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Strategic information sharing in a supply chain
European Journal of Operational Research 174 (2006) 1567–1579 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Strategic information sharing in a supply chain Wai Hung Julius Chu a
a,*
, Ching Chyi Lee
b
Emerson Electric Asia-Pacific, Suite 6701, Central Plaza, 18 Harbour Road, Wanchai, Hong Kong Faculty of Business Administration, Department of Decision Sciences and Managerial Economics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
b
Received 29 April 2003; accepted 16 February 2005 Available online 17 May 2005
Abstract We consider a two-member supply chain that manufactures and sells newsboy-type products and comprises a downstream retailer and an upstream vendor. In this supply chain, the vendor is responsible for making stock-level decisions and holding the inventory, and the retailer is better informed about market demand. In each period, the retailer receives a signal about market demand before the actual demand is realized, and must decide whether to reveal the information to the vendor, at a cost, before the vendor starts production. We assume that any information that the retailer reveals is truthful. We model the situation as a Bayesian game, and find that, in equilibrium, whether the retailer reveals or withholds the information depends on two things—the cost of revealing the information and the nature of market demand signal that the retailer receives. If the cost of sharing the information is sufficiently large, then the retailer will withhold the information from the vendor regardless of the type of signal that is received. If the cost of sharing the information is small, then the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it from the vendor if a low demand is signaled. In general, reducing the cost of sharing information and increasing the profit margin of either the retailer or the vendor (or reducing the cost of the vendor or retailer) will facilitate information sharing. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Information sharing; Supply chain; Asymmetric information; Game theory
1. Introduction Information sharing is an important issue in supply chain management, particularly in some of the new supply chain practices that have recently become popular, such as vendor managed inventories (VMI), click and mortar, drop shipping, and vendor hubs. These new supply chain practices have one thing in *
Corresponding author. Fax: +852 28272168. E-mail address: [email protected] (W.H.J. Chu).
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.053
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common—the stock level of the supply chain is determined by the upstream members of the supply chain, who are usually less informed than the downstream members about market demand. Thus, to guarantee the success of these new supply chain management practices, it is essential that the better-informed downstream members of the chain share their demand information effectively and efficiently with the lessinformed upstream members. Advocates of these new practices often emphasize the importance of information sharing in the supply chain. There is no doubt that effective information sharing allows a supply chain to operate more efficiently, and hence generate a higher overall supply chain profit. However, information sharing often involves cost. If the cost of information sharing is solely born by the informed party, and if, in addition, there is no pre-defined mechanism to distribute some of the additional profit that is generated through the information sharing to the informed party, then it is debatable whether the informed party has any incentive to share information with the uninformed party. In fact, researchers have noticed that many members of supply chains are doubtful about information sharing. Clark and Hammond (1997) report that: Retailers generally acknowledged that providing additional information to manufacturers would offer some savings to the manufacturers, but many retailers were skeptical about the benefits for their firms in sharing information with manufacturers. Cost is one of the key factors that may hinder information sharing. The main purpose of this paper is therefore to study how the incentive to share information is related to the cost of that sharing in the specific context of a supply chain that manufactures and sells newsboy-type products. The supply chain has two members— the upstream vendor (V) and the downstream retailer (R). In this supply chain, there is no cost for the retailer for overstocking because the vendor makes the stock level decisions and keeps the inventory. In each period, the retailer receives a signal about market demand before the actual demand is realized, and must then decide whether to reveal it voluntarily to the vendor at a cost before production begins, or whether to withhold the information. We assume any information that the retailer decides to reveal is truthful. We model the situation as a Bayesian game and find that, in equilibrium, whether or not the retailer reveals the information to the vendor depends on two things—the cost of revealing the information, and the nature of the market demand signal that is received. If the cost of sharing the information is sufficiently large, then the retailer will withhold the demand information from the vendor regardless of the type of signal that is received. When the cost of sharing information is smaller, the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it if a low demand is indicated. The latter result is driven by the fact that the retailer bears no overstock cost, and hence would like the vendor to stock as much inventory as possible. If the retailer knows that the demand is likely to be high, then he would want to ensure that the vendor receives the signal correctly and consequently stocks a high level of inventory. If the retailer knows that the demand is likely to be low, then there will be no incentive to reveal the information to the vendor, even though the vendor may be able to partially infer from the retailerÕs action that the demand is low. Basically, there is a cutoff value of market demand that separates these two cases. Further study on this cutoff yields some interesting managerial implications. In general, reducing information sharing cost and increasing the profit margin of either the vendor or the retailer (or reducing their costs) will facilitate information sharing. The rest of this paper is organized as follows. The next section provides a brief survey of the related literature. The model is described in Section 3, and the analysis of the model is given in Section 4. Section 5 offers some concluding remarks.
2. Related literature There is no lack of literature about information sharing in supply chains. Lee and Whang (2000) provide some real life examples of information sharing in a supply chain, and Bourland et al. (1996) and Chen
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(1998) develop theoretical models of sharing stationary stochastic downstream demand and inventory data to improve stock level decisions of the upstream. Lee et al. (1997a,b) address the bullwhip effect and Lee et al. (2000) consider the value of information sharing under a known autoregressive demand process. Gavirneni et al. (1999) and Cachon and Fisher (2000) study and quantify the value of sharing sales information to overcome demand distortion due to order batching, and Chen et al. (2000) analyze the effects of the forecasting process on information sharing. All of these papers report that there are some benefits to sharing demand information, although these benefits vary substantially. Gavirneni et al. (1999) conclude that information sharing is more beneficial when the capacity of the upstream members of a supply chain is unlimited. Moreover, Lee et al. (2000) suggest that the value of sharing demand information can be quite high, especially when the underlying demand is significantly correlated over time, when demand is highly variable, or when there is a long replenishment lead-time. However, Raghunathan (2001) revisits their model and shows that the value of information sharing is insignificant when the upstream members reduce the variance of the demand forecast by analyzing the entire order history. Hence, information sharing is valuable only if the information cannot be inferred by the receiving party. In another group of studies, the sharing of accurate demand information through contracts is examined, and readers are referred to Tsay (2000) for a recent review of these studies. Cachon and Lariviere (2001) study contracts that induce credible information sharing when there is asymmetric information. Even when the downstream members of a supply chain have an incentive to provide an overly optimistic demand forecast, they identify contracts that ensure credible information sharing. Our model differs from those that have been developed in previous research in one major aspect. We do not take information sharing in a supply chain for granted. We recognize the fact that even though information sharing may be beneficial to the supply chain as a whole, the better-informed party will have no incentive to share the information with the uninformed party if there is no benefit for them in doing so. In our model, the informed party (the retailer) bears the cost of the information sharing alone, and the passing of information from the informed party to the uninformed party (the vendor) is undertaken voluntarily and truthfully. The informed party does not receive any side payment from the uninformed party for the information transfer. The only benefit that the informed party may obtain from sharing the information is a possible increase in payoff due to the better stock level decisions that would be made by the uninformed party as a result of the information. However, such a benefit must be at least greater in value than the cost of sharing the information to justify the action. Hence, the decision of whether or not to reveal the information to the uninformed party is made strategically by the informed party.
3. The model We consider a supply chain that consists of two members—the downstream retailer R and the upstream vendor V. V supplies R with newsboy-type products to be sold to the end market. This means that the following assumptions hold. First, the market demand in all periods is independent and identically distributed. Second, all of the cost and price parameters remain fixed from period to period. Third, backordering is not allowed, and finally, no unsold item in a given period has any value in future periods. These assumptions effectively reduce a multi-period problem to a single period problem. We assume that all members are risk neutral, and have a zero-order setup cost. The market demand is denoted as Y = X + H, where X and H are independent variables. X is a random variable having support on [0, 1) and is governed by a continuously differentiable probability density function f(x), with the corresponding cumulative probability function being F(x). H, which is a random variable that represents the minimum market demand, is governed by a continuously differentiable probability
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density function g(h) with support on [a, b], where a P 0. The corresponding cumulative function of g(h) is G(h). In each period, R receives a signal that indicates the actual value of H before the realization of Y, and must decide whether to reveal this information to V before V makes a stock decision. Assuming that the signal received by R is h, then the conditional probability of Y given h is denoted as Fh(y), with the corresponding density function being fh(y). Note that Fh(y) = Prob (yjh) = Prob (x + hjh) = F(x) = F(y h), and similarly that fh(y) = f(y h). Also note that once it is known that H = h, then E(Y) is known to be equal to E(X) + h. Thus, although our model can be interpreted as R receiving a signal about minimum demand, it can also be interpreted as R receiving a signal about mean demand. Except for the actual value of H which is only known by R before the realization of the actual demand, all of the parameters and probability distribution functions are assumed to be common knowledge.
3.1. Chronology of events For any given period, the chronology of events is as follows: 1. R receives a signal h and must then decide whether to reveal it to V. We assume that any information that R reveals is truthful. Moreover, we assume that there is a non-negative operating cost for R in revealing the information. The cost of revealing the information is denoted as k. 2. If R chooses to reveal the actual value of h to V in the previous step, then V will become fully informed of the distribution of Y, and will then choose a stock level qh. Note here that the subscript h of qh indicates that VÕs stock level decision in this case is dependent on h. If, on the other hand, R chooses to withhold the information, then V will make the stock level decision based on incomplete information about Y. We denote the stock level that is chosen by V in this case as ~q. Note that as V does not know the actual value of h when ~ q is chosen, ~ q does not depend on h. Instead, ~q depends on VÕs posterior belief about h. The unit production cost of V is cV. 3. Demand y is realized. If VÕs stock level is higher than or equal to the demand, then V supplies R with y units of goods at a unit price of pV, and the excess inventory has a residual value of rV per unit. If VÕs stock level is lower than the demand, then V supplies R with all of the units of goods that are produced at a unit price pV. 4. R sells the goods to the market at a unit price pR. The unit cost of R is cR = pV + wR, where wR can be interpreted generally as the unit transaction cost that is necessary to bring the goods from the upstream to the downstream.
3.2. Assumptions To make the analysis non-trivial, we assume that pi > ci > ri P 0 (i = V, R) and that wR P 0. The former is simply a standard assumption of the newsboy problem and the latter ensures that RÕs transaction cost is non-negative. For technical reason and to simplify the analysis, we further assume that the probability function for H satisfies the following: A1 g(h)/G(h) is strictly decreasing in h. This assumption, which is quite similar to the standard monotone hazard rate assumption, is often referred to as the monotone reversed hazard rate assumption (Shaked and Shanthikumar, 1994). Among the commonly used distribution functions, the exponential function and the family of power functions, xa m F ðxÞ ¼ ðba Þ , where m P 1 and x 2 [a, b], both satisfy this assumption. Note that uniform distribution is a member of the power function family (m = 1).
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Finally, as H is assumed to be a continuous variable, we impose the following tiebreak rule. A2 If R is indifferent between revealing and withholding the information, then he will withhold the demand signal from V. This sequence of events is very typical in environments such as the supply chains of fast moving consumer goods (FMCG), consumer packaged goods (Kumar, 2004), and passive electronic components (Kopczak, 1997, 1998). In these kinds of supply chains, the products that are involved are typically short-lived (newsboy-type products) and the upstream members of the chain are responsible for the stock level decisions. 4. The analysis As the model that has been described is an incomplete information game, in this section, we will characterize the perfect Bayesian equilibrium of the game. For a detailed definition of perfect Bayesian equilibrium, we refer the reader to Fudenberg and Tirole (1991). We focus here only on pure strategy equilibrium. We first consider the case in which R reveals information to V. If R chooses to reveal h to V, then V will become fully informed of the distribution of Y, and will then choose the stock level qh to maximize his expected profit PV(qh), where PV ðqh Þ ¼ pV Ey minðqh ; yÞ cV qh þ rV Ey ðqh yÞþ þ
¼ pV fh þ Ex min ½ðqh hÞ; xg cV qh þ rV Ex ½qh ðx þ hÞ .
ð1Þ
Throughout the paper, the operator (a)+ is used to indicate the maximum of a and zero, and hence [qh (x + h)]+ represents the excess inventory. This is basically a newsboy problem, and VÕs optimal choice of qh in this case is K VU 1 p V cV 1 ð2Þ þh¼F þ h ¼ F 1 ðnV Þ þ h. qh ¼ F pV rV K VU þ K VO Here, KVO = (cV rV) is the cost of overstocking and KVU = (pV cV) the cost of understocking for V, and nV = (KVU/KVU + KVO) is the critical fractile ratio. Substituting qh into (1), and after simplification, we obtain VÕs maximum expected profit when R reveals h Z ðq hÞ h F ðxÞ dx. ð3Þ PV ðqh Þ ¼ K VU qh ðK VO þ K VU Þ 0
In addition, RÕs expected profit in this case is PR ðqh ; hÞ
¼ ðpR
cR ÞEy minðqh ; yÞ
Z k ¼ K RU qh
ðqh hÞ
F ðxÞ dx k
ð4Þ
0
where KRU = (pR cR) is RÕs cost of understocking. Before we proceed to the analysis of the case in which R withholds the information, we introduce the following lemma. Lemma 1. In any pure strategy equilibrium, if R is supposed to reveal the information when he receives a demand signal h 0 , then he will also reveal it when he receives any other demand signal h00 > h 0 . On the other hand, if R is supposed to withhold the information when he receives a demand signal h 0 , then he will also withhold the information when he receives any other demand signal h00 < h 0 . Proof. Please refer to Appendix 1 for the proof.
h
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Lemma 1 has an important implication that greatly simplifies our later analysis. If, in any equilibrium, RÕs revelation of the signal to V depends on the signal h that is received, then it must be the case that R will reveal the information only when h has a relatively large value. Let ^h be the specific value of the demand signal that leads R to be indifferent between revealing and withholding the information. Then, by Lemma 1, R will reveal h to V if and only if h > ^ h. Note that there may not exist any ^h in the interval [a, b] when k is sufficiently large, because ^ h would have to be greater than b, the upper bound of its support, for the indifference to hold. This means that, in equilibrium, R will always withhold the information, regardless of the signal h that is received. Let us assume that an ^ h exists in the interval [a, b]. When R withholds the information, V can then infer that the actual value of h must be between a and ^h, and VÕs posterior belief about h is therefore updated to g^h ðhÞ ¼ gðhÞ=Gð^ hÞ 8^ h 2 ½a; b.
ð5Þ
The corresponding cumulative function is hÞ 8^ h 2 ½a; b. G^h ðhÞ ¼ GðhÞ=Gð^
ð6Þ
Thus, if R withholds the information, then given ^h 2 ½a; b and the posterior belief g^h ðhÞ, V will choose the e V ð~qÞ, where stock level ~ q to maximize his expected profit P e V ð~ P qÞ ¼ Eh pV Ey minð~ q; yÞ cV ~ q yÞþ q þ rV Ey ð~ Z ^h Z ~q Z 1 ¼ ½K VU y K VO ð~ q yÞfh ðyÞdy þ K VU ~qfh ðyÞdy g^h ðhÞdh. ð7Þ a
~ q
h
e V ð~qÞ is The first-order condition for the maximum of P Z ^h e V ð~ dP qÞ ¼ ½K VU ðK VO þ K VU ÞF h ð~ qÞg^h ðhÞdh ¼ 0. d~ q a
ð8Þ
Define U^h ðyÞ ¼
Z
^ h
F h ðyÞg^h ðhÞdh.
ð9Þ
a
Note that U^h ðyÞ is simply a conjugated probability distribution function, and thus it is easy to verify that U^h ðyÞ satisfies all of the basic probability axioms. Substituting (9) into (8), we find that VÕs optimal choice of stock level ~ q^h must satisfy p V cV ~ q^h ¼ U1 ðnV Þ. ð10Þ ¼ U1 ^ ^ h h pV rV Using (9), the condition that is given in (10) can also be written as ~^h ¼ U^h q
Z a
^ h
~^h g^h ðhÞdh ¼ nV . Fh q
ð11Þ
~^h is used to indicate that q~^h is actually dependent on ^h. Here, the subscript ^ h of q Substituting ~ q^h into (7), and after simplification, VÕs maximum expected profit can be shown to be equal to Z ^h Z ~q ^h e PV ~ F h ðyÞg^h ðhÞdy dh. ð12Þ q^h ¼ K VU ~ q^h ðK VO þ K VU Þ a
h
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Thus, RÕs expected profit if the information is withheld is 2 3
Z ~q h ^ h 6 7 eR ~ q^h ; y ¼ K RU 4~q^h F ðxÞ dx5. q^h ; h ¼ ðpR cR ÞEy min ~ P
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ð13Þ
0
By comparing (4) and (13), we can see that R will withhold the demand signal h from V if and only if Z ð~q hÞ Z ðq hÞ ^h h e P R ð~ q^h ; hÞ P PR ðqh ; hÞ () K RU ~ F ðxÞ dx P K RU qh F ðxÞ dx k q^h 0
() k P K RU
Z
qh h
~^ h q
0
½1 F ðxÞdx ¼ K RU
h
Z
F 1 ðnV
Þ
½1 F ðxÞdx.
ð14Þ
~^ h q h
The last equality results from the fact that, by (2), qh ¼ F 1 ðnV Þ þ h. There is an intuitive explanation for (14). As qh is the level of stock that R expects V to choose when the information is revealed, ~ q^h is the level of stock that R expects V to choose when the information is withheld, and KRU is RÕs cost of understocking, then the right-hand side of (14) is the reduction in RÕs expected cost of understocking if the information is revealed (compared with the case if the information is withheld). Thus, if the cost of revealing information, k, is higher than the reduction in the expected cost of understocking, then clearly, R will withhold the information. Note that ~q^h may be larger than qh . In this case, (14) will clearly hold, because the expected cost of understocking increases rather than decreases when R reveals the information. This happens when R expects V to choose a higher stock level when the demand information is withheld than when the information is revealed. As R has no cost associated with overstocking, R clearly prefers a higher stock level, and thus in this case will withhold the information. When h ¼ ^ h, by the definition of ^ h, R is indifferent between revealing and withholding the information. Hence, ^ h must satisfy k ¼ K RU
Z
q^ ^h h
~ h q^ ^
½1 F ðxÞdx ¼ K RU
Z
h
F 1 ðnV Þ
½1 F ðxÞdx.
ð15Þ
~ h q^ ^ h
We now introduce the following lemma. ~^h ^ Lemma 2. Given assumption A1, q h is continuous and strictly decreasing in ^h for all ^h 2 ½a; b. Proof. Please refer to Appendix 2 for the proof.
h
We are now ready for our first result, which states that when k is sufficiently large, then R will never reveal the actual value of h to V in equilibrium. Proposition 1. If the cost of revealing information k is sufficiently large such that Z F 1 ðnV Þ k P K RU ½1 F ðxÞ dx; ~b b q
then there exists a unique pure strategy perfect Bayesian equilibrium in which R always withholds the information, regardless of the signal h that is received. Proof. Note that, by (14), k P K RU
R F 1 ðnV Þ ~ qb b
e R ð~q ; bÞ P PR ðq ; bÞ. ½1 F ðxÞ dx implies that P b b
This means that if R expects V to choose the stock level ~qb when the information is withheld, then R will not deem it worthwhile to reveal the information even when the demand signal is h = b. By Lemma 1, as R
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will withhold the information when the signal is h = b, R will also withhold it when the signal is h < b. Thus, for the given condition, R will withhold the information regardless of the value of h. Now, given that R will always withhold the information regardless of the value of h, VÕs posterior belief about h remains the same as his prior belief when he finds that the information is being withheld by R. Thus, believing that h 2 [a, b] (* ^ h ¼ bÞ, V chooses the optimal stock level accordingly, which is ~qb . So, the given condition indeed leads to a pure strategy equilibrium. To check that this is the only equilibrium, let us assume that another equilibrium exists in which R will reveal the information if and only if h > ^ h, where a 6 ^h < b. In such an equilibrium, as is explained in (15), because R is indifferent between revealing and withholding the information when the signal h ¼ ^h is received, the following must hold: k ¼ K RU
Z
F 1 ðnV Þ
½1 F ðxÞdx.
~^ ^h q h
As
~ qb
b<
~ q^h
^ h by Lemma 2 (* ^ h < b), we therefore have
Z k < K RU
F 1 ðnV Þ
½1 F ðxÞ dx;
qb b ~
which violates the given condition. Thus, such an equilibrium cannot exist, and the equilibrium that is described in the proposition is indeed the only possible pure strategy equilibrium. h Proposition 1 describes the condition under which R always withholds the information in equilibrium. We now consider the case in which Z k < K RU
F 1 ðnV Þ
½1 F ðxÞ dx.
ð16Þ
~ qb b
In this case, the condition that is given in Proposition 1 is violated, and hence the equilibrium that is described in the proposition is no longer valid. Lemma 2 states that ~ q^h ^ h is continuous and strictly decreasing in ^h for all ^h 2 ½a; b. Hence, if (16) holds, then there must exist a unique ^ h 2 ½a; b such that (15) holds, i.e., Z q ^h Z F 1 ðnV Þ ^ h k ¼ K RU ½1 F ðxÞ dx ¼ K RU ½1 F ðxÞ dx. ~ q^ ^h h
~ q^ ^ h h
As has been explained, given that R expects V to choose stock level ~q^h if the information is withheld, and expects V to choose qh if the information is revealed, then the aforementioned equality means that R is indifferent between revealing and withholding the information when the signal is h ¼ ^h. Hence, by Lemma 1, R will withhold the information if the signal is h 6 ^h, and will reveal the information if the signal is h > ^h. Given the strategy of R that has just been described, VÕs posterior belief about h becomes G^h ðhÞ when R withholds the demand information. In this case, believing that h 2 ½a; ^h, V chooses the optimal stock level ~ q^h accordingly. Thus, condition (16) also leads to a pure strategy equilibrium. In addition, given that there is only one ^ h that satisfies (15), the equilibrium described above must be unique. We now summarize the above discussions into the following proposition. Proposition 2. If the cost of revealing information, k, satisfies Z F 1 ðnV Þ k < K RU ½1 F ðxÞ dx; ~ qb b
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then there exists a unique pure strategy perfect Bayesian equilibrium in which R will reveal the demand signal h to V if and only if h > ^ h, where ^ h satisfies (15). h
Proof. See the previous discussions. 4.1. Properties of ^ h
Note that, in equilibrium, ^ h and ~ q^h must satisfy (15). As KRU = pRcR = pR (pV + wR), and letting 1 x* = F (nV), Eq. (15) can be rewritten as: k ¼ K RU
Z
F 1 ðnV Þ ~ h q^ ^
½1 F ðxÞ dx ¼ ðpR pV wR Þ
h
Z
x
½1 F ðxÞ dx.
ð17Þ
~ h q^ ^ h
We now investigate the way in which ^ h changes with k, pV, cV, rV, pR, and wR. Differentiating both sides of Eq. (17) with respect to k, we obtain h
i1 d~ q^h 1 d^ h 1 ^ ¼ K RU 1 F ~ q^h h 1 . dk d^ h d~ q
ð18Þ
^
Because d^h^h < 1 by Lemma 2, it must be the case that ddkh > 0. Differentiating both sides of Eq. (17) with respect to pV, we obtain ( ) Z x d~q^h dx d^h ^ 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h hÞ 1 . dpV dpV d^h h q^ ^ ~ h
V V rV ; dx ¼ f ðxcÞðp Note that as F ðx Þ ¼ nV ¼ ppV c rV dp r V
(19) is negative, the sign of d~q^ h dh^ dh^ dpV
V
d^ h dpV
V
V
Þ2
> 0ð* cV > rV Þ. As the first term on the right-hand side of
dx cannot be determined immediately, because pR pV wR > 0; dp > 0 and V
^
2
dx < 1 (by Lemma 2). However, it can be shown that dpdh ? 0 if k ðpR pV wR Þ ½1 F ðx Þ dp ? 0. Thus, V
> 0 for a relatively large k and
d^ h dpV
V
< 0 for a relatively small k.
Differentiating both sides of Eq. (17) with respect to cV, we obtain ( ) ^h d~ q d ^ dx h 0 ¼ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h ^hÞ 1 . dcV dcV d^h As
dx dcV
1 ¼ f ðx Þðp < 0ð* pV > rV Þ and rV Þ V
d~ q^ h ^ dh
< 1, we must have
d^ h dcV
ð20Þ
> 0.
Differentiating both sides of Eq. (17) with respect to rV, we obtain ( ) d~q^h d^h dx ^ 0 ¼ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h hÞ 1 . drV drV d^h As
ð19Þ
dx drV
V cV ¼ f ðxpÞðp r V
V
Þ
2
> 0 ð* pV > cV Þ and
d~ q^ h d^ h
< 1, we must have
d^ h drV
ð21Þ
< 0.
Differentiating both sides of Eq. (17) with respect to pR, we obtain Z x d~q^ d^h 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ½1 F ð~q^h ^hÞ 1 h . dpR d^h ~ h q^ ^
ð22Þ
h
As the first term on the right-hand side of (22) is positive, and
d~ q^ h ^ dh
< 1, we must have
d^ h dpR
< 0.
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Finally, differentiating both sides of Eq. (17) with respect to wR, we obtain Z x d~q^h d^h ^ 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ½1 F ð~q^h hÞ 1 . dwR d^h h q^ ^ ~
ð23Þ
h
As the first term on the right-hand side of (23) is negative and We can now summarize these results.
d~ q^ h d^ h
< 1, we must have
d^ h dwR
< 0.
1. Because d^ h=dk > 0, as k increases, the likelihood that R will reveal the demand information decreases. By contrast, as k decreases, the likelihood that R will reveal the demand information increases. 2. As d^ h=dcV > 0, an increase in VÕs production cost will decrease the likelihood that R will reveal the demand information, or to put it differently, a decrease in VÕs production cost can increase the likelihood that R will reveal the demand information. 3. As d^ h=drV < 0, an increase in VÕs salvage value will increase the likelihood that R will reveal the demand information. 4. As d^ h=dpR < 0, an increase in RÕs price (retailer price) will increase the likelihood that R will reveal the demand information. 5. As d^ h=dwR > 0, an increase in RÕs transaction cost will decrease the likelihood that R will reveal the demand information, or to put it differently, a decrease in RÕs transaction cost will increase the likelihood that R will reveal the demand information. 6. As a change in pV will affect the profit margins of both V and R, the effect of a change in pV on the likelihood that R will reveal the demand information is less clear. Depending on the values of the other parameters, a change in pV may increase or decrease the likelihood that R will reveal the demand information. In general, with the exception of a change in pV, we can conclude that a decrease in information sharing cost or an increase in the profit margin of either V or R (a decrease in cV, an increase in rV, an increase in pR, or a decrease in wR) will increase the likelihood that R will reveal the demand information.
5. Conclusion In the preceding sections, we consider voluntary information sharing between a pair of asymmetrically informed members of a supply chain in which the less-informed upstream vendor V makes the stock level decision, and the better-informed downstream retailer R must decide whether or not to reveal the market demand information to the vendor before the vendor makes the stock level decision. We model the situation as a Bayesian game and find that, in equilibrium, whether the retailer reveals or withholds the information depends on two things—the cost of revealing the information, and the nature of the market demand signal that is received. If the cost of information sharing is sufficiently large, then the retailer will withhold the information from vendor regardless of the nature of the signal that is received. If the cost of sharing information is smaller, then the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it if a low demand is signaled. In general, reducing the cost of information sharing or increasing the profit margin of either the retailer or the vendor (or reducing the costs of either) will facilitate information sharing. Because the vendor can expect to make better stock level decisions and achieve higher expected profits if better information about the market demand can be obtained, then the vendor should have an incentive to help to reduce the cost to the retailer of revealing information. The adoption of information technology (IT) with adequate system integration, for example, can clearly help to reduce the cost to retailers of reveal-
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ing information, and would hence promote information sharing. This may explain why IT has become an important enabler in many new supply chain practices. However, IT investment must be economically justifiable. To help reduce the retailerÕs cost, the vendor could share part of the IT implementation cost. Our model can be used to help evaluate the benefit of information sharing to justify an investment decision. For example, if a company intends to install Radio Frequency Identification (RFID) to facilitate the transfer of information, then our model can be used to evaluate its cost and benefit. If the cost outweighs the benefit, then our model can be used to evaluate the extent to which the vendor could help to share the cost to make RFID investment an economically viable option. Furthermore, some types of information cost more to reveal than others. If there are different types of information that all serve the same purpose, then the information that is the least costly should be used. For example, it is often more costly to share point-of-sale (POS) data than aggregated data. Thus, if aggregated data is sufficient for the purpose, then there is little value in sharing POS data. To conclude, by reducing the cost of revealing information, the incentive for the downstream members of a supply chain to share information will increase, and thus the upstream members will have a better chance of obtaining valuable information from the downstream members. Our analysis shows that a lower operating cost facilitates information sharing, and thus the upstream members of a supply chain should develop lean business processes to reduce internal operating cost, which in turn will encourage voluntary information sharing on the part of the downstream members. Finally, there are some possible extensions to this paper. Our model can also be generalized to consider a case in which the delivery from the upstream members of a supply chain to the downstream members is not instantaneous. In such a case, the single period demand distribution of our model needs to be transformed to a demand distribution that covers the demand during the lead-time. Although the algebra may be more tedious, we expect to obtain similar results. Another possible extension would be to consider the different types of information that may be shared. In our model, we assume that the information to be shared is the minimum demand (or the mean demand), but in reality, there are many different types of information, such as POS data, aggregated demand data, aggregated forecasts, promotion plans, and process constraints, that downstream members can share with upstream members. As different types of shared information will clearly have different impacts on the decisions that take place upstream, the incentive for the downstream members to share information may also depend on the type of information that is being shared.
Appendix 1. Proof of Lemma 1 In any pure strategy equilibrium, if R reveals the information when the signal h is received, then his expected payoff, as defined in (4), is " # Z ðq hÞ h PR ðqh ; hÞ ¼ K RU qh F ðxÞ dx k; 0
qh ,
where as defined in (2), is VÕs choice of optimal stock level if R reveals the information about the signal h that is received. Let ~ q be VÕs choice of optimal stock level if R withholds the information. Note that ~q does not depend on h because V does not know the value of h when ~q is chosen. Thus, RÕs expected payoff when the information is withheld is Z ð~qhÞ e R ð~ P q; hÞ ¼ ðpR cR ÞEy minð~ q; yÞ ¼ K RU ~ F ðxÞ dx . q 0
e R ð~q; hÞ > 0; otherwise, the information will be In equilibrium, R will reveal the information if PR ðqh ; hÞ P e R ð~ withheld. We now show that PR ðqh ; hÞ P q; hÞ is monotonically increasing in h.
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" e R ð~ PR ðqh ; hÞ P q; hÞ ¼ K RU qh ~ q
Z
ðqh hÞ
# F ðxÞ dx k:
ð~ qhÞ
By (2), qh ¼ F 1 ðnV Þ þ h. Let x* = F1(nV), and hence qh ¼ x þ h. We can then rewrite the above expression as " # Z x e R ð~ q; hÞ ¼ K RU x þ h ~ q F ðxÞ dx k. PR ðqh ; hÞ P ð~ qhÞ
By LeibnizÕs rule, the first derivative of the above function with respect to h is K RU ½1 F ð~q hÞ, which e R ð~ is greater than zero, and thus PR ðqh ; hÞ P q; hÞ is monotonically increasing in h. Hence, PR ðqh0 ; h0 Þ 0 00 e R ð~ e R ð~ P q; h Þ > 0 implies that PR ðqh00 ; h Þ P q; h00 Þ > 0 for all h00 > h 0 . That is, in equilibrium, if R is sup0 posed to reveal the information when he receives the signal h , then he will also reveal it when he receives 0 0 e R ð~ e R ð~q; h00 Þ < 0 for all h00 < h 0 , the signal h00 . However, PR ðqh0 ; h Þ P q; h Þ < 0 implies that PR ðqh00 ; h00 Þ P or in other words, if R is supposed to withhold the information in equilibrium when he receives the signal h 0 , then he will also withhold the information when he receives the signal h00 . h Appendix 2. Proof of Lemma 2 For the equilibrium in which R does not reveal if the signal h that he received is less than or equal to ^h, VÕs optimal stock level ~ q^h , as defined in (11), must satisfy Z ^h U^h ð~ q^h Þ ¼ Fh ~ ðiÞ q^h g^h ðhÞdh ¼ nV . a
d~ q d~ q h is decreasing in ^ h, it suffices to show that d^h^h < 1. Note that d^h^h exits because, as can To show that ~ q^h ^ be seen from (i), U^h ð~ qÞ is obviously differentiable at every point in the domain of ~q, and dU is never zero. This d~ q ^ ^ ^ ~ also means that q^ h is continuous in h for h 2 ½a; b. h
gðhÞ As g^h ðhÞ ¼ Gð , Eq. (i) can be rewritten as ^ hÞ Z ^h F ð~ q^h hÞgðhÞdh ¼ nV Gð^ hÞ.
ðiiÞ
a
^ we obtain Taking the derivative of (ii) with respect to h, Z ^h d~ q^ F ð~ q^h ^ hÞgð^ hÞ þ f ð~ q^h hÞgðhÞ h dh ¼ nV gð^hÞ. d^ h a Hence, ^ V F ð~ d~ q^h gðhÞ½n q^h ^ hÞ ¼ R^ . h ^ dh f ð~ q hÞgðhÞdh ^ a h
ðiiiÞ
By (i), q^h Þ ¼ U^h ð~
Z a
^ h
~^h g^h ðhÞdh ¼ Fh q
() nV ¼ ()
Z a
^ h
F ð~ q^h
^ hÞ þ
Z a ^ h
Z a
^ h
F ð~ q^h hÞg^h ðhÞdh ¼ nV G^h ðhÞf ð~ q^h hÞdh
GðhÞ f ð~ q^h hÞdh ¼ nV F ð~q^h ^hÞ. ^ GðhÞ
ðivÞ
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Substituting (iv) into (iii), we have R ^h R ^h hÞ a GðhÞ f ð~ q^h hÞdh d~ qh^ gð^ gð^ hÞGðhÞf ð~q^h hÞdh Gð^ hÞ ¼ R^ ¼ Ra^ < 1. h h d^ h f ð~q^h hÞgðhÞdh gðhÞGð^ hÞf ð~q^h hÞdh a a The last inequality holds because of Assumption A1. As g(h)/G(h) is strictly decreasing in h, gðhÞ=GðhÞ > gð^ hÞ=Gð^ hÞ for all h < ^ h. Therefore, gð^hÞGðhÞ < gðhÞGð^hÞ. h
References Bourland, K., Powell, S., Pyke, D., 1996. Exploiting timely demand information to reduce inventories. European Journal of Operational Research 92, 239–253. Cachon, G., Fisher, M., 2000. Supply chain inventory management and the value of shared information. Management Science 46 (8), 1032–1048. Cachon, G., Lariviere, M., 2001. Contracting to assure supply: How to share demand forecasts in a supply chain. Management Science 47 (5), 629–646. Chen, F., 1998. Echelon reorder points, installation reorder points, and the value of centralized demand information. Management Sciences 44 (12), S221–S234. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000. Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information. Management Science 46 (3), 436–443. Clark, T.H., Hammond, J.H., 1997. Reengineering channel reordering processes to improve total supply chain performance. Production and Operations Management 6 (3), 248–265. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. Gavirneni, S., Kapuscinski, R., Tayur, S., 1999. The value of information in a capacitated supply chain. Management Science 45 (1), 16–24. Kopczak, L.R., 1997. Logistics partnerships and supply chain restructuring: Survey results from the U.S. computer industry. Production and Operations Management 6 (3), 226–247. Kopczak, L.R., 1998. How computer companies use logistics partnerships: A snapshot of the present and a view of the future. Global Supply Chain and Technology Management 1, 91–111. Kumar, D., 2004. Beyond the low-hanging fruit. Industrial Engineer (May), 44–48. Lee, H.L., Padmanabhan, V., Whang, S., 1997a. Information distortion in a supply chain: The bullwhip effect. Management Science 43 (4), 546–558. Lee, H.L., Padmanabhan, V., Whang, S., 1997b. The bullwhip effect in supply chains. Sloan Management Review 38 (3), 93–102. Lee, H.L., So, K.C., Tang, C., 2000. The value of information sharing in a two-level supply chain. Management Science 46 (5), 626– 643. Lee, H.L., Whang, S., 2000. Information sharing in a supply chain. International Journal of Manufacturing Technology and Management 1 (1), 79–93. Raghunathan, S., 2001. Information sharing in a supply chain: A note on its value when demand is nonstationary. Management Science 47 (4), 605–610. Shaked, M., Shanthikumar, J., 1994. Stochastic Orders and Their Applications. Academic Press, London. Tsay, A., 2000. The quantity flexibility contract and supplier–customer incentives. Management Science 46 (10), 1509–1522.
Production, Manufacturing and Logistics
Strategic information sharing in a supply chain Wai Hung Julius Chu a
a,*
, Ching Chyi Lee
b
Emerson Electric Asia-Pacific, Suite 6701, Central Plaza, 18 Harbour Road, Wanchai, Hong Kong Faculty of Business Administration, Department of Decision Sciences and Managerial Economics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
b
Received 29 April 2003; accepted 16 February 2005 Available online 17 May 2005
Abstract We consider a two-member supply chain that manufactures and sells newsboy-type products and comprises a downstream retailer and an upstream vendor. In this supply chain, the vendor is responsible for making stock-level decisions and holding the inventory, and the retailer is better informed about market demand. In each period, the retailer receives a signal about market demand before the actual demand is realized, and must decide whether to reveal the information to the vendor, at a cost, before the vendor starts production. We assume that any information that the retailer reveals is truthful. We model the situation as a Bayesian game, and find that, in equilibrium, whether the retailer reveals or withholds the information depends on two things—the cost of revealing the information and the nature of market demand signal that the retailer receives. If the cost of sharing the information is sufficiently large, then the retailer will withhold the information from the vendor regardless of the type of signal that is received. If the cost of sharing the information is small, then the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it from the vendor if a low demand is signaled. In general, reducing the cost of sharing information and increasing the profit margin of either the retailer or the vendor (or reducing the cost of the vendor or retailer) will facilitate information sharing. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Information sharing; Supply chain; Asymmetric information; Game theory
1. Introduction Information sharing is an important issue in supply chain management, particularly in some of the new supply chain practices that have recently become popular, such as vendor managed inventories (VMI), click and mortar, drop shipping, and vendor hubs. These new supply chain practices have one thing in *
Corresponding author. Fax: +852 28272168. E-mail address: [email protected] (W.H.J. Chu).
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.053
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common—the stock level of the supply chain is determined by the upstream members of the supply chain, who are usually less informed than the downstream members about market demand. Thus, to guarantee the success of these new supply chain management practices, it is essential that the better-informed downstream members of the chain share their demand information effectively and efficiently with the lessinformed upstream members. Advocates of these new practices often emphasize the importance of information sharing in the supply chain. There is no doubt that effective information sharing allows a supply chain to operate more efficiently, and hence generate a higher overall supply chain profit. However, information sharing often involves cost. If the cost of information sharing is solely born by the informed party, and if, in addition, there is no pre-defined mechanism to distribute some of the additional profit that is generated through the information sharing to the informed party, then it is debatable whether the informed party has any incentive to share information with the uninformed party. In fact, researchers have noticed that many members of supply chains are doubtful about information sharing. Clark and Hammond (1997) report that: Retailers generally acknowledged that providing additional information to manufacturers would offer some savings to the manufacturers, but many retailers were skeptical about the benefits for their firms in sharing information with manufacturers. Cost is one of the key factors that may hinder information sharing. The main purpose of this paper is therefore to study how the incentive to share information is related to the cost of that sharing in the specific context of a supply chain that manufactures and sells newsboy-type products. The supply chain has two members— the upstream vendor (V) and the downstream retailer (R). In this supply chain, there is no cost for the retailer for overstocking because the vendor makes the stock level decisions and keeps the inventory. In each period, the retailer receives a signal about market demand before the actual demand is realized, and must then decide whether to reveal it voluntarily to the vendor at a cost before production begins, or whether to withhold the information. We assume any information that the retailer decides to reveal is truthful. We model the situation as a Bayesian game and find that, in equilibrium, whether or not the retailer reveals the information to the vendor depends on two things—the cost of revealing the information, and the nature of the market demand signal that is received. If the cost of sharing the information is sufficiently large, then the retailer will withhold the demand information from the vendor regardless of the type of signal that is received. When the cost of sharing information is smaller, the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it if a low demand is indicated. The latter result is driven by the fact that the retailer bears no overstock cost, and hence would like the vendor to stock as much inventory as possible. If the retailer knows that the demand is likely to be high, then he would want to ensure that the vendor receives the signal correctly and consequently stocks a high level of inventory. If the retailer knows that the demand is likely to be low, then there will be no incentive to reveal the information to the vendor, even though the vendor may be able to partially infer from the retailerÕs action that the demand is low. Basically, there is a cutoff value of market demand that separates these two cases. Further study on this cutoff yields some interesting managerial implications. In general, reducing information sharing cost and increasing the profit margin of either the vendor or the retailer (or reducing their costs) will facilitate information sharing. The rest of this paper is organized as follows. The next section provides a brief survey of the related literature. The model is described in Section 3, and the analysis of the model is given in Section 4. Section 5 offers some concluding remarks.
2. Related literature There is no lack of literature about information sharing in supply chains. Lee and Whang (2000) provide some real life examples of information sharing in a supply chain, and Bourland et al. (1996) and Chen
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(1998) develop theoretical models of sharing stationary stochastic downstream demand and inventory data to improve stock level decisions of the upstream. Lee et al. (1997a,b) address the bullwhip effect and Lee et al. (2000) consider the value of information sharing under a known autoregressive demand process. Gavirneni et al. (1999) and Cachon and Fisher (2000) study and quantify the value of sharing sales information to overcome demand distortion due to order batching, and Chen et al. (2000) analyze the effects of the forecasting process on information sharing. All of these papers report that there are some benefits to sharing demand information, although these benefits vary substantially. Gavirneni et al. (1999) conclude that information sharing is more beneficial when the capacity of the upstream members of a supply chain is unlimited. Moreover, Lee et al. (2000) suggest that the value of sharing demand information can be quite high, especially when the underlying demand is significantly correlated over time, when demand is highly variable, or when there is a long replenishment lead-time. However, Raghunathan (2001) revisits their model and shows that the value of information sharing is insignificant when the upstream members reduce the variance of the demand forecast by analyzing the entire order history. Hence, information sharing is valuable only if the information cannot be inferred by the receiving party. In another group of studies, the sharing of accurate demand information through contracts is examined, and readers are referred to Tsay (2000) for a recent review of these studies. Cachon and Lariviere (2001) study contracts that induce credible information sharing when there is asymmetric information. Even when the downstream members of a supply chain have an incentive to provide an overly optimistic demand forecast, they identify contracts that ensure credible information sharing. Our model differs from those that have been developed in previous research in one major aspect. We do not take information sharing in a supply chain for granted. We recognize the fact that even though information sharing may be beneficial to the supply chain as a whole, the better-informed party will have no incentive to share the information with the uninformed party if there is no benefit for them in doing so. In our model, the informed party (the retailer) bears the cost of the information sharing alone, and the passing of information from the informed party to the uninformed party (the vendor) is undertaken voluntarily and truthfully. The informed party does not receive any side payment from the uninformed party for the information transfer. The only benefit that the informed party may obtain from sharing the information is a possible increase in payoff due to the better stock level decisions that would be made by the uninformed party as a result of the information. However, such a benefit must be at least greater in value than the cost of sharing the information to justify the action. Hence, the decision of whether or not to reveal the information to the uninformed party is made strategically by the informed party.
3. The model We consider a supply chain that consists of two members—the downstream retailer R and the upstream vendor V. V supplies R with newsboy-type products to be sold to the end market. This means that the following assumptions hold. First, the market demand in all periods is independent and identically distributed. Second, all of the cost and price parameters remain fixed from period to period. Third, backordering is not allowed, and finally, no unsold item in a given period has any value in future periods. These assumptions effectively reduce a multi-period problem to a single period problem. We assume that all members are risk neutral, and have a zero-order setup cost. The market demand is denoted as Y = X + H, where X and H are independent variables. X is a random variable having support on [0, 1) and is governed by a continuously differentiable probability density function f(x), with the corresponding cumulative probability function being F(x). H, which is a random variable that represents the minimum market demand, is governed by a continuously differentiable probability
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density function g(h) with support on [a, b], where a P 0. The corresponding cumulative function of g(h) is G(h). In each period, R receives a signal that indicates the actual value of H before the realization of Y, and must decide whether to reveal this information to V before V makes a stock decision. Assuming that the signal received by R is h, then the conditional probability of Y given h is denoted as Fh(y), with the corresponding density function being fh(y). Note that Fh(y) = Prob (yjh) = Prob (x + hjh) = F(x) = F(y h), and similarly that fh(y) = f(y h). Also note that once it is known that H = h, then E(Y) is known to be equal to E(X) + h. Thus, although our model can be interpreted as R receiving a signal about minimum demand, it can also be interpreted as R receiving a signal about mean demand. Except for the actual value of H which is only known by R before the realization of the actual demand, all of the parameters and probability distribution functions are assumed to be common knowledge.
3.1. Chronology of events For any given period, the chronology of events is as follows: 1. R receives a signal h and must then decide whether to reveal it to V. We assume that any information that R reveals is truthful. Moreover, we assume that there is a non-negative operating cost for R in revealing the information. The cost of revealing the information is denoted as k. 2. If R chooses to reveal the actual value of h to V in the previous step, then V will become fully informed of the distribution of Y, and will then choose a stock level qh. Note here that the subscript h of qh indicates that VÕs stock level decision in this case is dependent on h. If, on the other hand, R chooses to withhold the information, then V will make the stock level decision based on incomplete information about Y. We denote the stock level that is chosen by V in this case as ~q. Note that as V does not know the actual value of h when ~ q is chosen, ~ q does not depend on h. Instead, ~q depends on VÕs posterior belief about h. The unit production cost of V is cV. 3. Demand y is realized. If VÕs stock level is higher than or equal to the demand, then V supplies R with y units of goods at a unit price of pV, and the excess inventory has a residual value of rV per unit. If VÕs stock level is lower than the demand, then V supplies R with all of the units of goods that are produced at a unit price pV. 4. R sells the goods to the market at a unit price pR. The unit cost of R is cR = pV + wR, where wR can be interpreted generally as the unit transaction cost that is necessary to bring the goods from the upstream to the downstream.
3.2. Assumptions To make the analysis non-trivial, we assume that pi > ci > ri P 0 (i = V, R) and that wR P 0. The former is simply a standard assumption of the newsboy problem and the latter ensures that RÕs transaction cost is non-negative. For technical reason and to simplify the analysis, we further assume that the probability function for H satisfies the following: A1 g(h)/G(h) is strictly decreasing in h. This assumption, which is quite similar to the standard monotone hazard rate assumption, is often referred to as the monotone reversed hazard rate assumption (Shaked and Shanthikumar, 1994). Among the commonly used distribution functions, the exponential function and the family of power functions, xa m F ðxÞ ¼ ðba Þ , where m P 1 and x 2 [a, b], both satisfy this assumption. Note that uniform distribution is a member of the power function family (m = 1).
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Finally, as H is assumed to be a continuous variable, we impose the following tiebreak rule. A2 If R is indifferent between revealing and withholding the information, then he will withhold the demand signal from V. This sequence of events is very typical in environments such as the supply chains of fast moving consumer goods (FMCG), consumer packaged goods (Kumar, 2004), and passive electronic components (Kopczak, 1997, 1998). In these kinds of supply chains, the products that are involved are typically short-lived (newsboy-type products) and the upstream members of the chain are responsible for the stock level decisions. 4. The analysis As the model that has been described is an incomplete information game, in this section, we will characterize the perfect Bayesian equilibrium of the game. For a detailed definition of perfect Bayesian equilibrium, we refer the reader to Fudenberg and Tirole (1991). We focus here only on pure strategy equilibrium. We first consider the case in which R reveals information to V. If R chooses to reveal h to V, then V will become fully informed of the distribution of Y, and will then choose the stock level qh to maximize his expected profit PV(qh), where PV ðqh Þ ¼ pV Ey minðqh ; yÞ cV qh þ rV Ey ðqh yÞþ þ
¼ pV fh þ Ex min ½ðqh hÞ; xg cV qh þ rV Ex ½qh ðx þ hÞ .
ð1Þ
Throughout the paper, the operator (a)+ is used to indicate the maximum of a and zero, and hence [qh (x + h)]+ represents the excess inventory. This is basically a newsboy problem, and VÕs optimal choice of qh in this case is K VU 1 p V cV 1 ð2Þ þh¼F þ h ¼ F 1 ðnV Þ þ h. qh ¼ F pV rV K VU þ K VO Here, KVO = (cV rV) is the cost of overstocking and KVU = (pV cV) the cost of understocking for V, and nV = (KVU/KVU + KVO) is the critical fractile ratio. Substituting qh into (1), and after simplification, we obtain VÕs maximum expected profit when R reveals h Z ðq hÞ h F ðxÞ dx. ð3Þ PV ðqh Þ ¼ K VU qh ðK VO þ K VU Þ 0
In addition, RÕs expected profit in this case is PR ðqh ; hÞ
¼ ðpR
cR ÞEy minðqh ; yÞ
Z k ¼ K RU qh
ðqh hÞ
F ðxÞ dx k
ð4Þ
0
where KRU = (pR cR) is RÕs cost of understocking. Before we proceed to the analysis of the case in which R withholds the information, we introduce the following lemma. Lemma 1. In any pure strategy equilibrium, if R is supposed to reveal the information when he receives a demand signal h 0 , then he will also reveal it when he receives any other demand signal h00 > h 0 . On the other hand, if R is supposed to withhold the information when he receives a demand signal h 0 , then he will also withhold the information when he receives any other demand signal h00 < h 0 . Proof. Please refer to Appendix 1 for the proof.
h
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Lemma 1 has an important implication that greatly simplifies our later analysis. If, in any equilibrium, RÕs revelation of the signal to V depends on the signal h that is received, then it must be the case that R will reveal the information only when h has a relatively large value. Let ^h be the specific value of the demand signal that leads R to be indifferent between revealing and withholding the information. Then, by Lemma 1, R will reveal h to V if and only if h > ^ h. Note that there may not exist any ^h in the interval [a, b] when k is sufficiently large, because ^ h would have to be greater than b, the upper bound of its support, for the indifference to hold. This means that, in equilibrium, R will always withhold the information, regardless of the signal h that is received. Let us assume that an ^ h exists in the interval [a, b]. When R withholds the information, V can then infer that the actual value of h must be between a and ^h, and VÕs posterior belief about h is therefore updated to g^h ðhÞ ¼ gðhÞ=Gð^ hÞ 8^ h 2 ½a; b.
ð5Þ
The corresponding cumulative function is hÞ 8^ h 2 ½a; b. G^h ðhÞ ¼ GðhÞ=Gð^
ð6Þ
Thus, if R withholds the information, then given ^h 2 ½a; b and the posterior belief g^h ðhÞ, V will choose the e V ð~qÞ, where stock level ~ q to maximize his expected profit P e V ð~ P qÞ ¼ Eh pV Ey minð~ q; yÞ cV ~ q yÞþ q þ rV Ey ð~ Z ^h Z ~q Z 1 ¼ ½K VU y K VO ð~ q yÞfh ðyÞdy þ K VU ~qfh ðyÞdy g^h ðhÞdh. ð7Þ a
~ q
h
e V ð~qÞ is The first-order condition for the maximum of P Z ^h e V ð~ dP qÞ ¼ ½K VU ðK VO þ K VU ÞF h ð~ qÞg^h ðhÞdh ¼ 0. d~ q a
ð8Þ
Define U^h ðyÞ ¼
Z
^ h
F h ðyÞg^h ðhÞdh.
ð9Þ
a
Note that U^h ðyÞ is simply a conjugated probability distribution function, and thus it is easy to verify that U^h ðyÞ satisfies all of the basic probability axioms. Substituting (9) into (8), we find that VÕs optimal choice of stock level ~ q^h must satisfy p V cV ~ q^h ¼ U1 ðnV Þ. ð10Þ ¼ U1 ^ ^ h h pV rV Using (9), the condition that is given in (10) can also be written as ~^h ¼ U^h q
Z a
^ h
~^h g^h ðhÞdh ¼ nV . Fh q
ð11Þ
~^h is used to indicate that q~^h is actually dependent on ^h. Here, the subscript ^ h of q Substituting ~ q^h into (7), and after simplification, VÕs maximum expected profit can be shown to be equal to Z ^h Z ~q ^h e PV ~ F h ðyÞg^h ðhÞdy dh. ð12Þ q^h ¼ K VU ~ q^h ðK VO þ K VU Þ a
h
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Thus, RÕs expected profit if the information is withheld is 2 3
Z ~q h ^ h 6 7 eR ~ q^h ; y ¼ K RU 4~q^h F ðxÞ dx5. q^h ; h ¼ ðpR cR ÞEy min ~ P
1573
ð13Þ
0
By comparing (4) and (13), we can see that R will withhold the demand signal h from V if and only if Z ð~q hÞ Z ðq hÞ ^h h e P R ð~ q^h ; hÞ P PR ðqh ; hÞ () K RU ~ F ðxÞ dx P K RU qh F ðxÞ dx k q^h 0
() k P K RU
Z
qh h
~^ h q
0
½1 F ðxÞdx ¼ K RU
h
Z
F 1 ðnV
Þ
½1 F ðxÞdx.
ð14Þ
~^ h q h
The last equality results from the fact that, by (2), qh ¼ F 1 ðnV Þ þ h. There is an intuitive explanation for (14). As qh is the level of stock that R expects V to choose when the information is revealed, ~ q^h is the level of stock that R expects V to choose when the information is withheld, and KRU is RÕs cost of understocking, then the right-hand side of (14) is the reduction in RÕs expected cost of understocking if the information is revealed (compared with the case if the information is withheld). Thus, if the cost of revealing information, k, is higher than the reduction in the expected cost of understocking, then clearly, R will withhold the information. Note that ~q^h may be larger than qh . In this case, (14) will clearly hold, because the expected cost of understocking increases rather than decreases when R reveals the information. This happens when R expects V to choose a higher stock level when the demand information is withheld than when the information is revealed. As R has no cost associated with overstocking, R clearly prefers a higher stock level, and thus in this case will withhold the information. When h ¼ ^ h, by the definition of ^ h, R is indifferent between revealing and withholding the information. Hence, ^ h must satisfy k ¼ K RU
Z
q^ ^h h
~ h q^ ^
½1 F ðxÞdx ¼ K RU
Z
h
F 1 ðnV Þ
½1 F ðxÞdx.
ð15Þ
~ h q^ ^ h
We now introduce the following lemma. ~^h ^ Lemma 2. Given assumption A1, q h is continuous and strictly decreasing in ^h for all ^h 2 ½a; b. Proof. Please refer to Appendix 2 for the proof.
h
We are now ready for our first result, which states that when k is sufficiently large, then R will never reveal the actual value of h to V in equilibrium. Proposition 1. If the cost of revealing information k is sufficiently large such that Z F 1 ðnV Þ k P K RU ½1 F ðxÞ dx; ~b b q
then there exists a unique pure strategy perfect Bayesian equilibrium in which R always withholds the information, regardless of the signal h that is received. Proof. Note that, by (14), k P K RU
R F 1 ðnV Þ ~ qb b
e R ð~q ; bÞ P PR ðq ; bÞ. ½1 F ðxÞ dx implies that P b b
This means that if R expects V to choose the stock level ~qb when the information is withheld, then R will not deem it worthwhile to reveal the information even when the demand signal is h = b. By Lemma 1, as R
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will withhold the information when the signal is h = b, R will also withhold it when the signal is h < b. Thus, for the given condition, R will withhold the information regardless of the value of h. Now, given that R will always withhold the information regardless of the value of h, VÕs posterior belief about h remains the same as his prior belief when he finds that the information is being withheld by R. Thus, believing that h 2 [a, b] (* ^ h ¼ bÞ, V chooses the optimal stock level accordingly, which is ~qb . So, the given condition indeed leads to a pure strategy equilibrium. To check that this is the only equilibrium, let us assume that another equilibrium exists in which R will reveal the information if and only if h > ^ h, where a 6 ^h < b. In such an equilibrium, as is explained in (15), because R is indifferent between revealing and withholding the information when the signal h ¼ ^h is received, the following must hold: k ¼ K RU
Z
F 1 ðnV Þ
½1 F ðxÞdx.
~^ ^h q h
As
~ qb
b<
~ q^h
^ h by Lemma 2 (* ^ h < b), we therefore have
Z k < K RU
F 1 ðnV Þ
½1 F ðxÞ dx;
qb b ~
which violates the given condition. Thus, such an equilibrium cannot exist, and the equilibrium that is described in the proposition is indeed the only possible pure strategy equilibrium. h Proposition 1 describes the condition under which R always withholds the information in equilibrium. We now consider the case in which Z k < K RU
F 1 ðnV Þ
½1 F ðxÞ dx.
ð16Þ
~ qb b
In this case, the condition that is given in Proposition 1 is violated, and hence the equilibrium that is described in the proposition is no longer valid. Lemma 2 states that ~ q^h ^ h is continuous and strictly decreasing in ^h for all ^h 2 ½a; b. Hence, if (16) holds, then there must exist a unique ^ h 2 ½a; b such that (15) holds, i.e., Z q ^h Z F 1 ðnV Þ ^ h k ¼ K RU ½1 F ðxÞ dx ¼ K RU ½1 F ðxÞ dx. ~ q^ ^h h
~ q^ ^ h h
As has been explained, given that R expects V to choose stock level ~q^h if the information is withheld, and expects V to choose qh if the information is revealed, then the aforementioned equality means that R is indifferent between revealing and withholding the information when the signal is h ¼ ^h. Hence, by Lemma 1, R will withhold the information if the signal is h 6 ^h, and will reveal the information if the signal is h > ^h. Given the strategy of R that has just been described, VÕs posterior belief about h becomes G^h ðhÞ when R withholds the demand information. In this case, believing that h 2 ½a; ^h, V chooses the optimal stock level ~ q^h accordingly. Thus, condition (16) also leads to a pure strategy equilibrium. In addition, given that there is only one ^ h that satisfies (15), the equilibrium described above must be unique. We now summarize the above discussions into the following proposition. Proposition 2. If the cost of revealing information, k, satisfies Z F 1 ðnV Þ k < K RU ½1 F ðxÞ dx; ~ qb b
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1575
then there exists a unique pure strategy perfect Bayesian equilibrium in which R will reveal the demand signal h to V if and only if h > ^ h, where ^ h satisfies (15). h
Proof. See the previous discussions. 4.1. Properties of ^ h
Note that, in equilibrium, ^ h and ~ q^h must satisfy (15). As KRU = pRcR = pR (pV + wR), and letting 1 x* = F (nV), Eq. (15) can be rewritten as: k ¼ K RU
Z
F 1 ðnV Þ ~ h q^ ^
½1 F ðxÞ dx ¼ ðpR pV wR Þ
h
Z
x
½1 F ðxÞ dx.
ð17Þ
~ h q^ ^ h
We now investigate the way in which ^ h changes with k, pV, cV, rV, pR, and wR. Differentiating both sides of Eq. (17) with respect to k, we obtain h
i1 d~ q^h 1 d^ h 1 ^ ¼ K RU 1 F ~ q^h h 1 . dk d^ h d~ q
ð18Þ
^
Because d^h^h < 1 by Lemma 2, it must be the case that ddkh > 0. Differentiating both sides of Eq. (17) with respect to pV, we obtain ( ) Z x d~q^h dx d^h ^ 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h hÞ 1 . dpV dpV d^h h q^ ^ ~ h
V V rV ; dx ¼ f ðxcÞðp Note that as F ðx Þ ¼ nV ¼ ppV c rV dp r V
(19) is negative, the sign of d~q^ h dh^ dh^ dpV
V
d^ h dpV
V
V
Þ2
> 0ð* cV > rV Þ. As the first term on the right-hand side of
dx cannot be determined immediately, because pR pV wR > 0; dp > 0 and V
^
2
dx < 1 (by Lemma 2). However, it can be shown that dpdh ? 0 if k ðpR pV wR Þ ½1 F ðx Þ dp ? 0. Thus, V
> 0 for a relatively large k and
d^ h dpV
V
< 0 for a relatively small k.
Differentiating both sides of Eq. (17) with respect to cV, we obtain ( ) ^h d~ q d ^ dx h 0 ¼ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h ^hÞ 1 . dcV dcV d^h As
dx dcV
1 ¼ f ðx Þðp < 0ð* pV > rV Þ and rV Þ V
d~ q^ h ^ dh
< 1, we must have
d^ h dcV
ð20Þ
> 0.
Differentiating both sides of Eq. (17) with respect to rV, we obtain ( ) d~q^h d^h dx ^ 0 ¼ ðpR pV wR Þ ½1 F ðx Þ þ ½1 F ð~q^h hÞ 1 . drV drV d^h As
ð19Þ
dx drV
V cV ¼ f ðxpÞðp r V
V
Þ
2
> 0 ð* pV > cV Þ and
d~ q^ h d^ h
< 1, we must have
d^ h drV
ð21Þ
< 0.
Differentiating both sides of Eq. (17) with respect to pR, we obtain Z x d~q^ d^h 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ½1 F ð~q^h ^hÞ 1 h . dpR d^h ~ h q^ ^
ð22Þ
h
As the first term on the right-hand side of (22) is positive, and
d~ q^ h ^ dh
< 1, we must have
d^ h dpR
< 0.
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Finally, differentiating both sides of Eq. (17) with respect to wR, we obtain Z x d~q^h d^h ^ 0¼ ½1 F ðxÞ dx þ ðpR pV wR Þ½1 F ð~q^h hÞ 1 . dwR d^h h q^ ^ ~
ð23Þ
h
As the first term on the right-hand side of (23) is negative and We can now summarize these results.
d~ q^ h d^ h
< 1, we must have
d^ h dwR
< 0.
1. Because d^ h=dk > 0, as k increases, the likelihood that R will reveal the demand information decreases. By contrast, as k decreases, the likelihood that R will reveal the demand information increases. 2. As d^ h=dcV > 0, an increase in VÕs production cost will decrease the likelihood that R will reveal the demand information, or to put it differently, a decrease in VÕs production cost can increase the likelihood that R will reveal the demand information. 3. As d^ h=drV < 0, an increase in VÕs salvage value will increase the likelihood that R will reveal the demand information. 4. As d^ h=dpR < 0, an increase in RÕs price (retailer price) will increase the likelihood that R will reveal the demand information. 5. As d^ h=dwR > 0, an increase in RÕs transaction cost will decrease the likelihood that R will reveal the demand information, or to put it differently, a decrease in RÕs transaction cost will increase the likelihood that R will reveal the demand information. 6. As a change in pV will affect the profit margins of both V and R, the effect of a change in pV on the likelihood that R will reveal the demand information is less clear. Depending on the values of the other parameters, a change in pV may increase or decrease the likelihood that R will reveal the demand information. In general, with the exception of a change in pV, we can conclude that a decrease in information sharing cost or an increase in the profit margin of either V or R (a decrease in cV, an increase in rV, an increase in pR, or a decrease in wR) will increase the likelihood that R will reveal the demand information.
5. Conclusion In the preceding sections, we consider voluntary information sharing between a pair of asymmetrically informed members of a supply chain in which the less-informed upstream vendor V makes the stock level decision, and the better-informed downstream retailer R must decide whether or not to reveal the market demand information to the vendor before the vendor makes the stock level decision. We model the situation as a Bayesian game and find that, in equilibrium, whether the retailer reveals or withholds the information depends on two things—the cost of revealing the information, and the nature of the market demand signal that is received. If the cost of information sharing is sufficiently large, then the retailer will withhold the information from vendor regardless of the nature of the signal that is received. If the cost of sharing information is smaller, then the retailer will reveal the information to the vendor if a high demand is signaled, but will withhold it if a low demand is signaled. In general, reducing the cost of information sharing or increasing the profit margin of either the retailer or the vendor (or reducing the costs of either) will facilitate information sharing. Because the vendor can expect to make better stock level decisions and achieve higher expected profits if better information about the market demand can be obtained, then the vendor should have an incentive to help to reduce the cost to the retailer of revealing information. The adoption of information technology (IT) with adequate system integration, for example, can clearly help to reduce the cost to retailers of reveal-
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1577
ing information, and would hence promote information sharing. This may explain why IT has become an important enabler in many new supply chain practices. However, IT investment must be economically justifiable. To help reduce the retailerÕs cost, the vendor could share part of the IT implementation cost. Our model can be used to help evaluate the benefit of information sharing to justify an investment decision. For example, if a company intends to install Radio Frequency Identification (RFID) to facilitate the transfer of information, then our model can be used to evaluate its cost and benefit. If the cost outweighs the benefit, then our model can be used to evaluate the extent to which the vendor could help to share the cost to make RFID investment an economically viable option. Furthermore, some types of information cost more to reveal than others. If there are different types of information that all serve the same purpose, then the information that is the least costly should be used. For example, it is often more costly to share point-of-sale (POS) data than aggregated data. Thus, if aggregated data is sufficient for the purpose, then there is little value in sharing POS data. To conclude, by reducing the cost of revealing information, the incentive for the downstream members of a supply chain to share information will increase, and thus the upstream members will have a better chance of obtaining valuable information from the downstream members. Our analysis shows that a lower operating cost facilitates information sharing, and thus the upstream members of a supply chain should develop lean business processes to reduce internal operating cost, which in turn will encourage voluntary information sharing on the part of the downstream members. Finally, there are some possible extensions to this paper. Our model can also be generalized to consider a case in which the delivery from the upstream members of a supply chain to the downstream members is not instantaneous. In such a case, the single period demand distribution of our model needs to be transformed to a demand distribution that covers the demand during the lead-time. Although the algebra may be more tedious, we expect to obtain similar results. Another possible extension would be to consider the different types of information that may be shared. In our model, we assume that the information to be shared is the minimum demand (or the mean demand), but in reality, there are many different types of information, such as POS data, aggregated demand data, aggregated forecasts, promotion plans, and process constraints, that downstream members can share with upstream members. As different types of shared information will clearly have different impacts on the decisions that take place upstream, the incentive for the downstream members to share information may also depend on the type of information that is being shared.
Appendix 1. Proof of Lemma 1 In any pure strategy equilibrium, if R reveals the information when the signal h is received, then his expected payoff, as defined in (4), is " # Z ðq hÞ h PR ðqh ; hÞ ¼ K RU qh F ðxÞ dx k; 0
qh ,
where as defined in (2), is VÕs choice of optimal stock level if R reveals the information about the signal h that is received. Let ~ q be VÕs choice of optimal stock level if R withholds the information. Note that ~q does not depend on h because V does not know the value of h when ~q is chosen. Thus, RÕs expected payoff when the information is withheld is Z ð~qhÞ e R ð~ P q; hÞ ¼ ðpR cR ÞEy minð~ q; yÞ ¼ K RU ~ F ðxÞ dx . q 0
e R ð~q; hÞ > 0; otherwise, the information will be In equilibrium, R will reveal the information if PR ðqh ; hÞ P e R ð~ withheld. We now show that PR ðqh ; hÞ P q; hÞ is monotonically increasing in h.
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" e R ð~ PR ðqh ; hÞ P q; hÞ ¼ K RU qh ~ q
Z
ðqh hÞ
# F ðxÞ dx k:
ð~ qhÞ
By (2), qh ¼ F 1 ðnV Þ þ h. Let x* = F1(nV), and hence qh ¼ x þ h. We can then rewrite the above expression as " # Z x e R ð~ q; hÞ ¼ K RU x þ h ~ q F ðxÞ dx k. PR ðqh ; hÞ P ð~ qhÞ
By LeibnizÕs rule, the first derivative of the above function with respect to h is K RU ½1 F ð~q hÞ, which e R ð~ is greater than zero, and thus PR ðqh ; hÞ P q; hÞ is monotonically increasing in h. Hence, PR ðqh0 ; h0 Þ 0 00 e R ð~ e R ð~ P q; h Þ > 0 implies that PR ðqh00 ; h Þ P q; h00 Þ > 0 for all h00 > h 0 . That is, in equilibrium, if R is sup0 posed to reveal the information when he receives the signal h , then he will also reveal it when he receives 0 0 e R ð~ e R ð~q; h00 Þ < 0 for all h00 < h 0 , the signal h00 . However, PR ðqh0 ; h Þ P q; h Þ < 0 implies that PR ðqh00 ; h00 Þ P or in other words, if R is supposed to withhold the information in equilibrium when he receives the signal h 0 , then he will also withhold the information when he receives the signal h00 . h Appendix 2. Proof of Lemma 2 For the equilibrium in which R does not reveal if the signal h that he received is less than or equal to ^h, VÕs optimal stock level ~ q^h , as defined in (11), must satisfy Z ^h U^h ð~ q^h Þ ¼ Fh ~ ðiÞ q^h g^h ðhÞdh ¼ nV . a
d~ q d~ q h is decreasing in ^ h, it suffices to show that d^h^h < 1. Note that d^h^h exits because, as can To show that ~ q^h ^ be seen from (i), U^h ð~ qÞ is obviously differentiable at every point in the domain of ~q, and dU is never zero. This d~ q ^ ^ ^ ~ also means that q^ h is continuous in h for h 2 ½a; b. h
gðhÞ As g^h ðhÞ ¼ Gð , Eq. (i) can be rewritten as ^ hÞ Z ^h F ð~ q^h hÞgðhÞdh ¼ nV Gð^ hÞ.
ðiiÞ
a
^ we obtain Taking the derivative of (ii) with respect to h, Z ^h d~ q^ F ð~ q^h ^ hÞgð^ hÞ þ f ð~ q^h hÞgðhÞ h dh ¼ nV gð^hÞ. d^ h a Hence, ^ V F ð~ d~ q^h gðhÞ½n q^h ^ hÞ ¼ R^ . h ^ dh f ð~ q hÞgðhÞdh ^ a h
ðiiiÞ
By (i), q^h Þ ¼ U^h ð~
Z a
^ h
~^h g^h ðhÞdh ¼ Fh q
() nV ¼ ()
Z a
^ h
F ð~ q^h
^ hÞ þ
Z a ^ h
Z a
^ h
F ð~ q^h hÞg^h ðhÞdh ¼ nV G^h ðhÞf ð~ q^h hÞdh
GðhÞ f ð~ q^h hÞdh ¼ nV F ð~q^h ^hÞ. ^ GðhÞ
ðivÞ
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1579
Substituting (iv) into (iii), we have R ^h R ^h hÞ a GðhÞ f ð~ q^h hÞdh d~ qh^ gð^ gð^ hÞGðhÞf ð~q^h hÞdh Gð^ hÞ ¼ R^ ¼ Ra^ < 1. h h d^ h f ð~q^h hÞgðhÞdh gðhÞGð^ hÞf ð~q^h hÞdh a a The last inequality holds because of Assumption A1. As g(h)/G(h) is strictly decreasing in h, gðhÞ=GðhÞ > gð^ hÞ=Gð^ hÞ for all h < ^ h. Therefore, gð^hÞGðhÞ < gðhÞGð^hÞ. h
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