Private Information and Endogenous Matching in Supply Chains: Theory and Experiments

Transcription

1 Private Information and Endogenous Matching in Suly Chains: Theory and Exeriments Andrew M. Davis Samuel Curtis Johnson Graduate School of Management, Cornell SC Johnson College of Business, Cornell University, Kyle Hyndman Naveen Jindal School of Management, University of Texas at Dallas, htt:// We investigate a suly chain setting where a sulier s cost information may be rivate information (and they have the ability to disclose it) and buyers and suliers may endogenously match into airs. After forming airs, the two arties interact in a dynamic back-and-forth bargaining environment. We first derive theoretical redictions deending on whether the sulier s cost information is rivate or not, and the matching rotocol tye. We then test these redictions through a human-subject exeriment, which yields a number of insights. A key one is that suliers always make less than the normative theory redicts, whether their cost information is known or rivate information. This effect is esecially ronounced under rivate information for high cost suliers, because buyers make more aggressive bargaining offers in such a setting. Thus, contrary to theory, a second result is that higher cost suliers actually benefit from disclosing their rivate costs, in an effort to achieve a more favorable outcome while bargaining. A third imortant result is that endogenous matching leads to a higher variance of rofits, comared to an exogenous matching rotocol. This is due to the ability of higher quality buyers to choose to contract with higher quality suliers and vice-versa. Key words : behavioral oerations; suly chains; rivate information; endogenous matching; disclosure 1. Introduction A majority of the existing exerimental research on suly chain management assumes full information of cost, rice, and demand arameters. This literature, when investigating two-stage suly chains, also tends to focus on settings where buyer-sulier airs are exogenously (i.e. randomly) determined rior to any sort of contracting. Yet, in ractice it is common for certain information to be rivate to one arty and that buyers and suliers may endogenously decide with whom they want to contract. For examle, an executive for a large durable goods manufacturer recently told us, I usually don t know my sulier s cost structure exactly, but I have a rough estimate of what it might be, suggesting that the sulier s cost details may be rivate information. There is also evidence that buyers are interested in targeting secific suliers. For examle, a recent study by McKinsey & Comany showed that buyers should look for suliers interested in 1

2 2 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains collaborative relationshis: buyers that collaborated with their suliers achieved EBIT (earnings before interest and tax) growth rates that were double comared to those that did not (Noor et al. 2012). This same reort also details how these collaborative efforts include negotiations (which) are based on full transarency into costs. In our study, we investigate suly chain contracting in a dynamic bargaining environment where sulier costs may be rivate or full information, under endogenous and exogenous matching. Industry is relete with examles of buyers wanting transarency from their suliers. A famous one is Toyota. Its urchasing deartment encourages suliers to share their cost details with them so that they can attemt to identify joint gains (Mishina 1995). Yet, suliers may be reluctant to share this rivate information if they feel it will ut them at a disadvantage in subsequent negotiations. This creates a tension between disclosing costs and being able to contract (or match hereafter) with a higher quality buyer, versus having rivate information that may be advantageous in negotiations. The literature on suly chain management, combined with these observations in ractice, brings us to our main research questions. What is the effect of rivate sulier cost information on suly chain outcomes (e.g., contract terms and distribution of rofits) and bargaining dynamics? How are suly chain outcomes affected when buyers and suliers can endogenously choose with whom to contract? When is it in a sulier s best interest to disclose their rivate cost information? We address these questions in an unstructured bargaining environment with random demand. In articular, we deviate from the classical one-shot ultimatum setting and allow for both arties to make multile offers and send limited feedback. Thus, we study our main research questions in a setting where two arties engage in a more natural bargaining rocess, and both arties have relatively equal bargaining ower. We also focus exclusively on wholesale rice contracts such that the two arties negotiate a wholesale rice and stocking quantity simultaneously, and the sulier incurs the cost of any unsold inventory. 1 We begin by deriving theoretical redictions under full and rivate information regarding the sulier s cost, under both exogenous and endogenous matching. We refer to any buying firm in a B2B relationshi (e.g., manufacturers, retailers, distributors, assemblers, etc) as retailers for simlicity. In order to rovide heterogeneity amongst suliers and retailers, we assume that higher quality suliers have lower er unit roduction costs, and higher quality retailers have 1 This setting closely matches a droshiing, vendor-managed inventory (Cachon and Fisher 1997), or e-commerce environment. Randall et al. (6) estimate that between 23% and 33% of e-retailers use dro-shiing, and the U.S. Census estimates that sales by e-retailers totaled $340.4 billion in 2015 (United States Census Bureau 2015).

3 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 3 higher selling rices. Regardless of the roduction cost or selling rice, the quality of the roduct and demand distribution remain the same (i.e., a low roduction cost is not associated with low quality and a high selling rice is not associated with low demand). Regarding the sulier s cost details, under full information we rely on the Nash bargaining solution (Nash 1950) to generate theoretical redictions, whereas under rivate information we adat the solution concet from Myerson (1984), which seeks to generalize the full information Nash bargaining solution. With regards to matching, under exogenous matching we assume sulier-retailer airs are randomly formed and immediately begin bargaining, whereas our endogenous setting assumes that there is an initial stage where retailers and suliers can signal with whom they want to match. If two arties agree to match, then they form a air and roceed to the bargaining stage. In these settings, we generate oint redictions including distribution of rofits, efficiency, wholesale rices, and quantities. While the solution concets emloyed are general, in some cases we derive results for the exerimental arameters in order to derive clear, testable redictions. Some examles of the theoretical insights are as follows. First, while bargaining, all sulier tyes benefit from their cost information being rivate. This means that when a sulier s cost information is unknown to retailers, they can use this rivate information to their advantage while bargaining and always earn weakly higher (sometimes strictly) exected rofits comared to the full information case. Second, endogenous matching under full information should lead to lower cost suliers matching with higher riced retailers and vice versa, contributing to an increase in the variance of rofits across arties. And third, in an environment where a sulier s cost information remains rivate while bargaining if they did not disclose it while matching, then either all or none of the suliers should disclose their costs, deending on the secific cost and rice arameters. We formulate a number of hyotheses based on our theoretical redictions and test them through a controlled human-subjects exeriment. We accomlish this through a design which maniulates two dimensions. For the first dimension we vary the matching rotocol with two levels: exogenously matched airs versus endogenously matched airs (where retailers and suliers can choose with whom to bargain with). For the second dimension we vary the availability of the sulier s cost information across three levels. In the first, all of the suliers costs are full information and known by retailers. In the second, suliers costs are rivate and they have the otion of disclosing their costs to the other layers. Regardless of whether they disclose their costs, once they are matched with a retailer their cost becomes known by the retailer they are matched with. In the third variant, suliers costs are rivate and they also have the otion of disclosing

4 4 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains their costs to the other layers. However, if they do not choose to disclose their costs, then this information remains rivate while bargaining with their matched retailer. Our exeriments yield a number of insights. First, we reject nearly all of the theoretical redictions for the contract arameters. For examle, agreed wholesale rices are less than the equilibrium redictions for every airing of retailer rice and sulier cost, and regardless of whether the sulier costs are rivate or full information. We also observe that agreed quantities systematically differ from the theoretical redictions. Second, contrary to the normative theory, suly chain efficiency is significantly below 100% under full information and, because wholesale rices are below the equilibrium redictions, suliers earn considerably less than their retailer counterarts. Third, consistent with theory, endogenous matching between retailers and suliers leads to a higher variance of rofits, comared to the more common exogenous matching rotocol. Fourth, under rivate information while bargaining, retailers make offers as if they assume the sulier has the lowest cost. As a consequence, high cost suliers earn only a small fraction of the overall suly chain surlus under rivate information, between 13.99% and 21.95%, when the normative theory redicts that they should earn between 57.41% and 65.87%. Fifth, because high cost suliers are unable to take advantage of their rivate cost information while bargaining, they actually benefit from disclosing their rivate costs to retailers. Lastly, we find that both arties are suscetible to suerficial fairness and anchoring when bargaining, even in our rivate information setting, which accounts for many of our results, such as suliers earning less than their redicted shares of the suly chain surlus. 2. Related Literature The literature most related to our work includes suly chain research that considers wholesale rice contracts, dynamic bargaining rocesses, rivate information, and/or endogenous matching. In regards to contracting, there have been a number of theoretical and exerimental studies. From a theoretical standoint, Lariviere and Porteus (1) consider wholesale rice contracts in a two-stage suly chain and investigate how asects like demand variability affect rices and the distribution of rofits. Bernstein et al. (6) identify how wholesale rice contracts can coordinate a suly chain with a single sulier and multile retailers. Taylor and Plambeck (7) study how rice-only (and rice-and-quantity) contracts can induce investment in a roduct develoment setting between a sulier and buyer. For a summary of the theoretical features of suly chain contracts lease see Cachon (3). Exerimentally, some aers which investigate suly chain contracting include Ho and Zhang (8), who study how framing a fixed fee can affect overall suly chain efficiency and Kalkanci

5 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 5 et al. (2011), who demonstrate how simle rice-only contracts can erform well in a setting where the retailer has accurate information regarding demand. Davis et al. (2014) investigate wholesale rice contracts in three alternative inventory risk arrangements, while Zhang et al. (2015) comare buy-back and revenue-sharing contracts under alternative overage and underage costs with loss-averse suliers. For a comrehensive summary of the exerimental suly chain contracting literature we refer the interested reader to Chen and Wu (2018). A vast majority of the works mentioned above assume that one arty in the suly chain makes a one-shot ultimatum offer to the other arty. Some studies have extended this setting by allowing for a more natural bargaining rocess. Theoretically, one key framework for solving these roblems under full information is the Nash bargaining solution (Nash 1950). Exerimentally, the only suly chain aers that we are aware of which deviate from one-shot offers are Haruvy et al. (2014), Leider and Lovejoy (2016), Davis and Leider (2017) and Davis and Hyndman (2017). Haruvy et al. (2014) extend the one-shot scenario by ermitting only one arty to make multile offers. Leider and Lovejoy (2016) consider back-and-forth bargaining in a three-stage suly chain with chat box communication. Davis and Leider (2017) and Davis and Hyndman (2017) are similar to our work in that they allow for back-and-forth offers with the ability for layers to send limited feedback over a limited time frame. The suly chain literature frequently assumes full information of rice, demand, and cost arameters (e.g., Cachon (4)). There have been certainly some deviations from this, esecially in a shot-shot offer setting. For examle, some studies have examined a setting where the retailer may have rivate knowledge about consumer demand. Thus the issue of sharing forecast information is relevant (e.g., Cachon and Lariviere (1) from a theoretical ersective and Ozer et al. (2011) from an exerimental ersective). There are also studies which investigate how a sulier can obtain information about a buyer s cost structure (Corbett et al. 4). More relevant to our work are those aers in suly chain management which consider rivate information combined with a more natural bargaining interaction between two arties. Theoretically, one aer which satisfies this is Feng et al. (2015), who investigate multile alternating offers where both arties are imatient and the buyer has rivate information about their tye. There is also work in economics ertaining to bargaining with rivate information (for a summary lease see Muthoo (1999)). From an exerimental ersective, Raoort et al. (1995) investigate a setting where there is rivate information on one side with regards to a buyer s reservation rice, and a seller can make multile offers of rice, where each offer is discounted in a way that makes delay costly.

6 6 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Lastly, many exerimental suly chain studies exogenously match suliers and retailers into airs. Excetions to this include Beer et al. (2017), Fan et al. (2018), Hyndman and Honhon (2017) and Honhon and Hyndman (2017), though the latter two do not allow eole to choose new artners, only to terminate existing artnershis. Fan et al. (2018) look at entry into a costly-tojoin grou and as such layers cannot veto others from joining the grou. Only Beer et al. (2017) involves mutual consent to enter into a contractual relationshi. Overall, we believe our work extends the existing literature in three imortant dimensions: (1) we consider a more natural back-and-forth bargaining rocess between retailers and suliers which allows for multile offers and limited feedback, (2) we evaluate rivate versus full information with resect to the sulier s cost, and (3) we comare endogenous matching with exogenous matching between grous of retailers and suliers. In addition, we investigate these issues from both a theoretical and exerimental ersective. 3. Theoretical Background In this section we rovide a theoretical analysis for the bargaining and matching institutions that we will test in the lab. The basic framework consists of a set of retailers indexed by their retail rice, 1 > 2 >... > n and a set of suliers indexed by their er-unit cost of roduction, c 1 < c 2 <... < c n < n. We assume that the set of ossible costs for suliers and rices for retailers is common knowledge. We consider several settings deending on whether: sulier costs are rivate or full information, matching is exogenous (i.e., random) or endogenous, and, under endogenous matching with rivate costs, suliers can truthfully disclose their rivate cost information during the matching hase. For the full information case we rovide a more general analysis, while for rivate information we resent a general aroach to solving the roblem but generate redictions based on the arameters in our exeriments. 2 Since, under endogenous matching, it matters what subjects exect to haen in the bargaining stage, we first rovide results for bargaining and subsequently discuss endogenous matching Bargaining with Full Information We imlemented an unstructured bargaining rotocol in which a retailer with selling rice negotiates a wholesale rice, w, and order quantity, q, with a sulier with unit cost c <. We assume that the sulier bears the risk of unsold inventory. 3 We assume that both c and are 2 In articular, we considered markets of three retailers and three suliers. The ossible selling rices were 10, 11 or 12 and exactly one retailer had each selling rice. Similarly, the ossible er-unit costs for suliers were 3, 4 or 5 and exactly one sulier had each unit cost. 3 We oted for the sulier to incur the inventory risk because, unlike when the retailer incurs the risk, both arties face random demand. For instance, when the retailer incurs the inventory risk, the sulier can roduce exactly what the retailer orders, and avoid both inventory and demand risk.

7 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 7 common knowledge. In all cases, the underlying demand, D, is drawn uniformly from [a, b] where 0 a < b <, but the actual realization of demand is unknown at the time of bargaining. Because of our unstructured bargaining rotocol, the relevant theoretical lens for the full information case (i.e., the sulier s cost is known) is the Nash bargaining solution (see, Camerer (3, Ch. 4.1) as well as Davis and Hyndman (2017)). Denote by π i (w, q) the exected rofits for firm i {r(etailer), s(ulier)}, from an agreement with wholesale rice, w, and order quantity, q. The exected rofits can be exressed as: π r (w, q) = w b a b a min{q, x}dx; π s (w, q) = w b a b a min{q, x}dx cq. (1) The disagreement ayoff is 0 for both layers. The Nash bargaining solution is the solution to: max w,q π r (w, q) π s (w, q) s.t. c w and a q b. Since the full information bargaining environment is identical to Davis and Hyndman (2017), we state without roof the following result: Proosition 1. The Nash bargaining solution has the following roerties: (i) q = a + (b a) ( c) /, which means that the suly chain is coordinated; (ii) exected ayoffs for the retailer and sulier are equalized; and (iii) w = (3ac2 +a 2 3bc 2 +2bc+b 2 ) > 2(ac 2 +a 2 bc 2 +b 2 ) +c. 2 Because it will lay a role in our emirical analysis, we emhasize that the agreed wholesale rice, w, is strictly greater than the mid-oint between c and (i.e., (+c) /2). This follows because the sulier bears the inventory risk. Therefore, to equalize the exected ayoffs of the retailer and sulier, the wholesale rice must increase beyond the midoint between the retailer s rice and the sulier s cost Bargaining with Private Information When sulier costs are rivate information, we need a suitable generalization of the Nash bargaining solution. Using insights from mechanism design, Myerson (1984) rovides such a generalization. The general aroach is to find contract terms that maximize a weighted sum of the retailer s and sulier s exected rofits subject to incentive comatibility constraints (i.e., that the sulier wants to reveal her cost tye) and so-called warrant conditions, which are the minimum amounts that each layer tye warrants in a fair division. In addition to the warrant conditions, the other extra comlication is that the weights must also be derived as art of the solution. Secifically, let π s (w, q, c i ) denote the exected rofits of sulier tye c i when faced with the contract (w, q).

8 8 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Similarly, let π r (w, q, ) denote the exected rofits of a retailer with selling rice facing the contract (w, q). Then we must solve: 3 max λ i π s (w i, q i, c i ) π r (w i, q i, ) λ i,q i,w i 0 3 i=1 i=1 s.t. π s (w i, q i, c i ) π s (w i+1, q i+1, c i ), i = 1, 2. The inequalities state that sulier tye c i refers to reort her cost tye, rather than the cost tye c i+1, which would lead to contract (w i+1, q i+1 ). Without loss of generality, we can take λ 3 = 1 λ 1 λ 2. Letting α i denote the Lagrange multilier on sulier tye i s incentive comatibility constraint and substituting in the exressions for exected rofits, the Lagrangean can be written as: L = 1 ( 3(λ 1 + α 1 ) ( 3(λ 2 + α 2 ) ( w1 ( w2 ( 3(1 λ 1 λ 2 ) ( ) q1 q 2 1) c1 q 1 + w 1 ( ) ) q1 q 2 1 ( ) ( q2 q 2 w2 ( ) 2) c2 q 2 3α 1 q2 q 2 2) c1 q 2 + w 2 ( ) ) q2 q 2 2 ( w3 ( ) ( q3 q 2 w3 ( ) 3) c3 q 3 3α 2 q3 q 2 3) c2 q 3 + w 3 ( ) ) q3 q 2 3 Observe that the wholesale rice is effectively a linear transfer between the retailer and the sulier. Therefore, we can imose the constraints that λ 1 +α 1 = 1 /3 and λ 2 +α 2 α 1 = 1 /3. This temorarily obviates the need to solve for w i in the otimization roblem and allows us to focus on q i. We will return to solve for the wholesale rices in the last ste of the roblem. Notice also that each line of the above equation reresents the total virtual surlus if sulier tye c i interacts with the retailer. We rovide more details in Aendix A; here, we outline the stes to obtain the solution. In the first ste, we find the otimal quantities for each sulier cost tye as a function of the λ s and α s. In the second ste, using the exressions for q i and also the relationshi between λ and α, we can write the Lagrangean as a function of c i, and the λ s. Doing so yields: L = 50 ( ( c1 ) 2 + ( + c 1 3λ 1 c 1 + c 2 (3λ 1 2)) 2 + ( + c ) 2(2 3λ 1 3λ 2 ) 3c 3 (1 λ 1 λ 2 )) 2 3 where again, each of the three terms reresents the total surlus assuming that given sulier cost tye is drawn. The next ste is to solve the warrant conditions, which rovides a virtual utility that each sulier tye warrants akin to the standard Nash bargaining solution, half the total virtual surlus generated by the interaction between the retailer and the articular sulier cost tye. In articular, according to Myerson (1984), we must solve: (λ 1 + α 1 )W 1 = 1 ( ) 50 ( c1 ) 2 2 3

9 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 9 for W i, i = 1, 2, 3. (λ 2 + α 2 )W 2 α 1 W 1 = 1 ( ) 50 ( + c1 3λ 1 c 1 + c 2 (3λ 1 2)) (1 λ 1 λ 2 )W 3 α 2 W 2 = 1 ( ) 50 ( + c2 (2 3λ 1 3λ 2 ) 3c 3 (1 λ 1 λ 2 )) The final ste is to find values of λ i and w i such that: π s (w i, q i, c i ) W i, with equality if λ i > 0, π s (w i, q i, c i ) π s (w i+1, q i+1, c i ), for i = 1, 2 and with equality if α i > 0. With three sulier cost tyes and two incentive comatibility constraints this is, otentially, a system of five equations in five unknown variables: (w 1, w 2, w 3, λ 1, λ 2 ). Moreover, one must check boundary conditions on the λ s and α s, making it necessary to consider seven different systems of equations to find the valid solution. Given the arameters of our exeriment, it turns out that the bargaining solution involves λ 1 = λ 2 = 0, and the incentive constraints on the c 1 and c 2 sulier tyes are binding (the equilibrium contract arameters are rovided in Table 2b and will be discussed when we review our exerimental design). We can summarize the result as: Proosition 2. Based on the arameters of the exeriment, (i) all sulier cost tyes benefit from their cost being rivate information; (ii) The suly chain is coordinated only for the lowest cost sulier (c 1 ), with inefficiency for the other sulier tyes (c 2 and c 3 ); and (iii) the wholesale rice increases in sulier cost but less steely than under full information Endogenous Matching with Full Information We now generate redictions about how a grou of n retailers and n suliers would form matches if given the chance to do so, in contrast to the case in which retailers and suliers are exogenously matched into airs. We first consider the case in which suliers costs are full information at the time of matching. Therefore, in the bargaining hase, each retailer-sulier air will maximize the suly chain surlus, with each arty receiving an equal share. Given our assumtions on demand, and further secializing to the case of a = 0 and b = 100, the first-best suly chain surlus is: π(, c) = 50 ( c)2. Notice that this exression is increasing in and decreasing in c. Therefore, π( 1, c 1 ) yields the highest suly chain surlus of all the ossible matchings. Moreover, π( i, c i ) > π( i, c j ) for any j > i. Similarly, π( i, c i ) > π( k, c i ) for any k > i.

10 10 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains We define a matching, M, as a relation from { 1,..., n } to {c 1,..., c n } and we say that a retailer i and sulier c j are matched if ( i, c j ) M. Let π r ( i, c j ) denote retailer i s rofit when matched with sulier c j. Similarly, let π s ( i, c j ) denote sulier c j s rofit when matched with retailer i, assuming that they will bargain as in 3.1. A matching, M, is stable if there does not exist a retailer k and a sulier c j such that ( k, c j) M and ( k, c j ) M for which u r ( k, c j ) > u r ( k, c j) and u s ( k, c j ) > u s ( k, c j ). That is, retailer k and sulier c j both refer to be matched with each other rather than remain matched with their resective artners under the matching M. Proosition 3. Under full information about sulier costs and retailer rices, the unique stable matching must be assortative. That is M = {( i, c i ) i = 1, 2,..., n}. Proof. Let M denote a stable matching. We first show that ( 1, c 1 ) M. Suose that this is not the case. Then there exists j, k > 1 such that ( 1, c j ) M and ( k, c 1 ) M. Then retailer 1 can exect rofits of 0.5π( 1, c j ) and sulier c 1 can exect rofits of 0.5π( k, c 1 ). However, if they broke their matches to match together, then they would each earn 0.5π( 1, c 1 ), which is strictly larger. This contradicts that M was stable. Having shown that ( 1, c 1 ) must be art of any stable matching, we can roceed iteratively and show that ( 2, c 2 ), ( 3, c 3 ),... must be art of any stable matching. Q.E.D Endogenous Matching with Private Information The next case we consider is one in which, during matching, sulier costs are rivate information but suliers have the ability to truthfully disclose their costs. Moreover, in this case we assume that regardless of whether or not a sulier discloses during matching, in the bargaining stage, the sulier s cost becomes full information for the retailer. Under these assumtions, we can show the following: Proosition 4. When suliers have the otion to disclose their costs, all suliers, excet ossibly the highest cost sulier, disclose and the unique stable matching is assortative. That is, M = {( i, c i ) i = 1, 2,..., n}. Proof. First, observe that sulier c 1 will certainly disclose as this ensures that he will match with retailer 1 and earn 0.5π( 1, c 1 ). If he did not disclose then there is a strictly ositive robability that he will not be matched with retailer 1, which means his exected earnings would be strictly less than 0.5π( 1, c 1 ). Thus, sulier c 1 discloses. Similar arguments show that sulier c 2 will disclose and be matched with retailer 2, and so on. The last sulier c n need not disclose but since

11 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 11 all other suliers strictly refer to disclose, his cost will be correctly inferred to be the highest, which will lead to a matching of ( n, c n ). Thus, all suliers excet ossibly c n disclose and the matching is assortative, denoted by M. Q.E.D. Finally, we consider the case in which, during matching, suliers costs are rivate information. We still ermit suliers to truthfully disclose their cost to the other arties; however, if they do not disclose, then their cost will remain rivate information during the bargaining hase. In this case, there is a tension. As Proosition 4 shows, disclosure facilitates assortative matching, which is beneficial for the lowest cost sulier. However, Proosition 2 showed that, for the arameters of our exeriment, all sulier tyes benefit from rivate information. Thus whether disclosure haens deends on whether the benefits from disclosing to imrove one s match outweigh the resulting cost (in terms of bargaining surlus) from disclosing. Restricting attention to the arameters used in our exeriment (c {3, 4, 5} and {10, 11, 12}), all sulier tyes are strictly better off not disclosing their rivate cost (which effectively leads to random matching and bargaining with rivate information) rather than disclosing costs and being assortatively matched. For examle, using the numbers in Table 2b (which illustrate the redictions for our exeriment), if the sulier with lowest cost c = 3 discloses, she will be matched with the highest riced retailer = 12 and can exect to earn On the other hand, if she does not disclose her cost, then she has an equal chance to be matched with any retailer, leading to exected earnings of ( 1 /3)( ) = > Therefore, the c = 3 sulier would not choose to disclose. We can go further and show that there is no disclosure in equilibrium. First, observe that even if the sulier with highest cost c = 5 disclosed and the other two tyes did not, no retailer would choose to match with this sulier, referring to take their chances with one of the other lower cost suliers. Therefore, this sulier cannot disclose her information in the hoe of being matched with a better retailer. Thus there can be no equilibrium in which only the c = 5 tye discloses. Last consider the case in which the c = 4 sulier disclosed. In the absence of any further disclosure, the = 12 retailer would strictly refer to match with the c = 4 sulier who disclosed. However, this cannot be stable because the c = 3 sulier would be strictly better off by immediately disclosing in order to initiate a match with the = 12 sulier, which the = 12 retailer would refer. Therefore, the c = 4 tye will not unilaterally disclose because it would set off a chain of disclosure which would end with assortative matching. For the c = 4 sulier, this assortative outcome is worse than not disclosing, being matched with a random retailer, and bargaining with rivate cost information. Thus, given the arameters of our exeriment, we have:

12 12 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Proosition 5. If sulier cost information remains rivate during bargaining unless reviously disclosed, no sulier tyes disclose their cost, leading to random matching between suliers and retailers. Remark 1. To be sure, this result deends on the arameters of the exeriment. We can construct examles in which the lowest cost sulier refers to disclose her rivate information. This occurs when, in the solution to the bargaining roblem, λ 1 (the weight given to the lowest cost sulier in the otimization roblem) is strictly greater than zero. In this case, the sulier does not earn any information rents from rivate information. Therefore, she would choose to disclose in order to imrove the quality of her match. One such set of arameters is {10, 11, 12} and c {3, 6.5, 8.5}. However, this generates imlausibly large variation in sulier costs, which is why we did not imlement such a setting. 4. Exerimental Design In our exeriment, articiants were assigned a role of sulier or retailer and laced into a matching grou of six, three retailers and three suliers. Both roles and matching grous remained fixed for the duration of the exeriment. In every round, each retailer was randomly assigned a selling rice er unit, {10, 11, 12} where each retailer had a unique rice (i.e., one retailer had = 10, another = 11, and the third = 12). Similarly, in every round each sulier was randomly assigned a roduction cost er unit, c {3, 4, 5}, where each sulier had a unique cost. The ossible rices and costs were common knowledge to both arties. Each round consisted of u to two stages, a matching stage and a bargaining stage. In the matching stage articiants would form retailer-sulier airs and then, in the bargaining stage, they would bargain over contract terms for a roduct with uncertain demand (more details of each stage to follow). Regardless of the retailer s selling rice and sulier s roduction cost, demand for the roduct was always a random draw from the discrete uniform distribution on {1, 2,..., 100}. If the two arties came to an agreement while bargaining, demand would be realized, and retailers would satisfy demand by sourcing roduct directly from the sulier, such that the sulier incurred the cost of any unsold inventory. Our overall exerimental design maniulated two dimensions: the matching rotocol in which retailer-sulier airs were formed and the availability of the suliers cost information. For the matching rotocol, we imlemented two levels. The first, Exo(genous), randomly matched retailers and suliers into airs such that each round began with the bargaining stage. The second, End(ogenous), began with a matching stage where retailers and suliers could signal as to with

13 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 13 Table 1 Exerimental Design and Number of Particiating Subjects Matching Protocol Match-Barg Exo(genous) End(ogenous) F(ull)-F(ull) Cost Information D(isc)-F(ull) - 54 D(isc)-P(rivate) Note: Match reresents cost information during the matching stage (for the endogenous treatments), Barg reresents cost information during the bargaining stage, and Disc reresents a scenario when the sulier had the ability to disclose their rivate cost while matching. Note that Private cost information, during the bargaining stage under endogenous matching (End-D-P), reresents that if a sulier did not disclose their cost while matching, then this cost information would remain rivate while bargaining. whom they wanted to be matched. In these latter treatments, if a retailer and sulier both agreed to be matched together, then they would be aired for the subsequent bargaining stage. Any subjects who remained unmatched at the end of the matching stage would be randomly matched from within the grou of unmatched subjects into airs. Regarding the suliers cost information, we evaluated two levels for the exogenous treatments and three levels for the endogenous treatments. The two levels for the exogenous treatments varied as to whether the suliers cost information was available, Exo-F(ull), or not, Exo-P(rivate), to retailers while bargaining (since there was no formal matching stage in these treatments). Turning to the endogenous treatments, in the first of the three levels, End-F(ull)-F(ull), each sulier s cost was known to the other suliers and retailers in the matching stage and the bargaining stage. In the next two levels, each sulier s cost was rivate information in the matching stage. Secifically, in the second level, End-D(isc)-F(ull), each sulier knew her own rivate cost information while matching, but had the otion of disclosing this information to the other layers. If a sulier chose to disclose her cost, the other suliers and retailers would see this information. Then, while bargaining, regardless of the sulier s disclosure decision, the sulier s cost information would be known by the retailer. Lastly, in the third variant, End-D(isc)-P(rivate), once again, each sulier knew her own rivate cost information while matching and had the otion of disclosing this cost information to the other layers. However, if they did not choose to disclose this information while matching, then in the bargaining stage it remained as rivate information and was thus unknown to the retailer. Overall, the exeriment consisted of a (2 3) 1 between-subjects design with 258 total articiants, outlined in Table 1. For details ertaining to the matching stage in the endogenous treatments, we gave articiants two and a half minutes to form airs. During this time, any retailer or sulier could roose to be matched with a layer of the oosite role, and they could roose to more than one layer at a time. We also allowed layers to rescind a roosal that had not been acceted. If a layer received a formal roosal and chose to accet it, then those two layers formed a matched air,

14 14 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains were removed from the screen, and unavailable for matching with the other articiants. Also, in the two treatments where the sulier s cost information was rivate, we allowed each sulier to disclose it to the other layers by clicking a button. We rovide a screenshot of the matching stage with this disclosure otion in Aendix B. Lastly, as mentioned reviously, in the exogenous treatments retailer-sulier airings were randomly assigned at the beginning of each round such that they skied the matching stage and began directly in the bargaining stage. After any otential matching, articiants roceeded to the bargaining stage. In this stage, each retailer-sulier air was given five minutes to negotiate a contract which consisted of two terms, a wholesale rice, w, and a quantity, q. During this time, retailers and suliers were ermitted to make offers at any oint and to make as many offers as they desired. If a air was unable to reach an agreement after five minutes then both layers would receive a ayoff of zero. If the air reached an agreement (which occurred when one layer acceted the other layer s most recent offer) then demand would be realized and layers would receive feedback that included realized rofits. While bargaining, we allowed articiants to rovide feedback about the most recent offer received. In articular, they could reject any of the roosed terms through a button for each contract term, which they could click at any time for the most recent offer received. This feedback would then be dislayed on the rooser s screen. Note that a articiant could later accet the offer even if they signaled disaroval with it, so long as a more recent offer was not received. We oted for this tye of feedback to simulate a more natural bargaining rocess, while also allowing us to monitor offers and feedback. Lastly, offers were not required to imrove uon a revious offer and only the most recent offer could be acceted. We chose the former because we did not want to inform subjects what constitutes a better offer, while the latter kees bargaining straightforward. To reduce comlexity throughout the exeriment, in both the matching stage (if alicable) and the bargaining stage, we rovided articiants with a decision suort tool. In this tool they could enter hyothetical values for w and q, which would generate a grah showing the rofit for both layers as a function of demand. This is also seen in the screenshot available in Aendix B. Overall, subjects articiated in six rounds. For each of the five treatments we ran three sessions, each of which had 12 or 18 subjects. 4 The exerimental software was rogrammed in z-tree (Fischbacher 7), and all sessions took lace in the exerimental laboratory at a large northeast university. Sessions took between 60 and 75 minutes, with average earnings of $37, a maximum of $90, and a minimum of $7. Subjects were comensated for all rounds of decisions. Samle instructions and additional screenshots are available uon request. 4 For the End-D-F treatment, we also had a fourth session of six subjects to balance all endogenous treatments at 54 articiants.

15 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 15 Table 2 Exerimental Predictions (a) Full Information (Exo-F, End-F-F, End-D-F) Retailer/Sulier Profit Wholesale Price (w) Quantity (q) = 10 = 11 = 12 = 10 = 11 = 12 = 10 = 11 = 12 c = c = c = (b) Private Information (Exo-P, End-D-P) Sulier Profit Wholesale Price (w) Quantity (q) = 10 = 11 = 12 = 10 = 11 = 12 = 10 = 11 = 12 c = c = c = Note: The ex ante exected rofits of the retailer under rivate information (i.e., before learning, via the bargaining mechanism, which cost sulier the retailer is bargaining with) coincide with the highest cost tye sulier Predictions and Hyotheses In Tables 2a and 2b we rovide oint redictions based on our revious theoretical analysis and exerimental arameters. Table 2a deicts the oint redictions when the suliers s costs are full information while bargaining (Exo-F, End-F-F, End-D-F), whereas Table 2b illustrates the oint redictions when the suliers costs should be rivate information while bargaining (Exo-P, End-D-P). A combination of these metrics and our roositions in 3 lead us to the following exerimental hyotheses: Hyothesis 1. Under full information, 100% efficiency is achieved for every (, c) combination, and retailers and suliers earn 50%/50% slits of total suly chain exected rofit. Hyothesis 2. Under rivate information: (i) 100% efficiency is only achieved for the air with the lowest cost sulier. (ii) Suliers earn at least 50% of the total suly chain exected rofit. (iii) The agreed wholesale rice is always higher comared to when there is full information. Hyothesis 3. Under endogenous matching and suliers have the otion to disclose their costs, but their costs will be known during bargaining (End-D-F), the two lowest cost suliers should disclose their costs. Hyothesis 4. When there is endogenous matching and suliers costs may remain rivate while bargaining (End-D-P), no sulier tye should disclose her cost.

16 16 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 5. Results We resent our exerimental results in three subsections. In 5.1 we investigate how our data comares to the exerimental redictions and hyotheses from 4.1. In 5.2, because the bargaining solutions do not rovide details ertaining to the rocess itself, we analyze the bargaining dynamics in our data. In 5.3 we investigate detailed behavior during the matching stage. For all hyothesis tests we use a matching grou of six as an indeendent observation, and run all regressions with random effects and clustered standard errors at the matching grou level. Before roceeding to our detailed results, we rovide a snashot of some key insights. First, we reject nearly all of the normative oint redictions for the contract arameters in Tables 2a and 2b. For examle, wholesale rices are consistently below the equilibrium values, in both full and rivate information, and quantities tend to gravitate towards mean demand. Second, suly chain efficiency is roughly 90% in all treatments, whereas it should always be 100% under full information and 100% for (c = 3) under rivate information. Third, suliers earn significantly less than the redicted 50% (or more) of the exected suly chain rofit, in both full and rivate information. This is esecially true for high cost suliers under rivate information, where they only earn between 13.99% and 21.95% of total rofits. Fourth, under endogenous matching, matching is artially assortative, with higher (res. lower) riced retailers frequently matching with lower (res. higher) cost suliers. This increases the variance in earnings, with high quality retailers and suliers earning more and low quality retailers and suliers earning less under endogenous matching than under exogenous matching. Fifth, under rivate information, high-cost suliers earnings are much lower than redicted by theory so much so that that it is actually advantageous for them to disclose their cost during matching to be able to credibly demand a higher wholesale rice. Sixth, when sulier costs may remain rivate while bargaining, a substantial ortion of suliers choose to disclose their costs while matching. As just stated, this is actually emirically otimal for higher-cost suliers but is (weakly) disadvantageous for the lowest-cost sulier. Seventh, regardless of whether sulier costs are full or rivate information, subjects are suscetible to suerficial fairness and anchoring biases during bargaining. This is consistent with ast studies, which documented such biases under full information Comarison to Predictions and Hyotheses The average agreement rates while bargaining in the five exerimental treatments were similar to one another: 93.06% in Exo-P, 90.28% in Exo-F, 88.89% in End-F-F, 90.12% in End-D-F, and 90.12% in End-D-P. As a consequence, we will reort all summary statistics conditional on agreement in this section.

17 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Contract Parameters We first comare the agreed uon contract arameters to the redictions in Tables 2a and 2b. To this end, Tables 3a and 3b deict the average wholesale rices and quantities in the full information treatments (Exo-F, End-F-F, and End-D-F) and rivate information treatments (Exo-P and End-D-P), for each combination of c and. One can recognize that there are deviations between the actual contract terms and the normative redictions. Regarding wholesale rices, for every combination of c and, the wholesale rice is significantly different and lower than the normative redictions under both full and rivate information (t tests, all < 0.01). Moreover, contrary to Hyothesis 2(iii), which states that the wholesale rice is always higher under rivate information versus full information, there is no difference in wholesale rices between full and rivate information. In fact, for only one of the nine combinations of (, c), ( = 10, c = 3), is the difference statistically significant in the direction redicted by theory ( < 0.05). Theory also redicts that the agreed wholesale rice should increase in the suliers costs. The data are suortive of this rediction, but the magnitude is less than theory. For examle, if = 10, and c increases from c = 3 to c = 5, then redicted wholesale rices should increase by 0.70 (w = 7.89 to w = 8.59). Yet for our data in this scenario, the change in wholesale rices is only 0.46 (w = 7.17 to w = 7.63). The fact that wholesale rices are not higher under rivate information, suggests that suliers earn less than theory, while this result imlies that the difference is articularly large for the highest-cost suliers. Continuing with the agreed uon quantities in Tables 3a and 3b, one can also discern a number of deviations with resect to the redictions in Tables 2a and 2b. In general, comared to the normative benchmarks, the observed quantities are often closer to the mean demand of 50. We will exlore anchoring in both wholesale rices and order quantities in more detail later. Overall, these results lead to the following: Result 1 Under both full and rivate information wholesale rices are significantly lower than the normative redictions. Also contrary to theory, there is no difference in average wholesale rices between full and rivate information. Under rivate information, increases in wholesale rices are flatter than redicted with resect to increases in sulier costs. Regarding quantities, under both full and rivate information, there is bias towards the mean demand of Efficiency and Distribution of Profits We now focus on Hyotheses 1 2, which ertain to suly chain efficiency and distribution of exected rofits under full and rivate information. In both the full and rivate information conditions, observed channel efficiency is nearly

18 18 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Table 3 Wholesale Prices and Quantities, Conditional on Agreement (a) Full Info Wholesale Price (w) Quantity (q) = 10 = 11 = 12 = 10 = 11 = 12 c = (.47) (.50) (.68) (15.64) (13.13) (12.88) c = (.50) (.69) (.68) (14.08) (14.51) (10.27) c = (.52) (.45) (.60) (14.64) (11.52) (15.83) (b) Private Info Wholesale Price (w) Quantity (q) = 10 = 11 = 12 = 10 = 11 = (.74) (.62) (.65) (10.79) (11.78) (9.60) (.87) (.61) (.98) (17.41) (16.20) (13.44) (.71) (.60) (.56) (10.68) (12.85) (15.40) Note: Standard deviations (based on matching grou averages) are reorted in arentheses below each mean. Significance of t-tests versus the normative theory given by < 0.01, < 0.05, and < %, with a minimum of 88.97% and a maximum of 93.09% across all eighteen (information, c, ) scenarios. In all instances where efficiency is redicted to be 100% (all full information scenarios and c = 3 under rivate information), we strongly reject the theoretical rediction (t-tests, all < 0.01). For the rivate information case, when c = 4, the observed channel efficiencies are also significantly less than the normative rediction ( < 0.01 for = 10; < 0.05 for = 11; and < 0.10 for = 12). As we have mentioned, and will discuss in greater detail below, these results aear to be driven by an anchoring bias in order quantities. For the distribution of rofits, the normative theory redicts a slit between the two arties in the full information treatments. In Figure 1a, which deicts the sulier s share of total suly chain exected rofit, we observe that suliers earn significantly less than 50%, with a minimum share of 23.06% and a maximum share of 45.43% (eight t tests, < 0.01, and one t test, < 0.05). Combined with the above-mentioned results on efficiency we can reject Hyothesis 1. Turning to rivate information, deicted in Figure 1b, the normative rediction is that suliers should earn more than 50% of the total suly chain exected rofit. Yet the differences between the actual sulier share of rofits versus the redictions are even more stark than the full information scenario. Thus we reject Hyothesis 2(i) (ii). One can also discern, in both Figures 1a and 1b, that as c increases, suliers earn a disroortionately low slit of the suly chain exected rofits (e.g., sulier share is between 13.99% and 29.78% when c = 5). In fact, when sulier s costs are rivate information, high cost suliers only earn between 13.99% and 21.95% of the total suly chain exected rofit. This extremely low share is driven by two reviously noted results. First, under rivate information, suliers do not receive the redicted higher wholesale rice than under full information. Second, the increase in agreed wholesale rice as sulier cost increases is flatter than redicted. We will further exlore the driver of these wholesale rices when we investigate the bargaining data.

19 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 19 Figure 1 Sulier Share (%) of Suly Chain Exected Profit, Conditional on Agreement (a) Full Info (b) Private Info Sulier Share (%) Normative Predictions =10 =11 =12 Sulier Share (%) Normative Predictions =10 =11 = c=3 c=4 c=5 c=3 c=4 c=5 We can summarize these results as follows: Result 2 We reject Hyotheses 1 and 2. Under both the full and rivate information channel efficiency is generally less than redicted. Also, because of unfavorable wholesale rices suliers earn less esecially higher cost suliers than their redicted slit of suly chain rofits, whereas retailers earn more. In addition, high cost suliers under rivate information earn the lowest slit of suly chain rofits Disclosure and Endogenous Matching We now turn our attention to Hyotheses 3 4, which concern cost information disclosure by suliers and endogenous matching. Figure 2(a) rovides the frequency of disclosure by each cost tye, and shows that decisions do not erfectly conform to the redictions in the End-D-F or End-D-P treatments. In articular, in End-D-F a sulier with c = 3 discloses 46.30% of the time and c = 4 discloses 31.48% of the time, which is non-negligible but less than the rediction of 100%. On the other hand, in End-D-P, we see that all suliers disclose more than theoretical rediction of 0%. This latter result could be because suliers do not anticiate that the bargaining advantage of rivate information should out-weigh the matching benefits of disclosure (at least for low-cost suliers). Indeed, our earlier results for suliers share of rofits under rivate information suggest that this may not be an irrational strategy, articularly for higher cost suliers, as they seem to earn lower rofits when their costs are rivate. Lastly, in Figure 2(b) we observe that the lowest cost suliers are the first to disclose: roughly 80% of the time in both the End-D-F and End-D-P settings. Thus we have the following result:

20 20 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Result 3 We reject Hyotheses 3 and 4. Suliers disclose between 20.37% % in the End- D-F and End-D-P treatments, where theory redicts the two lower cost suliers should disclose 100% in End-D-F, and no sulier should ever disclose in End-D-P (i.e., 0%). Low cost suliers disclose most frequently and are most likely to be the first to disclose. Figure 2 Frequency of Disclosure Overall and Frequency of Disclosing First by Cost Tye (a) Frequency Disclosure (%) (b) Frequency Discloses First (%) Frequency Disclosure (%) End-D-F End-D-P Frequency Discloses First (%) End-D-F End-D-P 8.00 c=3 c=4 c=5 c=3 c=4 c=5 Note: We count a subject as having disclosed if the other two suliers disclosed their cost, thereby automatically revealing the remaining layer s cost. As redicted by the theory, with one excetion, this affects only the highest cost sulier. Next we consider the realized imact of disclosure on sulier exected rofits. Our theoretical redictions were derived assuming that matching and bargaining would conform to the equilibrium. However, we have already seen clear deviations from the normative theory such that suliers disclosure decisions might be emirically rational. Table 4 examines the imact of disclosure for different sulier cost tyes, where sulier exected rofit is the deendent variable in a random effects regression. As can be seen, for the c = 3 sulier, disclosure is weakly rofitable under full information (i.e., End-D-F) while it is quite harmful for the c = 4 and c = 5 suliers. In contrast, under rivate information, disclosure harms the c = 3 sulier (but not significantly so) and is beneficial for both the c = 4 and c = 5 suliers. Although this contradicts the theory for our arameterization, it is consistent with our exerimental result that high cost suliers earn a disroortionately low slit of total suly chain rofits under rivate information. That is, under rivate information, the high cost sulier may wish to disclose her cost to credibly convey her need for a relatively high wholesale rice.

21 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 21 Table 4 Effects on Sulier Exected Profits: Private Information, Disclosure, Endogenous Matching c = 3 c = 4 c = 5 Disclosed (7.522) (12.956) (39.930) Private Treatment (6.516) (13.204) (7.649) Disclosed Private Treatment (17.993) (27.930) (45.480) End-F-F Treatment (8.525) (8.322) (8.177) Constant (4.555) (7.109) (4.818) R N Note: Significance given by < 0.01, < 0.05, and < Table 5 Assortative Matching Details Matching of Subjects Treatment value End-F-F End-D-F End-D-P {(3, 12), (4, 11), (5, 10)} {(3, 12), (4, 10), (5, 11)} {(3, 11), (4, 12), (5, 10)} {(3, 11), (4, 10), (5, 12)} {(3, 10), (4, 12), (5, 11)} {(3, 10), (4, 11), (5, 12)} Note 1: The value is from a joint test that all coefficients in a regression of frequency on indicator variables for each matching (where we cluster at the matching grou level). Under random matching, all frequencies would be equal and equal to 1 /6. Note 2: The highlighted column indicates fully assortative matching. Result 4 Under full information, disclosure is beneficial for the lowest cost sulier and detrimental to the higher cost suliers. Under rivate information, the effects are reversed such that higher cost suliers benefit from disclosing their costs. We now turn our attention to endogenous matching. Table 5 shows the frequency that subjects were matched assortatively (first column) as well as the frequency of all other combinations (columns 2 6). The frequency of fully assortative matching is significantly higher than the ( 1 /6) random chance of occurring in the End-F-F treatment. Thus, endogenous matching, by itself, facilitates assortative matching, though the rates of assortative matching are well below 100%. For the two disclosure treatments, neither rates of assortative matching are significantly different from random chance, though for the End-D-P treatment, we strongly reject that matching is random. These insights lead to the following result: Result 5 Matching is artially assortative in the End-F-F treatment, but below 100%. In addition, the rates of fully assortative matching in both disclosure treatments (End-D-F and End-D-P) are not significantly different from random chance. Table 6 reorts a series of random effects regressions where the deendent variable is the retailer

22 22 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains rice that each sulier tye is matched with, and the exlanatory variables are indicators for whether each sulier tye disclosed. Disclosure by the c = 3 sulier significantly increases the quality of retailer that she is matched with (i.e., higher retailer ) and significantly decreases the quality of retailer that the other two sulier tyes are matched with. Disclosure by any other sulier tye does not significantly influence match quality. Table 6 Effects on Matching with Better Retailers: Sulier Disclosure c = 3 c = 4 c = 5 c = 3 Disclosed (0.150) (0.187) (0.122) c = 4 Disclosed (0.193) (0.287) (0.204) c = 5 Disclosed (0.277) (0.233) (0.190) Constant (0.101) (0.101) (0.131) R N Note: Deendent variable is the rice of the retailer that sulier gets matched with. Significance given by < 0.01, < 0.05, and < Finally, we consider more global consequences of endogenous matching and disclosure. Since endogenous matching facilitates best-with-best and worst-with-worst matching, and since there comlementarities from such matching, a final rediction of our theory on matching is that the variance of earnings should be higher under endogenous matching. In line with this rediction, Figure 3 shows that the average rofits were virtually identical between the exogenous and endogenous treatments ( versus ), but the variance increases from to when allowing retailers and suliers to endogenously match (one-sided test based on matching grou averages, = 0.025). Thus we have: Result 6 As redicted by theory, endogenous matching does not change the average earnings but does significantly increase the variance of earnings Suerficial Fairness and Anchoring Previous research has shown that when a retailer and sulier engage in a more natural back-and-forth bargaining rocess, subjects tend to anchor on a suerficially fair wholesale rice and insufficiently adjust to account for asymmetries in exosure to risk (Davis and Hyndman (2017) and Davis and Leider (2017)). However, these studies are somewhat limited because they each consider only one (, c) combination and assume full information, whereas our study has nine such combinations and also considers both full and rivate cost information. We can also better test for anchoring the order quantity on mean demand; in our rivate information treatments there are cases where the otimal order quantity is less than

23 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 23 Figure 3 Average and Variance of Profits for Exogenous and Endogenous Settings, Conditional on Agreement Average and Variance of Exected Profits Exogenous Average Endogenous Variance 50. The next result demonstrates anchoring on both suerficially fair wholesale rices and the mean demand in our study. Result 7 There is evidence of anchoring for both wholesale rices with anchoring on the suerficially fair rice of (+c) /2 and order quantities, with anchoring on mean demand of 50. Suort for this result can be seen in Figure 4, with anel (a) showing agreed wholesale rices, normalized so that 0 is the suerficially fair wholesale rice of (+c) /2 and 100 is the equilibrium rice. If the theoretical rediction was correct, then we would exect the distribution to be centered around 100. However, the data clearly show that this rediction is not born out. Fully 25% of agreed wholesale rices are exactly at (+c) /2, and 62.57% are in the suerficially fair inclusive region between (+c) /2 and w. Furthermore, another 25% of agreements actually have wholesale rices less than (+c) /2, which would give suliers less than half the surlus even in the absence of risk. A t test that the mean is equal to 100 is rejected at < To show anchoring in order quantities, we define the variable: { Q Q A Qbr, Q = br 50 Q br Q, Q br > 50, where Q br reresents the sulier s otimal quantity given the agreed wholesale rice. 5 If anchoring is not resent, then we would exect the distribution deicted in anel (b) to be centered on 0, while evidence of mean anchoring would manifest as a rightward shift of the distribution. As one can see, such a rightward shift is resent and the mean is Moreover, a t test that Q A equals zero is rejected at < Strictly seaking, in the bargaining solution, the order quantity need not be a best resonse to the wholesale rice. For examle, in the full information case, the order quantity is set to maximize channel surlus, which is different from the quantity that would maximize the sulier s exected rofits given the agreed wholesale rice. A similar figure, based uon the theoretical redictions is qualitatively similar.

24 24 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Figure 4 Suerficial Fairness in Wholesale Prices and Mean Anchoring in Quantities, Conditional on Agreement (a) Suerficial Fairness and Wholesale Prices (b) Mean Anchoring in Quantity Fraction of Agreements % < >100 Relative Position of Wholesale Price Agreement Note: 0 = Suerficially Fair Wholesale Price; 100 = Equilibrium Price Fraction of Agreements Mean= Q A : Difference Between Agreed Quantity and Sulier Best Resonse Note: Variable on x-axis calculated as Q - Qbr if Qbr 50 and Qbr - Q if Qbr > 50. If no anchoring, distribution should be centered around 0 While order quantities are closer to the mean demand, the mechanism is not necessarily the same as in traditional newsvendor exeriments. Since the retailer is not exosed to inventory risk, her exected rofits are strictly increasing in the order quantity. Therefore, when negotiating, she should try to ush the agreement to higher order quantities. Indeed, if we consider searately those cases where Q br < 50 and Q br > 50, then we see very strong evidence of anchoring in the former case, but almost no evidence in the latter case. This suggests that the retailer is able to exacerbate any anchoring bias that the sulier may have for low otimal order quantities and to mitigate the bias for high otimal order quantities Bargaining Dynamics The revious subsection documented outcomes which suggest that anchoring biases might be resent when bargaining. In an effort to better understand how these outcomes were achieved, we now turn our attention to bargaining dynamics. Secifically, roviding results on bargaining duration, the anchoring effect of first offers, and the concession rocess over time Bargaining Duration Our first result is: Result 8 Bargaining takes longer when the sulier s cost is rivate information. Suort for this result is in Table 7 which shows the average time remaining when an agreement was reached. As one can see, agreements are reached with less time remaining when suliers costs are rivate information. This is true for both the Exo-P treatment and for the End-D-P treatment (for those suliers who did not disclose). Both random-effects regressions and t tests based on matching grou averages indicate that the result is statistically significant ( < 0.01).

25 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 25 Thus, although the agreements do not differ substantially deending on whether the sulier s cost is known or not, it takes longer to reach an agreement. It is also interesting to note that bargaining duration does not aear to be influenced either by the retailer s rice or the sulier s cost (results available uon request). Table 7 Time Remaining (in seconds) When Agreement is Reached Sulier s Cost Treatment Full Private Exo-F End-F-F End-D-F End-D-P Exo-P First Offers and Anchoring Oening offers are of interest for two reasons. First, in the resence of rivate information, if different cost suliers make different oening offers, it serves to artially reveal their rivate information to retailers. Second, first offers have been shown to have an anchoring effect on negotiations (Galinsky and Mussweiler 1), ultimately influencing the final agrement that is reached. Table 8 deicts two random effects regressions with first offers for wholesale rices as the deendent variable for retailers and suliers, and documents the following result: Result 9 Suliers first offers for wholesale rices are the same whether their cost is full or rivate information, while retailers first offers for wholesale rices become more aggressive under rivate information. Table 8 Effects on First Offers for Wholesale Prices by Role Suliers Retailers c = (0.102) (0.096) c = (0.083) (0.070) Private Cost (0.186) (0.183) (c = 4) (Private Cost) (0.151) (0.140) (c = 5) (Private Cost) (0.146) (0.123) = (0.070) (0.069) = (0.086) (0.105) Constant (0.096) (0.102) R N Note: Significance given by < 0.01, < 0.05, and < 0.10.

26 26 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains In Table 8 for the full information case, both the suliers and retailers first offers are increasing in both the sulier s cost and the retailer s rice. Of course, suliers first offers are nearly 2 units higher than retailers, observed by comaring the intercet (Constant) term between suliers and retailers. The main distinction between retailers and suliers arises when we consider rivate information. Suliers first wholesale rice offers under rivate information are statistically indistinguishable from those under full information (e.g., coefficients on the three Private Cost terms are insignificant for the sulier). The imlication is that, uon observing the oening wholesale rice, retailers can udate their belief about the tye of sulier they are matched with. In contrast, for retailers, their first offer under rivate information is always identical to the oening offer made to a c = 3 sulier under full information. For instance, for retailers, the coefficient on c = 4 is and for c = 5 it is under full information. Under rivate information, the interaction terms ( for c = 4 and for c = 5) comletely wash away the effect. This, combined with our next result on the anchoring effect of first offers, goes a long way to exlain why the high cost suliers earn a disroortionately small share of the suly chain surlus, articularly under rivate information. Table 9, which shows a series of random effects regressions with agreed uon contract terms as the deendent variable by sulier and retailer, documents the following result: Result 10 Final agreements, for both the wholesale rice and the order quantity, are significantly influenced by first offers. Table 9 Effects on Agreed Terms: Anchoring on First Offers Wholesale Price Order Quantity Suliers Retailers Suliers Retailers First w Offer (0.048) (0.034) First q Offer (0.043) (0.046) c = (0.062) (0.050) (1.229) (1.286) c = (0.070) (0.064) (0.936) (1.358) Private Cost (0.079) (0.113) (1.317) (1.996) = (0.051) (0.059) (1.301) (1.223) = (0.082) (0.071) (1.331) (1.481) Constant (0.348) (0.224) (2.328) (2.783) R N Note: The deendent variable is the agreed wholesale rice for the first two regressions and the agreed order quantity for the last two regressions. Significance given by < 0.01, < 0.05, and < In Table 9, for both the wholesale rice and order quantity, the final agreements are significantly

27 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 27 and ositively associated with each arty s first offer. For both contract arameters, the coefficient on the first offer is larger for the sulier, which indicates that their first offer is more determinative of the final outcome, but this difference is only significant for the order quantity (wholesale rice versus 0.307, = 0.15; order quantity versus 0.361, < 0.01). This is not surrising because, given that they are exosed to inventory risk and that they generally earn less than half of the surlus in our data, they have more to gain by sticking more closely to their initial demands. It also seaks to the imortance of making a good initial demand, which we now turn to. Because layers negotiate over both the wholesale rice and order quantity, we can evaluate the exected rofits of each offer to both layers. Figure 5 deicts the suliers average share of the total suly chain rofit by offer number for those negotiations in which a layer made at least five offers and an agreement was eventually reached. We rovide three lots, one for each sulier cost tye. 6 As can be seen, for each sulier cost tye, the retailer s first offer rovides very little in two cases even negative exected rofit for the sulier. In contrast, suliers first offer demands between 56% and 66% of the surlus. What is interesting to note is that for c = 4 and c = 5, suliers eventually concede, on average, more than half the surlus to the retailer. Remarkably, for the c = 5 sulier this actually haens at the second offer. While retailers generally imrove their offers, the average share of surlus to suliers never exceeds 40% and for c = 5, it is not even 10% Matching Dynamics We now turn our attention to the rocess by which matchings were formed in our endogenous matching treatments. The rocess itself was unrestricted excet that, after 150 seconds, any layers unmatched would be randomly matched to a layer of the oosite role who was also unmatched. Secifically, a retailer of tye x {10, 11, 12} could roose to a sulier of tye y {3, 4, 5}, denoted by x y and vice-versa, denoted by y x. In the two disclosure treatments, sulier tye y could disclose, denoted by Dy, y {3, 4, 5}. Lastly, a roosed matching could be acceted, either by the retailer or sulier. This is denoted by A (y, x), where A {R(etailer), S(ulier)}. 7 We define each such roosal, disclosure decision, or matching, to be a matching event. In Figure 6 we lot the frequency of each tye of matching event, restricting attention to the first three such events in each (matching grou, eriod). Panel (a) rovides the results for the End-F-F 6 We also generated lots for each of the nine (, c) combinations but no additional insights arise so we omit them in the interest of sace. 7 We also allowed roosed matchings to be rescinded. These were relatively rare and so we omit them from our discussion to avoid clutter.

28 28 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Figure 5 The Evolution of Surlus Division: Share of Suly Chain Exected Profit to Suliers (%) (a) c = 3 (b) c = 4 Share of Suly Chain Profit (%) Offer Number Share of Suly Chain Profit (%) Offer Number Retailer Offer Sulier Offer Retailer Offer Sulier Offer (c) c = 5 Share of Suly Chain Profit (%) Offer Number Retailer Offer Sulier Offer Note: Includes subject-eriods in which 5 or more offers were made. treatment. There are two interesting results: first, the rocess, at least at the start, is driven by retailers roosing to suliers. Second, as one might exect, the modal roosal is to the lowest cost sulier. The latter result should not be too surrising since all retailer tyes benefit from being matched to a low cost sulier. It is also the case that the modal matching roosal by all sulier cost tyes is to the highest rice retailer. In the two disclosure treatments, we see an oosite attern with matching roosals being driven by the suliers, which makes sense because sulier costs are rivate information unless they choose to disclose. As before, we see that the modal sulier roosal is to the highest rice retailer. Suorting earlier results, we also see that the lowest cost sulier has the greatest frequency of disclosure among suliers over the first three matching events.

29 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 29 Figure 6 Distribution of Matching Events (First 3 Events, %) (a) End-F-F (b) End-D-F & End-D-P Frequency of Each Matching Event (%) >3 11->3 12->3 3->10 3->11 3->12 10->4 11->4 12->4 4->11 4->12 11->5 12->5 5->10 5->11 5->12 S-(3,12) c=3 c=4 c=5 Match Matching Event Frequency of Each Matching Event (%) >3 11->3 12->3 3->10 3->11 3->12 10->4 11->4 12->4 4->10 4->11 4->12 10->5 11->5 12->5 5->10 5->11 5->12 D3 D4D5 R-(3,12) R-(4,12) S-(3,10) S-(4,11) S-(4,12) c=3 c=4 c=5 Disc. Match Matching Event In Figure 7, we focus only on roosals to match and we reort the average number of roosals each layer tye made and received. There is a clear monotone relationshi in all four ossible comarisons. In articular, the better the layer tye (i.e., lower cost sulier or higher rice retailer), the fewer roosals made but the more roosals received. Non-arametric trend tests easily reject that the average number of roosals made/received are the same across retailer and sulier tyes. Thus, the more desirable (in terms of suly chain surlus) of a match a layer tye is, the more flexibility he/she has to choose the artner that she wants to be matched with. Conversely, less desirable artners must make more roosals and are at the mercy of the other layer role. Figure 7 Average Number of Matching Offers Made and Received Average Number of Matching Offers c=3 c=4 c=5 =10 =11 =12 Sulier Retailer Offers Made Offers Received Finally, in Table 10, we look at the timing at which the ossible matchings form, reorting

30 30 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains both the average time remaining when each matching was formed and the frequency that each articular matching is the first to form in a (matching grou, eriod). For the latter we reort overall frequencies as well as the frequencies over the last half of the exeriment. As can be seen, the best matching (3, 12) forms earliest on average and is the first matching to form nearly 40% overall and over 50% of the time in the last half of the exeriment. The next earliest matching to form is (4, 11), which is next in the assortative matching chain, occurring aroximately 2.5 seconds later, on average. Thus, it aears that the formation of the (3, 12) matching sets off a chain of events, frequently leading to assortative matching. 8 We summarize this discussion as: Result 11 Retailers (res. suliers) are more likely to roose matching to the best quality sulier (res. retailer). Assortative matches in articular, (3, 12), form earlier on average, and are more likely to be the first matching to form. There is a tendency towards assortative matching over the last half of the exeriment. Table 10 Time and Order of Matching Average Time Frequency First To Occur Matching Remaining Overall Last Half (3, 10) (3, 11) (3, 12) (4, 10) (4, 11) (4, 12) (5, 10) (5, 11) (5, 12) Note: The shaded cells are those matchings which are consistent with assortative matching. 6. Discussion and Concluding Remarks In this study we investigate rivate cost information and endogenous matching in a two-stage suly chain with dynamic bargaining. In ractice it is common for buyers in a B2B setting to have only a rough estimate of their sulier s cost. It is also natural that retailers and suliers can choose with whom to contract with. Our study is unique in evaluating this setting, where suliers also have the otion to disclose their rivate cost information. 8 Although not shown in the table, (4, 11) is most frequently the second matching to form.

31 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 31 We first examine this setting theoretically and find a number of redictions. For instance, given our exerimental arameters, so long as costs remain rivate information while bargaining, suliers should ot to kee their cost information rivate during matching because the informational rents from rivate information outweigh any gain from matching with a better artner due to disclosure. However, desite the vastly different redictions on disclosure deending on whether or not costs remain rivate information, disclosure atterns did not differ in these two treatments. At least art of this was likely due to imortant deviations from the theoretical redictions during bargaining, which saw suliers earning a disroortionately small share of the surlus articularly under rivate information and even more so for high-cost sulier tyes. Given these deviations, we showed that it was actually in higher-cost suliers best interest to disclose their costs and credibly convey the need for a more equitable wholesale rice. As reviously mentioned, much of the existing exerimental suly chain literature exogenously matches retailers and suliers into airs. Therefore, we consider our study an imortant methodological ste in understanding the imlications that endogenous matching has on overall suly chain efficiency and distribution of rofits. One key result we observe relating to this is that endogenous matching does not lead to significant differences in average exected rofits. However, in terms of second order effects, we find that endogenous matching leads to a higher variance in rofits: lower cost (i.e., higher quality) suliers often choose to contract with higher riced (i.e., higher quality) retailers, and vice versa. Although in our exeriment, subject tyes were drawn indeendently each eriod, in reality sulier costs and retailer quality are likely ersistent. Thus endogenous matching may have imortant long-run consequences in the overall market, with gains being concentrated amongst the very best. This is an interesting avenue for future research. Aside from these main results, we also observed a number of other decisions and outcomes that differed from the normative theory. For instance, suly chain efficiency was below the redicted values, wholesale rices were consistently lower than the normative redictions, and quantities were anchored towards mean demand. One advantage of our exerimental design is that we were able to track various roosals and feedback, in both the bargaining and matching stages. This allows us to better understand the mechanisms that lead to any outcomes. Our analyses of these data suggest that suerficial fairness, defined as the tendency to anchor wholesale rices on the midoint between the retailer s rice and sulier s cost and the equilibrium rediction, exists in our full information treatments. More imortantly, our study shows that suerficial fairness ersists even in the absence of a salient sulier cost value, i.e., when a sulier s cost is rivate information.

32 32 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Desite roviding extensive decision suort in both the matching and bargaining stages, some of our results are indicative of bounded rationality. For instance, disclosure decisions in End-D-F and End-D-P did not significantly differ from one another, whereas the normative theory redicts that lowest cost suliers should always disclose in the former and never in the latter. However, as we noted in our theoretical analysis, when a sulier chooses not to disclose and their costs remain rivate, the redictions for disclosure are somewhat nuanced: for our exerimental arameters no sulier should disclose, but certain arameters exist such that all suliers should disclose. Thus, it is not entirely surrising that human articiants did not conform to these disclosure redictions. In addition, while bounded rationality is a lausible reason for observing deviations relative to normative theory, it is unlikely to be the rimary exlanation for differences across treatments. In articular, bounded rationality is robably resent to a similar extent in all treatments. While we believe suerficial fairness and anchoring are lausible drivers for our results, we admit that other behavioral biases may be resent as well. Some of these, but not all, can be ruled out in our data. For examle, social references are unlikely to be a driver in that the full information treatments redict equal slits of exected rofits between the two arties, whereas we observed retailers earning significantly more than suliers. Davis and Hyndman (2017) discusses this in the context of full information. Nevertheless, it may be interesting to investigate how such biases may interact with endogenous matching and rivate information. In summary, we find that suliers do not always benefit from rivate information while bargaining. Instead, we find that lower quality suliers actually benefit from revealing their cost structures to retailers, in an effort to achieve more equitable gains while bargaining. Further, our study demonstrates that endogenous matching is an imortant feature of exeriments that can affect outcomes such as the variance of rofits. Secifically, higher quality suliers and higher quality retailers benefit from endogenous matching, whereas lower quality suliers and lower quality retailers are disadvantaged by it. As discussed, this may have imortant consequences in the market over the long run. Overall, our study rovides imortant insights for retailer and suliers, deending on whether they are high or low quality with resect to their cometition. Acknowledgments We thank Anyan Qi for fruitful discussions. We gratefully acknowledge the financial suort of Cornell University. References Beer, Ruth, Hyun-Soo Ahn, Stehen Leider awards. Working Paer. The informational and incentives effects of sulier

33 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 33 Bernstein, Fernando, Fangruo Chen, Awi Federgruen. 6. Coordinating suly chains with simle ricing schemes: The role of vendor-managed inventories. Management Science 52(10) Cachon, Gerard P. 3. Suly chain coordination with contracts. Steven Graves, Ton de Kok, eds., Handbooks in Oerations Research and Management Science: Suly Chain Management. North-Holland, Amsterdam. Cachon, Gerard P. 4. The allocation of inventory risk in a suly chain: Push, ull and advance-urchase discount contracts. Management Science 50(2) Cachon, Gerard P., M. Fisher Cambell s sou s continuous relenishment rogram: Evaluation and enhanced inventory decision rules. Production and Oerations Management 6(3) Cachon, Gerard P., M. Lariviere. 1. Contracting to assure suly: How to share demand forecasts in a suly chain. Management Science 47(5) Camerer, Colin F. 3. Behavioral Game Theory: Exeriments in Strategic Interaction. Princeton University Press. Chen, Kay-Yut, Diana Wu Buyer-sulier interactions. Karen Donohue, Elena Katok, Stehen Leider, eds., The Handbook of Behavioral Oerations. Wiley. Corbett, Charles J., Deming Zhou, Chistoher S. Tang. 4. Designing suly contracts: Contract tye and information asymmetry. Management Science 50(4) Davis, Andrew, Kyle Hyndman Multidimensional bargaining and inventory risk in suly chains: An exerimental study. Forthcoming, Management Science. Davis, Andrew, Elena Katok, Natalia Santamaría Push, ull, or both? a behavioral study of how the allocation of inventory risk affects channel efficiency. Management Science 60(11) Davis, Andrew M., Stehen G. Leider Contracts and caacity investment in suly chains. Forthcoming, Manufacturing & Service Oerations Management. Fan, James, Anthony Kwasnica, Douglas Thomas Paying for teamwork: Sulier coordination with endogenously selected grous. Forthcoming, Production and Oerations Management. Feng, Qi, Guoming Lai, Lauren Xiaoyuan Lu Dynamic bargaining in a suly chain with asymmetric demand infinitely. Management Science 61(2) Fischbacher, Urs. 7. z-tree: Zurich toolbox for ready-made economic exeriments. Exerimental Economics 10(2) Galinsky, Adam D., Thomas Mussweiler. 1. First offers as anchors: The role of ersective-taking and negotiator focus. Journal of Personality and Social Psychology 81(4) Haruvy, Ernan, Elena Katok, Valery Pavlov Can structured bargaining imrove channel efficiency. Working Paer.

34 34 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Ho, Teck-Hua, Juanjuan Zhang. 8. Designing ricing contracts for boundedly rational customers: Does the framing of the fixed fee matter? Management Science 54(4) Honhon, Dorothée, Kyle Hyndman Flexibility and reutation in reeated risoner s dilemma games. Working Paer. Hyndman, Kyle, Dorothée Honhon Working Paer. Flexibility in long-term relationshis: An exerimental study. Kalkanci, Basak, Kay-Yut Chen, Feryal Erhun Contract comlexity and erformance under asymmetric demand information: An exerimental evaluation. Management Science 57(4) Lariviere, Martin, Evan L. Porteus. 1. Selling to the newsvendor: An analysis of rice-only contracts. Manufacturing & Service Oerations Management 3(4) Leider, Stehen, William S. Lovejoy Bargaining in suly chains. Management Science 62(10) Mishina, Kazuhiro Toyota Motor Manufacturing, U.S.A., Inc. (Case Study Boston. Harvard Business Publishing). Muthoo, Abhinay Bargaining Theory with Alications. Cambridge: Cambridge University Press. Myerson, Roger B Two-erson bargaining roblems with incomlete information. Econometrica 52(2) Nash, John F The bargaining roblem. Econometrica 2(18) Noor, Jehanzeb, Aurobind Satathy, Jeff Shulman, Jan Wullenwebe The ower of successful sulier collaboration. McKinsey & Comany, htts: // www. mckinsey. com/ ractice-clients/ oerations/ the-ower-of-successful-sulier-collaboration. Ozer, Ozal, Yanchong Zheng, Kay-Yut Chen Trust in forecast information sharing. Management Science 57(6) Randall, Taylor, Serguie Netessine, Nils Rudi. 6. An emirical examination of the decision to invest in fulfillment caabilities: A study of internet retailers. Management Science 52(4) Raoort, Amnon, Ido Erev, Rami Zwick An exerimental study of buyer-seller negotiation with one-sided incomlete information and time discounting. Management Science 41(3) Taylor, Terry A., Erica L. Plambeck. 7. Simle relational contracts to motivate caacity investment: Price only vs. rice and quantity. Manufacturing & Service Oerations Management 9(1) United States Census Bureau E-stats 2015: Measuring the electronic economy. htts: // www. census. gov/ library/ ublications/ 2017/ econ/ 2015-e-stats. html. Zhang, Yinghao, Karen Donohue, Tony Cui Contract references and erformance for the loss averse sulier: Buyback versus revenue sharing. Management Science 62(6)

35 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 35 Aendix A. Bargaining with Private Information For this analysis we secialize to the case of a = 0 and b = 100, to avoid any extraneous algebra. The exected rofits of the retailer can be written as: π r (, c, w, q) = w while the exected rofits of the sulier can be written as: A.1. Two Sulier Cost Tyes ( q q 2 ), π s (, c, w, q) = w ( ) q q 2 cq. We first consider the case in which there are two equally likely sulier tyes indexed by their cost of roduction, c 1 < c 2. This is less comutationally intensive and so is useful to build intuition. It is also relevant because, in order to say whether suliers will disclose their cost in End-D-P (where we have three sulier cost tyes), we require redictions for rivate information bargaining if only one sulier tye discloses. Following Myerson (1984), a mechanism, M, assigns a contract, (w i, q i ), for each ossible reort c i, i {1, 2} of the sulier. We can set u the objective function as: max λπ s (, c 1, w 1, q 1 ) + (1 λ)π s (, c 2, w 2, q 2 ) + 1 q i,w i 2 (π r(, c 1, w 1, q 1 ) + π r (, c 2, w 2, q 2 )) subject to the incentive comatibility constraint that the low cost-sulier does not mimic the high-cost sulier: π s (, c 1, w 1, q 1 ) π s (, c 1, w 2, q 2 ). Substituting in for the rofit exressions, we can exress the Lagrangean as: ( w1 ( ) ( L = λ q1 q 2 w2 ( ) 1) c1 q 1 + (1 λ) q2 q 2 2) c2 q ( w1 ( q1 q 2 w 2 ( ) 2 1) ) + q2 q 2 2 ( w1 ( + α q1 q 2 1) c1 q 1 w 2 ( ) q2 q 2 2) + c1 q 2 We can now rearrange to grou those factors involving the contract for the low- and high-tye sulier resectively. Doing so, we obtain: L = 1 ( ( w1 ( ) 2(λ + α) q1 q 2 2 1) c1 q 1 + w 1 ( ( ) 2(1 λ) q2 q2) 2 c2 q 2 2α ( w2 ( w2 ( ) ) q1 q 2 1 ( q2 q 2 2) c1 q 2 ) + w 2 ( ) ) q2 q 2 2 Note that the exression is linear in w, which means that it effectively acts as a transfer ayment between retailer and sulier. Therefore, we know that λ + α = 1 /2. Solving for q 1 and q 2 yields: q 1 = 100( w 1 + 2(λ + α)(w 1 c 1 )) + (2(α + λ) 1)w 1

36 36 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains and q 2 = 100( 2c 2 + w 2 (1 2(α + λ)) + 2αc 1 + 2λc 2 ). + w 2 (1 2(α + λ)) If we substitute these exressions into the objective function and imose the constraint that λ = 1 /2 α, then the dual roblem becomes: 50 ((4α 2 + 1) c 2 1 2(2α + 1)c 2 (2αc 1 + ) + (2αc 2 + c 2 ) 2 + 2(2α 1)c ) min α [0,1/2] Observe that the derivative of this exression with resect to α is (ignoring the in the denominator): 50(8αc 2 1 4(1 + 2α)c 1 c 2 + 4c 2 (c 2 + 2αc 2 ) + 4c 1 4c 2 (2αc 1 + )). Setting the exression equal to 0 and solving for α yields: α = c 2 2(c 2 c 1 ). Observe that for the arameters of the exeriment, the smallest that this can be is 5 /4 > 1 /2. Therefore, the minimum of this constrained otimization roblem occurs at α = 1 /2. 9 and Using α = 1 /2 λ and lugging into the values for q 1 and q 2, yields: The otimized objective function is: q 1 = 100( c 1) q 2 = 100( 2c 2 + c 1 + 2λ(c 2 c 1 )). 25( c 1 ) 2 The warrant conditions imly that: + 25( 2c 2 + c 1 + 2λ(c 2 c 1 )) 2. Solving yields: (λ + 1 /2 λ)w 1 = 1 25( c 1 ) 2 2 (1 λ)w 2 ( 1 /2 λ)w 1 = 1 25( 2c 2 + c 1 + 2λ(c 2 c 1 )) 2. 2 W 1 = 25( c 1) 2 W 2 = 25 (2c2 2(λ 1) + 2c 2 ( 2c 1 λ + c 1 + ) + c 2 1(2λ 1) 2 ). Finally, we can solve for w 1, w 2 and λ. For λ interior, we have the following system: w 1 ( q1 q 2 1) c1 q 1 = 25( c 1) 2 w 2 ( q2 q 2 2) c2 q 2 = 25 (2c2 2(λ 1) + 2c 2 ( 2c 1 λ + c 1 + ) + c 2 1(2λ 1) 2 ) 9 Note that this is, indeed, the minimum, because the second derivative is everywhere ositive, making this a strictly convex function.

37 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 37 w 1 ( q1 q 2 1) c1 q 1 = w 2 ( ) q2 q 2 2 c1 q 2, where we use the exressions for q 1 and q 2 above. The solution to this is: λ = + 2c 1 3c 2 3(c 2 c 1 ) w 1 = ( + 3c 1) 2( + c 1 ) w 2 = 3(3 + c 1) 2(5 + c 1 ). This requires λ (0, 1 /2), which is equivalent to: (0.5(3c 2 c 1 ), 3c 2 2c 1 ). Note that this is never the case for the exerimental arameters used in this aer. The relevant region for is 3c 2 2c 1, in which case λ = 0 and we must solve: This yields: w 1 ( q1 q 2 1) c1 q 1 25( c 1) 2 w 2 ( q2 q 2 2) c2 q 2 = 25 (2c2 2(λ 1) + 2c 2 ( 2c 1 λ + c 1 + ) + c 2 1(2λ 1) 2 ) w 1 ( q1 q 2 1) c1 q 1 = w 2 ( ) q2 q 2 2 c1 q 2. Note that the inequality above is satisfied so long as: w 1 = ( 7c c 1 c 2 6c c ) 2 ( 2 c 2 1) w 2 = (c c 1 c 2 6c c ) 2 ( c c 1 c 2 4c ) 50(c 2 c 1 )(2c 1 3c 2 + ) 0, which is equivalent to 3c 2 2c 1, which is the case we are considering. In Table A.1, we rovide the theoretical redictions on wholesale rices, order quantities and exected rofits for the three combinations of sulier cost tyes and the three different retailer rices. A.2. Three Cost Tyes We can set u the objective function as: max = λ 1 π s (, c 1, w 1, q 1 ) + λ 2 π s (, c 2, w 2, q 2 ) + (1 λ 1 λ 2 )π s (, c 3, w 3, q 3 ) q i,w i (π r(, c 1, w 1, q 1 ) + π r (, c 2, w 2, q 2 ) + π r (, c 3, w 3, q 3 )) subject to the incentive comatibility constraint that the sulier tye c i does not mimic the sulier tye c i+1 for i = 1, 2: π s (, c 1, w 1, q 1 ) π s (, c 1, w 2, q 2 ) π s (, c 2, w 2, q 2 ) π s (, c 2, w 3, q 3 ). The Lagrangean, rewritten as in the two-tye case to collect certain terms, is: L = 1 ( ( w1 ( ) 3(λ 1 + α 1 ) q1 q 2 3 1) c1 q 1 + w 1 ( ) ) q1 q 2 1

38 38 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Table A.1 Theoretical Predictions For Two Sulier Cost Tyes (a) Sulier Cost Tyes: {3, 4} Retailer s Price Cost (7.75, 70.00, ) (8.35, 72.73, ) (8.93, 75.00, ) 4 (7.80, 50.00, 92.50) (8.36, 54.55, ) (8.92, 58.33, ) (b) Sulier Cost Tyes: {3, 5} Retailer s Price Cost (7.53, 70.00, ) (8.25, 72.73, ) (8.93, 75.00, ) 5 (8.73, 30.00, 72.50) (9.17, 36.36, 90.91) (9.66, 41.67, ) (c) Sulier Cost Tyes: {4, 5} Retailer s Price Cost (8.21, 60.00, ) (8.85, 63.64, ) (9.47, 66.67, ) 5 (8.28, 40.00, 65.00) (8.86, 45.45, 84.09) (9.44, 50.00, ) Note: The numbers in each trile (x, y, z) corresond to the redicted wholesale rice, order quantity and sulier exected rofits, resectively. The exected rofits of the retailer coincide with the high cost tye sulier s exected rofits. + 1 ( 3(λ 2 + α 2 ) ( w2 ( 3(1 λ 1 λ 2 ) ( ) ( q2 q 2 w2 ( ) 2) c2 q 2 3α 1 q2 q 2 2) c1 q 2 + w 2 ( w3 ( ) ( q3 q 2 w3 ( ) 3) c3 q 3 3α 2 q3 q 2 3) c2 q 3 + w 3 ( ) ) q2 q 2 2 ( ) ) q3 q 2 3 Observe that the linearity in w imlies that λ 1 + α 1 = 1 /3 and λ 2 + α 2 α 1 = 1 /3. The next ste is to otimize with resect to q i. This yields: q 1 = 100( w 1 + 3(λ 1 + α 1 )(w 1 c 1 )) w 1 + 3(λ 1 + α 1 )w 1 q 2 = 100( w 2(3(λ 2 + α 2 α 1 ) 1) 3(λ 2 + α 2 )c 2 + 3α 1 c 1 ) + w 2 (3(λ 2 + α 2 α 1 ) 1) q 3 = 100( + w 3(3(1 λ 1 λ 2 ) 3α 2 1) 3(1 λ 1 λ 2 )c 3 + 3α 2 c 2 ). + w 3 (3(1 λ 1 λ 2 ) 3α 2 1) With these values for quantities and also using the relationshi between the α and λ variables, we obtain the value of the objective function (as a function of the λs): ( 50 ( c1 ) 2 + ( + c 1 3λ 1 c 1 + c 2 (3λ 1 2)) 2 + ( + c ) 2(2 3λ 1 3λ 2 ) 3c 3 (1 λ 1 λ 2 )) 2 3 The warrant conditions imly that: (λ 1 + α 1 )W 1 = 1 ( ) 50 ( c1 ) (λ 2 + α 2 )W 2 α 1 W 1 = 1 ( ) 50 ( + c1 3λ 1 c 1 + c 2 (3λ 1 2)) (1 λ 1 λ 2 )W 3 α 2 W 2 = 1 ( ) 50 ( + c2 (2 3λ 1 3λ 2 ) 3c 3 (1 λ 1 λ 2 ))

39 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains 39 Table A.2 Equilibrium Predictions Under Private Information (Three Cost Tyes) Sulier Profit Wholesale Price (w) Quantity (q) = 10 = 11 = 12 = 10 = 11 = 12 = 10 = 11 = 12 c = c = c = This yields: W 1 = 25( c 1) 2 W 2 = 25(2 + c 2 1(1 3λ 1 ) + c 2 2(2 3λ 1 ) 2c 1 c 2 (1 3λ 1 ) 2c 2 ) W 3 = 25 (c2 1(3λ 1 1)(3λ 1 + 3λ 2 2) 2c 1 c 2 (3λ 1 1)(3λ 1 + 3λ 2 2) + c 2 2 (18λ λ 1 (9λ 2 8) + 9λ λ 2 + 8)) 3(1 λ 1 λ 2 ) + 25 ( 6c 2c 3 (3λ λ 1 (6λ 2 5) + 3λ 2 2 5λ 2 + 2) + 3(λ 1 + λ 2 1) (3c 2 3(λ 1 + λ 2 1) + 2c 3 2 )). 3(1 λ 1 λ 2 ) It remains to solve for λ 1, λ 2, w 1, w 2 and w 3. To do this, we use the exressions for W i, where we imose that W i is greater than or equal to the exected rofits for sulier tye i (with equality if λ i is interior) and the incentive comatibility constraints (which must be satisfied with equality if α i > 0). This leads to many cases to consider. Using the values that we imlemented in the exeriment, the relevant case is λ 1 = λ 2 = 0. This yields the following contract arameters for each ossible rice of the retailer (which was always known with certainty) deicted in Table A.2. As we discuss in the main text, one can use the results in Tables A.1 and A.2, to conclude that suliers will not disclose their cost in the End-D-P treatment. B. Samle Exerimental Screenshot

40 40 Davis and Hyndman: Private Information and Endogenous Matching in Suly Chains Figure B.1 Screenshot of the Matching Stage for the Endogenous Treatments