Forecasting and Information Sharing in Supply Chains Under Quasi-ARMA Demand

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1 Forecasing and Informaion Sharing in Supply Chains Under Quasi-ARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in a muli-sage supply chain in which he reailer observes ARMA demand. In conras o previous work, we show how each player consrucs he order based upon is bes linear forecas of leadime demand given is available informaion. In order o characerize how demand propagaes hrough he supply chain we consruc a new process which we call quasi-arma or QUARMA. QUARMA is a generalizaion of he ARMA model. We show ha he ypical player observes QUARMA demand and places orders ha are also QUARMA. Thus, he demand propagaion model is QUARMA-in-QUARMA-ou. We sudy he value of informaion sharing beween adjacen players in he supply chain. We demonsrae ha under cerain condiions informaion sharing can have unbounded benefis. Our analysis hence reverses and sharpens several previous resuls in he lieraure involving informaion sharing and also opens up many quesions for fuure research. KEYWORDS: Supply Chain Managemen, Informaion Sharing, Time Series, ARMA, Inveribiliy. Sy Syms School of Business, Yeshiva Universiy, BH-428, 500 Wes 185h S., New York, NY Deparmen of Informaion, Operaions, and Managemen Science, Leonard N. Sern School of Business, New York Universiy, Suie 8-52, 44 Wes 4h S., New York, NY IROM B6500, 1 Universiy Saion, Ausin, TX

2 1 Inroducion We sudy demand propagaion in a muli-sage supply chain in which he reailer observes ARMA demand wih Gaussian whie noise (shocks). Similar o previous research, we assume each supply chain player consrucs a linear forecas for he leadime demand and uses i o deermine he order quaniy via a periodic review myopic order-up-o policy. The new feaure in our paper is he derivaion of he bes linear forecas of leadime demand for each supply chain player given he paricular informaion ha is available o ha player. This is imporan since under he assumpion of Gaussian whie noise he bes linear forecas is he bes forecas. Therefore, in he absence of he bes linear forecas, he order quaniy deermined by a player would be sub-opimal, see for example Johnson and Thompson (1975). In order o consruc is bes linear forecas, we describe how a supply chain player mus firs, via a radiional ime-series mehodology, characerize he informaion available o he player. We assume ha informaion sharing occurs only beween coniguous players. We show ha under cerain assumpions here arise only four muually exclusive and exhausive informaion ses when coniguous players eiher share or do no share heir informaion (shocks). This permis us o exacly consruc a recursive process ha characerizes how demand propagaes from player o player when each player uses his or her bes linear forecas. Under each of he four informaion ses, we show ha he demand propagaion model is QUARMA-in-QUARMA-ou where a QUARMA model is defined in (3) and is a more general ime series model han an ARMA model. This resul generalizes he ARMA-in-ARMA-ou propery proposed by Zhang (2004). Furhermore, we show ha under cerain condiions informaion sharing can have remendous benefis. Our analysis hence reverses and sharpens several of he previous resuls in he lieraure involving informaion sharing and also opens up many quesions for fuure research. Our paper is an exension of he work of Lee, So and Tang (2000) and Raghunahan (2001), Zhang (2004) and Gaur, Giloni, and Seshadri (GGS, 2005), who sudy he value of informaion 1

3 sharing in supply chains. Zhang and GGS exend he original work of Lee, So and Tang (2000) and Raghunahan (2001) by sudying he value of informaion sharing in supply chains where he reailer serves an ARMA(p, q) demand as opposed o AR(1) demand. In each of hese papers, he reailer places orders wih a supplier using a periodic review order-up-o policy. Boh he supplier and he reailer know he parameers of he demand process, however, he reailer may or may no choose o share informaion abou he acual realizaions of demand wih he supplier. Zhang (2004) sudies how he order process propagaes upsream in a supply chain under he assumpion ha ARMA demand o he reailer is inverible. Zhang does no consider he case when demand becomes non-inverible a any sage of he chain, a phenomenon ha GGS poin ou can happen even hough he reailer s demand is inverible. The concep of inveribiliy is cenral o an undersanding of he value of informaion sharing, and also o an undersanding of how demand propagaes hrough a supply chain. In any ARMA(p, q) model for demand a a given sage of he supply chain, a linear combinaion of he presen observaion and p pas demand observaions is se equal o a consan plus a linear combinaion of he presen and q pas values of a paricular whie noise (shock) series. The given player in he supply chain does no direcly observe he whie noise series, bu only he presen and he pas values of his or her own demand. To faciliae he consrucion of he opimal forecas and he evaluaion of he corresponding mean squared forecas errors, i is useful o represen a given demand series as a consan plus a (poenially infinie) linear combinaion of presen and pas shocks. Zhang assumes ha he curren shock can be obained from presen and pas demand observaions, and uses hese shocks o consruc forecass for leadime demand (equaion (11) in Zhang). Unforunaely, as poined ou by GGS, his is no always possible due o lack of inveribiliy. Informally, inveribiliy of a demand series wih respec o a paricular shock series is he propery under which i is possible o obain he curren shock by linear operaions on he presen and pas demand observaions. In Secion 2, we provide a precise definiion of inveribiliy, as well 2

4 as he necessary and sufficien condiions for inveribiliy of an ARMA(p, q) model wih respec o he shocks in is defining equaion, in erms of he q moving average parameers in ha equaion. GGS show ha he supplier s demand may no be inverible wih respec o he reailer s shocks, even when he reailer s demand is inverible wih respec o is own shocks. Therefore, in his case, Zhang s order process canno be uilized since i would require informaion ha is no available o he supplier, while he order process proposed by GGS can be used. The GGS order process for his case uilizes a sub-opimal forecas as opposed o he bes linear forecas of leadime demand, and hence, resuls in larger invenory relaed coss compared o hose under he bes linear forecas. In his paper, we provide an order process for he reailer as well as (under some resricions) for each upsream supply chain player based on is bes linear forecas of leadime demand where is demand is non-inverible wih respec o he reailer s shocks, and informaion is no shared beween he reailer and he supplier. We also derive he resuling opimal mean squared forecas error. In his case, he demand observed by he supplier is less informaive han he reailer s demand, because he supplier can recover neiher he reailer s shocks nor he reailer s demand based on he supplier s own demand. Here, here is a benefi o he supplier when he reailer shares is demand informaion wih he supplier, because hen he supplier can base he forecas of leadime demand on he reailer s shocks. We show ha in some circumsances he improvemen from demand sharing can be unbounded, as measured by he raio of he mean squared errors of he resuling forecass. The implicaions of our research, boh for wo-sage and muli-sage supply chains, are in conras o he managerial implicaions suggesed by boh Zhang and GGS. For example, Zhang concludes ha informaion sharing is generally no beneficial o he supplier, nor o any oher upsream player, since he saes ha he player in quesion will be able o infer he previous player s demand and ulimaely he reailer s shocks. GGS concludes ha under an addiional condiion, once he supplier s demand is no inverible wih respec o he reailer s shocks, informaion sharing 3

5 would be required by all upsream players in order o consruc heir order processes. In conras o Zhang, we show ha in some circumsances here can be grea benefi from informaion sharing o a supply chain player when he demand observed by he player is no inverible wih respec o he previous player s full informaion shocks, see Definiion 2. Informally, a player s full informaion shocks are hose shocks ha convey he full informaion ha is available o he player. This informaion is based upon eiher he player s own curren and hisorical demand or a sharing arrangemen wih he previous player. Thus, even hough a cerain se of shocks migh be driving he order process of he previous player, he curren player migh no have access o hose shocks unless he previous player shares hem. Furhermore, we show ha when a supply chain player is able o obain he previous player s full informaion shocks, is order process is similar o ha suggesed by Zhang. If a supply chain player is no able o obain or consruc he previous player s full informaion shocks, we show ha he forecas proposed by Zhang canno be used. For he majoriy of our paper, we assume he supply chain is such ha a each individual sage, eiher he equivalen of full informaion shocks are shared or here is no sharing. In conras o GGS, we show ha under such assumpions, even if a supply chain player s demand is no inverible wih respec o he previous player s shocks and he previous player does no share informaion i is possible o consruc is bes linear forecas of leadime demand. In oher words, informaion sharing is never needed o deermine a player s order process and hence he propagaion of demand can be described precisely wih or wihou informaion sharing a any sage of he chain. Even hough we allow only for he possible sharing of informaion beween coniguous players in he chain, he paerns of he effecs of informaion sharing on mean squared forecasing error can be quie complex. As we show in Secion 6, here can be a difference beween sharing of demand and sharing of shocks beween players k 1 and k when k 3. Unil now, researchers have sudied demand sharing, whereas we show ha shock sharing a imes provides greaer informaion. Furher, if one were o relax he assumpion ha informaion is shared only beween coniguous 4

6 players, here exiss fuure work on how o characerize he informaion ses for each player in he supply chain. Our research is valuable o managers because i even perains o he simple case where a reailer observes AR(1) demand, since even in his case, he reailer s order process can be a non-inverible ARMA(1, 1) process wih respec o he reailer s shocks. As noed by oher researchers, anoher reason why our research is valuable o managers is ha acual demand paerns ofen follow higher order auoregressive processes due o he presence of seasonaliy and business cycles. For example, he monhly demand for a seasonal iem in is simples form is approximaely an AR(12) process. More general ARMA(p, q) processes are found o fi demand for long life cycle goods such as fuel, food producs, machine ools, ec. as observed in Chopra and Meindl (2001) and Nahmias (1993). Aviv (2003) proposes an adapive invenory replenishmen policy ha uilizes he Kalman filer echnique in which he ypes of demand considered are ARMA models as well as more general demand processes. The srucure of our paper is as follows. In Secion 2 we discuss he model seup wih regard o our assumpions on all supply chain players, define he concep of inveribiliy and coninue our discussion of is imporance in supply chain managemen under ARMA demand. In Secion 3 we discuss he reailer s order process along wih he mean square forecas error of aggregaed demand over he lead ime. In Secion 4, we demonsrae how a player observing demand ha is no inverible wih respec o he previous player s full informaion shocks can represen is demand as an inverible ARMA process wih respec o a new se of shocks. In Secion 5, we show how demand propagaes upsream in a supply chain under he assumpion ha all players eiher shared he equivalen of heir full informaion shocks or did no share any informaion. We also discuss he value of informaion sharing for he various scenarios discussed previously. In Secion 6, we demonsrae ha he value o player k of receiving demand informaion shared by player k 1 may be differen han he value o player k of receiving shocks shared by player k 1. In Secion 7, we 5

7 provide a summary of our resuls and include some suggesions for fuure research. 2 The Model; Inveribiliy We consider a K-sage supply chain where a discree equally-spaced ime periods, he reailer (assumed o be a sage 1) faces exernal demand {D 1, }, for a single iem. Le {D 1, } follow a covariance saionary ARMA (p, q 1 ) process wih p 0, q 1 0, D 1, = d + φ 1 D 1, 1 + φ 2 D 1, φ p D 1, p + ɛ 1, θ 1,1 ɛ 1, 1 θ 1,2 ɛ 1, 2 θ 1,q1 ɛ 1, q1, (1) where {ɛ 1, } is a sequence of uncorrelaed random variables wih mean zero and variance σ 2 1, and φ 1,..., φ p and θ 1,1,..., θ 1,q1 are known consans such ha φ p 0 and all roos of he polynomial 1 φ 1 z φ p z p are ouside he uni circle. We also assume ha all roos of he polynomial 1 θ 1,1 z θ 1,q1 z q 1 are on or ouside he uni circle. This ensures ha he reailer s demand is inverible wih respec o is full informaion shocks. (See he discussion below.) If θ 1,1,..., θ 1,q1 are all equal o zero, hen {D 1, } is an AR(p) process. I is ofen useful o express (1) in erms of he backshif operaor, B, where B s (ɛ 1, ) = ɛ 1, s. In order o do so, le φ(b) = 1 φ 1 B φ 2 B 2 φ p B p and θ 1 (B) = 1 θ 1,1 B θ 1,2 B 2 θ 1,q1 B q 1. Then {D 1, } in (1) can be expressed as φ(b)d 1, = d + θ 1 (B)ɛ 1,. (2) Since {D 1, } is covariance saionary, E[D 1, ] exiss and is consan for all, V ar[d 1, ] is a finie consan, and he covariance beween D 1, and D 1,+h depends on h bu no on. Following Lee, So and Tang (2000) and Zhang (2004), we assume ha he shocks {ɛ 1, } are Gaussian whie noise. We noe ha his implies ha full informaion shocks (when hey exis) a all subsequen sages are also Gaussian due o he manner in which demand and full informaion shocks propagae upsream a supply chain (see Theorem 1). Le he replenishmen leadime from he reailer s supplier o he reailer be l 1 periods. Similarly, le he replenishmen leadime from he player a sage k + 1 o sage k be l k periods. We assume ha all supply chain players use a 6

8 myopic order-up-o invenory policy where negaive order quaniies are allowed, and d is sufficienly large so ha he probabiliy of negaive demand or negaive orders is negligible. In ime period, afer demand D 1, has been realized, he reailer observes he invenory posiion and places order D 2, wih is supplier. The reailer receives he shipmen of his order a he beginning of period + l 1, where l 1 1. The sequence of evens a all supply chain players is similar. We assume ha he l k period lead ime guaranee holds, i.e., if he player a sage k + 1 does no have enough sock o fill an order from he player a sage k, hen he player a sage k +1 will mee he shorfall from an alernaive source, possibly by incurring an addiional cos. See, for example, Lee So and Tang or Chen (2003) for a discussion of his assumpion. Excess demand a he reailer is backlogged. Wih respec o he informaion srucure, we assume, as did GGS and ohers, ha he form and parameers of he model generaing player k 1 s demand are known by player k, bu player k 1 s demand realizaions and/or full informaion shocks may be privae knowledge. When here is no informaion sharing, he reailer s supplier receives an order of D 2, in ime period from he reailer. On he oher hand, when here is informaion sharing, he reailer s supplier receives he order D 2, as well as informaion abou D 1, and/or ɛ 1, in ime period from he reailer. As we show laer in he paper, if he reailer s supplier receives informaion abou D 1, in ime period from he reailer, he will be able o consruc he shock series {ɛ 1, } as described in (1). Unforunaely, as we show in Secion 6, his is no necessarily he case for upsream player k who receives informaion abou D k 1, in ime period from player k 1 (ha is, i migh no be possible for player k o consruc he shock series {ɛ k 1, }). Therefore, he poenial informaion sharing arrangemens beween upsream supply chain players a coniguous sages can be quie complex. Cenral o our sudy of informaion sharing and how demand {D 1, } propagaes upsream in a supply chain is he concep of inveribiliy, for which we provide a definiion below. Firs, we define he quasi-arma model. In Theorem 1, we show ha under cerain assumpions, ARMA demand o he reailer propagaes o quasi-arma or QUARMA demand for upsream players even when 7

9 a given player s demand is non-inverible wih respec o he previous player s shocks. Hence, he QUARMA model is cenral o our sudy of he propagaion of demand in a supply chain. One manner in which a given player s demand is non-inverible wih respec o he previous player s shocks is when he given player s demand, represened in erms of he previous player s shocks, has a coefficien of zero on he curren shock. This case is imporan, as i is here ha he poenial for he improvemen from shock sharing is unbounded. In oher words, when presened wih such demand, if he previous player shares is shocks, he given player may have a perfec forecas of leadime demand, whereas wihou informaion sharing he forecas will no be perfec. Consider a demand series {D } expressed in erms of he shocks {ɛ }, D = d + φ 1 D 1 + φ 2 D φ p D p + ɛ J θ 1 ɛ J 1 θ 2 ɛ J 2 θ q ɛ J q, (3) where J 0 enails a lagging of he shocks by J ime unis. We show ha he scenario J > 0 in (3) can arise a any sage beyond he reailer s, even hough he reailer s demand has J = 0, wihou loss of generaliy. We denoe he model (3) wih J 0 by QUARMA(p, q, J). By convenion, we say ha a consan demand series (see Remark 1) is QUARMA wih J =. In erms of he backshif operaor, he QUARMA(p, q, J) model may be expressed as φ(b)d = d + B J θ(b)ɛ, where θ(b) = 1 q j=1 θ jb j if J < and θ(b) = 0 if J =. When J = 0, {D } is ARMA(p, q) wih respec o {ɛ }, so ha he QUARMA(p, q, 0) model reduces o an ARMA(p, q) model. The AR and MA polynomials of he QUARMA(p, q, J) model in (3) may have common roos. A QUARMA(p, q, J) model in which he AR and MA polynomials do no share any common roos is said o be in minimal form. Definiion 1 {D } is inverible wih respec o shocks {ɛ } in (3) wih J < if he curren and previous shocks can be recovered by a consan plus a linear combinaion of presen and pas values of he demand process {D } or as a consan plus he limi of a sequence of linear combinaions of presen and pas values of he demand process {D }. 8

10 Quesions of inveribiliy are no relevan in (3) wih J = (since {D } does no depend on he shocks {ɛ }), nor are quesions of informaion sharing since if a player observes consan demand, here is no need for informaion sharing. Thus, henceforh when we discuss inveribiliy or informaion sharing, we implicily assume (unless saed oherwise) ha he upsream player s demand is no consan. I is clear from he definiion ha if 0 < J <, {D } canno be inverible wih respec o {ɛ }. I is well-known ha for he ARMA case J = 0 (see Brockwell and Davis 1991, pp ), {D } is inverible wih respec o shocks {ɛ } in (3) if and only if all he roos of he equaion 1 θ 1 z θ 2 z 2 θ q z q = 0 (4) lie ouside or on he uni circle. I follows ha for he QUARMA(p, q, J) model wih any 0 J <, {D } is inverible wih respec o he lagged shocks {B J ɛ } if and only if all roos of (4) lie on or ouside he uni circle. In all of he QUARMA represenaions of demand {D } wih respec o shocks {ɛ } considered in his paper, he AR polynomial of upsream players is he same as he AR polynomial of he reailer. Since i was assumed ha he reailer s AR polynomial has all of is roos ouside he uni circle, his is also he case for upsream players. Alhough i is furher assumed ha he reailer s demand is in minimal form, i is possible for an upsream player o observe QUARMA demand whose AR and MA polynomials share a leas one common roo. Such a common roo mus lie ouside he uni circle since all he roos of he AR polynomials in his paper are ouside he uni circle. Therefore, in he even ha {D } is QUARMA(p, q, J) wih respec o {ɛ } where φ(z) and θ(z) share a leas one common roo, consider he associaed QUARMA represenaion of {D } wih respec o {ɛ } in minimal form wih AR polynomial φ m (z) and MA polynomial θ m (z). Based upon our assumpions on φ(b), i follows ha {D } is inverible wih respec o {ɛ } if and only if all roos of he original MA polynomial θ(z) lie on or ouside he uni circle. This is because alhough inveribiliy is based upon he roos of θ m (z), he roos of θ m (z) lie on or ouside he uni 9

11 circle if and only if he roos of he original MA polynomial θ(z) lie on or ouside he uni circle. Under our definiion of inveribiliy, he demand series is inverible wih respec o shocks {ɛ } if and only if he presen and pas demand observaions generae he same linear space (up o an addiive consan) as he presen and pas shocks, so ha from he poin of view of linear operaions he shock series conains no more informaion han he demand series. We noe ha our definiion of inveribiliy coincides wih he exended definiion considered by Brockwell and Davis (1991, p. 128). The exended definiion includes he case where he shocks {ɛ } can only be recovered as a consan plus he limi of a sequence of linear combinaions of presen and pas values of he demand process {D } as opposed o a consan plus a linear combinaion of presen and pas values of {D }. This corresponds o he case where a leas one roo of (4) lies on he uni circle (see Brockwell and Davis 1991, Problem 3.8, p. 111) and can resul in he nex player s demand being consan (see Remark 1). As noed in he inroducion, inveribiliy of player k s demand wih respec o {ɛ k 1, }, he full informaion shocks of player k 1, is cenral o our discussion of he value of informaion sharing in a supply chain. If {D k, } is inverible wih respec o {ɛ k 1, }, hen player k can uilize presen and pas values of {ɛ k 1, } in is forecas of is leadime demand. On he oher hand, if {D k, } is no inverible wih respec o {ɛ k 1, }, hen player k can uilize ɛ k 1, in is forecas of is leadime demand if player k 1 shares informaion equivalen o is full informaion shocks. If {D k, } has a QUARMA represenaion wih respec o {ɛ k 1, } where J > 0 hen {D k, } is no inverible wih respec o {ɛ k 1, } since he curren shock ɛ k 1, canno be observed by player k. In he nex secion, we discuss he reailer s order process and hus provide a saring poin for undersanding how demand propagaes upsream in a supply chain. 10

12 3 Deermining he Reailer s Order As menioned in he previous secion, we assume ha he reailer observes ARMA(p, q) demand as given by (1). We assume ha he ARMA(p, q) model of he reailer s demand is in minimal form, i.e., he values of p and q are such ha here are no common roos of he AR and MA polynomials. We furher assume, as in Lee, So, and Tang, Raghunahan, Zhang, and GGS, ha he reailer uilizes a periodic review sysem. Le S 1, be he order up o level ha minimizes he reailer s oal expeced holding and shorage coss in period + l 1. Le M D 1 = sp{1, D 1,, D 1, 1,...}, i.e., he Hilber space generaed by {1, D 1,, D 1, 1,...} wih inner produc given by he covariance. We refer o M D 1 as he linear pas of {D 1, }. Similarly, le M ɛ 1 = sp{1, ɛ 1,, ɛ 1, 1,...}. For he linear pas of wo ime series, for example, {D 1, } and {D 2, }, we wrie M D 1,D 2 = sp{1, D 1,, D 2,, D 1, 1, D 2, 1,...}. In his paper, we only consider linear forecasing since i follows from our assumpions ha all random variables appearing in he paper are Gaussian, in which case he bes possible forecas is a linear one. In his conex, a given player s informaion se can always be represened as he linear pas of some collecion of one or more ime series. We denoe he full informaion se available o player 1 as M 1. We assumed in Secion 2 ha he reailer s demand {D 1, } is inverible wih respec o he shocks {ɛ 1, }. This enails no loss of generaliy, since he only shocks observable o he reailer are hose obained from an inverible ARMA(p, q) represenaion of is demand. Therefore, M 1 = M D 1 = M ɛ 1. This is he reailer s informaion se available a ime afer D 1, has been observed. We refer o {ɛ 1, } as he reailer s full informaion shocks. Le σ 1 be he sandard deviaion of ɛ 1,. Therefore, he reailer s order-up-o level a ime is given by S 1, = m 1, + kσ 1 v1 where m 1, is he bes linear forecas of l 1 i=1 D 1,+i in he space M 1 and v 1 is he associaed mean squared forecas error. Thus, he reailer s order o is supplier is D 2, = D 1, + S 1, S 1, 1 = D 1, + m 1, m 1, 1. 11

13 Since i is assumed ha {D 1, } is inverible wih respec o {ɛ 1, }, we have ɛ 1, M 1. I follows ha ɛ 1, is observable o he reailer, as are ɛ 1, 1, ɛ 1, 2,.... Furhermore, since {D 1, } is ARMA wih respec o {ɛ 1, } wih all roos of φ(z) ouside he uni circle, i has a one-sided MA( ) represenaion wih respec o hese shocks. Proposiion 1 (Zhang, 2004). The reailer s demand and is bes linear forecas of lead-ime demand can be represened as MA( ) processes wih respec o he reailer s full informaion shocks {ɛ 1, }. Is order {D 2, } o is supplier is given by (5) wih respec o he reailer s full informaion shocks {ɛ 1, }. In paricular wih µ d = d/(1 φ 1 φ p ), he reailer s demand is D 1, = i=0 ψ 1,iɛ 1, i + µ d, where he {ψ 1,i } are given in (19). Furhermore, he reailer s order is D 2, = ( l 1 j=0 ψ 1,j )ɛ 1, + ψ 1,i+l1 ɛ 1, i + µ d. (5) See he Appendix for a version of he proof ha lays he groundwork for our developmen. i=1 Equaion (5) provides a represenaion of he reailer s order as a consan plus a linear combinaion of presen and pas values of {ɛ 1, }, ha are in he reailer s informaion se. Alhough he reailer s order is equivalen o is supplier s demand, {D 2, } can be expressed in erms of various ses of shocks. As will be shown in Theorem 1, {D 2, } has a QUARMA represenaion wih respec o {ɛ 1, }. However, i is possible ha {ɛ 1, } may no be in he supplier s informaion se in he absence of informaion sharing. This occurs when {D 2, } is no inverible wih respec o {ɛ 1, } and hence he supplier will no be able o obain he reailer s curren shock from he supplier s presen and pas demand observaions. Thus, in order o sudy how ARMA demand propagaes upsream in a supply chain, i is necessary for us o undersand he poenial informaion ses faced by a given supply chain player. In he above case of he reailer, is informaion se is sraighforward. However, he informaion se ha is available o supply chain player k wih k 2 can be quie complex. We discuss some of hese informaion ses in Secion 5. Bu, before doing so, we firs describe how a player observing demand ha is no inverible wih respec o he previous player s full informaion shocks can represen is demand as an inverible ARMA process wih respec o a new se of shocks. 12

14 Remark 1 I is possible ha {D 2, } in (5) reduces o a consan, D 2, = µ d for all. This case arises when 1 + l 1 j=1 ψ 1,j = 0 and ψ 1,i+l1 = 0 for i 1. I can be easily seen ha hese condiions are saisfied when {D 1, } is an MA(q 1 ) process wih respec o {ɛ 1, } wih q 1 l 1 and he MA polynomial has a roo a 1. In Theorem 1, we provide condiions when {D k, } reduces o a consan when player k 1 s demand is QUARMA wih respec o {ɛ k 1, }. 4 Represenaion of Noninverible ARMA Demand Consider an ARMA(p, q) model for a demand series {D } wih respec o a shock series {ɛ }, ha is, φ(b)d = d + θ(b)ɛ where θ(b) = 1 θ 1 B θ q B q. As discussed in Secion 2, he series {D } is non-inverible wih respec o {ɛ } if and only if a leas one roo of θ(z) lies inside he uni circle, where z is a complex variable. This siuaion may be faced by a given supply chain player wih respec o he previous player s full informaion shocks. If he previous player (k 1) does no share any informaion wih he given player (k), hen in order o forecas is leadime demand, under he assumpions in Secion 5 along wih he assumpion ha player k s demand is no consan, player k firs represens is demand as ARMA(p, q) wih respec o a new se of shocks ha are observable in he sense ha hey generae he same linear pas as he given player s demand series, such ha he new ARMA(p, q) model yields he same variance and auocorrelaions as he original one. We now explain how his is done for he ARMA series {D } originally expressed wih respec o {ɛ }. Since {D } is ARMA(p, q) wih respec o shocks ɛ, p q (1 a 1 s B)D = d + (1 b 1 s B)ɛ, (6) s=1 s=1 where a s, 1 s p are he roos of he polynomial equaion 1 φ 1 z φ 2 z 2 φ p z p, b s, 1 s q are he roos of he polynomial equaion 1 θ 1 z θ 2 z 2 θ q z q. 13

15 Since {D } is no inverible wih respec o shocks {ɛ }, here exiss a leas one 1 r q such ha b r < 1. Wihou resricion of generaliy, we assume ha b s > 1, 1 s h, and b s < 1, h < s q. I follows (see for example Brockwell and Davis (1991, pp )) ha {D } can be represened as an inverible ARMA(p, q) wih respec o a new se of shocks {γ } wih variance σ 2 γ = ( h<s q b s ) 2 )σ 2 ɛ. Since b s < 1, for h < s q, q s=h+1 b s 2 > 1 and hence σ 2 γ > σ 2 ɛ. This inverible represenaion has form p (1 a 1 s B)D = d + s=1 where b s is he complex conjugae of b s. Thus h (1 b 1 s B) s=1 q s=h+1 (1 b s B)γ, (7) φ(b)d = d + θ(b)ɛ = d + θ (B)γ (8) where h q θ (B) = (1 b 1 s B) (1 b s B) (9) s=1 s=h+1 has all of is roos ouside he uni circle. Thus, he ARMA(p,q) represenaion of {D } in (8) is inverible wih respec o {γ } and hence γ, γ 1,... are observable o he player observing demand {D }, i.e., M γ = MD. Furhermore, in such a case, {γ } is he only se of shocks wih he following properies: (i) {D } is inverible wih respec o {γ }, (ii) he roos of φ(b) are all ouside he uni circle, and (iii) he coefficien in fron of γ is one in he ARMA represenaion of {D }. The above mehodology is widely available in he ime series lieraure, bu o he bes of our knowledge has no been noed in he OM lieraure. This mehodology is cenral o our paper as i demonsraes how a supply chain player can consruc shocks ha include as much informaion as is available o he player wihou informaion sharing. In conras, GGS did no use his mehodology when considering a supplier s demand ha is no inverible wih respec o he reailer s shocks and hence is proposed forecas is sub-opimal from he supplier s perspecive. Furhermore, he fac ha σγ 2 > σɛ 2 provides he poenial for value of informaion sharing. 14

16 5 The Propagaion of Demand Under a Resricion on Informaion Sharing Player k mus forecas is leadime demand, using all informaion available o i. A variey of informaion ses can arise here, depending on sharing arrangemens and on consideraions of inveribiliy. We denoe he full informaion se available o player k as M k. This enails M D k possibly ogeher wih informaion obained hrough sharing wih player k 1 when k 2. Definiion 2 We say ha {ɛ k, } are player k s full informaion shocks if here exiss a whie noise sequence, {ɛ k, }, such ha M k = M ɛ k. In his secion, we show how demand propagaes upsream (and how player k can consruc is full informaion shocks) in a supply chain under he assumpion ha all players eiher shared he equivalen of heir full informaion shocks or did no share any informaion. In order o accomplish his ask, we need o undersand he various informaion ses ha a supply chain player may face. In Theorem 1 below, we show ha here are four such ses, which are muually exclusive and exhausive. The heorem hus needs o esablish he exisence and consrucion of each player s full informaion shocks which iself is coningen upon demonsraing hose informaion ses ha are available o a supply chain player. Furhermore, i is only possible o deermine which informaion ses are available o a supply chain player when one models how demand propagaes upsream he supply chain. Hence, Theorem 1 necessarily includes hese hree pars along wih a fourh ha alhough appears o only be a special case would resul in he heorem being incomplee if lef ou. In oher words, in order o deermine is order o player k, player k 1 mus firs represen is demand, and hen is bes linear forecas of is leadime demand, each as a consan plus a linear combinaion of is presen and pas full informaion shocks, {ɛ k 1, }. Finally, using he order-up-o-policy, player k 1 consrucs is order {D k, } o player k again as a consan plus a linear combinaion of is presen and pas full informaion shocks, {ɛ k 1, }. The forecasing and 15

17 order consrucion is performed using hese represenaions. Theorem 1 esablishes he exisence of such represenaions by consrucing player k s full informaion shocks based on is informaion se. There are several properies which propagae from one sage o he nex, as described in he heorem. These are proved by inducion. One of hese properies is he QUARMA srucure of he demand a each sage (wih respec o boh he given player s and he previous player s full informaion shocks), leading o a QUARMA-in-QUARMA-ou propery. This srucure is used o deermine wheher or no a given demand series is inverible wih respec o is own full informaion shocks, he previous player s full informaion shocks, or o lagged versions of eiher se of shocks. The QUARMA srucure also allows a given player o represen is observed demand as a consan plus a linear combinaion of is own presen and pas full informaion shocks. The four informaion ses ha supply chain player k may face are he following, wih denoing sric subspace here and hroughou he paper. (i)m k = M k 1 and M D k R (ii)m k = M D k = M ɛ k 1 J where 1 J <, and MD k R (iii)m k = M D k where M D k M k 1 and M D k M ɛ k 1 J, for J 0, and MD k R (iv)m D k = R. (10) In case (i), player k has access o he full informaion shocks of player k 1. This can occur eiher because i is possible o represen {D k, } as an inverible ARMA process wih respec o {ɛ k 1, } or becase {D k, } is no inverible wih respec o {ɛ k 1, } bu player k 1 shared he equivalen of is full informaion shocks wih player k. In case (ii), player k s informaion se is equivalen o he linear pas of is demand which in urn is equivalen o he linear pas of a lagged se of player k 1 s full informaion shocks. This occurs when β k 1 = 0 in Equaion (12) below, player k 1 has no shared is full informaion shocks wih player k, bu player k s demand is inverible wih respec o he lagged se of player 16

18 k 1 s shocks. In case (iii), player k s informaion se is equivalen o he linear pas of is demand which in urn is a sric subse of player k 1 s informaion se. This occurs when player k s demand is no inverible wih respec o player k 1 s full informaion shocks nor wih respec o any lagged version of player k 1 s full informaion shocks. The forecasing in his case is no done based on he original shocks. Insead a more general mehodology as described in Secion 4 mus be used. In case (iv), player k s demand is consan. Some hough reveals ha inuiively hese four ses are muually exclusive and collecively exhausive for he informaion sharing assumpions made in his secion. This is proven in Theorem 1 below, our main resul on he propagaion of demand. To iniialize his propagaion, we hink of he player previous o he reailer as he reailer iself. Tha is, we define {D 0, } = {D 1, }, {ɛ 0, } = {ɛ 1, }, and l 0 = l 1. We also noe ha if {D k 1, } is QUARMA(p, q k 1, J k 1 ) wih respec o {ɛ k 1, } wih J k 1 < and MA parameers θ k 1,1,, θ k 1,qk 1, hen {D k 1, } can be represened as a consan plus a linear combinaion of is presen and pas full informaion shocks, i.e., D k 1, = µ d + i=0 ψ k 1,iɛ k 1, i where 0 i < J k 1 ψ k 1,i = 1 i = J k 1 (11) p j=1 φ jψ k 1,i j θ k 1,i Jk 1 i > J k 1 and θ k 1,i = 0 if i > q k 1 + J k 1. If {D k 1, } is QUARMA(p, 0, ) wih respec o {ɛ k 1, }, hen ψ k 1,i = 0 for all i and {D k 1, } reduces o a consan. From (11), i can be seen ha J k 1 is an ineger equal o he index of he firs ψ k i,i ha is nonzero. Jk and J k have similar inerpreaions where J k refers o he QUARMA represenaion of {D k, } wih respec o ɛ k 1, and J k refers o he QUARMA represenaion of {D k, } wih respec o ɛ k,. An imporan quaniy for he compuaion of player k 1 s order o player k, as well as he values of J k and J k is 17

19 l k 1 β k 1 = ψ k 1,i. (12) i=0 Theorem 1 Le k 1. Assume ha player k 1 shares wih player k eiher nohing or he equivalen of player k 1 s full informaion shocks, and ha similar arrangemens hold beween all previous players. In he case where a player s demand is consan, informaion sharing o or from he player is irrelevan. Then (I) Player k 1 s order o player k wih respec o player k 1 s full informaion shocks: {D k, } has a QUARMA(p, q k, J k ) represenaion wih respec o {λ k, Jk ɛ k 1, } of form φ(b)d k, = d + B J k θk (B)λ k, Jk ɛ k 1,, where {λ k,i } are defined in (32) and (33), J0 = J 1 = 0, Jk 0 is formally defined in (31) and q k = max{max(p, q k 1 l k 1 + J k 1 ) J k, 0}. If Jk <, hen θ k (B) is a polynomial in B of order q k wih leading coefficien 1 and θ k,j = λ k, J k +j λ k, Jk, j = 1,, q k. If J k =, hen θ k (B) = 0. (II) The four informaion ses: The four ses in (10) are he muually exclusive and exhausive possibiliies for player k s informaion se. These deermine player k s full informaion shocks {ɛ k, } as follows: In case (i), ɛ k, = λ k, ɛ Jk k 1, ; in case (ii), ɛ k, = λ k, ɛ Jk k 1, ; in case (iii), Jk ɛ k, = λ k, ( θ Jk k ) 1 (B)[φ(B)D k, d]; in case (iv), ɛ k, = 0. (III) Player k s demand wih respec o is full informaion shcks: {D k, } has a QUARMA(p, q k, J k ) represenaion wih respec o {ɛ k, } of form φ(b)d k, = d + B J kθ k (B)ɛ k,. If Jk <, θ k (B) is a polynomial in B of order q k wih leading coefficien 1. In case (i), θ k (B) = θ k (B), J k = J k < ; in case (ii), θ k (B) = θ k (B), J k = 0 and 0 < J k < ; in case (iii), θ k (B) = θ k (B), J k = 0 and 0 J k < ; in case (iv), θ k (B) = θ k B = 0 and J k = J k =. (IV) Consan demand: For k 2, if {D k 1, } is QUARMA(0, qk 1 m, J k 1) in minimal form wih respec o {ɛ k 1, } wih an MA polynomial ha has a roo a 1 and l k 1 J k 1 +qk 1 m, hen J k = and {D k, } is consan. The full informaion shocks in par (II) of Theorem 1 have a specific muliplicaive consan, chosen in order o ensure ha he coefficien of he firs shock wih a nonzero coefficien (if here exiss such a shock) in he QUARMA represenaion in Par (III) of he heorem is one. This value of λ k, in (32) and (33), and he variance σ 2 Jk k of player k s full informaion shocks ɛ k,, depends upon sharing arrangemens beween player k 1 and player k as well as consideraions of inveribiliy 18

20 of player k s demand wih respec o player k 1 s full informaion shocks. Each disinc scenario (excep for he case of consan demand) corresponds o a paricular subcase of (10) as described in Corollary 1 below, which follows from he proof of Theorem 1 and from Lemma 2. The QUARMA models in pars (I) and (III) of heorem 1 can have MA and AR polynomials wih common roos even hough his is no he case a he reailer. This is imporan because in such a case a QUARMA represenaion of {D k 1, } wih respec o {ɛ k 1, } in minimal form will have p m k 1 < p. Furhermore, he exisence of common roos may resul in pm k 1 = 0 which ogeher wih he oher condiions in par (IV) of Theorem 1 will resul in {D k, } being consan. In order o inerpre he heorem and ease he discussion of he value of informaion sharing, Corollary 1 is included as i maps he resuls of Theorem 1 o he possible siuaions ha a supply chain player may face. Once a player s full informaion shocks and heir variance are deermined, we show below in Secion 5.1 how a player s bes linear forecas and hence MSFE can be consruced. Finally, he MSFE of he no informaion sharing case can be compared o ha of he informaion sharing case in order o deermine he value of informaion sharing which we show below in Secion 5.2. Corollary 1 Under he condiions of Theorem 1, if {D k, } is inverible wih respec o {ɛ k 1, }, hen case (i) holds, β k 1 0, λ k, = β Jk k 1 and σk 2 = β2 k 1 σ2 k 1 ; if {D k, } is no inverible wih respec o {ɛ k 1, }, β k 1 0, and player k 1 shared he equivalen of is full informaion shocks wih player k, hen case (i) holds, λ k, Jk σ 2 k = β2 k 1 σ2 k 1 ; = β k 1 and if {D k, } is no inverible wih respec o {ɛ k 1, }, β k 1 = 0, and player k 1 shared he equivalen of is shocks wih player k, hen case (i) holds, λ k, Jk = ψ k 1,lk 1 + J k and σk 2 = ψ 2 k 1,l k 1 + J σ 2 k k 1 ; if {D k, } is no inverible wih respec o {ɛ k 1, }, β k 1 = 0, player k 1 did no share wih player k, and θ k (B) has all roos ouside he uni circle, hen case (ii) holds, λ k, = Jk 19

21 ψ k 1,lk 1 + J k and σ 2 k = ψ2 k 1,l k 1 + J k σ 2 k 1 ; if {D k, } is no inverible wih respec o {ɛ k 1, }, β k 1 0, and player k 1 did no share wih player k, hen case (iii) holds, λ k, Jk = β k 1 and σ 2 k = β2 k 1 ( h<s q k b s ) 2 )σ 2 k 1 ; if {D k, } is no inverible wih respec o {ɛ k 1, }, β k 1 = 0, player k 1 did no share wih player k, and θ k (B) has a leas one roo inside he uni circle, hen λ k, Jk = ψ k 1,lk 1 + J k and σ 2 k = ψ2 k 1,l k 1 + J k ( h<s q k b s ) 2 )σ 2 k 1 ; 5.1 The Bes Linear Forecas We now urn o a discussion of forecasing leadime demand from he poin of view of player k, and is associaed mean squared forecas error. We sae a lemma on forecasing of a general demand series {D } ha can be represened as a consan plus a linear combinaion of presen and pas values of a whie noise sequence {ɛ } wih M ɛ assumed known. This lemma and is proof were uilized in he proof of Theorem 1. Lemma 1 If demand {D } can be represened as a consan plus a linear combinaion of presen and pas values of a whie noise series {ɛ } wih M ɛ known of form D = µ d + j=0 ψ jɛ j, he bes linear forecas of l-sep leadime demand (he projecion ino M ɛ ), is given by m = i=0 ω i+lɛ i + lµ d and he associaed mean squared forecas error is given by σ 2 ɛ l 1 i=0 ω2 i, where ω i are defined in (24) and σ 2 ɛ = var(ɛ ). We now discuss how o use his lemma in conjuncion wih Corollary 1. Firs, in all siuaions considered in his secion, player k 1 s order o player k, {D k, }, can be represened as a consan plus a linear combinaion of presen and pas {ɛ k 1, }. However, player k will only be able o use his represenaion o forecas is leadime demand if M k = M ɛ k 1, ha is, in case (i) of (10). Zhang (2004) implicily assumes ha presen and pas shocks ɛ k 1,, ɛ k 1, 1, are always observable o player k. Unforunaely, his assumpion does no hold in cases (ii) and (iii) of (10). Second, by Theorem 1 par (III), player k has a QUARMA represenaion of is demand wih respec o is full informaion shocks {ɛ k, }. However, player k s full informaion shocks and hence 20

22 is QUARMA represenaion will differ depending upon which scenario lised in Corollary 1 player k may face. The scenario faced by player k depends upon inveribiliy of {D k, } wih respec o {ɛ k 1, } and wheher or no here exiss a sharing arrangemen beween player k 1 and player k. Lemma 1 provides player k s mean squared error of is forecas of leadime demand in erms of he variance of is full informaion shocks, σk 2 = var(ɛ k,). Specifically, we le ω k,i = 0 for i < 0, ω k,0 = ψ k,0, ω k,i = ω k,i 1 + ψ k,i for 0 < i < l k and ω k,i = ω k,i 1 + ψ k,i ψ k,i lk for i l k, where ψ k,i follows from (11) wih k 1 replaced by k. Player k s bes linear forecas of leadime demand and associaed mean squared forecas error are given by m k, and σ 2 k lk 1 i=0 ω2 k,i respecively, where m k, = ω k,i+lk ɛ k, i + l k µ d. (13) i=0 A his sage, we show how i is possible o recursively consruc he propagaion of demand hrough he supply chain. Player k 1 places an order o player k ha is QUARMA wih respec o is full informaion shocks as per par (I) of Theorem 1. Nex, one needs o deermine if player k s demand is inverible wih respec o {ɛ k 1, }. In he even ha player k s demand is no inverible wih respec o {ɛ k 1, }, one needs o consider if here is a sharing arrangemen beween player k 1 and player k. Based upon hese wo deerminaions, one can consruc player k s full informaion shocks and can deermine which specific case of Corollary 1 player k faces. Then, player k s QUARMA represenaion of is demand can be deermined as in par (III) of Theorem 1 along wih is bes linear forecas of leadime demand as in (13). Player k s order o player k + 1 follows he myopic order-up-o policy and he propagaion of demand coninues upsream similarly. 5.2 The Value of Informaion Sharing Wheher or no here is an informaion sharing agreemen beween players k 1 and k, and wheher or no player k s demand is inverible wih respec o player k 1 s full informaion shocks, player k is able o use Theorem 1 in conjuncion wih Corollary 1 as well as he proof of Lemma 2 o consruc is bes linear forecas of leadime demand given he informaion available o i. Below, we 21

23 deermine he value of informaion sharing, by comparing he mean square forecas error of each no informaion sharing scenario o he associaed informaion sharing scenario. In oher words, player k s mean squared forecas error depends on he variance of is full informaion shocks as well as on he values of ω k,i. However, boh of hese quaniies depend on which scenario from Corollary 1 player k faces. We propose o measure he value of informaion sharing by he raio of player k s mean square forecas error where player k 1 did no share is shocks o where player k 1 did share is shocks. In oher words, V = MSF ENS k. I is clear ha V 1 and when V = 1 here is no value o MSF Ek S informaion sharing. Below, we discuss he value of informaion sharing by considering hose cases from Corollary 1. If {D k, } is inverible wih respec o {ɛ k 1, }, hen here is no value of informaion sharing beween player k 1 and player k since he informaion available o player k from he linear pas of is demand is equivalen o player k 1 s full informaion se, i.e., V = 1. If {D k, } is no inverible wih respec o {ɛ k 1, }, hen here generally is value of informaion sharing beween player k 1 and player k. See Proposiion 2 below showing he possibiliy of unbounded gain from informaion sharing. We noe ha in hose cases where M D k M k 1, i is possible ha here may be no value o informaion sharing (see Pierce, 1975). Typically, hough, if M D k M k 1, hen if player k 1 does no share is full informaion shocks wih player k, here is informaion loss and player k s MSFE of is aggregaed demand over he lead ime is larger han if full informaion shocks had been shared. We numerically demonsrae he value of informaion sharing and he poenial for a simple AR(1) demand o resul in benefi of informaion sharing for an upsream player. In all figures in his secion, we assume ha he reailer observes ARMA(1,1) demand wih leadime l 1 = 1. In Figure 1, he enclosed region shows hose values of φ and θ 1 of he reailer s demand model such ha he supplier s demand will no be inverible wih respec o he reailer s shocks. We poin ou ha 22

24 he enclosed region consiss of one quarer of all possible ARMA(1,1) demand configuraions for he reailer if he leadime is 1. In Figure 2, we sudy he same siuaion as menioned above excep here we measure he value of informaion sharing for he supplier in erms of ln(v ) where V = MSF ENS MSF E S and we assume ha he reailer observes ARMA(1,1) demand wih φ =.7. As θ 1 approaches.3, he value of informaion sharing approaches infiniy since β 1 = 1 + φ θ 1 approaches 0 (See Proposiion 2). From Figure 1, i can be seen ha even when he reailer observes AR(1) demand, i.e,. when θ 1 = 0, he supplier s demand will no be inverible wih respec o he reailer s shocks if φ <.5. Hence our paper is crucial o undersanding how a simple AR(1) demand propagaes along a supply chain and he value of informaion sharing wihin i. For example, if φ =.6 and θ 1 = 0, if l 1 = 1, he reailer s order o he supplier is given by D 2, =.6D 2, 1 + d + ɛ 1, 1.5ɛ 1, 1. Hence {D 2, } is no inverible wih respec o {ɛ 1, }. In he even he reailer does no share wih he supplier and l 2 = 1, hen he supplier s MSFE is.36σ 2 1 whereas if he supplier would sub-opimally use he GGS forecas, he MSFE would be.52σ1 2 MSF EGGS, i.e. MSF E = Proposiion 2 If l k J k, hen here exiss unbounded gain o player k from player k 1 sharing is full informaion shocks. Proof. Since l k J k, i follows ha if player k 1 shares is full informaion shocks wih player k, hen ω k,i = 0 for 0 i l k 1 and hence he mean square forecas error of leadime demand, σ 2 k lk 1 i=0 ω2 k,i, under informaion sharing is 0. On he oher hand, if player k 1 does no share wih player k, hen by Theorem 1 (III), J k = 0, ω k,0 = 1 and hence MSF E NS k σ 2 k > 0. Remark 2 Proposiion 2 demonsraes where here exiss unbounded gain o player k from player k 1 sharing is full informaion shocks. However, as can be seen from Figure 2, here can be grea value o informaion sharing even in a neighborhood around such a configuraion where here exiss unbounded value of informaion sharing. In he nex secion, we relax he assumpion ha every supply chain player shares eiher is 23

25 full informaion shocks or shares nohing. There, we demonsrae ha i is possible for player k s demand no o be inverible wih respec o player k 1 s shocks and ye here is no value o player k for player k 1 o share is demand. Besides being mahemaically ineresing, his also is in conras o resuls of GGS. 6 Shock Sharing versus Demand Sharing In his secion, we assume ha he reailer hrough player k 2 shared eiher nohing or he equivalen of heir full informaion shocks. We show ha under hese assumpions he value o player k of receiving demand informaion shared by player k 1 may be differen han he value o player k of receiving shocks shared by player k 1. However, we leave he sudy of he srucure of he propagaion of demand when sharing of demand informaion occurs as well as research on he relaed opic of sharing beween non-coniguous players o fuure work. Here, we discuss specific scenarios where {D k, } is no inverible wih respec o {ɛ k 1, } under which here are various possible relaionships among M D k 1, M D k, M D k 1,D k, and M ɛ k 1. We show ha he value o player k of receiving demand informaion shared by player k 1 may be : nonexisen; posiive, bu less han ha obained from player k 1 sharing is shocks {ɛ k 1, }; or posiive and equivalen o ha obained from player k 1 sharing is shocks {ɛ k 1, }. The firs scenario we consider is where k = 2 and {D 2, } is no inverible wih respec o {ɛ 1, }. Here, i is equivalen for he reailer o share is demand or is full informaion shocks wih he supplier since {D 1, } is always inverible wih respec o {ɛ 1, } and hence M D 2 M D 1 = M ɛ 1. For he res of his secion, we consider he scenario where k = 3 and {D 2, } is no inverible wih respec o {ɛ 1, } bu he reailer shared he equivalen of is full informaion shocks wih he supplier (player 2). We furher assume ha {D 3, } is no inverible wih respec o {ɛ 2, }, J 2 = J 3 = 0 and boh θ 2 (B) as well as θ 3 (B) have no roos on he uni circle. Since he reailer 24

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