is inceasing in s i. Hence, the winne is the buye with the highest signal, which, again unde standad assumptions, implies that he

6 346 QUARTERLY JOURNAL OF ECONOMICS is the buye with the highest valuation. Howeve, this conclusion ests heavily on symmety. 9 To see this, conside a simple twobuye example with pivate values in which s 1 is dawn fom a continuous distibution on [0,1] wheeas s 2 is dawn (independently) fom a continuous distibution on [0,10]. It is easy to see that the equilibium bid functions (b 1 ( ),b 2 ( )) in the high-bid auction satisfy b 1 (1) b 2 (10), whee b 2 ( ) is stictly inceasing at s But these popeties imply that buye 2 with valuation slightly less than 10 will bid stictly less than b 1 (1) and so will lose to buye 1 with valuation 1. The equilibium is thus inefficient. The second-pice auction is efficient in the case of pivate values. This is because, in that case, it is a dominant stategy fo a buye to bid his valuation, and so the winne will be the buye with the highest valuation. Howeve, once we dop the pivate values assumption, the efficiency of the second-pice auction beaks down. To see this, conside the following example. Example 3. Suppose that thee ae thee buyes, whose valuations ae v 1 (s 1,s 2,s 3 ) s 1 1 2s 2 1 4s 3 v 2 (s 1,s 2,s 3 ) s 2 1 4s 1 1 2s 3 v 3 (s 1,s 2,s 3 ) s 3. Assume that s 1 s 2 1. Note that if s 3 is slightly less than 1, then v 1 is the biggest valuation, but if s 3 is slightly geate than 1, v 2 is biggest. Thus, in a neighbohood of (s 1,s 2,s 3 ) (1,1,1), efficient allocation of the good between buyes 1 and 2 (it is not efficient fo buye 3 to be allocated the good) depends on the value of s 3. But in the second-pice auction, a buye s bid can depend only on his own signal (the othes signals ae pivate infomation). Hence, when s 1 s 2 1, which of buyes 1 and 2 wins cannot depend on 9. Back and Zende [1993] and Ausubel and Camton [1998a], show that, even with symmety, the natual extension of the high-bid auction to the case of multiple goods fails to be efficient. 10. To see that b 1 (1) b 2 (10), note that if instead b 2 (10) b 1 (1), buye 2 with valuation 10 could educe his bid slightly fom b 2 (10) and still win with pobability 1, contadicting the assumption that b 2 (10) is an equilibium bid. If b 2 ( ) is not stictly inceasing at s 2 10, then thee is an inteval of signal values fo buye 2 fo all of which he bids b 2 (10), that is, he bids b 2 (10) with positive ex ante pobability. But then buye 1 is stictly bette off bidding slightly moe than b 1 (1) than b 1 (1) itself, since with positive pobability he only ties as winne with b 1 (1), wheeas he wins with pobability 1 with a bid moe than b 1 (1) 1.

7 EFFICIENT AUCTIONS 347 whethe s 3 is geate o less than 1, and so the second-pice auction is inefficient. One staightfowad way to constuct an efficient auction fo a setting like that of Example 3 would be to invoke the methods of the mechanism-design liteatue and conside diect evelation mechanisms in which each buye i epots a signal value ŝ i, the good is awaded to the buye i fo whom v i (ŝ 1,...,ŝ n ) max j i v j (ŝ 1,...,ŝ n ), and in equilibium, ŝ i equals the tue value s i. A seious objection to such an appoach, howeve, is that it would, in effect, equie the mechanism designe (o auctionee) to know the physical signal spaces S 1, S 2, and S 3 and the functional foms of the valuation functions v 1 ( ), v 2 ( ), and v 3 ( ), a stong assumption (indeed, in ou view it emains a stong assumption even to suppose that the buyes themselves know all this infomation; see Remak 2 following Poposition 1). Instead, we will seek auction ules that ae independent of the details such as functional foms o distibutions of signals of any paticula application and that wok well (i.e., attain efficiency o constained efficiency) in a boad ange of cicumstances. 11 We will show, in fact, that the Vickey auction can be extended (in a detail-fee way) to ensue efficiency when, as in Example 3, thee ae common values. To do this, we equie, in addition to (1), the following condition on valuations: 12 (6) fo all i and j i, v i (s s 1,...,s n ) v j (s i s 1,...,s n ) i at any point whee v i (s 1,...,s n ) v j (s 1,...,s n ) max v k (s 1,...,s n ). k Condition (6) says that (if buyes i and j have equal and maximal valuations) buye i s signal must have a geate maginal effect on his own valuation than on that of buye j. Notice that this condition is satisfied tivially in the case of pivate values (in which buyes valuation functions satisfy (2)), since the ight-hand 11. The insistence that an auction institution be detail-fee has been called the Wilson Doctine afte R. Wilson. 12. Fomula (6) is a single-cossing condition in the sense of Milees [1971] o Spence [1973]. We ae indebted to P. Milgom, who uged us to adopt essentially this condition in place of an ealie, moe stingent equiement. Gesik [1993] intoduced a stonge vesion of this condition in his study of tading mechanisms with common values.

8 348 QUARTERLY JOURNAL OF ECONOMICS side of the inequality in (6) is then zeo. Moeove, it holds fo Example 2. In that model, s i is a moe infomative signal about y i than about y j, j i, since s i conveys infomation about both the idiosyncatic and common components (z i and z) ofy i but only about the common component z of y j. Hence, a small change in s i will affect the expected value of y i moe than that of y j. Condition (6) is in fact necessay if any auction, let alone ou genealized Vickey auction, is to be efficient. 13 To see this, conside the following example. Example 4. Conside two wildcattes who ae competing fo the ight to dill fo oil on a given tact of land. The wildcattes costs of dilling diffe. Wildcatte 1 has a fixed cost of 1 and a maginal cost of 2 (pe unit of oil extacted). Wildcatte 2 s fixed cost is 2 and maginal cost is 1. Oil can be sold at a pice of 4. Wildcatte 1 pefoms a (pivate) test and discoves that the expected size of the oil eseve is s 1 units. Wildcatte 2 s pivate infomation s 2 does not affect eithe dille s payoff. We have and Notice that v 1 (s 1,s 2 ) (4 2)s 1 1 2s 1 1 v 2 (s 1,s 2 ) (4 1)s 1 2 3s 1 2. v 1 v 2, s 1 s 1 and so (6) is violated. Moeove, we claim that thee is no way to induce wildcatte 1 to eveal his infomation while maintaining efficiency (assuming that, even ex post, s 1 cannot be measued diectly and that nobody but the winning wildcatte can monito how much oil thee tuns out to be). Efficiency dictates that wildcatte 1 get the dilling ights if 1 2 s 1 1 and that wildcatte 2 get the dilling ights if s 1 1 (if s 1 1 2, it is inefficient to dill at all; and because of the fixed costs, it would always be inefficient to give both wildcattes dilling ights). 13. Moe pecisely, if (6) fails to hold, then we can find a joint distibution F(s 1,...,s n ) with espect to which thee is no auction that is efficient. Signals in this joint distibution will be independent; othewise we could use the methods of Céme and McLean [1988] to constuct a fully efficient auction egadless of whethe (6) holds.

9 Suppose that wildcatte 1 is given a ewad R(ŝ 1 ) if he claims that thee ae ŝ 1 units of oil. Then if s 1 1 s 1 1 2, incentive compatability and efficiency demand that (7) R (s 1 ) 2s 1 1 R(s 1 ) and (8) 2s 1 1 R (s 1 ) R(s 1 ). Subtacting (7) fom (8), we obtain EFFICIENT AUCTIONS 349 2(s 1 s 1 ) 0, a contadiction. Hence efficiency is impossible. It is easy to see what is going wong hee. As s 1 ises, the dilling ights become moe valuable to wildcatte 1. But, fom the standpoint of efficiency, they ae inceasingly likely to be awaded to 2, thanks to the violation of (6). It is this conflict between 1 s pesonal objective and oveall efficiency that ceates the incentive poblem. Notice that the thee buyes valuations in Example 3 satisfy condition (6). We saw, howeve, that the Vickey auction fails to be efficient in that example because buyes 1 and 2 cannot embody infomation about buye 3 s valuation in thei bids. This suggests that genealizing the Vickey auction to allow buyes to make contingent bids bids that depend on othe buyes valuations may ovecome this poblem. Fo simplicity of exposition, we fist conside the case of two buyes. II.A.2. Auctions with Two Buyes Instead of a single bid, we will have each buye i epot a bid function, bˆ i : Vˆ j =, whee j i and Vˆ j is an inteval [0,v j ]in. Fo each v j Vˆ j we can intepet bˆ i(v j ) as buye i s bid if the othe buye s valuation tuns out to be v j. Given the bid functions (bˆ 1( ),bˆ 2( )), let us look fo a fixed point, i.e., a pai (v 1,v 2 ) such that (9) (v 1,v 2 ) (bˆ 1(v 2 ),bˆ 2(v 1 )).

10 350 QUARTERLY JOURNAL OF ECONOMICS (We shall deal with the issue of nonexistent o multiple fixed points below.) Then we will suppose that (10) buye i is the winne bˆ i(v j ) bˆ j(v i ) (beak ties by flipping a coin). To see that this allocation ule is the ight one, conside what happens when buyes bid tuthfully. In the standad Vickey auction with pivate values, bidding tuthfully means bidding one s tue valuation. In ou setting, a buye does not actually know his valuation because he does not know the othe buye s signal value. So, bidding tuthfully means making a bid contingent on the othe buye s valuation so that, whateve that othe signal value (and hence othe valuation) tuns out to be, his coesponding bid will equal his own valuation. That is, if buye 1 s signal value is s 1, the tuthful bid function is b 1 ( ) such that (11a) b 1 (v 2 (s 1,s 2 )) v 1 (s 1,s 2 ) fo all s 2. Similaly, (11b) b 2 (v 1 (s 1,s 2 )) v 2 (s 1,s 2 ) fo all s 1 (note that b 1 ( ) is well-defined, i.e., if v 2 (s 1,s 2 ) v 2, thee cannot exist s 2 s 2 such that v 2 (s 1,s 2 ) v 2 because v 2 ( ) satisfies (1); similaly, b 2 ( ) is well-defined). Obseve that (12) (v 1,v 2 ) (v 1 (s 1,s 2 ),v 2 (s 1,s 2 )) is a fixed point of the mapping (13) (v 1,v 2 ) p (b 1 (v 2 ),b 2 (v 1 )), fo (v 1,v 2 ) V 1 V 2, whee (14) V 1 5v 1 0thee exists s 1 such that v 1 (s 1,s 2 ) v 1 with stict inequality only if s 1 s 1 6 and V 2 5v 2 0thee exists s 2 such that v 2 (s 1,s 2 ) v 2 with stict inequality only if s 2 s 2 6 (note that V 1 consists of all possible valuations v 1 that buye 1 could have that ae consistent with some signal value s 1 ; fo completeness, it also includes valuations that ae smalle than

11 EFFICIENT AUCTIONS 351 those consistent with any signal value. V 2 is defined symmetically). This means that, if buyes bid tuthfully, ou allocation ule (9) (10) ensues that buye 1 wins if and only if v 1 (s 1,s 2 ) v 2 (s 1,s 2 ), which is pecisely what is entailed by efficiency. We conclude that, if buyes make tuthful epots, the allocation ule (9) (10) is efficient. Thee is, howeve, one technical caveat to this conclusion: conditions (1) and (6) ae not stong enough to ule out othe fixed points of (13) besides (12). 14 We shall confont the issue of multiple fixed points in detail below (see Poposition 2). Fo now, let us assume that the inequality in (6) holds at all points (s 1,s 2 ) and that its ight-hand side is nonnegative. Then (12) is indeed the unique fixed point of (13). To see this, note fom (11a) and (11b) that db 2 (v dv 1 (s 1,s 2 )) v 1 (s 1 s 1,s 2 ) v 2 (s 1 s 1,s 2 ), fo all (s 1,s 2 ), 1 and so, fom (1) and the stonge vesion of (6), we obtain 0 db 2 dv 1 (v 1 ) 0 1 fo all v 1 (15) and 0 db 1 dv 2 (v 2 ) 0 1 fo all v 2. The contaction mapping popety (15) ensues that any fixed point is unique. We have shown that the outcome is efficient if buyes bid tuthfully, but it emains to establish that thee exists a payment scheme that induces tuthful bidding. The way that the Vickey auction induces tuthfulness in the pivate-values case is to make a winning buye s payment equal to the lowest bid that he could have made fo which he would still have won the auction. So, fo example, if buye 1 bids 5 and buye 2 bids 3, buye 1 should win but pay 3, since that is the lowest bid (ignoing ties) that would have won him the good. Let us ty to adhee to this pinciple as 14. Suppose, fo example, that v 1 (s 1,s 2 ) s 1 2 s 1 s 2 s 2 2 s 1 2s 2 24 and v 2 (s 1,s 2 ) s 2 2 s 1 s 2 s 1 2 9s Then if (s 1,s 2 ) (2,3), one fixed point is the pai of tue valuations (v 1 (2,3),v 2 (2,3)) (21,6). Howeve, fo these signal values, (v 1 (2,4),v 2 (1,3)) (14,15) also constitutes a fixed point, because v i (2,4) v i (1,3), i 1,2, and so v 2 (1,3) b 2 (v 1 (1,3)), and v 1 (2,4) b 1 (v 2 (2,4)).

12 352 QUARTERLY JOURNAL OF ECONOMICS closely as possible. In the two-buye case this means that, if buye 1 is the winne (i.e., bˆ 1(v 2 ) bˆ 2(v 1 ), whee (v 1,v 2 ) (bˆ 1(v 2 ), bˆ 2(v 1 )), then he should pay (16) bˆ 2(v* 1 ) whee 15 (17) v* 1 bˆ 2(v* 1 ). This is because if buye 1 wee esticted to constant bids, v* 1 would be the lowest such bid fo which, unde ou allocation ule, buye 1 would still win the auction, given buye 2 s epoted bid schedule bˆ 2( ). Note that if 1 s valuation actually wee v* 1, his net payoff fom winning would be zeo. To see that, povided that (1) and the stonge vesion of (6) hold, buye 1 has an incentive to bid tuthfully in equilibium unde payment ules (16) and (17), suppose that buye 2 is tuthful, i.e., he sets bˆ 2( ) b 2 ( ), whee b 2 ( ) satisfies (11b). Then, if buye 1 wins, his payoff is (18) v 1 (s 1,s 2 ) b 2 (v* 1 ), whee 16 (19) v* 1 b 2 (v* 1 ). It suffices to show that if buye 1 sets bˆ 1( ) b 1 ( ) satisfying (11), then he wins if and only if (18) is positive. (This is because buye 1 s payoff if he wins (i.e., (18)) is independent of how much he bids, and so the best he can do is to ensue that he wins pecisely in those cases in which this payoff is positive.) Now fom (15), (18) is positive if and only if, fo any v 1, (20) v 1 (s 1,s 2 ) v* 1 db 2 (v dv 1 )(v 1 (s 1,s 2 ) v* 1 ). 1 Fom the intemediate value theoem, thee exists a value of v 1 such that b 2 (v 1 (s 1,s 2 )) b 2 (v* 1 ) db 2 dv 1 (v 1 )(v 1 (s 1,s 2 ) v* 1 ). 15. Because Vˆ 1 [0,v 1 ] and v 1 bˆ 2(v 1 ), thee exists (at least) one point v* 1 satisfying (16) and (17). We will deal with the issue of multiple solutions in Poposition Because (15) holds and v 1 b 2 (v 1 ), thee exists a unique point v* 1 satisfying (17).

13 EFFICIENT AUCTIONS 353 Hence, (20) holds if and only if 17 (21) v 1 (s 1,s 2 ) v* 1 b 2 (v 1 (s 1,s 2 )) b 2 (v* 1 ). Now, fom (11b), b 2 (v 1 (s 1,s 2 )) v 2 (s 1,s 2 ); and fom (19), v* 1 b 2 (v* 1 ). Hence (21) holds if and only if (22) v 1 (s 1,s 2 ) v 2 (s 1,s 2 ). But, when he is tuthful, buye 1 wins if and only if (22) holds. Hence, if buye 1 bids tuthfully, (18) is indeed positive if and only if buye 1 wins. To summaize, we have shown PROPOSITION 1. Conside the two-buye auction in which, fo i 1,2, (i) buye i epots Vˆ j [0,v j ](j i) and a contingent bid function bˆ i : Vˆ j =, with the stipulation that (23) 0 dbˆ i dv j0 1; (ii) a fixed point (v 1,v 2 ) is taken accoding to (9) (the estiction (23) ensues that thee can be at most one fixed point; if thee is no fixed point at all, the good is not allocated); (iii) the winne is detemined accoding to (10); (iv) if buye 1 is the winne, he makes a payment accoding to (16) (17) (buye 2 s payment when he is the winne is symmetic). If (1) holds, the inequality of (6) is satisfied at all points (s 1,s 2 ), and the ight-hand side of this inequality is nonnegative, then this auction is efficient. That is, it is an equilibium fo each buye i to bid tuthfully, i.e., to set Vˆ j V j and bˆ i( ) b i ( ), whee V j satisfies (14) and b i ( ) satisfies (11a) o (11b). Moeove, if both buyes do so, the auction esults in an efficient outcome If (21) holds, then, fom the intemediate value theoem, (20) holds fo some value of v 1. Hence, fom (15), we must have v 1 (s 1,s 2 ) v* 1 0. But then (20) holds fo all values of v Note that tuthful bidding is not a dominant stategy in the auction of Poposition 1 (except in the case of pivate values). Howeve, it constitutes a Bayesian equilibium that is obust in the sense that, given v 1 ( ) and v 2 ( ), we can change the distibution of signals F abitaily without affecting equilibium behavio. (Indeed, notice that none of ou analysis has efeed to F at all.) Thus, equilibium stategies ae invaiant to changes in the distibution of signals. We have not uled out the possibility of untuthful equilibia in Poposition 1. Howeve, we ae confident that, by using techniques fom the Bayesian

14 354 QUARTERLY JOURNAL OF ECONOMICS Remak 1. It may seem vey demanding to insist that a buye make his bid a function of the othe buye s valuation. In this two-buye case, howeve, thee is only a single point of each buye i s schedule bˆ i( ) that has to be coect (i.e., tuthful) in ode to ensue an efficient outcome, 19 namely, the point whee bˆ i( ) intesects the 45 degee line: (24) v* j bˆ i(v* j ). That is, when s i is buye i s signal value, all that we equie of buye i is that he epot bˆ i( ) so that, if (24) holds, then thee exists s j such that (25) v* j v j (s i,s j ) v i (s i,s j ). It can eadily be checked that, given that bˆ 2( ) satisfies (23) (25), buye 1 will win if and only if his payoff fom winning is positive, povided that he chooses bˆ 1( ) so that it too satisfies (23) (25). Hence, this behavio constitutes an equilibium. Admittedly, even calculating the v* j that satisfies (25) equies buye i to know something about buye j s valuation function v j ( ), and, fo an efficient equilibium to occu, this knowledge must be common to the two buyes. Howeve, such common knowledge is not necessay fo the auction and buyes to pefom easonably well. Indeed, to take the opposite exteme, suppose that buye 1 knew nothing about the natue of v 2 ( ). He could, nevetheless, make an uncontingent bid b 1 ( ) b 1, fo some b 1. With such a bid he would win if and only if b 1 bˆ 2(b 1 ), in which case he would pay bˆ 2(v* 1 ) v* 1 b 1. In othe wods, by submitting a constant function, he can induce an outcome and payment vey much like those in the odinay second-pice auction. And this places a lowe bound on how badly he can do in ou auction. But if he has even a vague idea of how his and buye 2 s valuations ae intedependent, buye 1 should be able to do stictly bette than this by submitting a bid function with nonzeo slope. In any case, because the fixed point is continuous in his epot, he cannot go too badly wong by doing so. implementation liteatue (see Palfey [1993] fo a suvey), one could modify the auction to eliminate such equilibia (albeit at the cost of a moe complex set of ules). 19. Indeed, if n 2, the odinay Vickey auction is efficient (see Maskin [1992]). Howeve, as Example 3 shows, its efficiency fails fo n 3; wheeas ou genealized Vickey auction emains efficient fo that case.

15 EFFICIENT AUCTIONS 355 In this sense, having buyes epot contingent bids should be viewed as giving them an oppotunity to expess thei intedependencies an oppotunity that they can exploit to any degee that they wish o thei knowledge pemits athe than as imposing an oneous equiement on them. Remak 2. As noted in the pevious emak, some degee of common knowledge about valuation functions is needed to ensue that playes can calculate equilibium (this is tue not only of ou auction, but of any othe auction high-bid, second-pice, etc. as well). One may inquie, theefoe, why we do not go all the way and have each buye i epot a pai of valuation functions (vˆ 1( ), vˆ 2( )) and then (i) use a diect evelation mechanism (in which each buye epots his signal value and these ae then plugged into the epoted valuation functions) if the two buyes epots agee, and (ii) punish buyes in some way if thei epots disagee, athe than esoting to the indiect device of having them make contingent bids. Thee is a difficulty, howeve, with having buye 1 epot vˆ 2( ), namely, he may not even know what buye 2 s physical signal space is (we have modeled S 2 as an inteval of numbes, but, to buye 2, the signal s 2 pesumably coesponds to something physical). To complicate mattes futhe, s 2 may in fact be only a sufficient statistic fo a vaiety of infomation paametes that buye 2 eceives. Notice that thee is no contadiction in supposing that buye 1 does not know v 2 ( ) but does know v* 2 satisfying (25); the latte equies less knowledge. 20 Indeed, that it is common knowledge that, fo i 1,2, buye i can calculate v* j can be thought of as the weakest hypothesis that ensues efficiency in equilibium. In that sense, the efficient auction we ae poposing is the simplest possible one. 21 II.A.3. Auctions with Moe than Two Buyes Ou two-buye auction can eadily be extended to the case of thee o moe buyes. Thee ae two mino complications that aise in this case. Fist, it may no longe suffice fo buyes to submit single-valued bid functions. In the two-buye case, thee is 20. Imagine, fo example, that fom pevious expeience with simila goods buye 1 has leaned that, when buye 2 s valuation is less than v* 2 his own valuation v 1 is geate than v 2, wheeas when v 2 v* 2, v 1 v 2. Notice that this infomation entails knowledge neithe of the functional fom v 2 ( ) no of buye 2 s signal space. 21. It also has the desiable popety that equilibium stategies emain in equilibium even ex post when buyes signal values have become common knowledge.

16 356 QUARTERLY JOURNAL OF ECONOMICS a unique tuthful bid b 1 (v 2 ) by buye 1 fo each possible buye 2 valuation v 2 (because, given s 1, thee is a unique s 2 such that v 2 (s 1,s 2 ) v 2 ). In the case of thee buyes, howeve, thee could exist, fo given s 1, two diffeent pais (s 2,s 3 ) and (s 2,s 3 ) such that v 2 (s 1,s 2,s 3 ) v 2 (s 1,s 2,s 3 ) v 2 and v 3 (s 1,s 2,s 3 ) v 3 (s 1,s 2,s 3 ) v 3. Then, a unique tuthful bid b 1 (v 2,v 3 ) would not be well-defined if v 1 (s 1,s 2,s 3 ) v 1 (s 1,s 2,s 3 ). Accodingly, we shall have buyes epot bid coespondences. Second, fo n 3, conditions to ule out the possibility of multiple fixed points become too estictive to be palatable. With two buyes, the stong vesion of condition (6), which is still faily mild, suffices. But this condition is no longe sufficient with thee o moe buyes, even if all buyes bid tuthfully (with tuthful bidding one fixed point will be the tue valuations, but thee could be othes). To deal with this poblem, we will intoduce a potential second stage of the auction in which buyes, in effect, choose among the diffeent fixed points that may have aisen in the fist stage. To simplify mattes, let us assume that, fo all i, s i and that 22 (26) fo all s i S i thee exists s i S i such that v i (s i,s i ) max v j (s i,s i ). j i That is, egadless of the othe buyes signal values, buye i has a signal value that gives him the highest valuation. (This assumption is not equied, but helps keep complications to a minimum.) Conside the following auction defined in six steps (steps (a) (d) ae the heat of the auction; steps (e) and (f) seve only to deal with multiple fixed points and can be ignoed by eades uninteested in these technicalities): (a) each buye i (i 1,...,n) submits a bid coespondence bˆ i : Vˆ i ==, whee Vˆ i n 1 ; (b) a fixed point (v 1,...,v n ) is calculated so that (27) v i bˆ i(v i ) fo all i (if thee is no fixed point, the good is not allocated, and no buye makes a payment; if thee ae multiple fixed points, go to (e) below); (c) if v i max j i v j, the good is awaded to buye i; 22. The notation s i denotes a vecto of signals fo all buyes othe than i.

17 EFFICIENT AUCTIONS 357 (d) if buye i is the winne, he makes a payment (28) max v* j, j i whee (v* 1,...,v* n ) is a vecto such that and (29) v* i max j i v* j (30) v* k bˆ k(v* k ) fo all k i (if thee is no vecto (v* 1,...,v* n ) satisfying (29) and (30), the good is not awaded, and no buye makes a payment; if thee ae multiple such vectos, go to (f)); (e) if thee ae multiple vectos (v 11,...,v n1 ),...,(v 1 J,...,v n J ) satisfying (27), then the submitted bid functions (bˆ 1( ),...,bˆ n( )) ae made public, and each buye i chooses v i 5v i 1,...,v i J 6; if thee exists j 51,...,J6 such that v i v i j fo all i, then the auction etuns to step (c) using (v 1 j,...,v n j )to detemine the winne; othewise the good is not awaded and no buye makes a payment; ( f ) if thee ae multiple vectos (v* 11,...,v* n1 ),..., (v* 1 K,...,v* n K ) satisfying (29) and (30), then the schedules (bˆ 1( ),...,bˆ n( )) ae made public, and each buye j i chooses v* j 5v* j 1,...,v* j K 6; if thee exists k 51,...,K6 such that v* j v* j k fo all j i, then the auction etuns to step (d) using (v* 1 k,...,v* n k )to detemine the winne s payment; othewise, the good is not awaded, and no buye makes a payment. Steps (a) (c) exactly mio steps (i) (iii) of Poposition 1 in the case of two buyes. As fo step (d), we ae attempting, once again, to have the winne pay the lowest bid fo which he would still have won the auction. Fo expositional puposes, let us suppose that the bid schedules bˆ j( ) ae single-valued. If buye i is the winne, let us educe v i (stating fom v i v i ) until it is equal to the second-highest bid. Note, howeve, that as we educe v i, the

18 358 QUARTERLY JOURNAL OF ECONOMICS othe buyes bids change (because they ae functions of v i ). We theefoe seek a vecto (v* 1,...,v* n ) fo which v* i max bˆ j(v* j ) j i and that is consistent in the sense that v* k bˆ k(v* k ) fo all k i. (These conditions genealize to (29) and (30) when the bid schedules need not be single-valued.) Suppose that thee ae multiple fixed points (v 1,...,v n )at step (b). Step (e) is intended to esolve the indeteminacy (eades uninteested in multiple fixed points may wish to skip diectly to the discussion of tuthful bidding below). Afte the multiplicity aises, a second stage is played in which buyes epoted bid coespondences ae fist made public, and then (simultaneously) each buye chooses fom among the fixed points that have aisen. If buyes have bid tuthfully (we will make pecise what this means below), then, povided that a weak condition on valuation functions (see (38)) is satisfied, they will have complete infomation at this second stage about othe buyes valuations and so, in paticula, will be able to identify the fixed point coesponding to tue valuations (as in the two-buye case, the tue valuations will always constitute a fixed point, although as noted above thee could also be othes). Thus, the auction will induce an equilibium in which each buye chooses his tue valuation in this second stage if the ules stipulate that, should any buye i choose a component v i that is not consistent with v i, the good is not allocated at all. 23 Finally, step (f) is meant to esolve the indeteminancy ceated by multiple fixed points (v* 1,...,v* n ) at step (d). Again, this is accomplished by having a second stage in which, afte leaning one anothe s bid schedules (and so, povided that condition (38) holds, obtaining complete infomation about one anothe s valuations), buyes choose among the fixed points 23. Of couse, this theat would povide only a neutal incentive to be tuthful to a buye who (fom the evelation of the othes bid functions) knows that he is not going to win (i.e., he would be indiffeent between being tuthful and untuthful). But, as in the case of eliminating multiple equilibia (see footnote 18), we could modify the auction somewhat to give all buyes a stict incentive to tell tuth (at the cost of making the auction moe complex). See also the discussion of dynamic auctions in Section IV.

19 satisfying (29) and (30). In paticula, they can identify the tuthful fixed point (v i (s i,s i ),v i (s i,s i )), whee s i is sufficiently less than the tue value s i so that v i (s i,s i ) max j i v j (s i,s i ). What does it mean fo buye i to bid tuthfully in this auction? By analogy with the two-buye case, we shall say that buye i s schedule bˆ i( ) is tuthful fo signal value s i if he sets bˆ i( ) b i ( ), whee is such that b i : V i == (31) V i 5v i 0thee exists s i S i [s j,s j ] such that, j i fo all j i, v j (s i,s i ) v j, with equality if s j s j 6 and, fo all v i V i, EFFICIENT AUCTIONS 359 (32) b i (v i ) 5v i 0thee exists s i S i such that v i v i (s i,s i ) and, fo all j i, v j (s i,s i ) v j, with equality if s j s j 6. In othe wods, buye i sets the domain V i to consist of all possible valuation vectos v i that the othe buyes could have, i.e., the valuation vectos that ae consistent with some vecto of signal values fo those buyes (fo completeness, V i also includes valuation values that ae smalle than those consistent with any signal value). And b i (v i ) consists of those valuations fo buye i that ae consistent with the othes having valuations v i.itis easy to see that, if buyes ae tuthful in this sense, the tue valuations (v 1 (s 1,...,s n ),...,v n (s 1,...,s n )) constitute a fixed point satisfying (27) and that, when buye i is the winne, the payment max j i v j (s i,s i ), whee v i (s i,s i ) max j i v j (s i,s i ), satisfies (28) and (29). (Reades not inteested in the issue of multiple fixed points may wish to skip to the statement of Poposition 2 at this point.) If the auction moves to step (e), whee buyes have to choose among diffeent fixed points satisfying (27), we shall say that buye i with signal s i is tuthful if, assuming that thee exists a

20 360 QUARTERLY JOURNAL OF ECONOMICS unique vecto s i such that, fo all j i and all v j Vˆ j, 24 (33) bˆ j(v j ) 5v j 0thee exist s j S j, such that v j v j (s j,s j ) then he chooses 25 and, fo all k j, v k (s j,s j ) v k with equality if s k s k 6, (34) v i v i (s i,s i ). If thee exist eithe no such vectos s i o multiple such vectos, then buye i andomizes unifomly ove all fixed points satisfying (27). If the auction moves to step (f), whee, given that buye i is the winne, each buye j i has to choose among diffeent vectos satisfying (29) and (30), we shall say that buye j with signal s j is tuthful if, assuming that thee exists a unique vecto s j such that fo all k j,i and all v k Vˆ k (35) bˆ k(v k ) 5v k 0 thee exists s k such that v k v k (s k,s k ) and and, fo all l k, v l (s k,s k ) v l with equality if s l s l 6 (36) v i (s j,s j ) max v l (s j,s j ), l i whee (36) holds with equality if s i s i, he chooses v* j such that (37) v* j v j (s j,s j ). If thee exists eithe no such vecto s j o multiple such vectos, then buye j andomizes unifomly ove all fixed points satisfying (29) and (30). Steps (e) and (f) ely on the popety that if buye i bids tuthfully, his bid function b i ( ) will be consistent with a unique signal value, namely, the tue value s i. In fact, this uniqueness popety need not always hold. 26 Nevetheless, a faily weak 24. Fomula (33) implies that each buye j i has bid as though his tue signal wee s j. 25. Notice that if buye i is tuthful and, fo all j i, bˆ j( ) satisfies (33), then (v 1 (s i,s i ),...,v n (s i,s i )) is a fixed point satisfying (27). 26. Fo example, suppose that v 1 (s 1,s 2 ) s 1 2s 2 5 and v 2 (s 1,s 2 ) s 2 1 2s 1 5. Then b 1 (v 2 ) 15 2v 2 egadless of the value of s 1.

21 condition, namely, EFFICIENT AUCTIONS 361 v 1 v1 m s 1 s 1 (38) det m2 0, fo all m 1,...,n, v 1 v m s m s ensues that it will obtain. Notice that (38) automatically holds in the case of pivate values, since then only the main diagonal enties ae nonzeo. 27 PROPOSITION 2. Assume that, fo all i 1,...,n, buye i s valuation function satisfies (1), (6), and (26) and that buyes valuation functions collectively satisfy (38). Then, the n-buye genealization of the Vickey auction given by steps (a) (f) above is efficient. Specifically, it is an equilibium fo each buye i to bid tuthfully, i.e., to set bˆ i( ) b i ( ) satisfying (31) and (32) (and to choose, if need be, v i and v* i accoding to (34) and (37)). Moeove, if buyes ae tuthful, the auction esults in an efficient allocation. Poof. See the Appendix. In the Appendix it is shown how the auction of Poposition 2 applies to the model of Example 3. II.B. Multidimensional Signals Thee ae some cicumstances in which it is natual to think of buyes signals as being one-dimensional. The case of pivate values (Example 1) is one such instance. 28 Ou noisy signal model (Example 2) is anothe. Howeve, thee ae many othe cases in which a buye s infomation cannot be educed to one dimension. Conside the following example. Example 5. Thee ae two wildcattes competing fo the ight to dill fo oil on a tact of land consisting of an easten and 27. Note that the example in the peceding footnote violates (38). 28. Even with pivate values, a buye may eceive many signals. But as long as his pivate signals ae uncoelated with those of othe buyes, his valuation (which is one-dimensional) will be a sufficient statistic fo all his infomation. Thus, the pivate-values case is inheently one-dimensional.

22 362 QUARTERLY JOURNAL OF ECONOMICS westen egion. Wildcatte 1 has a (fixed) cost of dilling c 1, which is pivate infomation. She also pefoms a pivate test that tells he that the expected quantity of oil in the easten egion is q 1. Similaly, wildcatte 2 has a pivate fixed cost c 2 and obseves the expected quantity of oil, q 2, in the westen egion. All fou signals c 1,c 2,q 1,q 2, ae independently distibuted. The pice of oil is 1. Hence, wildcatte i s expected payoff conditional on all signals (and goss of any payment she must make) is (i) q 1 q 2 c i. Note that this model combines elements of pue pivate values (the signal c i ) with those of pue common values (the signal q i ). Note too, fom (i), that wildcatte 1 s infomation can be summaized, fom he own standpoint, by the one-dimensional signal t 1 q 1 c 1. Indeed, fo any, he pefeences ae exactly the same when he signal values ae (q 1,c 1 ) as when they ae (q 1,c 1 ). Howeve, t 1 is not an adequate summay of 1 s infomation fom wildcatte 2 s standpoint: 2 caes about q 1, but not about c 1 (so, in paticula, he would bid moe aggessively if he knew that signal values wee (q 1,c 1 ) athe than (q 1,c 1 ) fo 0). Hence, buye 1 s infomation cannot be summaized by a one-dimensional signal. Futhemoe, no auction can be fully efficient in this example. To show this, let us focus hencefoth on equilibia having the popety that if, egadless of othe playes behavio, playe i s decision poblem is the same fo two diffeent signal values s i and s i, then his equilibium behavio is the same fo eithe value (call such equilibia egula). Conside c 1, c 2 and such that (ii) c 1 c 2 c 1. Fo full efficiency, (ii) implies that dilling ights should be awaded to wildcatte 1 if he costs ae c 1 but not if they ae c 1. Howeve, he decision poblem is the same fo signal values (q 1,c 1 ) as fo (q 1,c 1 ), and so in any egula equilibium she will behave the same way in eithe case. Thus, thee is no way of devising an auction in which the outcome diffes between (q 1,c 1 ) and (q 1,c 1 ). That is, thee is no way to attain full efficiency. Moe geneally, we can expess the difficulty illustated by Example 3 as follows (Maskin [1992] establishes a vesion of the

23 following Poposition 3; see Jehiel and Moldovanu [1998] fo a closely elated impossibility esult). PROPOSITION 3. Suppose that the model of subsection II.A is genealized so that fo at least one buye i, s i is multidimensional, i.e., s i (s 1 i,..., s m i ) m fo m 1. Suppose that s i is distibuted independently of s i. If thee exist signal values s i, s i and s i such that (39) v i (s i, ) v i (s i, ), but (40) ag max j EFFICIENT AUCTIONS 363 v j (s i,s i ) ag max j v j (s i,s i ), then thee is no efficient auction with egula equilibia. Poof. In any egula equilibium (39) and the fact that s i is independent of s i imply that equilibium play when the signal values ae (s i,s i ) emains as equilibium play when the signal values ae (s i,s i ). But fom (40) the winning buye of an efficient auction cannot be the same fo (s i,s i ) as fo (s i,s i ). Hence, thee is no efficient auction. QED Notice that hypothesis (39) can eadily be satisfied if, as in Example 5, buye i s valuation is sepaable between his own signal and those of othes. That is, thee exist functions i : m = and i ( ) such that (41) v i (s i,s i ) i ( i (s i ),s i ) fo all (s i,s i ). Poposition 3 tells us that full efficiency is too much to expect in the multidimensional case. 29 Thus, all we can easonably hope fo is efficiency subject to the constaints of incentive-compatibility. It is staightfowad to show, howeve, that ou genealized Vickey auction is efficient in this constained sense, unde faily weak assumptions. Let us assume that, fo all i, s i is multidimensional (i.e., s i S i, whee S i m and m 1) and that signals ae independently distibuted acoss buyes (the independence assumption can be 29. Jehiel and Moldovanu [1998] show that full efficiency may not be attainable even if, fo each possible allocation of esouces, only a one-dimensional component of each buye s signal is petinent to buyes payoffs fom that allocation. By contast, in Example 5, both c 1 and q 1 ae petinent to wildcatte 1 s payoff if she wins the dilling ights.

24 364 QUARTERLY JOURNAL OF ECONOMICS elaxed consideably; see footnote). 30 Also suppose that, fo all i, buye i s valuation function is sepaable in the sense of (41). Fo each vecto (t 1,...,t n ) in the ange of ( 1 ( ),..., n ( )), define (42) w i (t 1,...,t n ) E s i [ i (t i,s i )0 j (s j ) t j, fo all j i]. Thus, w i (t 1,...,t n ) is buye i s expected valuation when his own signal value s i is such that i (s i ) t i (notice that, given t i, buye i is indiffeent between all signal values s i fo which i (s i ) i, and so he will behave in the same way fo any of them) and each othe buye j s signal value s j is such that j (s j ) t j (independence implies that the expectation does not depend on s i ). Because, in any egula equilibium, each buye i will behave the same way fo all signal values leading to the same t i, constained efficiency means awading the good to the buye fo whom w i (t 1,...,t n ) is highest. But as long as the deived valuation functions w i ( ) satisfy conditions (1) and (6), all conclusions fom the one-dimensional case go though. 30 We have PROPOSITION 4. Suppose that, fo all i, buye i s signal s i is multidimensional and that signals ae independently distibuted acoss buyes. 31 Assume that each buye i s valuation function is sepaable in the sense of (41), and define the deived valuation function w i (t i,...,t n ) as in (42). Then, esticting to egula equilibia, the genealized Vickey auction of Poposition 2 will be constained efficient if, fo all i, w i ( ) satisfies (1), (6), (26), and (38). In the Appendix we show how Poposition 4 applies to the model of Example 5. III. MULTIPLE GOODS III.A. Fomulation The efficient auction of Popositions 2 and 4 can be extended to multiple goods though an appopiate genealization of the 30. Jehiel, Moldovanu, and Stacchetti [1996] give conditions unde which one can educe a multidimensional poblem to one dimension fo the pupose of evenue maximization. 31. Actually, independence is much stonge than necessay. All that is needed in that, fo all t i and s i,s i S i such that i (s i ) i (s i ) t i, the distibution ove s i conditional on s i is the same as that conditional on s i. Notice that this weake condition is automatically satisfied when buyes signals ae onedimensional.