This paper studies a procurement problem with one buyer and multiple potential suppliers who hold private

MANAGEMENT SCIENCE Vol. 53, No. 10, Otober 2007, pp. 1562 1576 issn 0025-1909 eissn 1526-5501 07 5310 1562 informs doi 10.1287/mns.1070.0716 2007 INFORMS Autioning Supply Contrats Fangruo Chen Graduate Shool of Business, Columbia University, New York, New York 10027 and Shool of Management, Shanghai Jiao Tong University, Shanghai 200052, China, f26@olumbia.edu This paper studies a prourement problem with one buyer and multiple potential suppliers who hold private information about their own prodution osts. Both the purhase quantity and the prie need to be determined. An optimal prourement strategy for the buyer requires the buyer to first design a supply ontrat that speifies a payment for eah possible purhase quantity and then invites the suppliers to bid for this ontrat. The aution an be onduted in many formats suh as the English aution, the Duth aution, the first-pried, sealed-bid aution, and the Vikrey aution. The winner is the supplier with the highest bid, and is given the deision right for the quantity produed and delivered. Applying this theory to a newsvendor model with supply-side ompetition, this paper establishes a onnetion between the above optimal prourement strategy and a ommon pratie in the retail industry, namely, the use of slotting allowanes and vendor-managed inventory. Also disussed in the newsvendor ontext are the role of well-known supply ontrats suh as returns ontrats and revenue-sharing ontrats in prourement autions, the senarios where the buyer and suppliers may possess asymmetri information about the demand distribution, and how the ost of supply-demand mismath is affeted by supply-side ompetition. Finally, this paper ompares the optimal prourement strategy with a simpler but suboptimal strategy where the buyer first determines a purhase quantity and then seeks the lowest-ost supplier for the quantity in an aution. Key words: prourement strategies; deision rights; newsvendor model; autions; ompetitive bidding; slotting allowanes History: Aepted by William S. Lovejoy, operations and supply hain management; reeived Marh 13, 2003. This paper was with the author 2 years for 3 revisions. Published online in Artiles in Advane August 31, 2007. 1. Introdution Mathematial models in operations management addressing prourement deisions (i.e., when and how muh to buy) often make the simplisti assumption that the pries for the items to be purhased are exogenously given. For example, the elebrated newsvendor model typially assumes a given purhase prie. The plethora of inventory models in the literature also make a similar assumption. In reality, prourement managers often need to disover the pries for the items they want to buy, and the disovery proess typially involves market researh, negotiations, autions/bidding, et. Prourement models in eonomis, on the other hand, tend to fous on prie disovery, fixing quantity deisions. The purpose of this paper is to understand how prie disovery an be integrated with quantity deisions to reah an optimal prourement strategy. We onsider a model with one buyer and a number of potential suppliers. The question faing the buyer is how muh of a produt should be purhased and from whih supplier. The buyer s revenue is a onave, inreasing funtion of the purhased quantity. Eah supplier an produe the produt at a onstant marginal ost and without any apaity limit. The marginal osts of different suppliers all ome from a ommon probability distribution and eah supplier is privately informed about his own marginal ost. The suppliers are risk neutral. We study the prourement problem from the buyer s standpoint, seeking an optimal prourement strategy that maximizes her expeted profit, whih is her revenue (as a funtion of the purhased quantity) minus the prourement ost. The struture of suh a strategy shows how prie disovery should be integrated with the quantity deision. A key result of this paper is that the following supply ontrat aution is optimal. Here the buyer first ommits to a supply ontrat, whih speifies a payment for eah quantity the buyer may purhase and delegates the quantity deision to the winning supplier. Eah potential supplier views this ontrat as a business opportunity, taking the payment funtion offered by the buyer as their revenue funtion. Based on their own marginal prodution ost, they eah determine an optimal quantity to produe and deliver to the buyer to maximize their own profit. Note that different suppliers derive different values from the ontrat, with the lowest-marginal-ost supplier deriving the most value. In an aution, the suppliers ompete for the ontrat by submitting an up-front, lump-sum fee they are willing to pay, with the winner being the supplier offering the highest fee. Only the 1562

Chen: Autioning Supply Contrats Management Siene 53(10), pp. 1562 1576, 2007 INFORMS 1563 winning supplier pays an up-front fee. 1 The aution an be onduted in many different forms, inluding the English aution, the Duth aution, the first-prie, sealed-bid aution, and the Vikrey aution. The integration of prie disovery and the quantity deision therefore requires the design of a supply ontrat (a quantity-to-payment mapping) followed by an aution that awards the ontrat to a supplier, who is given the deision right to determine the prourement quantity. This is intuitive. The aution hooses the most effiient supplier. Moreover, the lowest marginal ost among the suppliers together with the supply ontrat offered by the buyer determine the ultimate quantity deision. In other words, the quantity deision omes from both sides of the trade. The supply ontrat aution fits well with a prevalent pratie in the retail industry, i.e., the use of slotting allowanes and vendor-managed inventory (VMI). A slotting allowane is an up-front, lumpsum fee that a manufaturer pays to a retail store when it introdues a new produt to the store. It an also be a fee paid to keep an existing produt in the store, although in this ase, it is often referred to as a pay-to-stay fee. The total of slotting allowanes is estimated at between $6 and $9 billion a year (Rao and Mahi 2003). The reasons given for using slotting allowanes often fous on the need to signal to the retailer the manufaturer s belief about the produt s market potential or the need to ompensate the retailer for the risks of arrying a new produt. On the other hand, VMI refers to the pratie of delegating inventory deisions to the supplier. The arguments for this realloation of stoking deision rights are multifaeted, inluding the bargaining power of the retailer (e.g., Wal-Mart), the vendor s expertise in managing inventories, the vendor s ability to oordinate the prodution and distribution of multiple produts, modern ommuniation tehnologies, et. While VMI has been hailed as an important strategy for improving supply hain effiieny, the use of slotting allowanes is more ontroversial, attrating antitrust inquiries (see, e.g., Federal Trade Commission 1999). It is therefore interesting that our partiular optimal prourement strategy atually requires the use of a slotting allowane (i.e., the lump-sum fee) and the delegation of the prodution/inventory deision to the hosen supplier. The lump-sum fee results from the suppliers ompetition for the retailer s shelf spae, while the alloation of deision rights is to take advantage of the suppliers private ost information. 1 An alternative way to implement the supply ontrat aution is to require eah supplier to submit a two-dimensional bid onsisting of a fee and a quantity, but the winner is hosen solely based on the fee. This implementation eliminates any unertainty about what a supplier may deliver after being hosen. Thanks to a referee for making this suggestion. Our results therefore have the potential to ontribute to the on-going debate on the purposes and onsequenes of slotting allowanes. Interpreting the buyer in our prourement model as a newsvendor faing an inventory deision at the beginning of a selling season with unertain demand, we arrive at a newsvendor model with supply-side ompetition. When demand is unertain, we often see supply ontrats that are demand dependent. For example, a returns ontrat alls for a wholesale prie paid for eah unit delivered to the newsvendor at the beginning of the selling season, and a rebate for eah unit of exess inventory at the end of the season. On the other hand, a revenue-sharing ontrat ontains a wholesale prie for eah unit of inventory at the beginning of the selling season, and an agreement on how the realized revenues are to be split between the newsvendor and the seleted supplier. Note that under either a returns ontrat or a revenue-sharing ontrat, the (net) transfer payment for any given purhase quantity (or initial inventory) is unertain as it depends on the realized demand. This is learly different from the ontrat onsidered so far, whereby the transfer payment is a deterministi funtion of the purhase quantity, but for our risk-neutral suppliers, all that matters is the expeted transfer payment as a funtion of the purhase quantity. It is shown that under fairly general onditions, a returns ontrat or a revenue-sharing ontrat an indeed be found that, when autioned off among the suppliers, onstitutes an optimal prourement strategy for the newsvendor. We also disuss the advantages and disadvantages of these demand-dependent ontrats if the newsvendor and the suppliers have asymmetri information about the probability distribution of the demand. An alternative prourement strategy is for the firm to hoose a purhase quantity and then run an aution to identify a supplier that an deliver the quantity at the lowest ost. We refer to this strategy as the fixedquantity aution. It is suboptimal beause the quantity deision only takes into aount the distributional knowledge the buyer has about the suppliers osts (the atual prodution osts do not play a role in the quantity deision), but it is simpler than the optimal strategy beause it does not involve the speifiation of a supply ontrat. There are two deision variables that need to be determined: the purhase quantity and the reserve prie for the aution. For the newsvendor setting, we develop an algorithm to determine the optimal values of these variables that maximize the buyer s expeted profit. Numerial examples are used to illustrate the differene in the newsvendor s expeted profit between the fixed-quantity aution and the optimal prourement strategy. Closely related to this paper is Dasgupta and Spulber (1990; hereafter DS for brevity), who have provided a different solution to the above proure-

Chen: Autioning Supply Contrats 1564 Management Siene 53(10), pp. 1562 1576, 2007 INFORMS ment problem. We refer to their solution as the quantity aution, whih works as follows. As in the supply ontrat aution, the buyer first announes a supply ontrat (a payment funtion). The suppliers then submit quantity offers in a sealed, high-bid aution. The supplier with the highest bid (i.e., quantity offer) wins the aution, produes and delivers his bid, and reeives a payment aording to the preannouned ontrat. With a properly hosen supply ontrat, the quantity aution maximizes the buyer s expeted profit, but unlike the supply ontrat aution, the quantity aution must be onduted in the sealed-bid fashion, and would lose its optimality when implemented in other formats. Another distintion between the quantity aution and the supply ontrat aution is the amount of detail required to determine the optimal payment funtion: the quantity aution requires the number of potential suppliers, whereas the supply ontrat aution does not. 2 Also losely related to this paper is the literature on multidimensional autions (see, e.g., Che 1993 and Brano 1997). Here the suppliers ompete for a prourement ontrat on prie and quality. For example, the Department of Defense, in prouring a weapons system, ares about the system s performane as well as prie. The optimal design of a multidimensional aution typially speifies a soring rule that ombines the multiple dimensions of a bid (e.g., a quality speifiation and a prie) into a single sore, whih is then used to determine a winner. As mentioned earlier, the supply ontrat aution disussed in this paper an also be implemented as a multidimensional aution, with an up-front fee and a quantity, and a soring rule that puts the weight entirely on the former. For further disussions on the relationship between the supply ontrat aution and Che s soring rule approah, see 2.2. This paper is at the intersetion of aution theory and operations management. Aution theory has grown tremendously sine Vikrey s (1961) seminal work; MAfee and MMillan (1987a) and Klemperer (1999) provide omprehensive surveys of the theory. Supply ontrats have reently reeived muh researh attention in operations management and the researh effort is entered around their role in ahieving sup- 2 An alternative way to implement the quantity aution is via a wholesale prie aution. Here the buyer first announes a purhasing plan, whih speifies the quantity the buyer is ommitted to purhase as a funtion of the realized wholesale prie. The suppliers then engage in a sealed, low-bid aution, where the supplier submitting the lowest wholesale prie wins. The transation is ompleted at the lowest bid (wholesale prie) and the orresponding purhase quantity. The purhasing plan an be shown to be a quantity disount sheme, i.e., lower wholesale pries are assoiated with larger purhase quantities. Wholesale prie autions have been studied by Hansen (1988) and Jin and Wu (2000), where the purhasing plan is exogenously given. ply hain oordination (see Cahon 2003 and Chen 2003b). The ontribution of this paper is therefore to demonstrate how supply ontrats an be integrated with aution mehanisms to obtain an optimal prourement strategy (for one party of the supply hain). Researh efforts to introdue aution theory to the field of operations management have been on the rise. Many papers have started to inlude an aution mehanism or some other prie-disovery mehanism in an operations ontext (see Mendelson and Tuna 2000, Gallien and Wein 2000, Lee and Whang 2002, et.). Elmaghraby (2000) provides a survey of this area. Whereas our paper fouses on integrating a supply ontrat with an aution mehanism, others have studied the use of inentive ontrats in autions (see, e.g., Laffont and Tirole 1987; MAfee and MMillan 1986, 1987b; and Riordan and Sappington 1987). The setting typially inludes one prinipal and multiple agents. The prinipal has a projet for whih the agents ompete. Eah agent possesses private information and his ation is unobservable to the prinipal. The solution is to aution off an inentive ontrat among the agents. This part of the literature is also disussed in Laffont and Tirole (1993). The remainder of this paper is organized as follows: 2 desribes and develops different optimal prourement strategies; 3 applies these strategies to a newsvendor model with supply-side ompetition and ompares them with a suboptimal but simpler prourement strategy; 4 ontains onluding remarks. 2. Optimal Prourement Strategies Consider the following prourement problem with one buyer and multiple potential suppliers. The buyer an purhase any quantity of a produt from any of the suppliers. Let Q be the total quantity purhased and R Q the value the buyer attahes to the total purhase. Assume that R is stritly onave and inreasing with R 0 = 0. There are n, n 2, potential suppliers and eah of them is apable of produing the produt with a onstant marginal ost and an unlimited apaity. Let i be the marginal ost for supplier i, i = 1 n. Eah supplier is privately informed of his own ost. These osts are independent draws from a ommon probability distribution F,, whih is differentiable with F = 0 and F = 1. Define H x = x + F x (1) F x and assume H is inreasing. 3 The suppliers are risk neutral. We seek an optimal prourement strategy that maximizes the buyer s expeted profit. 3 This is a regularity ondition often assumed in aution ontexts (see, e.g., Myerson 1981). The assumption is learly true if F x /F x is inreasing in x, or equivalently, F is logonave. Many

Chen: Autioning Supply Contrats Management Siene 53(10), pp. 1562 1576, 2007 INFORMS 1565 2.1. Quantity Aution DS provides an optimal strategy for the above prourement problem. It requires the buyer to first announe a ontrat P, whereby the buyer pays P Q to a supplier if Q units are purhased from the supplier for any possible value of Q. Knowing P, the suppliers eah name a quantity in a sealed bid. The supplier who bids the maximum quantity wins the ontrat, produes and delivers his bid, and, in return, reeives a payment from the buyer aording to P. (The other suppliers do not produe and do not reeive any payment.) With a properly hosen payment funtion P, the above aution maximizes the buyer s expeted profit. We shall refer to the above prourement strategy as the quantity aution onduted in the sealed, high-bid format. (For brevity, we may say the quantity aution without speifying the format. In suh ases, we usually mean the sealed, high-bid format, but other formats are possible. The exat meaning should be lear from the ontext.) DS first onsidered the diret revelation game. The solution to this game is the optimal diret mehanism. They then suggested that the quantity aution implements the optimal diret mehanism and is thus optimal among all possible prourement strategies due to the revelation priniple (see, e.g., Kreps 1990). Below we provide a more detailed analysis of this prourement strategy. Consider the quantity aution (sealed, high-bid). Given the payment funtion P, the suppliers play a game of inomplete information (due to their private ost information) for whih the Bayesian-Nash equilibrium is an appropriate solution onept. Assume that there is a symmetri Bayesian-Nash equilibrium strategy; this is plausible beause the suppliers are ex ante symmetri (with independent and identially distributed (i.i.d.) osts). Denote this strategy by Q : a supplier with marginal ost bids Q,. Assume Q is stritly dereasing in for, for some, and Q = 0 for >. Note that there is a trade if and only if C 1, where C 1 is the lowest marginal ost among the suppliers. Beause the lowest-ost supplier always wins the aution, we know that R Q C 1 is the system s revenue and C 1 Q C 1 is the system s prodution ost. As a result, the expeted systemwide profit is R Q x xq x f 1 x dx (2) where f 1 is the probability density funtion (p.d.f.) probability distributions are logonave, inluding the beta family of whih the uniform distribution is a member, and the normal distribution trunated and saled to a finite interval. See Rosling (2002) for an extensive disussion on logonave probability distributions and further referenes on the topi. of C 1. To derive the buyer s expeted profit, it suffies to determine the suppliers expeted profits. Take any i = 1 n. Suppose that all the suppliers but supplier i play the strategy Q. Consider the problem faing supplier i. For Q to be an equilibrium strategy, it must be the ase that supplier i an do no better than bidding Q i for all i.in partiular, supplier i gains nothing by bidding Q x for some x i, whih is the same as playing the strategy Q, but pretending that his marginal ost is x. Take any and suppose that i =. Let i x be supplier i s expeted profit if he bids Q x while his marginal ost is, given that all the other suppliers play Q. Note that i x = P Q x Q x 1 F x n 1 (3) where P Q x Q x is the supplier s profit if he wins the aution (by bidding Q x ) and 1 F x n 1 is the probability of winning, whih ours if and only if every other supplier s marginal ost is greater than supplier i s reported marginal ost x. For i x to be maximized at x =, it is neessary that i x / x x= = 0, i.e., P Q Q Q 1 F n 1 P Q Q n 1 1 F n 2 F = 0 Using this equation in the expression of i, where i def = i for any, we have i = Q 1 F n 1. Setting i = i = 0, we have i = x dx = Q x 1 F x n 1 dx (4) with E i = = = { { x } Q x 1 F x n 1 dx F d } Q x 1 F x n 1 F d dx Q x 1 F x n 1 F x dx Beause the suppliers are symmetri, the sum of the expeted profits of the suppliers is simply n times the above expression. Subtrating ne i from (2) gives the buyer s expeted profit R Q x H x Q x f 1 x dx (5) Define Q x = arg max R Q H x Q x (6) Q 0 Beause R is onave and H is inreasing, Q x is dereasing in x. Set equal to the minimum x with Q x = 0; if no suh x exists, set =. Itis lear from (5) that the buyer s expeted profit is maximized if Q arises as a Bayesian-Nash equilibrium in the bidding game.

Chen: Autioning Supply Contrats 1566 Management Siene 53(10), pp. 1562 1576, 2007 INFORMS Suppose that Q is a Bayesian-Nash equilibrium. Take any. From (4), i = Q z 1 F z n 1 dz Beause i = i, we have from (3), P Q Q 1 F n 1 = Q z 1 F z n 1 dz Therefore, Q z 1 F z n 1 dz P Q = Q + (7) 1 F n 1 Denote by P the payment funtion that satisfies the above equation for all. The above derivation for P is entirely based on neessary onditions for Q to be a Bayesian-Nash equilibrium. It remains to verify that under P, Q indeed arises as suh. (DS did not address this issue.) To see this, suppose P is the payment funtion and assume that all players but player i follow strategy Q. Now in (3), replae P with P and Q with Q. We have i x = x Q x 1 F x n 1 + Note that i x x Q z 1 F z n 1 dz x = x x Q x 1 F x n 1 x Beause both Q x and 1 F x n 1 are dereasing in x, the partial derivative on the right side is negative. Therefore, i x / x > (respetively, <) 0 for x<(respetively, >). Consequently, i x is maximized at x =. The following theorem, whih we attribute to DS, summarizes the above development. Theorem 1 (Dasgupta and Spulber 1990). In the quantity aution defined by the payment funtion P and onduted in the sealed, high-bid format, Q arises as a ommon Bayesian-Nash equilibrium strategy for the suppliers, and the buyer s expeted profit is E R Q C 1 H C 1 Q C 1 This is also the highest expeted profit the buyer an ahieve among all feasible prourement strategies. The expression for the buyer s maximum expeted profit in Theorem 1 indiates that the buyer s lak of ost information is refleted in the virtual ost she pays to the winning supplier, i.e., she pays H x to the winning supplier when in fat his marginal prodution ost is x. The differene H x x is thus the information rent. The outome is as if the buyer had omplete information about the suppliers osts, but agreed to pay virtual osts. Several observations are immediate. The quantity traded is Q C 1, a dereasing funtion of C 1. Therefore, as ompetition intensifies, i.e., as n grows, C 1 beomes stohastially smaller, leading to a stohastially larger trade. Moreover, as expeted, the buyer s expeted profit inreases with supply-side ompetition. To formally show this, define x = max Q 0 R Q H x Q, a dereasing funtion of x beause H x is inreasing in x. Asn grows, C 1 beomes stohastially smaller, inreasing E C 1, the buyer s expeted profit. Finally, note that the effiient trade volume between the lowest-ost supplier and the buyer, i.e., the one that maximizes their joint gains, is Q 0 x def = arg max Q 0 R Q xq. Note that Q 0 x > Q x for all x. Hene, asymmetri ost information auses supply hain ineffiienies by reduing trade. This is reminisent of the well-known doublemarginalization phenomenon (see, e.g., Tirole 1988): the marginal ost faing the buyer, i.e., the virtual ost H C 1, is higher than the system s marginal ost, C 1. For independent, private-values autions, a wellknown result is the revenue equivalene theorem, whih states that the autioneer is indifferent among many ommonly used aution formats suh as the English aution, the Duth aution, the first-prie, sealed-bid aution, and the Vikrey aution. Does a similar result hold for the quantity aution? In other words, if the buyer an freely modify the payment funtion to suit the aution format used, an she still ahieve the optimal expeted profit by using an aution format other than sealed, high-bid? For example, the buyer an run the quantity aution in the Vikrey fashion: the suppliers submit quantity offers in sealed bids, the winner is the highest bidder, but the quantity produed by the winning supplier and delivered to the buyer is equal to the seond-highest bid. What is the buyer s maximum expeted profit in this ase? The following theorem shows that revenue equivalene does not hold for the quantity aution. The proof is in Chen (2003a). Theorem 2. The aution format used in the quantity aution matters:while the sealed, high-bid aution and the Duth aution are optimal (maximizing the buyer s expeted profit), the English aution and the Vikrey aution are not. Moreover, the buyer prefers the English aution to the Vikrey aution. 2.2. Supply Contrat Aution An important feature of the quantity aution is that the buyer first ommits to a payment funtion (supply ontrat). This payment funtion represents a potential soure of revenue for eah supplier. This business opportunity (of trading with the buyer) is likely to be valued differently by different suppliers, with the lowest-ost supplier ahieving the highest value. Therefore, an alternative way to selet a supplier is to

Chen: Autioning Supply Contrats Management Siene 53(10), pp. 1562 1576, 2007 INFORMS 1567 ask them to bid in terms of an up-front, lump-sum fee. The winner is the supplier offering to pay the highest fee, and only the winning supplier pays an up-front fee. The winning supplier then determines his prodution quantity (to maximize his profit), delivers it to the buyer, and reeives a payment from the buyer aording to the supply ontrat. This prourement strategy will be referred to as the supply ontrat aution. It is also optimal for the buyer, as we show next. Take any payment funtion P with P 0 = 0. Consider the problem of supplier seletion. Define v = max P Q Q (8) Q 0 Let Q = arg max Q 0 P Q Q. 4 Therefore, supplier i values the business opportunity at v i def = v i, i = 1 2 n. Beause the suppliers marginal osts are independent draws from a ommon distribution, the values v i n i=1 are i.i.d. random variables. Consequently, the problem of hoosing a supplier an be thought of as selling an objet (i.e., the business opportunity) to the highest bidder, where the bidders have i.i.d. valuations. From the revenue equivalene theorem, the buyer obtains the same expeted lumpsum fee (and selets the same supplier) if she uses the English aution, the Duth aution, the sealed, highbid aution, or the Vikrey aution. Suppose the buyer uses the English aution for supplier seletion. That is, the suppliers openly bid on the fee they are willing to pay for the privilege to trade, and the supplier with the highest bid wins and pays his bid. Clearly, the supplier with the highest valuation (and the lowest marginal ost) wins the aution and pays (to the buyer) a lump-sum fee equal to the valuation of the seond-lowest-ost supplier. Let V k = v C k, k = 1 n, where C k is the kth lowest ost. Thus, the lump-sum fee the buyer reeives is V 2. 5 We now proeed to determine the optimal payment funtion. Consider the buyer s ash flow. First, the buyer ollets a lump-sum fee, V 2, from the lowestost supplier. This supplier determines the quantity to maximize his profit, i.e., maximizing P Q C 1 Q over Q. The optimal quantity is Q C 1. The trade gives the buyer revenues in the amount of R Q C 1, but osts her P Q C 1. Consequently, the buyer s profit is def = R Q C 1 P Q C 1 + V 2 4 In the previous subsetion, Q was used for a Bayesian-Nash equilibrium strategy in the quantity aution. 5 A reader versed in aution theory may think that the buyer is better off with a reserve prie (see, e.g., Riley and Samuelson 1981). This is atually not the ase, as the subsequent development (summarized in Theorem 3) shows. The reason is that the aution is not regular in the sense of Myerson (1981). Chen (2003a) illustrates this with an example. Beause V 1 = P Q C 1 C 1 Q C 1, = R Q C 1 C 1 Q C 1 V 1 V 2 (9) We next obtain a onvenient expression for the expeted value of V 1 V 2, whih is the winning supplier s profit. Note from the optimization problem in (8) that Q is dereasing in. (This is true for any P.) Let 0 be the minimum with Q = 0. If Q > 0 for all, then set 0 =. Take any < 0. Thus, Q > 0. Writing v = P Q Q and differentiating, v = P Q Q Q Q Beause P Q = (the first-order ondition for the optimization problem in (8)), we have v = Q. On the other hand, for any > 0, Q = 0 and thus v = 0 beause P 0 = 0. Consequently, for all > 0, v = 0 = Q. Combining the two ases, we have V 1 V 2 = v C 1 v C 2 = C2 C 1 Q x dx Using the onditional probability density funtion of C 2 given C 1 =, we have [ C2 ] E Q x dx C 1 [ C2 ] = E C1 E Q x dx C 1 = = = C 1 nf 1 F n 1 d ( y ) n 1 F y 1 F y n 2 Q x dx dy 1 F n 1 ( y ) nf d Q x dx n 1 y= y= F y 1 F y n 2 dy whih, after hanging the order of integration twie (first between x and y and then between and x), beomes Q x nf x 1 F x n 1 dx (10) Substituting (10) for the expeted value of V 1 V 2 in (9), we have E = R Q x H x Q x nf x 1 F x n 1 dx whih is exatly the same as (5), the buyer s expeted profit in the quantity aution. (Note the different meanings of Q in the two plaes.) Although the above expression is obtained under the assumption that the buyer uses the English aution to selet a supplier, we know, from the revenue equivalene theorem, that the same expression holds if she instead uses the Duth aution, the first-prie, sealed-bid aution, or the Vikrey aution.

Chen: Autioning Supply Contrats 1568 Management Siene 53(10), pp. 1562 1576, 2007 INFORMS Note that if Q x = Q x, whih is defined in (6), for all x, then E equals the buyer s maximum expeted profit (Theorem 1). This is indeed possible. The payment funtion that ahieves this, denoted by P, is the solution to P Q = (11) for all with Q > 0. Note that this optimal payment funtion is inreasing in Q, onave beause Q is dereasing in, and independent of the number of bidders. Also note that it is independent of the aution format. Finally, adding a onstant to the payment funtion does not hange its optimality. Theorem 3. The buyer maximizes her expeted profit if she uses the supply ontrat aution with P as the payment funtion. Moreover, this payment funtion is inreasing, onave, and independent of the number of potential suppliers. We pause here to establish onnetions between the supply ontrat aution developed here and the soring rule aution developed by Che (1993) for multidimensional autions. It should be evident by now that the prourement problem disussed in this paper has two objetives, i.e., prie disovery and quantity deision. The idea of a soring rule is to ask the suppliers to bid in both prie and quantity, and then for eah supplier, ombine the prie bid and the quantity bid into a single number (the supplier s sore). The soring rule is the mehanism that does the seond step. The supplier with the highest sore wins the aution. Let p and q be the prie bid and quantity bid, respetively, submitted by a supplier. Che (1993) gave a soring rule of the following form: the sore for a supplier bidding p q is s = s q p for some funtion s. (Che onsidered prie and quality, but this is just a osmeti differene.) The supply ontrat aution with payment funtion P is losely related to Che s firstsore aution, with s P. (Under the first-sore aution, the winning supplier delivers his quantity bid to, and reeives his prie bid from, the buyer.) Note that under Che s soring rule, a supplier bidding p q is given sore s = P q p, whih orresponds to the up-front, lump-sum fee the supplier offers to pay in the supply ontrat aution. The two approahes imply different ash flows between the buyer and the winning supplier: under the supply ontrat aution, the supplier first makes an up-front payment to the buyer (to seure the business), then produes and delivers the quantity, and finally reeives a payment from the buyer for the quantity delivered (aording to a previously given ontrat), whereas under Che s design, there is no up-front payment, and the terms of trade (the quantity to be delivered and the orresponding prie) are fully speified in the winning bid. The basi idea of the supply ontrat aution is to treat a supply ontrat as an objet to be autioned off among the suppliers. The main task is to design this objet properly, and then invite singledimensional bids. The supply ontrat (the objet ) serves to onvert a quantity deision into a prie figure, effetively aggregating prie and quantity dimensions into one dimension (prie). In ontrast, the soring-rule approah is to first invite multidimensional bids and then aggregate the different dimensions into one. Note that both approahes ahieve aggregation, one before bidding and the other after. Interestingly, ahieving aggregation before bidding has real benefits: simpler analysis with more general results. For example, with the supply ontrat aution approah, one ould use the revenue equivalene theorem to effortlessly establish the equivalene of different aution formats, whereas Che was onfined to sealed bidding. Moreover, the new approah produes an optimal prourement strategy that fits well with a pratie in the retail industry ( 3.1). 2.3. Comparisons Although both the quantity and the supply ontrat autions are optimal for the buyer, they are different in many ways. Under the supply ontrat aution, the buyer has the flexibility of using any of the ommon aution forms mentioned earlier. This flexibility is lost for the quantity aution (Theorem 2). Moreover, the optimal payment funtion for the supply ontrat aution is onave, inreasing, and independent of the number of bidders, but for the quantity aution, it may atually derease (a rather unpleasant feature), and it generally depends on the number of bidders. In other words, the optimal payment funtion in the supply ontrat aution is more intuitive (a supplier is paid more if he delivers more) and more detail-free (the buyer does not have to know the exat number of potential suppliers). 3. A Newsvendor Model with Supply-Side Competition One of the most elebrated models in operations management is the newsvendor model, whih suintly aptures the trade-off between buying too muh and buying too little. The model assumes omplete ertainty on the supply side, where, typially, an unlimited quantity an be proured at an exogenously given per-unit wholesale prie. In reality, however, most industrial buyers fae multiple potential suppliers with private information about their prodution osts. As a result, the purhase prie needs to be disovered, whih, of ourse, influenes the purhase quantity. In this setion, we introdue a newsvendor model with supply-side ompetition and haraterize the optimal prourement strategies in this setting.

Chen: Autioning Supply Contrats Management Siene 53(10), pp. 1562 1576, 2007 INFORMS 1569 Toward the end of the setion, we study a simpler, but suboptimal, prourement strategy and numerially ompare it with the optimal strategies. Consider the following newsvendor model with supply-side ompetition. A firm (the buyer) must determine how muh inventory of a produt to stok in advane of a selling season. The total demand for the produt over the entire selling season, D, is a nonnegative random variable, with p.d.f. g and.d.f. G. The selling prie is p per unit, whih is exogenously given. (Later, we will disuss the ase where the selling prie is a deision variable.) If the buyer runs out of stok during the selling season, there are no replenishment opportunities and the exess demand will be lost. On the other hand, if there is exess inventory at the end of the season, it an be salvaged at v per unit. Let e be the proessing ost per unit of purhased quantity, suh as expenses inurred for spae and handling. On the supply side, there are n 2 potential suppliers for the produt. We shall retain all the assumptions made earlier about the suppliers in 2. That is, eah supplier is apable of produing an unlimited quantity of the produt at a onstant but supplier-speifi marginal ost. Reall that i is supplier i s marginal ost, i = 1 n, and that these marginal osts are i.i.d. draws from a ommon probability distribution over the finite interval, with p.d.f. f and.d.f. F satisfying F = 0 and F = 1. Eah supplier is privately informed of his own marginal ost. Finally, the virtual ost funtion, H, as defined in (1), is stritly inreasing. The buyer and the suppliers are all expeted-profits maximizers. We seek an optimal prourement strategy for the buyer. Let Q be the level of inventory at the beginning of the selling season. The buyer s profit, exluding the osts inurred to purhase the inventory, an be expressed as pmin Q D + v Q D + eq Beause min Q D = Q Q D +, the expeted profit is R Q def = p e Q p v Q 0 G y dy For onveniene, we will refer to R as the buyer s revenue funtion. Clearly, this funtion is onave with R 0 = 0. For the rest of the setion, we will fous on the ase where the revenue funtion is stritly onave. Note that lim Q + R Q = v e, whih may beome negative (unlike the revenue funtion used in 2). We assume that p>v p e> and e + >v (12) where the first inequality is natural, the seond indiates that a profitable supply hain may exist, and the third eliminates the possibility of an arbitrage. 3.1. Optimal Prourement Strategies The prourement problem faing the newsvendor is idential to the one onsidered in 2. 6 Therefore, an optimal prourement strategy is to use either the quantity aution (sealed, high-bid) or the supply ontrat aution (multiple formats). Reall that this requires the buyer to first speify a payment funtion and then invite the suppliers to bid in quantity or in fee. To determine an optimal prourement strategy, first obtain Q from (6) (with the newsvendor revenue funtion). Note that the range of H is +1/F and that R 0 = p e (assuming G 0 = 0). Beause p e> (see (12)), there are only two possibilities. If p e> + 1/F, then Q x > 0 for all x, and by definition, =. Otherwise, is the solution to p e = H. Using the speifi form of the newsvendor revenue funtion, we have Q x = G 1 ( p e H x p v ) x where G 1 is the inverse of the demand distribution funtion G. Clearly, Q x is dereasing in x. Let Q Q denote the range of Q. 3.1.1. Quantity Aution. Consider the quantity aution. The optimal payment funtion is given in (7). Although there is in general not a losed-form solution for P, it an be easily omputed numerially. Note that P Q is atually a quantity disount sheme beause P Q /Q, the average per-unit wholesale prie, is dereasing in Q. To see this, simply note from (7) that P Q Q z 1 F z n 1 dz = + Q Q 1 F n 1 and that the right side is inreasing in. (The proof, omitted here, does not require the speifi form of the newsvendor revenue funtion.) Hene, under the quantity aution, the buyer demands idential quantity disounts from the suppliers and piks a winner based on the quantities they are willing to supply. 7 The above observation provides a new rationale for using quantity disounts. Dolan (1987) gives an exel- 6 The only differene is that the revenue funtion here may be unimodal, but this is learly inonsequential, as there is no loss of optimality if one restrits the prourement quantity to the range where the funtion is inreasing. 7 The quantity aution an also be implemented by inviting the suppliers to bid in wholesale prie. Define w Q = P Q for all Q Q Q a dereasing funtion. Invert this funtion to obtain a purhase shedule for the buyer. Consider the following sealed-bid aution. Eah supplier submits a per-unit wholesale prie that they would harge, and the supplier with the lowest wholesale prie wins the aution. It is straightforward to show that the Bayesian-Nash equilibrium in this aution, w for, is suh that w Q = w for all. Therefore, bidding in wholesale prie, with the

Chen: Autioning Supply Contrats 1570 Management Siene 53(10), pp. 1562 1576, 2007 INFORMS lent review of the different forms of quantity disounts. He identified logistis system effiieny (e.g., full truk-load eonomies), prie disrimination, and hannel oordination as potential reasons for using quantity disounts. Toward the end of the paper, Dolan used ompetitive bidding among eletroni omponents suppliers (the buyer being IBM) as an example to illustrate a new motivation for quantity disounts. He said even with onstant unit ost, quantity disounts may be optimal if the buyer has a desire to multisoure prourement (p. 14). In Dolan s example, the buyer s purhase quantity is fixed, and quantity disounts emerge as a rational bidding strategy when the buyer attahes a large enough penalty to single souring. (As the purhase quantity inreases, the ompetition for eah additional unit intensifies, triggering disounts.) Note that in our setting, the buyer s purhase quantity is not fixed a priori and there is no penalty for single souring. Consequently, the need to integrate quantity deision with prie disovery in variable-quantity prourement is a new reason for quantity disounts. The intuition is lear: the more effiient the winning supplier, the larger the purhase quantity, indiating that the per-unit prie the buyer pays should derease as she buys more. In short, quantity ontains ost information. 3.1.2. Supply Contrat Aution. Another optimal prourement strategy, whih is in many ways more attrative than the quantity aution (see 2.3), is the supply ontrat aution. Here, the buyer offers a payment funtion (a supply ontrat) and invites the suppliers to bid for the right to trade. The supplier willing to pay the most for the right to trade wins and is given the deision right to hoose the quantity to deliver to the buyer. The optimal supply ontrat, P, is haraterized by (11), whih has a losed-form solution for the newsvendor model. Take any Q Q Q. Thus, there exists an x with Q = Q x. Note that P Q = x and R Q = H x. Beause R Q = p e p v G Q, we have P Q = H 1 p e p v G Q where the inverse funtion H 1 is well defined beause H is inreasing. Therefore, P Q = Q Q H 1 p e p v G z dz Q Q Q (13) Note that adding a onstant to the above payment funtion does not alter the marginal alulations faing the suppliers and only shifts their bids by the purhase shedule properly hosen, is also optimal. Hansen (1988) studied a variable-quantity aution where the purhase quantity is an exogenously given dereasing funtion of the winning wholesale prie. same onstant. Therefore, autioning off P + K for any onstant K onstitutes an optimal prourement strategy. (Adding a onstant allows us to establish onnetions with some well-known supply ontrats, as we will see later.) The supply ontrat aution fits well with the use of slotting allowanes and VMI, both of whih are prevalent in the retail industry. Slotting allowanes are lump-sum, up-front payments from a supplier to a retailer when the supplier introdues a new produt to the retailer s stores. Sometimes, suh payments are made to keep an existing produt on the retailer s shelves (also alled pay-to-stay fees). Therefore, a slotting allowane orresponds to the right-to-trade fee olleted by the newsvendor in our model. On the other hand, VMI is a pratie where the inventory replenishment deision is delegated to the supplier. In our model, this means that the supplier that has been seleted is given the deision right to determine the level of inventory to stok at the beginning of the selling season. Of ourse, the supplier should not be given omplete freedom in making the inventory deision. Instead, the deision is guided by the payment funtion speified by the buyer. It is interesting that in our framework, slotting allowanes and VMI arise as two distint features of an optimal prourement strategy. The existing literature on the reasons for using slotting allowanes tends to fous on the signaling effet (when the supplier has more information about the produt s potential in the marketplae) or the risk-sharing effet (the retailer often inurs real, out-of-poket osts in new-produt introdutions and many new produts fail) (see Federal Trade Commission 2001 and the many aademi papers on slotting allowanes suh as Shaffer 1991, Chu 1992, Lariviere and Padmanabhan 1997, Sullivan 1997, Bloom et al. 2000, and Desai 2000). Our interpretation is simply that slotting allowanes are fees resulting from suppliers ompeting for the sare shelf spae and that they an be part of an optimal prourement strategy for the retailer. This observation adds a new dimension to the ongoing debate on the purposes and onsequenes of slotting allowanes. The above interpretation of the supply ontrat aution relies on a partiular method of bookkeeping, i.e., the winning supplier first pays a lump-sum fee to the buyer to gain the right to trade, followed by the supplier s prodution/delivery deision, and finally ending with a payment from the buyer to the supplier for the delivered quantity. As mentioned earlier, one ould also ask eah of the suppliers to submit a twodimensional bid, onsisting of a prie and a quantity. The suppliers are then ranked aording to a given soring rule and the supplier having the highest sore wins the aution. Under this implementation, we do not require the winning supplier to pay the buyer

Chen: Autioning Supply Contrats Management Siene 53(10), pp. 1562 1576, 2007 INFORMS 1571 an up-front, lump-sum fee. Consequently, whether or not we observe a slotting allowane depends on the bookkeeping method we use to implement the optimal prourement strategy. The fat that slotting allowanes are prevalent in pratie suggests that other fators may be at play here. For example, the buyer, given her lak of information about the new produt s market potential, may want some insurane before ommitting her shelf spae (a very important resoure) to a partiular supplier. An up-front fee serves just that purpose. Inorporating the buyer s aversion to risks may therefore tilt the balane toward the use of slotting allowanes. This is an interesting topi for future researh. 3.1.3. Returns Contrats and Revenue-Sharing Contrats. In both the quantity and the supply ontrat autions, the payment funtion ontains no unertainty, i.e., the payment a supplier reeives is a deterministi funtion of the quantity he delivers to the buyer. In pratie, we often see supply ontrats where the transfer payment also depends on the realized demand (see Cahon 2003 for a omprehensive review of the supply ontrat literature). Next, we disuss the impliations of using some of these supply ontrats in our prourement setting. Let P Q D be the transfer payment (from the buyer to a supplier) if Q units are delivered to the buyer and the realized demand is D. Consider, e.g., the returns ontrat, whereby the buyer pays a perunit wholesale prie w for every unit of inventory stoked at the beginning of the selling season, and if there is leftover inventory at the end of the selling season, the buyer obtains a rebate of b per unit of exess inventory. (The buyer does not physially return the leftover inventory to the supplier, but salvages the exess inventory for v per unit. Other arrangements are possible, but are not onsidered here.) Under suh a ontrat, P Q D = wq b Q D + Another example is the revenue-sharing ontrat. Here the buyer pays per-unit wholesale prie w for the initial inventory and promises to transfer 0 1 fration of her revenues (regular sales and salvage sales) to the supplier. In this ase, P Q D = wq + p min Q D + v Q D + Suppose our newsvendor autions off the supply ontrat P Q D. Beause the suppliers are risk neutral, they value the ontrat at E D P Q D def = P Q. If P = P (respetively, P ), then autioning off the ontrat by inviting the suppliers to bid in quantity (respetively, up-front fee) maximizes the newsvendor s expeted profit. Below we provide an example where an optimal prourement strategy an be implemented by autioning off either a returns ontrat or a revenue-sharing ontrat. Assume F x = x / for all x, i.e., the marginal prodution ost is uniformly distributed. Hene, H x = 2x, x. From (13), P Q = Q 0 1 p e p v G z + dz 2 (Here we have added a onstant, Q 0, to the payment funtion. As noted earlier, the resulting payment funtion is still optimal.) It an be easily verified that a returns ontrat with w = p e + 2 and b = p v 2 satisfies P = P for any G. 8 On the other hand, an alternative way to express the above payment funtion is P Q = e 2 Q + p 2 E D min Q D + v 2 E D Q D + Therefore, a revenue-sharing ontrat with w = e and = 1 2 2 also ahieves P = P for any G. 9 An optimal prourement strategy for the newsvendor is, therefore, to aution off a returns ontrat or a revenue-sharing ontrat with the above parameters in the supply ontrat aution. Note that the ontrat speifiation does not require the knowledge of the demand distribution. Now we have seen an example where speifying the payment funtion either as P Q or P Q D makes no differene for the parties of the trade. This is no longer true if there is information asymmetry regarding the demand distribution or there are differenes in risk attitudes. Suppose that only the buyer knows the demand distribution G. In this ase, the buyer an ompute the optimal payment funtion P for either the quantity aution or the supply ontrat aution. Beause the payment is independent of demand, there is no need for the suppliers to know the demand distribution. The buyer s expeted profits are maximized (as if the suppliers had full knowledge about the demand distribution). In ontrast, if the buyer hooses to speify the payment funtion as P Q D (through, e.g., a returns or a revenue-sharing ontrat), then the suppliers are 8 Note that b may be higher than w. But this is fine beause the supplier hooses the quantity to deliver to the buyer in the supply ontrat aution and beause b +v < p, whih means the buyer does not want to turn away demand. 9 If w<0, then the supplier subsidizes the initial inventory before taking a share of the buyer s revenues.

Chen: Autioning Supply Contrats 1572 Management Siene 53(10), pp. 1562 1576, 2007 INFORMS fored to evaluate this ontrat based on their imperfet information about demand. As a result, their bidding strategies will be different from their strategies under full knowledge about the demand distribution. The buyer s expeted profits depend on the nature of the suppliers imperfet knowledge about demand. If the suppliers happen to believe that demand is higher than it really is, then their evaluation of the supply ontrat may be inflated and the buyer may even be better off (relative to her expeted profits under the optimal prourement strategy assuming symmetri demand information). On the other hand, suppose that the buyer does not know the demand distribution, but the suppliers do. In this ase, if the buyer hooses to offer a payment funtion in the form of P Q, then she must first alulate the revenue funtion R Q based on her faulty knowledge of the demand distribution. The resulting payment funtion is likely to be different from what she would offer given true demand distribution. The outome is that the payment funtion is suboptimal and the buyer s expeted profit is not maximized. (In this ase, the suppliers knowledge of the demand distribution goes wasted beause, as mentioned above, they have no use for this knowledge when given a deterministi payment funtion.) Now onsider the alternative of speifying the payment funtion in the form of P Q D. As our earlier example demonstrates, when the marginal prodution osts are uniformly distributed, the design of an optimal prourement strategy does not require the knowledge of the demand distribution if the buyer hooses to write the payment funtion as a returns ontrat or a revenue-sharing ontrat. 10 Therefore, the buyer suffers no loss of optimality even without knowledge of the demand distribution. (When the suppliers evaluate the ontrat, i.e., E D P Q D, they use the true demand distribution.) This rather surprising result is due to the speifi form of the optimal payment funtion for the supply ontrat aution, P Q, and its lose onnetion to the returns or revenue-sharing ontrat. (The form of the optimal payment funtion for the quantity aution, however, does not seem to lend itself to this onnetion.) Note that speifying the payment funtion as P Q D instead of P Q makes the suppliers bear more risk. Of ourse, in the urrent model setup with risk neutrality, the suppliers are indifferent. An interesting future researh topi is to investigate the impat of risk aversion on the form of the optimal prourement strategy. 3.1.4. Impat of Supply and Demand Charateristis. We next onsider how the demand- and supply- 10 This is reminisent of Pasternak (1985), where a oordinating ontrat is found to be independent of the demand distribution. side harateristis affet the newsvendor s profit. To this end, suppose D is normally distributed with mean and standard deviation. As this assumption implies the possibility of negative demand, 11 we modify the revenue funtion as follows: Q R Q = p e Q p v G y dy (14) In this ase, we have Q x = + 1 ( p e H x p v ) x (15) where is the.d.f. of the standard normal and 1 its inverse. Reall that is determined as follows. If p e> H x for all x, then =. Otherwise, is the solution to p e = H. With the normal demand distribution, the buyer s maximum expeted profit, given in Theorem 1, an be expressed as = a b (16) where a = p e H x f 1 x dx and ( ( )) p e H x b = p v 1 f p v 1 x dx (17) with being the density funtion of the standard normal. Note from (16) that the impats of the demand- and supply-side harateristis on are deoupled : a and b depend on the number of suppliers and their marginal ost distribution, but are independent of and, whih are of ourse the demand-side harateristis. Note also that a would be the buyer s expeted profit if there were no demand unertainty (the expetation is with respet to the supply-side unertainty), where a is the buyer s expeted profit margin. 12 On the other hand, b is the ost of demand unertainty, whih inreases linearly in the demand standard deviation. The oeffiient b thus represents the buyer s marginal benefit from redution in demand unertainty. It is easy to see that a is inreasing in supplyside ompetition (i.e., n), whih also means that there is synergy between inreasing the number of potential suppliers and inreasing the mean demand, i.e., inreasing n inreases the benefit of inreasing 11 The likelihood of negative demand should be negligible for the normal assumption to be plausible. 12 Note that a is also the newsvendor s expeted profit under the make-to-order regime, whereby she waits until the demand is revealed and then proures the exat quantity needed from the lowest-ost supplier via an aution. Of ourse, this mode of operation is not always feasible: it depends on how patient the ustomers are and how fast the suppliers an deliver the produt.