Private Information and Dynamic Bargaining in Supply ... · Keywords: behavioral operations †supply chains bargaining † private information 1. Introduction Private information

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Private Information and Dynamic Bargaining in SupplyChains: An Experimental StudyAndrew M. Davis, Kyle Hyndman

To cite this article:Andrew M. Davis, Kyle Hyndman (2020) Private Information and Dynamic Bargaining in Supply Chains: An Experimental Study.Manufacturing & Service Operations Management

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MANUFACTURING & SERVICE OPERATIONS MANAGEMENTArticles in Advance, pp. 1–19

http://pubsonline.informs.org/journal/msom ISSN 1523-4614 (print), ISSN 1526-5498 (online)

Private Information and Dynamic Bargaining in Supply Chains: AnExperimental StudyAndrew M. Davis,a Kyle Hyndmanb

a Samuel Curtis Johnson Graduate School of Management, SC Johnson College of Business, Cornell University, Ithaca, New York 14853;bNaveen Jindal School of Management, University of Texas at Dallas, Richardson, Texas 75080Contact: [email protected], https://orcid.org/0000-0002-1689-2299 (AMD); [email protected],

https://orcid.org/0000-0003-3666-8734 (KH)

Received: May 17, 2019Revised: December 16, 2019; March 3, 2020;March 14, 2020Accepted: March 18, 2020Published Online in Articles in Advance:October 6, 2020


Copyright: © 2020 INFORMS

Abstract. Problem definition: We conduct a controlled human-subjects experiment in atwo-tier supply chain where a supplier’s per-unit production cost may be private infor-mation while bargaining with a buyer. Academic/practical relevance: Academically,supply chain studies often assume full-information or highly structured bargaining. Weconsider private information with dynamic, unstructured bargaining. In practice, a buyermay not know its supplier’s cost exactly and interact with its supplier in a back-and-forthbargaining environment. Thus, understanding how a supplier’s private cost informationaffects both supply chain outcomes and bargaining is new to the literature and relevant topractice. Methodology: We employ insights from mechanism design to generate restric-tions on the space of agreements and solve for a specific bargaining solution under privateinformation to generate precise predictions. These predictions are then tested through ahuman-subjects experiment. Results: In our experiment, theory predicts that all suppliertypes should earn at least 50% of total profits when their cost information is private.However, we find that high-cost suppliers earn a disproportionately low share of totalprofits under private information, 20.16%. We show that this is because buyers, underprivate information, act as if they are bargaining with the lowest-cost supplier andsuppliers do not appear to blame buyers for behaving this way. Based on these findings, weconduct an additional experiment where suppliers have the ability to communicate theirprivate costs to buyers and observe that verifiable disclosure significantly increases profitsfor high-cost suppliers. Managerial implications: High-cost suppliers actually suffer fromhaving their costs as private information, which runs counter to theory. However, if high-cost suppliers can credibly disclose their costs to buyers, they can significantly increaseprofits. Lastly, although private information does not lead to more disagreements, ne-gotiations do take longer, which can be costly to firms.

Funding: The authors acknowledge the financial support of Cornell University and the University ofTexas at Dallas.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2020.0896.

Keywords: behavioral operations • supply chains • bargaining • private information

1. IntroductionPrivate information is ubiquitous in many supplychain settings where a buyer wishes to contract withone or more suppliers to procure a product. For ex-ample, the buyer and supplier may have differentbeliefs or forecasts about demand. Even more com-monly, the buyer may not know the supplier’s coststructure. Indeed, an executive for a large durablegoods manufacturer recently told us, when discussinghis company’s procurement practices, “I usually don’tknowmy supplier’s cost structure exactly, but I have arough estimate of what it might be.”

Understanding how private information affectssupply chains is important from both academic andpractical perspectives. In an environment in whichthere aremany potential suppliers, the use of auctions

as a procurement strategy for dealing with privateinformation is well studied in operations manage-ment (for a survey of the experimental literature, seeElmaghraby and Katok 2019). In this paper, our in-terest lies in those situations in which the set ofsuppliers is small. In particular, we focus on the caseof a single buyer and supplier whomeet to negotiate acontract and study how private information affectsthe outcome of such an interaction.In studying buyer-supplier dyads, the supply chain

literature often considers a highly structured form ofbargaining, such as a powerful proposer making anultimatum offer to the responder. In the presence ofprivate information, this proposal is often extended toinclude amenu of contracts fromwhich the responderchooses which contract, if any, to accept (as in, e.g.,

1

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Corbett et al. 2004). Using the tools of mechanismdesign, these menus of contracts are carefully de-signed to screen the supplier types with private in-formation in an incentive-compatible manner so thateach supplier accepts the contract that was specifi-cally intended for that type. These approaches arenot without merit. First, from a practical perspec-tive, there are business-to-business (B2B) relation-ships where one party has considerable bargainingpower. Second, at a more abstract level, such frame-works often yield clean, testable, theoretical predic-tions. However, not all negotiations take this form inpractice. Instead, two companies may have relativelyequal bargaining power and partake in a more dy-namic, unstructured, back-and-forth negotiation. Suchsituations are the focus of our paper.

We investigate supply chain contracting in a dy-namic unstructured bargaining environment wheresupplier costs may be private or full information.Because negotiations are conducted by human man-agers, we employ a combination of theory and human-subjects experiments to address the following ques-tions. (1) What is the effect of private supplier costinformation on supply chain outcomes (e.g., profits,efficiency, and contract terms)? (2) How does privatesupplier cost information affect the bargaining dy-namics (e.g., agreements, duration, opening offers,and concessions)?

We operationalize more natural bargaining (be-tween players with equal bargaining power) by al-lowing both parties to make unlimited contract offersand send limited feedback over a fixed amount oftime. One advantage of this setting is that, in ourexperiment, we can observe more than simply out-comes. For instance, we can track each offer made byeither party while negotiating, along with any feed-back, over time. We focus exclusively on wholesaleprice contracts such that the two parties negotiate awholesale price and stocking quantity simultaneously.We also assume that the supplier incurs the cost ofany unsold inventory. This closely matches a drop-shipping, vendor-managed inventory or e-commerceenvironment. Randall et al. (2006) estimate that be-tween 23% and 33% of e-retailers use drop shipping,and the U.S. Census estimates that sales by e-retailerstotaled $389.1 billion in 2016 (United States CensusBureau 2016).

We begin by deriving theoretical predictions underfull and private information regarding the supplier’scost. We refer to any buying firm in a B2B relationship(e.g., manufacturers, retailers, distributors, etc.) asretailers for simplicity. In order to provide hetero-geneity amongst suppliers, we assume that “higher-quality” suppliers have lower per-unit productioncosts (i.e., low cost does not imply low quality). Togenerate predictions under full information, we rely

on the Nash bargaining solution (Nash 1950), whichhas been used in past studies. Under private infor-mation, we first use insights from mechanism designto see how incentive compatibility restricts the setof possible contracts. From the set of incentive-compatible contracts, we go further and use the pri-vate information generalization of Myerson (1984) togenerate more precise predictions.Under full and private information, we generate

point predictions including distribution of profits,supply chain efficiency, wholesale prices, and quan-tities. Interestingly, we show that incentive compat-ibility need not generate inefficiencies. That is, thereexist incentive-compatible mechanisms in which sup]plierswould truthfully reveal their private informationand in which the supply chain is coordinated for allsupplier cost types. This stands in contrast to thetypical mechanism design results in which inefficiencyis a necessary consequence of providing incentives fortruthful revelation of information. However, when wefocus on the Myerson bargaining solution, for theparameters of our experiment, the supply chain is onlycoordinated for the lowest-cost supplier.Another important theoretical insight from the

Myerson bargaining solution is that, for our experi-mental parameters (and for a broad range of otherparameters), all supplier types benefit from their costinformation being private. This means that supplierscan use this private information to their advantagewhile bargaining and earn strictly higher expectedprofits compared with the full-information case. Thisis one of the key theoretical predictions that we seekto test.In addition to the normative theoretical bench-

marks, we formulate a set of behavioral hypothesesthat we also test in our experiment. We accomplishthis through a two-treatment experimental design,which manipulates whether the supplier’s cost in-formation is known or not by the retailer while bar-gaining. Our experiments yield a number of insights.Of them, one important result is that high-cost sup-pliers, under private information, earn a dispropor-tionately low share of total supply chain expectedprofits: 20.16% on average. This stands in stark con-trast to the normative theoretical benchmark, whichpredicts that they should earn more than 50%.Upon further examination, our data suggest that

this result is largely because of the bargaining dy-namics under private information. For instance, re-tailers’ first offers to suppliers under private infor-mation are virtually identical to their first offers to thelowest-cost suppliers under full information. Indeed,many of the retailer offers under private informationwould actually provide higher-cost suppliers with anegative expected profit. In short, under private in-formation, retailers act as if they are bargaining with

Davis and Hyndman: Private Information and Dynamic Bargaining in Supply Chains2 Manufacturing & Service Operations Management, Articles in Advance, pp. 1–19, © 2020 INFORMS

the lowest-cost supplier, which translates into lowerwholesale prices and higher stocking quantities forhigh-cost suppliers. Suppliers, in turn, do not seem toblame retailers for behaving this way: aggressiveoffers by retailers do not decrease the likelihood ofcoming to an agreement under private information(but do under full information).

Given these findings, we then explore managerialinterventions with the aim of improving profits forhigh-cost suppliers under private information. Weaccomplish this through two additional experiments.In one, suppliers have the ability to verifiably disclosetheir private costs; in the other, suppliers have theoption to send a nonverifiable message about theircosts that need not be truthful. Supporting our pri-mary results, we observe that high-cost supplierschoose to disclose their costs nearly 75% of the timeunder verifiable disclosure and that this significantlyincreases their profits. Overall, these additional ex-periments indicate that high-cost suppliers can, coun-terintuitively, gain by revealing their private informa-tion, as long as they can do so credibly.

2. Related LiteratureThe literature most related to our study includes re-search that investigates wholesale price contracts,unstructured dynamic bargaining processes, and/orprivate information. We highlight a subset of im-portant works and refer the reader to more compre-hensive summaries.

Regarding wholesale price contracting theoreti-cally, Lariviere and Porteus (2001) consider a two-stage supply chain and investigate how demand vari-ability affects prices and the distribution of profits.Tomlin (2003) demonstrates how price-only contractscan allocate total supply chain profit between a man-ufacturer and a supplier who can both invest in ca-pacity. Bernstein et al. (2006) identify how wholesaleprice contracts can coordinate a supply chain with asingle supplier and multiple retailers. Cachon (2003)provides a more general summary of this theoreticalliterature. Turning to experiments, some papers thatinvestigate supply chain contracting include Ho andZhang (2008), who study how framing a fixed fee canaffect overall supply chain efficiency, and Kalkanciet al. (2011), who demonstrate how simple price-onlycontracts can perform well in a setting where theretailer has accurate information regarding demand.Davis et al. (2014) investigate wholesale price con-tracts in three alternative inventory risk arrange-ments, whereas Zhang et al. (2015) compare buy-backand revenue-sharing contracts under alternative over-age andunderage costswith loss-averse suppliers. For acomprehensive summary of the experimental supplychain contracting literature, we refer the interestedreader to Chen and Wu (2019).

A majority of the mentioned papers assume thatone party in the supply chain makes an ultimatumoffer to the other party. Some studies have extendedthis setting by allowing for a more natural bargainingprocess. Theoretically, an important framework forsolving these problems under full information is theNash bargaining solution (Nash 1950). Experimen-tally, the supply chain papers that we are aware ofwhich consider dynamic bargaining are Leider andLovejoy (2016), Davis and Leider (2018), and Davisand Hyndman (2019). Leider and Lovejoy (2016)consider back-and-forth bargaining in a three-stagesupply chain with chat box communication. Davisand Leider (2018) allow for unstructured bargainingand evaluate which contracts can alleviate underin-vestment in capacity by suppliers. Davis and Hyndman(2019) study which contract terms should be in-cluded in an unstructured negotiation with full in-formation. Inmanyways, ourwork can be consideredan extension of Davis and Hyndman (2019), with acritical difference being that we investigate one-sided private information, which is directly rele-vant to practice.The supply chain literature frequently assumes

full information of price, demand, and cost parame-ters, with some notable exceptions (especially in anultimatum-offers setting). One example includes pa-pers in which the retailer may have private knowl-edge about consumer demand (e.g., Cachon andLariviere 2001 from a theoretical perspective andOzer et al. 2011 from an experimental perspective).In a private information setting, Corbett et al. (2004)adopt a mechanism design framework in which apowerful supplier offers menus of contracts to screenbuyer cost types. More relevant to our work are thosepapers in supply chain management that considerprivate information combined with a more naturalbargaining interaction between two parties. Fenget al. (2015) is one example. They investigate multi-ple alternating offerswhere both parties are impatientand the buyer has private information about theirtype. They show how quantity distortion and infor-mation rents may or may not be avoided dependingon the patience of the parties involved.There is also a rich literature in experimental eco-

nomics pertaining to bargaining (see Roth 1985, 1995;Muthoo 1999; and Camerer 2003 for summaries).Early work demonstrates that pairs tend to agree on a50/50 split of a surplus (Nydegger and Owen 1974).However, a number of other experiments extend thisresearch by having pairs bargain not over payoffsbut lottery tickets and show that participants candiffer in their focal points for what they deem as fairoutcomes (Roth and Malouf 1979, Roth et al. 1981,Roth andMurnighan 1982). Other papers have shownsimilar effects of “self-serving” biases (Babcock and

Davis and Hyndman: Private Information and Dynamic Bargaining in Supply ChainsManufacturing & Service Operations Management, Articles in Advance, pp. 1–19, © 2020 INFORMS 3

Loewenstein 1997), which differ from more standardmodels of social preferences (Fehr and Schmidt 1999,Bolton and Ockenfels 2000). This is especially relevantto our study as retailers have some leeway as to howthey interpret unknown supplier costs. In anotherbargaining study, Roth et al. (1988) demonstrate arobust behavioral result in that pairs often come toagreements during the final seconds of a negotiation,which they deem the “deadline effect.” Regardingprivate information, Forsythe et al. (1994) conduct anovel unstructured bargaining experiment with one-sided private information and investigate when dis-agreements take place. Mitzkewitz and Nagel (1993)experimentally study the ultimatum game with pri-vate information and manipulate whether the pro-poser offers an amount to the receiver (i.e., the re-ceiver only knows their own earnings) or an amountfor themselves (i.e., the receiver only knows what theproposer earns). Valley et al. (2002) consider two-sided private information in a double-auction envi-ronment and show that preplay communication canhelp contribute to fewer disagreements. More re-cently, Camerer et al. (2019) apply insights frommechanism design (like us) and take a machine-learning approach to analyzing short unstructurednegotiations with one-sided private information.

3. Theoretical BackgroundIn this section, we provide a theoretical analysis forthe bargaining institutions that we will test in thelaboratory. The basic framework consists of a retailerwith selling price p and a supplier with per-unit costof production, c, whomust negotiate both awholesaleprice, w, and an order quantity, q. For ease of expo-sition and because it conforms to our experimentalparameters, underlying demand, D, is drawn uni-formly from [0, 100], but the actual realization ofdemand is unknown at the time of bargaining. Wealso assume that the supplier bears the risk of unsoldinventory. We include this last assumption for threereasons. First, it represents a scenario where bothparties face demand risk.1 Second, it is consistentwithsettings in practice, such as drop shipping ande-commerce. Third, in a related bargaining study onfull information, Davis and Hyndman (2019) findthat many behavioral results do not depend on whichparty holds the inventory risk. However, we recog-nize that this assumption does not represent all typesof supply chain risk environments.

We consider two settings that differ as to whetherthe supplier’s cost is full or private information. Forthe full-information case, we provide a more generalbut brief analysis. For the private information case,we present a general approach to solving the problemand provide some general insights based on the no-tion of incentive compatibility. Because of the highly

computational nature of the specific bargaining so-lution, some predictions are based on the parametersin our experiment, which will be outlined later (butsee Remark 1).

3.1. Bargaining with Full InformationUnder full information, the supplier’s cost, c, is com-mon knowledge while bargaining. Because our ex-periment implements an unstructured bargainingprotocol, the relevant theoretical lens for the full-information case is the Nash bargaining solution(Nash 1950). Denote by πi(w, q) the expected profitsfor firm i ∈ {r(etailer), s(upplier)} from an agreementwith wholesale price, w, and order quantity, q. Theexpected profits can be expressed as

πr w, q( ) � p − w

100

∫ 100

0min q, x

{ }dx;

πs w, q( ) � w

100

∫ 100

0min q, x

{ }dx − cq. (1)

The disagreement payoff is zero for both players. TheNash bargaining solution is the solution to

maxw,q

πr w, q( ) · πs w, q

( )

s.t. c ≤ w ≤ p and a ≤ q ≤ b.

Because the full-information bargaining environ-ment is identical to Davis and Hyndman (2019), westate without proof the following result.

Proposition 1. When bargaining under full information:i. The supply chain is coordinated, q∗ � 100(p−c)/p.ii. Expected profits for the retailer and supplier are split

equally, 50%/50%.iii. The wholesale price is w∗ � p(p+3c)

(2(p+c)) >(p+c)2 .

Note that the agreed wholesale price,w∗, under fullinformation is strictly greater than the midpoint be-tween c and p (i.e., (p+c)/2). This follows because thesupplier bears the inventory risk. Therefore, to equalizethe expected payoffs of the retailer and supplier, thewholesale price must increase beyond the midpointbetween the retailer’s price and the supplier’s cost.

3.2. Bargaining with Private InformationWhen supplier costs are private information whilebargaining (i.e., unknown to retailers), we assumethat the supplier has three, equally likely, possiblecost types c1 < c2 < c3 < p and that the set of possiblecosts is common knowledge. In such a setting, we needa suitable generalization of the Nash bargaining so-lution. Using insights from mechanism design, Myerson(1984) provides such a generalization that has beenlargely unexplored in the operations literature. Thebasic idea is that the players negotiate over “mech-anisms,” which consist of a menu of contracts—one


for each possible type of supplier—that are incentivecompatible, so that each supplier type truthfully re-veals her type.2 From the set of individually rationaland incentive-compatible mechanisms, the Myerson(1984) solution then seeks to maximize a weightedsum of the retailer’s and supplier’s expected profits.The solution must also respect so-called warrant con-ditions, which are the minimum amounts that eachplayer type “warrants” in a fair division. The finalcomplication is that the weights must be derived aspart of the solution.

Although we will leave most details deriving theMyerson (1984) bargaining solution to the onlineappendix, it is instructive to look at the incentivecompatibility constraints. Let}� {(γi,wi,qi), i� 1,2,3}denote a mechanism, where (γi,wi, qi) denotes a con-tract intended for supplier type ci. Thewholesale priceand order quantity are (wi, qi), and γi ∈ [0, 1] is theprobability of agreement. Given our assumption on thedemand distribution, we know that supplier expectedprofits, conditional on agreement, are πs(wi, qi, ci) �(wi/200)(200qi − q2i ) − ciqi. A mechanism is incentivecompatible if

γi wi/200( ) 200qi − q2i( ) − ciqi

( )≥ γj wj/200

( )200qj − q2j( )

− ciqj( )

∀i � 1, 2, 3 and ∀j �� i.

We first provide two preliminary results, the proofsof which are in Online Appendix A, which followfrom incentive compatibility. First, expected quanti-ties decline in supplier cost type. Specifically,

Lemma 1. Let } be an incentive-compatible mechanism.Then, γ1q1 ≥ γ2q2 ≥ γ3q3.

Second, we show that it is without loss of generalityto look only at mechanisms in which disagreementnever occurs. Specifically,

Lemma 2. Given any incentive-compatible mechanism,}� {(γi,wi,qi), i� 1,2,3}, there exists an alternative mech-anism, } � {(1, w̄i, q̄i), i � 1, 2, 3}, which (i) is incen-tive compatible, (ii) has no disagreement, and (iii) gen-erates the same expected profits for each supplier type andthe retailer.

This result is interesting because when bargainingwith private information, the chance of disagreementis often required to generate incentives for truthfulrevelation. However, because the parties bargainover two parameters: quantity—which determinesthe size of the pie—and wholesale price—which de-termines the division of the pie—we can do awaywiththe possibility of disagreement, while still main-taining incentive compatibility. Remark 3 providesfurther discussion.

Another typical result when contracting with pri-vate information is inefficiency. This can happen ei-ther because of the possibility of disagreement or thedistortion in key parameters (e.g., quantity) in orderto generate the necessary incentives for truthful rev-elation. However, in our setting, we can prove thatefficiency and incentive compatibility can simulta-neously coexist. That is,

Proposition 2. There exist incentive-compatible mecha-nisms in which the supply chain is coordinated for allpossible supplier types. That is, qi � 100((p−ci)/p) for all i.Therefore, if inefficiency arises, it must be because ofthe properties of the bargaining solution and notmerely because of incentive compatibility.With these preliminary results, we now provide a

brief discussion of the bargaining solution ofMyerson(1984). Recall that πs(w, q, ci) denotes the expectedprofits of supplier type ci when faced with the contract(w, q), whereas πr(w, q, p) denotes the expected profitsof a retailer with selling price p facing the contract(w, q). The general approach consists of solving

maxλi,qi,wi≥0

∑3i�1

λiπs wi, qi, ci( ) + 1

3

∑3i�1

πr wi, qi, p( )

s.t. πs wi, qi, ci( ) ≥ πs wi+1, qi+1, ci

( ), i � 1, 2.

That is, wemaximize aweighted sumof the supplier’sand retailer’s expected profits subject to incentivecompatibility constraints. One complicating factor isthat the solution must also determine the weights, λi,that we apply to each supplier type i’s profits. That is,the bargaining solution need not weight each sup-plier type equally. Without loss of generality, we cantake λ3 � 1 − λ1 − λ2.3

We outline the steps to obtain the solution, withtechnical details relegated to Online Appendix A.1. Let αi denote the Lagrange multiplier on sup-

plier type i’s incentive compatibility constraint. Ob-serve that the wholesale price, wi, is a linear transferbetween the retailer and the supplier (see (2) in OnlineAppendix A). Therefore, we can impose the con-straints that λ1 + α1 � 1/3 and λ2 + α2 − α1 � 1/3.2. Find the optimal order quantities as a function of

αi and λi (see (3)–(5) in Online Appendix A).3. Ensure that the warrant conditions are satisfied.

The warrant conditions dictate the virtual utility,W∗i ,

that each supplier type i warrants (i.e., “deserves”)in a bargaining solution. This is akin to the standardNash bargaining solution but with private informa-tion, corresponds to half the total virtual surplusgenerated by the interaction between the retailer andthe particular supplier cost type. The system of war-rant equations to solve is given by (6)–(8) and thesolution by (9)–(11) in Online Appendix A.


4. The final step is to find values of λi and wi

such that

πs wi, qi, ci( ) ≥ W∗

i , with equality if λi > 0,πs wi, qi, ci( ) ≥ πs wi+1, qi+1, ci

( ), for i � 1, 2 and

with equality if αi > 0.

With three supplier cost types and two incentivecompatibility constraints, this is, potentially, a systemof five equations in five unknown variables: (w1,w2,w3, λ1, λ2). Moreover, one must check boundary con-ditions on λ and α, making it necessary to considerseven different systems of equations to find the validsolution. Given the parameters of our experiment, thebargaining solution involves λ1 � λ2 � 0, and theincentive constraints on the c1 and c2 supplier typesare binding (the equilibrium contract parameters areprovided in Table 1 and are discussed when we re-view our experimental design). We can summarizethese results as:

Proposition 3. When bargaining under private informationand given the parameters of our experiment:

i. The supply chain is coordinated only for the lowest-cost supplier (c1).

ii. All suppliers benefit from private information. Inparticular, expected profits are strictly higher than underfull information, and suppliers earn at least half of theexpected supply chain profits.

iii. The wholesale price is higher and the order quantityis lower under private information than under full information.

As in the Nash bargaining solution with completeinformation, here too there is a notion of equity be-tween the players involved. In particular, it mustbe that

∑3i�1

λiπs wi, qi, ci( ) � ∑3

i�11/3( )πr wi, qi

( ).

Therefore, when λ1 � λ2 � 0, the bargaining solutiononly places weight on the highest-cost supplier type,and fairness between the retailer and supplier isjudged as being between the high-cost supplierand the expected retailer profits, averaging over all

possible supplier-type pairings. However, from theretailer’s perspective, she has a chance to bargainwith lower-cost suppliers, where she will earn more;this means that she will receive less than half of thesupply chain profits when matched with the highest-cost supplier.Aword of discussion on Proposition 3(ii) is merited

because it may seem counterintuitive that even thehigh-cost supplier benefits from private information.As we show in the online appendix, because of theincentive compatibility requirement, the warrant con-ditions between supplier types are deeply intertwined.Specifically, the warrant condition for the low-costsupplier, W∗

1 , is always half of the full-informationsurplus, and the warrant condition of the medium-cost supplier, W∗

2 , is increasing in W∗1 (and W∗

3 is in-creasing in W∗

2 ). Therefore, particularly in our casewhere λ1 � λ2 � 0, this serves to push up the earningsof even the high-cost supplier.

Remark 1. Although we focus here on the parametersthat we eventually implement in the experiment, wealso numerically solved for the Myerson bargainingsolution for a wide array of parameters (see OnlineAppendix A). If we restrict attention to the more em-pirically relevant high margin case (i.e., c3 ≤ p/2), thenour parameter choices appear to be representative.Specifically, in such cases, about 94% of the time wefind that all supplier types benefit relative to full in-formation, although it almost never happens that notypes benefit. When we allow that supplier costs mayexceed half the retailer’s price, then it is less commonthat all supplier types benefit (about 54%), but it is stilluncommon that no supplier types benefit (< 1%). Itmay be interesting in future work to study cases inwhich ci > p/2 is possible.

Remark 2. One might find it odd that the private in-formation extension of the Nash bargaining solution—which involves solving an additive optimizationproblem—is so different from the complete informa-tion Nash bargaining solution—which involves solv-ing a multiplicative optimization problem. An ear-lier extension by Harsanyi and Selten (1972) proposeda multiplicative version. However, as discussed inMyerson (1984), this necessarily violates a probabilityinvariance axiom. We refer the interested reader tothese papers for further details.

Remark 3. It is natural to wonder whether our re-striction to mechanisms without disagreement plays arole: in particular, whether there exists another bar-gaining solution that generates the same expectedpayoffs but has disagreement. We believe that theanswer is no. This is because, in our proof of Lemma 2,the alternative mechanism with no disagreement addsslack to the incentive constraints. On the other hand, in

Table 1. Normative Experimental Predictions

Suppliertype

Suppliershare (%)

Efficiency(%)

Wholesaleprice (w) Quantity (q)

Full Private Full Private Full Private Full Private

c � 3 50.00 61.29 100 100 7.85 8.57 72.58 72.58c � 4 50.00 56.83 100 97.87 8.43 8.62 63.43 54.29c � 5 50.00 61.23 100 88.24 8.93 9.08 54.29 36.01Average 50.00 59.78 100 95.37 8.41 8.76 63.43 54.29

Note. The supplier’s share under private information is based on theex ante expected profits.


the bargaining solution for our experimental parame-ters, the incentive constraints are binding. There-fore, we cannot reverse the direction and look for anequivalent mechanism with disagreement. To do sowould require removing slack from the incentiveconstraints, but no such slack exists because of thebinding constraints.

4. Experimental Design andHypothesis Development

In our experiment, participants were assigned a roleof supplier or retailer and placed into a group of six,three retailers and three suppliers. Both roles andgroups remained fixed for the duration of the experi-ment. In every round, each supplier was randomlyassigned a production cost per unit, c ∈ {3, 4, 5}, with-out replacement, and each retailer was randomlyassigned a selling price per unit, p ∈ {10, 11, 12}, with-out replacement. Retailer and supplier roles werefixed throughout the experiment, but a participant’scost (if supplier) or price (if retailer) could vary fromperiod to period. The distributions of prices and costswere common knowledge to both parties. This designallows us to test the robustness of results to changes insupplier cost and retailer price. We drew prices andcosts without replacement to ensure a more balancedcollection of different observations.

Each round began by randomly assigning retailersand suppliers into pairs. Each pair would then bar-gain over contract terms for a product with uncertaindemand. Regardless of the retailer’s selling price andsupplier’s production cost, demand for the productwas always a random draw from the discrete uniformdistribution on {1, 2, . . . , 100}. If the two parties cameto an agreement while bargaining, demand wouldbe realized, and retailers would satisfy demand bysourcing product directly from the supplier, such thatthe supplier incurred the cost of any unsold inventory.

For our unstructured bargaining, we followed aprotocol similar to Davis and Leider (2018) and Davisand Hyndman (2019). Specifically, each retailer-supplier pair was given five minutes to negotiate acontract that consisted of two terms, a wholesaleprice, w, and a quantity, q. During this time, retailersand suppliers could make as many offers as theydesired at any point in time. If either party chose toaccept the most recent offer of the other player, thendemand would be realized, and participants wouldreceive feedback that included realized profits. If apair was unable to reach an agreement after fiveminutes, then both players would receive a profitof zero.

While bargaining, we allowed participants to pro-vide feedback about the most recent offer received. Inparticular, they could “reject” either of the proposed

terms through a button for each contract term, whichthey could click at any time for a currently validproposal. This feedback would then be displayed onthe proposer’s screen. Note that a participant couldlater accept the offer even if he or she signaled dis-approvalwith it, so long as amore recent offerwas notreceived. We opted for this type of feedback tosimulate amore natural bargaining processwhile alsoallowing us to monitor offers and feedback. Lastly, toreduce complexity in the experiment, we providedparticipants with a decision support tool where theycould enter hypothetical values for w and q, whichwould generate a graph showing the profit for bothplayers as a function of demand.Consistent with our theory section, our experi-

mental design consists of two treatments: one wheresupplier’s costs were common knowledge (full in-formation) and one where the supplier’s cost infor-mation was unknown to retailers (private informa-tion). Each treatment consisted of six rounds andincluded 48 participants across three sessions. Thus,each treatment included eight groups of six partici-pants. The experimental software was programmedin z-Tree (Fischbacher 2007), and all sessions tookplace at a large northeast university. Sessions tookroughly 60 minutes with earnings varying consider-ably: average of $38, maximum of $78, and minimumof $7 (participants were compensated for all roundsof decisions).

4.1. Predictions and HypothesesIn Table 1, we provide point predictions for our ex-periment using our theoretical analysis, averagedacross all prices for ease of exposition.4 We willcompare our data directly with these predictions.Therefore, the point predictions can be consideredas a null hypothesis. In addition to this null hy-pothesis, we also develop three behavioral hypoth-eses. Because the full-information setting has beenexplored previously, each behavioral hypothesis fo-cuses on the private information case.Our first hypothesis considers the supplier’s share

of total expected profits. We know from past researchon full information that inventory risk holders (i.e.,suppliers in our study) typically earn less than thenormative predictions (Davis and Hyndman 2019).Hence, a similar result may exist under private in-formation. Further, although the normative theorypredicts that suppliers should always be weaklybetter off under private information, this may notbe true with human decision makers. For instance,Schelling (1960) posited that ignorance can be anadvantage in negotiations because the uninformedparty can bargainmore aggressively because she doesnot “know” what a fair allocation would be, whereasthe informed party may be more accepting of such


behavior because he recognizes the informationaldisadvantage of the uninformed party. In our ex-periment, this suggests that, under private informa-tion, higher-cost suppliers may be inclined to acceptless favorable offers compared with full information.In experiments that use abstract settings and/or ul-timatum offers, Schelling’s hypothesis has found em-pirical support. For instance, Siegel and Fouraker(1960) and Hamner and Harnett (1975) find that un-informed parties typically set high aspiration levelsunder private information (i.e., they assume the bestcase for themselves) and are able to achieve more fa-vorable outcomes compared with a full-informationscenario. In line with this, Knez and Camerer (1995)conduct an ultimatum game experiment and find thatpeople “egocentrically choose the view which bene-fits themmost”when there is room for interpretation.Overall, this research suggests that uninformed re-tailers may act as if they are bargaining with thelowest-cost supplier (c � 3) and that suppliers will notblame them for this. As a result, we expect that thec � 5 (and possibly, the c � 4) supplier will earn lesscompared with full information.

Hypothesis 1 (Supplier Share). In private information, theobserved suppliers’ share of expected profits will be less thanthe normative predictions for all supplier costs. High-cost(c � 5) suppliers will earn less than under full information.

We turn now to supply chain efficiency. Both“strategic ignorance” and anchoring may have aneffect, but the direction of the effect depends on thesupplier cost. Recall that anchoring is the tendency foragreed quantities to be closer to the midpoint of de-mand than normative theory predicts. Consider eachsupplier type separately. For c � 3 under private in-formation, the predicted average quantity is 72.58(same as full information). Anchoring will pull thisdown and drive efficiency below the normative pre-diction. However, strategic ignorance should notplay a role because retailers are likely to bargain as ifthey are matched with the low-cost supplier—whichthey are in this case. At the other extreme, for the c � 5supplier, both anchoring and strategic ignorance shouldwork to push quantities and therefore, efficiency up.This is because the predicted quantity for the c � 5supplier is below both the mean and the quantity forthe low-cost supplier. For the c � 4 supplier, an-choring and strategic ignorance work in oppositedirections. The predicted average quantity is 54.29,which is greater than the mean demand but less thanthe c � 3 quantity. Therefore, anchoring pushes thequantity down, whereas strategic ignorance pushes itup. It is not clearwhich effectwill dominate,making itdifficult to predict whether efficiency is higher orlower than the normative prediction. Fortunately, ourexperiment can assist with this.

Hypothesis 2 (Efficiency). In private information, the ob-served supply chain efficiency will be lower than the nor-mative theory for the c � 3 supplier and higher than thenormative theory for the c � 5 supplier.

Lastly, we consider bargaining agreements andduration under private information. Compared withthe normative theory, we hypothesize that agreementrates will be less than 100%. Comparing the two in-formation settings with one another, it is not clearhow agreement rates may vary. First, one mightsuspect that the larger contracting space under pri-vate information will lead to more disagreement.Malouf and Roth (1981) evaluate this notion througha simple bargaining experiment with full informa-tion and find mixed support. Second, Conrads andIrlenbusch (2013) conduct an ultimatum game ex-periment with private and full information and ob-serve that private information can lead to higheragreement rates. Because past experimental researchdoes not provide a clear prediction and Lemma 2states that players should not use disagreement asa tool for separating supplier cost types, we follow thenull hypothesis that agreement rates will be equalbetween full and private information. Lastly, re-garding bargaining duration, Malouf and Roth (1981)show that a larger contracting space increases bar-gaining duration. In addition, a study by Loewensteinand Moore (2004) shows that information revelationdecreases bargaining duration, provided that theinformation is not subject to interpretation. This sug-gests that bargaining duration should be shorter in ourfull-information treatments.

Hypothesis 3 (Bargaining). In private information, for allsupplier costs:i. Agreements will be less than 100% and equal to the

full-information case.ii. Bargaining duration will be longer than the full-

information case.

Ultimately, through our human-subjects experi-ment we can determine which of these hypotheses aresupported and simultaneously evaluate Myerson’snormative benchmarks.

5. ResultsWe present our experimental results in two subsec-tions. In Section 5.1, we investigate outcomes, and inSection 5.2, we analyze the bargaining dynamics.Because six participants interacted with each otherthroughout a single session (through randommatchingand random prices/costs each round), we take a con-servative approach for all hypothesis tests and use ttests where a group of six is an independent obser-vation. For instance, for a particular metric (e.g.,supplier share of profit for c � 3), we first calculate the


supplier’s share for all rounds and participants wherec � 3.We then average these values for each particulargroup of six, yielding eight independent observationsin each treatment. Also, we run all regressions withrandom effects and clustered standard errors at thegroup level. Lastly, we note that there is no significantdifference in agreement rates between treatments.Therefore, although we will investigate agreementsin detail later, all outcome results in Section 5.1 areconditional on a bargaining agreement.

5.1. Outcomes5.1.1. Supplier’s Share of Profits. Normative theorypredicts that suppliers should earn 50% of total ex-pected profits under full information and at least 50%under private information. Yet Hypothesis 1 positsthat suppliers will earn less than 50% for all suppliercosts. Moreover, because of strategic ignorance, c � 5suppliers will earn a significantly lower share underprivate information. Figure 1 depicts the supplier’sshare of total supply chain expected profit under fulland private information, averaged across all retailerprices and conditional on agreement.5 One can seethat suppliers do indeed earn less than 50% on av-erage in both treatments (overall average of 36.89%for full and 33.56% for private, both p < 0.01).6 There isalso a clear decreasing relationship between suppliercost and supplier share of total profits. Comparing thetwo treatments with one another, there are virtuallyno differences between full and private informationfor c � 3 and c � 4. However, there is a stark differencebetween the two treatments for c � 5. Although nor-mative theory predicts that c � 5 suppliers shouldearn a higher share of total profits compared with fullinformation (61.23% versus 50%) (see Table 1), weobserve that c � 5 suppliers earn a disproportionatelylower share of total profits in private information,20.19%. Overall, we find support for both parts of

Hypothesis 1: under private information, suppliers’shares of total supply chain expected profits aresignificantly below the normative predictions anddecreasing in cost. Also, high-cost (c � 5) suppliersearn lower profits in private information than fullinformation.

5.1.2. Efficiency. Table 2 provides results for supplychain efficiency and shows that it is 91.66% under fullinformation, on average. This is consistent with pastbargaining studies on full information that consideronly a single c and single p. It is also not statisticallydifferent from the average 92.20% efficiency underprivate information. Turning to Hypothesis 2, whichstates that efficiency under private information willbe lower than the normative prediction for the c � 3supplier and higher than the normative prediction forthe c � 5 supplier, we find support for both: observedefficiency is significantly lower for c � 3 (91.53% versus100%, p < 0.01) and marginally higher than the nor-mative prediction for c � 5 (91.05% versus 88.24%,p � 0.097). In addition, recall that a clear hypothesisfor the c � 4 supplier was unavailable. For this case,we observe that efficiency is significantly lower thanthe normative prediction (94.16% versus 97.87%,p � 0.011).

5.1.3. Contract Terms. To gain a better understandingas to what is driving our profit and efficiency results,we now focus on the agreed upon contract terms.Table 3 illustrates the agreed upon wholesale pricesand quantities in our experiment, averaged across allretailer prices. Beginning with wholesale prices, onecan see that average wholesale prices are too lowrelative to the normative predictions, by 0.75 underfull information (8.41 versus 7.66, p < 0.01) and by1.22 under private information (8.76 versus 7.54,

Figure 1. Observed Supplier Share (%) of Supply ChainExpected Profit, Conditional on Agreement

Table 2. Observed Efficiency (Percentage), Conditional onAgreement

Supplier type

Predicted Observed

Full Private Full Private

c � 3 100∗∗∗ 100∗∗∗ 90.96 91.53(1.85) (1.80)

c � 4 100∗∗∗ 97.87∗∗ 91.04 94.16(2.16) (1.10)

c � 5 100∗∗∗ 88.24∗ 92.83 91.25(1.57) (1.42)

Average 100∗∗∗ 95.37∗∗ 91.66 92.20(1.33) (1.01)

Notes. Standard errors across groups are reported in parentheses.Significance of t tests between observed values and the normativepredictions is given in columns 2 and 3. Significance of t tests betweenfull and private information is given in the “Observed Private”column. All differences are denoted by asterisks.∗p < 0.10; ∗∗p < 0.05; ∗∗∗p < 0.01.


p < 0.01). This result holds for all supplier costs aswell(all six p < 0.01), which goes a long way to explainingwhy suppliers earn less than the normative theorypredicts. Comparing wholesale prices under full versusprivate information,wefind that there are no significantdifferences. However, the difference is largest in the c � 5case, where average wholesale prices are 8.11 underfull information and 7.89 under private information.

For quantities, all of them are at least marginallysignificantly different than the normative predictions.The lone exception to this statement is in private in-formation for c � 4. In particular, recall that the dis-cussion around Hypothesis 2 suggested that anchor-ing should pull quantities down for c � 4, but ifretailers assume they are bargaining with a c � 3supplier, this may push quantities up. These effectsappear to be supported because observed quantitiesare not statistically different from the normativetheory for c � 4 (although it did not translate into anefficiency equal to the normative prediction). Also, incomparing quantities between the full and privateinformation treatments, there are no statistically sig-nificant differences.

Although there are no statistically significant dif-ferences between full and private information forwholesale prices and quantities, a combination ofsmall deviations can translate into meaningful dif-ferences in profits. Consider the case of c � 5 andp � 10, taken from Table B.3 in Online Appendix B.

The average wholesale prices are 7.68 and 7.54 in fullversus private information, respectively, and quan-tities are 49.34 and 56.86, respectively. Importantly,neither difference is statistically significant, but to-gether, they have a dramatic effect on a supplier’sshare of profits. In Table 4, we provide the details. Aslight decrease of 0.14 (7.68 − 7.54) in the wholesaleprice lowers the supplier’s share by 4.16%. On top ofthis, because the suppliers stock a larger quantity, itrequires them to take more risk and reduces theirshare by another 8.46%. Indeed, these differences inwholesale prices and quantities for c � 5 suppliers aredirectionally opposite to what the normative theorypredicts. In particular, Proposition 3(iii) showed thatwholesale prices should be higher and stocking quan-tities should be lower under private information, butwe observe the reverse for c � 5. In sum, for c � 5suppliers under private information, a combination ofslightly lower wholesale prices and higher quantitiescontributes directly to a disproportionately low splitof total supply chain expected profits.

5.2. Bargaining DynamicsIn an effort to understand how participants arrived atsuch contract terms and outcomes, we now turn ourattention to bargaining dynamics. Specifically, weprovide results on bargaining agreements and du-ration, anchoring on initial offers, and the concessionprocess over time.

Table 3. Observed Wholesale Prices and Quantities, Conditional on Agreement

Supplier type

Predicted Observed

Wholesale price(w) Quantity (q) Wholesale price (w) Quantity (q)

Full Private Full Private Full Private Full Private

c � 3 7.85∗∗ 8.57∗∗∗ 72.58∗∗∗ 72.58∗∗∗ 7.20 7.22 56.21 54.98(0.17) (0.18) (2.74) (3.30)

c � 4 8.43∗∗∗ 8.62∗∗∗ 63.43∗ 54.29 7.68 7.47 56.31 52.61(0.23) (0.21) (3.97) (2.13)

c � 5 8.93∗∗∗ 9.08∗∗∗ 54.29∗∗ 36.01∗∗∗ 8.11 7.89 49.09 50.47(0.10) (0.12) (1.68) (3.13)

Average 8.41∗∗∗ 8.76∗∗∗ 63.43∗∗∗ 54.29 7.66 7.54 54.07 52.65(0.13) (0.14) (2.61) (2.42)

Notes. Standard errors across groups are reported in parentheses. Significance of t tests betweenobserved values and the normative predictions is given in columns 2–5. Significance of t testsbetween full and private information is given in the “Observed Private” columns. All differences aredenoted by asterisks.∗p < 0.10; ∗∗p < 0.05; ∗∗∗p < 0.01.

Table 4. Impact of Small Contract Term Deviations on Supplier’s Share of Profits (c � 5and p � 10)

Full vs. private information Wholesale Price (w) Quantity (q) Supplier share, % Difference, %

Full information, baseline w = 7.68 q = 49.34 31.00 —Lower wholesale price w = 7.54 q = 49.34 26.84 4.16Higher quantity w = 7.54 q = 56.86 18.38 8.46


5.2.1. Agreements and Duration. Retailer-supplier pairscame to a bargaining agreement at a similar rate be-tween full and private information: 90.28% under fulland 93.06% under private (p � 0.525). This result val-idates Lemma 2 and shows that players do not usedisagreement as a tool for separating supplier costtypes. It is also consistent with the first part ofHypothesis 3, which stated that agreement rateswill be less than 100% and similar between full andprivate information.

Although the agreement rates are similar betweenthe two information conditions, we do observe dif-ferences in bargaining duration. Figure 2 shows ahistogram for the bargaining duration (in seconds)between full and private information. It also includesthe average number of offers. First, there is clearly adeadline effect in both conditions—under privateinformation, 37% of agreements are reached in the last10 seconds, whereas under full information, 24% ofagreements are reached in the last 10 seconds. This isconsistent with Roth et al. (1988). Second, one can seethat the bargaining duration is shorter under fullinformation, with an average of 207.2 seconds, versusprivate information,with an average of 241.7 seconds.Although this did not have ameaningful consequencein our experiment, in practice, longer negotiations canincur additional costs. Thus, we support the secondpart of Hypothesis 3 as well.

5.2.2. Opening Offers and Anchoring. Past studieshave shown that first offers can have an anchoringeffect on negotiations (Galinsky andMussweiler 2001)and influence the final agreement that is reached. Ourdata allow us to analyze opening offers and deter-mine if there is a similar effect in our experiments. Inthe interest of brevity, we focus on opening wholesaleprice offers. The results on opening quantity offershave a similar qualitative interpretation. Table 5provides the results. Specifically, it depicts two random

effects regressions with first offers for wholesale pricesas the dependent variable for retailers and suppliers. Inboth models, c � 3 and p � 10 under full informationare the baseline. For the full-information case, it isunsurprising to see that both the suppliers’ and re-tailers’ first offers are increasing in the supplier’s costand the retailer’s price.The main distinction between retailers and sup-

pliers arises when we consider private information.Suppliers’ first wholesale price offers under privateinformation are indistinguishable from those underfull information because the coefficients on the threePrivateInfo terms are insignificant for the supplier. Theimplication is that suppliers do not appear to ex-ploit their private information by proposing a higherwholesale price. In contrast, for retailers, their firstoffer under private information is virtually identicalto the opening offermade to a c � 3 supplier under fullinformation. Specifically, under full information forretailers, the coefficient on c � 4 is 0.371, and for c � 5,it is 0.981. However, under private information, thecoefficients on the interaction terms (−0.471 for c � 4and −0.965 for c � 5) completely wash away theseeffects. This means that under private information,retailers begin bargaining as if they are negotiatingwith the lowest-cost supplier.Building on this analysis, we examine how final

agreed upon contract terms are potentially anchoredon such opening offers. Although the detailed resultsof a series of random effects regressions of agreedcontract terms on opening offers are in Table B.4 inOnline Appendix B, they show that final agreementsare indeed significantly and positively associatedwith each party’s first offer. This helps explain whyhigh-cost suppliers earn a disproportionately smallshare of the supply chain expected profits underprivate information. That is, retailers make openingoffers as if they were matched with a low-cost sup-plier, and final agreements are anchored on open-ing offers.

Figure 2. Observed Frequency of Bargaining Duration

Table 5. The Determinants of Opening Wholesale PriceOffers by Role

Explanatory variables Suppliers Retailers

c � 4 0.511* (0.289) 0.371 (0.236)c � 5 0.538*** (0.093) 0.981*** (0.099)p � 11 0.836*** (0.127) 0.484*** (0.092)p � 12 1.174*** (0.166) 0.681*** (0.191)PrivateInfo 0.192 (0.268) −0.066 (0.304)(c � 4) × (PrivateInfo) −0.095 (0.319) −0.471 (0.297)(c � 5) × (PrivateInfo) 0.225 (0.176) −0.965*** (0.148)Period 0.068*** (0.026) 0.020 (0.039)Constant 7.037*** (0.213) 5.841*** (0.234)

Notes. The dependent variable is the first wholesale price offer, byrole. Standard errors, corrected for clustering at the session level, arein parentheses. Significance is given by asterisks.

*p < 0.10; ***p < 0.01.


5.2.3. Concession Process. Because players negotiateover both the wholesale price and order quantitysimultaneously, we can also evaluate the expectedprofits of each offer to both players and get a betterunderstanding as to how opening offers translate intofinal terms. Figure 3 depicts the supplier’s averageshare of the total supply chain expected profit, byoffer number, for those negotiations in which a playermade at least five offers and an agreement waseventually reached.

We provide plots for c � 3 and c � 5 under full andprivate information. In all four plots, there is a clearconcession pattern over time.What is especially strikingare the retailers’ offers (grey columns) when a supplierhas a higher cost, c � 5. In this case, retailers make

initial offers that would leave the supplier with anegative share of expected total profits. This effectis especially pronounced under private informationwhere the concession process is “flatter” for retaileroffers (Figure 3(d)): even after five offers the retailer isproviding the supplier with a negative share.These bargaining results demonstrate both the im-

portance of first offers and concessions on the bar-gaining outcome. One caveat to anchoring, however,is that aggressive opening offers could also reducethe likelihood that the players are able to reach anagreement. In Table 6, we report the results of arandom effects logit regression where the dependentvariable is an indicator for whether an agreement wasreached and the explanatory variables are the opening

Figure 3. The Concession Process Within a Bargaining Round: Supplier’s Share of Expected Supply Chain Surplus,Conditional on Agreement (c � 3 and c � 5)

Notes. Includes subject-periods in which five or more offers were made. A similar pattern is observed for c � 4.


offers of suppliers and retailers as well as indicatorsfor retailer price and supplier cost parameters. Underfull information, we see that the higher the retailer’sopening wholesale price is (i.e., the less aggressive itis), the higher the likelihood of agreement is. How-ever, under private information, there is no such ef-fect. Thus, under private information, we see thatretailers make more aggressive opening offers (Table 5),which lead to less favorable agreements for supplierswithout increasing the likelihood of disagreement(Table 6). This is consistent with the idea that high-cost suppliers do not blame retailers for being moreaggressive under private information, which ultimatelyleads to less favorable terms.

5.3. Results SummaryIn summary, our experimental results provide sup-port for all three hypotheses. First, consistent withHypothesis 1, all suppliers earn less than 50% of to-tal supply chain profits under private information.Moreover, high-cost suppliers earn a significantlylower share under private information than full in-formation. Second, in line with Hypothesis 2, supplychain efficiency under private information is signifi-cantly below the normative theory for low-cost suppliersand above the normative theory for high-cost suppliers.Third, consistent with Hypothesis 3, agreement ratesare less than 100% and similar between private andfull information, and bargaining duration does in-deed take longer under private information.

Our analyses shed light on the drivers of some ofthese hypothesis test results, particularly Hypothesis1. For instance, as in the full-information case, con-tract terms under private information do not ade-quately adjust for the risk of unsold inventory. Be-cause this risk is more consequential for high-costsuppliers, it is they who disproportionately suffer.Indeed, the bargaining dynamics exacerbate this painfurther because of the conflation between private

information and the anchoring of initial offers on finalagreements. Specifically, because of private infor-mation, retailers make significantly more aggressiveopening offers (as if they were paired with a low-costsupplier). Final agreements are anchored on theseopening offers, and consequently, high-cost suppliersearn lower profits. In what follows, we investigatewhether there is an intervention that can help increaseprofits of high-cost suppliers under private infor-mation. To this end, we explore the issue of com-munication—in particular, providing a supplier withthe ability to send a verifiable or nonverifiable mes-sage regarding its cost.

6. Verifiable Disclosure vs.Nonverifiable Communication

6.1. Normative TheoryWe now modify our setting by adding, before bar-gaining, a stage in which the supplier is able to senda message to the retailer. We consider two variants:(1) “verifiable” information disclosure, in which thesupplier has the option to disclose her true cost, and(2) “nonverifiable” communication, in which the sup-plier has the option to send a message about her costthat may not be true.7

Consider first the case of verifiable disclosure.Given Proposition 3(ii), because all supplier typesbenefit from private information, they would neverdisclose. Doing so would only give up their infor-mation rents with no compensating benefit. Thus,theory predicts no disclosure.Now consider the game with nonverifiable com-

munication, and suppose that supplier c sent a mes-sage that her unit cost is c′. If this is believed, then thesubsequent bargaining outcome must be the Nashbargaining solution under full information givencost c′ and price p—because otherwise, it would re-veal that the supplier had lied about her cost. Usingthewholesale price and quantity derived in Proposition 1,we can compute the supplier’s expected profits fromreporting c′ when her true cost is c:

πs c′|c( ) � 25 p + 3c′ − 4c( )

p − c′( )

p.

Observe that the partial derivative of profits withrespect to reported cost, c′, is

∂πs c′|c( )∂c′

� 50 p − 3c′ + 2c( )

p,

which will be positive so long as c′ ≤(p+2c)/3. Given theparameters of our experiment, this condition is al-ways satisfied. Therefore, all supplier types wouldstrictly prefer to report that they are the highest-costsupplier. Of course, this means that retailers will

Table 6. The Effect of First Offers on Agreements

Explanatory variables Full informationPrivate

information

Retailer wholesale price 0.873*** (0.320) 0.082 (0.210)Retailer quantity −0.005 (0.013) −0.026 (0.022)Supplier wholesale price −0.447 (0.436) −0.748 (0.480)Supplier quantity −0.031 (0.024) 0.020 (0.043)p � 4 −0.762 (1.240) −0.415 (0.774)c � 5 −2.071* (1.187) 1.910 (2.156)p � 11 −1.248 (0.775) −0.318 (1.212)p � 12 0.787 (1.668) −0.209 (1.159)Constant 3.681 (3.854) 9.964* (5.565)

Notes. The dependent variable is an indicator for whether anagreement was reached. Standard errors, corrected for clusteringat the session level, are in parentheses. Significance is given byasterisks.

*p < 0.10; ***p < 0.01.


ignore themessage that they receive, and the outcomeof the game corresponds to the private informationbargaining game without communication. We sum-marize this discussion as follows:

Proposition 4. Given the parameters of our experiment,when bargaining under private information but in whichsuppliers can communicate:

i. Under verifiable information disclosure, no suppliersdisclose their cost, and bargaining proceeds as underprivate information without communication.

ii. Under nonverifiable communication, suppliers havean incentive to inflate their cost. Consequently, retailersignore any messages, and bargaining proceeds as underprivate information without communication.

Before proceeding, we note that both of these set-tings exist in practice. Regarding verifiable infor-mation disclosure, there are a number of examples ofbuyers collaborating with suppliers who are trans-parent about their true costs. For instance, a study onToyota’s best practices with suppliers includes aquote from a supplier reporting that “[w]e are willingto share our cost structure with Toyota” (BostonConsulting Group 2007). Indeed, disclosure of costsgenerally requires opening one’s books and does notnecessarily pose the same challenges as crediblycommunicating more latent aspects of supply chains,such as environmental and working conditions(Bateman and Bonanni 2019). Turning to our non-verifiable variant, it is feasible that a company canstate their costs without providing supporting evi-dence. For further details on how companies ex-change sensitive information in practice, please seeLamming et al. (2005).

6.2. Experimental Design andHypothesis Development

We conducted new experiments that followed thesame procedures as the private information treat-ment, except that after observing their cost but priorto being matched, suppliers were given the option tosend a message to the retailer they will be matchedwith. As with our main experiment, each treatmentincluded 48 participants in groups of 6. In one treat-ment, messages were verifiable, and in the othertreatment, messages were nonverifiable (i.e., sup-pliers could report that their cost was three, four, orfive or not send any message, regardless of their truecost). The only other difference was that these treat-ments included eight rounds (rather than six) becausewe anticipated that it could take time for suppliers tosettle upon an optimal policy.

Moving to hypotheses, Proposition 4 represents thenull hypothesis based on normative theory. Specifi-cally, (a) no suppliers will disclose under verifiable

information disclosure, and (b) suppliers have anincentive to report the highest cost under nonverifi-able communication.However, our original experimentgives reason to be skeptical about these predictionswithhuman decision makers. Because high-cost suppliersare disproportionately hurt under private information,they have an incentive to disclose their cost whenmessages are verifiable. This provides a credible com-mitment to demand a better initial offer, which shouldtranslate into higher earnings. On the other hand,recalling Figure 1, because the lower-cost suppliersare not harmed by private information, we do notexpect them to disclose when messages are verifiable.One might question whether disclosure by the high-cost supplier would trigger a cascade effect in whichall suppliers disclose. In a previous working paper(Davis and Hyndman 2018), which this paper su-persedes, we show that even if the high-cost supplierdiscloses, the remaining supplier types still prefer notto disclose. Thus, we have:

Hypothesis 4 (Disclosure Alternative 1). When suppliers candisclose their costs through verifiable messages, only high-costsuppliers do so, which leads to higher profits for such suppliers.

There are also behavioral reasons for whywemightexpect disclosure to take place for all suppliers in theverifiable variant. Recall that the normative theorypredicts that, under private information, the supplychain is only coordinated for c � 3. Therefore, higher-cost suppliers with social preferences may opt todisclose in order to increase efficiency. To be sure, ifthey played the full-information Nash bargainingsolution, their payoff would suffer, but this would bemore than offset by the retailer’s gain. Past studieshave found that some participants are willing tosacrifice their own payoff (up to a point) if efficiency isenhanced (e.g., Charness and Rabin 2002). Addi-tionally, it is plausible that retailers could view dis-closure as a kind act and reciprocate by giving sup-pliers a greater share of the full-information surplus.Indeed, there is a nondegenerate set of contract pa-rameters that would represent a Pareto improvementover the private information outcome. Lastly, thereis evidence that even nonverifiable communicationcan be efficiency enhancing (e.g., Ozer et al. 2011,Hyndman et al. 2013, Siegenthaler 2017). Therefore,we include the following alternative hypothesis anduse our experimental data to determine which hy-pothesis, if either, is supported:

Hypothesis 5 (Disclosure Alternative 2). When supplierscan send a message to the retailer before bargaining, allsupplier types disclose their cost when verifiable and whennonverifiable, send truthful messages. In doing so, bothefficiency and supplier profits are increased.


6.3. ResultsTables 7 and 8 report supplier messages in bothtreatments. First, under verifiable information dis-closure, c � 3 suppliers rarely disclose their cost, andc � 5 suppliers disclose their cost nearly 75% of thetime. This is consistent with Hypothesis 4. Second,under nonverifiable communication, all supplier costtypes report that their cost is c � 5 a majority of thetime, and in only about 5% of the cases do supplierswith cost c ∈ {3, 4} report their true cost. This isconsistent with the null hypothesis (i.e., normativeprediction). Thus, we reject the first part of Hy-pothesis 5. We also note that, despite our prior beliefthat there may be learning for suppliers, we foundvery little: only under verifiable information disclo-sure did low-cost suppliers learn to disclose signifi-cantly less frequently over the course of the experiment.

We turn now to how suppliers’ profits are affectedby the variousmessages.8 Given that, under verifiableinformation disclosure, disclosure rates are consistentwith Hypothesis 4, we expect that such disclosureshurt the c � 3 supplier and help the c � 5 supplier.Figure 4(a) confirms this prediction (p � 0.017 for c � 3and p < 0.01 for c � 5). This further supports Hy-pothesis 4 and indicates that, in practice, high-costsuppliers should attempt to credibly reveal their costsin an effort to achieve more favorable outcomes. Wealso conducted an additional test (not shown inFigure 4(a)) comparing supplier profits in the origi-nal full-information treatment with participants whodisclosed in the verifiable disclosure treatment. Forno supplier cost types is the difference significant.

In contrast, under nonverifiable communication,given that messages are consistent with the null hy-pothesis, we expect that retailers should ignore them,which should lead to no differences in earnings de-pending on the message. However, as can be seen,when suppliers are truthful it hurts c ∈ {3, 4} suppliersand helps the c � 5 supplier, relative to not sending amessage. When suppliers lie about their cost, there isno significant effect for the c ∈ {3, 4} suppliers and notsurprisingly, a negative effect for the c � 5 supplier.Again, these results lend further support to the nullhypothesiswith respect tononverifiable communication.

Figure 5 shows the effects of supplier messages onthe contract parameters. The results are intuitive; forthe c ∈ {3, 4} suppliers, both disclosing and sending atruthful message lead to a lower wholesale price and

higher order quantity, which explains why they earnless. On the other hand, for the c � 5 supplier, bothdisclosing and sending a truthful message lead to ahigher wholesale price and a lower order quantity,which has a favorable compound effect on earnings.Our results on supplier messages and earnings are

largely supportive of Hypothesis 4. However, thiscould also coexist with the aforementioned behav-ioral factors, which suggest that disclosure andtruthfulmessages couldgrow thepie and allow retailersto benefit as well. However, as Table 9 shows, retailerssignificantly benefit following a message that sup-plier cost is three (whether verifiable or not) andsignificantly suffer following a message that suppliercost is five (whether verifiable or not). These wereprecisely the cases in which suppliers suffered andgained. The punch line is that suppliers’ gains orlosses are offset by nearly equivalent losses or gainsby retailers, rejecting Hypothesis 5. From a mana-gerial standpoint, this indicates that whereas truthfulmessages by lower-cost suppliers are rare in thenonverifiable condition (around 5%), when they dotake place, they are not beneficial to suppliers. In-stead, retailers use the information to squeeze sup-pliers. This result runs counter to related experimentson private information in operations management,such as Ozer et al. (2011), who consider an environ-ment where a buyer has private information about aforecast and find that messages are much more in-formative about the true state than theory pre-dicts, which leads to significantly higher supplychain efficiency.

7. ConclusionIn practice, it is not common for buyers in a B2Bsetting to have perfect information about their sup-plier’s cost. Further, many companies negotiate withone another in a more equitable back-and-forth bar-gaining environment. In this study, we investigateprivate supplier cost information in a two-stage supplychain with dynamic unstructured bargaining. We firstexamine this setting theoretically and find that, givenour experimental parameters, suppliers should benefitfrom their private cost information while bargaining.Despite this theoretical benefit, however, our human-

subjects experiment shows that certain types of suppliers

Table 7. Observed Frequency of Supplier Messages:Verifiable

Supplier type Disclosure frequency

c � 3 9.72c � 4 37.50c � 5 73.61

Table 8. Observed Frequency of Supplier Messages:Nonverifiable

Supplier type

Frequency of message

None c′ � 3 c′ � 4 c′ � 5

c � 3 31.25 6.25 7.81 54.69c � 4 23.44 0.00 4.69 71.88c � 5 23.44 1.56 3.13 71.88


are actually disadvantaged by having private cost in-formation. In particular, high-cost suppliers under pri-vate information earn the lowest share of overall supplychain expected profit. Through our experimental data,we are able to determine that this is driven by a num-ber of behavioral bargaining tendencies by buyers andsuppliers, rather than, say, social preferences. Indeed,fairness concerns cannot be the main driver because ourresult that high-cost suppliers earn less than half thesurplus holds under full information as well, where wewould expect a 50/50 norm to hold for all costs.

Instead, we see that under private information,buyers make opening offers that are similar to theirinitial offers to the lowest-cost supplier under fullinformation and that final agreements are anchoredon these opening offers. As a result, high-cost sup-pliers not only end up with slightly lower wholesaleprices but also, higher quantities, which contributedirectly to a low share of the overall supply chainexpected profits. Lastly, under private information,buyers appear to be able to hide behind a veil of ig-norance: aggressive opening offers by buyers do notincrease the chance of disagreement like they dounder full information.

Our study provides a number of insights for man-agers in practice. First, although some suppliers maybe reluctant to share their private cost informationwith buyers, our work indicates that such an action isbeneficial for certain types of suppliers. In particular,our additional experiments find that high-cost sup-pliers earn significantly higher profits when they cancredibly disclose their private cost information. Sec-ond, private information does not necessarily lead to ahigher rate of disagreement; however, it does lead to

longer negotiation times. In practice, where longernegotiations can require additional expenses, bothparties may benefit from a supplier’s cost being fullinformation in the hopes of coming to an agreementin a timely manner. Both of these recommendationscould foster further collaborative interactions betweenbuyers and suppliers and yield additional benefits.One limitation of our study relates to the timing of

disclosure. Specifically, our results suggest that, for agiven buyer, the high-cost supplier earns more andthe low-cost supplier earns less when they disclosetheir cost. That is, disclosure does not appear to creategoodwill in our setting, which could limit its value.However, if disclosure occurs before the retailer andsupplier are committed to a relationship, then boththe low-cost and high-cost suppliers face a tension:for a given buyer, she will earn less (low-cost sup-plier) or more (high-cost supplier) during bargaining,but by disclosing one’s cost, the supplier reveals herdesirability as a matching partner. Specifically, thelow-cost supplier reveals herself to be desirable,whereas the high-cost supplier is revealed to be un-desirable. Thus, the type of retailer a supplier canmatchwith may change, which affects the overall size of thepie. Davis and Hyndman (2018) study this tensionbetween matching and disclosure in supply chains.Therefore, our results suggest that high-cost suppliersshould delay their disclosure decision until they arebargaining with a buyer. In addition, although webelieve that it is possible for companies to opentheir books and credibly reveal their private costs(e.g., Boston Consulting Group 2007, Jackson andPfitzmann 2007), we admit that this may not be fea-sible for all suppliers.

Figure 4. Observed Supplier Expected Profits and Messages

Notes. The numbers on top of the bars represent p−values of the estimated marginal effects of sending the given message against the baseline ofnot sending any message.


AcknowledgmentsThe authors thank Anyan Qi and Ruthy Beer for fruitfuldiscussions. They also thank seminar participants at Duke

University; the University of Virginia; Stanford University;INCAE Business School; Baruch College; the 2018 and 2019INFORMS Conferences; the 2018 Bargaining: Experiments,Empirics, and Theory Conference; and the 2018 BehavioralOperations Conference for their helpful comments.

Endnotes1When the retailer incurs the cost of unsold inventory, the supplieravoids both inventory and random demand risk.2To be sure, in actual bargaining, players do not actually negotiateover mechanisms, but the underlying assumption is that the un-structured bargaining process provides an indirect mechanism toimplement the bargaining solution.3Because there is only one retailer type, there are no weights appliedto the retailer’s payoff, beyond taking expectations over the type ofsupplier she is matched with. Observe also that we only considerincentive compatibility constraints for the ci type to report truthfully,rather than the next highest cost (i.e., ci+1). This is without lossof generality.

Figure 5. Observed Wholesale Price, Quantity, and Messages, Conditional on Agreement

Notes. The numbers on top are p−values of the marginal effects of the given message against the baseline of not sending any message.

Table 9. The Effects of Messages on Retailer ExpectedProfits

Explanatory variables Verifiable Nonverifiable

Message � 3 37.611*** (10.058) 57.398*** (14.355)Message � 4 11.004 (7.718) 16.305 (20.794)Message � 5 −29.583*** (6.975) −13.773** (6.820)p � 11 21.017*** (6.027) 19.269*** (5.848)p � 12 34.680*** (4.129) 54.326*** (7.777)Constant 101.980*** (6.142) 105.167*** (6.462)

Notes. The dependent variable is the retailer's expected profits.Standard errors, corrected for clustering at the session level, are inparentheses. Significance is given by asterisks.

*p < 0.05; ***p < 0.01.


4We provide the individual point predictions for each pair of c and pin Online Appendix B.5One observation was removed before creating this figure, where aparticipant’s supplier share was −100% of the supply chain expectedprofit, which occurred in the first period. Also, please see OnlineAppendix B for figures and tables that report all results for eachcombination of c and p.6 In full information, this result is a minor robustness check for Davisand Hyndman (2019), who only considered p � 15 and c � 3 andfound that the inventory risk holder consistently earns less than 50%of the total profits.7This differs from the classical “cheap talk” experiments, which testvariants of the Crawford and Sobel (1982) model of strategic infor-mation transmission. Specifically, we provide suppliers with theoption of not sending any message. Beyond this, after any message ispotentially sent, the players engage in a dynamic bargaining processversus a unilateral decision by the receiver of the message in theclassical environment.8Agreement rates are similar to each other and our main experiment:90.74% in verifiable and 91.67% in nonverifiable.

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Private Information and Dynamic Bargaining in Supply ... · Keywords: behavioral operations †supply chains bargaining † private information 1. Introduction Private information