A Bargaining Framework in Supply Chains- The Assembly Problem.pdf

MANAGEMENT SCIENCEVol. 54, No. 8, August 2008, pp. 14821496issn 0025-1909 eissn 1526-5501 08 5408 1482

informs doi 10.1287/mnsc.1080.0880

2008 INFORMS

A Bargaining Framework in Supply Chains:The Assembly Problem

Mahesh NagarajanSauder School of Business, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada,

[email protected]

Yehuda BassokMarshall School of Business, University of Southern California, Los Angeles, California 90089,

[email protected]

We examine a decentralized supply chain in which a single assembler buys complementary componentsfrom n suppliers and assembles the final product in anticipation of demand. Players take actions inthe following sequence. First (stage 1), the suppliers form coalitions among themselves. Second (stage 2), thecoalitions compete for a position in the negotiation sequence. Finally (stage 3), the coalitions negotiate withthe assembler on allocations of the supply chains profit. We model the multilateral negotiations between thesuppliers and the assembler sequentially, i.e., the assembler negotiates with one coalition at a time. Each ofthese negotiations is modeled using the Nash bargaining concept. Further, in forming coalitions we assumethat players are farsighted. We then predict at equilibrium the structure of the supply chain as a function ofthe players relative negotiation powers. In particular, we show that the assembler always prefers the outcomewhere suppliers do not form coalitions. However, when the assembler is weak (low negotiation power) thesuppliers join forces as a grand coalition, but when the assembler is powerful the suppliers stay independent,which is the preferred outcome to the assembler.

Key words : decentralized assembly systems; Nash bargaining; negotiation power; commitment tactics;farsighted stable coalitions

History : Accepted by Candace A. Yano, operations and supply chain management; received October 26, 2005.This paper was with the authors 1 year and 2 months for 4 revisions. Published online in Articles in AdvanceJune 20, 2008.

1. IntroductionConsider a decentralized assembly system in whicha single assembler buys complementary componentsfrom n suppliers and assembles the final product inanticipation of demand in a single period. We assumethat all the players are risk neutral, that the final prod-uct requires a single unit from each of the suppli-ers (the components are perfect complements), andthat there is a single source for each component. Weanalyze this decentralized supply chain by modelingthe interactions among the agents using a negotiationframework. In particular, we analyze profit allocationsand the issue of stable supplier alliances in this assem-bly system.Adopting a negotiation framework to study these

two issues seems natural. A substantial part of therecent operations management literature examinescontracting issues in decentralized supply chains.The parameters of a contract define the allocation ofthe supply chains profit among its players. Muchof the contracting literature uses mechanisms suchas auctions and take-it-or-leave-it offers and ignoresnegotiation, even though in practice negotiation is avery widely observed mechanism. Indeed, anecdotal

evidence and articles in the academic literature haveoverwhelmingly indicated that contractual relation-ships between agents in a supply chain are char-acterized by negotiating over the terms of trade.Sellers and buyers often negotiate price, quantity,delivery schedules, etc. Examples span service indus-tries (Bajari et al. 2002), manufacturing (Bonaccorsiet al. 2000, Worley et al. 2000), raw material markets(Elyakime et al. 2000, Iskow and Sexton 1992), etc. Animportant aspect of supply chain interactions is nego-tiation power. It is important to understand the effectsof the negotiation power of individual firms and thestrategies they pursue to enhance their relative cloutin a supply chain. For instance, it is generally believedthat one of the reasons that players join forces byforming strategic alliances is to increase their negoti-ation power. Alliances directly affect the structure ofthe supply chain, and this in turn may have repercus-sions on prices, profit allocations, etc. We analyze thestructure of supplier alliances as a function of theirrelative negotiation power.In this paper, we analyze this decentralized as-

sembly system using the following steps: First, the

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Nagarajan and Bassok: A Bargaining Framework in Supply Chains: The Assembly ProblemManagement Science 54(8), pp. 14821496, 2008 INFORMS 1483

suppliers form coalitions, i.e., seller alliances (weuse alliances and coalitions interchangeably) amongthemselves. Thus, each coalition now sells a kit ofproducts to the assembler. Next, supplier coalitionsnegotiate sequentially with the assembler on theterms of trade. These negotiations consist of a coop-erative bargaining procedure (modeled using Nashbargaining, Nash 1950) followed by a competitivegame (modeled as an assignment problem). At theend of these negotiations, the profit allocation of everyplayer is determined. Using a dynamic notion ofcoalition formation (modeled using a notion of far-sightedness by Chwe 1994), we calculate the stablesupplier coalition structure. Thus, we predict, in equi-librium, the structure of this decentralized assemblysystem as a function of the relative bargaining powerof the players.We believe that the above steps capture several

aspects of supply chain interactions. In what follows,we discuss this and the appropriateness of some theassumptions we make in our analysis.Typically, economic studies of negotiations focus

on a fixed pie that has to be allocated between twoplayers. This is not the case in this study. Here,the pie has to be allocated among the n + 1 play-ers, i.e., the n suppliers and the assembler. To exam-ine this multilateral negotiation problem, we needto make modeling assumptions regarding the formatin which the negotiation takes place. One can imag-ine a process in which all suppliers negotiate withthe assembler simultaneously. In contrast, we adopta process in which the assembler negotiates with thesuppliers, one at a time, in a sequential manner. Thesequential process reflects the reality of negotiationparadigms in several instances. The following exam-ples, although not in assembly situations, are illus-trative. A recent article (Wall Street Journal 2003a),describes how state health officials meet with drugcompanies (GlaxoSmithKline, Merck & Co, Pfizer,etc.) sequentially to negotiate prices of drugs. Anotherarticle (Wall Street Journal 2003b) describes the sequen-tial approach used by Iberia Airlines in negotiatingwith Airbus and Boeing when shopping for jetliners.Unions in the automobile industry and the construc-tion industry routinely engage in pattern bargain-ing, which is essentially a sequential approach. Twointeresting and elegant papers that deal with issuesof bargaining between a buyer and two sellers andthe sequence of negotiations are Marx and Shaffer(2003, 2004). In their papers, they show that the bar-gaining outcomes are affected by the position in thenegotiation sequence. We discuss their relevance toour paper after our analysis of sequential negotia-tions. However, the spirit of our model and style ofresults are completely different from theirs. They ana-lyze the case where negotiations are conducted with

two sellers. Their primary interest is in the effect ofsequential negotiation and power on profit allocationsand contracting. We look at the case with n suppliersand focus on the issue of supplier coalitions and thestructure of the supply chain.We argue that negotiation power of agents affects

the structure of the supply chain. One way that thismanifests itself in supply chains is in the formation ofselling (procurement) alliances. By forming alliances,each member of the alliance expects to improve hisor her position. Examples of selling alliances in sup-ply chains are numerous and encompass variousindustries, such as agribusiness (Hueth and Marcoul2003), commodity producers (Krasner 1974), and retailfirms (Lass et al. 2001). In what follows we pro-vide some examples of alliances between componentmanufacturers in supply chains that are similar tothe assembly systems that we analyze. Greene (2002)indicates several instances of alliances between com-ponent manufacturers in the semiconductor indus-try. Reasons for such alliances are manifold, such ascapacity sharing, savings due to economies of scale,increased competitiveness, and other strategic rea-sons, as well as to improve their relative negotiat-ing position. Examples include SMIC, one of Chinaslargest chip manufacturing companies, and IMEC, theBelgian research company, forming a supplier alliancethat sells 90 nm chips to Texas Instruments. Otherexamples include alliances between Asyst and Shinkoin semiconductor equipment manufacturing, etc.Stallkamp (2001) discusses alliance formations

among auto part suppliers with a view towardincreasing supplier power. In his article he mentionsa recent move in which Delphi and Lear (interior trimmanufacturers) have independently displayed inter-est in forming strategic alliances with suppliers ofwiring, carpets, and molded plastic, with the aim ofbeing major cockpit suppliers to the original equip-ment manufacturers (in this case, one of the big threeauto manufacturers). In yet another example, Sym-bol Technologies, a leading scanner manufacturer, hasformed a strategic partnership with Paxar/Monarch,a leader in bar code labeling. Together, they havebecome a major supplier of a bundle that includes abar code labeler and scanner.We assume that all cost parameters and demand

information are common knowledge to all players.We also assume that actions that players take arepublic knowledge. In particular, this implies thatat any stage players are fully aware of the resultsof previous negotiations. Information symmetry andcommon knowledge are indeed strong assumptions.However, these assumptions are not always unrealis-tic. Cost and demand information are fairly commonknowledge in the biopharmaceutical and semiconduc-tor manufacturing sectors (see Plambeck and Taylor

Nagarajan and Bassok: A Bargaining Framework in Supply Chains: The Assembly Problem1484 Management Science 54(8), pp. 14821496, 2008 INFORMS

2007b). They analyze contract renegotiation assum-ing common public information. In the automo-bile industry, MySAP SRM, a supply chain softwarethat is commonly used, allows for a public platformwhere costs, demand, and other relevant informa-tion are shared between automakers and their suppli-ers. Further, an article (MySAP 2002) mentions thatthe state-of-the-art platform allows for informationsharing that includes auction results, bids, and pur-chase quotes of components. This can be thought ofas publicly available information (to supply chainmembers) that indicates allocations obtained throughsome negotiation-type process. EPIQ, an eNegotia-tion platform, allows buyers and suppliers to negoti-ate through an electronic marketplace. Information onprevious tradesi.e., the results of negotiationsispublicly available to registered EPIQ suppliers. In thewage-bargaining literature, where a firm negotiatessequentially with multiple unions, concessions givento a union earlier in the sequence is public knowledgeand is used by negotiating parties in subsequent deal-ings (see, for instance, Abu 1998). Krauss and Freitsis(2006) discuss sequential negotiations used in pollu-tion sharing agreements among firms. They claim thatan agent responding to an offer is informed of offersfrom preceding agents. Finally, we note that most ofthe extensive literature on multilateral bargaining ineconomics makes strong information assumptions.To summarize using the above framework, we are

interested in addressing the following two issues:(1) How are the profit allocations of the playersdetermined and how are they affected by the rela-tive negotiation power of the different players? (2) Inequilibrium, what are the stable supplier coalitionsand how are they affected by the relative nego-tiation power of the players? We provide a briefreview of topics that bear relevance to our paper andits objectives. In this process we wish to highlightour contribution to the supply chain managementliterature.The literature on supply chain contracting and

profit sharing is vast. Papers by Pasternack (1985),Cachon and Lariviere (2004), Barnes-Schuster et al.(2002), Gerchak and Wang (2004), Padmanabhan andPng (1997), and Deneckere et al. (1997) are someexamples. It is not our intention to provide a com-prehensive review of these studies. For an excellentreview, please refer to Cachon (2003). In almost all ofthese papers, although the mechanisms and structureof the supply chains differ, there are certain common-alities. Typically, the agents function noncooperativelyand maximize their own expected profits. These stud-ies identify various mechanisms that coordinate thechannel. In addition, some of these papers have madethe observation that the channel can be coordinated in

many different ways. That is, the mechanism is flexi-ble in that it allows the parameters to be set to allowarbitrary allocations of the first-best profit among theplayers.In many of these studies, a Stackelberg game is

used to model the supply chain. Stackelberg gamesusually endow players with disproportionate negoti-ation power. For instance, the agent who is the leaderusually enjoys the first-mover advantage, thereby hav-ing control over extracting shares of profits. It is nat-ural to expect environments in which the Stackelberggame does not capture the interactions between dif-ferent players. Indeed, one could argue that there areenvironments where negotiation power is not as dis-crete or extreme as dictated by a Stackelberg game.In such circumstances, it may be more desirable tomodel negotiation power as a continuous parameter.The negotiation framework used in this paper allowsus to model this aspect.We now review some papers that examine the

structure of a supply chain as pertaining to for-mation of strategic alliances. In the operations litera-ture, Corbett and Karmarkar (2001) study the effectsof competition in serial supply chains. They pro-vide a framework for studying a variety of sup-ply chain structures and the effect of cost structuresand entrants at different levels of the chain. In amore recent work, Carr and Karmarkar (2005) lookat competition in a multiechelon supply chain withan assembly structure. Their focus is on establish-ing equilibrium prices, given an assembly structure.Granot and Sosic (2005) predict the formation of sta-ble alliances in an electronic market place under pricecompetition. They use the idea of farsightedness,a dynamic concept of coalition formation. In a recentwork, Majumder and Srinivasan (2008) study a serialsupply chain in which different players adopt therole of a Stackelberg leader. They focus on determin-ing the equilibrium prices and on the optimal loca-tion of the Stackelberg leader(s). Modeling negotiationbetween the agents, or understanding their relativenegotiation powers and their impact on the supplychain structure, is not the main objective. With theexception of Granot and Sosic (2005), these papers donot examine the issue of coalitional stability. A fewpapers analyze coalition formation in supply chains.We refer the reader to Nagarajan and Sosic (2006)for a survey. However, these papers do not use thenotion of negotiation power, and are thus not directlyrelevant to this work. Our approach is completelydifferent. In our model, agents form coalitions toenhance their negotiating positions. In addition, weallow for players to be farsighted in forming alliances.This implies that players consider all possible futuredefections from any coalitions when making theirdecisions.


The operations management literature has seenvery little work that uses economic bargaining theoryto model relations between agents in a supply chain.Notable among these are van Mieghem (1999), whichlooks at negotiations on incomplete contracts, wherecontract parameters are left unspecified and the sur-plus is divided by players based on their ex post bar-gaining power. Gurnani and Shi (2006) and Ertograland Wu (2001) look at profit allocation using bilat-eral negotiations in supply chains with two players.Plambeck and Taylor (2007a, b) consider the effects ofrenegotiation on contracts. A recent paper that exam-ines supply chain profit allocations in the contextof bargaining power is Bernstein and Marx (2005).They model power using reservation profit levels ina Stackelberg game framework. Although their paperbears similarities to ours, especially in stage 1, theirfocus is more on contracting issues and less on thestructure of the supply chain. Finally, these papers donot consider assembly systems and assume symmet-ric information, with the exception of van Mieghem,whose focus is neither on the exact bargaining gamenor coalition formation.The paper is organized as follows: In 2, we

describe the Nash bargaining game as it applies to ourproblem. In 3, we describe how we model negotia-tion power in this paper. In 4, we describe the basicassembly bargaining problem (A-B-P) and solve forthe profit allocations obtained through the multilat-eral negotiations. In 5, we first discuss the conceptof farsighted coalitional stability and then character-ize the stable coalitional structures that suppliers willform. At the end of 5, we examine the issue of costsincurred by the agents due to multiple negotiationsand its impact on the stable outcomes. Future researchand conclusions are in 6. All relevant proofs aregiven in a separate online technical appendix (pro-vided in the e-companion).1

2. Negotiation ProcessTo put things into perspective, we now look at theissues that arise when we try to establish a negotia-tion framework for the assembly problem. We assumethat the shares of the profit pie that the suppli-ers and the assembler receive are calculated throughsome sort of negotiations among the parties. Also,recall our assumption that the negotiation process issequential. Thus, at any point in time the assemblernegotiates with only one supplier (or a supplier coali-tion). The negotiation between the assembler and anysupplier is captured by what we call the basic nego-tiation process, in which the two players negotiate to

1 An electronic companion to this paper is available as part ofthe online version that can be found at http://mansci.journal.informs.org/.

determine their allocation of the profit (pie). We firstspecify this basic model and then describe the sequen-tial multilateral negotiations used in the assemblyproblem.Negotiations between agents in a supply chain can

be quite complex. We do not presume to model theseexact dynamics of negotiations. To model the basicnegotiation process, we use the cooperative bargain-ing process initiated by Nash (1950) (referred to asNB). Nash proposed a basic framework to modelnegotiation among players. He assumes four axiomsthat every bargaining solution must satisfy. Nashwas able to prove that there is a unique solutionthat satisfies the four axioms. The bargaining solu-tion is obtained by solving the following optimizationproblem:

max4x11x25d11d2x1+x2

4x1 d154x2 d251

where is the pie to be allocated between the twoplayers, x = 4x11x25 are the shares (the set of all pos-sible shares is called the feasible set) of the pie thateach player obtains, and d = 4d11d25 (referred to thedisagreement points) are the utilities obtained by theplayers when they fail to reach an agreement.The NB solution is simple and robust. Rubinstein

(1982) has proved that when parameters are assignedappropriate values, noncooperative models in whichplayers make alternating offers yield results identicalto the NB solution concept. Moreover, experimentalbargaining theory indicates stronger empirical evi-dence to support Nashs bargaining theory than anyother. For an excellent review of the experimental lit-erature, we refer the reader to Roth (1995). The two-person NB game has another very nice property inthat it is somewhat robust to bargaining games withasymmetric information. Indeed, one of the limita-tions of our paper is the assumption that the play-ers have symmetric information. This means thatall players know the demand distribution for thefinal product and the cost structures of the otherplayers, including the disagreement points. However,Harsanyi and Selten (1972) show the robustness of theNB concept in situations where information is incom-plete. They show that bargaining games with asym-metric information can be thought of as a suitablydefined generalized Nash Bargaining game. Thus,we believe that despite the limiting assumption, thestructure of the results presented in this study pro-vides valuable insights in understanding the effectsand outcomes of negotiation in supply chains.In the next section, we develop the specifics of the

model that are relevant to this paper. For a detaileddescription of the general NB problem, we refer theinterested reader to Roth (1979).


2.1. Nash Bargaining Problem in Our SettingWe start by describing the basic negotiation modelin which a risk-neutral supplier and a risk-neutralassembler negotiate the profit allocations. The NBgame requires us to identify a feasible set of payoffsthat describe the possible allocations that the playerscan obtain through negotiations and a disagreementpoint (say d), which describes the outcome (payoffs) ifthe players fail to reach an agreement. The economictheory of bargaining intuits from the problem ofdividing a fixed pie, with the players having fixed dis-agreements. It is also assumed that the size of the pieand the disagreement payoffs are predetermined andare independent of the negotiations and actions of theplayers. Naturally, in our setting we assume that theplayers negotiate on the share of the expected pie. Weobserve that due to Pareto optimality of the NB solu-tion, players recognize that it is in their interest tonegotiate on a pie that corresponds to the first-bestsolution. In particular, we note that this observationholds in the rest of the paper and will not be repeated.Throughout this paper, we denote the channel coordi-nated pie by C . Thus, using a cooperative approachmeans that the players do not go through the motionsof a Stackelberg-type game (a noncooperative solu-tion) and the well-known phenomenon of doublemarginalization (for instance, see Pasternack 1985) isnot relevant to our analysis. This approach echoes thesentiments of Brandenburg et al. (1997), who in theirpopular book, Co-opetition, introduce the concept ofvalue net, in which supply chain players cooper-ate and are partners in creating value by maximizingthe supply chains profit, thereby avoiding certain pit-falls of competition. Thus, they stress the value of acooperative symbiosis between the various agents ina supply chain. However, they also indicate that thesame players may compete for their shares of the sup-ply chain profit.Before concluding this section, we note that we

assume that the parameters of our problems are suchthat the channel earns positive expected profit. Thisensures that the feasible set of the bargaining problemis not empty. Thus, because the players are rational,they will find it advantageous to agree on some allo-cation of the pie and will never choose to disagree.

3. Negotiation PowerIn this paper, we allude to negotiation power or bar-gaining power several times. There are two aspectsof negotiation power. The first is specific to the eco-nomic model that is used to describe the negotiationprocess. This could translate to specific characteristicsthat the players may possess (e.g., utility of the play-ers, risk preferences, credibility of threats, etc.) Thesecond aspect is the negotiation power that arises due

to the structure of the supply chain. Examples wouldbe the number of suppliers in the supply chain, thetiming of their negotiation, etc. This aspect is muchmore complex, and we will defer elaboration on thissecond aspect to the very end, when we will have putforth a complete description of the assembly problemand the results thereof.Throughout this paper, we make two assumptions.

First, we assume that players are risk neutral. Second,we normalize disagreement points to zero. The firstassumption is for tractability. In our setting, risk aver-sion makes it very hard, if not impossible, to obtainclosed-form expressions for allocations of profits. Thisdistracts us from the objective of getting tractableexpressions and structural results, which makes itimperative to preserve the linear structure offered byrisk neutrality. The second assumption does not affectthe quality of our results when we use the NB solu-tion. To see this, let us define the net profit of aplayer to be the difference between the allocation thata player receives and his disagreement point. It iseasy to see that when players are risk neutral, despitehaving unequal disagreement points, the NB solu-tion allocates them equal net profits. This resonateswith the general idea that risk-neutral players in anNB game are equally powerful. Indeed, this resultis strongly supported by empirical evidence. See, forinstance, Raiffa (1985) and Stone (1958).

3.1. Generalized Nash Bargaining (GNB) GameRisk preference is an approach to model negotia-tion power. The NB solution allocates a higher pay-off to the player who is less risk averse than hisopponent. However, as mentioned, we want to retainrisk neutrality. The GNB solution provides a recipefor capturing negotiation power without relaxing riskneutrality. It is obtained by ignoring the axiom ofsymmetry in the NB game. It can be shown (Roth1979) that the remaining axioms determine a familyof solutions of the form:

argmax4x11x25d11d2x1+x2

4x1 d154x2 d251

where is a pie to be allocated and += 1. Theindices and are loosely representative of the indi-vidual powers of the agents. For instance, when twoplayers with zero disagreement values negotiate on apie of size one, the GNB solution implies that the firstplayer gets x1 = and the second player receives x2 =1= . Our assumption normalizing the disagree-ment points to zero may seem restrictive when theindices are unequal. A natural interpretation of theseindices is given by Muthoo (1996). This interpreta-tion uses what are referred to as commitment tactics.Commitment tactics allows us to preserve linearity,


and thus ensure that our assumption on the disagree-ment points is not restrictive, even when the indicesare unequal. We now describe the use of commitmenttactics in the context of Nash bargaining, when theassembler (A) negotiates with a supplier (S).As before, C is the pie, the allocation of which

the players are negotiating. Let zi i 2 8S1A9 be theindependent choices of the two players, and play-ers commit to receiving no less than these choices.However, these commitments are partial, and playerscan revoke their commitments at a cost. For instance,the cost may be a loss of credibility. Let Ui4xi1zi5 =xi Ci4xi1zi5 be the actual utility to player i fromreceiving a share xi of the pie and making a commit-ment to receive no less than zi. This utility, as seenfrom the above expression, is equal to the actual allo-cation the player receives, less the cost of revoking hiscommitment. Muthoo assumes a linear cost, given by:

Ci4xi1zi5=(ki4zi xi5 if xi zi10 if xi > zi1

ki > 00

Muthoo proves that 4zA1zs5 = 441 + kA5C/42 + kA + kS5, 41 + kS5C/42 + kS + kA55 is the uniqueNash equilibrium of the players commitments. Thus,a players partial commitment, which is identical tothe equilibrium share of the pie that he obtains, isincreasing in his cost of revoking a commitment anddecreasing in the corresponding cost of his opponent.In other words, a player whose revoking costs arehigh is considered more credible when he makes acommitment, and, consequently, is more powerfuland is awarded a larger share of the pie. Such com-mitment tactics are not unusual and often signal aplayers negotiation power. Bacharach and Lawler(1981), Schelling (1960), and Cutcher-Gershenfeldet al. (1995) discuss the role of these tactics andgive examples from industry. Further, Muthoo (1999)shows that = 41+ kA5/42+ kA + kS5, where is theparameter in the GNB the argmaxxA+xSC 4xA5

4xS5,

+= 1. Thus, we think of as the negotiationpower of the assembler. Using this result as the basicoutcome of any two-person negotiation, we can ana-lyze stage 2 of the game, where the assembler nego-tiates sequentially with suppliers having revokingcosts ki, i = 11 0 0 0n. In what follows, we perform thisanalysis for the case where all suppliers have the samerevoking costs (we discuss the implications of thisassumption in the end of 4). Thus, we let ki = ks , 8 iand define = 41+ kA5/42+ kA+ ks5 and = 41+ ks5/42+kA+ks5 consistent with the discussion of the GNBgame. Thus, henceforth, at each negotiation step, weuse the GNB with parameters as described above.

4. Single Assembler n SuppliersAssembly Bargaining Problem

Having described the two-person bargaining game,we are now in a position to describe the generalassembly problem in which there are n suppliers.Our model is a three-stage game. In stage 1 (the

first stage), the suppliers form coalitions. In stage 2,supplier coalitions compete for positions in the nego-tiation sequence. Finally, in stage 3 (the last stage), theassembler negotiates with the different suppliers orcoalitions of suppliers. To determine the stable out-come of stage 1, we work backward, starting fromstage 3. In this section, we discuss the exact negotia-tion process between the assembler and the individ-ual suppliersi.e., stage 3. The reader, without anyloss of generality, may substitute a supplier by a coali-tion of suppliers.We formulate stage 3 of the A-B-P, where the

assembler negotiates sequentially with n suppliersas an n-substage problem. A sequential frameworkraises several interesting issues. If the profit alloca-tions obtained by the bargaining solution is a func-tion of the negotiation sequence, i.e., the sequence inwhich the negotiations are conducted, then it is nec-essary to decide how the sequence is determined. Weassume that the assembler determines the negotiationsequence (to be discussed in stage 2). We first describestage 3 for any given negotiation sequence and thendetermine the equilibrium sequence in stage 2. Atthe first substage of stage 3, the first supplier getshis share of the profit. The assembler gets the shareof the profit that will be further allocated betweenhim and the remaining suppliers. Similarly, at the ithsubstage, the assembler and the ith supplier negotiateover allocations of the profit carried over from negoti-ations with the i1th supplier. The negotiation at thissubstage determines: (i) the profit of the ith supplier,and (ii) the portion of the pie that the assembler gets,which he will carry over for himself and the remain-ing suppliers (i+ 1 through n).For our analysis to work, we need a few simpli-

fying assumptions in our sequential approach. Weassume that once the assembler negotiates the termsof a contract with a supplier, they sign a conditionalcontract. The contingency in the contract is that theactual transaction will take place only if an agreementis reached with all suppliers. The reason for such anagreement is to prevent any kind of hold up prob-lems that may arise. For instance, if the assemblerpays suppliers as soon as each negotiation ends, thelast supplier in the negotiation sequence can extractgreater profits from the assembler who has alreadypaid for the components from the previous n1 sup-pliers, but cannot assemble the final product withoutthe components provided by the last supplier.


To describe the feasible set in this game, we firstintroduce a few notations. The negotiations betweenthe suppliers and assembler determine a share of Cfor each player. Also, throughout this discussion, weuse x (with suitable subscripts) to denote variablesin the bargaining game and (with suitable sub-scripts) to denote the solution of the bargaining. Con-sider now the negotiations between the ith supplierand the assembler, i.e., the ith substage. Thus, let xA1 ibe the pie that the assembler and suppliers i, i +11 0 0 0 1n divide among themselves. Thus xA11, the pieto be allocated between the assembler and the suppli-ers 11 0 0 0n is nothing but C . Let xi denote the shareof the channel profit that the ith supplier receives.Because the GNB solution is Pareto optimal, xA1 i+1 +xi = xA1 i. The feasible set for each substage is thusdefined recursively. At the first substage, define F1 =84xA121x152 xA12+x1 =C9 to be the feasible set. At theith substage, feasible set, defined recursively is Fi =84xA1 i+11xi52 xA1 i+1+xi = xA1 i9. At the ith substage, theith bargaining game is constrained by the ith feasibleset and is given by max4xA1 i+11xi52Fi 4xA1 i+15

4xi5. Thus,

at the nth substage, the bargaining problem reduces tomax4xA1xn52Fn4xA5

4xn5. Working backward and solv-

ing constrained problems at each step, we can solvethese recurrence relations to obtain the shares of Cof the assembler and suppliers. The following is theresult of this process, which we state without a proof:

Theorem 4.1. Thesolu ion of heA-B-P wi h n suppli-ers implies ha hepro of heassembler and he i h sup-plier are, respec ively, A = nC and i = i1C .The above allocations are for any given negotia-

tion sequence. Notice that the assembler is indifferentto the sequence of negotiation. Moreover, it is alsoclear that suppliers prefer to negotiate earlier in thesequence because their profits decrease with i. Thus,even though all suppliers are equally powerful, theyreceive unequal shares in view of their position inthe negotiation sequence. In stage 2, described next,we resolve the issue of what any ultimate negotia-tion sequence would look like. We use a model ofsupplier competition and call this stage the assemblybargaining problem with competition (A-B-P-C).

4.1. A-B-P-C GameIn the second stage of the game, we let the suppli-ers compete for a position in the negotiation sequence.Because suppliers prefer to bargain earlier and theassembler is indifferent to the sequence, it is rea-sonable to expect that the suppliers would competefor positions in the negotiation sequence. We haveassumed that the assembler chooses the sequence.Hence, to resolve this competition, we assume that thesuppliers are willing to make payments to the assem-bler to obtain a favorable position. Payments made

for favorable positions could be through discountson wholesale prices, etc. There is empirical and the-oretical evidence to suggest that to secure favorablepositions in negotiation outcomes, players take actionsexplicitly in the form of concessions, discounts, etc.Marx and Shaffer (2001, 2003, 2004) discuss a situa-tion in which a buyer negotiates sequentially with twosellers. Similar to our model, sellers have a preferenceon positions in the negotiation sequence. In their 2001paper, they resolve this by calculating payments thatsellers are willing to make for favorable positions. Intheir 2004 paper, they tackle this issue by using rentshifting contracts that can take the form of discountcontracts contingent on terms of trade with both sell-ers. In the debt restructuring literature, Noe and Wang(2000) examine a situation in which a firm negotiat-ing with several indebted creditors may in some casesbe indifferent to the sequence of negotiations, but thecreditors may be sensitive to the negotiation sequence.They claim that this situation may lead to debtorsobtaining concessions from the creditors. In the U.S.cable television market, the business press claims thatmultiple-system operators (MSOs) prefer to negotiatefirst with the broadcasters and offer concessions todo so, which the smaller local cable-operating compa-nies could not afford. The Federal Cable Act of 1992ordered the FCC to put in place restrictions on the sizeof MSOs and to monitor the concessions that MSOsnegotiate with broadcasters. In the computer scienceliterature, where protocols are designed for B2B andB2C marketplaces where bilateral sequential negotia-tions (BSNs) are commonplace, it is recognized thatprotocols need to take into account the advantagesthat a position in the negotiation sequence may createfor an agent. Protocols are available that allow for bidson the positions in the sequence in certain BSN sys-tems. We refer the reader to Bazzan and Labidi (2004)and Endriss (2006). In what follows, we describe thegame and the resulting allocations.To avoid any confusion between indices, we index

suppliers by i, i = 1121 0 0 0 1n and the positions inthe sequence by j , j = 1121 0 0 0 1n. The strategy ofthe ith supplier is denoted by STi. We define STi =84pi11p

i21 0 0 0 1p

in5 2 Rn9, where pij is the payment that

supplier i is willing to make in order to be in position j .Thus, we assume that every supplier announces a pay-ment vector, whose components are the payments heis willing to pay for each position in the negotiationsequence. Further, let ij be the profit that the ith sup-plier will make by occupying the jth position in thenegotiation sequence, prior to making the payment tothe assembler. Thus, ij = j1C , 8 i1 j . Hence, hisnet profit, after making the payment, will be ij pij .The supplier who is the last member of the negoti-ation sequence will make a profit of n1C . Thisis the lowest profit that any supplier can possibly


earn. Thus, pin = 0, 8 i. In fact, extending this lineof reasoning, if pj = ij n1C , it is sufficient toconsider the strategy space of supplier i to be STi =84pi11p

i21 0 0 0 1p

in5 2Rn9; 0 pij pj .

To summarize, in the A-B-P-C game, the n sup-pliers, by simultaneously offering the assembler thepayment vector STi, compete with each other fora position in the negotiation sequence. The assem-bler determines the position of each supplier in thesequence by maximizing his profit, which requireshim to solve an assignment problem. Thus, the sup-pliers payoff by announcing a payment scheme isdetermined by the position allotted to them by theassembler. For the same reason, we do not considerthe assembler as an active player in the A-B-P-Cgame, and limit his role to that of an automaticmachine that solves an assignment problem, therebyassigning positions to suppliers. Thus, a unilateraldeviation by a player occurs if a supplier changes hispayment strategy, i.e., changes at least one elementin his payment vector. We continue to assume perfectand complete information among the players.

Theorem 4.2. The A-B-P-C game possesses a Nashequilibrium. A every Nash equilibrium, he following arerue21. The supplier in posi ion j makes a paymen pj .2. Every supplier earns a ne pro equal o n1C .3. Theassembler 's ne pro equalsC41nn15.Notice that the ability of the assembler to deter-

mine the negotiation sequence forces all the suppliersto accept net profits equal to the net profit of the lastsupplier in the negotiation sequence. Thus, the assem-blers ability to determine the negotiation sequenceendows him with a greater level of negotiation powerthan that derived from his ability to make commit-ments. The above result indicates that if, for instance,the supply chains costs are bounded (i.e., the costsincurred by the supply chain are uniformly boundedwith respect to n), the assembler always prefers tohave a large number of suppliers. This is in contrastto Theorem 4.1, where the assembler preferred fewersuppliers. The difference is due to the fact that inthe second stage of the game, the assembler benefitsfrom downstream supplier competition for negotia-tion position. Competing suppliers lose some of theirnegotiation power, thereby allowing the assembler tocapture a larger share of the profits. This result isalso driven by the fact that we have not accountedfor any kind of negotiation and supplier managementcosts incurred by having a large number of suppliers.We defer our analysis of the model with negotiationcosts to 6, at which time the impact of such costson the resulting coalitional structures can be fullyappreciated. Note that the suppliers can reduce thecompetition by forming alliances among themselves.

Forming alliances among themselves and negotiatingwith the assembler may result in increased allocationsfor the alliance members. Alliance formation raisesbasic questions of stability. In the next section, weassume that suppliers form alliances among them-selves, thereby perhaps improving their profits, anddiscuss coalitional stability.Finally, we point out that our assumption that all

suppliers have the same revoking costs enables us toovercome two sets of difficulties. First are the diffi-culties associated with the competitive stage 2 of theanalysis. When supplier revoking costs are unequalin the bargaining stage, suppliers not only preferearly negotiating positions, but also prefer to fol-low weaker suppliers. This immediately implies thatto analyze the competitive game, one needs to con-sider a somewhat complicated and perhaps unrealis-tic strategy space. However, even when players haveunequal revoking costs, with suitable assumptions,we can analyze this stage of the game. We discussthis in greater detail in 6. The second set of difficul-ties relates to the analysis of supplier coalitions. Whenrevoking powers are identical, one can make a formalargument that the revoking cost of a coalition of sup-pliers is simply equal to their average revoking costand players in a coalition divide the gains equally.This solves two problems: (i) the negotiation powerof the coalitions is now known and (ii) individualallocations of the coalitions profit are easily obtain-able. However, when players with unequal negotia-tion powers form a coalition, there is no formal recipeto arrive at the negotiation power of the coalition.This has been an issue that has troubled game theo-rists and economists for quite some time (for a dis-cussion, see Chae and Heidhues 2001) and is yet tobe suitably resolved. Nevertheless, one may arguethat, in reality, the negotiation power of a coalitionis in some way related to its size, perhaps increas-ing with the size of its membership. In 6, we discussthis aspect of the model for the case where playershave equal revoking costs and the robustness of ourresults.

5. Analysis of Supplier CoalitionsFinally, we reach stage 1 of the game. Here the sup-pliers may form coalitions to increase their profits.We make a few simplifying assumptions. First, weassume that coalitions do not benefit from economiesof scale. Thus, the production cost of a coalition ismerely the sum of the costs of its individual members.Second, we assume (for reasons of tractability dis-cussed in 4) that every supplier has the same cost ofrevoking a partial commitment, and thus equal nego-tiation power. We make no assumption on the assem-blers revoking cost. Thus, we let the assemblers


negotiation power be arbitrary (0 1), and exam-ine cases where he is more, as well as less, powerfulthan any one supplier. We assume that the revokingcost of a coalition is the average revoking cost of thecoalitions members. In this case, the revoking costof every coalition is exactly the same, and is equalto the revoking cost of each of its members. Finally,we assume that every coalition member gets an equalshare of the coalitions profit. Because the suppli-ers have identical revoking costs, they thus have thesame negotiation power. Thus, obtaining equal sharesof the coalitions profit is consistent with the idea thatthe members negotiate (modeled using NB) and splitthe coalitions profits. Notice that although each coali-tion has the same negotiation power as any one ofthe suppliers, forming coalitions alters the structureof supply chain, thus possibly increasing the profits ofthe individual coalitions and the suppliers thereof.As before, we denote the set of suppliers by

8112131 0 0 0 1n9 = N . A coalition formation game(in our context) is defined by G = 4N 1Z1 8


emphasis on the word consider. Clearly, when play-ers form coalitions, they reach some kind of an agree-ment. Thus, one can think of a contract that is signedby players binding them to their affiliation. Indeed,even if there are no legal obstacles, defections andre-formations are costly. The LCS uses the logic thatplayers take into account all possible defections byfellow players before agreeing on an outcome. Thus,if the n players are involved in some kind of a negoti-ation process that will eventually determine a mutu-ally amicable outcome (a coalitional structure), theLCS takes into account the possible scenarios threatsand counterthreats involving defections and counterdefections that the players will consider before sign-ing a binding agreement. Thus, the LCS should not beconstrued as a concept in which players defect in var-ious time periods, and in doing so, accrue payoffs ateach step. Neither does the farsightedness idea implythat the dynamics used in the analysis are meant toreflect and explain the exact dynamics of splits andmergers of firms in the real world. The spirit of thefarsighted stable concept lies in the fact that decisionmakers may account for long-term trade-offs whenforming coalitions. Without further ado, it is definedbelow, and is used as a stability criterion in our anal-ysis of stable alliance structures. We begin by con-sidering the game G as described above, with a fewadditional notations and definitions that will help usin describing the LCS.An outcome a is indirectly dominated by b4a G b5

if there exist sequences a = a01a11a21 0 0 0 1an = b andS11S21 0 0 0 1Sn such that 4ai !Si+1 ai+15 and ai


the player {1} who defected from the grand coali-tion contemplates the possibility that his initial defec-tion may pave the way for a further defection from{1}, 82139 to {1}, {2}, {3}, which may lead back to thegrand coalition, thus yielding him no benefits fromhis initial move. This deters the initial defection fromthe grand coalition. However, notice the evolution ofother stable outcomes with two unequal coalitions. Inparticular, part (3) of the theorem implies that whenfacing a more powerful assembler than in the previ-ous instance, the suppliers may form stable outcomesin which there is more than one coalition. However,not all two-coalition outcomes are stable. For instance,one can show that for even values of n> 4, outcomeswith two equal-sized coalitions are not stable.In what follows, we demonstrate the structure of

certain stable outcomes for > 005. We show thatas the assembler becomes stronger, outcomes withincreasing numbers of coalitions become stable. Inparticular, we are interested in showing the stabilityof approximately equal-sized coalitions. To this end,we need a few definitions and notations before statingour results.Define =3 as the outcome with three equal coali-

tions. When nmod3= 0, =3 contains three equal coali-tions. When n= k+ 1, where kmod3= 0, =3 has twocoalitions of size k/3 and one coalition size k/3+ 1.Finally, when n= k+ 2, kmod3= 0, =3 has two coali-tions of size k/3 + 1 and one coalition of size k/3.Thus, every coalition in =3 has less than or equal to4n+ 25/3 members. For some of the above notationsto be well defined and for several of the followingresults, we require that n be large and sometimeseven. From the proofs, it will be clear that extensionsto odd n are straightforward and require very similartechniques.We begin with a result that is very useful in

characterizing stable outcomes for > 005.

Proposition 5.1. Le > 005. For every ou come aand for any i 2N if a!i b, hen a 005, it is

always myopically beneficial for a single player todefect from a coalition. Moreover, it highlights theimportance of using a farsighted concept to charac-terize stable outcomes. A myopic concept of stabil-ity (the coalitional structure core, for instance) willpredict that for > 005, no supplier coalition is stable.

Theorem 5.2.1. Le n be even and large. Le 005< < 2/3. The ou -

come x= 81121 0 0 0 1n/291 8n/2+ 11 0 0 0 1n9 is s able.2. Le n be largeand even. Le A= 81121 0 0 0 1n/2191

8n/21 0 0 0 1n9. There exis s 4b11 b25 400512/35 such ha8 2 4b11 b25, he ou comeA is s able.3. Le n be large. There exis 2/3< 1 < 2 < 3/4 such

ha for 2 41125, =3 is s able.

Recall that we observed stable outcomes with twocoalitions (of unequal size) when < 005. When theassembler becomes stronger, we observe that out-comes with two coalitions are still stable, but unlikebefore, the sizes of the coalitions become more equal.Indeed, part (1) of the above theorem demonstratesthe stability of an outcome with two coalitions ofequal size, and part (2) predicts that coalitions withsizes being close to equal are also stable. Finally,part (3) demonstrates the stability of outcomes withthree coalitions (of equal or close-to-equal sizes) as theassembler becomes even more powerful.Our results in the above theorem hold for appropri-

ate large values of n. For small values of n, one cancompute stable outcomes by brute force. However,large values of n allow us to observe the evolution ofcoalition structures and to derive some structural pat-terns. Thus, we focus on larger values of n. In fact, forthe following theorem, by large, we mean valuesof n as small as eight.A natural prediction would be the evolution of

these equal-sized coalition structures as the value of increases. Let =k be defined in a similar fashion asabove. Using similar proof techniques as in the aboveanalysis, we can demonstrate the following theorem(which we state without proof), which is quite generalin its prediction of stable outcomes.

Theorem 5.3. For > 005 and n large, here exis005 = 1 < 2 < < n = 1, such ha =k is s able for 2 4k11 k5.Theorem 5.3 demonstrates the evolution of stable

structures with increasing number of coalitions as theassembler becomes increasingly powerful. One mayhypothesize that a natural culmination would be thatwhen facing an extremely strong assembler, the sup-pliers would not form any coalitions (or one couldalternatively say that they form n coalitions) amongthemselves. Indeed, we have the following result.

Theorem 5.4. Le n> 2 and > 400551/4n25. Then u=8191 8291 0 0 0 1 8n9 is he only s able ou come.This result is quite interesting. Assume that there

are five suppliers. Also, assume that the assembler ispowerful (= 008). In this case it is in the best inter-est of the suppliers to act independently and not toform any coalition. By staying independent, each sup-plier gets 8.1% of the total profit. This is in contrastto the 4% that each member of the grand coalitionobtains. By staying independent the suppliers are ableto double their profits as compared to the profits ofmembers of the grand coalition. This is somewhatsurprising. One would expect that the suppliers willalways benefit by joining forces. Although this is truewhen the assembler is weak, it is not true when theassembler is powerful.


We have thus exhibited stable outcomes for differentvalues of . In some cases we can show uniqueness.In certain cases, although we cannot show unique-ness, we can rule out certain possibilities. For instance,when 2 41/212/35, we know that =2 is stable. At thispoint, we can eliminate g as a possible stable outcomeusing the following rationale. Suppose g is farsightedstable, but we have, in the given range of , =2 >i gfor every i. Thus, it stands to reason that, in this case,g is unlikely to be a stable outcome.An aspect of negotiation that we have ignored is the

associated cost. Consider the case where the assem-bler suffers a cost b per supplier (or coalition) withwhom he negotiates. Indeed, the assembler is bur-dened with a certain cost that is proportional to thenumber of negotiations he conducts. Thus, if he hasto deal with n supplier coalitions, his total costs dueto the negotiation process is bn. In general, the cost bcan be allowed to reflect any kind of transaction costs.Solving the A-B-P game with this additional cost,we get very similar outcomes for all three stages ofthe games. Indeed, the structures of the stable out-comes are quite similar, except that the correspond-ing ranges of are now shifted. Notice that when acost of negotiation is applied, the normalized profitshrinks from 1 to (1 tb), where t is the number ofcoalitions. Notice that even though we assume thatthe bargaining cost is incurred only by the assem-bler, it also affects the profits of the suppliers. Thus,when all suppliers form a single coalition, the profitof each player is 41541b5/n. The maximum profitof any supplier in the outcome 8191 82131 0 0 0 1n9 is41541 2b5. As a result, when 41541 b5/n >41 541 2b5, the grand coalition directly domi-nates every other outcome for every player. Thus,we have the following proposition, corresponding toTheorems 5.1 and 5.4, which we state without proof.

Proposition 5.2. (1) For < 41 b5/6n41 2b57 hegrand coali ion g is uniquely s able.(2) Le nb < 1. If > 641 2b5/241 nb571/4n25, hen

u is uniquely s able.As expected, we observe that due to the intro-

duction of the cost of negotiation, it is now morelikely that the grand coalition will be the only sta-ble outcome. Similarly, to reduce the negotiation cost(thereby increasing the size of the pie), the suppliersare likely to form fewer coalitions. This is reflected inthe extreme case where nb approaches one. Here theoutcome u = 8191 0 0 0 1 8n9 is stable only when > 1,which is never true.

6. DiscussionIn this paper, we have developed a general frame-work for negotiations in a supply chain. We are ableto extend the two-person GNB solution to a sequential

multilateral negotiation process, and are able to modelnegotiation power as a function of the cost of revokinga partial commitment. We apply this framework to theassembly bargaining problem. The assembler is giventhe power to choose the sequence in which he negoti-ates with the suppliers for allocations of the profit pie.In a sequential setting, we show that the assemblersprofit is not affected by the sequence of negotiation,but is a function of the number of suppliers and theirrelative negotiation powers. The profit of the suppliersis a function of their position in the negotiation processand their negotiation power. Assuming suppliers withidentical negotiation power, we model the competitionamong the suppliers for a position in the negotiationsequence. We show that there is a Nash equilibrium inpayments, and at every equilibrium the effective prof-its of the suppliers are identical. The assembler wouldideally prefer to have as many suppliers as possibleto extract the entire profit. This is an artifact of ignor-ing costs incurred by the assembler in negotiating witha supplier. We then empower the suppliers to formcoalitions among themselves.We borrow the concept of farsighted stability to

analyze the suppliers responses. Doing so, we arealso able to determine the stable coalition structureof the supply chain. The coalition structure is a func-tion of the number of suppliers and their negotiationpower. We find that it is not always in the best inter-est of the suppliers to join forces and form the grandcoalition. In fact, the structure seems to have a cer-tain interesting dynamic with respect to the assem-blers negotiation power. When the assembler is veryweak, the suppliers unite as one grand coalition andnegotiate for their share. Further, we show that as theassemblers negotiation power increases, the suppli-ers tend to form smaller coalitions. Thus, at the otherextreme, when the assembler is immensely powerful,the suppliers organize themselves individually anddo not form any coalitions. In intermediate situations,we see stable outcomes that differ from the above twooutcomes.We now pause and reflect on the comments made

in 3. When speaking of negotiation power, we men-tioned two aspects. The first is an artifact of the eco-nomic model that is used to capture the bargainingprocess. Our choice of Nash bargaining implied thatthe ability to make credible commitments affectedthe individual negotiation power of the players. Thesecond aspect of negotiation power was extraneousto the choice of the specific model that we use todescribe negotiations. Having completed our analy-sis of the supply chain, we are now in a position toexpound on this feature. The assembler at the firststage chooses a sequential negotiation process. Theoutcome dictates that he receives a rather small por-tion of the profits. Indeed, recall that at this stage, the


assembler prefers to have as few suppliers as possible.However, this immediately gives rise to a situationwhere the suppliers compete for positions in thenegotiation sequence. Competition among the suppli-ers immediately increases the negotiation power ofthe assembler. Indeed, the fact that the suppliers com-pete reverses the assemblers preference on the num-ber of suppliers. Determinants of negotiation powerin a supply chain are manifold. Clearly, competition isone of them, and we demonstrated its impact on oursupply chain. The ability to form alliances is another.The third stage of our game specifically addresses thisaspect. The suppliers retaliate by forming coalitions.We assumed that the effective revoking cost of thecoalition is exactly the average, and thus equal to therevoking cost of any individual supplier. However, byforming coalitions, the suppliers affect the structureof the supply chain. They traded off the benefits ofcompeting for a position in the negotiation sequencewith the number of times they would negotiate withthe assembler. Once again, this changes the negoti-ation power of the suppliers as well as that of theassembler.We have illustrated examples of supplier coali-

tions in various parts of the paper (bar code labelersPaxar/Monarch, automobile suppliers, etc.). Com-panies such as Cisco Systems (Souza 2003) oftenpurchase components not only from individual sup-pliers, but also from sellers of kits of components(e.g., Solectron). In another example, the TaiwaneseSemiconductor Industry Association (http://www.tsia.org.tw/) reports that it provides a platform forsmaller suppliers to come together and negotiatetrade terms with manufacturers from overseas. Theseplatforms usually involve a coalition of semicon-ductor component manufacturers who trade with abuyer. We are not suggesting that outsourcing or theexistence of virtual coalitions can be fully explainedthrough our model. The reasons may be manifold. Forinstance, coalitions may also serve coordinating andrisk-pooling functionsWe now discuss two of our assumptions mentioned

at the end of 4, and inferences that may be drawnwhen relaxing them. The first one of these relates tothe game where suppliers compete for positions whenthey have unequal revoking costs. When suppliershave unequal revoking costs, we can show, using asimilar analysis as in the proof of Theorem 4.1, thata suppliers profit allocation, for a given position inthe negotiation sequence, depends on the set of sup-pliers negotiating ahead of him in the sequence. Asbefore, the assemblers profit remains independent ofthe sequence. Assume that suppliers are willing tomake payments for positions that are conditioned onthe set of players negotiating ahead of them. We canshow, similarly to Theorem 4.2, that this game has

a Nash equilibrium, and in every equilibrium, everysupplier receives exactly the amount he would get ifhe negotiates in the last position in the sequence. Eventhough this is similar to our earlier results, the anal-ysis crucially depends on the assumptions regardingthe feasible strategy space of the suppliers.The second assumption relates to the issue of the

negotiation power of the coalition when players haveequal revoking costs. We assume that the negotiationpower of a coalition is the average of the negotia-tion power of its members, and thus, the negotiationpower of a coalition is equal to the negotiation powerof each of its members. Although accurate from thegame-theoretic perspective, there is merit in the argu-ment that, in reality, negotiation power of a group mayincrease with its size. To incorporate this realism intoour discussion, we do the following. We continue toassume that players have equal revoking costs. Recallfrom our discussion in 4 that there is no formalframework to calculate the coalitions utility/powerwhen its members have unequal negotiation power.However, now we let the negotiation power of a coali-tion depend on its size. Assume that the negotiationpower of a coalition is 1/md , where is the negotia-tion power of a supplier, m is the number of suppli-ers in the coalition, and d is an exogenously chosenparameter that controls the rate of increase of negotia-tion power of the coalition as a function of its size. Forsmall values of n, we can analytically check the stableoutcomes. To illustrate, for instance, when n= 3, =001 (the assembler is powerful), and d= 002, we get thefollowing results: If each player forms the stand-alonecoalition then each of them gets 22 + 3 = 00081.Thus each supplier will get around 8% of the pie. If, onthe other hand, the three players form the grand coali-tion then each of them will get 1/3002/3 = 00052. Thisindicates that when the assembler is strong, the stableoutcome is one in which each supplier forms a stand-alone coalition. On the other hand, when = 009,each of the suppliers in the grand coalition gets 30.3%of the pie (the assembler gets 9%), greater than the0.9% that suppliers receive when negotiating alone.For larger values of n, when values are not extreme,we find other equal-sized coalitions emerging as sta-ble outcomes. In general, we empirically found thesame trend that we observed from our earlier analy-sis. When the assembler is weak, the grand coalitionis stable and as the assembler becomes more power-ful, other stable coalitions emerge. We also find that asthe negotiation power increases more rapidly with thesize, there is a greater propensity for the grand coali-tion to emerge as the stable outcome. Moreover, thisalso implies that the marketplace sees large suppliercoalitions. This seems to indicate that there is robust-ness to our predictions.


We believe that future research should focus onmodeling negotiation power as a function of riskaversion, competition among suppliers manufactur-ing similar components, etc. It will also be interest-ing to model and understand negotiation situations inwhich the different parties possess asymmetric infor-mation. An issue of interest is to model negotiationpower exogenously as a function of the position inthe sequence. Indeed, one could argue that a supplierwho goes last in the sequence may have more negoti-ation power, because his ability to veto may be quitecostly to the assembler. This can be accommodated inour framework by letting the cost of revoking com-mitments be a function of the position. Indeed, thequestion of an equilibrium set of payments (position)is to be reckoned with. This is an important and aninteresting extension. Our results are sensitive to thedefinition of stability. A broader notion of stabilityand the examination of the results thereof are left forfuture research.

7. Electronic CompanionAn electronic companion to this paper is available aspart of the online version that can be found at http://mansci.journal.informs.org/.

ReferencesAbu, P. B. 1998. A multilateral bargaining approach towards

effective dispute resolution in the public sector. Ph.D thesis,University of Ibadan, Nigeria.

Aumann, R. J. 1959. Acceptable points in general cooperativen-person games. A. W. Tucker, R. D. Luce, eds. Contributions tothe Theory of Games IV. Princeton University Press, Princeton,NJ, 287324.

Aumann, R. J., J. H. Dreze. 1974. Cooperative games with coalitionstructures. Internat. J. Game Theory 3 217237.

Bacharach, S. B., E. J. Lawler. 1981. Bargaining, Power, Tactics andOutcomes. Jossey-Bass Press, San Francisco.

Bajari, P., R. McMillan, S. Tadelis. 2002. Auctions versus negotia-tions in procurement. An empirical analysis. Working paper,Department of Economics, Stanford University, Palo Alto, CA.

Barnes-Schuster, D., Y. Bassok, R. Anupindi. 2002. Coordination andflexibility in supply contracts with options. Manufacturing Ser-vice Oper. Management 4(3) 01710207.

Bazzan, A., S. Labidi. 2004. Advances in Artificial IntelligenceSBIA 2004: 17th Brazilian Sympos. Artificial Intelligence, Sao Luis,Maranhao, Brazil, 2004 Proceedings. Springer Verlag, Germany.

Bernheim, B. D., B. Peleg, M. D. Whinston. 1987. Coalition-proofNash equilibria, I. Concepts. J. Econom. Theory 42 112.

Bernstein, F., L. Marx. 2005. Reservation profit level and the divi-sion of supply chain profit. Working paper, Duke University,Raleigh, NC.

Bonaccorsi, A., T. P. Lyon, F. Pammolli, G. Turchetti. 2000. Auctionsversus bargaining: An empirical analysis of medical deviceprocurement. LEM working paper series, Sant Anna School ofAdvanced Study, Pisa, Italy.

Brandenburg, A. M., B. J. Nalebuff, A. Brandenburg. 1997.Co-opetition. Doubleday Publishers, New York.

Cachon, G. P. 2003. Supply chain coordination with contracts.S. Graves, T. de Kok, eds. Handbooks in Operations Research andManagement Science2 Supply Chain Management. North Holland,Amsterdam.

Cachon, G. P., M. A. Lariviere. 2004. Supply chain coordinationwith revenue sharing contracts. Management Sci. 51(1) 3044.

Carr, S. M., U. S. Karmarkar. 2005. Competition in multiechelonassembly supply chains. Management Sci. 51(1) 4559.

Chae, S., P. Heidhues. 2001. Bargaining between groups. Workingpaper, Rice University, Houston.

Chwe, M. 1994. Farsighted coalitional stability. J. Econom. Theory63(2) 259325.

Corbett, C. J., U. S. Karmarkar. 2001. Competition and structure inserial supply chains with deterministic demand. ManagementSci. 47(7) 09660978.

Cutcher-Gershenfeld, J., R. B. McKersie, R. E. Walton. 1995.Pathways to change2 Case studies of strategic negotiations.W. E. Upjohn Institute for Employment Research, Kalamazoo,MI.

Deneckere, R., H. P. Marvel, J. Peck. 1997. Demand uncertaintyand price maintenance: Markdowns as destructive competi-tion. Amer. Econom. Rev. 87(4) 619641.

Elyakime, B., J. J. Laffont, P. Loisel, Q. Vuong. 2000. Auctionand bargaining: An econometric study of timber auctionswith secret reservation prices. J. Bus. Econom. Statist. 15(2)209220.

Endriss, U. 2006. Monotonic concession protocols for multilateralnegotiation. Proc. Fifth Internat. Joint Conf. Autonomous Agentand Multiagent Systems, Hakodate, Japan.

Ertogral, K., S. D. Wu. 2001. A bargaining game for supply chaincontracting. Working paper, Department of Industrial and Sys-tems Engineering, Lehigh University, Bethlehem, PA.

Gerchak, Y., Y. Wang. 2004. Revenue sharing vs. wholesale pricecontracts in assembly systems with random demand. Produc-tion Oper. Management 13(1) 2333.

Gillies, D. B. 1959. Solutions to general non-zero sum games. A. W.Tucker, R. D. Luce, eds. Contributions to the Theory of Games IV.Princeton University Press, Princeton, NJ, 4783.

Granot, D., G. Sosic. 2005. Formation of alliances in Internet-basedsupply exchanges. Management Sci. 51(1) 92105.

Greene, D. 2002. JVS, alliances, consortia on path to survival formany. Semiconductor Magazine 3(6) 1216.

Gurnani, H., M. Shi. 2006. A bargaining model for a first-time inter-action under asymmetric beliefs of supply reliability. Manage-ment Sci. 52(6) 865880.

Harsanyi, J. C. 1974. An equilibrium-point interpretation of sta-ble sets and a proposed alternative definition. Management Sci.20(11) 14721495.

Harsanyi, J. C., R. Selten. 1972. A generalized solution for two-person bargaining games with incomplete information. Man-agement Sci. 18(5) 80106.

Hueth, B., P. Marcoul. 2003. An essay on cooperative bargainingin U.S. agricultural markets. J. Agricultural Food Indust. Organ.1(1) 134152.

Iskow, J., R. Sexton. 1992. Bargaining associations in grower-processor markets for fruits and vegetables. TechnicalReport 104, Agricultural Cooperative Service, U.S. Departmentof Agriculture.

Krasner, S. D. 1974. Trade in raw materials: The benefits of capitalistalliances. S. Rosen, J. R. Kurth, eds. Testing Theories of EconomicImperialism. D.C. Heath, Lexington, MA, 77121.

Krauss, S., E. Freitsis. 2006. Strategic Negotiation in Multiagent Envi-ronments. MIT Press, Boston, Forthcoming.

Lass, D. A., M. Adanu, P. G. Allen. 2001. Impact of the northeastdairy compact on retail prices. Agricultural Resources Econom.Rev. 30(1) 8392.

Majumder, P., A. Srinivasan. 2008. Leadership and competition innetwork supply chains. Management Sci. 54(6) 11891204.


Marx, L. M., G. E. Shaffer. 2001. Bargaining power in sequen-tial contracting. Working Paper FR-01-09. The Bradley PolicyResearch Center, University of Rochester, Rochester, NY.

Marx, L. M., G. E. Shaffer. 2003. Bargaining power in sequentialcontracting. Working paper, Duke University, Raleigh, NC.

Marx, L. M., G. Shaffer. 2004. Rent shifting, exclusion and market-share discounts. Working paper, Duke University, Raleigh,NC.

Muthoo, A. 1996. A bargaining model based on the commitmenttactic. J. Econom. Theory 69(2) 134152.

Muthoo, A. 1999. Bargaining Theory with Applications. CambridgeUniversity Press, Cambridge, UK.

MySap. 2002. MySap supplier relationship management for theautomotive industry. (June) Article 50 077 844.

Nagarajan, M., G. Sosic. 2006. Game-theoretic analysis of coopera-tion among supply chain agents: Review and extensions. Eur.J. Oper. Res. Forthcoming.

Nash, J. F. 1950. The bargaining game. Econometrica 18(2) 155162.Noe, T. H., J. Wang. 2000. Strategic debt restructuring. Rev. Financial

Stud. 13(4) 9851015.Padmanabhan, V., I. P. L. Png. 1997. Manufacturers return policy

and retail competition. Marketing Sci. 16(1) 8194.Pasternack, B. A. 1985. Optimal pricing and return policies for per-

ishable commodities. Marketing Sci. 4(2) 8187.Plambeck, E. L., T. A. Taylor. 2007a. Implications of breach remedy

and renegotiation design for innovation and capacity. Manage-ment Sci. 53(12) 18591871.

Plambeck, E. L., T. A. Taylor. 2007b. Implications of renegotiationfor optimal contract flexibility and investment.Management Sci.53(12) 18721886.

Raiffa, H. 1985. The Art and Science of Negotiations. Belknap Press,Boston.

Roth, A. 1979. Axiomatic Models in Bargaining. Springer-Verlag,Germany.

Roth, A. 1995. Handbook of Experimental Economics. Princeton Uni-versity Press, Princeton, NJ.

Rubinstein, A. 1982. Perfect equilibrium in a bargaining model.Econometrica 50(1) 97110.

Schelling, T. 1960. The Strategy of Conflict. Harvard University Press,Boston.

Souza, C. 2003. Cisco regulates inventory intake. Electronics SupplyManufacturing (February 3).

Stallkamp, T. T. 2001. Fixing a broken economic model2 A case forsupplier alliances. Management Briefing Seminars, MSX Inter-national, Traverse City, MI.

Stone, J. 1958. An experiment in bargaining games. Econometrica26(2) 286296.

van Mieghem, J. 1999. Coordinating investment production andsubcontracting. Management Sci. 45(7) 954971.

Wall Street Journal. 2003a. States go to court in bid to rein in priceof medicine. Wall Street Journal (May 21) 3.

Wall Street Journal. 2003b. Airbus and Boeing duke it out to winlucrative Iberia deal. Wall Street Jounal (March 10) 1.

Weber, R. J. 1976. Absolutely stable games. Proc. Amer. Math. Soc.55(1) 116118.

Worley, T., R. Folwell, J. Foltz, A. Jacqua. 2000. Management ofa cooperative bargaining association: A case in the PacificNorthwest asparagus industry. Rev. Agricultural Econom. 22(2)548565.

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