Necessary Of A Retaler-Operator

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1 Decentralzed Inventory Sharng wth Asymmetrc Informaton Xnghao Yan Hu Zhao 1 xyan@vey.uwo.ca zhaoh@purdue.edu Rchard Ivey School of Busness The Unversty of Western Ontaro Krannert School of Management Purdue Unversty 1151 Rchmond Street North W. Lafayette, IN London, ON, N6A 3K7 Canada Aprl 2008, Revsed December 2008, July 2009, Aprl, September, November 2010 Abstract We study the nformaton asymmetry ssues n a decentralzed nventory sharng system consstng of a manufacturer and two ndependent retalers, who prvately hold demand nformaton, non-cooperatvely place ther orders, but cooperatvely share nventores wth each other. We fnd that whle the manufacturer needs retalers mean demand and standard devaton for her wholesale prce decson, each retaler only needs to know the other retaler s demand standard devaton for hs order quantty decson. However, an ncentve compatblty analyss shows that retalers have ncentves to share ther demand nformaton untruthfully. Although a truth-nducng scheme can be developed for a system wth symmetrc retalers who share nformaton between themselves, no such scheme can be developed to ensure truth-tellng to the manufacturer. Further, we develop a coordnaton mechansm (CIS) for the decentralzed nventory sharng system, consderng nformaton asymmetry. We show that CIS coordnates the manufacturer-retalers system and leads to an all-wn stuaton under complete nformaton. More mportantly, CIS mnmzes the value of nformaton such that each party can obtan expected profts very close to ther frst-best profts even under asymmetrc nformaton, hence ndrectly solves the nformaton asymmetry problem. To our knowledge, ths work s the frst to study decentralzed nventory sharng and ts coordnaton consderng asymmetrc nformaton. Key Words: Decentralzed nventory sharng, asymmetrc nformaton, nformaton sharng, ncentve compatblty, coordnaton. 1 Correspondng author

2 1 Introducton and Lterature Revew Inventory sharng (transshpment) among retalers has drawn ncreased attenton from retalers and manufacturers as they seek to succeed n a hghly compettve market. The practce of nventory transshpment from a retaler wth surplus nventory to a retaler who stocks out has become prevalent n the automotve and machne tool ndustres (Narus and Anderson, 1996) and s also routnely performed n fashon ndustry (Dong and Rud, 2004) and wholesale/retal ndustry (e.g., Gallagher 2002). Ths work s motvated by our nteractons wth a leadng heavy-machne manufacturer, whose stuaton s very typcal n ths ndustry. Recognzng the mportance of helpng ts many ndependent dealers provde better servces to ther customers, the manufacturer has set up a dealer servce parts nventory sharng program and even provdes the nformaton system and ncentves for partcpaton. Whle dealers share servce parts nventory, they are very senstve about sharng demand nformaton due to varous reasons. Thus, several nterestng questons arse: What pece(s) of demand nformaton s mportant to the manufacturer and each dealer/retaler for her/hs decson? How should the manufacturer/dealers make ther wholesale prce/orderng decsons f the dealers do not share demand nformaton wth other partes? If the dealers share demand nformaton, wll they share t truthfully? Can the decentralzed system be coordnated wth asymmetrc nformaton? What s the mpact of nformaton asymmetry under the coordnaton mechansm? In ths note, we am to answer these questons by studyng a decentralzed nventory sharng system consstng of a manufacturer (she) and two ndependent retalers (he). The manufacturer, consderng the nventory sharng opportuntes between retalers, determnes her wholesale prce. The retalers prvately hold demand nformaton, non-cooperatvely determne ther order quanttes, but cooperatvely share nventores wth each other. The frst part of the work s focused on nformaton asymmetry ssues n the decentralzed system (secton 2). We frst analyze the full/complete nformaton scenaro (FIS) to see what nformaton s necessary for each player s decson. We then study two scenaros under asymmetrc nformaton, one n whch retalers do not share nformaton (NIS), hence we 1

3 study a Bayesan game, and the other n whch retalers reveal/share ther nformaton (IRS), hence we conduct an ncentve compatblty analyss to nvestgate whether retalers wll share true nformaton and then have a dscusson of truth-nducng. The second focus of ths work s to develop a coordnaton mechansm (CIS) for the decentralzed nventory sharng system, wth the consderaton of nformaton asymmetry (secton 3). We frst analyze CIS under the same three scenaros, complete nformaton (CFIS), asymmetrc nformaton wth no nformaton sharng (CNIS), and asymmetrc nformaton wth nformaton revealng/sharng (CIRS), as those analyzed under the decentralzed system. We then conduct an extensve numercal study to demonstrate the value of coordnaton and the value of nformaton n the decentralzed system and under CIS (secton 3.2). When takng nto consderaton nventory transshpment opportuntes, retalers wll modfy ther order quanttes to gan maxmum proft from satsfyng ther own demand as well as the nventory sharng requests from other retalers. Much lterature has focused on nventory stockng and/or transshpment decsons n these systems (e.g., Tagaras (1989), Anupnd, et al. (2001), Rud, et al. (2001), Zhao, et al. (2005, 2006,2008)). However, few have consdered manufacturer s response to the retalers nventory sharng opportuntes n her wholesale prce decsons (except Dong and Rud (2004) and Shao, et al. (2009)). And even fewer, f any, have consdered the nformaton ssues n the system. All the prevous work on nventory sharng has assumed retalers demand nformaton known to all partes n the network, an assumpton that can be far from realty for decentralzed supply chans n whch each ndependent retaler has hs own objectve. As for coordnaton, there has been lmted work on coordnaton mechansms n nventory sharng systems and none of whch consders asymmetrc nformaton. Anupnd, et al. (1999) and Rud, et al. (2001), consderng an exogenous wholesale prce, use transfer prces for the shared unts to coordnate the retalers. Such coordnatng prces do not always exst (Anupnd, et al. (1999) and Hu, et al. (2007)) and such mechansm runs nto sgnfcant ncentve compatblty ssues when appled to asymmetrc nformaton case. One bg dfferental advantage of the mechansm we develop (CIS) s that t mnmzes the mpact of nformaton such that each party can obtan 2

4 expected profts very close to what they would obtan n a complete nformaton scenaro (.e., ther frst-best profts) even under asymmetrc nformaton, hence ndrectly solves the nformaton asymmetry problems. In addton to the above contrbutons, compared wth the large amount of lterature on asymmetrc nformaton and nformaton sharng, our work also possesses some other dstngushng characterstcs. Frst, whle most of the prevous lterature studes asymmetrc nformaton between vertcal partes n the supply chan, e.g., L (2002), Özer and We (2006), we study both vertcal (between the retalers and the manufacturer) and horzontal (between the retalers themselves) nformaton asymmetry. Further, whle most of the lterature on nformaton sharng assumes players reveal true nformaton, we consder nformaton credblty ssue whch s non-trval n ths nventory sharng case. We also nvestgate truth-nducng schemes. 2 Decentralzed Inventory Sharng System wth Asymmetrc Informaton Consder a decentralzed system consstng of a manufacturer (she) and two ndependent retalers (he), each servng hs own customer base at two dstnct locatons, ndexed by, j = 1, 2 ( j), wthout competng for ther demands. We study a sngle-perod model wth each retaler (say retaler ) facng uncertan demand, D, represented by a cdf F ( ) wth a mean of µ and a standard devaton (SD) of, ndependently drawn from what we refer to as the standardzable dstrbutons, where Z = D µ are..d. wth E[Z ] = 0 and V ar[z ] = 1, e.g., normal dstrbuton, unform dstrbuton, etc. Ths group of dstrbutons s also used n Zhang (2005). We assume µ >> so that the probablty of negatve demand s very small. We assume that retaler has prvate and accurate nformaton on hs true demand dstrbuton parameters, {µ, }, because of hs exclusve proxmty to hs market, whle others only have pror nformaton (detaled later) of what µ and may be. Retalers order from the manufacturer whose producton cost s c per unt and who sells to 3

5 the retalers at a unt wholesale prce, w. Retaler obtans revenue r > w for each unt sold and ncurs a penalty cost, p, for each unt of unsatsfed demand. Unsold nventory has a unt salvage value s < w. To focus on demand nformaton, we assume all cost parameters are common knowledge to all players. The sequence of events s as follows. Frst, the manufacturer announces the wholesale prce (w) to the two retalers, who separately place ther orders (Q ). Then, demand realzes at both retalers. After each retaler satsfes hs demand wth hs own nventory, the retalers share (transshp) nventory f one faces shortage (referred to as the buyer) and the other has surplus nventory (referred to as the seller). Wth τ j beng the cost of transshppng one unt from retaler to retaler j, u j = r j τ j s +p j represents the net proft from transshppng one unt from retaler to retaler j. We assume u j > 0 such that transshpment s mutually proftable to the retalers. In addton, we also assume s < s j + τ j and r + p < r j + p j + τ j, whch ensure transshpment occur only when one retaler has surplus and the other has shortage (Tagaras (1989)). The two retalers cooperatvely dvde the transshpment profts accordng to pre-specfed proportons, (α, α j ), wth α [0, 1] representng the proporton of profts allocated to retaler for each unt transshpped from to j. There s a one-to-one correspondence between (α, α j ) and (C j, C j ) used n Rud, et al. (2001) where C j s the transfer prce pad by retaler j to retaler for each unt of nventory transshpped from retaler to j 2. Snce each retaler () receves a constant proft, α u j, for each unt shared from to j, retalers are sure to truthfully report ther surplus or shortage, the retalers have asymmetrc nformaton regardng ex-ante demand and ex-post vsblty of excess demand/stock that s automatcally guaranteed. Inventory Sharng Game Under Full/Complete Informaton Scenaro (FIS) Ths benchmark scenaro, where retalers demand dstrbuton parameters are known to all partes, corresponds to a two-stage stackelberg game whch can be solved backwards. Specfcally, gven the manufacturer s wholesale prce, w, each retaler,, chooses hs order 2 Wthout loss of generalty, f the seller pays for the transportaton cost, C j s τ j = α u j. 4

6 quantty, Q (w), expectng the nventory sharng opportuntes at the end of the perod. Ths s the same as the nventory sharng game analyzed n Rud, et al. (2001). Usng our notaton, retaler s expected proft, Π F (Q, Q j ), can be wrtten as Π F (Q, Q j ) = E [ r mn(d, Q ) + s (Q D ) + p (D Q ) + +α u j T j + (1 α j )u j T j ] wq, where T j = mn[(q D ) +, (D j Q j ) + ] and the retalers best response order quanttes, (Q F (w), Q F j (w)), can be solved from the followng two frst order equatons: F D (Q ) = r + p w + α u j P {Q (D j Q j ) + < D < Q } r + p s r + p s (1 α j) u j r + p s P {Q < D < Q + (Q j D j ) + },, j = 1, 2. (1) Then, gven (Q F (w), Q F j (w)), the manufacturer s optmal wholesale prce, w F, can be solved from maxmzng her expected proft, Π F M(w) = (w c) (Q F (w) + Q F j (w)). Although there do not exst closed-form solutons to (Q F (w), Q F j (w)) or w F, we can show (proof n appendx) that there exsts the manufacturer s optmal wholesale prce w F and a unque par of retalers correspondng equlbrum order quanttes, {Q F (w F ), Q F j (w F )}. The followng theorem helps us understand the mpact of retalers demand parameters on the manufacturer s optmal wholesale prce. Theorem 1 As retaler s mean demand µ ncreases, the manufacturer s wholesale prce w F ncreases. As retaler s demand SD ncreases, the manufacturer s wholesale prce w F can ether ncrease or decrease. Theorem 1 ndcates that the manufacturer wll choose a hgher wholesale prce for a hgher mean demand at a retaler, because wth the hgher mean demand she can offset the lower order quantty caused by the hgher prce. On the other hand, the nfluence of s less predctable, dependng on the transshpment beneft allocaton proporton α and the magntudes of retalers demand parameters. 5

7 Although the manufacturer s decson, w F, depends on both retalers {µ, } (Theorem 1), we show next that gven w, a retaler only needs to know the other retaler s SD n determnng hs order quantty. To see ths, we use a transformaton of varables by standardzng the demand varable D wth Z = D µ F Z (z ) = r + p w r + p s + and Q wth z = Q µ. Thus, equaton (1) can be rewrtten as α u j P {z σ j (Z j z j ) + < Z < z } r + p s (1 α j) u j P {z < Z < z + σ j (z j Z j ) + },, j = 1, 2. (2) r + p s Therefore, solvng for (Q F (w), Q F j (w)) n equatons (1) s equvalent to solvng for (z F (w), z F j (w)) n equatons (2) due to the followng theorem. Theorem 2 Q F (w) = µ + z F (w), where, of the retalers demand parameters, z F (w) s a functon of only σ j,.e., z F (w) = z F ( σ j, w),, j = 1, 2. Theorem 2 reveals a rather nterestng fact: Gven a wholesale prce, w, each retaler s equlbrum order quantty (say, Q F ) s not nfluenced by the other retaler s mean demand (µ j ), but nfluenced only by the other retaler s SD (σ j ). In other words, a retaler only needs the other retaler s demand SD for hs order quantty decson. Ths result can be explaned by the fact that retalers equlbrum order quanttes, wth the consderaton of the nventory sharng opportuntes at the end of the perod, depend on the dstrbuton of the shortage or surplus at each retaler, whch s nfluenced only by the varance of the demand (and not the mean demand). Ths result provdes us mportant nsghts nto the nformaton asymmetry problem n decentralzed nventory sharng systems. Next, we consder the asymmetrc nformaton case n whch each retaler,, holds prvate nformaton of hs demand dstrbuton parameters, {µ, }. Specfcally, we assume µ takes one of m possble values, and takes one of n possble values,.e., µ O = {µ,k, k = 1, 2,...m }, and S = {,l, l = 1, 2,...n }, wth P,µ (µ ) and P,σ ( ) beng the probabltes that retaler s mean demand and demand SD take the value of µ and, respectvely. We assume {O, S }, and the probablty dstrbutons, {P,µ (µ )}, and {P,σ ( )}, = 1, 2, are common knowledge to all partes. 6

8 Inventory Sharng Game Under No Informaton Sharng (NIS) For ths scenaro, we solve each player s equlbrum decson when each retaler,, holds prvate nformaton, {µ, }, and does not share ths nformaton. Hence, the decentralzed nventory sharng system under ths scenaro s a Bayesan Stackelberg game. Gven w, the two retalers calculate ther Bayesan Nash equlbrum order quanttes based on ther pror nformaton of the other s prvate nformaton. Defne {Q B (µ,, w)}, µ O, S as retaler s Bayesan Nash equlbrum order quanttes for each par of hs demand parameter (µ, ), gven w. Followng the same transformaton n Secton 2 and omttng all the techncal detals, we can show that Q B (µ,, w) = µ + z B (, w), where z B (, w), solved from the n + n j equatons, (3), only depends on all possble ratos of the two retalers standard devatons, { σ j, S, σ j S j }. F Z (z ( )) = r + p w r + p s + + (1 α j) u j r + p s α u j r + p s σ j S j P j,σ (σ j )P {z ( ) σ j (Z j z j (σ j )) + < Z < z ( )} σ j S j P j,σ (σ j )P {z ( ) < Z < z ( ) + σ j (z j (σ j ) Z j ) + }, j = 1, 2, S. (3) Then, based on her pror nformaton of the retalers demand parameters, the manufacturer can solve for her optmal wholesale prce w B that maxmzes her expected proft E[Π B M(w)] = (w c) P,µ (µ ) P,σ ( ) P j,µ (µ j ) P j,σ (σ j ) ( Q B (µ,, w) + Q B j (µ j, σ j, w) ). µ,,µ j,σ j We can show (detals omtted) that there exsts the manufacturer s optmal wholesale prce, w B, and a unque set of correspondng Bayesan Nash equlbrum order quanttes for retaler, Q B (µ,, w B ), for each par of hs demand dstrbuton parameters, {µ, }, µ O, S. Informaton Revealng Scenaro (IRS) and Informaton Credblty 7

9 In ths scenaro, the retalers who have prvate demand nformaton reveal/share ths nformaton to other partes. The queston s: Wll the retalers reveal true nformaton? We assume each retaler wll share only the necessary nformaton,.e., retaler wll share {µ, } wth the manufacturer before she determnes her wholesale prce and wll share wth retaler j upon recevng the wholesale prce before they each make the order quantty decsons 3. Followng Hammond (1979) and Mas-Colell and Vves (1993), we conduct the IC analyss by checkng whether under IRS, retaler wll report hs nformaton truthfully, gven that others trust what retaler reports as true nformaton and that retaler j reports hs demand dstrbuton parameters truthfully. If not, then truth-tellng s not an equlbrum strategy. Let {µ t, σ} t and {µ r, σ r } denote retaler s true and reported demand dstrbuton parameters, respectvely. Snce retaler s order quantty (observable to the manufacturer) should be consstent wth {µ r, σ r }, retalers order quanttes can be calculated as Q (µ r, σ r, w) = µ r + σ r z F (w) and Q j(µ r, σ r, w) = µ t j + σjz t j F (w), where {z F (w), zj F (w)} are solved from equatons (2) wth = σ r and σ j = σj. t Gven {Q (µ r, σ r, w), Q j(µ r, σ r, w)}, we can calculate the manufacturer s optmal w gven retaler reports {µ r, σ r }, w (µ r, σ r ). Therefore, IC s guaranteed f and only f reportng true demand nformaton maxmzes a retaler s expected proft. Defne {µ r, σ r } = argmax {µ r O,σ r S }Π t (Q (µ r, σ r, w (µ r, σ r )), Q j(µ r, σ r, w (µ r, σ r ))). The followng theorem presents the results of the ncentve compatblty analyss. Theorem 3 Truth-tellng s not retalers equlbrum reportng strategy when they share demand nformaton wth the manufacturer and the other retaler,.e., {µ r, σ r } {µ t, σ},. t Theorem 3 demonstrates that, under IRS, a retaler has ncentves to report demand nformaton untruthfully to dstort the manufacturer s wholesale prce and the other retaler s order quantty such that he can obtan a hgher proft. Next, s t possble to develop truthnducng schemes? The answer s yes, to some extent. Usng a reward/penalty contngent 3 We assume each retaler (say ) reports the same demand SD to the manufacturer and retaler j because the manufacturer can easly fnd out any nconsstency n the nformaton by observng the retalers order quanttes and can always obtan hgher profts f the retalers shared the same nformaton wth them (by adjustng w). 8

10 upon the nformaton reported by the retalers, we can develop a truth-nducng scheme for a system wth symmetrc retalers (dentcal cost parameters and α = 0.5) who share nformaton between themselves (detals n Yan and Zhao (2008)). However, due to the complex nteractons among the manufacturer s wholesale prce, retalers order quanttes, and the reported nformaton, no such schemes can be developed to ensure truth-tellng to the manufacturer. 3 A Coordnated Inventory Sharng (CIS) Mechansm Wth Asymmetrc Demand Informaton In ths secton, we develop and analyze a coordnaton mechansm of the nventory sharng system, wth the consderaton of asymmetrc nformaton. Ths mechansm s developed through the nvolvement of the manufacturer and can coordnate just the two retalers or the manufacturer-retalers system and can apply to a system wth ether complete nformaton or asymmetrc nformaton. The coordnaton mechansm we propose (CIS) specfes how to operate the nventory sharng system, wth a set of certan payments. The sequence of events under CIS s as follows: 1. At the begnnng of the game (sellng season), a fxed fee (premum) to jon the nventory sharng program for ths season, δ, s collected from each retaler by the manufacturer. 2. Accordng to her best nformaton of the retalers demand, the manufacturer determnes the wholesale prce and announces t to the retalers. 3. Upon recevng the wholesale prce, each retaler separately places hs order accordng to hs best nformaton of the other retaler s demand. 4. Demand realzes at each retaler s locaton. After satsfyng demand wth hs own nventory, each retaler reports to the manufacturer how much surplus/shortage he has. 5. If there s surplus at one retaler and shortage at the other, nventory s shared and the 9

11 payments are arranged as follows 4 : Wthout loss of generalty, assumng the manufacturer pays the transportaton cost, τ j, the manufacturer wll charge the buyer s + τ j and pay the seller r j τ j + p j for each unt shpped from to j. There are a few ponts that wll help us understand the ratonale behnd CIS. Frst, CIS descrbed above s smlar to a two-part tarff (a fxed premum and a per-unt wholesale prce pad to the manufacturer from each retaler) plus a set of transshpment payments handled through the manufacturer, whch s key to the coordnaton. Second, dependng on who pays for the transportaton cost, we may have dfferent sets of payments for the transshpped unts. The bottom lne s to ensure each retaler obtan u j as the net proft for each unt shared from to j, as accomplshed by the set of payments descrbed n step 5. Thrd, recall that the system gans a net of u j for each unt shared. By allowng each retaler to have a net proft of u j, the manufacturer needs to pay a net u j for each unt shared from to j. Ths amount s provded from the premum charged at the begnnng of the game. Fnally, under CIS, snce each retaler obtans a constant net proft of u j for each unt shared from to j, we can be sure that each retaler wll truthfully report hs surplus or shortage to the manufacturer. 3.1 Analytcal Investgaton of CIS In ths secton, we analyze CIS under three scenaros, complete nformaton (CFIS), prvate nformaton wth no nformaton sharng (CNIS), and prvate nformaton wth nformaton revealng/sharng (CIRS), correspondng to FIS, NIS, and IRS n the decentralzed system, respectvely. Theorem 4 Under complete nformaton, gven any w, CIS coordnates the two retalers. Further, when settng ŵ = c, CIS coordnates the two-level manufacturer-retalers system. In ether case, (1) there exsts a unque equlbrum of the retalers order quanttes that equal to the respectve centralzed order quanttes, and (2) there exsts at least one par of {δ, δ j } 4 Note that the physcal nventory does not have to be routed through the manufacturer, but the money needs to n order to ensure that each retaler obtans u j for each unt shared from to j. 10

12 such that each player can gan hgher expected proft compared to a decentralzed system (e.g., FIS, NIS). Theorem 4 shows that under complete nformaton, CIS maxmzes the two retalers total proft for any gven w and further, t maxmzes the total proft of the manufacturer-retalers system f the manufacturer sets w = c. Wth proper allocaton of the maxmzed total proft, CIS can lead to an all-wn stuaton for all the players. Under asymmetrc demand nformaton, CIS can stll be mplemented followng the steps lsted earler n the secton. Notce that the manufacturer does not need the retalers nformaton under CIS, leavng only the demand SD the useful nformaton for each other. Under CNIS, n step 2 of CIS, each retaler determnes hs order quantty, denoted as ˆQ B ( ), based on hs pror nformaton of the other retaler s demand SD, σ j, just as n the Bayesan game. Specfcally, ˆQ B ( ) = µ + ẑ B ( ), where ẑ B (, w) s solved from the followng equatons: u j F Z (z ( )) = r + p w + r + p s r + p s u j + r + p s σ j S j P j,σ (σ j )P {z ( ) σ j (Z j z j (σ j )) + < Z < z ( )} σ j S j P j,σ (σ j )P {z ( ) < Z < z ( ) + σ j (z j (σ j ) Z j ) + }, j = 1, 2, S. (4) Under CIRS, each retaler shares/reveals hs SD to the other retaler. We conduct an IC analyss agan to see whether retalers wll share true nformaton. Snce nformaton s only shared between the retalers and one retaler s order quantty s not observable to the other, the IC analyss (a double moral hazard verson wth detals n Appendx??) s very dfferent from the one n Secton 2. The followng theorem presents the results. Theorem 5 Under CIRS, truth-tellng s not retalers equlbrum strategy,.e., ˆσ r σ t. Theorem 5 demonstrates that under CIS, truth-tellng s stll not retalers equlbrum reportng strategy. However, n the next secton, we wll see that CIS mnmzes the mpact 11

13 of nformaton, hence, the system can enjoy the coordnaton benefts wthout havng to worry about nformaton sharng or truth-nducng. 3.2 Numercal Investgaton of CIS In ths secton, we conduct an extensve numercal study to nvestgate the value of coordnaton and the mpact of nformaton asymmetry. Specfcally, by comparng NIS and CNIS, we see the value of coordnaton (CIS) n the decentralzed nventory sharng system under asymmetrc demand nformaton. Further, by comparng NIS wth FIS and CNIS wth CFIS, we see the value of nformaton n the decentralzed system and under CIS, respectvely. As shown, retalers have ncentves to untruthfully share ther demand parameter nformaton. Hence, the value of nformaton n a decentralzed system and under CIS also provdes an ndcator of the benefts of untruthful sharng n the decentralzed system and under CIS, respectvely. To focus on the mpact of demand nformaton, we assume dentcal cost parameters at the two retalers. We use values n Rud, et al. (2001) for these parameters,.e., r = 40, τ = 2, s = 10, p = 0 and set c = 15. We assume retalers demand follow normal dstrbuton wth mean and SD each takng two levels of values,.e., {µ H, µ L } and {σ H, σ L }, wth a chance of beng hgh or low. To see the mpact of the mean demand, we choose three levels of the expected mean demand, µ = 80, 100, 120, and for each level, we test three pars of {µ H, µ L } wth the rato of µ H µl = 1.2, 1.5, 1.8. For example, n the case of µ = 100, the three pars of {µ H, µ L } are {109, 91}, {120, 80}, and {129, 71}. Smlarly, we choose two levels for the expected SD, σ = 30, 18, wth σ H σl = 1.0, 2.0, 3.0 for each level of σ. For NIS whose results are affected by α, we test α = 1.0, 0.5, 0.0. For the total of = 162 cases, we calculate the manufacturer s optmal wholesale prce, the retalers expected order quanttes, system profts and ndvdual profts when approprate. We frst nvestgate the mpact of nformaton asymmetry n the decentralzed system. Ths wll also serve as a benchmark to compare wth the mpact of nformaton asymmetry under CIS. Table 1 shows the manufacturer s and retalers expected profts under FIS 12

14 and NIS, as well as the value of nformaton measured by the percentage dfference between them. Snce the change of µ H µl does not brng addtonal nsghts, we only show the results wth µ H µl = 1.8. Several mportant observatons stand out: (1) Whle knowng retalers demand nformaton always benefts the manufacturer, t does not always beneft the retalers. (2) Informaton asymmetry has a sgnfcant mpact n the decentralzed system and t has a bgger mpact on the retalers than on the manufacturer (the percentage dfference n retalers proft n the absolute value s hgher (up to 17.59%) than that of the manufacturer (up to 4.83%)). Ths s because the manufacturer can adjust her wholesale prce whch offsets her dsadvantage of nformaton. (3) Whle α (how the transshpment beneft s dvded between the retalers) has a sgnfcant mpact on the mpact of nformaton asymmetry, ts mpact s qute uncertan. Therefore, usng transfer prces between retalers (equvalent to α) to coordnate the nventory sharng system wll lead to consderable ncentve compatblty ssues under asymmetrc nformaton and there do not seem to exst truth-nducng mechansms for ths mechansm (Yan and Zhao (2008)). Next, we explore the value of the coordnaton mechansm proposed n ths work under asymmetrc nformaton. Table 2 compares CNIS and NIS to obtan the value of coordnaton measured by the percentage ncrease n the system proft under CNIS over NIS. We make a few nterestng observatons. Frst, the value of coordnaton s sgnfcant. For the cases we tested, CIS ncreases the system proft by 13.61% to 42.07%, dependng on the values of dfferent parameters. Second, whle the value of coordnaton ncreases as µ ncreases (because w under NIS ncreases), t does not depend on the dfference between µ H and µ L,.e., µ H µl (hence results are only lsted for dfferent values of µ). Ths s because retalers expected order quanttes (hence expected profts) and correspondngly w reman the same for fxed µ (recall a retaler s equlbrum order quantty depends only on hs own mean demand and both retalers demand SD). On the other hand, the value of coordnaton depends on both σ and σ H σl. As ncreases, the value of coordnaton ncreases because the dfference between the retalers expected order quanttes under NIS and CNIS ncreases. However, as σ H σl ncreases, t may ncrease or decrease the value of coordnaton due to ts 13

15 uncertan mpact on the retalers expected order quanttes and the manufacturer s w. Fnally, the value of coordnaton decreases as α ncreases. Ths s because as α ncreases, more transshpment profts wll be allocated to the retaler wth extra unts (the seller). Hence, retalers have ncentves to order more. In other words, n a vertcal context, the ncrease of α counter-balances double margnalzaton,.e., the value of coordnaton goes down. Ths observaton s also artculated n Jang and Anupnd (2010). After we see the sgnfcant value of the coordnaton mechansm CIS, we nvestgate the mpact of nformaton asymmetry under CIS. Knowng that CIS coordnates the manufacturerretalers system when retalers demand nformaton s common knowledge, we would lke to see how CIS behaves under asymmetrc nformaton? But before we see ths, we frst want to take a look at the mpact of nformaton asymmetry n the decentralzed system for the completeness of our analyss of the decentralzed system and as a bench mark to compare wth the mpact of nformaton asymmetry under the coordnaton mechansm CIS. Gven the sgnfcant value of the coordnaton mechansm CIS, we next nvestgate the mpact of nformaton asymmetry under CIS. We would lke to see how CIS behaves under asymmetrc nformaton. Recall from Table 1 that the mpact of nformaton may be sgnfcant but uncertan, the followng results shows the advantage of CIS n dealng wth asymmetrc nformaton. Table 3 shows the system profts under CFIS and CNIS, as well as the value of nformaton under CIS measured by the percentage dfference between them. Snce under both CFIS and CNIS, each player s net proft can be determned as a proporton of the system proft through adjustng the premum, a comparson of the system profts s suffcent. The results agan do not depend on µ H µl, but only depends on µ. Table 3 demonstrates that n all cases we test, the dfferences between the system profts (and hence each player s proft) under CNIS and CFIS are very small (close to 0.01%). Therefore, even f retalers do not share ther prvate nformaton, under CIS, each party and the system can obtan profts very close to what they can obtan under complete nformaton (CFIS),.e., frst-best profts. Ths also ndcates that the beneft of untruthful reportng s nsgnfcant under CIS. Therefore, 14

16 under CIS, all players can enjoy the frst-best profts wthout requestng the retalers to share ther nformaton or truth-nducng! Ths advantage of CIS may not be expected from other coordnaton mechansms. Therefore, t s mportant to consder nformaton asymmetry when developng coordnaton mechansms n a decentralzed system. 4 Concluson and Dscusson of Future Work We consder a supply chan wth a manufacturer supplyng two ndependent retalers who may share nventory wth each other at the end of a sngle perod. The man contrbutons of our work nclude: (1) To the best of our knowledge, t s the frst to study nventory sharng systems wth asymmetrc nformaton. (2) We not only consder the ssues of asymmetrc nformaton and nformaton sharng, but also nformaton credblty and truth-nducng n the nformaton sharng case. (3) It s also the frst to provde a coordnaton mechansm the decentralzed nventory sharng system wth the consderaton of asymmetrc nformaton. It s worth notng that although the analyss was presented for the standardzable demand dstrbuton, results n all theorems except Theorem 2 apply to any general dstrbuton. Below we summarze the manageral nsghts obtaned from our work: 1. In an nventory sharng system, dfferent partes need dfferent peces of demand nformaton for ther decsons. Specfcally, the manufacturer needs both the retalers mean demand and standard devaton to determne the wholesale prce, whle retalers only need the other retaler s demand standard devaton for hs order quantty decson. 2. Truth-tellng cannot be expected voluntarly from ndependent retalers n an nventory sharng system when they share ther prvate demand nformaton. A truth-nducng scheme can be developed for symmetrc retalers (wth same cost parameters) who share nformaton wth each other. However, no such or smlar schemes can be developed to ensure truth-tellng to the manufacturer. 3. Informaton asymmetry has sgnfcant mpact n the decentralzed system. Whle knowng retalers demand nformaton always benefts the manufacturer, t may not always 15

17 beneft the retalers. Further, nformaton asymmetry has a bgger mpact on the retalers than on the manufacturer. 4. A coordnated nventory sharng mechansm (CIS) can be developed for a decentralzed system and acheve sgnfcant proft ncrease for each player and the whole supply chan. One dfferental advantage of CIS s that t also mnmzes the mpact of asymmetrc nformaton,.e., each party can obtan an expected proft very close to what they would obtan under complete nformaton scenaro (.e., ther frst-best proft), even f the retalers do not share nformaton wth each other. 5. It s very mportant to take nto consderaton the nformaton ssues n desgnng coordnaton mechansms. Otherwse, a system may run nto sgnfcant ncentve compatblty ssues that are very complcated to solve when the coordnaton mechansm s mplemented under nformaton asymmetry. Fnally, the current research has opened up nterestng future research opportuntes. A natural extenson to ths work s to analyze an n-retaler nventory sharng system (n > 2) wth asymmetrc nformaton (Zhao and Yan 2010). Many complcated ssues arse when there are n decentralzed retalers n the nventory sharng game (n > 2), e.g., coalton and coordnaton. Further, ths work, as the frst mover to study nformaton asymmetry n the nventory sharng area, studes a sngle-perod model. An nterestng future work s to study the demand nformaton asymmetry n a multple-perod settng n whch retalers may choose to or not to share ther nventory, based on ther expectaton of future demand. As the retalers have more optons n sharng ther nventory, t would be nterestng to see what mpact nformaton asymmetry wll brng n ths even more complcated settng. Acknowledgement The authors thank the Department Edtor, Assocate Edtor and the anonymous referees for ther suggestons and comments to mprove the paper. REFERENCES 16

18 Anupnd, R., Bassok, Y., and Zemel, Etan A General Framework for the Study of Decentralzed Dstrbuton Systems. M&SOM, 3, No. 4, Fall Anupnd, R., Bassok, Y., and Zemel, Etan Study of Decentralzed Dstrbuton Systems: Part II - Applcatons, Workng paper, Ross school of busness, Unversty of Mchgan, Ann Arbor, MI. Cachon, G. P. and M. Larvere Contractng to assure supply: how to share demand forecasts n a supply chan. Management Scence 47 (5), Dong, L. and Rud, N Who benefts from transshpment? Exogenous vs. endogenous wholesale prces. Management Sc., 50(5), Gallagher, E. (2002), One Stop Co-op, dscover how Johnstone Supply, a lead wholesale dstrbuton company, uses a cooperatve busness model and succeeds. US Bus. Rev., December, 1-6. Granot D. and G. Sosc A Three-Stage Model For a Decentralzed Dstrbuton System of Retalers. Operatons Research, 51(5), Ha, A Suppler-Buyer Contractng: Asymmetrc Cost Informaton and the Cut-off Level Polcy for Buyer Partcpaton. Naval Research Logstcs, 48(1), Hammond, P. J Straghtforward Indvdual Incentve Compatblty n Large Economcs. Revew of Economc Studes. 46(2), Hu, X., I. Duenyas, and R. Kapuscnsk Exstence of Coordnatng Transshpment Prces n a Two-Locaton Inventory Model. Management Scence. 53(8), Jang, L. and R. Anupnd Customer-Drven vs. Retaler-Drven Search: Channel Performance and Implcatons, Manufacturng & Servce Operatons Management. 12(1), Wnter. L, L Cournot olgopoly wth nformaton sharng. Rand J. Econom. 16(4), L, L Informaton sharng n a supply chan wth horzontal competton. Management Scence, 48(9), Mas-Colell, A. and X. Vves Implementaton n Economcs wth a Contnuum of Agents, Revew of Economc Studes, July, 60(3),

19 Narus, J. A., and Anderson, J. C. (1996), Rethnkng Dstrbuton: Adaptve Channels. Harvard Busness Revew, July-August, Özer, O. and We, W. Strategc Commtments for an Optmal Capacty Decson Under Asymmetrc Forecast Informaton. Management Scence. 52(8) Rud, N. Kapur, S. and Pyke, D. F A two-locaton nventory model wth transshpment and local decson makng. Management Sc. 47 (12) Shao, J., Krshnan, H., McCormck, T.S Incentves for transshpment n a supply chan wth decentralzed retalers. Workng paper, Unversty of Brtsh Columba, Vancouver BC. Shapro, C Exchange of cost nformaton n olgopoly. Rev. Econ. Stud. 53(3), Tagaras, G. (1989), Effects of Poolng on the Optmzaton and Servce Levels of Two-locaton Inventory Systems. IIE Transactons, 21(3), Zhang, J Transshpment and Its Impact on Supply Chan Members Performance. Management Scence, 51(10) Zhao, H., Deshpande, V., and Ryan, J.K Inventory Sharng and Ratonng n Decentralzed Dealer Networks. Management Scence, 51(4), Zhao, H., Deshpande, V., and Ryan, J.K An Analyss of Emergency Transshpments n Decentralzed Dealer Networks. Naval Research Logstcs, 53(6), pp H. Zhao, J.K. Ryan, and V. Deshpande, Optmal Dynamc Producton and Inventory Transshpment Polces for Mult-Locaton Make-to-Stock Systems, Operatons Research, 56 (2), Yan, X. and Zhao, H Informaton Issues n Decentralzed Inventory Sharng Systems. Workng paper. Krannert School of Management, Purdue Unversty, West Lafayette, IN Zhao, H. and Yan, X Inventory Collaboraton and Coordnaton Among n Independent Retalers wth Asymmetry Demand Informaton. Workng paper. Krannert School of Management, Purdue Unversty, West Lafayette, IN

20 5 Proofs Proofs of Theorems 4 and 5 are shown n Onlne Appendx. 5.1 Proof of Theorem 1 and Theorem 2 If we use the transformaton Z = D µ, we can see that z F (w) are the solutons to equatons (2) f and only f µ + z F (w) are the solutons to equatons (1), = 1, 2. From equatons (2), we also see that, for gven cost parameters and (α, α j ), z F (w), = 1, 2, are functons of only σ j. Therefore, we have the best responses Q F (w) = µ + z F (w) and z F (w) = z F (w, σ j ). Ths proves the results n Theorem 2. Now we prove the two bullets n theorem 1. We can rewrte (??) as follows, dπ F M(w) dw = µ + z F (w) + µ j + σ j z F j (w) (w c) A j r +p s + A r j +p j s j (A A j + A B j + A j B ). (5) Suppose we have two dfferent µ : µ,h and µ,l wth µ,h > µ,l. We denote the manufacturer s correspondng optmal wholesale prces are w F (µ,h ) and w F (µ,l ), respectvely. Therefore, when µ = µ,l, we must have dπ F M(w) dw w=w F (µ,l ) = µ,l + z F (w F (µ,l )) + µ j + σ j z F j (w F (µ,l )) (w F (µ,l ) c) A j r +p s + A r j +p j s j (A A j + A B j + A j B ) = 0. (6) Now we consder the case when µ = µ,h. If we stll set w = w F (µ,l ), then dπ F M(w) dw w=w F (µ,l ) = µ,h + z F (w F (µ,l )) + µ j + σ j z F j (w F (µ,l )) (w F (µ,l ) c) A j r +p s + A r j +p j s j (A A j + A B j + A j B ). (7) We prove that z F (w F (µ,l )), z F j (w F (µ,l )), and other terms A, B, = 1, 2 are the same as the ones n the case when µ = µ,l. 19

21 Frst, from the prevous proof, we know that under the same wholesale prce w F (µ,l ), Z F (w) only depends on σ j. Thus, we know that z F (w F (µ,l )) does not depend on µ, = 1, 2 and hence z F (w F (µ,l )) under µ = µ,h are the same as the one under the case when µ = µ,l. Second, we prove that A and B under µ = µ,h are the same as the one under the case when µ = µ,l. We can rewrte A and B as follows (here we omt the arguments of z F for brevty), A = 1 exp zf 2πσ 2 [1 α u j r + p s 2 F (z F j ) (1 α j) u j r + p s F (z F j )], B = α u j r + p s + (1 α j) u j r + p s z F j z F j f(z j ) f(z j ) 1 2πσ exp (zf 1 2πσ exp (zf + σ j (z F j Z j )) σ j (z F j Z j )) 2 2 where f( ) and F ( ) are pdf and cdf functons of standard normal dstrbuton. We can see that A and B do not depend on µ and µ j and hence are the same under µ = µ,h as the ones under the case when µ = µ,l. Therefore, when µ = µ,h, dπf M (w)) dw w=w F (µ,l ) n equaton (7) s dπ F M(w) dw w=w F (µ,l ) = µ,h µ,l + [µ,l + z F (w F (µ,l )) + µ j + σ j z F j (w F (µ,l )) (w F (µ,l ) c) A j r +p s + A r j +p j s j (A A j + A B j + A j B ) ] dz j dz j, = µ,h µ,l > 0, (8) where the second equalty s true because of equaton (6). Thus, dπ F M (w) dw w=w F (µ,l ) > 0 when µ = µ,h, whch means w F (µ,h ) > w F (µ,l ),.e., when µ ncreases, w F ncreases. However, as ncreases, t s not clear how dπf M (w) dw wll be changed snce z F, A, B, = 1, 2, all have complcated relatonshps wth and may ncrease or decrease. Therefore, the manufacturer s optmal wholesale prce may also ether ncrease or decrease. 20

22 5.2 Proof of Theorem 3 To show truth-tellng s not retaler s equlbrum reportng strategy, we only need to show that gven retaler j reports hs demand nformaton truthfully,.e., {µ r j = µ t j, σ r j = σ t j}, revealng true nformaton of hs demand parameters, {µ, }, to the other partes, s not s best strategy. To smplfy the analyss, we frst focus on the mean demand nformaton revealng,.e., assumng σ r = σ t. We wll show that retaler has ncentves to report hs mean demand nformaton untruthfully. Smlarly, we can show that retaler also has ncentves to report hs demand standard devaton untruthfully. Ths proves that retalers have ncentves to report ther demand parameters untruthfully. Gven σ r = σ t, denote retaler s true proft as Π t (Q (µ r, w (µ r )), Q j(µ r, w (µ r ))). We now check dπt (Q (µr,w (µ r )),Q j (µr,w (µ r ))) µ r µ r =µ t. If µr = µ t maxmzes Π t (, ), then we must have dπ t (, ) dµ r µ r =µ t = 0 snce t must be a crtcal pont, otherwse µr = µ t cannot be a maxmzer. For brevty, we omt the arguments of Q, Q j, and w when obvous. dπ t (Q, Q j) dµ r µ r =µ t = Πt dw w dµ r µ r =µ t + Πt dq Q dµ r µ r =µ t + Πt dq j Q j dµ r µ r =µ t = Q dw dµ r µ r =µ t + [(1 α j)u j βj(q t, Q j) α u j γj(q t, Q j)]σ t dzj dµ r µ r =µ t, where Q = µ r + σ t z, Q j = µ t j + σ t jz j, and β j (Q, Q j ) and γ j (Q, Q j ) are defned as, β j (Q, Q j ) = P (Q j (D Q ) + < D j < Q j ), γ j (Q, Q j ) = P (Q j < D j < Q j + (Q D ) + ), (9) and {β t j(q, Q j), γ t j(q, Q j)} are defned accordngly wth D = D t, D j = D t j, Q = Q, and Q j = Q j. Now we solve dw dµ r and dz j. Defne O = dµ r A j A + r +p s r j +p j s j (A A j +A B j +A j B ) dervatve of the rght hand sde of equaton (??) w.r.t. µ r, we have n equaton (??). Takng the σ[1 t + (w c) O ] dz Q dµ r + σj[1 t + (w c) O ] dz j Q j dµ r + O dw dµ r = [1 + (w c) O ]. Q 21

23 Takng the dervatve of both sdes of equatons (2) w.r.t. µ r, we have (A + B ) dz dµ r + σ j B dz j dµ r 1 dw =, = 1, 2. r + p s dµ r From the above equatons, we can solve dz dµ r A j + B j σt j B σ = t (A A j + A B j + A j B )(r + p s ) dw, = 1, 2, dµ r and dw dµ r = σ t (1 + (w c) O Q ) A j +B j σt j σ t B 1 + (w c) O Q (A A j +A B j +A j B )(r +p s ) + σt j(1 + (w c) O Q j A +B σt σ t j B j ) + O (A A j +A B j +A j B )(r +p s ) Therefore, dπ t (Q, Q j) dµ r µ r =µ t = [Q + (α u j γj(q t, Q j) (1 α j )u j βj(q t, Q σj(a t + B ) σb t j j)) (A A j + A B j + A j B )(r j + p j s j ) ]dw dµ r µ r =µ t. σj t(a +B ) σ tb j (A A j +A B j +A j B )(r j +p j s j ) Snce n general, dw 0 and Q dµ r +(α u j γj(q t, Q j) (1 α j )u j βj(q t, Q j)) 0 at µ r = µ t, we have dπt (Q,Q j ) dµ r µ r =µ t 0,.e., truth tellng s not retaler s equlbrum reportng strategy. 22