European Journal of Operational Research 233 (2014) 557–565

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

Sustaining long-term supply chain partnerships using price-onlycontracts

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.09.020

⇑ Corresponding author. Tel.: +1 3129066527; fax: +1 312 906 6549.E-mail addresses: jsun22@iit.edu (J. Sun), ldebo@chicagogsb.edu (L. Debo).

Jiong Sun a,⇑, Laurens Debo b

a Stuart School of Business, Illinois Institute of Technology, Chicago, IL 60661, United Statesb Booth School of Business, University of Chicago, Chicago, IL 60637, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 May 2013Accepted 13 September 2013Available online 26 September 2013

Keywords:Supply chain managementGame theoryRepeated interactionTurbulent market

In this paper, we study how an informal, long-term relationship between a manufacturer and a retailerperforms in turbulent market environments characterized by uncertain demand. We show that thelong-term partnership based on repeated interaction is sustainable under price-only contracts whenthe supply chain partners are sufficiently patient. That is, the channel can be coordinated over a long timehorizon when the factor whereby the members discount the future value of this trusting relationship issufficiently high. Second, above the minimum discount factor, a range of wholesale prices exists that cansustain the long-term partnership, and there are different possible profit divisions between the two play-ers. Third, when the market is turbulent, i.e., either the expected demand or the demand variance changesfrom period to period according to a probabilistic law, it is typically less possible to sustain the long-termpartnership in a booming market or in a market with low demand variability. Finally, obtaining moreinformation about future market fluctuation may not help the supply chain to sustain the long-term part-nership, due to partners’ strategic considerations. With the availability of the market signal, total supplychain profits increase, but the retailer may even be worse-off.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

It is well-known that due to the double-marginalization effect, asupply chain governed by a wholesale-price contract cannot becoordinated in a single-shot interaction, i.e., joint maximum pay-offs cannot be achieved (Lariviere & Porteus, 2001). More elaboratecontracts must be designed to achieve coordination (see Tayur,Magazine, & Ganeshan, 1999 or a review in Cachon (2003)). How-ever, in practice, firms often interact repeatedly with each other.Pyke and Johnson (2003) argue that critical, high-value-addedcomponents or components with complex interfaces are often bet-ter handled through long-term supply chain partnerships. Theanticipation of future interaction may restrain firms’ opportunisticbehavior in a single interaction (Taylor & Plambeck, 2007). Whenanticipating repeating business, firms can adopt an informal agree-ment that can be sustained by the future value of a trusting, coop-erative relationship. Second, price-only contracts are easy toimplement and are still fairly common in industries. In practice,the actual implementation of those more elaborate contracts maydiffer from what is stipulated in the sophisticated contracts (Neu-ville, 1997). Moreover, these contracts have additional administra-tive and handling costs, or might create additional moral hazardproblems (Krishnan, Kapuscinski, & Butz, 2004). We fill the gap

in the literature by studying, in the case of repeating business,how external environmental turbulence influences their motiva-tion to cooperate under price-only contracts.

In addition, today’s markets are often characterized by fluctuat-ing demand due to financial crises, emerging markets, natural disas-ters, unstable geopolitics, changing consumption patterns, andemerging technologies. These random events may shift market de-mand and/or increase demand volatility, which we refer to as turbu-lent markets in this paper. For example, the pharmaceutical marketfluctuates greatly because of changing government regulations and/or the introduction of new drugs or diagnostics. The former usuallyshifts demand, while the latter injects more variability. Anotherexample of a turbulent market is the pork industry. The mean de-mand for pork in the US has remained constant since 1985. How-ever, its variability has increased due to changing consumptionpatterns towards greater variety and more value-added products.The demand for pork in China is mainly characterized by shifting de-mand due to strong economic growth coupled with a slowing pop-ulation growth rate (Pan & Kinsey, 2002). Firms may thus face abooming or a busting market if random events shift the demand,or they may face a stable or a volatile market if random events im-pact variability. These situations may impact the downstream firmdifferently from the upstream firm and, hence, the temptation tobreak an ‘‘informal’’ partnership will be different. This poses to sup-ply chain partners the following questions: How will their partner-ships survive turbulent environments? and Which types of

558 J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565

environments will present a higher temptation to deviate fromcoordination?

In this paper, we study how an informal relationship between amanufacturer (she) and a retailer (he), based on repeated interac-tion, performs in turbulent market environments. Players may bemotivated to cooperate with each other if the gains from coopera-tion over the long run exceed the short run gains that they wouldobtain from not cooperating. During an economic boom, attractiveshort run gains may make deviation from cooperation more tempt-ing. We may thus expect that market turbulence adversely impactstheir motivation to coordinate the supply chain through repeatedinteraction. We develop a mathematical model that allows us toaddress the following questions: (1) Under which circumstances,will jointly maximal payoffs be achieved through repeated interac-tion? (2) How will the jointly maximal profits be divided and whatwill the cooperating wholesale price be? (3) What will be theimpact of turbulent markets on their motivation to sustain thelong-term partnership? (4) What types of turbulent markets willconstrain the long-term partnership? and (5) Who benefits fromthe availability of the information about market turbulence?

We offer the following insights. First, we show that the long-term partnership based on repeated interaction is sustainable un-der wholesale-price contracts when the supply chain partners aresufficiently patient, i.e., the factor whereby the members discountthe future value of this trusting relationship is sufficiently high. Atthe minimum discount factor, the manufacturer’s expected profitin each period is not higher than the profit the manufacturer wouldobtain in a single-stage game. All the additional gains generated bythe long-term partnership flow to the retailer. This is because dueto repeated interactions, the disadvantage of the retailer as a sec-ond mover is mitigated, and his ability to punish a deviating man-ufacturer in the same period gains him some power in the supplychain. At higher discount factors, a range of wholesale prices existsthat can sustain the long-term partnership, and there are differentpossible profit divisions between the two players. With increasingdiscount factor, it becomes more likely that the manufacturercould obtain more than her stage-game profit.

Second, in a turbulent environment, where the market turbu-lence is modeled as a parameter of the market demand distributionthat changes from period to period according to a probabilistic law,we find that the ability to sustain the partnership is restricted com-pared with stationary environments. If a signal concerning turbu-lence is observable at the beginning of each period, then theability to sustain the partnership can be restricted or enhanced,depending on the signal content (e.g., mean or variance). The avail-ability of the market signal increases the manufacturer’s expectedprofits, but may decrease the retailer’s profits. Furthermore, wefind that the supply chain partners have the highest temptationto deviate in a booming market or in a market with low variance.

In Section 2, related literature is briefly discussed. In Section 3the single-period constituent game (the stage game) is brieflydiscussed. In Section 4.1, the stage game is repeatedly played instationary markets where the demand is independently andidentically distributed in each period. In Section 4.2, the game isrepeatedly played in turbulent markets, where the demand distri-bution changes from one period to another. In Section 5, points forfurther research are developed and conclusions are drawn. Proofsand additional results are provided in Appendix.

2. Related literature

Japanese techniques such as Total Quality Management (TQM)or Just-In-Time (JIT) production have triggered considerable inter-est in supplier-manufacturer relationships. The traditional strategyliterature has mainly focused on the exploitation of bargainingpower (Porter, 1980). The quality management practitioners,

however, have argued that the cost of close coordination withmanufacturers is less than the added benefits of better quality, re-duced inventories, etc., that it provides (Deming, 1986). Kahn,Kalwani, and Morrison (1986) identified the impact of adversarialmanufacturer relations on purchasing costs. Ali, Smith, and Saker(1997), for example, strongly advocate the adoption of partnershiprelationships with manufacturers. A number of authors comparethe advantages and disadvantages of partnership sourcing versuscompetitive sourcing. Richardson and Roumasset (1995) comparesole sourcing, competitive sourcing, and parallel sourcing (solesourcing, but limited to a particular product category) and findthe optimal sourcing arrangement in different environments.Taylor and Wiggins (1997) compare the cost performance of theAmerican system, which involves competitive bidding, largebatches, and quality inspection of an incoming order, to the Japa-nese system of repeat purchases from one manufacturer, smallbatches, and no inspection. They conclude that, when using flexiblemanufacturing technology and producing complex products, theJapanese system performs better. Parker and Hartley (1997) cri-tique the partnership-sourcing approach by adopting a transac-tion-cost framework and point out the existence of a continuumof relationships between adversaries and partners.

Lariviere and Porteus (2001) analyze a wholesale-price contractin a decentralized manufacturer-newsvendor supply chain in aStackelberg game setting, and find that supply chain coordinationcannot be achieved (i.e., the retailer orders fewer items than in acentralized supply chain) due to the double marginalization effect.Perakis and Roels (2007) quantify the loss of efficiency of decen-tralized supply chains that use price-only contracts relative tothe centralized supply chain.

In the economics, marketing, and operations management liter-ature, more elaborate contracts that do allow for supply chain coor-dination are studied, including franchising contracts, buy-back,revenue-sharing, quantity flexibility and sales-rebate contracts(see Cachon, 2003; Tsay, Nahmias, & Agrawal, 1998 for comprehen-sive reviews). When contract parameters are carefully set, the real-location of the inventory risk between the two supply chainpartners can induce the retailer to order the supply chain optimalquantity. However, these more elaborate contracts are morecomplex to administer (Krishnan et al., 2004). In practice, it is fairlycommon for many supply chain transactions to be governed bysimple contracts defined by a per-unit wholesale price (Lariviere& Porteus, 2001). Taking into account the retailer’s ordering oppor-tunities both before and after the supplier’s production decision,Cachon (2004) and Dong and Zhu (2007) show that supply chainefficiency can be improved using wholesale-price contracts. As withelaborate contracts, efficiency is improved by reallocating inven-tory risks. Martinez de Albeniz and Simchi-Levi (2007) considerre-negotiation (i.e., there are multiple quoting-and-ordering inter-actions between the two players) before a market demand is real-ized, and show that the supply chain efficiency improves as thelength of negotiation process extends. By dividing one orderingopportunity into multiple ones, the disadvantage of the retailer asa second mover is mitigated, and it gains some power to ‘‘force’’the supplier to reduce its price in the following interaction.

From a different perspective, we study how the value of theongoing relationship can create an incentive for the supply chainpartners to cooperate under wholesale-price contracts. Repeatedinteraction plays an important role in the coordination mechanism,as evidenced in Martinez de Albeniz and Simchi-Levi (2007). Wehowever study a different situation in which there is a marketopportunity following each interaction. The impacts of repeatedinteraction, each followed by a market opportunity, on firms’ deci-sions have been studied by researchers, but not for the purpose ofaligning individuals’ objectives with that of the supply chain. Forexample, Nagarajan and Sošic (2008) and Huang and Sošic (2010)

J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565 559

show that, in an infinitely repeated newsvendor game with multi-ple retailers, when the discount factor is large enough, the retailerswill share all of their leftover inventories with each other, which isnot possible in the single-stage (one-shot) game. These studiesshow that repeated interactions present significantly differentdynamics than do one-shot supply chain relationships. This is alsoconfirmed in our study with respect to supply chain coordinationin a long-term partnership. However, the impact of turbulent mar-kets on the long-term partnership is generally ignored in both thesingle-stage and repeated-interaction models.

Taylor and Plambeck (2007) and Atkins, Krishnan, and Zhao(2006) study supply chain relationships with repeated interactionusing informal agreements, i.e., ‘‘relational contracts’’. These con-tracts specify the interaction between an upstream and a down-stream player after each possible interaction. Relational contractsare sustained, not by the court system, but by the threat of lossof future payoff from non-cooperation. Taylor and Plambeck studyhow relational contracts can be used to govern a supplier’s capac-ity investment for a downstream manufacturer. Atkins et al. pro-pose that a unilateral deviation by the retailer would result inthe breakdown of the relationship for a certain length of time.We study here how such informal supply chain partnerships, basedon repeated and future interaction, perform in turbulent marketenvironments. Rotemberg and Saloner (1986) study collusion inan environment with a fluctuating market. They show that positivedemand shocks (booming markets) increase the temptation todeviate from a collusive pricing agreement. Similarly, in a marketof fluctuating demand mean or variability, we study situations(typically, in periods of high demand mean or low uncertainty, asshown later) in which the supply chain partners face a strongtemptation to deviate from coordination.

Many authors have extended wholesale-price contracts in anumber of ways in studying supply chain performance, by incorpo-rating transshipment (e.g., Huang & Sošic, 2010; Shao, Krishnan, &McCormick, 2011; Slikkera, Fransooa, & Wouters, 2005; Zhang,2005), asymmetric information (e.g., Cachon & Lariviere, 2001;Corbett, Zhou, & Tang, 2004; Zhang, Nagarajan, & Sosic, 2010), sub-contracting (e.g., Van Mieghem, 1999), options (e.g., Barnes-Schus-ter, Bassok, & Anupindi, 2002), strategic customers (e.g., Su, 2009;Su & Zhang, 2008), quick response (e.g., Krishnan, Kapuscinki, &Butz, 2010), and promotional efforts (e.g., Krishnan et al., 2004).The coordination issues are also studied in more complex supplychain structures, such as multiple retailers (e.g., Bernstein & Feder-gruen, 2007; Jain, Moinzadeh, & Zhou, 2012; Krishnan & Winter,2007), competing manufacturers (e.g., Cachon & Kok, 2010), andassembly systems (e.g., Gerchak & Wang, 2004; Jiang & Wang,2010; Leng & Parlar, 2010). In contrast with these studies, we focuson the impacts of turbulent markets on informal supply chain rela-tionships based on repeated interaction.

3. The single-period game

In this section, we briefly summarize the interaction between aretailer and a manufacturer in a single period (the ‘‘stage game’’)that has been widely discussed in the literature. Consider a central-ized supply chain with a downstream market to which a productcan be sold at unit retail price, r. The market demand, n, is uncertain,characterized by means of a density function, /(n), with n 2 ½n; �n�,0 < n < �n <1. The product is produced at a unit cost of c. Produc-tion must start before the actual demand is observed. If the realizeddemand is less than the produced quantity, y, demand is completelysatisfied and the remaining items are salvaged. Otherwise, all yitems are sold and the excess demand is lost. Without loss of gen-erality, we set the salvage value equal to zero. We assume r P c P 0to avoid trivial solutions. In the case that the supply chain is ownedand operated by a single agent (the ‘newsvendor’), the

newsvendor’s profit as a function of production quantity y is givenby PI(y) ¼: rD(y) � cy, where, DðyÞ ¼ Enðminðn; yÞÞ ¼

R yn n/ðnÞdnþ

yUðyÞ. The well-known newsvendor production quantity thatmaximizes the total profit is:

yI¼: U�1 cr

� �: ð1Þ

The integrated supply chain’s profits are given byPI = rD(yI) � cyI.

In a decentralized supply chain governed by a wholesale-price-only contract, the game sequence is as follows. First, the manufac-turer proposes a wholesale price w. After observing w, the retailerdecides on the quantity, y, of items that he will order from themanufacturer before the demand, n, is realized. Then the demandis realized and sales occur. The retailer’s profit as a function ofthe wholesale price w and order quantity y is given by PR(-w,y) ¼: rD(y) � wy, and the manufacturer’s profit is given by PM(-w,y) ¼: (w � c)y. According to Lariviere and Porteus (2001), when/(y) satisfies the increasing general failure rate (IGFR) property(i.e., y /ðyÞ

UðyÞis increasing), the equilibrium quantity y⁄ is uniquely

determined by cr ¼ UðyÞ 1� /ðyÞy

UðyÞ

� �, and the equilibrium wholesale

price is given by w� ¼ rUðy�Þ. If y� R ½n; �n�, then it is set to the clos-

est boundary (n or �n) (Lariviere & Porteus, 2001). As a notationalconvention, we use superscript ⁄ to refer to the stage game equilib-rium. (Lariviere & Porteus, 2001) showed that the order quantity inthe Stackelberg game, y⁄, is less than the jointly optimal orderquantity, yI. That is, too few items are ordered by the retailer inthe stage game with a wholesale-price contract. This is due tothe double-marginalization principle (Spengler, 1950). It followsthat the supply chain profits P�R þP�M with P�i ¼ Piðw�; y�Þ, fori 2 {R,M}are less than the integrated supply chain profits, PI. Inthe next section, we discuss how the supply chain can be coordi-nated with a wholesale-price contract in a repeated-game context,i.e., when a long-term supply chain partnership can be sustained.

4. The infinitely repeated game

In this section, we consider the Stackelberg game detailed aboveas the stage game of an infinitely repeated game. We first set upthe game, describe the strategies (i.e., trigger strategies), and dis-cuss the main results for repeated games (i.e., the Folk Theorem).In the first subsection, we analyze the repeated game in the caseof identical demand distributions in each period. In the secondsubsection, we analyze the repeated game in the case of fluctuatingdistributions.

The infinitely repeated game. We embed the single-periodStackelberg game in an infinitely repeated game. We focus on per-ishable goods, or on high-tech products with a short product lifecycle which corresponds to one period in the repeated game. Inneither case can left-over inventory be carried over from periodto period. Furthermore, we assume that the market demand in aperiod is independent of the market demand in any other period.In each period t, the complete history of interactions, ht = ((w0,y0),(w1,y1), . . . , (wt�1,yt�1)), is observable to both players. Let Ht be theset of all possible period t histories. A strategy consists of a map-ping from every possible history of actions, ht, to the player’s actionspace; Ht ! ðwtðhtÞ; ytðhtÞÞ 2 ½c; r� � ½n; �n�. The payoffs for both play-ers are determined by the normalized discounted revenue streamover an infinite horizon: VM ¼ ð1� dÞ

P1t¼0d

tPMðwtðhtÞ; ytðhtÞÞ andVR ¼ ð1� dÞ

P1t¼0d

tPRðwtðhtÞ; ytðhtÞÞ, where d 2 (0,1) is the discountfactor.

The Nash-threats folk theorem. Infinitely repeated games typ-ically have many equilibria (Fudenberg & Tirole, 1991). We focuson subgame perfect equilibria. Friedman (1971) showed that a

560 J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565

minimum discount factor d exists, so that any payoff profile may berealized by means of a subgame perfect equilibrium that Paretodominates the stage game equilibrium payoff for d P d, i.e.,Vi P P�i with i 2 {R,M}. Consider the following ‘‘Grim trigger strat-egy’’ characterized by (w,y):

In the first period, the manufacturer sets the wholesale price to w.In the following periods, if (w,y) was played in all previous periods,then the manufacturer sets the price to w. If in one of the previousperiods (w,y) was not played, then the manufacturer sets the whole-sale price to the stage-game wholesale price w⁄. In the first period,the retailer orders y. In the following periods, if (w,y) was played inall previous periods and the manufacturer sets w in this period, thenthe retailer orders y. If (w,y) was played in all previous periodsand the manufacturer does not set w (but w0) in this period, thenthe retailer orders the stage-game best-response quantity,yðw0Þ¼: U�1 w0

r

� �. If in one of the previous periods (w,y) was not played,

then the retailer orders the stage-game equilibrium quantity y⁄.It can be proven (see, e.g., Gibbons, 1992; Laffont & Tirole, 1991)

that a trigger strategy is a subgame perfect equilibrium if during asingle period, no player may have an incentive to deviate from (w,y).

Note that other versions of the folk theorem with a larger spaceof possible payoffs may exist. The strategies for realizing these pay-off profiles are more complicated. As our objective is to obtain in-sights in a long-term partnership, we focus on Grim triggerstrategies. As a notational convention, we use superscript � to referto the equilibrium trigger strategy of the repeated game. In the fol-lowing subsection, we determine the values of (w�,y�) that are sup-ported in equilibrium as a function of the discount factor d, the costand demand parameters. The infinitely repeated game can be fullycoordinated, i.e., the long-term partnership can be sustained, whenthe coordinated supply chain order quantity, yI, is achieved in equi-librium, i.e., y� = yI. That is, the partnership’s per-period profit is thesame as the integrated supply chain’s profit, i.e., P� = PI. We areinterested in the conditions such that y� = yI.

4.1. Repeated games with stationary demands

4.1.1. AnalysisThe equilibrium condition that no player may have an incentive

to deviate from (y,w) can be stated for the retailer as:

Maxyd

fPRðw; ydÞg þd

1� dP�R 6

11� d

PRðw; yÞ; ð2Þ

i.e., for a given w, the sum of the maximum profit attainable duringthis period by deviating and the discounted punishment payoffsfrom the next period on, must be less than the discounted payoffsof not deviating from the trigger strategy. Similarly, for the manu-facturer, the equilibrium condition can be stated as:

Maxwd

fPMðwd; yðwdÞÞg þd

1� dP�M 6

11� d

PMðw; yÞ: ð3Þ

Note that the optimal deviation price of the manufacturer is thewholesale price of the stage game, i.e., w�d ¼ w� ¼ arg maxwd

fPMðwd; yðwdÞÞg. Hence, (3) reduces to:

P�M 6 PMðw; yÞ: ð4Þ

Furthermore, MaxydfPRðw; ydÞg in (2) reduces to PR(w,y(w)). Let (w�

,y�) characterize the equilibrium strategies satisfying (2) and (4).Denote the partners’ per-period profits under such a long-termpartnership by Pyi ¼ Piðwy; yyÞ, for i 2 {R,M}, and the supply chain’sper-period profits by Py ¼ PyR þPyM .

Define Dd as the set of y such that $w such that (w,y) satisfiesequilibrium conditions (2) and (4). Full coordination (i.e., P� = PI)is possible if and only if yI 2 Dd. If full coordination is possible fora particular d, then it follows from the Folk Theorem that for anydiscount factor above the particular d, full coordination is also

possible. Proposition 1 characterizes the minimum discount factorabove which full coordination is possible:

Proposition 1.

(1) There exists a discount factor d > 0, at which the long-termpartnership can be sustained, i.e., P� = PI, at a single wholesaleprice, w satisfying P�M ¼ PMðw; yIÞ, and d is given by

d ¼WRKR

1þ WRKR

;

where, WR ¼ PdR �PR, KR ¼ PR �P�R;P

dR ¼ PRðw; yðwÞÞ with

w ¼ c þ P�MyI .

(2) At the minimum discount factor d, in each period, the manufac-turer gains her stage-game profit, i.e., PyM ¼ P�M, and theretailer gains the rest, i.e., PyR ¼ PI �P�M.

Proposition 1 suggests that at the minimum discount factor d,the long-term partnership can be sustained at a single wholesaleprice, w. The minimum discount factor d is determined by the ratioWRKR

, where WR ¼ PdR �PR is the increase in profits that the retailer

would obtain if he deviates from the coordinated supply chain or-der quantity yI, and KR ¼ PR �P�R is the increase in his profits thatmay be achieved by coordination. In other words, the former mea-sures the temptation for the retailer to deviate from coordination,and the latter measures the ability of the supply chain to punish adeviating retailer. Therefore, the ratio WR

KRcan thus be thought of as a

relative measure of temptation for the retailer to deviate. The lar-ger his temptation, or the more lenient the punishment, the higherthe minimum discount factor for which the long-term partnershipis possible.

Proposition 1 also suggests that at the lowest discount factor d,the (coordinated) supply chain profits are split in a unique way. Inaddition, the long-term partnership does not benefit the manufac-turer and all gains from the long-term partnership flow to the re-tailer. That is, the manufacturer obtains the stage-game profit ineach period, i.e., PyM ¼ PMðw; yIÞ ¼ P�M . Interestingly, Propsition 2below suggests that at higher discount factors, the manufacturercould gain more than her stage-game profit from the long-termpartnership.

Proposition 2.

(1) When d 6 d < 1, the long-term partnership is sustainable at arange of price schedules, wy 2 ½w; �w�, where w and �w are deter-mined by:

w : P�M ¼ PMðw; yIÞ;

�w : PRðw; yðwÞÞ þd

1� dP�R ¼

11� d

PRðw; yIÞ;

the lower bound w is independent of d, and the upper bound �w isincreasing in d, i.e., @w

@d ¼ 0 and @ �w@d > 0.

(2) A range of profit divisions is possible in the long-termpartnership:

(a) The lower bound of the retailer’s profit is decreasing in d, i.e.,

@PRð�w;yIÞ@d < 0, and the upper bound of his profit is a constant,

PI �P�M, i.e., @PRðw;yIÞ@d ¼ 0. When d ? 1, the lower bound of

his profit goes to P�R.(b) The lower bound of the manufacturer’s profit is a constant,

P�M, i.e., @PM ðw;yIÞ@d ¼ 0, and the upper bound of her profit is

increasing in d, i.e., @PM ð�w;yIÞ@d > 0. When d ? 1, the upper

bound of her profit goes to PI �P�R.

J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565 561

For any discount factor above d, full coordination can beachieved, i.e., yI 2Dd, in a range of price schedules above the whole-sale price achieving the full coordination at the minimum discountfactor d. Proposition 2 gives bounds on the wholesale price: �w (w) isthe highest (lowest) possible wholesale price at which the retailer(manufacturer) in the repeated game receives at least his (her) prof-its in the single-period game. In other words, �wðwÞ is the wholesaleprice that makes the retailer (manufacturer) indifferent to devia-tion or not. Proposition 2 states that full supply chain coordinationby repeated interaction is possible for high discount factors.

Compared to the single-period Stackelberg interaction, fullcooperation is made feasible by a reduction in the wholesale price,as wy 6 �w < w�. This is beneficial for the retailer. In exchange, theretailer orders the jointly optimal order quantity allowing forsupply chain coordination. However, the retailer faces strongincentives in each period to order less, i.e., his best response y(w�

) < yI (as y(w) is decreasing in w, w� > c and yI = y(c)). Taking futureinteraction with the manufacturer into account, the retailer doesnot want to deviate from the joint optimal order quantity yI be-cause the net gains of deviation in the current period are less thanthe profit reduction caused by entering the punishment phase.Similarly, the manufacturer has no incentive to increase the lowwholesale price, because the potential gains are greater than theprofit reduction caused by the immediate reaction of the retailer(who will order less) and the subsequent punishment phase.

Second, the price range, ½w; �w�, for the supply chain to sustainthe long-term partnership enlarges with the discount factor. Thelower bound w, which is the unique price in the case of d, is a con-stant. The constant lower bound indicates a constant upper boundof the retailer’s profit, or a constant lower bound of manufacturer’sprofit, which are the same as in the case of d. The upper bound �w isincreasing in d. This is because with increasing discount factor, fu-ture profits matter more in the retailer’s consideration of whetherto deviate or not, which in turn increases the maximum possiblewholesale price at which the retailer would not deviate for a highprofit in the current period. Therefore, the retailer’s minimumpossible profit decreases with d, or the manufacturer’s maximumpossible profit she could obtain from the long-term partnership in-creases with d. That is, for d < d < 1, the manufacturer’s profits mayrange from P�M (which are the profits at d) to some value greaterthan P�M (but less than PI �P�R).

In the case of Uniform demand distribution, the followingCorollary 1 summarizes how market conditions influence the min-imum discount factor.

Corollary 1. For the uniform distribution, /ðnÞ ¼ 1�n�n

for n 2 ½n; �n�, wehave

d ¼

15 if r�c

r > 1�nn�1

1

1þ 1þ �nn�1

� �r�c

r

h i2 otherwise

8>><>>: :

Corollary 1 implies that the long-term supply chain partnershipis more sustainable with increasing market variance or increasingproduct margin. That is, the minimum discount factor d is decreas-ing (weakly) in the product margin (measured by r�c

r ) or the demandvariance (measured by �n

n). Specifically, d first decreases as the mar-ket variance increases or as the product margin increases. When theproduct margin and/or the demand variance is sufficiently high, ddoes not depend on the cost or revenue structure, nor on the ex-pected value and standard deviation of the demand distribution.

The above model assumes that the manufacturer or the retailerhas no outside option. That is, the manufacturer is the exclusivesupplier of the product or close substitutes, and the retailer is

the gatekeeper of accessing the consumer market. The punishmentfollowing deviation is that the supply chain partners continue theirbusiness with each other, and determine the wholesale price andorder quantity as if they were in a stage game. We extend theabove model in Appendix by incorporating an outside option forboth partners. Our results suggest that the manufacturer’s outsideoption has an impact on the ability of the supply chain to sustainthe long-term partnership, i.e., the minimum discount factor isincreasing in the value of the manufacturer’s outside option. Inother words, more intense the downstream competition makes itless possible to sustain the long-term partnership. However, theretailer’s outside option or the upstream competition has no im-pact on the minimum discount factor. The retailer’s relatively goodposition of punishing a deviating manufacturer in the same periodmakes external opportunities less attractive to him. Intuitively,stronger either partner’s outside option, higher possible profit divi-sion the partner is able to gain from the partnership.

4.2. Repeated games in turbulent markets

In the previous subsection, the demand distribution in each per-iod was the same and the demand realization is independent of allprevious realizations. It is possible that the demand distributionchanges from one period to another, which we refer to as turbulentmarkets. In periods with high demand, higher profits may be made.It may then be more tempting to deviate from the equilibriumstrategy. We analyze the impact of demand fluctuation on the re-tailer’s and manufacturer’s incentive to coordinate. We modelexogenous demand fluctuation in the same spirit as in Rotembergand Saloner (1986). In each period, the demand distribution willdepend on the realization of a stochastic variable, j, with density/j(n) and CDF Uj(n) over ½n; �n�. The market signal j can be the stan-dard deviation, or the mean of the demand distribution. For ouranalysis, j has density h(j) and distribution HðjÞ¼:

R jj hðjÞdj over

½j; �j�. At the beginning of each period, the Nature draws a realiza-tion of ej independently from the previous periods. ej is observableto both the retailer and the manufacturer. The stage game thenproceeds as in the previous subsection: The manufacturer setsthe wholesale price, communicates it to the retailer, the retailerdetermines the order quantity, and finally, the Nature draws thedemand realization from the distribution /ejðnÞ. With fluctuatingdemand, the trigger strategy for the retailer and manufacturer de-pends on the realization of j and is thus determined by a price andproduction schedule: (w(j),y(j)). For notational convenience, weuse the operator Ej½�� to indicate

R jj �dHðjÞ.

4.2.1. AnalysisConditional on the realization of a signal j, denote the inte-

grated (stage game) order quantity by yI(j)(y⁄(j)) and, for a givenwholesale price, w, denote the optimal order quantity as: y(w,j).Denote eP�

R ¼ Ej P�RðjÞ� �

, and eP�M ¼ Ej P�MðjÞ

� �, which are the ex-

pected stage game profits. For each signal realization, equilibriumconditions have to be determined. For the retailer, the equilibriumcondition can be stated as:

MaxydfPRðwðjÞ; yd;jÞg þ

d1� d

eP�R

6 PRðwðjÞ; yðjÞ;jÞ þd

1� dEj½PRðwðjÞ; yðjÞ;jÞ� for j 2 ½j; �j�;

ð5Þ

and for the manufacturer:

MaxwdfPMðwd; yðwd;jÞ;jÞg þ

d1� d

eP�M

6 PMðwðjÞ; yðjÞ;jÞ þd

1� dEj½PMðwðjÞ; yðjÞ;jÞ� for j 2 ½j; �j�:

ð6Þ

562 J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565

Let (w�(j), y�(j)) characterize an equilibrium trigger strategysatisfying (5) and (6) for all j 2 ½j; �j�. Similarly, as in the stationarydemand case, the optimal deviation price of the manufacturer isthe wholesale price of the stage game, w�dðjÞ ¼ w�ðjÞ. In each per-iod, the profits and temptation to deviate depend on j. As punish-ment for the manufacturer can only start in the period followingthe deviation, the punishment ability does not depend on j, buton the expectation of punishment ability in the following periods.

Lemma 1 in Appendix characterizes the necessary and sufficientcondition for the existence of yI(j). We then are able to determinein Proposition 3 the minimum discount factor above which thelong-term partnership is sustainable:

Proposition 3. The minimum discount factor above which thelong-term supply chain partner can be sustained is determined by:

d ¼WRKR

1þ WRKR

;

where WR ¼maxjWRðjÞ ¼maxj PdRðjÞ �PRðjÞ

h i, and KR ¼ ePR

� eP�R, ıd

RðjÞ ¼ PRðwðjÞ; yðwðjÞ;jÞ;jÞ, PR(j) = PR(w(j), yI(j), j) with

wðjÞ ¼ c þ P�M ðjÞyIðjÞ , ePR ¼ ePI � eP�

M, and ePM ¼ eP�M.

As in the stationary demand case, the long-term partnership isachieved by a decrease in the wholesale price and an increase inthe order quantity (compared with the stage game price and quan-tity). Also, the minimum discount factor is determined by the retai-ler’s temptation to deviate and his punishment for deviation. Thelarger his temptation, or the more lenient his punishment, the higherthe minimum discount factor above which the long-term partnershipis possible. In the fluctuating demand case, however, the incentives todeviate in each period depend on the market signal, j. Therefore,there is a market signal, to which we refer to as j0, for which the devi-ation temptation WR is the highest to the retailer, or the relativetemptation WR

KRis the highest. The highest temptation determines

the lowest possible discount factor for cooperation. Furthermore, atthe lowest discount factor, the long-term partnership does not in-crease the manufacturer’s profit and all gains flow to the retailer.

When d > d, full coordination can be achieved. In Proposition 4below, we can determine bounds of the equilibrium profits:

Proposition 4.

(1) When d 6 d < 1, the long-term partnership is sustainable at arange of price schedules, wy 2 ½w; �w�

(2) A range of profit divisions is possible in the long-termpartnership:

(a) The lower bound of the retailer’s profit is decreasing in d,

i.e., @ePRð �w;yIÞ@d < 0, and the upper bound of his profit is a con-

stant, ePI � eP�M, i.e., @ePRðw;yIÞ

@d ¼ 0 . When d ? 1, the lower

bound of his profit goes to eP�R.

(b) The lower bound of the manufacturer’s profit is a constant,

eP�M, i.e., @

ePMðw;yIÞ@d ¼ 0, and the upper bound of her profit is

increasing in d, i.e., @ePM ð�w;yIÞ@d > 0. When d ? 1, the upper

bound of her profit goes to ePI � eP�R.

Fig. 1. Illustration of the retailer’s expected profit range as a function of d, and thearea where the long-term partnership is sustainable (d P d). In this example,d ¼ 0:2527; c

r ¼ 0:1, l = 1, j ¼ 0:1; �j ¼ 0:4, and j is the standard deviation of thedemand distribution.

(3) There exists ðd <Þ�dð< 1Þ such that: When d < d < �d , the mini-mum equilibrium profits for the retailer are such that there exista j� 2 ½j; �j� for which (10) is binding, and when �d 6 d < 1, theminimum equilibrium profits for the retailer are such that (11)is binding.

(4) In a turbulent environment, the ability to sustain the long-termpartnership is restricted: d is higher than when the environmentis stationary.

As in the stationary demand case, for d = d, the (coordinated)supply chain profits are split in a unique way. When d < d < 1, acontinuum of strategies becomes sustainable in equilibrium, anddifferent possible profit divisions exist between the two players.The maximum profit that the retailer can obtain is the same ashis profit at d, while the minimum profit is decreasing in d. Whendis close to 1, the stage game equilibrium profit of the retailer isagain sustainable in equilibrium, and the maximum profit thatthe manufacturer can get is ePI � eP�

R. The range of their profits de-creases as d decreases and collapses at d.

Fig. 1 shows a possible region of ePR when the long-term part-nership is possible, denoted by ‘‘Long-term Partnership.’’ As illus-trated, the minimum profit that the retailer can obtain isdecreasing in d, and is defined by two curves. On the first curve(for d < d < �d), there exists a strong market signal j⁄ such thatthe retailer and the manufacturer are indifferent as to whether ornot to maintain the long-term partnership. In other words, belowthis curve there does not exist a price schedule such that for allmarket signals neither of them will deviate. The second curve(for d < d < �d) is the boundary below which all ePR’s are lower thanthe lowest acceptable profits for the retailer.

4.2.2. Numerical study: availability of the signal about marketfluctuation

In this section, we study how the availability of the market sig-nal about market fluctuation and the information content of thesignal impact the long-term partnership sustainability and theprofit division between the two players. Due to its complexity,we conduct a numerical study by assuming that both j anddemand n follow uniform distributions.

Fluctuation of the market demand variance. First we assumethat the market signal j is the standard deviation of the demanddistribution, i.e., Var[n] = j2. Suppose E[n] = l. Then, /jðnÞ ¼ 1

2ffiffi3p

j

over l�ffiffiffi3p

j;lþffiffiffi3p

jh i

, and hðjÞ ¼ 1�j�j over ½j; �j�. We consider

two scenarios: one in which the market signal j is observableand one in which it is not. In the case that the market signal j isnot observable, let UðnÞ ¼ EjðUjðnÞÞ be the expected demand dis-

tribution. The expected value of n is l(for any value of j and �j).

The standard deviation is denoted by rn and is a function of j

J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565 563

and �j. Note that the expected standard deviation of the market de-mand is rnjj ¼

�jþj2 . Then, rn

rnjjis a measure of the informativeness of

the signal. If rn

rnjj¼ 1, then the signal does not contain any informa-

tion. Higher values of rn

rnjjindicate a higher variance reduction by

the signal. We denote the minimum discount factor d(d0) without(with) observable market signals. Table 1 shows all these valuesunder different parameter settings, with c/r = 0.1 and l = 1 for a

series of j and �j such that jþ�j2 ¼ 0:25. Note that for

j ¼ �j ¼ 0:25, the signal does not contain any informationrn

rnjj¼ 1

� �, and we have that c = 2.527 and 1� c

r ¼ 0:9 >

1c�1 ¼ 0:6547 such that from Corollary 1 we obtain that d = 0.2.

Define Dj ¼ �j� j.From Table 1, we have the following observations:

� The impact of market demand variance fluctuation on thelong-term partnership. When the information about marketfluctuation is observable to both players, the long-term supply

chain partnership may either be restricted (d0 > d) or enhanced

(d0 < d). If the signal is less informative ð rn

rnjj6 1:026Þ, then the

observed market fluctuation decreases the range of discountfactors for which the long-term partnership is possible. Here,

the critical market signal is j0 = j. As jis the market uncer-tainty variance, supply chain profits decrease as a function ofj. Thus, it is expected that temptation to deviate will be highest

for j0 = j. However, if the signal is informative rn

rnjjP 1:040

� �,

then the observed market fluctuation increases the range of dis-count factors for which the long-term partnership is possible.Here, the critical market signal, j0, is interior in ½j; �j�.� The impact of market demand variance fluctuation and sig-

nal information on the long-term partnership. The higherthe information content of the market signal (higher rn

rnjj), the

more difficult it is to sustain the long-term partnership: d0

increases as Dj increases. When the market signal is not obser-vable, the higher the market uncertainty (rn), the more difficultit is to sustain the long-term partnership. Remember from Cor-ollary 1 that the opposite is true for the uniform distribution forwhich 1� c

r

� �ðc� 1Þ < 1. Thus, without fluctuating demand,

market uncertainty may play an ambiguous role in sustainingthe long-term partnership.� The impact of market demand variance fluctuation and sig-

nal information on the players’ profits. The availability of amarket signal increases the supply chain profit, i.e.,ePI �PI P 0. Furthermore, as the information content of the

signal increases higher rn

rnjj

� �, the system profit gain increases.

The availability of a market signal increases the manufacturer’sprofit. Furthermore, as the information content of the signal

increases higher rn

rnjj

� �, the manufacturer’s profit increases.

Table 1Supply chain performance with Var[n] = j2, l = 1, rnjj = 0.25 and c/r = 0.1.

Dj d d0j0�j�j�j

rn

rnjjePI �PI

ePR �PRePM �PM

0.40 0.3174 0.2686 0.23 1.101 1.040E�2 �0.971E�2 2.011E�20.35 0.2949 0.2603 0.19 1.078 0.858E�2 �0.649E�2 0.150E�10.30 0.2726 0.2527 0.14 1.058 0.681E�2 �0.367E�2 1.049E�20.25 0.2510 0.2456 0.07 1.040 0.511E�2 �0.135E�2 0.647E�20.20 0.2303 0.2391 0.00 1.026 0.352E�2 0.029E�2 0.323E�20.15 0.2111 0.2298 0.00 1.014 0.208E�2 0.1003E�2 0.108E�20.10 0.2000 0.2182 0.00 1.006 0.093E�2 0.047E�2 0.046E�20.05 0.2000 0.2083 0.00 1.001 0.023E�2 0.011E�2 0.011E�20.00 0.2000 0.2000 0.00 1.000 0 0 0

However, the availability of a market signal may decrease theretailer’s profit. When the market signal is very informative

high rn

rnjj

� �, the retailer may have lower than expected profits

with a market signal than without one. In this case, the manu-facturer’s profit gains are not only due to the system efficiencyimprovement, but also due to a better strategic position withrespect to the retailer. Only with moderately informative sig-nals can the manufacturer share the increase in system profitswith the retailer. Obviously, the profit gains due to marketinformation decrease as the signal becomes less informative.

Markets with fluctuation of the mean demand. In the follow-ing, we assume that the market signal j is the average of thedemand distribution, i.e., E[n] = j. Suppose Var[n] = r2. Then,

/jðnÞ ¼ 12ffiffi3p

r over j�ffiffiffi3p

r;jþffiffiffi3p

rh i

, and hðjÞ ¼ 1�j�j over ½j; �j�.

We still consider two scenarios as above. In the case that the mar-ket signal j is not observable, let the standard deviation be de-noted by rn.

rn

r is a measure of the informativeness of the signal.Table 2 shows results under different parameter settings, with c/

r = 0.1 and r = 0.25 for a series of j and �j such that jþ�j2 ¼ 1:0. Note

that for j ¼ �j ¼ 1:0, the signal does not contain any informationrn

rnjj¼ 1

� �and we have that c = 2.527 and 1� c

r ¼ 0:9 >

1c�1 ¼ 0:6547 such that from Corollary 1 we obtain that d = 0.2.

From Table 2, we have the following observations:

� The impact of mean demand fluctuation on the long-termpartnership. When the information about market fluctuationis observable to both players, the long-term supply chain part-nership is always restricted, i.e., d0 > d. Here, the critical marketsignal is always j0 ¼ �j. As jis the average market demand, it isexpected that the temptation to deviate will be the highest forj ¼ �j.� The impact of mean demand fluctuation and signal informa-

tion on the long-term partnership and the players’ profits.The two observations obtained for j being the standard devia-tion also apply here.

In conclusion, extra market information may not always beused to implement more efficient long-term partnerships, due tostrategic considerations. The manufacturer seems to benefit morefrom such information.

5. Conclusions and further research

While single-shot newsvendor games cannot be coordinatedwith price-only contracts (see Cachon, 2003), repeated interactionover a long horizon using ‘‘informal’’ contracts allows a long-termpartnership when the supply chain members are sufficientlypatient. In today’s turbulent environments, an important question

Table 2Supply chain performance with E[n] = j, r = 0.25 and c/r = 0.1.

Dj d d0j0�j�j�j

rn

rePI �PI

ePR �PRePM �PM

0.40 0.2334 0.2466 1 1.1015 0.678E�2 �0.19E�2 0.86E�20.35 0.2252 0.2409 1 1.0786 0.54E�2 �0.28E�3 0.57E�20.30 0.2165 0.2352 1 1.0583 0.41E�2 0.78E�3 0.34E�20.25 0.2075 0.2294 1 1.0408 0.30E�2 0.12E�2 0.17E�20.20 0.2003 0.2236 1 1.0263 0.19E�2 0.96E�3 0.96E�30.15 0.2000 0.2177 1 1.0149 0.11E�2 0.54E�3 0.54E�30.10 0.2000 0.2118 1 1.0066 0.48E�3 0.24E�3 0.24E�30.05 0.2000 0.2059 1 1.0017 0.12E�3 0.60E�4 0.60E�40.00 0.2000 0.2000 1 1 0 0 0

564 J. Sun, L. Debo / European Journal of Operational Research 233 (2014) 557–565

is whether such informal partnerships are sustainable. In our re-search, market turbulence is modeled as a parameter of the marketdemand distribution that changes from period to period accordingto a probabilistic law. We find that the informal partnership is sus-tainable under the wholesale-price contracts with a Grim triggerstrategy, provided that the players discount the future stream ofprofits with a factor of at least d. When the discount value is ex-actly equal to d, the manufacturer’s expected profit is not higherthan the profit the manufacturer would obtain in a single-stagegame. The retailer however reaps the highest possible expectedprofit, which is equal to the total supply chain profit minus themanufacturer’s profit. For values larger than d, a range of wholesaleprices exists that can sustain the long-term partnership, and thereare different possible profit divisions between the two players. In aturbulent environment, the ability to sustain the partnership is re-stricted: d is higher than when the environment is stationary. If thesignal concerning turbulence is observable at the beginning of eachperiod, then the ability to sustain the partnership is typically re-stricted: d is higher than when the signal is unavailable. But theability may be enhanced when the signal is about the variance ofthe demand distribution. Even though the supply chain profitsincrease, it may be that such a signal decreases the retailer’sexpected profits. Furthermore, we show that it is not trivial toidentify the critical market condition that constrains the partner-ship. Typically, the supply chain partners have the highest tempta-tion to deviate in a booming market or in a market with lowvariance.

We believe that these conclusions will be of interest to a supplychain manager. Suppose that his interaction in the supply chain isrepeated, with flexibility to change order quantities and wholesaleprices at every interaction, and the discount factor is not extremelylow, then the manager does not need to use sophisticated con-tracts, but instead can use the trigger inventory policy. Under thispolicy, the manager always chooses a jointly optimizing action, butthreatens to order less, as soon as the manufacturer initiates aprice increase. Note that for production environments with longlead times, the discount factors will be low. For example, in thefashion industry, interaction might be on a biyearly basis. In a pro-duction environment with perishable inventory (e.g., food, news-papers), the discount factor will be substantially higher. Themodel developed in this paper would thus predict that it is easierin the latter production environment to sustain a long-term part-nership than in the former, and that the long-term partnership inthe former environment will be more sensitive to demandfluctuations.

Future research should generalize our conclusions to more com-plex environments. First, the retailer and manufacturer could havea different discount factor, dM and dR, which reflects the character-istics of the respective industries to which they belong. Second, thecarrot and stick strategy with a finite punishment length might beanalyzed (see Gibbons, 1992), instead of the trigger strategy. Thismore refined strategy, which is more complex to analyze, would al-low for the long-term partnership for lower discount factors. Third,a competitive downstream or upstream channel can be modeled toredress the power imbalance between manufacturer and retailer.Fourth, physical links between the periods may be studied. Forexample, what if the market demand depends on the past servicethat is offered by the supply chain? What if inventories can be car-ried over from one period to another? A decentralized two-echeloninventory system may be studied where the order quantities arethe strategic actions that are taken repeatedly by the actor at eachechelon. In this setting, the Markov equilibrium concept can beused (see Tirole, 1988). Finally, empirical research can be done totest the impact of market turbulence on long-term supply chainpartnerships. We hope that this paper will generate further inter-

est in studying repeated supply chain interaction in turbulentmarkets.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ejor.2013.09.020.

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