Stable Group Purchasing Organizations

Transcription

1 Stable Group Purchasng Organzatons September 2010 Abstract In ths paper, we study the stablty of Group Purchasng Organzatons (GPOs). GPOs exst n several sectors and beneft ts members through quantty dscounts when dealng wth supplers. However, despte several obvous benefts, GPOs suffer from member dssatsfacton due to allocatons of the accrued savngs among ts members. We frst explore the benefts of allocaton rules that are commonly reported as beng used n practce. We characterze stable coaltonal outcomes when these rules are used and provde condtons under whch the grand coalton emerges as a tenable outcome. These condtons are somewhat restrctve. We then study an allocaton mechansm based on the margnal value of a member s contrbuton and fnd that ths leads to stable GPOs n many scenaros of nterest. In ths analyss, we look at dscount schedules that encompass a large class of practcal schedules and analyze cases when purchasng requrements of the members are both exogenous as well as endogenous. We use a concept of stablty that allows for players to be farsghted,.e., players wll consder the possblty that once they act (say by causng a defecton), another coalton may react, and a thrd coalton mght n turn react, and so on, nullfyng ther orgnal advantage n makng the ntal move.

2 1. Introducton Group purchasng organzatons (GPOs) are coaltons of several frms (buyers) who pool ther purchasng requrements and buy large quanttes of a partcular product from a seller. The advantage of belongng to a GPO s evdent. GPOs are able to take advantage of sgnfcant quantty dscounts from the seller and transacton costs may be lowered by bundlng dfferent orders. In certan sectors, GPOs have hgh negotaton power and receve preferred terms of trade. GPOs, henceforth also referred to as purchasng coaltons, are seen n varous ndustry sectors. The orgns of such coaltons can be traced back to the evoluton of cooperatves, where cooperatves acted as purchasng and sellng coaltons. Among non-commodty ndustres, health care was one of the earlest to see the formaton of large GPOs. The frst healthcare GPO was created n 1910 by the Hosptal Bureau of New York. Today, 97% of all not-for-proft, non-governmental hosptals n the Unted States partcpate n some form of group purchasng, wth Provsta, Novaton, and Innovatx beng some leadng purchasng consorta. Purchasng coaltons can be seen n vrtually every sector spannng health care (Doucette 1997, Mtchell 2002), educaton (Doucette 1997), retal (Zentes and Swoboda 2000, Dana 2006), etc. Examples are FoodBuy n the grocery ndustry and PrmeAdvantage, a large ndustral manufacturers GPO. The advent of electronc commerce has added to ths trend nternatonally. Currently, there are several avenues on the web where ndvdual buyers can vrtually aggregate orders to aval dscounts (known as group buyng; see Anand and Aron 2003). The now defunct Mercata s one such example n the Unted States; a related and extremely popular shoppng strategy n Chna s Tuangou. Despte the touted benefts of formng such organzatons, the busness and academc lterature lament the fact that GPOs do not sustan themselves, often due to dsagreements between members. GPOs see membershp numbers often fluctuate. Further, problems often arse due to unequal member contrbutons, wth powerful members buldng barrers to keep out smaller partcpants from enterng the GPO (Bloch et al. 2008). Consder a group of buyers, each wth a certan purchasng requrement, who decde to form a coalton. The dscount from the seller s based on the aggregated purchasng quantty. Thus, each member of the coalton s able to receve a lower prce than what he would have receved based solely on hs ndvdual order. However, when a set of buyers wth heterogeneous requrements form a coalton, t s not mmedately clear how the benefts of ths lowered prce are to be allocated among coalton members. Ths dscord leads to a lack of commtment among coalton members and to potental nstablty (Hejboer 2003). A comprehensve study of several purchasng consorta n Europe ndcates that over a quarter of such coaltons are acutely aware of nherent unfarness n splttng the savngs obtaned by the coalton. An often-used mechansm that allocates savngs nternally to the coalton, known as equal prce (where all members of the coalton pay the same prce per tem), s an excellent example that opens tself to ths crtcsm. The common wsdom that gans accrued by the coaltons may far surpass any cause for dscord s ncreasngly questoned by coalton members and ths may explan the short lfe of several such consorta (Aylesworth 2003). Thus, an mportant strategc ssue to be consdered when settng up a GPO seems to be the 1

3 allocaton of the gans realzed by such a coalton. Evdently, the ssue of allocatons of gans s ntmately ted to the eventual stablty of these coaltons. Despte recognzng the mportance of ths topc, nether the lteratures n operatons management that deals wth purchasng nor the ones n economcs analyze a comprehensve model of purchasng consorta and offer any robust remedes. In ths paper, we examne how allocatons among group members n a purchasng coalton should be desgned (especally when contrbutons by ndvdual members may not be equal) and analyze the related ssues of stablty of purchasng coaltons. In terms of dvdng the gans of a coalton, we use and suggest a few allocaton rules that make sense from both a practcal perspectve as well as are related to theoretcal concepts from cooperatve games. We frst look at the two allocaton rules used n practce. These are () the equal allocaton (EA) prncple, whch allocates to each member the same porton of accrued savngs; and () the quantty-based proportonal allocaton (QPA) rule, whch allocates the savngs proportonally to the buyers orderng quanttes. We show that these rules are often problematc and nherently cause member dssatsfacton that usually lead to defectons by members. Wth the am of sustanng GPOs, we propose the Shapley value (Shapley 1953), whch computes the margnal contrbuton of an ndvdual member of the coalton as an allocaton rule. We show that usng the margnal contrbuton leads to a greater set of nstances n whch the GPO of all buyers s stable. The ssue of allocatons, as mentoned earler, s ted to the queston of stablty of the allance. A commonly used concept of stablty, popular n the operatons lterature, s the core (Glles 1959). The core dstngushes allocaton rules that yeld a stable allance of all players (roughly speakng, no set of players have an mmedate ncentve to defect from ther jont allance when an allocaton s n the core), but suffers from the problem of myopa. That s, t precludes the possblty that players and coaltons may consder the possblty that once they act (say, by causng a defecton), another coalton may react, and a thrd coalton mght n turn react, and so on, nullfyng ther advantage n makng the ntal move. A concept of stablty that takes a farsghted vew of players s the Largest Consstent Set (LCS, Chwe 1994). In our analyss, we allow for players to be farsghted and prmarly use the LCS to evaluate the stablty of coaltons. Our man fndngs when consderng the aforementoned allocaton rules are as follows. We frst look at a scenaro n whch the requrement of each frm (ts order quantty) s exogenous to the dscusson and thus ndependent of the specfcs of the dscount schedule. Such a scenaro s realstc n several publc-sector GPOs (the healthcare sector beng a good example). Here, purchase orders are perodcally determned based on needs, and less on other factors such as proft maxmzaton or re-sellng decsons. When one looks at pecewse lnear dscount schedules, the analyss quckly becomes complcated. We llustrate some of the underlyng ssues usng specfc examples wth a few players. However, when one looks at a stuaton wth a large number of players, we show an asymptotc result that says that for a sgnfcant class of pecewse lnear quantty dscount schedules, the Shapley value allocaton s the unque allocaton rule that produces a stable grand coalton (allance of all buyers). Ths result, though somewhat techncal n nature, has consderable practcal mplcatons. We demonstrate ths by emprcally examnng the scale of problems for whch the above result holds and show that for a large set of realstc problem parameters, the Shapley value 2

4 nduces a stable grand coalton. If we assume that dscount schedules are contnuous, when buyers are homogeneous, the grand coalton s stable and sustans tself ndependent of the allocaton rules. Ths result s not surprsng. However, when the buyers are heterogeneous, the Shapley value alone produces far allocatons that ensure the stablty of the grand coalton. When ether EA or QPA are used and one looks at the farsghted stablty of coaltons, there s a strong tendency n whch the buyers wth large orders splt to form ther own GPO, leavng buyers wth smaller contrbutons to themselves. Ths leads to the followng nsght f one needs to sustan a GPO wth heterogeneous buyers, contracts that allocate savngs between buyers need to be carefully arrved at, usng, for nstance, the Shapley value allocatons. That s, allocaton rules that are drawn, before membershp contracts are wrtten up, need to do a calculaton that takes nto account the margnal contrbuton of members of the coalton. Several of the currently followed allocaton rules that fal to do ths wll result n eventual nstablty. Next, we look at a scenaro n whch the quantty requred by each buyer s endogenous to the model. An llustratve case s that of a proft-maxmzng prce-settng frm whch decdes how much ts requrements are by consderng both the quantty dscount schedule offered by the seller and ts own downstream prce-drven demand. The frm under queston smply sets ts demand curve and the dscount schedule (supply curve) equal to each other and derves ts requrements based on ths ntersecton. As one can magne, ths analyss can be algebracally tedous, as solvng for the order quantty may not be smple. When we consder lnear demand and dscount schedules, we are able to show that several of our results and nsghts from prevous analyss wth exogenous demands stll hold. In partcular, we show that t s stll advsable to use a Shapley-value-based allocaton to guarantee the stablty of the GPO. We also show that when quantty decsons are endogenous and one uses ether the EA or the QPA, under certan assumptons our earler result contnues to hold (.e., large buyers coalesce together and leave the smaller members out). Ths predcton of ours s seen wdely n practce and s also the topc of several studes and reports on purchasng organzatons (see, e.g., Bloch et al. 2008). Usng numercal examples we show that when one looks at endogenous settng, the games under queston n general lose several of the nce propertes that facltate an elegant analyss. Games such as these may nether be convex nor concave. The mplcaton s that, n the presence of non-lnear schedules and demands (for nstance, thnk of a demand wth constant elastcty), or n nstances where the buyers face sgnfcant nternal decsons such as the retal prce of ther product, one needs to be very careful n prescrbng allocaton rules. In general, allocaton rules requred to preserve the allance may be far too complcated to mplement, and n some nstances may not even exst. Thus, n such nstances, t may well be n the health of the GPO to negotate less complcated dscount schedules that may lead to slghtly lower overall savngs. An mportant ssue that arses n group purchasng s perceptons of farness n allocatons. There s a large lterature n the feld of appled behavoral economcs that consders the ssue of farness and justce when gans/savngs are dvded between a group of ndvduals. The early studes assumed axomatcally that gans and nvestments depend on ther proportons accordng to a certan 3

5 functonal form. However, expermentally, there were sgnfcant devatons from these axoms. An excellent reference on some of these early ssues on dstrbutve justce s Selten (1972). Later work bulds on the axomatc approaches and uses observatons from the behavoral theory of farness. For a revew of some of these deas, please see Guth and Tetz (1985). The man dvergence between the expermental lterature and the theoretcal papers was n decdng functonal forms between the payoff a player receves and hs contrbuton. The ntal attempts used smplstc notons that looked at smple proportons between the two (ths corresponds to our QPA), whch ether faled axomatc requrements of the theoretcal lterature or dd not produce convncng results n experments where contrbutons and gans were not homogeneous. To settle ths ssue of equty regardng gans and contrbutons, later lterature looked at solutons that proposed equty based on relatve contrbutons and the value created by the ablty to defect. Ths resulted n the emergence of tests of farness, wthout ever formally defnng what farness s. Examples of such tests are the no-envy test, the stand-alone test, and the unanmty test. These tests are used to justfy soluton concepts as well as pck equlbra n games wth multple focal ponts. Not all of these apply to our settng, as our game s essentally one n whch value created by savngs s dvded between players n a coalton. For a dscusson of the dfferent varants of these tests, please see Thomson and Varan (1985) and Mouln (1993). The Shapley value satsfes the no-envy and stand-alone tests for savngs games such as ours. The nucleolus (Schmedler 1969) s an alternatve soluton concept that s sometmes proposed to address farness. Accordng to the nucleolus, one attends frst and as much as possble to the coaltons most adversely affected by feasble allocatons. One attractve property that the nucleolus has s that t mnmzes the components of the strct Chebyshev vector whose components are the value of a coalton mnus ts allocaton; t s also a core allocaton whenever the core s nonempty. However, t s well recognzed that the nucleolus s not an easy concept to mplement, and t volates the no-envy test. For ths reason, we do not use the nucleolus. In summary, our central message n ths paper s that usng the Shapley value to allocate savngs to the ndvdual members of a GPO s essentally robust to stablty concepts, heterogenety of buyer requrements, and percepton of farness, and thus yelds the best chance to sustan a GPO. We beleve our fndngs above yeld mportant nsghts to frms that contemplate jonng purchasng coaltons, or to ntermedares who wsh to create successful and effcent GPOs. Snce the creaton of GPOs has substantal consequences to socal surplus and creates effcences n many supply chans, we beleve t s mportant for the relevant players to understand and pursue strateges that wll contrbute to ts success. Further, our results seem to be qute robust and hold n many stuatons wth contnuous and dscrete schedules and n several endogenous and exogenous quantty models. The rest of the paper s organzed as follows. In Secton 2, we study the model n whch the purchasng requrements of the ndvdual buyers are exogenous, ntroduce the three allocaton rules consdered n ths paper, and analyze ther propertes. In Secton 3, we ntroduce the farsghted stablty concept (the largest consstent set), whch we then apply to our model to obtan farsghtedstable outcomes n Secton 4. We analyze the model n whch the purchasng requrements are prce-dependent under a lnear dscount scheme n Secton 5, whle n Secton 6 we dscuss some 4

6 sgnfcant extensons of ths model whch allows for a fuller analyss of stuatons when requrements are endogenous. Fnally, we conclude n Secton 7 and provde some drectons for future work. 2. Model wth Exogenous Quanttes We begn our analyss n ths secton by descrbng the basc model that looks at the creaton of a GPO. Suppose that n buyers decde to form a purchasng consortum n order to beneft from avalable quantty dscounts or economes of scale that generate lower per unt prces. There are several reasons for buyers to unte and form such a coalton. In ths paper, we solate and focus exclusvely on the effect of prce dscounts. That s, n our settng, there s a seller who announces a dscount schedule that buyers can aval of. Thus, we assume that unt purchase prce per tem, p, s a functon of quantty purchased, q, and that the functon p(q) reflects a dscount. Dscrete-step-szed dscount schedules. We frst consder dscrete-step-szed schedules. A pecewse quantty dscount functon wth m prce breaks s defned as follows: suppose we are gven break ponts 0 < κ 1 < κ 2 <... < κ m and prces p 0 > p 1 >... > p m. We let κ 0 = 0, κ m+1 =. If κ 1 < q κ for some {1,..., m + 1}, then p(q) = p 1. We note that often n practce dscrete dscount schedules are malleable to some extent, whch may further justfy the use of a contnuous schedule. For nstance, f a dscount scheme requres the prce of $100 for quanttes 0-54, and $90 for quanttes greater than 55, a buyer who requres 50 tems may order 55 and pay less, or he may smply negotate a prce lower than $100 wth the seller. That s, players may smply ether change requrements or negotate a more malleable form of schedule. Contnuous dscount schedules. The approprateness of contnuous quantty dscounts s well known and used wdely n the operatons lterature (see, for nstance, Mtchell 2002). The economcs lterature also uses menus and dscounts that are usually contnuous. Thus, we later extend our analyss by assumng that p(q) s decreasng, contnuous, and convex n q. Schotanus (2007) argues, based on a large data set, that a functon of type p(q) = α + βq η, q 1, α > 0 (1) fts well wth dfferent types of quantty dscounts. He analyzes 66 dscount schedules and shows few dscrepances (n three cases, due to outlyng ponts). The form of the dscountng scheme (1) s rather general, but mposes some restrctons on ts parameters. When η > 0, the dscount functon has a postve steepness and we requre that β > 0; when η < 0, the dscount functon has a negatve steepness and thus one requres that β < 0, 1 η < 0. We mpose an addtonal requrement that the amount transferred to the seller, qp(q), s a concave ncreasng functon. Ths assumpton seems to hold n most practcal schedules dentfed n the lterature. Note that a lnear dscount scheme, p(q) = ᾱ βq, whch wll be used later n ths paper, s a specal case of (1) obtaned when η = 1 and β < 0; ths dscount schedule seems especally useful when the number of buyers s large. 5

7 We acknowledge and clarfy a subtle dfference n coaltonal games that employ dscrete versus contnuous nputs. Note that under a contnuous dscount functon each buyer makes a postve contrbuton to the cost pad by the coalton, whle ths may not be true when the buyers are faced wth a pecewse dscount functon. In many games, ths can be a sgnfcant ssue. However, as dscussed below, we concentrate on total savngs realzed by a coalton and the allocaton of these savngs, and not on the change n purchasng cost. Thus, even f a buyer does not contrbute to a reducton of purchasng cost, he may contrbute to savngs seen by the coalton and thus make nontrval contrbuton to the amount allocated among buyers. For nstance, suppose that there are three buyers, orderng 10, 20, and 30 unts respectvely, and that the seller charges $10 for up to 45 unts, and $9 for larger quanttes. When buyers 2 and 3 form coaltons, they acheve the prce of $9, and the addton of buyer 1 does not change the prce. However, buyers 2 and 3 together generate savngs of $50, whle after admttng buyer 1 to ther coalton total savngs go up to $60, ncreasng the amount to be allocated among buyers. We begn by analyzng a settng n whch the purchasng requrements of the ndvdual buyers are exogenous to the model. That s, buyer requres quantty q, and ths quantty decson s not drven by the dscount schedule avalable to hm. In the lterature on purchasng coaltons, ths seems to be a common assumpton. In a later secton, we relax ths assumpton and assume that each buyer faces a demand curve and makes hs purchasng decson based on the dscount schedule (supply curve) and hs demand curve. Contnung wth ths exogenous assumpton, we let buyer to requre q, and we denote q = (q 1, q 2,..., q n ). The total order quantty of the GPO wth n buyers n the allance s then n =1 q = Q. The ratonale for formng the GPO s clear even n ths smple settng. The per unt prce pad by the coalton s p(q), whch s smaller than p(q ), the per unt prce that buyer would get by transactng wth the seller on hs own. It s not uncommon that n such consorta each member pays the same per unt prce, p(q), and hence ths model seems to be often adopted n research papers n collaboratve purchasng (Chen and Roma 2008, Chen and Yn 2008, Kesknocak and Savasanerl 2008). The paper of Chen and Yn (2008) s of partcular nterest to our work. They choose a dfferent form for ther value functon (.e., usng the cost). Although ths allocaton rule volates the no-envy prncple, they demonstrate usng an elegant analyss that when lnear dscounts are used, the unform allocaton s equvalent to the Shapley value. They are less nterested n the farsghted stablty of the GPO. In our analyss, we have chosen to concentrate on savngs games nstead of cost games. In cost games, players usually share costs ncurred when producng some common goods and/or servces. In savngs games, players share savngs obtaned as results of ndvdual efforts. Evdently, the latter better fts our problem. The savngs observed by buyer due to allance membershp can be wrtten as q [p(q ) p(q)]. Let us denote by N the set of all buyers,.e., N = {1, 2,...n}. Then, usng termnology and notatons from cooperatve games, the value of the grand coalton, v(n), s smply the total savngs generated by the entre consortum, and s gven by v(n) = n j=1 q jp(q j ) Qp(Q). Note that the purchasng requrements of the buyers are not necessarly equal. Thus, buyers may contrbute unequal amounts to the total quantty, Q. A drect consequence s that some of the allance members may 6

8 justfably feel that ther share of the savngs does not reflect the magntude of ther contrbuton towards the savngs generated by the allance, because a buyer who contrbutes more may receve lower overall savngs than a buyer that contrbutes less to the value of Q. Ths phenomena motvates the dscusson that, n order to sustan a successful consortum, one may want to propose some alternatve methods for allocaton of savngs among the allance members. In the rest of ths secton, we put forward a framework that allows us to analyze the central queston of nterest; that s, what knd of allocaton mechansms should be employed to ensure the success of the GPO? We need to ensure that no sub-coalton of buyers, S N, wants to leave the GPO (the coalton N) and form ther own GPO. Buyers may decde to leave for a myrad of reasons; our focus s on two of them. Frst, the buyers may beleve that, by defectng, they can create and allocate the resultng savngs more effcently than the orgnal GPO. Second (whch s closely related to the frst reason), the buyers may feel that ndvdual gans from beng n the GPO are allocated unfarly wth respect to ther contrbuton. Applyng the concepts from game theory, we wll defne a savngs game as follows: for any coalton, S N, we denote by v(s) value generated by that coalton. In ths part of our analyss, we dentfy the value of coalton S wth the total savngs generated due to the combned orders of buyers n S. Then, v(s) can be wrtten as v(s) = j S q j p(q j ) p j S q j. We further denote by ϕ (v) IR allocaton of total savngs, v(n), receved by buyer. When t s clear from the context what value functon we are referrng to, we wrte ϕ nstead of ϕ (v), and we denote by ϕ IR n the allocaton vector. In the remnder of the paper, we wll concentrate on three allocaton rules the Shapley value, QPA, and EA that we descrbe n more detal below. We want to compare the effects of applyng these dfferent allocaton rules, to nvestgate when the jont purchasng allance of all members s a stable outcome, as well as what would be the resultng stable structures f the grand purchasng organzaton s unstable. As mentoned earler, the noton of stablty we use (the largest consstent set, LCS) allows for players to be farsghted. We descrbe ths n detal n a later secton. Equal Allocatons (EA). The smplest allocaton of savngs would be to gve an equal porton to each buyer, ϕ E = [ n j=1 q p(q ) Qp(Q)]/n, so that buyer s payng n c E j=1 (q) = q p(q ) q jp(q j ) Qp(Q) j = {[q p(q ) q p(q)] [q j p(q j ) q j p(q)]} + q p(q). n n Thus, the buyer who orders the most can pay more or less than q p(q), dependng on the range of the quanttes ordered by all buyers (see Example 1). Quantty-Based Proportonal Allocatons (QPA). Another smple way of allocatng savngs would be to dstrbute them n proporton wth the contrbuton of dfferent buyers, ϕ P = 7

9 ( [ n ]) q j=1 q p(q ) Qp(Q) /Q, so that buyer s payng c P (q) = q p(q ) [ n ] q j=1 q jp(q j ) Qp(Q) Q = q p(q) + q Q n q j [p(q ) p(q j )]. In ths case, a buyer who orders the most pays less than q p(q), whle a buyer who orders the least pays more than q p(q). Shapley Value Allocatons. Another possblty s to dstrbute the savngs accordng to the Shapley value allocatons. Consder all possble orderngs of players, and defne a margnal contrbuton of player wth respect to a gven orderng as hs margnal worth to the coalton formed by the players before hm n the order, v({1, 2,..., 1, }) v({1, 2,..., 1}), where 1, 2,..., 1 are the players precedng n the gven orderng. Shapley value s obtaned by averagng the margnal contrbutons for all possble orderngs. If we denote by S number of buyers n allance S, Shapley allocaton to player can then be wrtten as ϕ N (v) = {S: S} ( S 1)!(n S )! n! j=1 (v(s) v(s \ {})). We now descrbe the logc behnd the allocaton rules we analyze n ths paper. From a theoretcal perspectve, allocaton rules are often tested based on the number of attractve propertes they satsfy from a farly standard lst of requrements (see Myerson 1997). Ths lst ncludes several reasonable propertes such as symmetry, effcency, addtvty, ndvdual ratonalty, etc. Of the many popular allocaton rules from theory, such as the Shapley value (Shapley 1953), the nucleolus and the compromse value (Dressen 1985), the Shapley value turns out to be the most attractve for the savngs game. Suppose that the dscount schedule s contnuous; recall that we assumed that for ths case p(q) s a convex functon and qp(q) s concave and ncreasng n q 1. Ths seems to be a reasonable assumpton that holds for most schedules n practce. In partcular, the total payments made to the seller, as one would expect, ncrease as the quantty purchased becomes larger, and ths ncrease sees a dmnshng return. When ths property s factored n, the savngs game turns out to be convex; that s, v(t {}) v(t ) v(s {}) v(s) whenever S T N \ {}. A drect mplcaton of ths fact s that the Shapley value satsfes a myopc stablty property (.e., the Shapley value belongs to the core when the game s convex, and no subset of players wants to defect from N). Note that, as per our earler dscusson on dstrbutve justce and farness, the Shapley value wthstands commonly used tests of farness and equty. Moreover, common problems assocated wth GPOs, such as monotoncty of payoff wth respect to contrbuton and weak free rder ssues, are mnmal when the Shapley value s used. Further, as mentoned earler, the nucleolus and compromse value do not yeld the grand coalton n the farsghted sense, are less ntutve, and 1 Our dscount functon (1) satsfes ths assumpton when 0 < η < 1, or when η = 1 for q α/(2β). 8

10 therefore harder to mplement n practce. The Shapley value also has the advantage that t has robust approxmatons, whch s convenent for practcal applcatons. However, we also note that the Shapley value behaves dfferently under pecewse and under contnuous dscount schedules, as shown below. Proposton 1 When dscount functon s contnuous, larger orderng quantty corresponds to larger contrbuton to savngs, and the buyer wth the largest orderng quantty receves the largest allocaton under the Shapley value allocaton rule; ths s not true f the dscount functon s pecewse. Thus, when pecewse dscount functons are used, t may not be easy to dentfy whch buyer contrbutes to the savngs the most; however, the buyer wth largest contrbuton wll always receve the largest allocaton. In addton, the convexty of the game s not preserved under pecewse dscount functons. Proposton 2 Under a pecewse dscount functon, the savngs game s superaddtve, but t s non-convex n general. It follows from the result above that, f we consder a pecewse dscount functon, the Shapley value does not always belong to the core and we lose some of ts nce propertes that hold n the contnuous case. As we show later, ths sgnfcantly complcates the analyss of stable outcomes. However, the game remans superaddtve; that s, when any two dsjunct coaltons jon, the total savngs generated by ther members ncrease. In practce, EA and QPA are commonly used. We beleve that the reason for ths s the apparent smplcty of these rules. We also note that early notons of dstrbutve justce would pck the QPA as a canddate of farness, but as per our earler dscusson, ths vew has snce been revsed n the behavoral economcs lterature. We next provde an example to llustrate dfferences among the allocaton rules ntroduced above. Example 1. Suppose that N = {1, 2, 3}, κ 1 = 25, κ 2 = 45, κ 3 = 100, and p 0 = 10, p 1 = 8, p 2 = 7.5, p 3 = 7. Orderng quanttes, q, unt prces, p(q ), and costs before cooperaton, q p(q ), together wth costs, C, and savngs allocaton, ϕ, under dfferent allocaton rules are gven n Table 1. Superscrpts 0, EA, QP A, and S denote equal prce, EA, QPA, and Shapley value, respectvely. Note that when the buyers form a purchasng allance and jontly buy the product, Q = 150 and p(q) = 7. Thus, wth QPA and Shapley allocatons, the buyer who contrbutes the most s allocated the hghest share of savngs. Note that even ths smple analyss seems to ndcate that gven our earler dscusson on equty, all of the three proposed methods above seem farer than the oftenused scheme n whch each buyer pays equal prce (see ϕ 0 ). We also note that under contnuous dscount functons Shapley value allocates the hghest share of savngs to the buyer who orders the most, but that ths s not always true wth pecewse-dscount functons. The reason s that under 9

11 q p(q ) q p(q ) q p(q) C EA C QP A C S ϕ 0 ϕ EA ϕ QP A Table 1: Orderng quanttes, unt prces, cost, and savngs under dfferent allocaton rules ϕ S dscontnuous dscount schemes the largest orderng quantty does not always correspond to largest contrbuton to savngs (see Proposton 1). For nstance, f we replace the above dscount scheme wth κ 1 = 25, κ 2 = 45, κ 3 = 125, and p 0 = 10, p 1 = 9, p 2 = 8, p 3 = 7, Shapley allocatons would be 78, 73, and 58, respectvely, as buyers 1 and 2 contrbute more to savngs n varous coalton than buyer 3. The next queston that needs to be addressed s stablty of purchasng allances do all allance members have an ncentve to jontly purchase the tems, or could there exst a subset of buyers that benefts from purchasng separately? A popular game-theoretc concept of stablty used n the operatons management lterature s the core, whch s defned as follows. An allocaton ϕ s a member of the core of f t satsfes S ϕ v(s) S N, and n =1 ϕ (v) = v(n). When core allocatons are used, no subset of players has an ncentve to secede and form ts own coalton. The core was ntroduced to the operatons lterature n Hartman and Dror (1996) n the newsvendor context and has snce been wdely adopted (for example see Hartman et al. 2000, Hartman and Dror 2003, 2005, Chen and Zhang 2008). Thus, n order to nduce partcpaton of all buyers n the GPO, one may want to select core allocatons. The drawback of the allocatons proposed above s that, n general, they do not belong to the core. Consder, for nstance, Example 1 above when the QPA s appled. If buyers 1 and 2 form a purchasng allance on ther own, they would generate total savngs of 65, whle under the QPA they receve a total of 48. Thus, they would be better of by actng alone. Smlarly, suppose that EA are used and that buyer 2 orders 80 nstead of 30. Then, the total share of savngs allocated to players 1 and 3 when all buyers form a consortum equals 97, whle by purchasng wthout buyer 2 they generate savngs of 105. Thus, nether of the two practcal rules nduces all buyers to partcpate n the consortum, provdng that players consder only one-step defectons. The above analyss merts a dscusson. The frst pont to note s that the two practcal rules at the outset seem farer than one n whch all buyers pay the same prce. However, when one looks at stablty concept such as the core, nether of these two rules yeld stable allocatons. That s, f one were to use these rules, a subset of buyers wll defect from the GPO. The second pont to note s that when all buyers pay the same prce, one can show that the core constrants hold. That s, despte the apparent unfarness, no sub-coalton has an mmedate ncentve to defect. We need to be careful about nterpretng these results. The observaton that chargng everyone the same 10

12 prce yelds a core allocaton needs to be reconcled wth emprcal fact that ths approach leads to members of the GPO beng unsatsfed due to perceved unfarness. Consequently, t appears that among the practcal rules, one has to choose between allocaton rules that are less far (as they may gve the smallest allocatons to buyers who contrbute the most, thus falng some of the equty tests), but encourage partcpaton of all buyers, and allocaton rules that seem more justfable, but may ncentvze some buyers to defect, thereby destroyng the GPO. The Shapley value, as per ts defnton, does a calculaton that brngs to the table the margnal contrbuton of the players. Ths proposes a noton of farness that s easy to descrbe and explan n practce as well as consstent wth the general framework of dstrbutve justce. We conclude ths secton by notng that the concept of stablty used n the above dscusson, the core, s statc. That s, players are myopc and smply look at one-step devatons by other buyers. We want to nvestgate whether the same results hold n a dynamc settng, n whch buyers are farsghted and consder how others may react to ther actons. In order to study ths problem we frst need to ntroduce some addtonal concepts from game theory, whch look at stablty n a farsghted sense. We contnue to focus on the three allocaton schemes descrbed above and gnore the equal-prce mechansm n vew of the above dscusson. 3. Stablty Concepts In ths secton, we ntroduce some concepts used n our analyss of the stablty of buyer allances. Before we descrbe the exact methodology, we wll brefly try to motvate our framework. Game-theoretcal concepts of stablty are usually statc. In noncooperatve strategc-form games, the often-used concept s the Nash equlbrum, whch only consders devatons by ndvdual players. In our settng, we assume that all buyers (coaltons) can communcate among themselves and can jon or leave allances at ther wll. Thus, we may expect that they wll consder both unlateral and jont (mult-lateral) devatons from a gven coalton structure (a partton of the set {1, 2,..., n}). The strong Nash equlbrum (SNE) (Aumann, 1959) and the coalton structure core (Aumann and Dreze, 1974) admt ths extenson. However, these soluton concepts, along wth the majorty of soluton concepts commonly used n the analyss of coalton-structures stablty ncludng the core, share the same problem that afflcts all statc concepts. Ths can be descrbed as follows: consder Example 1 and assume that the status-quo s the allance of all buyers (the grand coalton). We have shown that t s benefcal for a subset of players {1, 2} to defect from the grand coalton under the QPA. The exstng statc concepts wll mmedately conclude that the grand coalton s not stable. There are potentally two fundamental problems wth ths logc. Frst, does ths mean that the resultng outcome, obtaned by a defecton of players {1, 2}, s stable? If not, why should we conclude that the move from the grand coalton wll ever happen? Secondly, the statc analyss does not check f a further defecton wll occur. It may possbly happen that an ntal defecton trggers a sequence of further defectons that eventually leads to an outcome n whch the defectng partes 11

13 accrue a lower payoff than the status quo. If ths were the case, farsghted players may not choose to defect n the frst place, and thus an outcome whch we thought was possbly not stable may actually be a canddate for stablty! A statc concept, by defnton, does not handle such trade-offs. A soluton concept that allows players to consder multple possble further devatons s the largest consstent set, ntroduced by Chwe (1994). It s defned below, and s used as a stablty crteron n our analyss of stable allance structures. Any partton of N,.e. Z = {Z 1,..., Z m }, m =1 Z = N, Z j Z k =, j k corresponds to a coalton structure, Z. For each buyer, let ϕ Z denotes buyer s share of savngs n the coalton structure Z. Let us denote by the players strong preference relatons, descrbed as follows: for two coalton structures, Z 1 and Z 2, Z 1 Z 2 ϕ Z 1 < ϕ Z 2, where ϕ Z s a buyer s allocaton of savng n the coalton structure Z. If Z 1 Z 2 for all S, we wrte Z 1 S Z 2. Denote by S the followng relaton: Z 1 S Z 2 f the coalton structure Z 2 s obtaned when S devates from the coalton structure Z 1. We say that Z 1 s drectly domnated by Z 2, denoted by Z 1 < Z 2, f there exsts an S such that Z 1 S Z 2, and Z 1 S Z 2. We say that Z 1 s ndrectly domnated by Z m, denoted by Z 1 Z m, f there exst Z 1, Z 2, Z 3,..., Z m and S 1, S 2, S 3,..., S m 1 such that Z S Z +1 and Z S Z m for = 1, 2, 3,..., m 1. A set Y s called consstent f Z Y f and only f for all V and S, such that Z S V, there s an B Y, where V = B or V B, such that Z S B. In fact, Chwe (1994) proves the exstence, unqueness, and non-emptness of the largest consstent set (LCS). Snce every coalton consders the possblty that, once t reacts, another coalton may react, and then yet another, and so on, the LCS ncorporates farsghted coaltonal stablty. The LCS descrbes all possble stable outcomes and has the mert of rulng out wth confdence. That s, f Z does not belong to the LCS, Z cannot be stable. For a more detaled analyss of farsghted coaltonal stablty, see Chwe (1994). Some applcatons of analyss of stablty usng Chwe s LCS crteron nclude Nagarajan and Sošć (2007) and Granot and Yn (2008). 4. Stable Outcomes for the Model wth Exogenous Quanttes In ths secton, stablty mples stablty n the farsghted sense. Thus, an outcome s stable f t belongs to the LCS; as mentoned, ths s dfferent from the core membershp, a myopc concept. In order to establsh whch outcomes can be stable, we frst need to establsh players preferences for dfferent coalton structures. We assume wthout loss of generalty that q 1 q 2... q n. We denote the consortum of all buyers by N. If, for nstance, we have fve buyers dvded n two consorta, one contanng buyers 1 and 3, and the other contanng the remanng buyers, we denote t by {(13), (245)}. To smplfy the notaton, we wll use Z(k 1,..., k j ) to denote a monotonc coalton structure of the form {(1... k 1 ), (k k 2 ),..., (k j k j ), (k j n)}, where ether k 1 < k 2 <... < k j < n or k 1 = n (n whch case Z(k 1,..., k j ) = Z(n) corresponds to the grand coalton). Note that k 1 > 1 mples that there s an non-trval allance of smallest buyers, 12

14 whle k j < n 1 mples that there s a non-trval allance of largest buyers. 4.1 Pecewse Quantty Dscount Functon Suppose that a pecewse quantty dscount functon s defned as descrbed earler, wth m prce breaks 0 < κ 1 < κ 2 <... < κ m and prces p 0 > p 1 >... > p m. As we mentoned earler, there are many analytcal dsadvantages of a pecewse quantty dscount functon; we llustrate some of them n ths subsecton. The use of ths type of functon causes sgnfcant tedum, so we lmt our ntal analyss to the smple case wth three buyers, and consder several dfferent dscount functons and quantty orderng optons. The followng examples are merely for llustratve purposes. We conclude ths secton wth a theoretcal result when the number of players grows large. In our examples, we assume that break ponts are chosen from set {15, 25, 35, 45, 55} and consder dscount schemes wth one, two, three, four, and fve break ponts. The prces start at $10 per unt, and decrease by $1 at each break pont. Thus, we consder a total of thrty-one dfferent dscount schemes. In addton, we look at three dfferent cases of quanttes ordered by ndvdual buyers (each coordnate n a trplet represents the orderng quantty of the correspondng buyer): (a) (10, 20, 30); (b) (10, 15, 35); and (c) (5, 25, 30). For each combnaton of dscount scheme and orderng quantty, we dentfy savngs and allocatons correspondng to the three allocaton methods dscussed above, and dentfy stable outcomes. Whle we cannot provde analytcal expressons, our analyss shows a certan level of consstency across the results (and wth the results for contnuous dscount functons derved later), but t also exhbts several dscrepances: Equal Allocatons. For each of three orderng quanttes, the grand coalton was stable under sxteen dscount schemes, whle the allance of the two larger buyers was stable under twelve dscount schemes, and the allance of two smaller buyers under one scheme. In general, the allance of the two larger buyers was more lkely to be stable when at least one of the break ponts was 35 or 45. None of these results depended on the quanttes ordered by ndvdual buyers; that s, the results were true for cases (a), (b), and (c). There were two cases, though, where the stable outcome was ether the grand coalton or the allance of two smaller buyers: wth a sngle break pont at 15, or two break ponts, at 15 and 25, the allance of two smaller buyers was stable n cases (a) and (c), whle the grand coalton was stable n case (b). Quantty-Based Proportonal Allocatons. In ths case, our results show more consstency wth the contnuous dscount model. The grand coalton s stable n most of the cases; the exceptons are the dscount schemes wth a sngle break pont at 15 or two break ponts, at 15 and 25, n all three cases, (a), (b), and (c), n whch the allance of two smaller buyers was stable; the same was true for the dscount schemes wth a sngle break pont at 25 under cases (a) and (c). Interestngly, there was also an nstance, under case (a) and a dscount scheme wth two break ponts, at 15 and 35, n whch the allance of the smallest and the largest buyer was stable. 13

15 Shapley Value Allocatons. Under ths allocaton rule, the grand coalton was not stable n case (a) wth a sngle break pont at 25, or two break ponts, at 15 and 25; and n case (b) wth a sngle break pont at 15, n whch the allance of the two smaller buyers was stable; t was the stable outcome under all remanng scenaros. As an llustraton of Proposton 1, t s nterestng to note that under Shapley allocatons dfferent buyers can receve the largest share n the grand coalton, dependng on ndvdual orderng quanttes and dscount schemes. For nstance, n case (a), buyer 2 (who orders 20) receves the largest share when there s a sngle break pont at 25, buyer 1 (who orders 10) receves the largest share when there s a sngle break pont at 15, whle they both receve the same share (larger than buyer 3) when there are two break ponts, at 15 and 25. On the other hand, n case (b), buyer 3 (who orders 35) receves the largest share when there s a sngle break pont at 25, whle buyer 2, who orders 15, receves the largest share when there s a sngle break pont at 15, or when there are two break ponts, at 15 and 25. The results above mert some dscusson, as they show the mpact of usng allocatons of savngs versus havng all coalton members payng the same prce. Suppose that the dscount scheme has a sngle break pont at 25, and the buyers orderng quanttes are gven by (10, 20, 30). Then, f all buyers pay the same prce, buyer 3 does not beneft at all by addng any of the smaller buyers, as they wll not be able to reduce the prce further. If, however, coalton members allocate savngs among themselves, then coalton of buyers 1 and 2 generates the same savngs as coalton of all three buyers. Under ths scenaro, buyer 3 would beneft from an allance wth buyers 1 and/or 2, whle buyers 1 and 2 would prefer to act alone and share the savngs among fewer partcpants. Now consder the model wth pecewse dscount schemes n whch breakponts are at relatvely small quanttes. Whle smaller buyers have less mportance when all members pay the same prce, the opposte s true when coalton members allocate savngs larger buyers contrbute less as they can generate savngs on ther own, and coalton partcpaton does not ncrease the savngs level that they generate. Ths explans why n the above examples allance of two smaller buyers appears as stable when the breakponts are at smaller quanttes. The above examples also llustrate the dffcultes of workng wth pecewse dscount schedules. The fact that the savngs can change n dscrete steps upon the addton of small quanttes result n further combnatoral dffcultes when evaluatng the Shapley value. Gven that even the evaluaton s dffcult, one can easly magne the addtonal dffcultes assocated wth computng the LCS. An approach that s used n optmzaton problems when one encounters combnatoral dffcultes (such as the curse of dmensonalty n stochastc optmzaton) s to look at lmtng regmes of the problem n queston and examne f one can prove structural propertes n such regmes. The motvaton behnd ths approach s that f the problem under queston has a smple structure n such regmes, one can use the ntuton obtaned to suggest smlar behavor n more reasonable nstances of the problem. Needless to say, ths approach has justfcaton only when the lmtng regme tself s reasonable n that many practcal nstances mtate the behavor one sees n the lmt. An area 14

16 or research that has used ths approach wth great success s queueng control, where problems are analyzed under heavy traffc regmes (see, for nstance, Harrson 1985). Ths approach s now seen n other areas of operatons management such as stochastc nventory control (Huh et. al. 2009) or dynamc adverse selecton (Zhang et al. 2009, Fong and Sannkov 2009). We employ ths phlosophy to study stepwse dscount schedules. In partcular, we specfy dscount schedules as before, and we let the number of players (each wth quantty requrements) to grow arbtrarly large, moderated by large enough dscount schedules. We then look at allocaton rules that wll yeld the grand coalton as a member of the LCS. We show that n a lmtng regme, the Shapley value allocaton produces the grand coalton of all players as the unque farsghted-stable outcome. The basc dea behnd ths analyss uses the approach n Lggett et al. (2009) and the nterpretaton of the Shapley value n Gul (1991). The Shapley value can be nterpreted as the effectve probablty that a player enters a coalton and earns a savng averaged over all possble coaltons. Ths, n a lmtng regme, allows us to get an expresson for the Shapley value usng the central lmt theorem and Donneker s result. We then use the fact that, usng ths expresson, the grand coalton of all players possesses the external stablty property. Our man result s as follows. Theorem 1 Let a schedule be gven by κ = (κ 0, κ 1,..., κ m ), p = (p 0, p 1,..., p m ) as before. Let N be the set of buyers, and let q = (q 1,..., q n ) be ther requrements. Let B = n =1 q be the market sze, and assume that there s an l suffcently large such that p 0 > p 1 >... p l = p l+1 =... Then, there exsts T < such that f lm m,n κ mb of the LCS under the Shapley allocatons. (0, T ), then the grand coalton s the unque element To understand the mplcaton of the above result, we conducted a large set of numercal exercses where we vared the pecewse schedule s parameters, the number of players, and ther purchasng requrements. The purpose of these exercses was to confrm f the above result s robust n nonlmtng regmes. The experments followed the logc descrbed below. Frst, we set the smallest quantty requred by any player to be at least 1 unt. The lowest prce n the dscount schedule was bounded below by 1. The smulaton would start by specfyng the number of players, n. The ntal start for n was never smaller than 7. Once the smulaton started, t would randomly buld a pecewse dscount schedule and purchasng requrements (wthn specfed controls) and would keep ncreasng the n tll the Shapley value produced the grand coalton n the LCS. Each smulaton smply reported the parameters of the dscount schedule, quanttes, and the value of n when ths was acheved. The average value of n over all the smulatons was 9. That s, on an average when we had over 9 players, the grand coalton sustaned tself as the GPO n a farsghted-stable sense. As may be evdent, the exact number depends on the order quanttes generated and the dscount schedules. For most reasonable dscount schedules, the number of players never exceeded 16. When one looks at the GPOs n practce, the member szes that one observes are sgnfcantly larger than what we get from our experments. Moreover, n practce, schedules and requrements are much less 15

17 pathologcal than what was generated n our experments. Ths has some strong mplcatons for the robustness of the above result, n partcular the effcacy of the Shapley value s ablty to sustan the grand coalton as the stable GPO. 4.2 Contnuous Quantty Dscount Functon We now assume that the dscount functon offered by the seller s contnuous; as dscussed above, ths approxmaton s reasonable n many real-lfe stuatons, an t leads to more analytcal results Equal Allocatons We frst consder EA and obtan the followng result. Theorem 2 Suppose that the savngs are allocated accordng to EA. Then, the LCS contans a unque outcome, whch has the form Z(k 1,..., k j ), and ether k 1 = n or k j < n 1. The above result says that f EA s used to dvde savngs, when players are farsghted, monotonc coalton structures wll emerge as beng unquely stable. From a practcal perspectve, ths resonates wth our earler dscusson where n many GPOs, the larger players edge out the smaller ones and the market place sees several GPOs for the same product category. From a techncal perspectve, we note that one can calculate the stable outcome effcently. Normally, ths computaton has to go through an exponental possbltes. We leave the detals of ths analyss to the appendx. Note that the grand coalton of all buyers s not stable n a farsghted sense when buyers n, n 1,..., n k order sgnfcantly more than the buyers who order smaller quanttes, whch corresponds to the case when the EA does not belong to the core. Ths result comes as a natural consequence of the fact that all buyers receve an equal share of savngs, hence buyers orderng a larger amount beneft by leavng the buyers who order less outsde the consortum. However, f the buyers quanttes are not too far apart, the grand coalton s stable, as llustrated n our next result. Proposton 3 Suppose that the savngs are allocated accordng to EA and the quantty dscount scheme s lnear, p = ᾱ βq. Then, the grand coalton s stable f and only f >j j>k q k q q j (n k) j>k q, j for all 1 k n 2. In partcular, t s stable f q n q n 1 + q n, q n q n 2 + q n 1 + q n, and q k j>k q j n k for all 1 k n 4. The above proposton provdes frst a necessary and suffcent condton, as well as a second suffcent (but not necessary) condton for the grand coalton to be stable. The former condton mples that, for nstance, when p = 500 q, q 1 = 30, q 2 = 35, q 3 = 100, and q 4 = 200, the grand 16