First-price equilibrium and revenue equivalence in a sequential procurement auction model

Transcription

1 Eon Theory (200) 43:99 4 DOI 0.007/s RESEARCH ARTICLE First-prie equilibrium and revenue equivalene in a sequential prourement aution model J. Philipp Reiß Jens Robert Shöndube Reeived: 20 Deember 2006 / Aepted: 3 November 2008 / Published online: 0 Deember 2008 The Author(s) This artile is published with open aess at Springerlink.om Abstrat We analyze first-prie equilibrium bidding behavior of apaityonstrained firms in a sequene of two prourement autions. In the model, firms with a ost advantage in ompleting the projet autioned off at the end of the sequene may enter the unfavored first aution hoping to lose it. Equilibrium bidding in both autions deviates from the standard Symmetri Independent Private Value aution model due to opportunity osts of bidding reated by possibly employed apaity. For this sequential aution model with non-idential objets, we show that revenue equivalene applies. Keywords Sequential first-prie autions Revenue equivalene Endogenous outside options Prourement aution Capaity onstraints JEL Classifiation C72 D44 L5 We thank Werner Güth and Ulf Shiller for helpful omments and stimulating disussions. The paper greatly benefited from very helpful omments of the assoiate editor and two reviewers. We gratefully aknowledge omments from Jeannette Brosig, Veronika Grimm, Alfred Luhmer, Barbara Shöndube-Pirhegger, Gerhard Shwödiauer, and audienes at Erfurt, Frankfurt, Fribourg, Heidelberg, Leuven, Maastriht, Magdeburg, Marseille, Mannheim, Munih, and Swansea. J. P. Reiß (B) Eonomis Department, Maastriht University, 6200 MD Maastriht, The Netherlands p.reiss@alge.unimaas.nl J. R. Shöndube Faulty of Eonomis and Management, University of Magdeburg, 3906 Magdeburg, Germany jens.shoendube@ww.uni-magdeburg.de

2 00 J. P. Reiß, J. R. Shöndube Introdution This paper studies first-prie equilibrium bidding behavior in a sequential prourement aution model with apaity-onstrained firms. Usually prourers differ aross regions, firms, and institutions and set independently of eah other different aution dates implying a sequene of autions. Often projets to be autioned off are similar and offered for exeution during the same window of time. Although the sequential nature of prourement autions is prevalent, many theoretial studies impliitly abstrat it away by foussing on a single prourement aution in isolation (e.g., Holt 980; Riordan and Sappington 987; MAfee and MMillan 987b; Dasgupta and Spulber 989; Celentani and Ganuza 2002). In sequential prourement aution theory, it is ommon to assume that any bidding firm has unlimited apaity to exeute all sequentially offered projets, see, e.g., Luton and MAfee (986). However, reent empirial studies on sequential prourement autions point to the relevane of apaity onstraints. De Silva et al. (2002, 2003) find that bids are positively orrelated with employed apaity. Jofre-Bonet and Pesendorfer (2000, 2003) report that a firm whih did not win a highway prourement ontrat earlier in a sequene of autions is twie as likely to enter a subsequent aution than a firm whih already won a (large) ontrat. This evidene suggests that firms are aware of their opportunity osts of bidding reated by employed apaity and might be hoosy if faing an aution sequene of non-idential prourement ontrats. We onsider a sequene of prourement autions in the first-prie sealed-bid design where projets are stohastially equivalent and bidding firms are apaityonstrained. Sine a potential bidding firm finds itself restrited to exeution of a subset of sequentially offered heterogenous projets, it must deide whih prourement autions to enter entailing a seletion of projets it wishes to possibly end up with. For a prourement sequene of idential projets, the analysis of Weber (983) suggests that equilibrium expeted payments of winning bidders oinide and, hene, any firm may want to begin bidding at the start of the sequene. If, however, firms prefer to exeute one projet to another due to different projet ompletion osts, it is a priori not lear if a firm with more favorable ompletion osts for projets to be autioned later in the sequene submit bids for projets autioned earlier. This paper studies this kind of entry deision and analyzes how firms refine their bidding strategies with opportunity osts of early bid submission. Our main findings are that the entry deision depends on relative projet ompletion ost values and equilibrium bidding in both aution stages deviates from the standard Symmetri Independent Private Value aution model (SIPV) and its sequential formulation with homogenous goods and single-unit demand. Firms with lower ompletion osts for the first projet autioned off always submit bids while firms with lower ompletion osts for the projet subsequently autioned off only partiipate if their opportunity osts are not too large. Eah firm entering the first aution inludes its option value of the seond projet in its bid for the first projet. Moreover, we derive revenue and payoff equivalene for the sequential model with non-idential projets. Exeptions inlude Elmaghraby (2003)andGale et al. (2000) where in the latter the ost funtion an be hosen to effetively allow for apaity onstraints.

3 Sequential prourement aution model 0 An experimental investigation of our sequential first-prie aution model, see Brosig and Reiß (2007), finds that the bidding deviations from the standard SIPV model predited by our model are, indeed, observed in the laboratory. This finding emphasizes that opportunity osts of early bid submissions are understood by bidders and form a ruial determinant of bidding behavior. In the remainder of this setion we relate our paper to the relevant literature on sequential autions. Setion 2 introdues our model and its symmetri equilibrium. In Set. 3, we derive payoff and revenue equivalene. Setion 4 investigates differenes between the stati SIPV model and the sequential model. In partiular, we illustrate how the option value of subsequent ontrat opportunities affets bidding deisions of firms. In Set. 5 we extend the model to allow for reserve pries and weak apaity onstraints, finally Set. 6 onludes.. Ties to the literature In reent years the literature has devoted inreasing attention to sequential autions (predominately in the non-prourement ontext). Inspired by Ashenfelter s (989) observation of delining pries in wine autions where idential ases were sold one after the other, part of the literature aimed at deriving a theoretial explanation for the violation of the law of one prie that the has been dubbed delining prie anomaly ever sine. 2 For example MAfee and Vinent (993) explain it by risk averse bidders, Blak and De Meza (992) show that a winner s option to buy further objets at the same prie leads to dereasing pries and Bernhardt and Soones (994) and Engelbreht-Wiggans (994) derive a falling prie trend by assuming stohastially equivalent objets. While reent work suggests that strategi behavior in an aution sequene is a driving fore for this anomaly, Deltas and Kosmopoulou (2004) demonstrate that part of the overall effet annot be explained by strategi onsiderations... Prie trends and ompetition Dereasing equilibrium pries in sequential autions often ome along with a redued degree of ompetition. Von der Fehr (994) onsiders a sequene of two English autions with partiipation osts. As the first aution provides a ranking of bidders valuations, bidders who have no hane to win the seond aution do not enter this aution if partiipation is ostly. Gale and Haush (994) employ a sequene of two objets in the right-to-hoose format with two bidders having unit demand. In equilibrium there is no ompetition in the seond aution suh that the loser of the first aution wins it for a prie of zero (the seller s reserve). Jeitshko (999) onsiders a sequene of two autions where the seond one only ours with some exogenous probability. He finds that the unertain prospet of the seond aution redirets ompetitive pressure to the first aution at the expense of ompetition in the seond aution. 2 It should be noted that inreasing prie paths have been derived in some papers, e.g., Blak and De Meza (992). In Engelbreht-Wiggans (994) depending on the distributional assumptions expeted equilibrium pries may inrease or derease. In Menezes and Monteiro (2003) the prie trend depends on whether objets have positive or negative synergies.

4 02 J. P. Reiß, J. R. Shöndube Brano (997) assumes omplementarities between objets in a sequene of two autions: Bidding behavior in the first aution stage is more aggressive than in the seond stage as there is only one bidder remaining after the first aution who an win the bundle of objets. 3 Engelbreht-Wiggans (994) analyzes a sequene of autions with stohastially equivalent objets and unit demand bidders. On the one hand there is a delining pries effet as the number of bidders shrinks as the ourse progresses in the aution sequene. On the other hand the number of remaining hanes to win the objet dereases over the sequene of autions failitating more aggressive bidding over time. While in Weber s (983) analysis with idential objets these two effets neutralize eah other with stohastially equivalent objets equilibrium pries an deline. We do not fous on prie trends in this paper, though, endogenous ompetition is one major onern. Compared to the models above in this paper the degree of ompetition in the seond aution depends on endogenous entry in the first aution. In equilibrium, depending on the option value of partiipation in the seond aution, no bidder may enter the first aution. However, even if there is ompetition in the seond aution we demonstrate that bidding behavior does not oinide with equilibrium bidding in the stati SIPV model. The reason is that all bidders on the seond aution stage reeive a signal on ompletion osts of their opponents. Unlike with seond prie autions studied elsewhere, first prie equilibrium bidding is affeted by updates of bidders beliefs about their ompetitors...2 Option value and endogenous entry A number of models analyzing the influene of the option value of future autions on bidding strategies assumes like this paper apaity onstrained (unit demand) bidders. Bernhardt and Soones (994), Engelbreht-Wiggans (994), and Von der Fehr and Riis (998) assume that eah bidder learns his valuation for a speifi objet immediately before the time of sale whereas in our model osts for both objets are realized at the beginning of the first aution. Letting bidders learn the osts for all objets at the beginning of the sequene onsiderably affets strategi partiipation in the first period. Zeithammer (2007) employs an overlapping generations model to analyze the interation of buyers bid shading strategies and the seller s selling deision. Antiipating future autions bidders have an inentive to shade their bids in the urrent aution but at the same time the seller- antiipating bid shading- will learn form observed pries on atual demand. Zeithammer shows that in equilibrium bid shading survives but it is self-regulating in the sense that it vanishes before the market breaks down. Consistent with the above finding that option values lead to less aggressive bidding behavior we show in the prourement ontext how first aution equilibrium bids are inreasing in the option value resulting from the seond aution. This result does not only hold in ase of strong apaity onstraints (in the sense that eah firm an at most 3 Menezes and Monteiro (2003) show that with negative synergies pries are inreasing.

5 Sequential prourement aution model 03 exeute one ontrat) but it arries over to the ase that firms an purhase additional apaity to exeute a seond projet onditional on winning the first one. While aounting for the option value is ubiquitous there are few papers that expliitly disuss endogenous partiipation. As mentioned above a partiipation ost may indue bidders not to partiipate in an aution if the hane to win the aution is rather low (Von der Fehr 994). In Caillaud and Mezetti (2004) the seller annot ex ante ommit to a reserve for the seond aution. In equilibrium bidders may deide not to partiipate in the first aution to avoid a high seond period reservation prie. In analyzing endogenous partiipation in terms of the option value of future aution with apaity onstrained bidders our model is losely related to Gale and Haush (994). They disuss entry behavior in sequential autions in the seond-prie format. In a two bidders-two autions model with unit demand Gale and Haush demonstrate that bottom fishing may show up. A bidder with a preferene for the seond projet may submit a low bid in the first aution: If he loses he wins the preferred objet, and if the ompetitor also prefers the seond objet he may win the first objet at a low prie. Gale and Haush also show that in equilibrium both bidders might deide to skip the first aution. 4 In our paper we emphasize the relation between the endogenous option value in the first aution and non-entry behavior. We also show that this relation does not depend on strong (unit demand) apaity onstraints...3 First-prie equilibrium and revenue equivalene While Gale and Haush (994) onsider the seond-prie sealed bid format we analyze endogenous entry in sequential prourement ontrats by presenting the omplete first-prie equilibrium solution of the game. Although the seond-prie aution format is analytially more tratable it is rare in prourement suh that deriving the non-trivial (see Katzman 999, p. 78) first-prie equilibrium is more preditive for real bidding behavior. Note that under the seond-prie format if there is ompetition in stage two, eah bidder s dominant strategy is to bid its true ompletion ost while under the first-prie design equilibrium strategies (inluding the entry strategy) for both autions must be determined simultaneously (see also Gale and Haush 994, p. 32). Revenue equivalene in one-shot autions is well known. From Weber (983) and Maskin and Riley (989) we also know that in multi-unit autions with idential objets and unit demand bidders revenue equivalene applies. For an aution setting with multiple non-idential (stohastially equivalent) objets and an endogenous number of bidders no suh result has been shown yet. We prove revenue equivalene and show that equilibrium entry behavior is invariant to the partiular aution design. Our result of revenue equivalene implies that all results on prie-trends derived by Gale and Haush (994) in their seond-prie aution model apply to our model too. 4 Zeithammer (2006) studies the ase of N > 2 bidders.

6 04 J. P. Reiß, J. R. Shöndube..4 Isobid urves in two-dimensional type spae A key feature of our sequential prourement aution model is that bidders types are two-dimensional (the ost value for the first aution along with the ost value for the seond aution). Any analysis of bidding behavior with one-dimensional bids aggregates two-dimensional types into a single bid (e.g., the bid in the first prourement aution) whih depends on some appropriately defined one-dimensional type. It follows that the same partiular single bid value an be submitted by a ontinuum of newly defined one-dimensional types eah generated by appropriate variations of two-dimensional types in both dimensions. From a geometri point of view, there arise isobid urves in two-dimensional type spae. This approah was introdued in Che and Gale (998). Che and Gale (998) inorporate finanial onstraints into the SIPV model where the first dimension of the type is the value of the autioned objet and the seond dimension is the finanial onstraint, e.g., the budget available for purhasing the objet. Interestingly, equilibrium aggregation of two-dimensional types in Che and Gale (998) leads to a (refined) minimum seletion of type dimensions, the result that revenue equivalene does not hold and Leontief isobid urves for the first-prie and seond-prie aution. In our paper, the seond dimension is the ost of exeuting a projet that is offered in a seond aution whih is related to the option value of winning the seond aution. Here, type aggregation is failitated by an affine funtion of both type dimensions, revenue equivalene prevails, and isobid urves take the form of straight lines...5 Asymmetri prourement autions De Silva et al. (2003) provide an asymmetri, stati bidding model to explore differenes in bidding patterns observed in road onstrution autions. In their model, firms distributions of ompletion osts differ from one another whih is motivated by the idea that firms are either inumbents or entrants implying that bidders with idential ompletion osts differ in bidding behavior; partiularly, entrants bid more aggressively than inumbents whih is statistially onfirmed in their empirial analysis. In our paper, we provide an alternative explanation for differing bid funtions in a symmetri, sequential aution model stemming from differenes in the option value that firms assign to subsequently autioned prourement ontrats. In this spirit, our model may be viewed as a way to endogenize asymmetries and differenes in bid funtions in stati models through heterogenous opportunity osts. 2 Model and equilibrium Following the literature (e.g., Arozamena and Estelle 2006, or Porter and Zona 993), we view first-prie prourement autions as low-bid autions instead of high-bid autions and interpret the winning bid as the prie at whih the servie of projet ompletion is sold to the prouring buyer. Our results would not have been substantially affeted if we had analyzed a sequene of standard high-bid autions with bidders faing unit demand.

7 Sequential prourement aution model 05 There are two risk-neutral firms, eah endowed with apaity to omplete a single projet. 5 Two projets, L and M, are sequentially autioned off. Subontrating is prohibitively ostly. 6 The firms ost of ompleting any of the two projets are their private information and known to them at the beginning of the dynami aution game. In order to formalize the similarity of both projets and the aspet that the ranking of ompletion osts is unknown 7 to ompetitors, we assume that projets are stohastially equivalent. In partiular, it is ommon knowledge that firm i s osts of ompletion are jointly drawn from f (l i, m i ) with domain [, ] 2 and stohastially equivalent in thesenseof f (l, m) = f (m, l) for every (l, m) [, ] 2 implying E[L i ]=E[M i ] where l i and m i denote the ost realization of firm i for projets L and M respetively. Although ompletion osts of a single firm may be orrelated aross projets, pairs of ompletion osts of different firms are independently distributed. 8 If ost realizations of firm i are suh that l i < m i, this firm is said to have a ost advantage for projet L, the reversed inequality indiates a ost advantage for projet M. In eah prourement aution, a partiipating firm may submit a sealed bid where the lowest bid wins the projet and the amount bid is paid in exhange for ompletion of the projet. 9 However, bids annot exeed maximum ompletion osts whih may be interpreted as the prourers outside option. We assume that the autioneer annot set a prie below maximum ompletion osts and that resale of projets is not feasible. If there happens to be a bidding tie, autioneers employ a fair hane mehanism to break it. The sequene of autions begins with the prourement aution of projet L where the winner, if there is any, is publily announed before projet M is autioned off. Thus, with two firms, any firm knows if it faes ompetition in aution M before it submits its bid. 5 Although the restrition that firms are required to omplete at most a single projet seems severe, loser observation reveals that this element is ommon in prourement autions. Firstly, prourers may stipulate exlusive projet ompletion to avoid that its ompetitors running a similar projet gain benefits through a ontrator working for both prourers. Seondly, a firm may fae apaity onstraints if projets run simultaneously and require relatively large amounts of its resoures. Finally, firms may voluntarily deide not to exeute simultaneously several risky projets to prevent hanges in the risk distribution of their entrepreneurial ativities. 6 Empirial studies on prourement bidding (e.g., Jofre-Bonet and Pesendorfer 2000, 2003) find that the probability that a firm partiipates in an aution and that a partiipating firm wins the aution dereases in its baklog. This points to the fat that firms regard subontrating as ostly and not always as a feasible option to weaken their apaity onstraints. 7 Unlike the seond-prie prourement aution model in Elmaghraby (2003) where it is assumed that the seond projet is always more ostly than the first one. 8 For any individual firm i (i =, 2), we allow for any type of orrelations between its random projet osts L i and M i by letting them be jointly distributed aording to the bivariate probability density funtion f (l i, m i ) suh that stohasti equivalene is satisfied. This generality is possible sine the analysis of the model does not rely on independent projet ost distributions (whih would imply f (l i, m i ) = f L (l i ) f M (m i )), but utilizes the joint probability distribution funtion as a whole throughout the paper. Before the aution sequene, it is ommon knowledge that nature independently draws projet ompletion ost pairs following f (l i, m i ) for eah firm and privately reveals them. Realizations of projet osts annot be altered during the aution sequene, exluding the possibility to ondition projet ompletion osts on events during the aution sequene. 9 Although the analysis of the seond-prie aution design is less demanding, we employ the first-prie design sine otherwise our model s bidding preditions annot explain real-life data.

8 06 J. P. Reiß, J. R. Shöndube The dynami aution game is a sequential game with inomplete information, hene, we onfine our attention to perfet Bayesian equilibria. Importantly, a perfet Bayesian equilibrium provides a link between equilibrium beliefs and equilibrium strategies suh that equilibrium strategies are optimal given equilibrium beliefs and equilibrium beliefs are onsistent with equilibrium strategies and atual play. This irular reasoning requires the simultaneous determination of beliefs and strategies implying that our sequential aution game annot be solved by bakward indution. Instead we solve the entire game at one: We determine the Bayesian equilibrium of the ontinuation game, i.e., the seond aution game, given any set of a posteriori beliefs and solve for the Bayesian equilibrium in the first aution game given properties of the equilibrium in the ontinuation game. The final step for simultaneous equilibrium solution is to establish the link between a posteriori beliefs that are subjet to Bayesian updating given entry behavior into the first aution and equilibrium entry behavior into the first aution. Sine both firms are ex ante symmetri, we restrit attention to the ase of symmetri equilibria. We refer to the representative firm as firm. The strategy of the representative firm is given by (b L, bm ) where the bid for aution L is given by bl R {no entry} and b M R is the firm s bid in aution M. In addition to equilibrium bid funtions for eah aution stage, a firm s strategy also inludes a deision to submit a bid in the first aution or skip bidding for projet L. Intuitively, there must be a region of ost types where firms rejet to bid in aution L sine a firm with ompletion ost l = annot make any profit by ompleting this projet and, moreover, is if it has won projet L exluded from partiipating in aution M where its expeted profits may be positive due to more favorable osts of ompletion m <. Obviously, these extreme ost pairs highlight that opportunity osts, whih oinide with expeted profits from skipping aution L, exeed expeted profits from bidding for projet L. In general, firm partiipates in aution L if its expeted profit from bidding exeeds opportunity osts arising from possibly being exluded from bidding for projet M, formally E[ L+M (l, m )] E[ M m ], where E[ L+M (l, m )] denotes firm s expeted profit if it bids in the prourement aution for projet L and if unsuessful ontinues bidding in aution M and E[ M m ] is its expeted profit if it skips the first aution and bids only for the subsequently autioned projet M. Profits are random sine ompletion osts of any ompetitor are unknown and determine its bidding behavior. The firm s deision to skip aution L depends on the relation of its ompletion osts. In order to formalize the entry deision we introdue the entry indifferene urve g :[, ] [, ] that assigns a value for the ompletion ost of projet L to eah ost value of projet M suh that the firm is indifferent between taking part in aution L and skipping it. The entry indifferene urve l rit = g (m ) is impliitly defined by the equality of expeted profits from entering aution L and orresponding opportunity ost:

9 Sequential prourement aution model 07 E [ ( )] [ ] L+M l rit, m = E M m. () Sine it annot be worthwhile for a firm to partiipate in aution L with l > l rit but it must be if l < l rit, the firm s deision rule to partiipate in aution L is given by ε (l, m ) = { Enter aution L if l g (m ), Skip aution L if l > g (m ). In Set. 2.2 we show that the entry indifferene urve defined here exists. Figure 3 illustrates the qualitative properties of the entry indifferene urve in equilibrium. Next we derive the equilibrium bid funtions for both prourement autions sine these determine expeted profits on whih the entry indifferene urve g (m ) depends. Sine the equilibrium bid funtions depend on the equilibrium entry indifferene urve themselves, we derive these for any entry indifferene urve g (m ). The equilibrium indifferene urve is denoted by g(m) and is identified given equilibrium bidding behavior. Thus equilibrium bidding and the entry indifferene urve are simultaneously determined. 2. Equilibrium bid funtions Both autions employ the first-prie sealed-bid aution format where the lowest bid wins. 0 With first-prie autions, the symmetri equilibrium strategy in a one-shot prourement aution is well-known, e.g., Holt (980) and Cohen and Loeb (990), and summarized in Lemma. Lemma (Cohen and Loeb 990) Equilibrium bidding in a first-prie prourement aution. Let projet ompletion osts of two risk-neutral firms that bid for a single projet ontrat be private information and independently and identially distributed aording to df H(), [, ]. Then, the symmetri equilibrium bid funtion is b() = + [ H(x)] dx/ [ H()]. For our sequential prourement aution game, we derive the equilibrium bid funtion for eah of the two projet autions by appliation of this lemma to our speifi ontext with additional strategi interation: In the first aution stage every firm knows that a seond aution follows. In the seond aution stage, eah bidder reeives information on the outome of the first aution L. Figure illustrates the equilibrium struture of the game. Consider first the aution for projet L. Any of the two firms that enters the first aution antiipates that in ase it does not win the first aution, it will be the only bidder in the subsequent aution M where it will reeive m. Thus, it might submit a relatively high bid for projet L, sine it is, at least partially, insured against losing the first aution. In partiular, a firm 0 Cf. Vikrey (96), MAfee and MMillan (987a)orMilgrom (989).

10 08 J. P. Reiß, J. R. Shöndube Fig. Equilibrium struture of the sequential prourement aution game with a ost advantage for projet M knows that the largest payoff it an reeive is m from being the only bidder for projet M sine its largest payoff in aution L is l whih must be smaller due to the firm s ost advantage. Thus, provided the firm deides to partiipate in aution L, it seeks to lose the first aution and minimizes its hanes of winning projet L by submitting the highest feasible bid whih simultaneously maximizes its payoff from aidentally winning it. In ontrast, if a firm has a ost advantage for projet L, then it tops its ompletion ost l with its ertain return from aution M and uses this revised ost parameter λ l + m in aution L. Intuitively it uses its total ost of exeuting projet L that inlude the diret projet ost l and the opportunity ost of winning projet L, m (=benefit of not-winning aution L): Any firm taking part in aution L treats m as a safe profit; if it wins projet L, it pays the ost of exeuting this projet and repays the amount m. If there is no bidding ompetition in the aution for projet M, then any firm bidding for it submits the maximum feasible bid to maximize profits. If it is not the only bidder then it reeives the additional information that its ompetitor did not enter aution L, too. In response it updates its belief about its ompetitor s ost parameter for projet M sine skipping aution L might not be equilibrium behavior for every type. The appropriate a posteriori pdf is denoted by f M skip (m) and gives the (equilibrium) density that a firm with ompletion ost realization m for projet M bids only in aution M. Put differently, f M skip (m) is the marginal pdf of f (l, m) onditional on the fat that ompletion ost pair (l, m) leads the firm to skip aution L. The losed-form equilibrium bid funtions for autions L and M are stated in parts (a) and (b) of Proposition 2. In equilibrium, ompetitive bids equal projet osts, inluding opportunity osts, augmented, i.e., shaded, by an amount that depends on a firm s beliefs about the total projet ost of its ompetitor, F ( ), and its own total projet ost, λ. It is easy to see that equilibrium bids stritly inrease in total ost value λ. To expliitly obtain the first-prie equilibrium bid funtion for the first aution using standard methods, it is essential that there is an additive relationship between projet ost l and opportunity ost m that manifests here in total ost parameter λ l + m. The probability density funtion of λ is derived from the joint Reall that the term ost advantage refers to a omparison aross projets rather than a omparison aross firms.

11 Sequential prourement aution model 09 Fig. 2 Obtaining probability density funtion f λ (λ) distribution f (l, m) by integrating over all ost pairs (l, m) that lead to the same total ost value λ. Figure 2 illustrates various isoquants of λ in the type spae. 2 Integrating joint density f (l, m) along isoquants yields the orresponding probability density for total ost value λ, in general f λ (λ) = + λ f (λ + m, m) dm for λ [, ]. The equilibrium bid funtion for the seond aution depends on the outome of the first aution. Whenever there is ompetition for projet M, the firm submits a ompetitive bid that is smaller than reserve prie. For the ompetitive equilibrium bid, the amount of shading depends on the marginal distribution of projet osts for projet M that is onditioned on the fat the ompetitor did not enter the first aution. We ompare equilibrium bid funtions to SIPV ounterparts in Set. 4. Proposition Equilibrium bid funtions in autions L and M. The equilibrium bid funtions of a firm with ompletion ost pair (l, m) [, ] 2 are given by: { if l m (a) b L (λ) = λ + λ [ F λ(x)] dx/ [ F λ (λ)] otherwise if it submits a bid for projet L where λ l + m and F λ (x) = x + λ f (m + { λ, m) dm dλ with x [, ]. if it is the only bidder (b) b M (m)= m+ [ m FM skip (x) ] dx/ [ F M skip (m) ] otherwise 2 Sine the equilibrium bid along any isoquant is onstant in general, they are also alled isobid lines, e.g., Che and Gale (998). In our ase, however, this : relationship between the type λ and the equilibrium bid breaks down whenever there is a ost advantage for the seond projet, i.e., λ, sine in this ase any type λ submits a bid of. In the ontext of Fig. 2 this means that any isoquant below λ 3 (where the one for λ 3 oinides with the 45 -line) orresponds to a unique isobid urve; in ontrast the isobid urve for the largest feasible bid does not orrespond to a unique isoquant, rather it is an area defined as the set of types that faes a ost advantage for the seond projet and enters the first aution, i.e., λ λ 3 and λ g(m).

12 0 J. P. Reiß, J. R. Shöndube [ ] if it submits a bid for projet M where f M skip (x) = g(x) [ f (l, x) dl ] / g(t) f (l, t) dl dt and F M skip (x) = x f M skip (s) ds and g(x) denotes the ompetitor s entry indifferene urve. Proof Consider first part (b): Before bidding for aution M, any firm knows if its ompetitor won aution L. If the ompetitor entered aution L, the firm remains the only bidder in aution M and maximizes its return by submitting the largest feasible bid equaling. If, however, its ompetitor skipped aution L, the firm infers that its ompetitor s ompletion ost pair must satisfy L > g(m) (to be determined later) and Bayesian updating of the firm s ost belief regarding its ompetitor leads to the a posteriori pdf f M skip (x). Appealing to Lemma leads to b M (m). For (a), note that a firm reeives m in aution M if it loses aution L. If its ompletion osts satisfy m l then it annot reeive a larger return in aution L, m l by assumption. Thus the firm hooses to reeive the largest feasible return from aution L by bidding whih also maximizes the frequeny it ends up with the larger return from aution M, provided it submits a bid for projet L. In ase the firm has a ost advantage for projet L, i.e., l < m, its expeted profit from partiipating in aution L with any bid b L [, ] and possibly in aution M is given by l < m ]=(b L l ) Pr(b L wins aution L) ] + ( m ) [ Pr(b L wins aution L). E[ L+M Using firm s total ost parameter λ l + m, this an be rewritten as E[ L+M l < m ]=(b L λ ) Pr(b L wins aution L) + m. From firm s perspetive m is a known onstant and its expeted profit from bidding in aution L, E[ L+M l < m ], is maximized if b L maximizes Z(λ ) := (b L λ ) Pr(b L wins aution L), (2) where l < m λ < by definition. Suppose there exists a symmetri equilibrium bid funtion b L (λ) that maximizes Z suh that it is stritly inreasing for λ<, b L (λ) < for λ<, and b L (λ) = for λ. Sine b L (λ) is stritly inreasing on [, ], there exists an inverse on that domain denoted by b,l (b L ). Given that firm s ompetitor adheres to this equilibrium bid funtion, firm (with a ost advantage for projet L) wins always the first round if its ompetitor has a ost advantage for projet M, λ 2. It wins projet L too, if it bids an amount that orresponds to a lower total ost type λ = b,l (b L ) than the one of its ompetitor λ 2. Denoting the df of total ost types by F λ (λ), (2) an be rewritten as 3 3 Here the fat is used that types with a ost advantage for projet L always enter the first aution. This is formally onfirmed in Lemma 3.

13 Sequential prourement aution model ] Z(λ ) = (b L λ ) [ F λ (b,l (b L )), (3) where f λ (λ) = + λ f (m + λ, m) dm and F λ( ) = /2. If b L maximizes Z then Z / b L = 0 and differentiation of (3) at the optimum w.r.t. λ yields dz [ ] = F λ (b,l (b L dλ ). Integration in the boundaries [λ, ] together with the fat that in a Nash equilibrium b L must oinide with the value of the equilibrium bid funtion at the true ost type λ leads to Z ( ) Z (λ ) = λ [ F λ (x)] dx. Sine b L ( ) =,wehavez ( ) = 0 and obtain with (3) at its optimum the equilibrium bid funtion for λ < : b L λ (λ ) = λ + [ F λ (x)] dx. F λ (λ ) 2.2 Entry deision In this setion, we derive the entry indifferene urve () given equilibrium bidding summarized in Proposition. Obviously the expeted profit from bidding in aution L and possibly in aution M, E[ L+M (l, m )], always exeeds the expeted profit from skipping aution L, E[ M m ] if firm has a ost advantage for projet L (i.e., l m ): the expeted profit from entering aution L and possibly M is at least as large as m. To see this suppose that the firm would bid in aution L. Then it reeives in expetation ( l ) Pr(won L) + ( m ) Pr(lost L) m. For any bid in aution M, the expeted profit from skipping L, E[ M m ], must be lower than m sine there is a positive probability of bidding ompetition in aution M. It follows that () an only hold if l rit > m and that without loss of generality the entry indifferene urve is defined by: E [ ( )] [ ] L+M l rit > m, m = E M m. (4) In order to expliitly state equation (4), onsider first its left-hand side. A firm with a ost advantage for projet M that enters aution L bids in aution L and if it loses in aution M. This strategy results in four events summarized in the next table where

14 2 J. P. Reiß, J. R. Shöndube the probabilities depend on firm s belief that firm 2 ats in aordane with the entry indifferene urve g 2 (M 2 ). Events if firm bids in aution L Payoff Probability A it is the only bidder l Pr(L 2 > g 2 (M 2 )) B the ompetitor bids andfirmwinsl l Pr(g 2 (M 2 ) L 2 M 2 ) 0.5 C the ompetitor bids and firm loses L m Pr(g 2 (M 2 ) L 2 M 2 ) 0.5 D the ompetitor bids less than m Pr(L 2 < M 2 ) Hene the expeted benefit to a firm with a ost advantage for projet M from starting bidding for projet L with b L = and then ontinuing bidding in aution M after losing aution L is given by E [ ] L+M (l > m, m ) = + + g 2 (m 2 ) g 2 (m 2 ) m 2 m 2 f (l 2, m 2 ) dl 2 dm 2 ( l ) f (l 2, m 2 ) 2 (A) dl 2 dm 2 [2 (l + m )] (B + C) f (l 2, m 2 ) dl 2 dm 2 ( m ). Now, onsider the right-hand side of (4). If firm skips aution L there are three events depending on the entry behavior of its ompetitor. Again firm assesses the probabilities of these events given its belief about the ompetitor s entry indifferene urve g(m 2 ). Events if firm skips aution L Payoff Probability E no bidding ompetition m Pr(L 2 g 2 (M 2 )) F firm wins projet M b M (m ) m Pr(M 2 > m L 2 > g 2 (M 2 )) G the ompetitor wins projet M 0 Pr(M 2 < m L 2 > g 2 (M 2 )) Thus the expeted benefit to firm with a ost advantage for projet M from skipping bidding for projet L an be written as: E[ M m ]=( m ) + m g 2 (m 2 ) g 2 (m 2 ) f (l 2, m 2 ) dl 2 dm 2 (D) (E) [b M (m ) m ] f (l 2, m 2 ) dl 2 dm 2. (F + G) A standard result in aution theory is that in first-prie sealed-bid autions bids are formed suh that they equal the expeted seond-order statisti from the relevant type

15 Sequential prourement aution model 3 pool (onditional on the own type being the first-order statisti). Thus, b M (m ) in the last term in E[ M m ] an be substituted by m 2. This is formally onfirmed by Lemma 2 implying: E[ M m ]=( m ) + m g 2 (m 2 ) g 2 (m 2 ) f (l 2, m 2 ) dl 2 dm 2 Clearly E[ M m ] dereases in m and is independent of l. (m 2 m ) f (l 2, m 2 ) dl 2 dm 2. (5) Lemma 2 Firm s expeted profit from equilibrium bidding in aution M if there is bidding ompetition and b M = b M (m ) is equal to the expeted profit from bidding the expeted seond-order statisti of ompletion osts given that firm s ompletion osts are the lowest, i.e., m g 2 (m 2 ) ( ) b M m f (l 2, m 2 ) dl 2 dm 2 = m g 2 (m 2 ) (m 2 m ) f (l 2, m 2 ) dl 2 dm 2. Proof See the Appendix. Lemma 3 Boundaries of the entry indifferene urve g(m) Let l rit = g (m ) be impliitly defined by E[ L+M (l rit, m )] =E[ M m ]. If g (m ) exists, then m < g (m )<form [, ) and g () =. Proof See the Appendix. The existene of the entry indifferene urve in symmetri equilibrium is verified in Proposition 2 where also its properties are given. Its proof ontains a differential equation whose solution is the symmetri equilibrium indifferene urve g(m). Figure 3 illustrates these results for a representative firm with ompletion ost pair (l, m). Proposition 2 For the symmetri perfet Bayesian equilibrium haraterized by the representative firm s strategy [b L (l, m), b M (m), ε(m)] and the density funtion f (l, m): (a) (b) There exists a (non-empty) ompat and onvex set of ompletion ost pairs where a firm bids for projet L although it has a ost advantage for ompleting projet M. This subset is defined by G ={(l, m) [, ] 2 m l g(m)}. The ritial value funtion g(m) exists and (i) m < g(m) <ifm [, ), g() =, g() >, (ii) g (m) >0, (iii) g (m) <0 if m [, ) and g () = 0.

16 4 J. P. Reiß, J. R. Shöndube Fig. 3 Equilibrium entry behavior Proof See the Appendix. Consideration of Proposition 2 leads to the onlusion that any firm always enters aution L if it faes a ost advantage for this projet, i.e., l m. Then it earns at least m while skipping aution L leaves it with running the risk of lower profits in ase its ompetitor also skipped the first aution resulting in lower expeted profits of this strategy. In ontrast, a ost advantage for ompleting projet M implies the impossibility of the firm to seure itself the same return in aution L as it ould earn in aution M being the only bidder. However, if the ompeting firm has a strong ost advantage for projet M and skipped aution L, then there is ompetition in the seond aution with the risk of low or even zero profits due to aggressive bidding. Therefore a firm may wish to partiipate in the first aution and win the unloved projet L at a high prie to insure itself against low profits resulting from fiere ompetition in the seond aution, although it atually prefers losing the aution for projet L. Although our sequential aution model is highly stylized in the sense that we assume just two bidders implying that there is the possibility of being the sole bidder in the seond aution, we onjeture that our results apply also for a more general number of bidders. 4 As prourement ontrats are usually autioned off in the first-prie format we derived equilibrium bid funtions and entry deisions for this aution format. An immediate question that arises is, how these results and their impliations depend on the speifi aution design employed. In the next setion, we show that revenue equivalene applies for this sequential aution model. 4 In partiular we onjeture that in symmetri equilibrium with N > 2 firms with first-prie autions, (a) the probability that a firm wins with a bid equal to the reserve prie is stritly positive and (b) the probability that no firm enters the first aution is stritly positive.

17 Sequential prourement aution model 5 When firms with apaity to omplete a single projet know that a sequene of prourement ontrats is offered, future ontrating opportunities reate an option value that influenes today s bidding behavior. In Set. 4 we analyze this option value adopting equilibrium bidding behavior derived in this setion. Speifially, we demonstrate how the option value hanges equilibrium bidding ompared to the one-shot SIPV model. 3 Payoff and revenue equivalene In this setion we state a revenue equivalene result for our sequential aution model with non-idential bidder osts aross projets. For a sequential aution of idential objets with bidders that demand a single objet, revenue equivalene holds (Weber 983; Maskin and Riley 989). For our aution model it is a priori unlear if a similar revenue equivalene result ontinues to apply sine in our model bidders ost types for various projets are non-idential. For a sequene of two ompletely unrelated standard autions, i.e., two subsequent one-shot autions, it is obvious that revenue equivalene holds in eah of the stage autions (Myerson 98; Riley and Samuelson 98). In our sequene of autions, the number of bidders in the first stage aution is not fixed and the number of buyers in the seond stage aution is unknown ex ante, 5 so that our result extends standard revenue equivalene. 3. Equivalene result Following Riley and Samuelson (98), we impose the following five properties on aution designs adapted to our ontext that guarantee revenue equivalene for a lass of autions. Property A5 is motivated by our prourement ontext where we require the buyer to ompensate any types ost but not more and not less. A The projet buyer aepts a bid of every firm unless it exeeds the largest feasible bid. A2 The projet is awarded to a firm that submits the lowest bid. A3 The aution design does only disriminate between bidders on the basis of submitted bids. A4 There exists a symmetri Bayesian equilibrium with an inreasing bid funtion. A5 The aution design pays only amounts in the interval [, ] to a bidding firm. Furthermore, it guarantees that the highest ost-type reeives a ompensation equal to its ost. Definition Aution lass Ɣ omprises all aution designs satisfying properties A A5. 5 In the first stage aution the number of bidders is 0, or 2; in the seond stage aution it is either or 2.

18 6 J. P. Reiß, J. R. Shöndube Lemma 4 Expeted profit if skipping aution L. (a) No ompetition in M: Suppose that there is a single bidder in the seond aution M. Then, the bidder s expeted profit is given by π M NC (m ) = m. (b) Competition in M: Suppose that there are two bidders in the seond aution M. Then, bidder s expeted profit is given by π M C (m g 2 (.)) = m [ F M skip (s)] ds where the distribution is defined as in Proposition. () Ex ante profit from skipping aution L: Suppose that a bidder onsiders skipping the first aution L. Then, his expeted profit is given by E[ M m ; g 2 (.)] = p skip g 2(.) π M C (m )+ [ p skip g 2(.) M NC ] π (m g 2 (.)) where p skip g2(.) denotes the probability that the ompetitor skips the first aution implying ompetition in the seond aution onditional on entry behavior of the ompetitor, i.e., p skip g2(.) = g 2 (m) f (l, m) dl dm. Proof (a) By A5, the largest buyer payment to a firm is sine the aution design guarantees that the projet ost of the highest ost-type is fully overed. If there is no ompetition, the only bidder bids suh that it pretends to have to inur maximum ost whih it preisely reeives. (b) In aution M with ompetition, there are two bidders that have observed that the ompetitor did not enter aution L. Eah bidder updates the distribution about ompetitor s ost for ompleting projet M resulting in F M skip. The last stage of the sequential aution game an be viewed as a standard 6 one-shot aution with a modified distribution, the result immediately follows. () The given expression is the definition of the expeted firm s profit if skipping aution L. Proposition 3 (Payoff equivalene) If the two aution designs utilized in autions L and M are in lass Ɣ, eah firm s expeted equilibrium profit is independent of the partiular aution design and there is invariane of equilibrium entry behavior. Proof We have to show that eah ost type (l, m ) reeives the same expeted profit in equilibrium. Our strategy of proof is as follows: first, we show that a firm with a ost advantage for the first projet, i.e., l < m, always enters the first aution and reeives the same expeted profit from its equilibrium strategy enter L and possibly M (where l < m ) independently of the aution designs utilized in L and M. Seond, we show that for a given entry indifferene urve, hanges in the aution design do not hange expeted profits from the strategies enter L and possibly M (where l m ) and skip L. 6 The sense in whih the seond aution is not standard is that the number of bidders varies, it is either one or two, whih is publily known before bids are submitted in the seond aution and that bidders hold posterior beliefs. If a firm realizes that it is the only remaining bidder, it is dominant to submit the largest feasible bid,, for eah possible posterior belief about its ompetitor (whih does not enter the aution). If, however, a firm finds that also its ompetitor bids in the seond aution (whih is known sine the first projet was not alloated to any firm), a similar situation as in the standard one-shot aution set-up unfolds with the exeption that eah firm reeived a signal (the ompetitor did not enter the first aution) about the type of the ompetitor. Given Bayesian updating of prior beliefs, the seond aution with ompetition is standard in the sense that the same equilibrium bid funtion albeit with a different, updated probability density is used as equilibrium bid funtion for the seond aution.

19 Sequential prourement aution model 7. If a firm with a ost advantage for the first projet, i.e., l < m, does not enter the first aution, its expeted profit annot exeed m due to the fat that is the largest payment that any aution design in lass Ɣ allows. However, sine l > m, entering the first aution always offers the possibility to earn a higher expeted profit. Now we derive the expeted equilibrium profit if a firm with a ost advantage for projet L enters the first aution. Let the speifi payment rule aording to aution design L to bidder be given by the funtion ρ L (b L, bl 2 ) where bl i speifies the bid of bidder i in aution L. Assuming that there exists an equilibrium bidding strategy b L : R R {no entry} that is stritly inreasing on [, ] and that firm 2 follows where λ 2 = l 2 + m 2, firm s expeted payment from imitating type x in aution L is R L (x ) = E[ρ L (b L (x ), b L (λ 2 ))] and the firm s equilibrium profit is: E[ L+M l < m ]=R L (x ) Pr{win L} l + [ Pr{win L}] ( m ), whih an be rewritten as E[ L+M l < m ]=R L (x ) [ F λ (x )] λ + m, where λ = l + m and F λ (.) is defined as in Proposition. E[ L+M l < m ] is maximized if Q = R L (x ) [ F λ (x )] λ is maximized. Q an be interpreted as the expeted profit if opportunity osts are inluded. Maximizing Q leads to the following FOC for all λ [, ) at the Nash optimum λ = x dr L (x ) dx = df λ(x ) dx λ. After substituting the Nash ondition λ = x and integration in the boundaries [λ, ] we obtain R L (λ ) = R L ( ) + λ s df λ (s). (6) Note that R L (λ ) is disontinuous at λ = and the left-side limit is lim λ = 2.7 If λ = the firm bids as if it faes maximum ost for projet L and reeives payment if it wins the aution. However, it wins the aution only if firm 2 has no ost advantage for projet L and skips the aution or if it is favored by a fair 7 Every type λ [, λ] with λ = 2, or equivalently eah type with no ost advantage for aution L, i.e., l m, either skips aution L or bids as if having maximum ost for projet L. It follows from the inreasing bidding funtion b L (.) that every firm with a ost advantage for aution L (λ < ) winsthe aution if the ompetitor has no ost advantage for projet L (λ ). The probability that the ompetitor has no ost advantage for projet L is /2. Sine the highest aution payment in aution L to a bidder with maximum ost is, the expeted payment from bidding in aution L onverges to /2 asafirm stype approahes.