Procurement auctions are sometimes plagued with a chosen supplier s failing to accomplish a project successfully.

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1 Deision Analysis Vol. 7, No. 1, Marh 2010, pp issn eissn informs doi /dea INFORMS Managing Projet Failure Risk Through Contingent Contrats in Prourement Autions Jianqing Chen Haskayne Shool of Business, University of Calgary, Calgary, Alberta T2N 1N4, Canada, Lizhen Xu, Andrew Whinston Red MCombs Shool of Business, The University of Texas at Austin, Austin, Texas Prourement autions are sometimes plagued with a hosen supplier s failing to aomplish a projet suessfully. The risk of projet failure is onsiderable, espeially when the buyer has inadequate information about suppliers ex ante and the projet an only be evaluated at the end. To manage suh unertainty, a model of ompetitive prourement and ontrating for a projet is presented in this paper. We study a setting in whih suppliers differ in both the osts to fulfill the projet and the types refleting their suess probabilities. To sreen suppliers, the buyer invites suppliers to speify a two-dimensional bid omposed of the proposed ost and a penalty payment if the delivered projet fails to meet the requirements. We find that a quasi-linear soring rule an effetively separate suppliers regarding their types. We then study the effiient and optimal design of the soring rule. The effiient design internalizes the inferred information on suppliers type and essentially ranks suppliers based on the expeted total ost to the buyer. In the optimal design, the buyer may or may not under-reward suppliers high suess probability, depending on the balane between suppliers suess probabilities and the assoiated ost distributions. Interestingly, it is always optimal for the buyer to possibly award the projet to suppliers with low suess probability to promote the ompetition, even when the differene in suppliers suess probabilities is huge. We show that, ompared to standard autions, the prourement autions with ontingent ontrats an signifiantly improve both soial welfare and the buyer s payoff. Key words: prourement autions; ontingent ontrats; soring rules History: Reeived on April 21, Aepted by L. Robin Keller and guest editors Robert F. Bordley and Elena Katok on July 2, 2009, after 1 revision. Published online in Artiles in Advane Otober 13, Introdution Prourement autions play an important role in eliiting suitable suppliers or ontrators for various projets (Beall et al. 2003). In prourement autions, buyers often spend most of their effort in finding the best ontrat with the lowest prie. However, another ritial issue the ability of a potential supplier to fulfill the projet and the assoiated unertainty, whih are typially unobservable and thus nonontratible should never be negleted, beause it would otherwise lead to disastrous outomes. In this paper, we propose prourement autions with ontingent ontrats and study the design of suh autions to handle this ritial issue. In 1993, the Oregon Department of Motor Vehiles (DMV) launhed an information tehnology (IT) projet aiming to omputerize its manual and paperbased daily work. Only two years after its initiation, this initially five-year and $50 million projet was shifted to an estimation of ompletion in eight years with the ost of $123 million. In 1996, a prototype of the new system was tested. In less than half a week, the DMV offie had its longest waiting line and most omplaints ever. The usability of the system was so bad that the system was onsidered a total failure. Eventually, state offiials had no hoie but to kill the projet ompletely. Ironially, the IT projet that was supposed to downsize the workfore, after tens of millions of dollars of waste, sueeded in nothing but downsizing the DMV offiials who witnessed the disaster (Martin 2002). Large projets (e.g., large-sale IT implementation and military weapon programs) have a ertain possibility of failure, partially beause of their omplexity. 23

2 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 24 Deision Analysis 7(1), pp , 2010 INFORMS More importantly, the suess of a prourement projet ritially depends on the ontrator s projet management and quality ontrol apabilities. Without onsidering the unertainty of the outome and hene the possibility of failure, it is impratial for buyers to eliminate the risk by simply speifying all the required outomes in the ontrat. One the projet fails, the loss inurred to the buyer is inevitable, and most times it is unbearable. Although buyers ould resort to the ourt for enfored ompensation one ontrators fail to fulfill the prespeified requirement, the huge operational ost assoiated with legal proedures and even the possibility of bankrupty of the ontrators make the feasibility and effetiveness of this approah quite limited. Alternatively, buyers may rely on renegotiation to ontrol the loss ex post. Nevertheless, as the ontrator typially has more information than the buyer regarding the status of the projet and the operations being followed, the buyer is often in a disadvantaged position in the renegotiation. Depending on the negotiation power, some projets an take longer than they should and ause enormous ost overrun problems. Aording to a report from a U.S. Congressional auditing ageny in 2008, nearly 70% of the Pentagon s 96 largest weapons programs suffered from huge ost overruns, for a ombined total of $296 billion more than the original estimates (Drew 2009). In many other ases, renegotiation may not even be possible. To better manage the risk assoiated with a projet, the buyers should take into aount the differene among potential suppliers ability to suessfully aomplish the projet in the first plae, in addition to the ost that the suppliers request. The potential suess rate an vary greatly aross different ontrators, depending on ontrators experiene and expertise. On the one hand, the buyer ould try to verify eah potential ontrator s qualifiation and to assess their expeted performane, but suh verifiation and assessment typially involve ase-by-ase investigation and perhaps on-site involvement, and are thus very ostly (Wan and Beil 2009). In some other ases, autioneers may be able to infer potential ontrators apability by their past performane information (Rothkopf and Whinston 2007), suh as the typial approah searh engines adopt in online keyword advertising autions. However, in general, suh information an be diffiult or even impossible to ollet (e.g., past projet history may not be available in sensitive industries like weapons manufaturing). On the other hand, potential suppliers have better knowledge of their own experiene, expertise, and apability of managing the projet, and thus are able to form a more preise estimation of the suess rate. It ould be most effiient to indue bidders to self-reveal private information on their suess probabilities in prourement autions. In this sense, it seems extremely appealing to devise a novel mehanism to sreen potential ontrators so that the most suitable ontrator an distinguish itself and win the projet as a result. Fortunately, as we show in this paper, by introduing ontingent payments into prourement autions, we are able to ahieve this goal. We study a prourement setting in whih both prie and nonprie attributes matter. In partiular, suppliers differ in both their osts to fulfill the projet and their types in terms of the probabilities of their suessfully aomplishing the projet. Suppliers may have a low or high suess probability. Different from prie-only autions or typial soring autions, suppliers types are nonontratible, and thus annot be inorporated in a soring funtion. We propose a ontingent ontrat in determining the winner. Suppliers plae bids on fulfillment ost and a penalty payment if they fail to meet the projet requirement at the end. The buyers use a soring rule to determine the winner. We then study the suppliers equilibrium bidding, investigate the effiient and optimal design of suh autions, and ompare the proposed aution approah with standard prourement autions. We find that attahing ontingeny terms with a soring rule has a signifiant impat on suppliers equilibrium bidding. First, a quasi-linear soring rule an effetively separate suppliers regarding their suess probabilities. Suppliers equilibrium bids on the penalty signal their onfidene and reveal their types. Seond, suppliers bids on ost take into aount the possible penalty in ase of failure and are, not surprisingly, higher than in standard autions. Furthermore, different soring rules influene how suppliers with different suess probabilities ompete with eah other in equilibrium and affet the alloation of the projet.

3 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 25 Despite the omplexity of equilibrium bidding strategies of the bidders, the effiient design is remarkably simple: It internalizes the inferred information on suppliers types and essentially ranks suppliers based on the expeted total ost to the buyer. Relative to the effiient design, the optimal design may or may not under-reward suppliers high suess probabilities, depending on the balane between suess probability and ost distribution. Interestingly, to promote the ompetition between different types of suppliers, the optimal design always ensures that some suppliers with low suess probability an win over some of those with high suess probability, regardless of the differene of their suess probabilities. We show that, ompared to standard autions, the prourement autions with ontingent ontrats an signifiantly improve both soial welfare and the buyer s payoff. The rest of this paper is organized as follows. In 2, we disuss the related literature. We set forth our model in 3. In 4 we explore suppliers equilibrium bidding behaviors. We examine the design of effiient soring rules and optimal soring rules in 5 and 6, respetively. Setion 7 disusses some extensions, and 8 onludes this paper. 2. Literature Review Two streams of literature are related to this work. The first stream is the study of soring rules and soring autions. Soring rules have long been disussed in deision siene literature (Kilgour and Gerhak 2004, Johnstone 2007, Bikel 2007, Abbas 2009). Good (1952) was among the first who interpreted soring rules as inentive devies to indue agents true belief for assessing a projet. Sine then, many soring rules, suh as quadrati, spherial, logarithmi, and the parimutuel Kelly soring rules (Johnstone 2007), have been introdued and investigated for different purposes, suh as prediting the probability of an event. Bikel (2007) ompares different soring rules and offers some guidane as to whih rule is the most appropriate for whih purpose. Unlike the settings studied in the above literature, our paper fouses on prourement autions using a soring rule to determine a winner. This researh is losely related to previous studies on soring autions, or autions in whih bidders plae multidimensional bids and are ranked by sores alulated from their bids (Che 1993, Bushnell and Oren 1994, Asker and Cantillon 2008). For example, Che (1993) studies a form of soring aution used in government prourement, where suppliers bid on both prie and quality, and bids are evaluated by a soring rule. The ritial differene of our study from theirs lies in the ontingeny terms in the aution rule and, fundamentally, the supplier-dependent suess rate, whih itself annot be used for soring rules. In partiular, in their setting, bids (on prie and nonprie attributes) fully determine the buyer s payoff, whereas in our setting, in addition to their bids, suppliers suess probabilities play an important role in the buyer s payoff. Soring rules also appear to be an important omponent in other aution settings. Bushnell and Oren (1995) onsider autions to selet an internal supplier for intermediate produts, in whih suppliers bid on a fixed ost plus a unit prie and the winner is determined by sores alulated from their twodimensional bids. Liu et al. (2009) study the soring rule under keyword autions. They onsider ontingent payment in aution design under the assumption that the autioneer an observe the nonprie information, and winner determination is based on both the observed information and the bid. In our ase, we assume both the prie and nonprie information is private information, and winner determination is aording to suppliers bids on both ost and penalty. In a sense, Liu et al. (2009) take a learning approah, and here we take an eonomi sreening approah. The seond stream of work related to ours is the study of autioning ontingent ontrats. Ewerhart and Fieseler (2003) study unit-prie-ontrat autions in a prourement setting, in whih bidders bid unit pries for labor and materials, and the final payment depends on the realized amount of input used. The buyer speifies an estimated amount of eah input to evaluate suppliers bids (using a weighted sum of unit-prie bids by the speified amounts), and the bidder with the lowest estimated ost wins. MAfee and MMillan (1986) onsider a ontrator s moral hazard problem of ost ontrol in prourement autions and haraterize the optimal linear ontrat. The aution model in this paper differs from the above in the appliation setting and ontrat formats, and thus differs in bid struture and equilibrium bidding

4 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 26 Deision Analysis 7(1), pp , 2010 INFORMS behavior. In addition, unlike the equilibrium bidding being determined by a single parameter in the above autions, equilibrium bidding in our paper is determined both by suppliers osts and by their suess probabilities. 3. Model We onsider a risk-neutral buyer seeking to award an indivisible projet to a supplier. There is risk assoiated with the projet suh that a hosen supplier may not be able to omplete the projet suessfully or meet essential requirements. Examples of suh risks inlude a hosen supplier who annot deliver the projet on time, or the projet delivered does not meet the prespeified quality. Therefore, the buyer ares about both the prie that a supplier asks and the supplier s type, whih reflets its suitability to or preparation for the projet. We fous on the setting where the outome of the projet is verifiable so that it is possible to write a ontingent ontrat on the outome. The buyer hene introdues a ontingent ontrat into the prourement autions to sreen potential suppliers. In partiular, the buyer requires bidders to propose a ontingent payment when the projet fails, in addition to their bids on the ost of ompleting the projet. There are n 2 potential suppliers available in the andidate pool. We assume that suppliers differ in two dimensions: One dimension is the expeted ost of ompleting the projet, i. Suppliers may differ in their fulfillment osts beause they may use different tehnologies, human apitals, or proedures. The other dimension is the type of supplier, refleting the probability of its suessfully ompleting the projet. We fous on the ase where the probability of suessful ompletion for bidder i, 1 q i, is an inherent harateristi of the bidder; that is, q i measures supplier i s probability of failing, whih haraterizes supplier i s type its suitability, preparation, or qualifiation for the projet. Throughout this paper, we use the term type to refer exlusively to suppliers probability of suess (rather than both dimensions). Suppliers may be different in the probability of failure beause they may have different levels of expertise, experiene, and apability of projet management and quality ontrol. In the extension, we disuss the ase where suppliers types are haraterized by distributions of the degree to whih they meet the buyer s requirement. Suppliers privately observe their own ost and type, but not those of the other suppliers. The buyer observes neither suppliers osts nor their types, so autions are naturally adopted to hoose a ontrator. However, the distribution of suppliers osts and types is ommon knowledge. Cost and probability measure q are jointly drawn from 0 1, following a joint probability density funtion fq. We assume that, for any given q i, the density funtion fq i is positive and ontinuous on the interval. The buyer invites potential suppliers to bid on b i t i, where b i is the ost of fulfilling the projet and t i is the penalty that a supplier is willing to bear if the projet delivered fails to meet the essential requirements. We assume suppliers are risk neutral so their payoff funtions an be formulated as Ubtq= b qtprwin (1) The buyer derives value v if the delivered projet sueeds in meeting all the requirements. Otherwise, the value is disounted we assume that the buyer bears loss z from v. Therefore, if supplier i who proposes b i t i wins the ontrat, the expeted payoff for the buyer is Vb i t i q i = v b i z t i q i One important feature of the buyer s payoff differentiates our setting from those of others (e.g., Che 1993, Asker and Cantillon 2008): In our ase the buyer not only ares about a supplier s bid (the same as in other settings), but is also onerned about the supplier s true type, refleting its probability of suess (whih itself, as a probability measure, is unobservable and nonontratible, and thus annot be made part of an agreement). The projet is assigned aording to some soring rule Sbt preannouned by the buyer. The soring funtion is inreasing in b and dereasing in t. The supplier with the lowest sore wins the projet. In partiular, we onsider a quasi-linear soring funtion in the form of Sbt = b t, where t is inreasing and onave (i.e., t > 0 and t < 0). In other words, we onsider a lass of soring rule in whih bids on prie and bids on penalty are separable and additive. This lass of soring rule enompasses many ommonly used soring rules. For example,

5 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 27 t = w t represents the simplest square-root soring rule, and t = w ln1 + t represents a logarithmi soring rule. Next, we start with a simple ase of two possible types: q l and q h, 0 <q l <q h < 1. We all the suppliers with q l the preferred suppliers and the suppliers with q h the nonpreferred suppliers, beause the former has a lower failure probability and, everything else being equal, is more desirable to the buyer. We will extend our model to a multiple-type ase in 7. We assume that the probabilities for an individual supplier s being preferred and being nonpreferred are and 1, respetively. To simplify the notation, we let f l fq l and f h fq h, whih have a fixed and ommon support. Correspondingly, we define the umulative distribution funtions F l = f lx dx and F h = f hx dx. The ommon support assumption is to simplify notation. The analysis and results an be easily generalized to the ases with different supports l l and h h for the preferred and the nonpreferred suppliers, respetively. To avoid distration from trivial equilibrium outomes, throughout this paper, we fous our disussion on the ase in whih the differene in suppliers suess probabilities is not too huge so that in equilibrium at least some low-ost nonpreferred suppliers an win over some high-ost preferred suppliers. In other words, we fous our disussion on the soring rules that satisfy 1 q l q l 1 q l 1 q h q h 1 q h < (2) The explanation of this ondition shall beome lear after Proposition 1. When ondition (2) is not satisfied, the equilibrium outome redues to a simple ase in whih suppliers only ompete with peers of the same type, and similar analysis remains to apply. 4. Equilibrium Analysis We first onsider suppliers bidding strategies. Throughout this paper, we onsider a symmetri, pure-strategy Bayesian-Nash equilibrium. By symmetri we mean that suppliers with the same ost and type will bid the same. Let b tq, whih is the expeted profit a supplier earns upon winning the aution. Given any suh supposed profit level, the supplier intends to maximize the winning probability or, equivalently, to minimize its sore by properly hoosing the bid on penalty t: min S + + qtt = + + qt t (3) t Beause of the onavity of, the first-order ondition haraterizes the optimal bid on the penalty: q = t. For example, for the lass of logarithmi soring rule with t = w ln1+t, it is easy to derive t = 1/q 1; for the lass of square-root soring rule with t = w t, t = w2 4q (4) 2 We denote the optimal bid on penalty as t q 1 q (5) Lemma 1. Under the soring rule Sbt, in equilibrium the suppliers bid t q on the penalty part as defined by Equation (5). Moreover, t q is dereasing in q. Proof. Beause t < 0, 1 q is well defined and dereasing in q. Therefore, t q is dereasing in q. The above lemma indiates that in equilibrium bidders with lower failure probability bid more on penalty. In this sense, the quasi-linear soring funtion an effetively sreen bidders in terms of their private types: The more suitable or better prepared suppliers bid more aggressively on penalty. We next examine suppliers bidding strategies on the ost. Based on the above lemma, we now rewrite the suppliers payoff funtion Equation (1) as Ubt q h q h = b q h t q h Prwin Ubt q l q l = b q l t q l Prwin We denote b l and b h as the equilibrium bids on the fulfilling ost for suppliers with q l and q h, respetively. We onjeture and will verify that both bidding funtions are inreasing in supplier s ost, and there exists suh that Sb l t q l = Sb h t q h. The onjeture on is intuitive beause, in general, suppliers with a lower failure rate have a greater hane to win the ontrat, and a preferred supplier with the highest possible ost may (6)

6 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 28 Deision Analysis 7(1), pp , 2010 INFORMS Figure 1 Preferred supplier s osts Nonpreferred supplier s osts Comparable Bidders m() * m() have the same winning probability as some nonpreferred supplier with a lower ost. Under this onjeture, for any nonpreferred supplier with ost below, there must be a preferred supplier with ost m who mathes the former s sore, where m is a mapping from a nonpreferred supplier to its ounterpart in the preferred group. We all this pair of suppliers omparable bidders. Figure 1 illustrates a pair of omparable bidders by a dashed line, with a nonpreferred supplier of ost at one end and a preferred supplier of ost m at the other. Tehnially, we simply define m = for >, and m 1 = for <m. Based on suh notations, we an formulate the suppliers equilibrium winning probabilities: l = 1 F l F h m 1 ] n 1 h = 1 F l m F h ] n 1 A nonpreferred supplier with ost an beat a ompetitor only if the ompetitor is a nonpreferred supplier with a higher ost, or if the ompetitor is a preferred supplier but with a ost higher than m. The nonpreferred supplier an win the ontrat only if it beats all the remaining n 1 ompetitors, whih explains h. The winning probability for the preferred suppliers l is derived along a similar line. Proposition 1. Under the soring rule Sbt, in equilibrium, suppliers plae bids on penalty as in Lemma 1 and plae bids on ost as follows. b h = + q h t q h + hx dx h b l = + q l t q l + h x dx + lx dx l where m = +, =, and is defined by (7) t q l q l t q l ] t q h q h t q h ] (8) Proof. All proofs of propositions are deferred to the appendix. We an easily verify that the above bidding funtions are indeed monotonially inreasing in ost by heking the first-order derivative. As in the standard autions, in equilibrium suppliers ask more than their true ost. Two features of the bidding funtions differ from that in the standard autions. First, in the bid on ost, we have an extra term of qt q, whih is exatly the expeted penalty. One suppliers promise to bear the penalty if they fail to meet the requirement, they take suh possible revenue loss into aount in formulating their bidding. So the buyer is not better off from the suppliers promised penalty itself. Instead, the equilibrium penalty serves like a sreening devie that separates their suppliers with different types. Seond, in the preferred suppliers bidding funtion, we observe one extra term in the fration h x dx, ompared to that of a standard aution. We all it the base payoff for preferred suppliers. Unlike in the standard autions, in equilibrium, the preferred supplier with even the highest possible ost an still earn a ertain level of payoff, beause it has a lower failure rate and is more desirable, from the buyer s perspetive, than some nonpreferred suppliers with lower ost. The revealed mathing pattern m = + indiates that a nonpreferred supplier with ost and a preferred supplier with ost + are omparable (i.e., they have the same winning probability). Intuitively, beause b qt represents a supplier s expeted revenue from the buyer, defined in Equation (8) an be interpreted as the differene between a preferred supplier s expeted revenue and a nonpreferred supplier s expeted revenue when they both bid the same sore s (beause = s + t q l q l t q l s + t q h q h t q h ). A preferred supplier of ost + and a nonpreferred supplier of ost thus have the same expeted profit (i.e., expeted revenue less fulfillment ost) upon winning, if they bid the same sore. Therefore, the two suppliers fae the same trade-off between the expeted profit upon winning and probability of winning, whih leads to the same hoie of sore in equilibrium. Tehnially, when a nonpreferred supplier with ost bids sore s (i.e., bid s + t q h on ost) and a preferred supplier

7 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 29 with ost + bids sore s (i.e., bid s + t q l on ost), we an rewrite their payoffs as U ( ) s + t q h t q h q h = ( s + t q h q h t q h ) Pr ( s is the lowest sore ) U ( ) s + t q l t q l + q l = ( s + t q l q l t q l ) Pr ( s is the lowest sore ) = ( s + t q h q h t q h ) Pr ( s is the lowest sore ) where the last equality is ahieved by substituting defined in Equation (8). Evidently, those two suppliers fae exatly the same optimization problem when hoosing the optimal sore. That is, if s is the best hoie for the preferred supplier with +, s must be the best hoie for the nonpreferred supplier with ost as well. Therefore, the nonpreferred supplier with ost has the same winning probability as the preferred supplier with ost + ; or those two are omparable in equilibrium. Notie that the term on the left-hand side of Equation (2) is simply the definition of in Equation (8). So ondition (2) is to ensure that <. When the ondition is not satisfied (i.e., ), in equilibrium, the preferred suppliers adjust their bids so that the preferred supplier with the highest ost ties in sore with the nonpreferred one with the lowest ost. This is beause, for the preferred supplier with the highest ost, bidding the same sore as the nonpreferred one with the lowest ost is the most profitable way to win over all realized nonpreferred suppliers. As a result, none of the nonpreferred suppliers an win over any preferred supplier. More expliitly, b h takes the same form as in Equation (7), and 1 b l =+q l t q l + lxdx+ hxdx+ + l (9) l 1 b h and b l are derived in a manner similar to that for Proposition 1 exept that Ub l t q l q l is determined by b l t q l = b h t q h instead. Figure 2 b Example of Equilibrium Bidding Strategies Nonpreferred suppliers Preferred suppliers where l = 1 F l + 1 n 1 and h = 1 1 F h n 1. The following example illustrates suppliers equilibrium bidding. Example 1. We onsider a setting in whih = 1/2, q l = 1/4, q h = 3/4, and n = 3. We assume the osts of the preferred and the nonpreferred suppliers are distributed over 0 1, with the umulative distribution funtions F l x = x 2 and F h x = x, respetively. We use the square-root soring rule with w = 1 (i.e., t = t) as an example. Aording to Equation (4), equilibrium bids on penalty are t q l = 4 and t q h = 4/9. Clearly, the preferred suppliers bid more than the nonpreferred suppliers on penalty. Aording to Equation (8), we an alulate = 2/3, and thus v = 1/3 and m = + 2/3. We plot suppliers equilibrium bids on ost as in Figure 2. It is worth highlighting the kink in eah bidding funtion. In Figure 2, we observe a kink at 2/3 inthe preferred suppliers bidding funtion and a kink at 1/3 in the nonpreferred suppliers bidding funtion. Intuitively, the presene of suh a kink is beause of the hange in ompetition suppliers fae. For example, preferred suppliers with low ost have no omparable nonpreferred suppliers, and thus the ompetition is mainly within the preferred group. One preferred suppliers ost reahes a ertain level ( = 2/3 in the above example), they fae the ompetition from the omparable nonpreferred suppliers, in addition to their own peers, whih hanges the ompetition they fae and results in the kink in their bidding funtion.

8 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 30 Deision Analysis 7(1), pp , 2010 INFORMS A similar explanation applies to the kink in the nonpreferred suppliers bidding funtion. 5. Effiieny and Soring Rule In this setion, we examine the effiieny of different soring rules. We measure the effiieny by the total value reated. The effiieny riterion, therefore, emphasizes the total pie, whih is espeially important for prourement in the publi setor (i.e., government prourement). Beause of the monotoniity in the equilibrium bidding on ost analyzed in the previous setion, for any given bidders with realized osts drawn from the ost distributions, there are only two winning andidates: the preferred bidder with the lowest ost among its peers, l, and the nonpreferred bidder with the lowest ost among its peers, h. The former generates a total value of v q l z l, and the latter generates v q h z h upon being assigned the projet. Therefore, an effiient alloation should ensure that the ontrat be assigned to the preferred lowest-ost supplier if and only if v q l z l >v q h z h or, equivalently, l h <zq h q l. As is disussed above, in equilibrium, the preferred bidder with ost l wins over its nonpreferred ounterpart with ost h if and only if l < h +. Therefore, as long as = zq h q l,we an ensure effiient alloation no matter what the realized ost distribution among all bidders looks like. We thus show that the soring rule satisfying the ondition = zq h q l is ex post effiient. In addition to ex post effiieny, naturally, we are also interested in the soring rule ahieving ex ante effiieny. We say a soring rule is ex ante soially effiient if it maximizes the expeted total value generated in equilibrium. Given that the probabilities of assigning the projet to preferred and nonpreferred suppliers are l and h, respetively, the expeted total value generated in equilibrium is n v q l z l f l d + n1 v q h z h f h d (10) Proposition 2. Any soring rules satisfying = zq h q l are both ex ante and ex post soially effiient. 2 2 Notie that eff < requires that zq h q l <. When zq h q l, any soring rule with, under whih none Reall that we define in Equation (8). Under the effiient soring rule, we have t q l t q l zq l = t q h t q h zq h If we denote the value of the above as a onstant, then t q l = t q l zq l + and t q h = t q h zq h +. So the winner is essentially determined by b t q j = b + z t q j q j j l h Notie that b + z t q j q j is just the expeted total ost to the buyer. So the effiient soring rule essentially internalizes the inferred information regarding suppliers suess probabilities and implements a rule that ranks suppliers by their total expeted ost to the buyers. Suh a soring rule is effiient beause the bidders are fairly evaluated in the sense that the winner is the one who an minimize the buyer s expeted total ost or maximize the buyer s expeted payoff. It is worth noting that the poliy presribed above is independent of the distribution of suppliers osts and types. This feature makes it easy to implement an effiient soring rule: The buyer only needs to estimate the failure rates of suppliers and ombine them with its own loss from a failure. In fat, the oeffiient is exatly the differene in expeted loss from different suppliers. Notie that beause we do not speify the form of t, many soring rules with properly set parameters an be effiient. For example, for the lass of squareroot soring rule with t = w t, we have t q = w 2 /4q 2 by Equation (4) and defined in Equation (8) beomes w 2 = 2q l w2 4q l ] ] w 2 w2 = w2 w2 (11) 2q h 4q h 4q l 4q h Beause the effiient soring rule requires = zq h q l, the effiient oeffiient is w = 2 zq h q l (12) For the logarithmi soring rule with t = w ln1 + t, t = 1/q 1, and thus the effiient oeffiient w is haraterized by q h + w ln q h + q l w ln q l = zq h q l. of the nonpreferred suppliers wins over the preferred ones in equilibrium (see the disussion around Equation (9)), and thus l and h are independent of, ahieves the effiieny.

9 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 31 Figure 3 Expeted soial welfare Example of Expeted Soial Welfare Welfare under effiient soring rule Welfare under standard prourement autions The following example ompares the expeted soial welfare under our autions with the ontingent payments to that under standard prourement autions. As we an see, the aution format we propose an signifiantly improve the soial welfare, espeially when there is onsiderable unertainty about the outome (i.e., is around the middle). Example 2. We let q l = 1/3, q h = 2/3, z = 2, v = 3, and n = 10, and assume the osts of preferred and nonpreferred suppliers are uniformly distributed on 0 1. We use the square-root soring rule (i.e., t = w t) as an example. By Equation (12), the effiient oeffiient w is 4/3. Figure 3 depits how the expeted soial welfare ahieved with the effiient soring funtion hanges with. We use the expeted soial welfare ahieved in standard prourement autions (orresponding to = 0 in our setting) as a benhmark for omparison. 6. Optimal Soring Rule In addition to the effiieny riterion, another useful design riterion is payoff maximization. Espeially in the private setor, firms are typially interested in minimizing their prourement ost. When some nonprie attributes also matter, as in our setting, firms often intend to maximize their expeted payoff or minimize the expeted total ost (inluding the possible loss from failure). Next, we examine how a buyer should hoose the soring rule to maximize its own expeted payoff. We an expliitly derive the buyer s expeted payoff (see the appendix for details) as v n q l z + + F ] ] l f l l f l d + h d n1 q h z + + F ] h f h h f h d (13) The seond term is the expeted ost from a preferred supplier and the third term is the expeted ost from a nonpreferred supplier. In traditional prourement aution settings (with suppliers being the same regarding their type), the revenue is typially in the form of v n + F /f f d, in whih + F /f is usually alled the virtual ost. Along a similar line, we an all q l z + + F l /f l and q h z + + F h /f h virtual osts from a preferred supplier and nonpreferred supplier, respetively. Virtual ost essentially measures eah supplier s marginal ontribution to the buyer s expeted payment. The virtual ost is different from atual ost by a term that aptures the externality imposed to other suppliers. Three features in our expeted payoff stand out from traditional prourement autions. First, in addition to their osts, suppliers differ in their types in terms of suess probability, and thus we have two terms (one for eah type). Seond, unlike in the traditional prourement aution setting, we have q l z and q h z in the virtual ost. This is beause suppliers may fail to meet the requirement, and the buyer thus inurs the expeted loss from its value v. Third, beause preferred suppliers naturally have an advantage, their base payoff also plays a role in the buyer s expeted payoff, as we an antiipate. We all a soring rule that maximizes the expeted payoff in Equation (13) an optimal soring rule. The optimal soring rule an be obtained from the firstorder ondition of the expeted payoff with respet to. Exept for some speial ases, the optimal annot be expressed in an expliit form. Next, we fous on two issues regarding the optimal soring rule. First, how is it different from the effiient design? Seond, how is it affeted by the underlying model primitives, espeially ost distributions? We introdue the following definition for the purpose of omparison. Definition 1. For two umulative distribution funtions F and F (with the density funtions

10 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 32 Deision Analysis 7(1), pp , 2010 INFORMS being f and f, respetively), we say F is a distributional upgrade of F if f F f F (14) for all where the above expressions are well defined. The requirement (14) is one of onditional stohasti dominane: Conditional on any maximum level of ost, Fis less likely to yield a higher ost than F. Similar onditions have often been used in the aution literature. For example, Maskin and Riley (2000) use suh a ondition when they study asymmetri autions. Many lasses of distributions or hanges in distribution an satisfy the ondition of distributional upgrade. For instane, shifts of distribution to the left generate a distributional upgrade. By simple algebra, we an verify that a uniform distribution on 1 3 is a distributional upgrade of a uniform distribution on 2 4. Definition 2. A distribution F satisfies the regularity ondition if F /f is inreasing in for all where the expression is well defined. The regularity ondition is ommonly adopted in many prourement studies (e.g., Che 1993). Saying that a distribution satisfies a regularity ondition is equivalent to saying that the distribution F is logonave. Many distributions, inluding uniform, normal, exponential, and logisti distributions, satisfy the regularity ondition. Proposition 3. The optimal soring rule is haraterized by ( first-order ondition) q h z q l z + F h + f h F ] l dl f l d f l d = h (15) or by = 0 (orner solution). If F h satisfies the regularity ondition and F l is a distributional upgrade of F h, then the optimal soring rule satisfies opt < eff. Reall that a nonpreferred supplier with ost is omparable to a preferred supplier with + in equilibrium. A lower means a greater hane for the nonpreferred supplier to win over a preferred supplier, and thus a higher equilibrium probability of winning. Under the regularity and distributional upgrade onditions, the above proposition suggests that soring rules should be biased to promote the nonpreferred suppliers (ompared to the fairness of the effiient design). In other words, in equilibrium, the buyer under-rewards suppliers with higher suess probability for the sake of maximizing its expeted payoff. The bias or under-rewarding an be seen even more learly from the soring rule. For example, under the square-root soring rule t = w t, by the derived in Equation (11), opt < eff is equivalent to w opt 2 wopt 2 < weff 2 weff 2 4q l 4q h 4q l 4q h whih leads to that w opt <w eff. Reall that preferred suppliers are willing to bear a high penalty t for projet failure in general (by Lemma 1). Reduing w (from w eff to w opt ) simply uts preferred suppliers advantage in the soring rule, and essentially underrewards preferred suppliers advantage of their low failure rate. The intuition is as follows. First, for any soring rule with eff, under the regularity and distributional upgrade onditions, we have ( q l z + + F ) l > f l ( q h z + + F h f h This is beause q l z + = q h z + eff and F l /f l F h /f h > F h /f h. In other words, under any soring rule with eff, the virtual ost for a preferred supplier to fulfill the projet is always higher than that of a omparable nonpreferred supplier. Notie that in Equation (13), in addition to the preferred suppliers base payoff, the total expeted payment is the expetation of suppliers weighted virtual osts (with the weights being their respetive winning probability). Thus, the buyer obtains higher expeted payoff by hoosing any soring rule with < eff. Seond, a soring rule also affets the preferred suppliers base payoff h d. In partiular, the base payoff is inreasing in. Beause the buyer hopes to minimize the preferred suppliers base payoff, a soring rule with < eff is more desirable than the one with a higher. Therefore, the optimal soring rule must satisfy opt < eff under the regularity and distributional upgrade onditions. It is worth pointing out that the optimal soring rule presribed by Proposition 3 is optimal among all )

11 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 33 possible (inluding ) and the optimal soring rule is always less than. In other words, it is always benefiial for the buyer to use the soring rule to leverage the ompetition between the two groups of suppliers by letting some low-ost nonpreferred suppliers win over some high-ost preferred ones in equilibrium. Intuitively, > annot be optimal beause preferred suppliers base payoff is inreasing in (see Equation (9)), and thus a larger means a higher base payoff for preferred suppliers and a lower expeted payoff for the buyer. The finding is interesting in that even when the differene between the two types of suppliers suess probabilities is so huge that hoosing any nonpreferred supplier is apparently ineffiient, the optimal hoie for the buyer is still to possibly award the projet to a nonpreferred supplier to promote the ompetition between the two groups. Notie that the distributional upgrade we defined is in a weak form that allows idential distributions. Therefore, for the above proposition, one speial ase is that suppliers types are independent of their osts, suh that F h = F l. With this independene, the only ondition needed is the regularity one. It is also worth noting that the properties of the optimal soring rule derived in Proposition 3 (i.e., the value of opt and opt < eff under ertain onditions) do not depend on the form of t, beause we do not speify the form of the soring rule in the analysis. The optimal an be alulated from Equation (15), whih, in general, is a funtion of the number of suppliers. For any given lass of soring rule (e.g., the square-root soring rule t = w t), the optimal soring rule (e.g., the optimal w in the square-root soring rule) an be omputed by substituting the optimal into Equation (8). We next use an example to illustrate the optimal soring rule. Example 3. We onsider a setting in whih q l = 1/4, q h = 1/2, z = 3, and v = 3. Assume the osts of preferred and nonpreferred suppliers are uniformly distributed on 0 1. Given and n, opt an be derived by the first-order ondition haraterized by Equation (15). For example, when = 1/2 and n = 3, we an obtain opt = 1/3, whih is less than eff = zq h q l = 3/4. Furthermore, we an hek the optimal parameters of different soring rules. For instane, under the square-root soring rule, by substituting opt and eff into Equation (11), we an derive w opt = 2/3 and w eff = 3/2, respetively. Under the soring rule with t = t w (0 <w 1), an be formulated as ( ) w/w 1 ( ) 1/w 1 ] ql ql = q l w ( qh w w ) w/w 1 q h ( qh w ) 1/w 1 ] (16) We an thus derive that w opt 042 and w eff 057. Clearly, w opt <w eff in both ases, whih implies that under the optimal weighting sheme the buyer understates the importane of the penalty to reate ompetition pressure for the preferred suppliers. Figure 4 Figure 4 w w Example of Optimal w Under (t) = wt 1/2 : Under (t) = t w : (a) Optimal w hanges with (n = 3) Effiient w Effiient w Optimal w Optimal w (b) Optimal w hanges with n ( = 1/2) Under (t) = wt 1/2 : Under (t) = t w : Effiient w Effiient w Optimal w Optimal w n

12 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 34 Deision Analysis 7(1), pp , 2010 INFORMS presents how the optimal parameter w opt hanges with and with the number of bidders n under different soring rules. We see that w opt inreases in n and in, beause the ompetition within preferred suppliers inreases in n and in, reduing the need to indue ompetition by promoting nonpreferred suppliers. In addition, by substituting opt into Equation (13), we an alulate the maximum expeted payoff under the optimal soring rule, 146, whih is higher than that under standard prourement autions, When the ost distributions for the preferred and nonpreferred suppliers do not satisfy the onditions in Proposition 3, however, the optimal soring rule may not have opt < eff, as is illustrated by the following example. Example 4. Continue with Example 3. For the osts of preferred suppliers, we assume that they are uniformly distributed on 2 3 instead. 3 Then we an derive the optimal soring rule opt = 3/2 > eff = 3/4. This example illustrates that it is not always profitable for the buyer to disriminate against the preferred suppliers. The reason is that, in our setting, suppliers ontributions in the bidding ompetition are determined by both their suess probabilities and their ost distributions. When preferred suppliers, who have an advantage in suess probability, are disadvantaged in terms of ost distribution, they may behave aggressively in the bidding ompetition. As a result, a preferred supplier does not neessarily ontribute less to the buyer than its nonpreferred ounterpart who has the same expeted total ost. 7. Extensions and Disussion In this setion, we onsider relaxing some of the model assumptions Degree of Failure In the baseline model, we fous on the setting with a zero-or-one outome: The winner eventually either sueeds or fails to meet the requirements speified by the buyer. In other words, the outome of the projet is either a suess or a failure. The punishment sheme is based on suh an outome. In some 3 Notie that in this ase, it is the preferred suppliers with ost within who are omparable with the nonpreferred suppliers with ost within 2 1, provided that >1. other ases, the buyer may also are about the degree (if measurable) to whih the winner does not meet the requirements. One example of measurable degree is the delay of delivery of the projet. The buyer is onerned not only about whether the winner an aomplish the projet on time, but also about how soon the winner an omplete the projet if it passes the due date. In publi setor prourement, suh as infrastruture onstrution projets, the degree of delay affets the pereived loss of soial welfare. In private setor prourement, the degree of delay ould affet business opportunity, suh as the loss of onsumer goodwill (e.g., resulting from delay in reeipt of the goods requested), and the timing of entering a new market. We denote the requirement level (e.g., quality level) speified by the buyer as r, r 0 1. Instead of assuming that suppliers an meet the requirement or not, we let the outome level x of supplier i satisfy some distribution haraterized by a probability density funtion g i x, x 0 1. If the realized outome x r, the projet meets the requirement; otherwise, it does not, and the differene x r measures the degree of failure. We follow the same aution model as in the baseline ase exept that the penalty t is now a unit payment for unit degree of failure. Similarly, we an formulate the payoff funtion for suppliers as ( 1 ) Ubtg= b t x rgxdx Prwin r For the buyer, the potential loss is generally a funtion of the degree of the failure. Therefore, instead of a onstant z, we introdue a loss funtion x r, whih is inreasing in its argument. Hene, the expeted payoff for the buyer when bidder i wins the ontrat is 1 Vb i t i g i = v b i x rg i x dx r 1 + t i x rg i x dx r Following the same approah, all the analysis in the baseline ase an be onduted and similar results an be derived. In partiular, if the loss funtion is linear (i.e., x r = zx r), we an simply denote q i 1 r x rg ix dx, and all the results are the same as in the baseline ase. The only differene is the interpretation of q: In the baseline ase, q is the probability of failing to meet the requirements, and here q is the expeted degree of failure.

13 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS Multiple Types of Suppliers The basi insight derived from our baseline setting holds not just for two supplier types but also for multiple types of suppliers. Now we suppose there are k types of suppliers, indexed by = 1 2k, and the orresponding failure probabilities are q 1 q 2 q k. Without loss of generality, we assume q 1 <q 2 < <q k. We denote the probability of any individual supplier s being type as and the assoiated ost distribution as F fxq dx. Everything else follows the baseline setting. Lemma 1 ontinues to hold beause the analysis is not affeted by the number of supplier types. So suppliers plae bids on penalty, t q, along the same priniple based on their private information of their failure probabilities. Similar to, for = 2k,we now introdue : t q 1 q 1 t q 1 t q q t q We also define m and suh that Sb t q = Sb 1 m t q 1. Based on these notations, we an formulate the equilibrium probability of winning for a supplier with type and ost : 1 = i 1 F i m i+1 m i+2 m + 1 F i=1 ] k n 1 + i 1 F i m 1 i m 1 i 1 m 1 +1 i=+1 We an then obtain k equilibrium bidding funtions on ost in the same way as in Proposition 1, one for eah type. Speifially, b k = + q k t q k + kx dx k k b = + q t i=+1 i q + i x dx + x dx = 1k 1 where m = + and =. Analogous to the two-type ase, any soring rules satisfying = zq q 1, = 2 3k, are soially effiient. The optimal soring rule an also be haraterized in a similar way (with more omplexity), and basi intuition ontinues to hold. For example, under similar regularity and distributional upgrade onditions, it is revenue-maximizing for the buyer to under-reward the penalty payment from the suppliers with the lowest failure rate (ompared to the suppliers with the highest failure rate). 8. Conlusion In this paper, we fous on an unexplored prourement setting in whih suppliers ability to suessfully aomplish the projet is important to the buyer. We onsider a model in whih suppliers differ in both their osts and in their types in terms of suess probabilities. The buyer introdues a ontingent ontrat on the outome of the projet to further sreen suppliers beyond the basis of ost. In partiular, suppliers are asked to speify a penalty if they fail to deliver the projet as required. We find that an easy-to-implement quasi-linear soring rule an effetively separate supplers in regard to their types. In equilibrium, suppliers bid different amounts of monetary penalty based on their own types. Suppliers bids on the ost take into aount the possible penalty in ase of failure, suh that the inferred information regarding suppliers suess probability does not diretly benefit the buyer. However, the inferred information an be used to find the most suitable supplier or even to leverage the ompetition. A properly designed soring rule an easily implement an effiient alloation in whih the supplier with the lowest expeted total ost to the buyer gets awarded the ontrat, and the inferred information is essentially internalized. To minimize the total prourement ost, the buyer may or may not under-reward suppliers with high suess probability, depending on the balane between suppliers suess probabilities and the assoiated ost distributions. Beause suppliers differ in two dimensions, the onventional priniple of under-rewarding the high type an work only if the suppliers with high suess probability also have distributional advantage on their ost; otherwise, over-rewarding the high type ould be optimal to minimize the prourement ost. In addition, to promote the ompetition between different types of suppliers, it is optimal for the buyer to always let at least some suppliers with low suess probability be able to win over some of those with high suess probability, even though suh alloation may appear rather ineffiient when the differene in their suess probabilities is huge.

14 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 36 Deision Analysis 7(1), pp , 2010 INFORMS Our analysis has several impliations. First, we illustrate the importane and effetiveness of introduing ontingent ontrats in sreening suppliers and all for prourement managers attention to properly use suh an eonomi devie in aution design. Most nonommodity prourements are assoiated with risk to some extent; suppliers may not deliver the projet on time, or the delivered projet might not satisfy the requirement prespeified. Buyers are thus enouraged to reognize suh risks and to take some preaution in hoosing the ontrator ex ante to avoid the onsiderable ost of possible lawsuits or renegotiation. We show that a simple soring rule based on suppliers bids on ost and penalty in ase of failure an effetively help sreen suppliers without any operational ost assoiated with any physial sreening (e.g., visiting the supplier and heking its failities). Compared to standard autions, the prourement autions with ontingent ontrats an signifiantly improve both soial welfare and the buyer s payoff. Seond, different designs of the soring rule are presribed. The soring rule to maximize the total soial welfare is remarkably simple and easy to implement, despite the omplexity of suppliers strategi behavior. Any soring rule that essentially hooses the supplier with the lowest total expeted ost to the buyer an ahieve the goal. Suh a rule is also very easy to implement: The buyer only needs to estimate the distribution of suppliers suess probabilities and its own loss from a failure. To minimize the prourement ost, the buyer should adjust the soring rule. Suh a rule may favor or disfavor suppliers with high suess probability. When suppliers with high and low suitability have the same ost distribution, the buyer may obtain a lower prourement ost by favoring the suppliers with low suitability. When the suppliers with lower suess probability have some advantage in their ost distribution, the best design may favor those suppliers less and possibly even disfavor them. Suh results suggest that we annot automatially assume that the low-type suppliers should be favored in a ost-minimizing design. It is also worth noting that the design of soring rules we presribe are independent of the forms of soring rules. Therefore, many ommonly used lasses of soring rules (e.g., square-root and logarithmi soring rules) with proper parameters an serve the purpose of maximizing soial welfare or minimizing prourement ost (within the lass of quasi-linear soring rules), and the hoie of the speifi soring rule should depend on other riteria. Third, our results also have impliations for suppliers bidding strategies. For example, suppliers with high suess probability should signal their onfidene by bidding aggressively on the penalty in ase of a failure. Meanwhile, they should antiipate the possible penalty and take that into aount when formulating the bid on ost. Several other issues may be interesting for future researh. First, the moral hazard problem of suppliers an be disussed and inorporated in a future study. In the urrent work, we fous on the adverse seletion problem in whih suppliers differ in their types, whih are their inherent nature, and hene have no ontrol over the ost or probability of suess. Although the framework of adverse seletion an address a lass of problems and serves as a good starting point, further investigation on suppliers inentive to exert effort to influene the ost and/or suess probability an be a good omplement to the urrent study. Seond, it would be interesting to further examine the format of ontingent ontrats and its effet. For example, we may allow suppliers to propose different ontingent ontrat formats to reflet their types. Aknowledgments The authors thank the review team for very helpful omments and suggestions. Andrew Whinston aknowledges the benefit of disussion with Mihael H. Rothkopf while they ollaborated on the paper On E-Autions for Prourement Operations (Rothkopf and Whinston 2007). Lizhen Xu is the orresponding author. Appendix A.1. Proof of Proposition 1 Proof. By the envelope theorem, we have dub h t q h q h /d = U / = h from Equation (6) by noting Prwin = h in equilibrium. Notie the boundary ondition Ub h t q h q h = 0 (i.e., the nonpreferred supplier with the highest ost earns zero profit in equilibrium). So we have Ub h t q h q h = h x dx

15 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 37 Combining Equation (6) in equilibrium (so Prwin = h ) with the above payoff funtion, we an obtain the bidding funtion b h. Applying the envelope theorem again to the preferred suppliers equilibrium payoff, we an derive and Ub l t q l q l = Ub l t q l q l + l x dx b l = + q l t q l + Ub l t q l q l + lx dx l Then notie that a nonpreferred supplier with ost ties with a preferred supplier with ost m. We have + q h t q h + hx dx t q h h = m + q l t q l + Ub l t q l q l + m lx dx t q l m l = m + q l t q l + Ub l t q l q l + hxm x dx h t q l (17) Multiplying both sides by h and taking the first-order derivative with respet to, we have h + h + q h t q h h h t q h h = m h + m h + q lt q l h h m t q l h Therefore, m = + and thus =. Substituting m into Equation (17), we have + q h t q h + hx dx t q h h = + + q l t q l + Ub l t q l q l + hx dx h t q l Therefore, we have Ub l t q l q l = h x dx To show that the above bidding funtions b h and b l are equilibrium bidding strategies, we need to verify that suppliers have no profitable deviation. For a nonpreferred supplier with ost, given everyone else s bidding strategy, deviating from the presumed bidding strategy is equivalent to reporting a different ost. The payoff differene is Ub h t q h q h Ub h t q h q h = h h x dx Beause h is dereasing in, the payoff differene is negative for any. Therefore, deviation is nonprofitable for the nonpreferred bidders. Similar arguments apply to the preferred bidders. A.2. Proof of Proposition 2 Proof. As disussed in the text, = zq h q l an ensure ex post effiieny. We next show that = zq h q l an also guarantee ex ante effiieny. First, note that l = h and d h d = f l 1 f h d l d (18) Notie that in the expression of the expeted soial welfare Equation (10), the soring rule is fully represented by. Take the first-order derivative of Equation (10) with respet to, n v q l z d l d f l d + n1 v q h z d h d f h d Noting that d h /d = 0 for > = and that d l /d = 0 for <m 1 = +, we an reorganize the above as v q l z d l + d f l d + 1 v q h z d h d f h d = v q l z d l + d f l d v q h z + d l + d f l d = q h z q l z d l + d f l d where the first equality is beause of integration by substitution and Equation (18). Beause d l /d > 0, the above first-order derivative is positive if <zq h q l and negative if >zq h q l.so = zq h q l maximizes the soial welfare. A.3. The Derivation of Expeted Payoff The buyer s expeted payoff from a supplier is equal to the value reated upon winning minus the supplier s expeted payoff: v q j z j Ub j t q j q j

16 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions 38 Deision Analysis 7(1), pp , 2010 INFORMS where j h l. Notie Ub h t q h q h = hx dx and Ub l t q l q l = h x dx + lx dx from the proof of Proposition 1. Then, the payoff from one supplier (with probability being preferred and with probability 1 being nonpreferred) is Ev q l z l Ub l t q l q l + 1 Ev q h z h Ub h t q h q h ] = v q l z l h x dx l x dx f l d ] + 1 v q h z h h x dx f h d = v q l z F l f l + 1 = 1 n v 1 ] l f l d v q h z F h f h q l z + + F l f l q h z + + F h f h ] h f h d ] l f l d ] h f h d h x dx h x dx where the seond equality is ahieved by exhanging the integration order and the third equality is beause lf l d + 1 hf h d = 1/n (i.e., the ex ante expeted probability of winning for any bidder is 1/n). The total expeted payoff for the buyer is thus n times the above. A.4. Proof of Proposition 3 Proof. Taking the first-order derivative of the expeted payoff Equation (13) with respet to yields the following: h n q l z + + F ] l dl f l d f l d n1 q h z + + F ] h dh f h d f h d Noting that d h /d = 0 for > and d l /d = 0 for < +, we an reorganize the above as h q l z + + F ] l dl + f l d f l d 1 q h z + + F ] h dh f h d f h d = h q l z + + F ] l dl + f l d f l d + q h z + + F ] h dl + f h d f l d = q h z q l z + F h + f h F ] l dl f l d f l d h where the first equality is beause of integration by substitution and Equation (18). Notie that the first-order derivative at = is negative, and thus = annot be a orner solution. Therefore, the optimal soring rule is either at the other orner = 0 (as we an show, annot be negative) or haraterized by the above first-order derivative being zero. (Notie that any > is suboptimal. This is beause when, based on Equation (9), the buyer s expeted payoff an be similarly formulated as v n q l z++ F ] l f l l f l d ] + h d + + l n1 q h z + + F ] h f h h f h d whih, by noting that l and h are independent of, is dereasing in.) If F h satisfies the regularity ondition and F l is a distributional upgrade of F h, F h f h < F h f h F l f l Beause d l /d > 0 and h > 0, the above firstorder derivative is negative for all zq h q l.so opt < zq h q l. Referenes Abbas, A. E A Kullbak-Leibler view of linear and log-linear pools. Deision Anal. 6(1) Asker, J., E. Cantillon Properties of soring autions. RAND J. Eonom. 39(1) Beall, S., C. Carter, P. L. Carter, T. Germer, T. Hendrik, S. Jap, L. Kaufmann, D. Maiejewski, R. Monzka, K. Petersen The role of reverse autions in strategi souring. Report, CAPS Researh, Tempe, AZ. Bikel, J. E Some omparisons among quadrati, spherial, and logarithmi soring rules. Deision Anal. 4(2) Bushnell, J. B., S. S. Oren Bidder ost revelation in eletri power autions. J. Regulatory Eonom. 6(1) Bushnell, J. B., S. S. Oren Internal autions for the effiient souring of intermediate produts. J. Oper. Management 12(3 4) Che, Y.-K Design ompetition through multidimensional autions. RAND J. Eonom. 24(4) Drew, C $296 billion in overruns in U.S. weapons programs. New York Times (Marh 31), /03/31/business/31defense.html. Ewerhart, C., K. Fieseler Prourement autions and unit-prie ontrats. RAND J. Eonom. 34(3) Good, I. J Rational deisions. J. Royal Statist. So. 14(1) Johnstone, D. J The parimutuel Kelly probability soring rule. Deision Anal. 4(2)

17 Chen, Xu, and Whinston: Contingent Contrats in Prourement Autions Deision Analysis 7(1), pp , 2010 INFORMS 39 Kilgour, D. M., Y. Gerhak Eliitation of probabilities using ompetitive soring rules. Deision Anal. 1(2) Liu, D., J. Chen, A. B. Whinston Ex ante information and the design of keyword autions. Inform. Systems Res., epub ahead of print November 13, gi/ontent/abstrat/isre v1. Martin, M. G The importane of the SOW in managing projets. Delivering Projet Exellene with the Statement of Work, Chap. 2. Management Conepts, Vienna, VA, Maskin, E., J. Riley Asymmetri autions. Rev. Eonom. Stud. 67(3) MAfee, R. P., J. MMillan Bidding for ontrats: A prinipalagent analysis. RAND J. Eonom. 17(3) Rothkopf, M. H., A. B. Whinston On e-autions for prourement operations. Prodution Oper. Management 16(4) Wan, Z., D. R. Beil RFQ autions with supplier qualifiation sreening. Oper. Res. 57(4)