Computers & Industrial Engineering 80 (2015) 12–22

Contents lists available at ScienceDirect

Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

Dynamic supply chain coordination games with repeated bargaining

http://dx.doi.org/10.1016/j.cie.2014.11.0110360-8352/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: xavier.brusset@essca.fr (X. Brusset), per.agrell@uclouvain.be

(P.J. Agrell).

Xavier Brusset a,⇑, Per J. Agrell b

a ESSCA School of Management, 55, Quai Alphonse Le Gallo, 92513 Boulogne Billancourt cedex, Franceb Louvain School of Management and CORE, Université catholique de Louvain, B-1348 Louvain la Neuve, Belgium

a r t i c l e i n f o

Article history:Received 9 May 2014Received in revised form 29 September 2014Accepted 14 November 2014Available online 26 November 2014

Keywords:Relationship-specific assetsContract designAsymmetric informationRent captureBayesian belief

a b s t r a c t

Coordination in a supply chains may require investment in relationship-specific assets (RSA) includinginformation systems and human resources from all or a subset of the partners. These investments aretypically partially non-verifiable, possibly based on internal resources or opportunity costs. A supplieroffers a single-price single-period contract to a downstream manufacturer who accepts or turns to anon-strategic outside option. Both parties invest in relationship-specific assets (RSA) accordingly. Usinga game theoretic framework of repeated single-period bargaining under asymmetric information andoutside options, we show how a supplier may behave opportunistically. We show how this rent extrac-tion threat is mitigated when the manufacturer mis-informs the supplier or hides information from her.As a result of both behaviors, our model explains how supply chain coordination and efficiency areimpaired. On a normative basis, we provide the manufacturer with new justifications for both dual sourc-ing and distorting information. Numerical examples illustrate the results.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

A supply chain is a network of connected and interdependentorganizations mutually and cooperatively working together to con-trol, manage and improve the flow of materials and informationfrom suppliers to end users (Aitken, 1998). There is consensus thatsupply chain optimization involves emphasis on intra-functionaland inter-organizational collaboration, leading to coordination ofprocesses, orders and information in areas such as customer ser-vice, production planning, logistics, and capacity utilization. Coor-dination, in particular, has been a research focus (CSC, 2009). RSAhas received intense attention in inter-firm relationship research,and has become an important subject in both marketing channel(Kang, Mahoney, & Tan, 2009) and supply chain management.Asset specificity is mainly researched within the framework oftransaction cost economics (Williamson, 1985) and relationalexchange theory in which it signals the desire to invest in anendured relationship (Anderson & Weitz, 1992). Most of thatresearch considers that the RSA are the object of an agreement oreven a contract between two successive partners in a supply chainwith the purpose of increasing the performance of that supplychain and protecting the investor from ex-post opportunisticbehavior. This paper does not consider the contracts which can

be set up to coordinate trading between the seller and the buyer.We cover the preliminary evaluation that both must conduct inorder to maximize their projected interaction, even before a coor-dinating contract for their operations can be considered. We areinterested here in the ‘selfish’ investment by the downstream part-ner which is relationship-specific and primarily enhances theinvestor’s performance.

It is observed frequently in buyer–supplier relationships of verydifferent types of goods and services that RSA are deployed by buy-ers in absence of agreement or even of knowledge of suppliers.Knemeyer, Corsi, and Murphy (2003) has surveyed the outsourcingpractice of logistic services and shown that it involves investmentsin specific assets and non-retrievable commitments of resources onthe part of outsourcing companies. Sucky (2007) point to the trendtowards outsourcing logistic activities as support to the argumentthat large firms are focusing their activities on their perceived corecompetencies. A classical case is that of steel-makers or electricity-generating utilities using sulphurous or other low-value coals. Forcoke-making, coal blends are required to have specified ranges ofvalues for volatile matter, ash and sulfur content (Adeleke &Onumanyi, 2007). The blend, which lowers the purchasing cost,is processed in special equipment which scrubs the sulfur content.The RSA consist in this specialized equipment. Ball and Loncar(1991) modeled the demand elasticity for Australian coal to pricefluctuations in the medium term taking into account the qualityof the coal, overall economic activity and the price of oil (a substi-tute for coal in thermal energy generation). The conclusion was

X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22 13

that demand was relatively inelastic to price increases in the shortbut not in the medium or long run which is consistent with thetime required by customers to redeploy their RSA to adapt theircleaning processes to accommodate coal from other suppliers.

Supply chain coordination is a process built on gradually fos-tered trust in combination with adequately designed, financed,deployed and monitored collaboration and information-sharinginstruments. Partners invest time, capital and human resourcesinto adjusting their operations to their supply chain partners’ cor-responding processes eg, by changing and coordinating productand packaging dimensions, information technology (IT) communi-cation protocols, bar codes or radio identification codes, productcatalogues, product development platforms, production planning(schedule, detail and systems). However, Kampstra, Ashayeri, andGattorna (2006) notice that progress towards deeper collaborationand coordination with upstream suppliers and downstream cus-tomers is slow and frequently disappointing in practice. Theauthors cite, among other reasons for failure, the lack of trust, fearof external competition, missing infrastructure and financial barri-ers for the sharing of resources and gains. In particular the infra-structure standard investments that Kampstra et al. (2006)identify as lagging or missing are relationship-specific eg, theadjustment to a given customer’s IT standards has little or no valueoutside of that supply chain.

We identify three essential characteristics of these investmentsthat influence the ability of the supply chain to achieve coordina-tion. First, the specificity of the investment gives rational reasonto fear hold-up (i.e, post-contractual opportunistic action) fromthe supplier. A recent example is the situation in which Ryanairhas found itself when Boeing, sole supplier of planes and technicalassistance to the company, decided to increase the price of long-term service contracts (O’Doherty, 2009). In June 2011, partly toescape from this situation, Ryanair signed a cooperation agreementwith the Chinese aircraft manufacturer, Comac (Odell, 2011). Weare interested in the investment incentives: what matters is thepotential exposure to a hold-up.

Second, investments may not be verifiable by the supply chainpartner. Moreover, even if the investment is verifiable and welldefined, the potential cost sharing among several supply chainaffiliations make ‘‘open book’’ procedures ineffective in allocatingcosts and returns.1

Third, the coordination investments are empirically subject tocontinuous and repeated financial negotiations within the supplychain, often over several product or contract generations (Cf.Kampstra et al., 2006). Hence the truthful revelation of RSA costsmay not occur.

In this paper, we address the question of why supply chains failto coordinate relationship-specific investments by a stylizeddynamic dyadic model of a supply chain. Whereas the literaturehas suggested a range of remedies to the hold-up problem, suchas in Hart and Moore (1990) and references below, most workaddress the problem from one or two of these perspectives. Ourcontribution, based upon a game theoretic framework, is bothpositive and normative. From a positive viewpoint, our modelexplains the delays in supply chain coordination and its elusive-ness as well as distortions in rent creation and attribution. The nor-mative contribution takes two different viewpoints. From adecision-making stance, two contributions are presented: (a) theBayesian updating mechanism proposed for the coordinator maybe used under more general settings to inform sequential biddingprocedures; (b) dual-sourcing is provided with additional justifica-tion on opportunistic behavior grounds to protect against holdup

1 A manufacturer invests in a vertical silo to store a liquid compound which onesupplier makes. However, once the silo is there, how is its cost spread among severalsuppliers who provide equivalent products which can be stored in it?

within an ongoing relationship.2 From a behavioral stance, our con-tribution provides a theoretical argument justifying the manufac-turer’s secrecy or communicating biased information about his RSAcosts, independently from his bargaining power.

We first set up a full information centralized benchmark; com-mon information is not only the reference point for efficiency esti-mations, but may also exist in vertically integrated organizations (aproduction division and a distribution organization). We theninvestigate the case where information about RSA cost is privateto the manufacturer (he); the supplier (she) only has some priorbelief about the cost.

In the following section, we give some elements of related liter-ature on the subject. We present in Section 3.3 the full informationcase and Section 3.4 covers the case where the supplier is unawareof the investment costs that the manufacturer faces. A numericalinstance positions the different tradeoffs in Section 4. We concludein Section 5.

2. Literature review

The holdup problem under incomplete contracting and asym-metric information has attracted considerable academic attentionin economics, marketing and supply chain management (Cachon& Netessine, 2004, chap. 2). The properties of hold-up, asymmetryof information, renegotiation, incompleteness of contracts, switch-ing costs (Klemperer, 1987) and lock-ins have all been investigated(Garcia Mariñoso, 2001). In González (2004), the agent faces ahold-up situation while making a cost-reducing specific invest-ment unobservable by the principal. To escape the hold-up, theagent randomizes the investment whereas the principal offersscreening contracts. The models explored as presented in the mar-keting literature (Farrell & Klemperer, 2007, chap. 31) are oftenrestricted to full-information, two-period settings with endoge-nous downstream prices for various market organizations. Withinsupply chain management, several models explore the influence ofa supplier’s offers on the buyer’s decision (Sucky, 2004; Sucky &June, 2006; Esmaeili, Aryanezhad, & Zeephongsekul, 2009);othersexplore how the supplier can tailor his offers to obtain informationprivate to the buyer (Li, Ritchken, & Wang, 2009). To address theissue of channel inefficiency, a typical approach is to design incen-tive contracts to provide the downstream partner with flexibility toadapt to volatile demand (Cvsa & Gilbert, 2002; Barnes-Schuster,Bassok, & Anupindi, 2002; Wang & Liu, 2007; Zhao, Wang, Cheng,Yang, & Huang, 2010; Zhao, Ma, Xie, & Cheng, 2013). With thisincentive, the whole supply chain gains without compromisingany of the supply chain members’ profits. In the case where thedownstream partner or partners is (are) endowed with operatingcosts and the possibility to invest, Cho and Gerchak (2005),Plambeck and Taylor (2007) provide several coordination mecha-nisms for the decentralized chain.

Bargaining Theory is a branch of Game Theory that deals withthe bargaining situations between two parties (Wu, 2004, chap.3). In particular, if the bargaining game is single shot, one maycharacterize its Nash equilibria. Note that in the above literature,renegotiation does not take place within the model.

Segal and Whinston (2002) provide a survey of mechanismdesign with renegotiation in settings like the current one, i.e., withhold-up risk and asymmetric information on ‘‘selfish investments’’.In Tirole (1986), the seller obtains an information rent whereas inour model the buyer does not disclose the investment cost to theseller so as to mitigate the hold-up risk in future periods. Hou

2 We wish to distinguish here the case of ongoing transactions between a buyerand supplier who know each other from the case where the dual-sourcing is decidedbecause the buyer incurs a high risk of receiving a ‘‘bad offer’’ or bad service from aunique supplier.

14 X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22

and Zeng (2011), Sucky and June (2006), Abreu and Pearce (2007),Dang (2008) model bargaining and information asymmetry in sin-gle or multi-period settings. In most of those papers, the mecha-nism design addresses problems which appear during theoperations in a trading relationship. The present paper deals withthe investments in relationship specific assets that are requiredbefore the trading begins.

Tirole (1986) and Edlin and Reichelstein (1996) deal withinvestment in cost reduction by the seller which results in anadvantage over the uninformed buyer. In the present model, it isthe buyer who realizes a selfish investment and the seller who isnot informed of the investment cost (‘‘selfish investments’’ as inHart & Moore, 1999). The multi-period case is addressed in Segaland Whinston (2000) where the seller invests in RSA and usesrenegotiable exclusivity contracts with the buyer being able tobuy from a third party. In Battaglini (2007), the model character-izes the optimal renegotiation-proof contract in a dynamic princi-pal-agent model in which the agent can change typesstochastically over time by means of contract menus offered bythe principal.

We consider that the supplier is a Bayesian rational player ableto sequentially update her belief of the manufacturer’s cost andoutside option using responses to past offers (called Bayesianupdating with cutoff). The updating of beliefs using cutoffs hasbeen studied in Hart and Tirole (1988) but that mechanism cannotbe applied ex abrupto here because of the difference in the buyer’sand seller’s motivations and utilities.

The current model draws on the static model in Brusset (2009b,chap. 2), It builds upon the single-period pricing problem when theseller does not know the willingness-to-pay of the buyer in Brusset(2014). The present setting resembles the sequential bargaining orrenegotiation of rental price with one buyer under asymmetricinformation about his willingness-to-pay seen in Fudenberg andTirole (1983) which characterizes the set of equilibria of two-per-iod bargaining games when the seller and buyer each have twopotential types (two-sided incomplete information). The single-buyer interpretation when the buyer is willing to trade and profit-able mutual interaction is given has been looked into byFudenberg, Levine, and Tirole (1985) which demonstrate that aperfect Bayesian equilibrium exists when the buyer’s type followsa smooth bounded density. Other models describe the rent sharinginduced by bargaining but do not investigate the possibility of theseller holding up the buyer.

4 The possibility of both resolving their needs through another counterparty ortaking recourse in a spot market which entails introducing a second random variable

3. Model

Two players, a manufacturer (he) and a supplier (she) mayengage in mutually beneficial trade over a finite horizon i periods,i 2 f1; . . . ;ng. In any period, both can trade with the other or with anon-strategic outside option. In supply chains represented usinggame theory several suppliers (or retailers) will interact with adownstream (upstream) player either as cooperating or non-coop-erating players (Benz, Jäger, & van Rooij, 2005; Gibbons, 1992; Leng& Parlar, 2005; Leng & Zhu, 2009). In our setting, the non-strategicplayer does not have the liberty to intervene in the game played bythe other two.

If the manufacturer trades with the supplier, he invests A(strictly positive, irreversible), the cost of some RSA which maxi-mizes his benefit of trading with the supplier. He invests a (non-negative, irreversible) if choosing to trade with his non-strategicoutside option, cost of the corresponding RSA. Symmetrically, thesupplier invests B, the cost of some RSA, if trading with the manu-

3 Without loss of generality, the third parties for the supplier and the manufacturerare assumed to be non-strategic players.

facturer or b if trading with an outside option.3 The effect of therespective investments is lasting, such that trade is enabled in anysuccessive period following the corresponding investment. Note thatthis setting means that the successive periods are interrelated andnot exchangeable. We assume that the value of trade is alwayshigher than the cost. The manufacturer minimizes his expected costC of trading either with the supplier or the outside option.

Typically, one can assume that the outside option for both par-ties is to turn for their needs to some marketplace or previous trad-ing partner for which the RSA might be considered to be either nullbecause sunk or insignificant (meaning that a ¼ 0 and, or b ¼ 0).4

In this case, the whole problem revolves around the values of Aand B for supplier and manufacturer.

The supplier’s objective is to maximize her partial profit func-tion Ps.

In time (see Fig. 1), the supplier, as Stackelberg leader, offers acontract. If the manufacturer accepts, both invest in the requiredspecific assets A and B, unless not done in any prior period. Serviceis performed and payout takes place.5 If the manufacturer rejectsthe offer, both turn to outside options: incurring the correspondingenabling investments a and b, respectively, unless already sunk (thisoption is not represented in Fig. 1).

At each period i, the supplier is offering a single-period take-it-or-leave-it price contract Ui with i 2 f1; . . . ;ng. The strictly positivecontract which the manufacturer may sign with some third party islabeled u and, without loss of generality, shall be considered to betime invariant. The supplier can also sign a strictly positive time-invariant contract v with a third party as outside option. We calldi

m 2 f0;1g the manufacturer’s participation decision variable inperiod i (notation in Table 1).

The investments made by the manufacturer are unobservableby the supplier. We first present in Section 3.2 the coordinatedbenchmark case where decisions and information are centralized.We then study two information scenarios. In Section 3.3, the sup-plier is informed of the cost to the manufacturer of the investmenthe realizes. In Section 3.4, the supplier does not possess this infor-mation and so relies on her beliefs.

3.1. Updating process for the supplier

We assume that the supplier is a Bayesian rationalist whobuilds her assumptions from her experience at the start of the firstperiod. Using these priors, she calculates her most profitable esti-mate of the parameters involved and makes the correspondingoffer to the manufacturer. Unless the manufacturer agrees to thefirst contract offered, in the following period the supplier mustupdate her belief about these unknowns in a dynamic updating pro-cess (Selten, 1975). This belief is characterized by a continuouscumulative distribution function (cdf) of the probability withincreasing failure rates (IFR) as defined in Barlow and Proschan(1965, chap. 1) and a range of possible values for the givenunknown. The supplier updates her beliefs using the manufac-turer’s refusal in what is called a ‘‘Bayesian updating with cutoff’’(Hart & Tirole, 1988).

We assume that the supplier, as Stackelberg leader, makes all theoffers and the manufacturer simply accepts or declines, i.e., we giveall the bargaining power to the buyer. This assumption is consistentwith conventional literature. Second, we want to abstract from theissues of renegotiation of a contract by an informed party.

to model price and availability has been dealt with in Brusset (2009a). We havechosen not to include this feature in the present model so as to reduce its complexity.

5 Since there is no new information revealed, compliance conditions are irrelevanthere.

Fig. 1. Timeline of events when manufacturer and supplier first agree on a contract and to a new relationship. If the manufacturer does not agree to the contract offered, tradewith third parties occur (not represented).

Table 1Table of notations.

Definition

Manufacturer A Specific asset investment with the supplierA� Optimal estimate of A by the suppliera Specific asset investment with outside optiona� Optimal estimate of a by the supplieru Contract available from third party

Ui Contract offered in period i by supplier to manufacturer

Cið:Þ Manufacturer’s cost function in period i

dim

Binary decision variable,

1 when agreeing with supplierv i

mManufac. trading strategy vectors in state i

Rim

Manufac. trading strategy set in state i

Supplier B Specific asset investment with the manufacturerb Specific investment in outside optionv Outside option contract available

Pisð:Þ Supplier’s profit function in period i

Supplier beliefs f AðAÞ pdf of belief of AFAðAÞ cdf of belief of Af aðaÞ pdf of supplier’s belief of aFaðaÞ cdf of supplier’s belief of a

f AiðAiÞ pdf of revised belief of A in period i

FAiðAiÞ cdf of revised belief of A in period i

Li FAiðAiÞððn� iþ 1Þu� AiÞþFAiðAiÞðv þ Liþ1Þ

v is

Supplier strategy vectors in state i

Ris

Supplier strategy set in state i

X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22 15

By analogy to Hart and Tirole (1988) in its sales model undernon-commitment, the supplier’s strategy is a sequence of contractprice offers U1; . . . ;Un. The manufacturer’s strategy consists inaccepting or rejecting the supplier’s offer, resulting in a series ofdecision dm; . . . ; di

m. The corresponding strategies and beliefs, bothfor the supplier and manufacturer, form a Perfect Bayesian equilib-rium (PBE): strategies are sequentially optimal given beliefs (per-fection) and beliefs are derived from equilibrium strategies andobserved actions using Bayes’ rule whenever possible (Bayesianupdating).

3.2. Coordinated benchmark

Consider two firms under common ownership. The per-periodbenefit of service is positive. The problem is a cost minimizationone. Denote the joint cost function when switching to a third partytrade for k � 0 periods by VðkÞ.

VðkÞ ¼ ðAþ BÞminf1;n� kg þ ðaþ bÞminf1; kg þ kðu� vÞ: ð1Þ

The integrated firm has two options: (i) performing the servicesinternally by the supplier, or (ii) relying on external service provi-sion from period 1. Naturally, the option of changing regime hasno value to the integrated firm as the external prices are stationary.

Trade together Vð0Þ ¼ Aþ B ¼ V�

Trade separately VðnÞ ¼ aþ bþ nðu� vÞ

�ð2Þ

By inspection, the shifting policy dominates iff aþ bþ nðu� vÞ <Aþ B.

We limit our interest to the ‘‘mutual beneficial trade’’ case,defined by the following two conditions:

nU þ A � nuþ a;nU � B � nv � b;

�

which lead to the following inequalities

16 X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22

uþ a�An � v þ B�b

n : Condition 1;u � v; Condition 2:

ð3Þ

The first condition provides a basis for evaluation of any switchingpolicy, say for k periods. The coordination loss then equals:VðkÞ � V� ¼ aþ bþ kðu� vÞ. If the second condition is violated,there is no marginal room for beneficial trade. Using Condition 2from (3), we can then state that the marginal cost of delay is posi-tive and that there are two sources of coordination losses: over-investment in relationship assets aþ b and inferior trade conditionsu� v .

3.3. Full information

In the full information case, the supplier presents offers usingfull information about the RSA costs A; a and u.

We use the durable-good deterministic branching process asdefined in (Hart & Tirole, 1988): the supplier’s message space ateach date is empty and the manufacturer’s message at date i is achoice between 0 (rejects) and 1 (accepts): the manufacturer deci-des sequentially on whether to accept or not. The total transferfrom manufacturer to supplier depends only upon the accept-reject path.

The game is composed of three states depending on the manu-facturer’s investment(s): (S) exclusive investment A for the sup-plier, (ST) investments for both the supplier and the third party,(T) investment a only in third party trade. Table 2 presents thetwo-period case according to the decisions in the first and secondperiod by the manufacturer. The manufacturer’s optimal contractfor each state are presented in Section A.1 of the Appendix.

We present the result of the common information game as aProposition.

Proposition 1 (Full information). Given mutual interaction potentialfor manufacturer and supplier given in (3), the following optimalcontract is a (weak) Nash Equilibrium (NE)

U1 ¼ u� A;Ui ¼ u; 2 � i < n;

Un ¼ uþ a:

8><>: ð4Þ

Because this equilibrium is weak, the supplier (but not the manufac-turer) is exposed to considerable risk for costly mis-coordination bythe manufacturer. This can be reworded in the following remark.

Remark 1. The NE in Proposition 1 is neither trembling handperfect nor a Pareto equilibrium.

The proofs are provided in Section A.2 of the Appendix.The remark has practical implications: when the manufacturer

changes to his outside option, the cost is the same to him but notthe profit to the supplier. We note (although not developed here insupplier-driven bargaining) that this condition naturally would beexploited by the manufacturer in an open bargaining process.

Table 2The manufacturer’s and supplier’s investment in RSA according to the former’sdecisions in first (horizontal) and second (vertical) period.

d1m

0 1

d2m

0 fa; bg fAþ a;Bþ bg(T) (ST)

1 faþ A; bþ Bg fA;Bg(ST) (S)

We now turn our attention to the case of asymmetricinformation.

3.4. Asymmetric information and renegotiation

We assume that the RSA costs A and a are now the manufac-turer’s private information. To make offers, the supplier relies ona priori information on the distribution for A as supported by½A;A�, following a cumulative density distribution FA, and for a assupported by ½a; a� and following a cumulative density distributionFa. We assume without loss of generality that 0 � A < A and0 � a < a and that these distributions have increasing failure rates(IFR). A property of the IFR density functions is that the setfxjf AðxÞ > 0g is connected. Further, 8x 2 ½A;A�; FAðxÞ < 1 but canbe arbitrarily close to 1. The distributions which enjoy this prop-erty are numerous and include such familiar ones as the normaland uniform distributions (Barlow & Proschan, 1975; Barlow,2003).

3.4.1. Updating mechanismThe supplier updates her belief about A and a, using the infor-

mation conveyed by the rejection of the offers in previous periods.Following a rejection, the supplier makes an estimate of the distri-bution: mean, range upper and lower limit and standard deviation.The distribution is truncated at a lower limit resulting from her apriori belief. The lower bound of the distribution is, in the follow-ing, named the seed of the calculations. For the offer of the upcom-ing period, the supplier makes an estimate of A starting with thelast period (i.e., n). That is, An� is estimated first. The quantities An

and Ln are evaluated using (25), (27) and the correspondinglyseeded distribution of belief. By bootstrapping, knowing An andLn, she then calculates all previous values of Ak and Lk for1 � k � n� 1 up to the initial estimate A1� for A using the proce-dure described in Section A.3.4. This estimate forms the basis forthe offer to the manufacturer in the first period. The manufactureraccepts or rejects. The supplier can now use this information toupdate her belief by increasing An to A1�, modify the estimated dis-tribution of An before repeating the bootstrapping exercise in thesecond period.

We spell out a new proposition applicable to the present sce-nario (proof in Section A.3 in the Appendix).

Proposition 2 (Asymmetric information). Given asymmetric infor-mation about investment specific costs A and a and mutual interactionpotential defined in Condition 2 of (3), the contracts offered are

U1 ¼ u� A1�;

Ui ¼ u; 1 < i < n;Un ¼ uþ a�:

8><>: ð5Þ

If A1�< A, the supplier offers in subsequent periods i contracts such that

Ui ¼ u� Ai�; i � n;

Ut ¼ u; i < t � n; iff dim ¼ 1;

(ð6Þ

with

Ai� þ FAiðAi�Þf AiðAi�Þ

¼ ðn� iþ 1Þu� v � Liþ1;

Li ¼ FAiðAiÞððn� iþ 1Þu� AiÞþFAiðAiÞðv þ Liþ1Þ;1 � i � n� j� 1Ln ¼ FAnðAnÞððn� 1Þu� AnÞþFAnðAnÞððn� 1ÞvÞAi�

> Aði�1Þ�; ðICÞ

8>>>>>>>><>>>>>>>>:where FAið:Þ represents 1� FAið:Þ.

Table 3First period belief (seed), period of manufacturer acceptance, cut-off value, mean,standard deviation and estimate of the investment cost A for that period.

Seed 1st Ai li ri Ai

10 4 46.7 77.1 41.2 103.720 4 47.5 97.9 46.7 89.830 3 36.6 94. 35.6 52.840 3 45.8 113.7 39.1 57.850 1 50. 80.7 10.2 50.460 1 60. 90.9 10.3 60.470 1 70. 101. 10.3 70.480 1 80. 111.2 10.4 80.490 1 90. 121.3 10.4 90.4100 1 100. 131.5 10.5 100.4110 1 110. 141.6 10.5 110.4120 1 120. 151.7 10.6 120.4

X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22 17

The rent which accrues to the supply chain composed of thesupplier and the manufacturer over the n periods is affected asper the following third proposition (proof in Section A.4 in theAppendix).

Proposition 3. Given asymmetric information about investmentspecific costs A and a and mutual interaction potential over the nperiods, there is a risk that interaction between the supplier andmanufacturer will not start in the first period, reducing the supplychain rent. There is also a risk of over-investment equal to aþ b byboth supplier and manufacturer in the event that the manufacturerdecides to fall back on his outside option in any period.

This proposition leads to the following two corollaries.

Corollary 1. Under asymmetric information, the supply chain jointcost function is increasing in the number of periods during which thesupplier under-estimates A. In the case of asymmetric information,two cases present themselves: either trade takes place in the firstperiod and we have

d1m ¼ 1;

and so

Cn ¼ nuþ A� A� þ a�;

Pn ¼ nu� A� þ a� � B;

�ð7Þ

with

Vð0Þ ¼ Aþ B;

which means that the joint cost function yields the same cost as in thefull information case. But, if joint trade starts in period k ¼ jþ 1, where1 < j � n� 1 then

d1m ¼ 0; dk

m ¼ 1;

Cn ¼ nuþ A� A� þ a

Pn ¼ jv þ ðn� jÞu� A� � B

�

VðjÞ ¼ Aþ Bþ aþ bþ jðu� vÞ: ð8Þ

The joint cost to the supply chain suffers a sudden jump when d1m ¼ 0

and then increases further for each period in which the imprecisionof the information makes the supplier under-estimate the cost A. Notethat the joint supply chain cost is not affected by the exact value of theestimate A�.

6 When a renown specialty minerals manufacturer invests in complementaryinformation systems so that it can work seamlessly with a software as a service web-based logistics service provider, he will disguise the exact cost or extent of the workdone to the supplier. This allows the manufacturer to be in a more advantageousposition when negotiating the necessary adjustment effort that the supplier has tomake to his own information system.

Corollary 2. The manufacturer’s rent from the information about Aand a which the supplier has to estimate is proportional to the preci-sion of the information available to the latter. This information rent isthe difference between the full information and asymmetric informa-tion scenarios and can be expressed as

R ¼ A� � A: ð9Þ

This rent increases with the supplier’s over-estimation of A. It disap-pears when the supplier under-estimates A because the manufac-turer turns to his outside option: there is no interaction with thesupplier.

Under opportunistic behavior on the manufacturer’s part, thisrent is not null and increases with his ability to influence the apriori beliefs of the supplier. This type of behavior is noticeablein negotiations between buyers and suppliers in very differentbusiness to business settings. In the least, the true costs will notbe revealed so that, when renegotiation occurs, renewed opportu-nistic behavior can yield information rents to the downstreampartner. It is the authors’ experience that even after workingtogether many years, this type of information withholding

behavior over coordinating investments will still take place.6 Thistype of behavior has not been addressed in the modeling literatureand does not seat easily with the truthfulness and collaborationparadigms.

4. Numerical illustration

A manufacturer and a supplier interact over n ¼ 20 periods.Let

A ¼ 50; u ¼ 100; a ¼ 60;v ¼ 80; b ¼ 55: ð10Þ

4.1. Scenario of full information

In the case of common information the offers are straightfor-ward and, according to Proposition 1, yield the following multi-period contract

U1 ¼ 100� 50;Ui ¼ 100; 8i;2 � i < 20;U20 ¼ 100þ 60:

8><>: ð11Þ

The manufacturer pays a total of 2060 over 20 periods (includingA ¼ 50), exactly the same cost than his outside option (2060). Thesupplier earns a larger profit than with her outside option (1545).The chain has no coordination loss.

4.2. Scenario of asymmetric information

We use the updating procedure described in Section 3.4.1adapted to the present numerical illustration. Initially, we haveto determine the last period L and Aj� as well as a�. To do so,the seed is set at 10 and the distribution function FAðn�jÞ followsa truncated normal distribution with lower bound Aj ¼ 10(the seed), ln�j ¼ seedþ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiseedþ 100p

and rn�j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiseedþ 100p

.If the ensuing estimation of A leads to a refusal by the manufac-turer, in following periods, these a priori beliefs are updated inthe following way:

Fig. 2. Evolution of the supplier’s profit (bottom thick line) and manufacturer’s cost according to the supplier’s initial belief. The dashed lines represent the profit and costwhen the initial estimate of a stays constant.

18 X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22

Aiþ1 ¼ Ai�

liþ1 ¼ li þ Ai

2ð12Þ

riþ1 ¼ ri þ 2:2ffiffiffiffiffiAi

p:

7 When the author wished to sell remote sensing equipment to a water distributionutility, much was made of the cost the telecommunication link to relay theinformation back from the field.

The distribution chosen here is the truncated normal distribution,bounded from below by the lower limit for the range of Ai.

To illustrate the sensitivity with respect to the initial belief, wehave replicated the whole exercise for a range of initial seeds(results in Table 3).

As can be seen, the manufacturer only accepts in the first periodwhen the seed is higher than 50. In the first four rows, the manu-facturer rejects the first period offer and invests a, thus deprivingthe supplier of the ability to extract a� in the last period.

In Fig. 2, we can observe how the supply chain rent constitutedby the difference between the supplier’s profit and the manufac-turer’s cost evolves according to the seed. The impact of Proposi-tion 3 can clearly be seen when A20 � 50. As this seed increasesthe overall rent increases quickly before reaching a plateau. Notethat the first contract offered when A20 � 100 is a payment bythe supplier to the manufacturer!

When the estimate of a is constant whereas the seed for Achanges, we see (dashed lines in Fig. 2) that the supplier is ableto increase substantially her return even though the cost is almostthe same for the manufacturer. Note however that the supplier’sprofit decreases as the overestimation of the cost A before the firstperiod increases.

The attentive reader will note that the manufacturer stands toreduce his cost by exaggerating the cost of A and even more so ifhe also exaggerates the cost of a to the supplier as this leads herto overestimate one or both (dashed line in Fig. 2). However, thecorrect evaluation or slight over-evaluation by the supplier coordi-nates better the supply chain. The supplier’s profit is maximizedwhen she slightly overestimates the manufacturer’s investmentcosts A and underestimates a in the first period. She should notconsider the manufacturer’s information about A or a as trustwor-thy. Both attitudes can be observed in practice. The risk of over-investment and potential for supply chain inefficiency are clearlyapparent when the supplier underestimates the manufacturer’sinvestment cost (left part in Fig. 2).

5. Conclusion

In this paper, we have presented a model of a manufacturer anda supplier when they have yet to invest in the RSA required to workwith each other and the manufacturer has private informationabout the cost of such investment. We characterize the NE whenthe supplier’s Bayesian belief about the manufacturer’s investmentcosts follow distributions which exhibit Increasing Failure Rates.Under full information, the ensuing NE are weak and not tremblinghand perfect. Under asymmetric information, the NE is tremblinghand perfect and the manufacturer’s best course is to agree withthe supplier’s offers whenever it is less onerous.

We show that the supplier initially has to sweeten her offer inorder to later hold up the manufacturer by the cost of RSA with athird party when engaging with the manufacturer in a multi-per-iod relationship. This hold-up is only partly mitigated when themanufacturer withholds the RSA cost information. We show how,to protect himself, the manufacturer invests in two different rela-tionship-specific assets, thus impairing supply chain efficiency.This provides another motivation for dual-sourcing as a form ofprotection against opportunistic behavior.

The supplier’s and manufacturer’s strategies induce a temporallink among periods: she is motivated to renew the relationship soas to hold up the manufacturer in the last period. The manufactureris also motivated to stay with the incumbent by the cost of invest-ing in another relationship.

The numerical illustration shows how the manufacturer’s costis reduced when the supplier over-estimates his RSA costs bothfor their relationship and the relationship with the outside option.This effect can be related to common anecdotic observations ofcustomers over-declaring the effort deployed to establish the rela-tionship with their suppliers.7 The manufacturer has an incentive tosignal a high RSA cost. The supplier downplays such a message tomaximize her profit. Both behaviors, observed in practice, have notbeen documented in literature. On the other hand, the supply chainefficiency is increased if she obtains precise information about themanufacturer’s RSA costs.

Table 4Outcome for the supplier when the manufacturer applies trembling-hand strategies.

Outcome State

Suppliernu + a � A � B State 1ðk� 1Þu� A� B� bþ ðn� kÞmaxðu;vÞ State 2ðk� 1Þv � bþ ðn� kþ 1Þu� A State 3

X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22 19

Appendix A

A.1. Contracts offered under full information scenario

We present the evaluations for the offers made by the supplieraccording to the state the manufacturer finds himself in under fullinformation (from Section 3.3).

STATE S:Manufacturer has made a single investment, A in period one.Under strict rationality, in period n the supplier’s dominant

offer must thus be: Un ¼ uþ a. In the last period n, the manufac-turer’s cost becomes Cn ¼minðUn; uþ aÞ. In period n, the suppliercan hold the manufacturer up by asking for uþ a without the latterdefecting. Can the supplier repeat this hold up in the previous per-iod n� 1? She could, but if she did the manufacturer now woulddefect and turn to the other supplier by investing a and payingu. If the manufacturer does not defect in period n� 1, he facesthe same hold up in the last period. This strategy is clearly notoptimal to him since he would have a cost of 2uþ 2a. Hencethe manufacturers cost for the 2 periods n and n� 1 is:Cn�1 þ Cn ¼minðUn�1 þ uþ a;2uþ aÞ, bounding Un�1 � u formutual trade.

It follows that in any period j; 1 < j < n, the manufacturer hasto solve Cj ¼minðUj þ ðn� jÞuþ a; ðn� jþ 1Þuþ aÞ. So, to ensureincentive compatibility, the supplier is limited to offeringUj � u; 8j; 2 � j < n.

In state S, the overall minimized cost function becomesC ¼ nuþ a. The supplier’s maximal profit is Ps ¼ nuþ a� A� B.

The supplier’s optimal strategy under Condition 2 in (3) consistof the profit maximizing contracts as an n-sized vectorvS

s ¼ ððu� AÞ;u; . . . ;u;uþ aÞ. Let us call the manufacturer’s strategywhen he is in state S the n-sized vector comprised of the decisionsdi

m in response to the offers as

vSm ¼ d1

m; d2m; . . . ; dn

m

� �; 8i;1 � i � n; di

m ¼ 1: ð13Þ

The strategy sets RSm and RS

s for each player are reduced to just onevector in each.

STATE ST:The manufacturer has invested A and a, the supplier has

invested B and b. In the last period n, the manufacturer’s cost func-tion is Cn ¼minðUn;uÞ ) Un � u for acceptance. By extension,Uj � u, in periods j;2 � j � n.

The threat strategy for the manufacturer is to accept the intro-ductory offer u� A in period 1 and then reject the renewal of thecontract in period 2, resulting in C2 ¼ uþ a, whether or not thesupplier offers u. The total cost for this strategy over n periods isC ¼ nuþ a. In the best case where the manufacturer switches backto the supplier from period 3 to n, the supplier’s correspondingprofit becomes

Ps ¼ ðn� 1Þuþ v � A� B� b ð14Þ

If in period k; 1 < k � n, the manufacturer invests a, the profitfunction becomes

P0s � ðk� 1Þu� A� B� bþ v þ ðn� kÞvP0s � ðk� 1Þu� A� B� bþ v þ ðn� kÞu;

(ð15Þ

and P0s � Ps.The strategies of manufacturer and supplier are now described

by the following sets of vectors

RSTs ¼ vST

s ;vSTs ¼ ððu� AÞ;u; . . . ;uÞ

� �RST

m ¼ vSTm ;v

STm ¼ d1

m; d2m; . . . ; dn

m

� �;

�ð16Þ

9i;1 < i < n; dim ¼ 0

o:

STATE T:The manufacturer has only invested a, supplier has invested b.

In period j, the minimum acceptable offer from the supplier wouldbe Uj ¼ u� A, if 8i 2 f1; . . . ; j� 1g; di

m ¼ 0.If the manufacturer agrees in period k; k � n� j the stream of

profits to the supplier over n periods is: Ps ¼ ðk� 1Þv � bþðn� kþ 1Þu� A� B.

In any case, the cost to the manufacturer remains C ¼ nuþ a. Inthis state, the sets of vectors representing the strategies availableto the players are

RTs ¼ vT

s ; vTs ¼ ððu� AÞ;u; . . . ;uÞ

� �RT

m ¼ vTm;v

Tm ¼ 0; d2

m; . . . ; dnm

� �;

�9i;2 � i � n; di

m ¼ 1o: ð17Þ

A.2. Full information: proof of Proposition 1

Proof 1. When comparing the strategies among the three states,we see that in the first state, the supplier exploits the incumbent’sadvantage of the relationship specific investment with the man-ufacturer’s outside option only in the very last game. If she were totry doing so before, the manufacturer would simply defect. If thesupplier’s participation constraint

nuþ a� A� B � nv � b; ð18Þ

is satisfied, the profit extracted in state 1 is larger than those ineither other states. This justifies that she offers in all periodsUi ¼ u; 8i; 1 < i < n.

Given that the supplier is the Stackelberg leader and themanufacturer is reduced to accepting or rejecting the offers andthat both work in a full information environment, the NE strategyis the one presented in the case of state 1: the manufacturer agreesto the contract offered in the first period and, under sequentialrationality (Watson, 2002), subsequently accepts all the offersmade by the supplier without deviating by working with hisoutside option.

We will now investigate the weakness of this NE. The cartesianset Rs � Rm represents all the available strategies of both players.This set is larger than the union of the three sets defined whendescribing the three states in which the manufacturer finds herself.However, all the strategies which do not belong to the sets RS

m;RSTm ,

and RTm are evidently not profit maximizing ones or do not satisfy

the participation constraints as binding constraints and shall bediscarded.

For the supplier, it can easily be seen that, with vSm as defined in

(13),

8vm 2 Rm and vm – vs1;Psðv s1Þ > PsðvmÞ: ð19Þ

For the manufacturer, evidently Rm ¼ RSm [ RST

m [ RTm, so 8vm 2 Rm;

CðvmÞ ¼ n uþ a;) 9=v�m 2 RmjC v�m� �

> CðvmÞ. Hence, the NE isweak.

If the manufacturer chooses other responses, these strategiescan be assimilated to the ‘‘Trembling Hand’’ ones (Selten, 1975).The three states presented above are in fact occurrences of this

nv � b No contract

20 X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22

Trembling Hand argument: as can be seen in Table 4, themanufacturer may costlessly play a different strategy which hurtsthe supplier. Due to the Stackelberg structure, we do not explorethe other rent-appropriation possibilities and conclude that the NEis not a trembling-hand perfect equilibrium. h

Fig. 3. States of nature according to the supplier’s thresholds Aj� and a� .

A.3. Asymmetric information: proofs of Proposition 2

We present proofs substantiating the second proposition inSection 3.4 when A and a are the manufacturer’s privateinformation.

The supplier must evaluate optimal offers for each of the man-ufacturer’s enabling investment(s) states. She then formulatesoffers as functions of thresholds A1�; a� and eventually the updatedthresholds Ai� in periods iji 2 f2; . . . ;ng so that she maximizes herprofit. We first evaluate the strategies open to both players beforepresenting in Section A.3.2 the calculations of those thresholdswhich help to determine the supplier’s bids.

A.3.1. Strategies and manufacturer’s statesSTATE S:This state is attained only if A1� � A to incite an acceptance by

the manufacturer. So, given the sequential rationality of the man-ufacturer, he works with the supplier for this and all periods up toperiod n� 1 with probability 1 as long as the reservation utility ismet. In the last period, the manufacturer accepts the final hold-uponly if a� � a.

The supplier’s strategy is structurally analogous to the fullinformation case in Section 3.3:

U1 ¼ u� A1�;

Ui ¼ u; 1 < i < n;

Un ¼ uþ a�;

8><>: ð20Þ

under the adjusted participation constraint for the supplier:nu� A1� þ a� � nv � bðPCÞ.

The manufacturer’s cost and supplier’s profit over the n periodsare C ¼ nu� A1� þ Aþ a�;Ps ¼ nu� A1� � Bþ a�.

When period n starts, the supplier must now show her offerusing a�. If a� > a, the manufacturer rejects the offer, dn

m ¼ 0. Themanufacturer’s cost and supplier’s profit when terminating in stateS are C ¼ nu� A1� þ Aþ a;Ps ¼ ðn� 1Þu� A1� � Bþ v � b.

STATE ST:The double-investment state is attained at the earliest in period

2 if the supplier’s threshold A1� � A made the manufacturer rejectthe initial offer and accept it in a subsequent period k, 1 < k � n.The supplier has incurred b and the manufacturer has incurredinvestments Aþ a up to this period.

The dominant strategy for state ST is trivial, there is no oppor-tunity for hold-up in the last period since the manufacturer has asunk investment with the third party. Thus, the resulting strategyfor the supplier for all subsequent periods kþ 1 � j � n is Uj ¼ u,which follows from Condition 2.

The manufacturer cost and supplier profit expressions are,respectively:

C ¼ nu� Ak� þ Aþ a

Ps ¼ ðn� kÞu� Ak� � Bþ kv � b: ð21Þ

STATE T:State T results from Ak�

< A in all periods from 1 to k. The man-ufacturer has invested a in a third-party relationship. In periodj; k < j < n the supplier may offer a new contract based upon Aj

that, if accepted by the manufacturer, would change the state to

8 Note that j < n because there is no interest for the supplier to make an offer inperiod n lower than u.

ST.8 However, an updated offer by the supplier is made only if themanufacturer’s incentive compatibility (IC) constraint AðjÞ� < Aðjþ1Þ�

;

k < j < n, is satisfied.9 In the following period, the supplier cannotmake an acceptable offer to the manufacturer because doing sowould violate her participation constraint.

If the game terminates in state T, the manufacturer and supplierhave never worked together: the manufacturer’s overall cost isC ¼ nuþ a and supplier’s overall profit is Ps ¼ nv � b.

Fig. 3 represents the three states and the manufacturer’sdecisions leading to them.

A.3.2. Bid determinationWe now present the evaluations of the thresholds that enable

the supplier to calibrate her offers. The supplier’s partial objectivefunction over the n periods is

maxPsðAjÞ¼ FA1ðA1Þ ðn�1Þu�A1�BþFaðaÞðuþaÞþFaðaÞðv�bÞh i

þFA1ðA1Þ½v�bþFA2ðA2Þfðn�1Þu�A2þBg

þFA2ðA2ÞðvþFA3ðA3Þ½ðn�2Þu�A3þB�

þ . . .FAðn�jÞðAn�jÞððjþ1Þu�An�jþBÞþv

þFAðn�jÞðAn�jÞððjþ1ÞvÞ . . .Þ�;ð22Þ

s:t:Aðn�jþ1Þ � Aðn�jÞ

; 1 < j < n; ðICÞnuþ a� A1 � B � nv � b; ðPCÞ:

(ð23Þ

9 Justification is presented in Proof 2 in Section A.3.4

10 We present such a procedure in our numerical illustration. Any Bayesian updatingwith cutoff procedure which satisfies Ai ¼ Aði�1Þ� and enables the supplier tocharacterize distribution functions of her belief over the different periods could beused.

X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22 21

The two above inequalities represent the participation constraint(PC) and incentive compatibility constraint (IC). We solve for a inthe next subsection and Ai in the following one.

A.3.3. Belief about aWe first turn to the belief about a which the supplier has to

make in period n (if at all).Since we have assumed that f a has an IFR and f aðaÞ – 0, there

exists an interior value a� such that a� � Faða�Þf aða�Þ

¼ v � b� u.The proof is presented in Brusset (2014).

A.3.4. Updated belief about the manufacturer’s RSA with the supplierAn�j

In period n� j with 1 < j < n the supplier updates An�j. Differen-tiating in An�j:

@Pðn�jÞs ðAn�jÞ@An�j

¼ f Aðn�jÞðAn�jÞððjþ 1Þðu� vÞ � An�j � BÞ

� FAðn�jÞðAn�jÞ: ð24Þ

The Second Order Condition (S.O.C.) requires that f 0Aðn�jÞðAðn�jÞ�Þ

ððjþ 1Þðu� vÞ � Aðn�jÞ� � BÞ � 2f Aðn�jÞðAðn�jÞ�Þ < 0. The proof is simi-

lar to the one used above.So, we can deduct the optimal threshold Aðn�jÞ� in period n� j as

solution to

Aðn�jÞ� þ Bþ FAðn�jÞðAðn�jÞ�Þf Aðn�jÞðA

ðn�jÞ�Þ¼ ðjþ 1Þðu� vÞ; ð25Þ

Using backward induction, we can now evaluate the thresholdfor A in the previous period starting with period n� j� 1. Theprofit function in preceding periods i; 1 � i � n� j� 1, can bewritten as

Li ¼ FAiðAiÞððn� iþ 1Þu� Ai � BÞ þ FAiðAiÞðv þ Liþ1Þ: ð26Þ

In period n� j, the last term of the series before the supplier’s Par-ticipation Constraint (PC) is violated, Ln�j is written as

Ln�j ¼ FAn�jðAn�jÞððjþ 1Þu� An�j � BÞ þ FAn�jðAn�jÞððjþ 1ÞvÞ: ð27Þ

The optimal threshold in each period i; 2 � i � n� j� 1, isthe result of evaluating the First Order Condition (F.O.C.) andS.O.C. of the expression Li differentiated in Ai. We obtain

Ai� þ Bþ FAiðAi�Þf AiðA

i�Þ¼ ðn� iþ 1Þðu� vÞ � Liþ1: ð28Þ

Proceeding in a bootstrapping iteration we evaluate all the preced-ing Li back to L2 and A1:

A1� þ Bþ FA1ðA1�Þf A1ðA

1�Þ¼ ðn� 2Þðu� vÞ � L2: ð29Þ

Hence the threshold A1� is a function of the distribution of the beliefabout A, the difference between the outside options’ prices, and thelength of the game.

Remark 2. Witness how each threshold is determined using thesupplier’s posterior a priori beliefs about cost A. However, thesupplier relies upon updating her belief using an ‘‘updating withcutoff’’ procedure. That is, using the information imparted by themanufacturer’s decision to turn down her previous offer, she setsAi ¼ Aði�1Þ�, and updates the cdf accordingly. Hence, given the wayour model is built, this means that the posterior beliefs rely uponupdating the current period’s belief. Theoretically our suppliershould not be able to resolve this circular reasoning. To get aroundthis contradiction, the supplier must initially set a range and a

distribution function for period n� j, use a procedure10 tobootstrap her a priori belief in period 1, make an offer and use themanufacturer’s decision to update the a priori belief of period n� j.In this way, as periods go by, the supplier updates sequentially the apriori belief about A in the last period which satisfy IC and PC in (23)and bootstraps back her range of beliefs fAi;Aig and consequentdistribution function FAi to the period she finds herself in.

To be incorporated in contracts to be offered to the manufac-turer, the thresholds evaluated above must follow the strategywhich we described when the manufacturer is in state ST or T,namely that Ai�

< Aðiþ1Þ� for the first periods.We prove that the supplier’s PC and manufacturer’s IC in Prop-

osition 2 and in (23) the supplier’s objective function (22) aresatisfied.

Proof 2. By definition, Li represents the expected profit to thesupplier going forward in periods iþ 1 to n when she erred in herevaluation of A. As i increases, this expected profit decreases sinceeach period’s profit is not negative, even if it involves choosing heroutside option. So Liþ1 < Li. Further, from (28), in each period, theperiod’s threshold is higher than the previous one’s. Moreover,since the supplier offers a sequence of monotonously increasingbids, we have A�1 < A�2 < . . . < Ai�. We reach a point where Ai

becomes large compared to ðn� iþ 1Þu within Li in (26). Byconstruction, after that point Ai can no longer increase and in factdecreases as can be seen in the numerical illustration presented inSection 4. So the supplier cannot make an offer more enticing thanthe previous one to the manufacturer. Instead, she turns to heroutside option. h

Note that period n� j becomes the true last period for mutualbeneficial trade.

A.4. Proof of Proposition 3

Proof 3. We have seen in the integrated supply chain that if themutual interaction is delayed by k periods, then the loss in rent tothe supply chain is given by the difference between the cost to themanufacturer and the profit to the supplier:

C ¼ aþ nuþ A� Akþ1�

Ps ¼ kv � b� Akþ1� þ ðn� kÞu;

() C �Ps ¼ aþ Aþ bþ sðu� vÞ:

ð30Þ

Whereas, if the interaction starts in period 1, the partners’ objectivefunction work out as

C ¼ A� A� þ nuþ a�

Ps ¼ nu� A� þ a�;

�) C �Ps ¼ A: ð31Þ

This leads to the difference between both cases ofD ¼ aþ bþ kðu� vÞ, which is strictly positive because v � uaccording to the PC in (3).

Note that this result is independent of the period in which themanufacturer decides to change supplier. It is also independent ofthe supplier’s estimates of a and A. h

References

Abreu, D., & Pearce, D. (2007). Bargaining, reputation and equilibrium selection inrepeated games with contracts. Econometrica, 75(3), 653–710.

Adeleke, A., & Onumanyi, P. (2007). Numerical computations to produce cokeablecoal blends at the Ajaokuta steel plant, Nigeria. Journal of Minerals & materials

22 X. Brusset, P.J. Agrell / Computers & Industrial Engineering 80 (2015) 12–22

characterization & engineering, 6(2), 121–134. URL http://www.imp.mtu.edu/jmmce/issue6-2/P121-134AdelkeNumerical.pdf.

Aitken, J. (1998). Supply chain integration within the context of a supplier association(Ph.D. thesis). Cranfield University.

Anderson, E., & Weitz, B. (1992). The use of pledges to build and sustaincommitment in distribution channels. Journal of Marketing Research, 29, 18–34.

Ball, K., & Loncar, T. (1991). Factors influencing the demand for australian coal.Project 47.102, Australian Bureau of Agricultural and Resource Economics, GPOBox 1563, Canberra, Australia. URL <http://www.abare.gov.au/publications_html/energy/archive/aus_coal.pdf>.

Barlow, R. B., & Proschan, F. (1975). Statistical theory of reliability and life testing. HoltRhinehart and Winston.

Barlow, R. E. (2003). Mathematical reliability theory: From the beginning to thepresent. In Mathematical and statistical methods in reliability. Vol. 7 of series onquality, reliability & engineering statistics (pp. 3–13). World Scientific.

Barlow, R. E., & Proschan, F. (1965). Mathematical theory of reliability. John Wiley &Sons, pp. 9–18.

Barnes-Schuster, D., Bassok, Y., & Anupindi, R. (2002). Coordination and flexibility insupply contracts with options. Manufacturing & Service Operations Management,4(3), 171–207. Summer.

Battaglini, M. (2007). Optimality and renegotiation in dynamic contracting. Gamesand Economic Behavior, 60, 213–246.

Benz, A., Jäger, G., & van Rooij, R. (2005). Game theory and pragmatics. PalgraveMcMillan.

Brusset, X. (2009a). Choosing a transport contract over multiple periods.International Journal Logistics Systems and Management, 5(2-3), 273–322.

Brusset, X. (2009b). Multi-period contracts in transport under asymmetricinformation and prior investments. In Business and Economics (pp. 37–54).Berlin, Heidelberg: Springer–Verlag.

Brusset, X. (2014). Estimating the supply chain efficiency loss when the seller has toestimate the buyer’s willingness to pay. RAIRO – Operations Research, 48(4),477–496. URL <http://www.rairo-ro.org/action/displayAbstract?fromPage=online&aid=9282796>.

Cachon, G., & Netessine, S. (2004). Game theory in supply chain analysis. In:Handbook of Quantitative Supply Chain Analysis – Modeling in the eBusiness Era(1st ed.). International Series in Operations Research & Management Science(pp. 13–66). New York, NY: Springer Science.

Cho, R., & Gerchak, Y. (2005). Supply chain coordination with downstreamoperating costs: Coordination and investment to improve downstreamoperating efficiency. European Journal of Operational Research, 162, 762–772.

CSC (2009). 2009 global survey of supply chain progress. Tech. rep., Supply ChainManagement Review (SCMR) and Computer Sciences Corporation. URL <http://www.csc.com/management_consulting/insights/33001-2009_global_survey_of_supply_chain_progress>.

Cvsa, V., & Gilbert, S. (2002). Strategic commitment versus postponement in a two-tier supply chain. European Journal of Operational Research, 141, 526–543.

Dang, T. V. (2008). Bargaining with endogenous information. Journal of EconomicTheory.

Edlin, A., & Reichelstein, S. (1996). Holdups, standard breach remedies, and optimalinvestment. American Economic Review, 86(3), 478–501.

Esmaeili, M., Aryanezhad, M.-B., & Zeephongsekul, P. (2009). A game theoryapproach in seller–buyer supply chain. European Journal of Operational Research,195, 442–448.

Farrell, J., & Klemperer, P. (2007). Coordination and lock-in: Competition withswitching costs and network effects. In Handbook of Industrial Organization(pp. 1970–2072). Elsevier B.V..

Fudenberg, D., Levine, D. K., & Tirole, J. (1985). Infinite-horizon models of bargainingwith one-sided incomplete information. In Game-theoretic models of bargaining.Cambridge University Press.

Fudenberg, D., & Tirole, J. (1983). Sequential bargaining with incompleteinformation. Review of Economic Studies, 50, 221–247.

Garcia Mariñoso, B. (2001). Technological incompatibility, endogenous switchingcosts and lock-in. The Journal of Industrial Economics, 49(3), 281–298.

Gibbons, R. (1992). Game theory for applied economists. Princeton, NJ, USA: PrincetonUniversity Press.

González, P. (2004). Investment and screening under asymmetric endogenousinformation. Rand Journal of Economics, 35(3), 502–519.

Hart, O., & Moore, J. (1990). Property rights and the nature of the firm. Journal ofPolitical Economy, 98(6), 1119–1158.

Hart, O., & Moore, J. (1999). Foundations of incomplete contracts. Review ofEconomic Studies, 66, 115–138.

Hart, O., & Tirole, J. (1988). Contract renegotiation and Coasian dynamics. TheReview of Economic Studies, 55, 509–540.

Hou, J., & Zeng, A. Z. (2011). Bargaining in a two-stage supply chain throughrevenue-sharing contract. In Supply chain coordination under uncertainty(pp. 347–378). Springer.

Kampstra, R., Ashayeri, J., & Gattorna, J. (2006). Realities of supply chaincollaboration. The International Journal of Logistics Management, 17(3), 312–330.

Kang, M., Mahoney, J., & Tan, D. (2009). Why firms make unilateral investmentsspecific to other firms: the case of oem suppliers. Strategic Management Journal,330, 117–135.

Klemperer, P. (1987). Markets with consumer switching costs. The Quarterly Journalof Economics, 102(2), 375–394.

Knemeyer, A. M., Corsi, T., & Murphy, P. R. (2003). Logistics outsourcingrelationships: Customer perspectives. Journal of Business Logistics, 24(1),77–109.

Leng, M., & Parlar, M. (2005). Game theoretic applications in supply chainmanagement: A review. Information Systems and Operational Research, 43(3),187–220.

Leng, M., & Zhu, A. (2009). Side-payment contracts in two-person nonzero-sumsupply chain games: Review, discussion and applications. European Journal ofOperational Research, 196, 600–618.

Li, H., Ritchken, P., & Wang, Y. (2009). Option and forward contracting withasymmetric information: Valuation issues in supply chains. European Journal ofOperational Research, 197(1), 134–148.

Odell, M. (2011). Ryanair slams Boeing 737 ’confusion’. Financial Times June 21st.URL <http://www.ft.com/intl/cms/s/0/4d2375b6-9c12-11e0-bef9-00144feabdc0.html#axzz1cX57hBT9>.

O’Doherty, J. (2009). Ryanair breaks off talks with Boeing. Financial Times.Plambeck, E., & Taylor, T. (2007). Implications of renegotiation for optimal contract

flexibility and investment. Management Science, 53, 1872–1886.Segal, I., & Whinston, M. (2000). Winter. Exclusive contracts and protection

ofinvestments. Rand Journal of Economics, 31(4), 603–633.Segal, I., & Whinston, M. (2002). The Mirrlees approach to mechanism design with

renegotiation (with application to hold-up and risk sharing). Econometrica,70(1), 1–45.

Selten, R. (1975). Reexamination of the perfectness concept for equilibrium pointsin extensive games. International Journal of Game Theory, 4, 25–55.

Sucky, E. (2004). Inventory management in supply chains: A bargaining problem.International Journal of Production Economics, 253–262.

Sucky, E. (2006). A bargaining model with asymmetric information for a singlesupplier – Single buyer problem. European Journal of Operational Research,171(2), 516–535.

Sucky, E. (2007). A model for dynamic strategic vendor selection. Computers andOperations Research, 34, 3638–3651.

Tirole, J. (1986). Procurement and renegotiation. Journal of Political Economy, 94(2),235–259.

Wang, X. L., & Liu, L. W. (2007). Coordination in a retailer-led supply chain throughoption contract. International Journal of Production Economics, 110, 115–127.

Watson, J. (2002). Strategy: An introduction to game theory. W.W. Norton, p. 139.Williamson, O. E. (1985). The economic institutions of capitalism. New York, NY, USA:

Free Press.Wu, S. (2004). Supply chain intermediation, a bargaining theoretic framework. In

Handbook of quantitative supply chain analysis: Modeling in the E-Business era.International series in operations research & management science (pp. 67–115).New York, NY, USA: Springer.

Zhao, Y., Ma, L., Xie, G., & Cheng, T. C. E. (2013). Coordination of supply chains withbidirectional option contracts. European Journal of Operational Research, 229,375–381.

Zhao, Y., Wang, S., Cheng, T. C. E., Yang, X., & Huang, Z. (2010). Coordination ofsupply chains by option contracts: A cooperative game theory approach.European Journal of Operational Research, 207, 668–675.