CORC Report TR : Short Version Optimal Procurement Mechanisms for Divisible Goods with Capacitated Suppliers

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1 CORC Report TR : Short Version Optimal Prourement Mehanisms for Divisible Goods with Capaitated Suppliers Garud Iyengar Anuj Kumar First version: June 30, 2006 This version: August 31, 2007 Abstrat The literature on prourement autions typially assumes that the suppliers are unapaitated (see, e.g. Dasgupta and Spulber, 1990; Che, 1993). Consequently, these aution mehanisms award the ontrat to a single supplier. We study mehanism design in a model where suppliers have limited prodution apaity, and both the marginal osts and the prodution apaities are private information. We provide a losed form solution for the revenue maximizing diret mehanism when the distribution of the ost and prodution apaities satisfies a modified regularity ondition (Myerson, 1981). We also present a sealed low bid implementation of the optimal diret mehanism for the speial ase of idential suppliers, i.e. symmetri environment. The results in this paper extend to other priniple-agent mehanism design problems where the agents have a privately known upper bound on alloation. Examples of suh problems inlude monopoly priing with adverse seletion and forward autions. Keywords: Prourement autions, Optimal diret mehanism, Capaity onstraints, Multiple Souring. JEL Classifiation: D24, D44. Industrial Engineering and Operations Researh Department, Columbia University, New York, NY garud@ieor.olumbia.edu. Researh partially supported by NSF grants CCR , DMS and ONR grant N Industrial Engineering and Operations Researh Department, Columbia University, New York, NY ak2108@olumbia.edu. Researh partially supported by NSF grant DMS and ONR grant N

2 1 Bakground and Motivation Using autions to award ontrats to supply goods and servies is now pervasive in many industries, e.g. eletronis industry prourements, government defene prourements (Naegelen, 2002; Dasgupta and Spulber, 1990, and referenes therein), supply hain prourements (Chen, 2004, and referenes therein). Sine the autioneer is the buyer, the bidders are the suppliers or sellers, and the objet being autioned is the right to supply, these prourement autions are also alled reverse autions. The use of reverse autions to award ontrats has been vigorously advoated sine ompetitive bidding results in lower prourement osts, failitates demand revelation, allows order quantities to be determined ex-post based on the bids and limits the influenes of nepotism and politial ties. Moreover, the advent of the Internet has signifiantly redued the transation osts involved in onduting suh autions. There is now a large body of literature detailing the growing importane of reverse autions in industrial prourement. Aording to Parente et al. (2001), the total value of the B2B online aution transations totaled 109 billion in 1999, and that number was expeted to grow to 2.7 trillion by Although aution design is a well-studied problem, the models analyzed thus far do not adequately address the fat that the private information of the bidders is typially multi-dimensional (ost, apaity, quality, lead times, et.) and the instruments available to the autioneer to sreen this private information is also multidimensional, e.g. multiple produts, multiple omponents, different prourement loations, et. This paper investigates mehanism design for a one-shot reverse aution with divisible goods and apaitated suppliers, i.e. suppliers with finite apaities. The prodution apaities, in addition to the prodution osts, are only known to the respetive suppliers and need to be sreened by an appropriate mehanism. Thus, in our model the private information of the supplier is two dimensional. However, we assume that the suppliers an only underbid apaity. We show how to onstrut the optimal revenue maximizing diret mehanism for this model. Although the general Bayesian mehanism design problem with 2-dimensional types whih is known to be hard, we are able to irumvent the diffiulties in the general problem by exploiting the speifi struture of the model, in partiular that the suppliers are only allowed to underbid apaity. The basi insight is that the optimal mehanism does not give any information rent to a supplier for revealing apaity information when the prodution ost is known. We also present a low bid implementation of the optimal aution in a symmetri environment. The paper is organized as follows. In 1.1 we disuss some of the relevant literature. In 2 we desribe the model preliminaries. In this setion, we also elaborate on the suppliers inentive to lie about apaity and onsider various speial ases of the prourement aution problem. In 3 we present the optimal diret aution mehanism and its implementation via pay as you bid reverse aution. In 4 we disuss limitations of our model and diretions for future researh. 1.1 Literature Review Myerson (1981) first used the indiret utility approah to derive the optimal aution in an independent private value (IPV) model. Che (1993) onsiders 2-dimensional (reverse) aution where the sellers bid prie and quality, and the buyer s utility is a funtion of both quality and prie. However, in this model only the osts are private information; thus, the bidder type spae is one-dimensional. Also, Che (1993) only onsiders souring from a single supplier; therefore, the problem redues to 2

3 one of determining the winning probability instead of the expeted alloation. Naegelen (2002) models reverse autions for department of defense (DoD) projets by a model where the quality of eah of the firms is fixed and is ommon knowledge. The preferene over quality in this setting results in virtual utilities whih are biased. Again, she only onsiders the single winner ase. Dasgupta and Spulber (1990) onsider a model very similar to the one disussed in this paper exept that the suppliers have unlimited apaity. They onstrut the optimal aution mehanism for both single souring and multiple souring (due to non-linearities in prodution osts) when the private information is one-dimensional. Chen (2004) presents an alternate two-stage implementation for the optimal mehanism in Dasgupta and Spulber (1990). In this alternate implementation the winning firm is first determined via ompetition on fixed fees, and then the winner is offered an optimal prie-quantity shedule. Laffont et al. (1987) solve the optimal nonlinear priing (single agent priniple-agent mehanism design) problem with a two-dimensional type spae. They expliitly fore the integrability onditions on the gradient of the indiret utility funtion. Surprisingly, the optimal priing mehanism (the bundle menus) is rather involved even when the prior is uniform. Rohet and Stole (2003) also provide an exellent survey of multi-dimensional sreening and the assoiated diffiulties. Vohra and Malakhov (2004) desribe the indiret utility approah in multi-dimensional disrete type spaes. They show that network-flow tehniques an be used to establish many of the known results in aution theory in a very elegant and easily interpretable manner. They also show how to simplify the assoiated optimization problem by identifying and relaxing provably redundant inentive ompatibility onstraints. In Vohra and Malakhov (2005), the authors use these tehniques to identify the optimal mehanisms for an aution with apaitated bidders where both the apaity and marginal values are private information and the bidders are only allowed to lie about apaities in one diretion. Thus, the model they onsider is idential to the one disussed here and to an extent their work influenes the results in this paper. The main methodologial ontributions that distinguishes our work are as follows. (a) In Vohra and Malakhov (2005), the authors restrit attention to only those alloation rules that are monotone in the apaity dimension (i.e. the speial type). We show that any alloation rule that is monotone in the marginal ost for a fixed apaity bid an be made inentive ompatible by offering a side-payment to the suppliers that is only a funtion of the apaity bid (see Lemma 1). Thus, the spae of all inentive ompatible mehanisms is muh larger than the one onsidered in Vohra and Malakhov (2005). Our haraterization result also implies that the transfer payment is no longer uniquely determined by the alloation rule. A reverse aution with apaitated suppliers is speial in that the objetive does not expliitly depend on apaity bid the apaity bid only ontrols the feasibility of an alloation rule. This speial struture allows one to onlude that, when the prior distribution is regular, the optimal alloation rule is monotone in the apaity bid and, therefore, the optimal side payment an be set to zero, i.e. the solution in Vohra and Malakhov (2005) is indeed optimal. When the objetive funtion expliitly depends on the speial type, e.g. in bin paking with privately known weights or sheduling with privately known deadlines, one annot regularize the prior distribution. Consequently, the mehanism design problem even with one-sided lying remains a hard problem. 3

4 (b) We develop a new ironing proedure whih allows us to haraterize the optimal mehanism under milder regularity onditions. See Setion 3.2 for details. () Vohra and Malakhov (2005) study fixed quantity autions in disrete type spae where all the bidders have a linear utility funtion. In ontrast, we study variable quantity reverse autions in a ontinuous type spae. This allows us the flexibility of working with more general utility strutures. Notation We denote vetors by boldfae lowerase letters, e.g. x. A vetor indexed by i, (for example x i ) denotes the vetor x with the i-th omponent exluded. We use the onvention x = (x i, x i ). Salar (resp. vetor) funtions are denoted by lowerase (resp. boldfae) letters, e.g. x i (θ i, θ i ) (resp. x(θ i, θ i )) and onditional expetation of funtions by the upperase of the same letter, e.g. X i (θ i ) Eθ i x i (θ i, θ i ) (resp. X i (θ i ) = E θ i [x i (θ i, θ i )]. The possible misreport of the true parameters are represented with a hat over the same variable, e.g. ˆθ. 2 Reverse autions with finite supplier apaities We onsider a single period model with one buyer (retailer, manufaturer, et.) and n suppliers. The buyer purhases a single ommodity from the suppliers and resells it in the onsumer market. The buyer reeives an expeted revenue, R(q) from selling q units of the produt in the onsumer market the expetation is over the random demand realization and any other randomness involved in the downstream market for the buyer that is not ontratible. Thus, the side-payment to the suppliers annot be ontingent on the demand realization. We assume R(q) is stritly onave with R(0) = 0, R (0) = and R ( ) = 0, so that quantity ordered by the buyer is non-zero and bounded. Without this assumption the results in this paper would remain qualitatively the same; however, the optimal mehanism would have a reservation ost above whih the buyer will not order anything. Charaterizing the optimal reserve ost is straightforward and is well-studied (see, e.g. Dasgupta and Spulber, 1990). Supplier i, i = 1,..., n, has a onstant marginal prodution ost i [, ] (0, ) and finite apaity q i [q, q] (0, ). The joint distribution funtion of marginal ost i and prodution apaity q i is denoted by F i. We assume that ( i, q i ) and ( j, q j ) are independently distributed when i j, i.e. our model is an independent private value (IPV) model. We assume that distribution funtions {F i } n are ommon knowledge; however, the realization ( i, q i ) is only known to supplier i. The buyer seeks a revenue maximizing prourement mehanism that ensures that all suppliers partiipate in the aution. We employ the diret mehanism approah, i.e. the buyer asks suppliers to diretly bid their private information ( i, q i ). The revelation priniple (see Myerson, 1981; Harris and Townsend, 1981) implies that for any given mehanism one an onstrut a diret mehanism that has the same point-wise alloation and transfer payment as the given mehanism. Sine both mehanisms result in the same expeted profit for the buyer, it follows that there is no loss of generality in restriting oneself to diret mehanisms. 4

5 We denote the true type of supplier by b i = ( i, q i ) and the supplier i bid by b i = (ĉ i, ˆq i ). Let b = (b 1,..., b n ) and b = ( b 1,..., b ( n n ). Let B [, ] [q, q]) denote the type spae. A prourement mehanism onsists of (a) an alloation funtion x : B R n + that for eah bid vetor b speifies the quantity to be ordered from eah of the suppliers, and (b) a transfer payment funtion t : B R n that maps eah bid vetor b to the transfer payment from the buyer to the suppliers. The buyer seeks an alloation funtion x and a transfer funtion t that maximizes the ex-ante expeted profit ( n ) ] n Π(x, t) E b [R x i (b) t i (b) subjet to the following onstraints. 1. feasibility: x i (b) q i for all i = 1,..., n, and b B, 2. inentive ompatibility (IC): Conditional on their beliefs about the private information of other bidders, truthfully revealing their private information is weakly dominant for all suppliers, i.e. ( i, q i ) argmax E b i {t i ((ĉ i, ˆq i ), b i ) i x i ((ĉ i, ˆq i ), b i )}, i = 1,..., n, (1) ĉ i [, ] ˆq i [q,q i ] The above definition of inentive ompatibility is alled Bayesian inentive ompatibility (see Appendix??). Note that the range for the apaity bid ˆq i is [q, q i ], i.e. we do not allow the supplier to overbid apaity. This an be justified by assuming that the supplier inurs a heavy penalty for not being able to deliver the alloated quantity. 3. individual rationality (IR): The expeted interim surplus of eah supplier firm is non-negative, for all i = 1,..., n, and b B, i.e. π i (b i ) E b i [t i (b) i x i (b)] = T i ( i, q i ) i X i ( i, q i ) 0. (2) Here we have assumed that the outside option available to the suppliers is onstant and is normalized to zero. For any prourement mehanism (x, t), the offered expeted surplus ρ i (ĉ i, ˆq i ) when supplier i bids (ĉ i, ˆq i ) is defined as follows ρ i (ĉ i, ˆq i ) = T i (ĉ i, ˆq i ) ĉ i X i (ĉ i, ˆq i ) The offered surplus is simply a onvenient way of expressing the expeted transfer payment. The expeted surplus π i ( i, q i ) of supplier i with true type ( i, q i ) when she bids (ĉ i, ˆq i ) is given by π i ( i, q i ) = T i (ĉ i, ˆq i ) i X i (ĉ i, ˆq i ) = ρ i (ĉ i, ˆq i ) + (ĉ i i )X i (ĉ i, ˆq i ). The true surplus π i equals the offered surplus ρ i if the mehanism (x, t) is IC. To further motivate the prourement mehanism design problem, we elaborate on a supplier s inentives to lie about apaity and then onsider some illustrative speial ases. 5

6 2.1 Inentive to underbid apaity In this setion we show that autions that ignore the apaity information are not inentive ompatible. In partiular, the suppliers have an inentive to underbid apaity. Suppose we ignore the private apaity information and implement the lassi K th prie aution where the marginal payment to the supplier is equal to the ost of the first losing supplier, i.e. lowest ost supplier among those that did not reeive any alloation. Then truthfully bidding the marginal ost is a dominant strategy. However, we show below that in this mehanism the suppliers have an inentive to underbid apaity. Underbidding reates a fake shortage resulting in an inrease in the transfer payment that an often more than ompensates the loss due to a derease in the alloation. The following example illustrates these inentives in dominant strategy and Bayesian framework. Example 1. Consider a prourement aution with three apaitated suppliers implemented as the K th prie aution. Let = 1, = 5, q =.01 and q = 6. Suppose the apaity realization is (q 1, q 2, q 3 ) = (5, 1, 5) and the marginal ost realization is ( 1, 2, 3 ) = (1, 1, 5). Suppose the buyer wants to proure 5 units and that the spot prie, i.e. the outside publily known ost at whih the buyer an proure unlimited quantity is equal to 10. (We need to have an outside market when modeling fixed quantity aution beause the realized total apaity of the suppliers an be less than the fixed quantity that needs to be proured.) Assume that suppliers 2 and 3 bid truthfully. Consider supplier 1. If she truthfully reveals her apaity, her surplus is $0; however, if she bids ˆq 1 = 4 ɛ, her surplus is equal to $9(4 ɛ). Thus, bidding truthfully is not a dominant strategy for supplier 1. Next, we show that for appropriately hosen asymmetri prior distributions supplier 1 has inentives to underbid apaity even in the Bayesian framework. Assume that the marginal ost and apaity are independently distributed. Let ( 1, q 1 ) = (1, 5). Thus P((1 2 ) (1 3 )) = 1. Let the apaity distribution F q i, i = 2, 3, be suh that P(q 2 + q 3 1) > 1 ɛ for some 0 < ɛ 1. Then the expeted surplus π 1 (1, 5), if supplier 1 bids her apaity truthfully, is upper bounded by 5 ( 1) = 20. On the other hand the expeted surplus if she bids 4 ɛ is lower bounded by 9 (4 ɛ) (1 ɛ). Thus, supplier 1 has ex-ante inentive to underbid apaity. Figure 1 shows two uniform prie aution mehanisms, the K th prie aution and the market learing mehanism. In our model, the suppliers an hange the supply ladder urve both in terms of loation of the jumps (by misreporting osts) and the magnitude of the jump (by misreporting apaity). We know that in a model with ommonly known apaities, the fixed quantity optimal aution an be implemented as K th prie aution. We showed in the example above that in the K th prie aution with privately known apaity, the suppliers an game the mehanism. This effet is also true if pries are determined by the market learing ondition. Suppose the suppliers truthfully reveal their marginal osts and the buyer aggregates these bids to form the supply urve Q(p) = n ˆq i1 {i p}. The demand urve D(p) in this ontext is given by D(p) = argmax [R(u) pu] = (R ) 1 (p). u 0 Thus, the equilibrium prie p is given by the solution of the market learing ondition (R ) 1 (p) = Q(p ) (see Figure 1). The model primitives ensure that the market learing prie p (0, ). 6

7 Demand K th Prie Cummulative Capaity Market Clearing Prie 10 Supply Marginal Cost Figure 1: Uniform prie autions: K th prie aution and market learing prie aution In suh a setting, as in the K-th prie aution, the supplier with low ost and high apaity an at times inrease surplus by underbidding apaity beause the inrease in the marginal (market learing) prie an offset the derease in alloation. The above disussion shows that both the K-th prie aution and the market-learing mehanism are not truth revealing. In we show that if the suppliers bid the ost truthfully for exogenous reasons, the buyer an extrat all the surplus, i.e. the buyer does not pay any information rent to the suppliers for the apaity information. In this mehanism the transfer payments are simply the true osts of the supplier and the quantity alloated is a monotonially dereasing funtion of the marginal ost. This optimal mehanism is disriminatory and unique. In partiular, with privately known apaities, there does not exist a uniform prie optimal aution. Ausubel (2004) shows that a modified market learing mehanism, where items are awarded at the prie that they are linhed, is effiient, i.e. soially optimal ((see, also Ausubel and Cramton, 2002)). 2.2 Relaxations In this setion we disuss some speial ases of the prourement mehanism design problem formulated in 2. 7

8 2.2.1 Full information (or first-best) solution Suppose all suppliers bid truthfully. It is lear that in this setting the surplus of eah supplier would be identially zero. Denote the marginal ost of supplier firm with i th lowest marginal ost by [i] and it s apaity by q [i]. Then the piee-wise onvex linear ost funtion faed by the buyer is given by i 1 i 1 i 1 i (x) = q [j] [i] + x q [j] [i] for q [j] x q [j] (3) j=1 j=1 The optimal prourement strategy for the buyer is the same as that of a buyer faing a single supplier with piee wise linear onvex prodution ost (x). Clearly, multi-souring is optimal with a number of lowest ost suppliers produing at apaity and at most one supplier produing below apaity. Multiple souring an also our in an unapaitated model when the prodution osts are nonlinear. We expet that a risk averse buyer would also find it advantageous to multi-soure to diversify the ex-ante risk due to the asymmetri information. Sine, to the best of our knowledge, the problem of optimal autions with a risk averse prinipal has not been fully explored in the literature, this remains a onjeture. j=1 j= Seond-degree prie disrimination with a single apaitated supplier Suppose there is a single supplier with privately known marginal ost and apaity. Suppose the apaity and ost are independently distributed. Let F () and f() denote, respetively, the umulative distribution funtion (CDF) and density of the marginal ost and suppose the hazard rate f() F () is monotonially dereasing, i.e. we are in the so-alled regular are (Myerson, 1981). Note that this setting is the prourement ounterpart of seond degree prie disrimination in the monopoly priing model. We will first review the optimal mehanism when the supplier is unapaitated. Using the indiret utility approah, the buyer s problem an formulated as follows. ( max E [R(x()) + F () ) ] x(). (4) x( ) 0 f() x( )monotone Let x () denote the optimal solution of the relaxation of (4) where one ignores the monotoniity assumption, i.e. { ( x () argmax R(x) + F () ) } x. x 0 f() Then, regularity implies that x is a monotone funtion of, and is, therefore, feasible for (4). The transfer payment t () that makes the optimal alloation x inentive ompatible is given by t () = x () + x (u)du. Sine the optimal alloation x () and the transfer payment t () are both monotone in, the ost parameter an be eliminated to obtain the transfer t diretly in terms of the alloation x, i.e. 8

9 a tariff t (x). The indiret tariff implementation is very appealing for implementation as it an posted and the supplers an simply self-selet the prodution quantity based on the posted tariff. Now onsider the ase of a apaitated supplier. Feasibility requires that for all [, ], 0 x() q. Suppose the supplier bids the apaity truthfully. (We justify this assumption below.) Then the buyer s problem is given by ( max E (,q) [R(x(, q)) + F () ) ] x(, q) (5) x(, ) 0 f() x(,q)monotone where F denotes the marginal distribution of the ost. Set the alloation ˆx(, q) = min{x (), q}, where x denotes the optimal solution of the unapaitated problem (4). Then ˆx is learly feasible for (5). Moreover, { ( ˆx(, q) argmax R(x) + F () ) } x. 0 x q f() Thus, ˆx is an optimal solution of (5). As before, set transfer payment ˆt(, q) = ˆx(, q)+ ˆx(u, q)du. Then, the supplier surplus in the solution (ˆx, ˆt) is non-dereasing in the apaity bid q. Therefore, it is weakly dominant for the supplier to bid the apaity truthfully, and our initial assumption is justified. Note that the supplier surplus ˆπ(, q) = ˆx(u, q)du. The fat that the apaitated solution ˆx(, q) = min{x (), q} is simply a trunation of the unapaitated solution x () allows one to implement it in a very simple manner. Suppose the buyer offers the seller the tariff t (x) orresponding to the unapaitated solution. Then the solution x of the seller s optimization problem max 0 x q {t (x) x} is given by x = min{x (), q} = ˆx(, q), i.e. the quantity supplied is the same as that ditated by the optimal apaitated mehanism. Define q = sup{ [, ] : x () q}. Then the monotoniity of x () implies that { x (), > q, x = ˆx(, q) = q, q. Then, for all > q, the supplier requests x () and reeives a surplus π() = t (x ()) x () = x (u)du = For q, the supplier request q and the surplus π() = t (q) q, = t (x ( q )) q q + ( q )q, = π ( q ) + ( q )q = q x (u)du + min{x (u), q}du = q qdu = ˆx(u, q)du = ˆπ(, q). ˆx(u, q)du = ˆπ(, q). Thus, the supplier surplus in the tariff implementation is ˆπ(, q), the surplus assoiated with optimal apaitated mehanism. Consequently, it follows that the full tariff implements the apaitated optimal mehanism! This immediately implies that the buyer does need to know the apaity of the supplier, and pays zero information rent for the apaity information. In the next setion we show that the assumption of independene of apaity and ost is ritial for this result. 9

10 2.2.3 Marginal Cost ommon knowledge Suppose the marginal osts are ommon knowledge and only the prodution apaities are privately known. Then the optimal prourement mehanism maximizes [ ] n ) n max E q R (q i x i (q) t i (q) (x,t) suh that the expeted supplier i surplus T i (q i ) i X i (q i ) is weakly inreasing in q i (IC) and nonnegative (IR). Not surprisingly, the first-best or the full-information solution works in this ase. Set the transfer payment equal to the prodution osts of the supplier, i.e. t i (q) = i x i (q). Then the supplier surplus is zero and the buyer s optimization problem redues to the full-information ase. Sine the full-information alloation x i (ˆq i, q i ) is weakly inreasing in ˆq i for all q i, bidding the true apaity is a weakly dominant strategy for the suppliers. Thus, the buyer an effetively ignore the IC onstraints above and follow the full information alloation sheme and extrat all the supplier surplus. The fat that, onditional on knowing the ost, the buyer does not offer any informational rent for the apaity information is ruial to the result in the next setion. 3 Charaterizing Optimal Diret Mehanism We use the standard indiret utility approah to haraterize all inentive ompatible and individually rational diret mehanisms and the minimal transfer payment funtion that implements a given inentive ompatible alloation rule (see Lemma 1). The haraterization of the transfer payment allows us to write the expeted profit of the buyer for a given inentive ompatible alloation rule as a funtion of the alloation rule and the offered surplus ρ i (, q) (see Theorem 1). To proeed further, we make the following assumption. Assumption 1. For all i = 1, 2,, n, the joint density f i ( i, q i ) has full support. Therefore, the onditional density f i ( i q i ) also has full support. Lemma 1. Prourement mehanisms with apaitated suppliers satisfy the following. (a) A feasible alloation rule x : B R n + is IC if, and only if, the expeted alloation X i ( i, q i ) is non-inreasing in the ost parameter i. (b) A mehanism (x, t) is IC and IR if, and only if, the alloation rule x satisfies (a) and the offered surplus ρ i (ĉ i, ˆq i ) when supplier i bids (ĉ i, ˆq i ) is of the form ρ i (ĉ i, ˆq i ) = ρ i (, ˆq i ) + with ρ i (ĉ i, ˆq i ) non-negative and non-dereasing in ˆq i for all ĉ i [, ]. ĉ i X i (u, ˆq i )du (6) Remark 1. Reall that the offered surplus ρ i is, in fat, equal to the surplus π i when the alloation rule x (and the assoiated transfer payment t) is IC. 10

11 Proof: Fix the mehanism (x, t). Then the supplier i expeted surplus π i ( i, q i ) is given by π i ( i, q i ) = max {T i (ĉ i, ˆq i ) i X i (ĉ i, ˆq i )}. (7) ĉ i [, ] ˆq i [q,q i ] Note that the apaity bid ˆq i q i, the true apaity. This plays an important role in the proof. From (7), it follows that for all fixed q [q, q], the surplus π i ( i, q i ) is onvex in the ost parameter i. (There is, however, no guarantee that π i ( i, q i ) is jointly onvex in ( i, q i ).) Consequently, for all fixed q [q, q], the funtion π i ( i, q i ) is absolutely ontinuous in and differentiable almost everywhere in. Sine x is IC, it follows that ( i, q i ) ahieves the maximum in (7). Thus, in partiular, i argmax {T i (ĉ i, q i ) i X i (ĉ i, q i )}, (8) ĉ i [, ] i.e. if supplier i bids apaity q truthfully, it is still optimal for her to bid the ost truthfully. Sine π i ( i, q i ) is onvex in i, (8) implies that π i (, q) = X i (, q), a.e. (9) Consequently, X i (, q) is non-inreasing in for all q [q, q]. This proves the forward diretion of the assertion in part (a). To prove the onverse of part (a), suppose X i ( i, q i ) is non-inreasing in i for all q i. Set the offered surplus ρ i (ĉ i, ˆq i ) = ρ i (ˆq i ) + X i (u, ˆq i )du where the funtion ρ i (ˆq i ) so that ρ i (ĉ i, ˆq i ) is non-dereasing in ˆq i for all ĉ i [, ]. There are many feasible hoies for ρ(ˆq i ). In partiular, if X i(,q) q exists a.e., one an set, ρ i (ˆq i ) = { q i sup i [, ] q For any suh hoie of ρ i, the supplier i surplus π i (ĉ i, ˆq i ) = ρ i (ĉ i, ˆq i ) + (ĉ i i )X i (ĉ i, ˆq i ), = ρ i (ˆq i ) + = ρ i (ˆq i ) + ρ i (ˆq i ) + i ( Xi (t, z) z ) dtdz }. ĉ i X i (u, ˆq i )du + (ĉ i i )X i (ĉ i, ˆq i ), i i X i (u, ˆq i )du + i ĉ i X i (u, ˆq i )du + (ĉ i i )X i (ĉ i, ˆq i ), i X i (u, ˆq i )du, (10) ρ i (q i ) + X i (u, q i )du, (11) i = T i ( i, q i ) i X i ( i, q i ) = π i ( i, q i ), 11

12 where (10) follows from the fat that X i (, q) in non-inreasing in for all fixed q and (11) follows from the ρ i (ĉ i, ˆq i ) is non-dereasing in ˆq i and ˆq i q i. Thus, we have established that it is weakly dominant for supplier i to bid truthfully, or equivalently x is an inentive ompatible alloation. From (9) we have that whenever x is IC we must have that the supplier surplus is of the form π i ( i, q i ) = π i (, q i ) + Sine x is IR, π i ( i, q i ) 0, and, sine x is IC, X i (u, q i )du. q i argmax {T i ( i, ˆq i ) i X i ( i, ˆq i )} = argmax {π i ( i, ˆq i )}. ˆq i q i ˆq i q i Thus, we must have that π i ( i, q i ) is non-dereasing in q i for all i [, ]. This establishes the forward diretion of part (b). Suppose the offered surplus if of the form (6) then (x, t) satisfies IR. Sine X i ( i, q i ) is noninreasing in i for all q i, it follows that π i ( i, q i ) is onvex in i for all q i and π i( i,q i ) i = X i ( i, q i ). Consequently, π i (ĉ i, ˆq i ) = ρ i (ĉ i, ˆq i ) + ( i ĉ i ) ( X i (ĉ i, ˆq i ) ) π i ( i, ˆq i ) π i ( i, q i ), where the last inequality follows from the fat that π i ( i, q i ) is non-dereasing in q i for all i and ˆq i q i. Thus, we have establishes that (x, t) is IC. Next, we use the results in Lemma 1 to haraterize the buyer s expeted profit. Theorem 1. Suppose Assumption 1 holds. Then the buyer profit Π(x, t) orresponding to any feasible alloation rule x : B R n + that satisfies IC and IR is given by ( n ) ] n n Π(x, ρ) = E b [R x i (b) x i (b)h i ( i, q i ) ρ i (q i ), (12) where ρ i (q i ) is the surplus offered when the supplier i bid is (, q i ) and H i (, q) denotes the virtual ost defined in Assumption 2. Remark 2. Theorem 1 implies that the buyer s profit is determined by both the alloation rule x and offered surplus ρ(q) when supplier i bid is (, q). We emphasize this by denoting the buyer profit by Π(x, ρ). Proof: From Lemma 1, we have that the offered supplier i surplus ρ i ( i, q i ) under any IC IR alloation rule x is of the form ρ i ( i, q i ) = ρ i (, q i ) + i X i (t, q i )dt and Thus, the buyer profit funtion is ( n ) n ( Π = E b [R x i (b) i x i (b) + ρ i (, q i ) )] ( n q 12 q i X i (u i, q i )du i f i ( i, q i )d i dq i ).

13 By interhanging the order of integration, we have d i f i ( i, q i ) i du i X i (u i, q i ) = du i X i (u i, q i ) Substituting this bak into the expression for profit, we get ( n ) n ( Π(x) = E b [R x i (b) i x i (b) + ρ i (, q i ) )] ( n ) = E b [R x i (b) ( n ) = E b [R x i (b) t n ( i x i (b) + ρ i (, q i ) )] n df i (, q i ) = ( i + F ) i( i q i ) x i (b) f i ( i q i ) ( n q n q X i ( i, q i )F i ( i q i )f i (q i )d i. X i ( i, q i )F i ( i q i )f i (q i )d i dq i ) [ E b x i (b) F ] i( i q i ) f i ( i q i ) ] n ρ i (, q i ). This establishes the result. The virtual marginal osts H i (, q) in our model are very similar to the virtual marginal osts in the unapaitated reverse aution model; exept that the virtual osts are now defined in terms of the distribution of the marginal ost i onditioned on the apaity bid q i. Thus, the apaity bid provides information only if the ost and apaity are orrelated. (See and for more on this issue). Next, we haraterize the optimal alloation rule under the regularity Assumption 2 and to a limited extent under general model primitives., 3.1 Optimal mehanism in the regular ase In this setion, we make the following additional assumption. Assumption 2 (Regularity). For all i = 1, 2,, n, the virtual ost funtion H i ( i, q i ) i + F i ( i q i ) f i ( i q i ) is non-dereasing in i and non-inreasing in q i. Assumption 2 is alled the regularity ondition. It is satisfied when the onditional density of the marginal ost given apaity, is log onave in i, and the prodution ost and apaity are, loosely speaking, negatively affiliated in suh a way that F i( i q i ) f i ( i q i ) is nondereasing in q i. This is true, for example, when the ost and apaity are independent. For b B, define x (b) argmax 0 x q { ( n ) R x i } n x i H i ( i, q i ), (13) where the inequality 0 x q is interpreted omponent-wise. We all x : B R n + the pointwise optimal alloation rule. Sine (13) is idential to the full information problem with the ost i replaed by the virtual ost H i ( i, q i ), it follows that (13) an be solved by aggregating all the suppliers into one meta-supplier. Denote the virtual ost of supplier with i th lowest virtual ost by h [i] and the orresponding apaity by q [i]. Then the buyer faes a piee-wise onvex linear ost funtion h(q) given by i 1 i 1 h(q) = q [j] h [i] + q [i], (14) j=1 13 j=1 q [j]

14 for i 1 j=1 q [j] q i j=1 q [j], i = 1,..., n, where 0 j=1 q [j] is set to zero. From the struture of the supply urve it follows that the optimal solution of (13) is of the form q [i], [i] < [i], x [i] = q [i], [i] = [i], (15) 0, otherwise, where 1 [i] n. Lemma 2. Suppose Assumption 2 holds. Let x : B R n + denote the point-wise optimal defined in (13). (a) x i (( i, q i ), b i ) is non-inreasing in i for all fixed q i and b i. non-inreasing in i for all q i. Consequently, X i ( i, q i ) is (b) x i (( i, q i ), b i ) is non-dereasing in q i for all fixed i and b i. Therefore, X i ( i, q i ) is nondereasing in q i for all fixed i. Proof: From (15) it is lear that x (( i, q i ), b i ) is non-inreasing in the virtual ost H i ( i, q i ). When Assumption 2 holds, the virtual ost H i ( i, q i ) is non-dereasing in i for fixed q i ; onsequently, the alloation x i is non-inreasing in the apaity bid q i for fixed i and b i. Part (a) is established by taking expetations of b i. A similar argument proves (b). We are now in position to prove the main result of this setion. Theorem 2. Suppose Assumption 1 and 2 hold. Let x denote the point-wise optimal solution defined in (13). For i = 1,..., n, set the transfer payment t i ( b) = ĉ i X i ( i, q i ) + ĉ i X i ((u, ˆq i ))du. (16) Then (x, t ) is Bayesian inentive ompatible revenue maximizing prourement mehanism. Proof: From (12), we have that the buyer profit [ { ( n ) Π(x, ρ) E b max R x i 0 x q }] n x i H i ( i, q i ) = Π(x, 0). Thus, all that remains to be shown is that the offered surplus ρ i orresponding to the transfer payment t satisfies ρ i (q i) = ρ i ( i, q i ) 0, and (x, t ) is IC and IR. From (16), it follows that the offered surplus Thus, ρ i (q i) = ρ i (, q i) 0. ρ i (ĉ i, ˆq i ) = ĉ i X i (u, ˆq i )du. (17) Next, Lemma 2 (a) implies that X i ((ĉ i, ˆq i ), b i ) is non-inreasing in i for all q i. From Lemma 2 (b), we have that X i (u, ˆq i ) is non-dereasing in ˆq i. From (17), it follows that π i ( i, q i ) is non-dereasing in q i for all i. Now, Lemma 1 (b) allows us to onlude that (x, t ) is IC. Sine (x, t ) satisfies IC, the offered surplus ρ i ( i, q i ) is, indeed, the supplier surplus. Then (17) implies that (x, t ) is IR. Next, we illustrate the optimal reverse aution on a simple example. 14

15 Example 2. Consider a prourement aution with two idential suppliers. Suppose the marginal ost i and apaity q i of eah of the suppliers are uniformly distributed over the unit square, Therefore, the virtual osts f i ( i, q i ) = 1 ( i, q i ) [0, 1] 2, i = 1, 2. H i ( i, q i ) = i + F i( i q i ) f i ( i q i ) = i + i = 2 i i [0, 1], i = 1, 2. It is lear that this example satisfies Assumption 1 and Assumption 2. Suppose the buyer revenue funtion R(q) = 4 q. Then, it follows that buyer s optimization problem redues to the point-wise problem x (, q) = argmax x q x i 2 i x i. The above onstrained problem an be easily solved using the Karush-Kuhn-Tuker (KKT) onditions whih are suffiient beause of strit onavity of the buyer s profit funtion. For i = 1, 2, the solution is given by, 1 2 i i, q i 1, i 2 i q i i i, q i < 1, x i (, q) = 2 i 0 i i, q i 1, { { } } 2 i min max 0, 1 q 2 i, q i otherwise. i where i, is the index of the supplier ompeting with supplier i. transfer payments are given by equation (16). The orresponding expeted In order for an alloation rule x to be Bayesian inentive ompatible it is suffiient that the expeted alloation X i ( i, q i ) be weakly monotone in i and q i. Assumption 2 ensures that the point-wise optimal alloation x i is weakly monotone in i and q i. This stronger property of x an be exploited to show that x an be implemented in the dominant strategy solution onept, i.e. there exist a transfer payment funtion under whih truth telling forms an dominant strategy equilibrium. Theorem 3. Suppose Assumption 1 and Assumption 2 hold. For i = 1,..., n, let the transfer payment be t i ( b) = ĉ i x i ( b) + x i ((u, ˆq i ), b i )du. (18) ĉ i Then, (x, t ) is an dominant strategy inentive ompatible individually rational revenue maximizing prourement mehanism. Proof: It is lear that the buyer profit under any dominant strategy IC and IR mehanism is upper bounded by the profit Π(x, 0) of the point-wise optimal alloation x. From (18), it follows that (x, t ) is ex-post (pointwise) IR. 15

16 Thus, all that remains is to show that (x, t ) is dominant strategy IC. Suppose supplier i bids (ĉ i, ˆq i ). Then, for all possible misreports ˆb of suppliers other than i, we have t i ((ĉ i, ˆq i ), b i ) ĉ i x i ((ĉ i, ˆq i ), b i ) = x i ((u, ˆq i ), b i )du i i + ĉ i x i ((u, ˆq i ), b i )du ( i ĉ i )x i ((ĉ i, ˆq i ), b i ), i x i ((u, ˆq i ), b i )du,, (19) i x i ((u, q i ), b i )du,, (20) = t i (b i, b i ) i x i (b i, b i ), where inequality (19) follows from the fat that x i (( i, q i ), b i ) is non-inreasing in i for all (q i, b i ) (see Lemma 2 (a)) and inequality (20) is a onsequene of the fat that x i (( i, q i ), b i ) is non-dereasing in q i for all ( i, b i ) (see Lemma 2 (b)). Thus, truth-telling forms a dominant strategy equilibrium. 3.2 Optimal Mehanism in the General Case In this setion, we onsider the ase when Assumption 2 does not hold, i.e. the distribution of the ost and apaity does not satisfy regularity. The optimal alloation rule is given by the solution to following optimal ontrol problem ] ( n ) n E b [R x i (b) H i ( i, q i )x i (b) + ρ i (q i ) max x(b), ρ(q) s.t 0 x i ( i, q i ) q i i, q i, i ĉ i i X i (ĉ i, q i ) X i ( i, q i ) q i, i, ĉ i, i (21) ˆq i q i (X i (z, q i ) X i (z, ˆq i ))dz ρ i (ˆq i ) ρ i (q i ) i i, q i, ˆq i, i 0 ρ i (q i ) q i, i This problem is a very large sale stohasti program and is, typially, very hard to solve numerially. We haraterize the solution, under a ondition weaker than regularity, whih we all semi-regularity. We adapt the standard one dimensional ironing proedure (see, e.g. Myerson, 1981) to our problem whih has a two-dimensional type spae. Let L( i, q i ) denote the umulative density along the ost dimension, i.e. L i ( i, q i ) = i f i (u, q i )du 16

17 Sine the density f i ( i, q i ) is assumed to be stritly positive, L i ( i, q i ) is inreasing in i, and hene, invertible in the i oordinate. Let K i (p i, q i ) = i H i (u, q i )f i (u, q i )dt where i = L i (, q i ) 1 (p i ). Let ˆK i denote the onvex envelop of K i along p i, i.e. ˆK i (p i, q i ) = inf {λk i (a, q i ) + (1 λ)k i (b, q i ) a, b [0, L i (, q i )], λ [0, 1], λa + (1 λ)b = p i }. Define ironed-out virtual ost funtion Ĥi( i, q i ) by setting it to Ĥ i ( i, q i ) = ˆK i (p, q) p pi =L i ( i,q i ),q i wherever the partial derivative is defined and extending it to [, ] by right ontinuity. Lemma 3. The funtion K i, the onvex envelop ˆK i and the ironed-out virtual osts Ĥ( i, q i ) satisfy the following properties. (a) Ĥi( i, q i ) is ontinuous and nondereasing in i for all fixed q i. (b) ˆK i (0, q i ) = K i (0, q i ), ˆKi (L i (, q i ), q i ) = K i (L i (, q i ), q i ), () For all q i and p i, ˆKi (p i, q i ) K i (p i, q i ). (d) Whenever ˆK i (p i, q i ) < K i (p i, q i ), there is an interval (a i, b i ) ontaining p i suh that p ˆK(p, q i ) =, a onstant, for all p (a i, b i ). Thus, Ĥ i ( i, q i ) is onstant with i L i (, q i ) 1 ((a i, b i )). See (Rokafeller, 1970) for the proofs of these assertions. Now, we are ready to state our weaker regularity assumption. Assumption 3 (Semi-Regularity). For all i = 1, 2,, n, the ironed out virtual marginal prodution ost, Ĥ i ( i, q i ) is non-inreasing in q i. From Lemma 3 (a) above, it follows that the semi-regularity implies the usual regularity of Ĥi, i.e. Ĥ i satisfies Assumption 2. Theorem 4 shows that if we use this ironed out virtual ost funtion in the buyer s profit funtion instead of the original virtual ost and then pointwise maximize to find the optimal alloation relaxing the monotoniity onstraints on the optimal alloation and the side payments ρ i, then the resulting mehanism is inentive ompatible with ρ i = 0 and revenue maximizing. Theorem 4. Suppose Assumption 3 holds. Let x I : B R n + denote any solution of the pointwise optimization problem { } ( n ) n R x i x i Ĥ i ( i, q i ). max 0 x q Set the transfer payment funtion t I i(b) = i x I i(b) + i x I i((u, q i ), b i )du. (22) Then (x I, t I ) is a revenue maximizing, dominant strategy inentive ompatible and individually rational prourement mehanism. 17

18 Proof: Let x be any IC alloation and let ρ denote the orresponding offered surplus. Define ( n ) ] n n Π(x, ρ) E b [R x i (b) x i (b)ĥi( i, q i ) ρ i (q i ), i.e. Π(x, ρ) denotes buyer profit when the virtual osts Hi ( i, q i ) are replaed by the ironed-out virtual osts Ĥi( i, q i ). Then q [ ] Π(x, ρ) Π(x, ρ) = (Ĥi ( i, q i ) H i ( i, q i )) X i ( i, q i )f i ( i, q i )d i dq i The inner integral = q (Ĥi ( i, q i ) H i ( i, q i )) X i ( i, q i )f i ( i, q i )d i = 0, ( L i ( i,q i ) ˆKi ( i, t) K i ( i, t)) 0 ( ˆKi ( i, q i ) K i ( i, q i )) f i ( i, q i )d i [X i ( i, q i )], ( ˆKi (L( i, q i ), q i ) K i (L( i, q i ), q i )) f i ( i, q i ) i X i ( i, q i ), (23) where (23) follows from Lemma 3 (b), and (24) follows from Lemma 3 () and the fat that i X i ( i, q i ) 0 for any IC alloation rule. Thus, we have the Π(x, ρ) Π(x, ρ). A proof tehnique idential to the one used to prove Theorem 3 establishes that (x I, t I ) is an dominant strategy IC and IR prourement mehanism that maximizes the ironed-out buyer profit Π. Note that the orresponding offered surplus ρ I (q) 0. Suppose ˆK i (L( i, q i ), q i ) < K i (L( i, q i ), q i ). Then Lemma 3 (d) implies that H i ( i, q i ) is a onstant for some neighborhood around i, i.e. i X i ( i, q i ) = 0 in some neighborhood of i. Consequently, the inequality (24) is an equality when x = x I, i.e. Π(x I, ρ I ) = Π(x I, ρ I ). This establishes the result. (24) Theorem 4 haraterizes the revenue maximization diret mehanism when the virtual osts H i ( i, q i ) satisfy semi-regularity, or equivalently, when the ironed-out virtual osts Ĥi( i, q i ) satisfy regularity. When semi-regularity does not hold, the optimal diret mehanism an still be omputed by numerially solving the stohasti program (21). Our numerial experiments lend support to the following onjeture. Conjeture 5. A revenue maximizing prourement mehanism has the following properties. (a) The side payments ρ 0. (b) There exist ompletely ironed-out virtual ost funtions H i suh that the orresponding pointwise solution x = argmax { R( i x i(b)) n H i ( i, q i ) x i (b) : 0 x q } is the revenue maximizing IC alloation rule. () The ironing proedure and the ompletely ironed-out virtual osts revenue funtion R, in addition to the joint prior. H i ( i, q i ) depend on the 18

19 Rohet and Chone (1998) presents a general approah for multidimensional sreening but in a model where the agents have both sided inentives. 3.3 Low-bid Implementation of the Optimal Aution In this setion, we assume that all the suppliers are idential, i.e. F i (, q) = F (, q), and that the distribution F (, q) satisfies Assumption 1 and Assumption 2. From (16) it follows that the expeted transfer payment T i ( i, q i ) = i X i ( i, q i ) + i X i (u, q i )du Note that Ti ( i, q i ) = 0, whenever Xi ( i, q i ) = 0. Define a new point-wise transfer payment t as follows. ( t i (b) = i + i Xi (t, q ) i)dt Xi ( x i ( i, q i ) (25) i, q i ) Then E ( i,q i ) [t i (, q)] = T i ( i, q i ), therefore, (x, t) is Bayesian IC and IR. We use the transfer funtion t to ompute the bidding strategies in a low bid implementation of the diret mehanism. The get-your-bid aution proeeds as follows: 1. Supplier i bids the apaity ˆq i q i, she is willing to provide and the marginal payment p i she is willing to aept. 2. The buyer s ations are as follows: (a) Solve for the true marginal ost i by setting 1 p i = φ( i, ˆq i ) = i + i Xi (t, ˆq i)dt Xi (z, ˆq. i) (b) Aggregates these bids and forms the virtual prourement ost funtion by setting (q) = i 1 j=1 ( ˆq [j] h [j] + i 1 q ˆq [j] )h [i] (26) for i 1 j=1 ˆq [j] q i j=1 ˆq [j], where as before h [i] denotes the i-th lowest virtual marginal ost and ˆq [i] is the apaity bid of the orresponding supplier. () Solve for the quantity q = argmax{r(q) (q)}. Set the alloation ˆq [i], i j=1 ˆq [j] q, x [i] = q i 1 j=1 ˆq i 1 [j], j=1 ˆq [j] q i j=1 ˆq [j], 0 otherwise. 1 We assume that φ( i, q i) is stritly inreasing in i, for all q i. This would be true, for example when the virtual osts H i are stritly inreasing in i. Note that previously, we had been working with alloations that were nondereasing. j=1 19

20 3. Supplier i produes x i and reeives p i x i. When all the suppliers are idential, the expeted alloation funtion Xi (, q) is independent of the supplier index i. We will, therefore, drop the index. Theorem 6. The bidding strategy q(, q) = q, p(, q) = φ(, q) + X (t, q)dt X, (, q) is a symmetri Bayesian Nash equilibrium for the get-your-bid prourement mehanism. Proof: Comparing (14) and (26), it is lear that, in equilibrium, x (b) = x(p, q). Assume that all suppliers exept supplier i use the bidding the proposed bidding strategy. Then the expeted profit π i (ˆp i, ˆq i ) of supplier i is given by π i (ˆp i, ˆq i ) = (ˆp i i ) X i (ˆp i, ˆq i ) = (ˆp i i )X i (ĉ i, ˆq i ), where ĉ i given by the solution of the equation ˆp i = φ(, ˆq i ). Thus, we have that π i (ˆp i, ˆq i ) (ˆp i i )X i (ĉ i, q i ), (27) = = ĉ i X (u, q i )du (ĉ i i )X ( i, q i ), i X (u, q i )du + ĉ i X (u, q i )du(ĉ i i )X ( i, q i ), i X (u, q i )du = π( p( i, q i ), q(q i )), (28) where (27) and (28) follows from, respetively, Lemma 2 (b) and (a). Thus, it is optimal for supplier i to bid aording to the proposed strategy. 3.4 Corollaries Sine the point-wise profit in (12) depends on the apaity q i only through the onditional distribution F i ( i q i ) of the ost i given apaity q i, the following result is immediate. Corollary 1. Suppose the marginal ost i and apaity q i are independently distributed. Then the optimal alloation rule (and the orresponding transfer funtion) is insensitive to the apaity distribution. Contrasting this result with the get-your-bid implementation in the last setion, we find that although the optimal aution mehanism is insensitive to the apaity, the supplier bidding strategies may depend on the apaity distribution. The following result haraterizes the buyers profit funtion when the suppliers apaity is ommon knowledge. 20

21 Corollary 2. Suppose suppliers apaity is ommon knowledge. Then the buyers expeted profit under any feasible, IC and IR alloation rule x is given by ( n ) n ( Π(x) = E [R x i () x i () i + F ) ] i( i ) (29) f i ( i ) Suppose the buyer wishes to proure a fixed quantity Q from the suppliers. Sine a given realization of the apaity vetor q an be insuffiient for the needs to the buyer, i.e. n q i < Q, we have to allow for the possibility of an exogenous prourement soure. We assume that the buyer is able to proure an unlimited quantity at a marginal ost 0 >. Let EC(Q) denote the expeted ost of prouring quantity Q by any optimal mehanism. Corollary 3 (Fixed Quantity Aution). Suppose Assumption 1 and 2 hold. Then min n x i(, q)q i H i ( i, q i ) + q 0 0 EC(Q) = E (,q) s.t. n x iq i + q 0 = Q. (30) 0 x q Results in this paper an be adapted to other priniple-agent mehanism design settings. Consider monopoly priing with apaitated onsumers. Suppose a monopolist seller with a stritly onvex prodution ost (x) faes a ontinuum of ustomers with utility of the form { θx t, x q, u(x, t; θ, q) =, x > q, where (θ, q) is the private information of the onsumers. The form of the utility funtion u(x, t; θ, q) prevents the ustomer from overbidding apaity. This is neessary for the seller to be able to hek individual rationality. As always the type distribution F : [θ, θ] [q, q] R ++ is ommon knowledge. Corollary 4. Suppose the distribution F (θ, q) satisfies the regularity assumption that ν(θ, q) = 1 F (θ q) θ f(θ q) is separately non-dereasing in both θ and q. Then the following holds for monopoly priing with apaitated buyers. (a) The seller profit Π(x) for any feasible, IC alloation rule x, the seller expeted profit is of the form [ ( Π(x) = E (θ,q) θ 1 F (θ q) ) ] x(θ, q) (x(θ, q)) f(θ q) (b) An optimal diret mehanism is given by the alloation rule and transfer payment x (θ, q) = argmax 0 x q t (θ, q) = [ ( θ 1 F (θ q) ) x (x) f(θ q) θ θ x (t, q)dt ] 21

22 Sine the type spae is two-dimensional, the optimal diret mehanism an be implement by a posted tariff only if the parameter θ and the apaity q are independently distributed. All our results in this setion easily extend to nonlinear onvex prodution ost i (θ, x), θ [θ, θ], that are super-linear, i.e. 2 i θ x > 0. In this ase, the virtual prodution ost H i(θ i, x) is given by 4 Conlusion and Extensions H i (θ i, x) = i (θ i, q i, x) + iθ (θ i, x) F i(θ i q i ) f i (θ i q i ). This paper proposes a prourement mehanism that is able to optimally sreen for both privately known apaities and privately known ost information. The results an be easily adapted to other priniple-agent mehanism design problems in whih agents have a privately known bounds on onsumption. In Iyengar and Kumar (2006) we show that our model extends to reverse aution with multiple produts when the private information about the ost is one dimensional. In this report, we also present an appliation of the multi-produt model to autioning multi-period supply ontrat in whih a buyer who faes the risk of variable apaities over time an effetively hedge this risk by ommitting to order from different suppliers in different periods. Two very simple natural extensions of the model proposed here lead to hard mehanism design problems: (a) Suppose the suppliers an purhase additional apaity at a ost. Then supplier s utility expliitly depends on the initial apaity and mehanism design problem is truly 2-dimensional. Thus, Lemma 1 fails to holds and the mehanism design problem does not appear to have any tratable formulation. (b) Suppose the private information about the produt ost in the multi-produt ase is multidimensional. In this ase even the unapaitated version of this problem remains unsolved. Referenes Ausubel, L. M. (2004). An effiient asending-bid aution for multiple objets. Amerian Eonomi Review, 94(5): Ausubel, L. M. and Cramton, P. (2002). Demand redution and ineffiieny in multi-unit autions. Working Paper, University of Maryland. Che, Y. K. (1993). Design ompetition through multidimensional autions. Rand Journal of Eonomis, 24(4): Chen, F. (2004). Autioning supply hain ontrats. Tehnial report, Deisions Risk and Operations, Columbia Business Shool. Dasgupta, S. and Spulber, D. F. (1990). Managing prourement autions. Information Eonomis and Poliy, (4):