An Optimal Auction with Identity-Dependent. Externalities

Transcription

1 An Optmal Aucton wth Identty-Dependent Externaltes Jorge Aseff Hector Chade Abstract We analyze the problem of a seller of multple dentcal unts of a good who faces a set of buyers wth unt demands, prvate nformaton, and dentty-dependent externaltes. We derve the seller s optmal mechansm and characterze ts man propertes. We show that the probablty that a buyer obtans a unt s an ncreasng functon of the externaltes he generates and enjoys. Also, the seller s allocaton of the unts of the good need not be ex-post effcent. As an llustraton, we apply the model to the problem faced by a developer of a shoppng mall who wants to allocate and prce ts retal space among anchor and non anchor stores. We show that a commonly used sequental mechansm s not optmal unless externaltes are large enough. We are grateful to Alejandro Manell, Ed Schlee, Isabelle Brocas, semnar partcpants at the Ffth Spansh Meetng n Game Theory, 2003 Mdwest Economc Theory Meetngs, 2004 Latn-Amercan Econometrc Socety Meetngs, an Edtor of ths journal and three anonymous referees for ther comments and suggestons. Fxed Income Research, Brown Brothers & Harrman, jorge.aseff@gmal.com. Department of Economcs, Arzona State Unversty, hector.chade@asu.edu.

2 1 Introducton Ths paper solves an optmal mult-unt aucton desgn problem wth prvate nformaton and dentty dependent externaltes. The model we study s best understood as a model of space allocaton n a mall (although we shall see below that t subsumes other applcatons as well). Consder a developer of a new shoppng mall who wants to sell ts retal space to a set of stores. An mportant constrant n ths allocaton problem s that each store has prvate nformaton about ts proft functon; e.g., the cost of producton or the demand for ts product. Another key feature of ths problem s the exstence of nter-store externaltes: the dentty of the stores located n the mall determnes ts customer traffc, whch n turn affects the stores volume of sales. Thus, a store s wllngness to pay for retal space depends on the dentty of the other stores that locate n the mall. If the developer wants to maxmze her profts, what s the optmal sellng procedure? We develop a smple model that captures the most salent aspects of the problem. We consder a seller who has two dentcal unts of a good and faces a set of potental buyers wth unt demands. A buyer s valuaton for the good depends on hs prvately known type and on an externalty parameter that depends upon the dentty of the buyer who obtans the other unt. In ths settng, we characterze the revenue-maxmzng mechansm for the seller. We fnd that the seller should allocate the unts of the good to the par of buyers that generates the largest sum of vrtual surpluses, weghted by the external effects they enjoy. The probablty that a buyer obtans a unt of the good s ncreasng n both the externalty he mposes on other buyers and the one that he enjoys. More mportantly, the allocaton that ensues need not be ex-post effcent for the followng reasons: () as n the case wthout externaltes, the seller sometmes keeps one or both unts of the good; () snce the presence of external effects makes buyers asymmetrc, those who receve the good need not have the largest sum of valuatons; () when externaltes are negatve, the seller may sell a unt when t s ex-post effcent for her to keep t. We also characterze an optmal payment rule that nternalzes the externaltes generated and enjoyed by a buyer, and makes the optmal mechansm a domnant strategy one. In partcular, we show that a buyer s payment s a decreasng functon of the externaltes he generates. As an applcaton, we elaborate on the shoppng mall problem. A standard procedure n practce s to sgn the anchor stores frst (.e., department stores), whch are the man externalty generators, 1

3 and then approach the remanng nterested stores. The emprcal evdence also suggests that anchors receve large dscounts that are ncreasng n the externaltes they generate. 1 In turn, the non anchor stores that enjoy these externaltes pay a premum that s ncreasng n ther magntude. We characterze the propertes of ths sequental sellng procedure, and show that t s not an optmal one for the seller unless externaltes are large enough. In order to smplfy the exposton, we focus on the case of two unts, postve externaltes, and buyers payoff functons that are multplcatvely separable n types and externalty parameters. We later show that all the results extend to the case of N-unts, negatve externaltes, and also to a more general class of complementartes n the buyers payoff functons. Related Lterature. To the best of our knowledge, ths s the frst paper to analyze an optmal mult-unt aucton problem wth prvate nformaton and dentty-dependent externaltes. The closest related papers are, Jehel et al. (1996), Das Varma (2002), Fgueroa and Skreta (2008), and Brocas (2007). Jehel et al. (1996) and Das Varma (2002) both analyze auctons wth externaltes. Jehel et al. (1996) characterze the optmal mechansm and allow the external effects to be prvate nformaton, albet buyers are ex-ante dentcal n ther model. Das Varma (2002) studes open ascendng-bd auctons wth commonly known dentty-dependent externaltes, and shows that when they are non-recprocal the open aucton yelds a hgher expected revenue than a sealed-bd aucton. Unlke our paper, these references deal wth sngle-unt auctons and focus on the case n whch the wnner mposes a negatve externalty on the losers through ther reservaton utlty. 2 Both Fgueroa and Skreta (2008) and Brocas (2007) analyze optmal auctons wth externaltes, wth the emphass placed on the role played by (prvately known) outsde optons n the optmal allocaton of a sngle unt of a good. 3 Fnally, the paper also relates to Segal (1999), who analyzes contractng stuatons wth externaltes under complete nformaton. Our model s a contractng problem wth externaltes, but 1 See Pashgan and Gould (1998) and Gould et al. (2005) for a detaled emprcal analyss of the sze and mpact of nter-store externaltes on the allocaton and prcng of retal space n shoppng malls. 2 Jehel and Moldovanu (2001) analyze a general model that can accommodate mult-unt auctons wth denttydependent externaltes. Ther focus, however, s on effcency rather than revenue maxmzaton. We are grateful to Benny Moldovanu for pontng ths reference out to us. 3 An earler verson of Fgueroa and Skreta (2008) allows for multple unts and more general externaltes, but they focus on the aforementoned outsde opton effect. 2

4 unlke Segal, we study the effects that prvate nformaton has on the optmal contract. The rest of the paper proceeds as follows. Secton 2 presents the model and some prelmnary results. Secton 3 contans the dervaton of the optmal mechansm and ts man propertes. Secton 4 apples the model to the shoppng mall problem. Secton 5 presents several extensons of the analyss. Secton 6 concludes. The Appendx contans the proofs omtted from the text. 2 The Model There are I + 1 rsk-neutral agents: a seller, whom we call agent 0, and I 2 potental buyers, numbered 1, 2,..., I. The seller owns two dentcal unts of an ndvsble good, and buyers have unt demands (.e., each one demands at most one unt of the good). 4 Valuatons and External Effects. Wlog, we assume that the seller derves no value from the two unts. The valuaton of buyer, = 1, 2,...I, for a unt of the good depends on two factors. Frst, t depends on a parameter (type) θ that s prvate nformaton and t s dstrbuted on Θ = [θ, θ ], 0 θ < θ, wth postve and atomless densty φ ( ) and cumulatve dstrbuton functon Φ ( ). Let J (θ ) = θ 1 Φ (θ ) φ (θ ) ; we assume that J ( ) s a strctly ncreasng functon. Moreover, buyers types are ndependently dstrbuted. Second, a buyer s valuaton depends also on who obtans the other unt of the good. We model ths feature by ntroducng a matrx of external effects {α j } 1 I,0 j I, whch s assumed to be common knowledge among the agents. 5 To smplfy the presentaton of the man results, we assume that externaltes are postve and that a buyer s type and the externalty parameter nteract multplcatvely n hs payoff functon. 6 Formally, f buyers and j each obtans a unt of the good, and pays the seller t, then buyer s payoff s α j θ +t ; moreover, 1 α j α <, α = α 0 = 1. For smplcty, the reservaton utlty of a buyer s assumed to be equal to zero. Notce that does not derve value from a second unt of the good (α = 1), and he does not enjoy a postve externalty f the seller keeps t (α 0 = 1). The Seller s Problem. The goal of the seller s to desgn a mechansm that maxmzes 4 The extenson to more than two unts s mmedate, albet notatonally cumbersome. See Secton 5 for detals. 5 A more complete model would allow for the external effects to be prvate nformaton. Notce, however, that n many applcatons of the model, e.g., the shoppng mall case, ths common knowledge assumpton s plausble. 6 In Secton 5, we extend the results to the case of negatve externaltes, and also allow for a more general nteracton between a buyer s type and the externalty parameter. 3

5 her expected revenue, takng nto account that () buyers have prvate nformaton, () ownershp entals external effects, and () partcpaton s voluntary. By the Revelaton Prncple we can restrct the search for the optmal sellng scheme to drect revelaton mechansms (DRM) that are ncentve compatble and ndvdually ratonal. In the present case, snce the two unts are dentcal and buyers have unt demands, we can descrbe a DRM as follows. Let Λ = {[, j], j = 0, 1,..., I} be the set consstng of the (I+2)(I+1) 2 unordered pars [, j] (the notaton s borrowed from Shryaev (1996));.e., (, j) s the same par as (j, ). Let y = (y [,j] ) 0,j I be a probablty dstrbuton over Λ;.e., y [,j] s nterpreted as the probablty that obtans one unt and the other one goes to j. A DRM s a par of functons (y(θ), t(θ)), consstng of an allocaton rule y( ) and a payment rule t( ), wth y(θ) = (y [,j] (θ)) 0,j I, t(θ) = (t 0 (θ),..., t I (θ)), t 0 ( ) = I t ( ), 1 y [0,0] ( ) = [,j] [0,0] y [,j]( ), and θ = (θ 1,..., θ I ). For example, f θ s the reported vector of types, then y [,j] (θ) s the probablty that and j obtan the two unts of the good, and t (θ), = 1,..., I, s the amount of money transferred to buyer. Defne v (θ, θ ) = I α jy [,j] (θ, θ ). Then the seller s problem can be wrtten as follows: max (y [,j] ( )) [,j] [0,0],(t ( )) 1 I E θ [ ] t (θ) (1) subject to U (θ ) θ v (ˆθ ) + t (ˆθ ) (, θ, ˆθ ) (2) 1 [,j] [0,0] U (θ ) 0 (, θ ) (3) y [,j] (θ) 0 ([, j], θ) (4) y [,j] (θ) 0 θ, (5) where v (ˆθ ) = E θ [v (ˆθ, θ )] = E θ [ I α jy [,j] (ˆθ, θ )] s the expected external effect buyer enjoys f he reports ˆθ ; t (ˆθ ) = E θ [t (ˆθ, θ )] s s expected transfer f he reports ˆθ ; and U (θ ) = θ v (θ ) + t (θ ) s buyer s expected utlty f hs type s θ and he reports t truthfully. Smplfcaton of the Problem. Usng standard arguments, we can smplfy the seller s optmal mechansm desgn problem as follows. The ncentve compatblty constrants (2) are tantamount to the followng condtons: Lemma 1 (Myerson) A DRM s ncentve compatble f and only f for = 1, 2,..., I 4

6 () v ( ) s ncreasng; 7 and () U (θ ) = U (θ ) + θ θ v (s)ds, θ Θ. Lemma 1 reveals that (3) holds f and only f U (θ ) 0. Snce t (θ ) = θ v (θ ) U (θ ), we can use condton () n Lemma 1 and rewrte the objectve functon as follows: ] E θ [ t (θ) = = = = ] θ E θ [θ v (θ ) v (s)ds θ [( E θ θ 1 Φ ) ] (θ ) v (θ ) φ (θ ) E θ E θ U (θ ) U (θ ) ( θ 1 Φ ) (θ ) α j y φ (θ ) [,j] (θ) U (θ ) J (θ ) α j y [,j] (θ) U (θ ), (6) where the second lne follows by ntegraton by parts, and the last lne follows from the defnton of v ( ) and J ( ). It s clear from (6) that U (θ ) = 0 at the optmum for = 1, 2,..., I. Summarzng, the seller s problem becomes: J (θ ) α j y [,j] (θ) (7) max (y [,j] ( )) [,j] [0,0] E θ subject to (4)-(5) and condton () n Lemma 1. Some Examples. We envson the externaltes among buyers as emergng from a downstream nteracton among the agents who acqure the unts of the goods, wthout mposng any notceable effect on the losers. Ths nteracton s modelled n reduced form as a multplcatve term that s dentty dependent. Notce also that the externaltes mposed and enjoyed by the buyers who obtan the unts of the good are determned only by ther dentty; the prces pad for those unts play no role n the buyers nteracton after the aucton. Despte these restrctons, our model encompasses several substantve economc applcatons, as the followng examples llustrate. 7 Throughout the paper, ncreasng and decreasng are used n the weak sense. 5

7 Allocaton of Retal Space n Shoppng Malls. Let the seller be a developer who owns two structures, and let the buyers be stores each wshng to acqure one of them. Assume that each store faces a separate lnear demand for ts product, and let the slope be prvate nformaton and the ntercept depend upon the dentty of the other store that acqures a structure. For nstance, suppose P = α j b Q and let the cost of producton be zero. Then, store s valuaton for one unt s ts proft functon (after choosng the optmal Q ) gven by α j 1 4b = α j θ, where θ = 1 4b. It s clear n ths case that the stores that end up locatng n the mall beneft from the traffc flow generated by the presence and the dentty of the other store, whch s the man source of nteracton n ths example. We wll revst ths applcaton n Secton 4. 8 Optmal Choce of Tenants by a Landlord. The seller s the owner of two apartments or rooms, and the buyers are potental tenants or roommates, who are heterogenous along two dmensons, one observable (the externalty they generate) and one that s prvate nformaton (a preference parameter for an apartment or a room). If ther utlty functon s quaslnear and multplcatvely separable n the two characterstcs, then our model subsumes ths one-sded matchng problem. In ths case, the externaltes can be thought of as emergng from complementartes n tasks and effort allocaton wthn the household, smlarty n tastes, habts, etc., whch are dentty dependent. Fllng Multple Postons n a Frm. Let the seller be a frm tryng to fll two postons n ts R&D department. Each applcant s prvately nformed about hs dsutlty of effort, whch also depends on who wll be the canddate j fllng the remanng poston. For nstance, s payoff could be gven by t α j θ. Here the dentty of the other canddate makes the workng envronment more pleasant for ; alternatvely, one could model the externalty as an dentty-dependent mprovement n the productvty of. In ether case, our model subsumes ths contractng problem. 3 Man Results Consder the relaxed problem of maxmzng (7) subject to (4)-(5) only. 8 Note that we could allow stores to make multple decsons (nvestment, advertsng, etc.) after the aucton of the two structures takes place, so long as the fnal profts are multplcatvely separable n the externalty parameter. 6

8 Rewrte the seller s objectve functon as follows: J (θ )α j y [,j] (θ). E θ By nspecton, the seller s relaxed problem s equvalent to solvng, for each θ = (θ 1,..., θ I ), subject to (4)-(5). max (y [,j] (θ)) [,j] [0,0] Straghtforward algebra shows that (recall α 0 = α = 1) J (θ )α j y [,j] (θ) (8) J (θ )α j y [,j] (θ) = = J (θ ) ( y [,0] (θ) + y [,] (θ) ) I 1 + S [,j] (θ, θ j )y [,j] (θ) J (θ ) ( y [,0] (θ) + y [,] (θ) ) + j=+1 1 <j I S [,j] (θ, θ j )y [,j] (θ), (9) where, to shorten the notaton, we have defned S [,j] (θ, θ j ) = α j J (θ ) + α j J j (θ j ). 3.1 The Optmal Mechansm Soluton to the Relaxed Problem. It s evdent from (9) that the followng allocaton rule y ( ) solves problem (8): for every par [, j], 1 < j I, set y[,j] (θ) = 1 f S [,j] (θ, θ j ) > max{0, max l J l (θ l ), max [l,k],l k,l,k 1 S [l,k] (θ l, θ k )} 0 otherwse; for every par [, 0], = 1,..., I, set y[,0] (θ) = 1 f J (θ ) > max{0, max l,l J l (θ l ), max [l,k],l k,l,k 1 S [l,k] (θ l, θ k )} 0 otherwse; and for every par [, ], = 1,...I, set y [,] (θ) = 0. 9 In words, the seller ranks all pars of buyers accordng to ther weghted sum of vrtual surpluses, where the weghts are gven by the external effects they mpose on each other. The best par obtans the unts of the goods f ther weghted sum s nonnegatve and greater than the vrtual surplus 9 Settng y [,] (θ) = 0 s wlog as the seller s ndfferent between gvng a second unt to buyer and keepng t herself. Note also that we gnore tes, as they occur wth zero probablty. 7

9 of every sngle buyer. Otherwse, the seller allocates one unt to the buyer who has the largest nonnegatve vrtual surplus. If no buyer or par of buyers satsfy the aforementoned condton, then the seller keeps the two unts of the good. Monotoncty of v ( ). A necessary condton for y ( ) to be part of an optmal mechansm s that t satsfy condton () of Lemma 1,.e., that v ( ) be ncreasng n θ for all = 1, 2,..., N. It s not obvous that y ( ) satsfes ths condton, for y[,j] ( ) need not be ncreasng n θ (unlke the case wth a sngle unt or wth multple unts and no externaltes). Example 1 (Non-Monotoncty of y ( )). There are three bdders wth valuatons dstrbuted unformly on [0, 1], so J (θ ) = 2θ 1. Let θ 1 = 0.7, θ 2 = 0.8, θ 3 = 0.6, α 21 = α 31 = 1, α 12 = 1.5, α 13 = 2, and α 23 = α 32 = 1.3. Exhaustve checkng reveals that y[1,2] (θ 1, θ 1 ) = 1. Consder θ 1 = 0.95 > 0.7. Now S [1,3](θ 1, θ 1) s the largest weghted sum of vrtual surpluses, and thus y [1,2] (θ 1, θ 1) = 0 and y [1,3] (θ 1, θ 1) = 1, thereby provng that y [1,2] (, θ 1) s not ncreasng n θ 1. Ths example notwthstandng, the allocaton rule y ( ) mples that v ( ) s ncreasng n θ for all = 1, 2,...N. Lemma 2 (Monotoncty) The allocaton rule y ( ) satsfes condton () of Lemma 1. Proof: We need to prove that condton () n Lemma 1 s satsfed;.e., that v ( ) s an ncreasng functon. Note that t suffces to prove that v (, θ ) = I α jy [,j] (, θ ) s ncreasng. Take θ > θ ; we wll show that v (θ, θ ) v (θ, θ ). The result follows trvally when v (θ, θ ) = 0, snce v (, θ ) s a nonnegatve functon. So suppose v (θ, θ ) > 0. The allocaton rule y ( ) and the defnton of v (, θ ) reveal that we can, wlog, set v (θ, θ ) = α l for some l. Note that t must be the case that v (θ, θ ) > 0, for α l J (θ ) + α lj l (θ l ) > α l J (θ ) + α lj l (θ l ) mples that, f the soluton to the relaxed problem allocates a unt to when he reports θ, t must do so when he reports θ (recall θ s kept fxed). Thus, v (θ, θ ) = α k for some k. To complete the proof, we show that α k α l. Snce y [,l] (θ, θ ) = 1 and y [,k] (θ, θ ) = 1, α l J (θ ) + α l J l (θ l ) α k J (θ ) + α k J k (θ k ) α k J (θ ) + α k J k (θ k ) α l J (θ ) + α l J l (θ l ). These nequaltes yeld (α l α k )(J (θ ) J (θ )) 0. But J (θ ) < J (θ ) and hence α k α l, thereby completng the proof. 8 snce J ( ) s strctly ncreasng,

10 The ntuton of the proof s as follows. Suppose that, gven hs type, buyer receves a unt of the good along wth buyer j. If buyer s type ncreases, then hs wllngness to pay for a unt ncreases regardless of the dentty of the other buyer. That s, the weghted sum of vrtual surpluses of buyer and any k ncreases. Hence, the probablty that he obtans a unt of the good must go up when hs type s hgher. But snce the external effect enjoyed by when he s pared wth k need not be the same for all k, the rankng of all pars contanng may change as buyer s type ncreases. Thus, (n the relaxed problem) the seller may fnd t optmal to par hm wth k j. 10 An Optmal Mechansm. It follows from Lemma 2 that y ( ), along wth any payment rule t( ) that satsfes t (θ ) = θ v (θ ) θ θ v (s)ds for all θ and = 1, 2,..., N (see condton () n Lemma 1), wll consttute an optmal mechansm for the seller. In partcular, ths holds f we set, for every and every θ = (θ, θ ), θ t (θ, θ ) = θ v (θ, θ ) v (s, θ )ds θ θ = θ α j y[,j] (θ, θ ) θ Summarzng, we have shown the followng result. α j y[,j] (s, θ ) ds. (10) Theorem 1 The mechansm (y ( ), t ( )) s an optmal sellng procedure for the seller. The No-Externalty Case. It s easy to show that the allocaton rule y ( ) reduces to the one derved n Maskn and Rley (1989) when there are no external effects;.e., when α j = 1 for all = 1, 2,..., I, j = 0, 1,..., I. In ths case, the optmal allocaton rule y ( ) s such that, for every par [, j], 1 < j I, y [,j] (θ) = 1 f and only f () J (θ ) 0, () J j (θ j ) 0, () J (θ ) J k (θ k ) for k j, and (v) J j (θ j ) J k (θ k ) for k. In other words, the seller smply allocates the unts of the good to the buyers wth the largest vrtual surpluses. It follows that f types are dentcally dstrbuted, then a standard aucton wth a reserve prce s an optmal mechansm. Interpretaton of the Payment Rule. It s llumnatng to manpulate the payment rule (10) further. Recall that v (, θ ) = I α jy [,j] (, θ ) s an ncreasng functon. Indeed, gven the shape of y [,j] ( ), v (, θ ) s actually a step functon that, wlog, can be assumed to be rght-contnuous. Let θ k (θ ), k = 1, 2,..., n, wth θ 1 (θ ) < θ 2 (θ ) <... < θ n (θ ), be the 10 Notce that ths also explans the ntuton underlyng Example 1. 9

11 ponts at whch the functon jumps as θ ncreases. Notce that θ 1 (θ ) s the smallest type that could report and stll obtan a unt of the good. It s evdent from (10) that f θ < θ 1 (θ ) then t (θ, θ ) = 0. Suppose θ θ 1 (θ ), and let j k, k = 1, 2,..., n, be the dentty of the buyer who, along wth, wll obtan a unt of the good when buyer s type reaches the pont θ k (θ ). Notce that α j 1 < α j 2 <... < α j n by the monotoncty of v (, θ ). Obvously, f buyer reports θ, then he obtans a unt of the good and j n obtans the other unt. Then (settng θ n+1 (θ ) = θ ), θ n ( ) v (s, θ )ds = α j p θ p+1 (θ ) θ p (θ ), θ and therefore, p=1 t (θ, θ ) = θ α j n n p=1 α j p p=1 ( θ p+1 ) (θ ) θ p (θ ) n 1 = θ 1 ( (θ )α j 1 + αj p+1 α j p) θ p+1 (θ ). Fgure 1 provdes an llustraton of the payment rule n whch, gven (θ, θ ), obtans a unt of the good along wth j 3, and there are two other buyers, j 1 and j 2, wth whom could be pared. Insert Fgure 1 here In summary, the optmal payment rule (10) s equvalent to t θ 1 (θ, θ ) = (θ )α j 1 + n 1 ( p=1 αj p+1 α j p) θ p+1 (θ ) f θ θ 1(θ ) 0 f θ < θ 1(θ ). (11) The nterpretaton of (11) s as follows. Suppose that, gven (θ, θ ), obtans a unt of the good and j n receves the other unt. Then the amount that pays s the sum of two terms. The frst term s α j 1θ 1(θ ), whch s the value the good would have had to hm had he submtted the lowest wnnng report gven θ, namely, θ 1(θ ). The second term s the sum of the ncrements n utlty derved from the hgher external benefts that would have enjoyed, had he submtted the lowest wnnng reports that would have pared hm wth, respectvely, j 2, j 3, etc. For nstance, the smallest wnnng report above θ 1(θ ) that pars wth j 2 s θ 2(θ ), and the ncremental utlty enjoys s (α j 2 α j 1)θ 2(θ ), whch s nternalzed by the optmal payment rule (11) Ths payment rule s smlar to the one n Levn (1997), who analyzes a optmal mult-unt aucton problem wth synerges among the objects nstead of externaltes among the buyers. He shows that a buyer s payment nternalzes the complementartes among the goods n an ncremental way that resembles (11). 10

12 3.2 Propertes of the Optmal Mechansm The optmal mechansm derved above exhbts the followng mportant propertes: Allocaton and External Effects. The probablty that a buyer obtans a unt of the good s ncreasng n both the externalty he enjoys and the one he generates. To see ths, let y (θ) = I y [,j](θ) be the probablty that buyer obtans a unt of the good, gven the vector of announced types θ. Careful nspecton of the optmal allocaton rule y ( ) reveals that y (θ) s larger the bgger are α j or α j for any j. Notce also that, unlke the case wth a sngle unt or wth multple unts but wthout externaltes, a buyer could obtan a unt of the good despte havng J (θ ) < 0, so long as he s pared wth another buyer who enjoys a large external effect from beng wth. For a numercal llustraton, consder Example 1 but wth θ 1 =.95, θ 2 = 0.6, and θ 3 = 0.4. Then t s optmal to allocate the unts of the good to buyers 1 and 3, even though J 3 (0.4) = 0.2. Payment and External Effects. The amount of money pad by a buyer who obtans a unt of the good s a decreasng functon of the externalty he mposes on the buyer who obtans the other unt. To prove ths asserton, consder s payment when he obtans a unt of the good along wth j n (see (11)). We wll show that t (θ) decreases n α j n. Recall that θ n (θ ) s the threshold type of buyer that pars hm wth j n nstead of wth j n 1. Formally, θ n (θ ) solves S [,j n ](θ n (θ ), θ j n) = S [,j n 1 ](θ n (θ ), θ j n 1), whch yelds θ n (θ ) = J 1 ( αj n 1 J j n 1(θ j n 1) α j n J j n(θ j n) α j n α j n 1 Now, α j n > α j n 1 and J ( ) ncreasng mply that θ n (θ ) s a decreasng functon of α j n. And snce t (θ) ncreases n θ n (θ ), t follows that t (θ) decreases n α j n, thus provng the result. Domnant Strategy Implementaton. It s easy to verfy that, under (y ( ), t ( )), t s a domnant strategy for buyers to report ther true types. For f a buyer s report were hgher than hs type, then he could be pared wth a buyer who generates a larger externalty, but any ncrement n hs utlty s pad to the seller (see (11)). Thus, he cannot gan by overstatng hs type. And f a buyer s report were lower than hs type, then he could ether cease to obtan a unt of the good, whch clearly reduces hs payoff, or be pared wth a buyer who generates a lower externalty, n whch case hs net payoff goes down as well. ). 11

13 As an llustraton, consder Fgure 1 and suppose buyer reports ˆθ > θ. Two thngs can happen. If ˆθ < θ 4, then wll stll be pared wth j3 and hs payment remans unaltered. If ˆθ θ 4, then wll now be pared wth a buyer who generates a larger externalty, say buyer jk. But (α j k α j 3)θ becomes part of buyer s payment. Hence, hs payoff s the same as when he reports hs true type θ. Smlarly, he cannot gan by reportng ˆθ < θ. For nstance, f θ 2 ˆθ < θ 3, then wll be pared wth j 2, and hs payoff s reduced by the amount (α j 3 α j 2)(θ θ 3 ). Ineffcent Allocaton of the Unts. The allocaton rule y ( ) need not yeld an ex-post effcent allocaton of the unts of the good. When types are common knowledge, the seller s optmal mechansm s straghtforward. To wt, she should allocate the unts of the good to the par [, j] wth the largest α j θ + α j θ j, and then charge α j θ to buyer and α j θ j to j, thereby extractng all the surplus from the two buyers. Obvously, the allocaton that ensues s ex-post effcent. Moreover, snce θ 0 for all and α j 1, the seller should always sell the two unts of the good to dfferent buyers. Under the optmal allocaton rule y ( ), however, the seller may keep one or both unts, and even when she sells both unts, she need not allocate them to the par [, j] wth the largest α j θ + α j θ j, as the followng example llustrates. Example 2 (Ineffcent Allocaton). There are three bdders wth valuatons dstrbuted unformly on [θ, θ], wth 2θ > θ. Thus, J (θ ) = 2θ θ > 0 for all θ and, and the seller always sells both unts. Let α 21 = α 31 = α > 1, α 12 = α 13 = α 23 = α 32 = 1;.e., only buyer 1 generates externaltes. Consder θ = (θ 1, θ 2, θ 3 ), wth θ 2 > θ 3. Notce that n ths case t s never optmal for the seller to allocate the goods to [1, 3]. It s easy to verfy that the two unts wll be sold to [1, 2] f and only f (θ 1 + αθ 2 ) (θ 2 + θ 3 ) θ 2 (α 1) > 0. Thus, t could happen that the seller allocates the two unts to [2, 3] even though (θ 1 + αθ 2 ) > (θ 2 + θ 3 );.e., the allocaton s neffcent. For a numercal llustraton, set θ = 1.05, θ = 2, α = 1.1, θ 1 = 1.07, θ 2 = 1.2, and θ 3 = 1.1. The ntuton underlyng the neffcency llustrated n Example 2 s as follows. The exstence of dentty-dependent externaltes ntroduces an asymmetry n the model, for they make buyers ex-ante heterogeneous even wth dentcally dstrbuted types. It s well-known that, wthout externaltes, the presence of asymmetrc bdders can lead to the type of neffcency portrayed n Example 2 (e.g., see Myerson (1981)). The analyss above sutably generalzes ths result to the case wth externaltes. Another mportant mplcaton of ths asymmetry s that t makes t extremely 12

14 dffcult to fnd an ndrect mechansm that mplements the optmal aucton (y ( ), t ( )). 4 Applcaton: Shoppng Malls and Inter-Store Externaltes As an llustraton, we apply the model to the shoppng mall example descrbed n Secton 2. Our ntenton s not to provde a complete analyss of ths problem, whch s qute complex n practce. Instead, our objectves are () to show that our model can shed lght on the emprcal evdence on the allocaton and prcng of shoppng mall space, and () to show that a sequental sellng procedure commonly used n practce need not be an optmal mechansm for the seller. Evdence on Inter-Store Externaltes. Pashgan and Gould (1998) and Gould et al. (2005) provde a detaled emprcal analyss of the mportance of nter-store externaltes n the prcng and composton of retal space n shoppng malls. Indeed, a common procedure used by developers s to frst sgn at a low prce (rent) per square foot the so-called anchor stores (.e., department stores), whch are the ones that generate the largest mall traffc that beneft all the stores. Then they offer the remanng space to non anchor stores (Pashgan and Gould (1998)). A large fracton of anchor stores pay no rent or a trval amount, and among the stores that pay rent, the average prce per square foot pad by non anchor stores s about seven tmes hgher than that pad by anchor stores (Gould et al. (2005)). Thus, the evdence shows that the probablty of sgnng a store s an ncreasng functon of the externaltes t generates, whle the prce pad decreases (ncreases) n the externaltes generated (enjoyed). A Smple Model wth an Anchor Store. To shed lght on the man features of ths problem, let us consder the case of three potental stores n whch only store 1 generates externaltes,.e., store 1 s an anchor store. Formally, let α 21 = α 31 = α > 1, α 12 = α 13 = α 23 = α 32 = 1. For smplcty, assume that θ s dstrbuted on [θ, θ] wth densty φ( ) for all and θφ(θ) > 1, thereby ensurng that J(θ) > 0 and hence J(θ ) > 0 for all θ, = 1, 2, 3. A Sequental Procedure. Suppose the seller uses the followng sequental procedure. In the frst stage, she makes a take-t-or-leave-t offer to store 1. If store 1 accepts, t obtans one structure and then the seller uses a frst-prce aucton to allocate the remanng one between stores 2 and 3 n the second stage. If store 1 rejects, then the seller sells one structure to store 2 and the other to store 3 at a prce θ per unt n the second stage. Notce that the seller uses an optmal 13

15 mechansm n each possble case n whch she deals wth stores 2 and Consder the second stage. If there are two structures left, stores 2 and 3 accept the offer and the seller s revenue s equal to 2θ. If there s only one structure left, then the store wth the hghest type between 2 and 3 obtans t. It s straghtforward to verfy that the seller s expected revenue n ths case s αe[mn{θ 2, θ 3 }] = 2α θ θ s(1 Φ(s))φ(s)ds. Let us now turn to the frst stage. The seller s problem s ( max p θ (1 Φ(p)) p + 2α θ θ s(1 Φ(s))φ(s)ds where p s the take-t-or-leave-t offer tendered to store 1. ) + Φ(p)2θ, The soluton p to ths problem s the followng: f α α S, then p = θ, whle f α < α S, then p s the unque soluton to θ J(p ) = 2θ 2α s(1 Φ(s))φ(s)ds. (12) θ The threshold value of the external effect, α S, s gven by α S = θφ(θ) + 1 2φ(θ). θ θ s(1 Φ(s))φ(s)ds The propertes of the sequental mechansm can be summarzed as follows. Dfferentaton of (12) reveals that the offer the seller makes to store 1 s decreasng n the sze of the external effect t generates. And f store 1 accepts the offer, then the prce pad for the remanng unt s an ncreasng functon of the externalty enjoyed by the store that acqures t. Also, when the sze of the externalty s relatvely small, there s a postve probablty that store 1 wll not obtan a structure. But f the externalty that store 1 generates s suffcently large, then the seller ensures that the anchor store receve a unt for sure, and her expected revenue s θ+2α θ θ s(1 Φ(s))φ(s)ds. Comparson wth the Optmal Mechansm. The propertes of the sequental mechansm are broadly consstent wth the emprcal evdence, and they are also closely related to those of the optmal mechansm dscussed n Secton 3. A natural queston to ask s whether the sequental mechansm s ndeed optmal for the seller. We wll compare t wth the optmal mechansm and show that the answer s negatve. The comparson reles on the followng result. 12 If J(θ) < 0, we only need to ntroduce a reserve prce θ r n the second stage. 14

16 Lemma 3 Store 1 obtans a structure wth probablty one n the optmal mechansm f and only f α α O = 2 θφ(θ) 1. θφ(θ) Proof: We need to fnd the smallest value of α such that, for every θ = (θ 1, θ 2, θ 3 ), ether J(θ 1 ) + αj(θ 2 ) J(θ 2 ) + J(θ 3 ) or J(θ 1 ) + αj(θ 3 ) J(θ 2 ) + J(θ 3 ). Equvalently, the followng condton on α must hold for every θ = (θ 1, θ 2, θ 3 ): { J(θ3 ) J(θ 1 ) α 1 + mn, J(θ } 2) J(θ 1 ). (13) J(θ 2 ) J(θ 3 ) Suppose θ 2 θ 3. Then the rght sde of (13) becomes 1 + J(θ 3) J(θ 1 ) J(θ 2 ). The largest value of ths expresson occurs when θ 1 = θ, θ 2 = θ 3, and θ 3 = θ, whch s equal to 2 θφ(θ) 1 θφ(θ). If α 2 θφ(θ) 1, θφ(θ) then (13) holds for all θ = (θ 1, θ 2, θ 3 ) wth θ 2 θ 3. And f (13) holds for all θ = (θ 1, θ 2, θ 3 ) wth θ 2 θ 3, then t holds for θ = (θ, θ, θ). The case wth θ 3 θ 2 s analogous. We have thus shown that store 1 obtans a structure wth probablty one f and only f α α O = 2 θφ(θ) 1. θφ(θ) It s straghtforward to calculate the payments when α α O. Snce store 1 always receves one structure, ts smallest wnnng report s θ, and ths s ts payment. Store 2 obtans the other structure f and only f θ 2 θ 3 ; the smallest report that allows t to receve the remanng unt s θ 3, and ts payment s then αθ 3. Thus, the seller s expected revenue s θ + αe[mn{θ 2, θ 3 }]. Notce that the allocaton and payments under the optmal mechansm when α α O are the same as n the sequental mechansm when α α S. Thus, the sequental mechansm s optmal f α max{α O, α S }. Ths s ntutve, for f the externalty generated by the anchor store s suffcently large, then t s optmal for the seller to ensure that ths store locates n the mall. Ths s accomplshed by sellng a structure to the anchor store at the lowest prce, and then proft from the sale of the other structure, whose prce ncreases due to the presence of the anchor store. More nterestngly, we now show that when externaltes are not large enough, the sequental mechansm does not maxmze the seller s expected revenue. Proposton 1 The threshold α O s strctly bgger than α S. Thus, whenever α S α α O, store 1 obtans a structure n the sequental mechansm more often than n the optmal mechansm. Proof: Smple algebra reveals that α O α S > 0 f and only f ( ) θ φ(θ)(2θ θ) s(1 Φ(s))φ(s)ds θθφ(θ) θ > 0. (14) θ 15

17 But 2 θ θ s(1 Φ(s))φ(s)ds = E[mn{θ 2, θ 3 }] > θ. (θφ(θ) 1)(θ θ) > 0, thereby provng that α O > α S. Hence, the left sde of (14) s greater than The analyss reveals that a common procedure used n practce s not an optmal mechansm for the seller unless the externalty generated by the anchor store s large enough to ensure that t obtans a structure wth probablty one. When the external effects generated by the anchor store are not suffcently strong, the seller can mprove her expected revenue over that of the sequental mechansm by ncreasng the probablty of sellng the two structures to stores 2 and 3. In ths way, she rases the amount that store 1 pays whenever t obtans a structure. 5 Extensons Thus far we have focused on the case of two unts of the good, postve externaltes, and multplcatve nteracton between a buyer s type and the externalty he enjoys. As we emphaszed n Secton 2, we made these assumptons wth the sole purpose of smplfyng the presentaton of the man results. We now show that none of the nsghts depend on them. Negatve Externaltes. The analyss of the optmal mechansm does not depend on the assumpton that externaltes are postve. To wt, nowhere n Secton 3 have we used the fact that α j 1 for all j 0. Thus, all the results n that secton also hold when α j (0, 1]. The only excepton s that, wth negatve externaltes, ex-post effcency sometmes requres the seller to sell only one unt of the good. Ths gves rse to an nterestng property, whch has no counterpart n the sngle or mult-unt cases wthout externaltes: under the optmal mechansm, the seller may sell the second unt of the good n cases n whch t s ex-post effcent for her to keep t. The followng example llustrates ths phenomenon. Example 3 (Seller Sells when She Shouldn t). There are three bdders wth valuatons dstrbuted unformly on [0, θ ], so J (θ ) = 2θ θ for all. Let θ 1 = 10, θ 2 = θ 3 = 5, α 12 = α 13 = 0.3, and α 21 = α 31 = α 23 = α 32 = 1. That s, buyers 2 and 3 mpose a negatve externalty on buyer 1. Consder types θ 1 = 6, θ 2 = 3.3, and θ 3 = 1. Under complete nformaton, the seller should sell one unt to buyer 1 for a payment of 6, and keep the other unt. To see ths, note that the best the seller can do f she sells both unts s to allocate them to buyers 1 and 2, whose combned 16

18 surplus s 0.3θ 1 + θ 2 = 5.1 < 6. Under ncomplete nformaton, however, J 1 (6) = 2, J 2 (3.3) = 1.6, and J 3 (1) = 3 mply that t s optmal for the seller to sell both unts to buyers 1 and 2, whose combned vrtual surplus s 0.3J 1 (6) + J 2 (3.3) = 2.2 > 2. In other words, the seller sells the second unt of the good when the (ex-post) effcent allocaton nstructs her to keep t. 13 N Unts of the Good. For smplcty, we have assumed that the seller owns only two unts of the good. Barrng notatonal complexty, the man nsghts readly extend to the N-unt case. We now outlne the modfcatons needed n order to accommodate ths case. Let I N, and defne Λ as the set of all unordered N-tuples λ n {0, 1,..., I} N. A DRM s a par (y(θ), t(θ)), where y(θ) = (y λ (θ)) λ Λ and t(θ) = (t 0 (θ),..., t I (θ)). For each, the externalty parameters are defned as 0 α j α for every j, where j s an unordered N 1-tuple such that λ = [, j] Λ. If j conssts of all zeros or all s, then α j = 1, and the same holds for = 0 and any j. All the sums wth respect to j n Sectons 2 and 3 should now be replaced by sums over all j such that [, j] Λ. Wth these modfcatons, all the results extend to the case n whch the seller owns N unts of the good. Supermodular Interacton of Types and Externaltes. We have assumed that a buyer s payoff s multplcatvely separable n hs type and the externalty he enjoys. The followng example llustrates the usefulness of allowng for more general complementartes between them. Example 4 (Shoppng Mall). Suppose that f store locates n the shoppng mall, then t faces a lnear demand P = θ + α j Q for ts product, where j denotes the dentty of the neghborng store. For smplcty, let the cost of producton be zero. It s easy to verfy that s proft functon s equal to (α j+θ ) 2 4. Notce that profts are ncreasng n α j and θ, (strctly) convex n θ, and (strctly) supermodular n (α j, θ ). 14 We wll show that the results extend to buyers payoff functons of the form u (α j, θ ) + t, wth u (, ) nonnegatve, u (α j, θ ) > u (α k, θ ) f α j > α k, u θ > 0, 2 u θ 2 0, and u (, ) supermodular 13 Ths neffcency also arses n the sngle-unt case n whch the wnner mposes an externalty on the losers (see Jehel et al. (1996), Brocas (2007), and Fgueroa and Skreta (2008)). The example extends ths result to the mult-unt case wth externaltes among those who obtan the unts of the good. 14 A functon f(x, y) s supermodular f, gven (x 1, y 1 ) and (x 2, y 2 ), f(x 1 x 2, y 1 y 2 ) + f(x 1 x 2, y 1 y 2 ) f(x 1, y 1 ) + f(x 2, y 2 ). If f(x, y) s C 2, ths s equvalent to 2 f(x,y) x y 0. 17

19 n (α j, θ ). 15 Note that α j θ and α j + θ satsfy all these propertes. For each buyer, let V (α j, θ ) = u (α j, θ ) (1 Φ (θ )) u (α j, θ ). φ (θ ) θ We assume that V (α j, θ ) s strctly ncreasng n θ and supermodular n (α j, θ ). In the multplcatvely separable case, V (α j, θ ) = α j J (θ ), whch clearly satsfes all these propertes. The Appendx contans the detals of the analyss of ths extenson. Under the assumptons made, an optmal mechansm (y ( ), t ( )) for the seller s gven by the followng allocaton and payment rules: for every par [, j], 1 < j I, set y[,j] (θ) = 1 f W [,j] (θ, θ j ) > max{0, max l V l (α l0, θ l ), max [l,k] W [l,k],l k,l,k 1 (θ l, θ k )} 0 otherwse, where W [,j] (θ, θ j ) = V (α j, θ ) + V j (α j, θ j ); for every par [, 0], = 1,..., I, set y[,0] (θ) = 1 f J (θ ) > max{0, max l V l (α l0, θ l ), max [l,k] W [l,k],l k,l,k 1 (θ l, θ k )} 0 otherwse; and for every par [, ], = 1,...I, set y [,] (θ) = 0. Fnally, set t (θ, θ ) = θ u (α j, θ )y[,j] (θ, θ ) θ u (α j, s) θ y[,j] (s, θ ) ds. Intutvely, the allocaton and payment rules are sutable generalzatons of those n the multplcatvely separable case. It s evdent by nspecton that the ntuton of the allocaton rule s the same as before, and a bt of work reveals that the same s true for the payment rule. 6 Concludng Remarks Ths paper studes the optmal mult-unt aucton desgn problem of a seller n the presence of buyers prvate nformaton and dentty-dependent externaltes. We show that t s optmal for the seller to allocate the unts of the good to the set of buyers that generates the largest weghted 15 An alternatve way of statng some of these propertes s as follows. Defne the order by j k f and only f α j α k ( s defned analogously). Then u (α j, θ ) > u (α k, θ ) f α j > α k s equvalent to u (α j, θ ) beng ncreasng n j n the order ; and u (, ) supermodular n (α j, θ ) s equvalent to u (, ) supermodular n (j, θ ). 18

20 sum of vrtual surpluses. Ths rule often leads to an ex-post neffcent allocaton as the seller () may keep one or more unts when t s ex-post effcent to sell them, () may sell to the set of buyers that does not maxmze the jont valuaton for the unts, () may sell a unt when t s ex-post effcent to keep t. We derve an optmal payment rule that llustrates how the seller can structure payments n such a way that buyers who obtan the goods pay accordng to the external benefts they enjoy and generate. As an applcaton, we analyze the sellng problem faced by a shoppng center s developer, and we flesh out the man propertes of a sequental mechansm commonly used n practce. It turns out that the sequental procedure s not an optmal sellng mechansm unless nter-store externaltes are suffcently large. We have conducted the analyss under the assumpton that the sze of the dentty-dependent externaltes was common knowledge among the agents nvolved. Albet ths s a plausble assumpton n many settngs such as the shoppng mall applcaton that motvated ths paper, t would be desrable to relax t and solve the model allowng for two dmensons of prvate nformaton. Gven the well-known dffcultes assocated wth mechansm desgn problems wth multdmensonal types, ths extenson s not only nterestng but also apt to be nontrval. Appendx In the general case wth buyers utlty functons u (α j, θ ) + t, the seller s problem s: [ ] max E θ t (θ) (15) (y [,j] ( )) [,j] [0,0],(t ( )) 1 I subject to U (θ ) E θ u (α j, θ )y [,j] ( ˆθ, θ ) + t (ˆθ ) (, θ, ˆθ ) (16) U (θ ) 0 (, θ ) (17) 1 y [,j] (θ) 0 ([, j], θ) (18) 1 y [,j] (θ) 0 θ, (19) [,j] [0,0] The followng condtons are necessary and suffcent for ncentve compatblty: () E θ [ I ] u (α j, ) θ y [,j] (, θ ) s ncreasng; and () U (θ ) = U (θ ) + [ θ I u θ E (α j,s) θ θ y [,j] (s, θ ) 19 ] ds, θ Θ.

21 To prove necessty, let (y( ), t( )) be ncentve compatble. Then () follows from an applcaton of the Envelope Theorem. Regardng (), let ˆθ > θ wlog; then U (θ ) U ( ˆθ ) + E θ (u (α j, θ ) u (α j, ˆθ ))y [,j] ( ˆθ, θ ) E θ U ( ˆθ ) U (θ ) + E θ (u (α j, ˆθ ) u (α j, θ ))y [,j] (θ, θ ). These nequaltes yeld (u (α j, ˆθ ) u (α j, θ ))y [,j] ( ˆθ, θ ) E θ (u (α j, ˆθ ) u (α j, θ ))y [,j] (θ, θ ), whch s equvalent to u (α j, ˆθ )y [,j] ( ˆθ, θ ) + E θ u (α j, θ )y [,j] (θ, θ ) E θ E θ u (α j, ˆθ )y [,j] (θ, θ ) + E θ u (α j, θ )y [,j] ( ˆθ, θ ), [ I thereby showng that E θ u (α j, θ )y [,j] ( ˆθ ], θ ) s supermodular n (θ, ˆθ ) or, equvalently, E θ u (α j, θ ) y θ [,j] (, θ ) (20) s ncreasng. Snce 2 u θ 2 E θ u (α j, θ ) y θ [,j] (θ, θ ) E θ 0, (20) s ncreasng n θ as well. Take θ > θ ; then E θ u (α j, θ ) y θ [,j] (θ, θ ) u (α j, θ ) y θ [,j] (θ, θ ), where the frst nequalty follows by 2 u 0 and the second by supermodularty. θ 2 To prove suffcency, suppose that () and () hold, and let ˆθ > θ. Then, ˆθ U ( ˆθ ) U (θ ) = E θ u (α j, s) y θ [,j] (s, θ ) ds θ ˆθ E θ θ 20 u (α j, s) y θ [,j] (θ, θ ) ds, (21)

22 where the nequalty follows from (20). Smlarly, u (α j, ˆθ )y [,j] (θ, θ ) + t (θ ) U (θ ) = E θ (u (α j, ˆθ ) u (α j, θ ))y [,j] (θ, θ ) E θ = ˆθ E θ θ Expressons (21) and (22) yeld U ( ˆθ ) E θ u (α j, ˆθ )y [,j] (θ, θ ) + t (θ ), whch completes the proof of suffcency. Let V (α j, θ ) = u (α j, θ ) (1 Φ (θ )) φ (θ ) max (y [,j] ( )) [,j] [0,0] subject to (18)-(19) and condton (). u (α j,θ ) θ E θ u (α j, s) y θ [,j] (θ, θ ) ds. (22) ; the seller s problem can be wrtten as follows: V (α j, θ )y [,j] (θ) (23) Consder the relaxed problem n whch condton () s gnored. It s mmedate to show that the soluton to ths problem s gven by the allocaton rule y ( ). If ths allocaton rule satsfed () then, along wth the payment rule t ( ) (whch can be derved followng the same steps that led to (10)), they would consttute an optmal mechansm for the seller. We now show that y ( ) satsfes (); t suffces to show that I u (α j, ) θ y[,j] (, θ ) s ncreasng n θ. Take θ > θ I u (α j,θ ) θ y[,j] (θ I and suppose y [,l] (θ, θ ) = 1 (the other case s trval);.e.,, θ ) = u (α l,θ ) θ for some l. As n Lemma 2, t s easy to show that u (α j,θ ) θ y [,j] (θ, θ ) > 0; w.l.o.g., suppose that ths sum s equal to u (α k,θ ) θ for some k. To complete the proof, we need to show that u (α k,θ ) θ u (α l,θ ) θ. By supermodularty and convexty, t suffces to show that α k α l. Snce y [,l] (θ, θ ) = 1 and y [,k] (θ, θ ) = 1, These nequaltes yeld V (α l, θ ) + V l (α l, θ l ) V (α k, θ ) + V k (α k, θ k ) V (α k, θ ) + V k (α k, θ k ) V (α l, θ ) + V l (α l, θ l ). V (α k, θ ) + V (α l, θ ) V (α k, θ ) + V (α l, θ ). (24) If α k < α l, then (24) would volate the supermodularty of V (α j, θ ). Thus, α k α l. 21

23 References Brocas, I., Auctons wth Type-Dependent and Negatve Externaltes: The Optmal Mechansm, mmeo, Unversty of Southern Calforna, (2007). Das Varma, G., Standard Auctons wth Identty-Dependent Externaltes, Rand Journal of Economcs (2002), 33(4), Fgueroa, N. and V. Skreta, The Role of Outsde Optons n Aucton Desgn, mmeo, Stern School of Busness, New York Unversty, (2008). Gould, E., P. Pashgan, and C. Prendergast Contracts, Externaltes, and Incentves n Shoppng Malls, The Revew of Economcs and Statstcs (2005), 87, Jehel, P. and B. Moldovanu, Effcent Desgn wth Interdependent Valuatons, Econometrca (2001), 69, Jehel, P., B. Moldovanu, and E. Stacchett, How (Not) to Sell Nuclear Weapons, Amercan Economc Revew (1996), 86, Levn, J., An Optmal Aucton for Complements, Games and Economc Behavor (1997), 18, Maskn, E. and J. Rley, Optmal Mult-Unt Auctons, n F. Hahn ed., The Economcs of Mssng Markets, Informaton, and Game, Clarendon Press: Oxford 1989, Myerson, R., Optmal Aucton Desgn, Mathematcs of Operatons Research (1981), 6, Pashgan, P. and E. Gould, Internalzng Externaltes: The Prcng of Space n Shoppng Malls, Journal of Law and Economcs (1998), XLI, Segal, I., Contractng wth Externaltes, Quarterly Journal of Economcs (1999), CXIV, Shryaev, A. N., Probablty. New York: Sprnger

24 α j 3 α j 2 t (θ, θ ) α j 1 θ = 0 θ 1 θ 2 θ 3 θ Fgure 1 23