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Information leakage and supply chain contracts
Accepted Manuscript
Information Leakage and Supply Chain Contracts Hao Liu , Wei Jiang , Gengzhong Feng , Kwai-Sang Chin PII: DOI: Reference:
S0305-0483(17)30659-X https://doi.org/10.1016/j.omega.2018.11.003 OME 1994
To appear in:
Omega
Received date: Accepted date:
8 July 2017 2 November 2018
Please cite this article as: Hao Liu , Wei Jiang , Gengzhong Feng , Kwai-Sang Chin , Information Leakage and Supply Chain Contracts, Omega (2018), doi: https://doi.org/10.1016/j.omega.2018.11.003
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Highlights We investigate information leakage under different contracts (WPC or RSC) with different retailers.
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When the supplier signs RSC and WPC respectively with the incumbent and entrant, there exists the non-leakage equilibrium (NLE) under certain parameter regions. When the supplier signs WPC and RSC respectively with the incumbent and entrant, NLE does not exist in high demand variation. Following the results in Anand and Goyal (2009) and Kong et al. (2013), we summarize that in case of high demand variations, NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract.
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Information Leakage and Supply Chain Contracts Hao Liu1,3, Wei Jiang2, Gengzhong Feng1, Kwai-Sang Chin3 1 2
Xi’an Jiaotong University
Dongbei University of Finance and Economics City University of Hong Kong
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Abstract: This paper investigates information leakage under different contract configurations in a supply chain. We consider two competing retailers, one of whom (incumbent) has private information about market demand while the other (entrant) has no access to acquire any market demand information. We assume that the supplier is the leader who may choose a wholesale price contract (WPC) or a
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revenue-sharing contract (RSC) with each retailer independently. We explore the effect of the supplier’s and incumbent’s incentives on non-leakage equilibrium and find out that there exists a non-leakage equilibrium only when the incumbent signs an RSC with the supplier under an appropriately given revenue-sharing rate in situations of high demand variation.
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Key words: information leakage; contract mechanisms; game theory; supply chain management
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1 Introduction Contemporary supply chains face uncertain demand due to long lead time, short product life cycle, and proliferation of products and categories, such as fashion apparel, popular toys, electronics, etc. (Burnetas et al. 2007). Demand uncertainty leads to a series of problems, e.g., the mismatch between suppliers and retailers, shortages of products, and reducing customer service levels. Information sharing has become a critical driver for improving supply chain performance and weakening the negative effect of asymmetric
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demand information between suppliers and retailers. Advanced information sharing technology promotes the collaboration between the enterprises in a supply chain, which results in various initiatives such as Collaborative Planning, Forecasting, and Replenishment (CPFR), VMI (Vendor Management Inventory), Efficient Consumer Response (ECR), and Forecast Information Sharing (FIS) (Ali et al. 2017).
However, these coordination initiatives have increased the likelihood of information leakage to
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competitors or third parties due to deliberate efforts to lift supplier’s own profits. In the UK clothing industry, many retailers realize that suppliers may divulge important information to other companies and they are reluctant to share information with those suppliers (Adewole 2005). A survey made by supplychainaccess.com shows that common suppliers sharing product/demand information with competing retailers have severe threats to the security of online supply chain activities (Zhang and Li 2006). In general, information leakage is pervasive across all sectors of industry and consequently
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becomes a major obstacle to information sharing between suppliers and retailers. There are multifarious types of contract mechanisms for suppliers to cooperate with retailers, e.g.,
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wholesale price contract (WPC), revenue-sharing contract (RSC), buy-back contract (BBC), quantity discount contract (QDC), quantity–flexibility contract (QFC), etc. (Cachon 2003). Information leakage
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may occur under contract mechanisms when horizontal competition exists between retailers. For example, when a supplier signs WPC with two retailers, Anand and Goyal (2009) develop a game theoretic model and show that there always exists information leakage. When a supplier signs RSC with two retailers,
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however, Kong et al. (2013) indicate that private information may be protected by an appropriately designed revenue-sharing rate.
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While WPC can be considered a special case of RSC with zero revenue-sharing rate, a supplier must be able to ex post verify a retailer’s revenue under an RSC and needs to balance the costs of running RSC with the profit sacrificed by using WPC (Cachon and Lariviere 2005). Since small retailers’ contribution to the supplier’s profit is relatively low, the supplier is more willing to choose a WPC with small retailers considering regulatory costs. In practice, suppliers may choose different types of contract with different retailers depending on the market, negotiation power of the retailers, and regulatory capability of the supplier (Pan et al. 2010). For example, Hanwha Techwin Co., Ltd signed a revenue-sharing contract with Pratt & Whitney, a subsidiary of United Technologies International Corporation-Asia Private Limited 3
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(Watson and Tan 2016). Hanwha can enhance delivery of critical parts more efficiently to Pratt & Whitney so as to meet Pratt & Whitney’s increased demand for orders. However, according to Hanwha Group Management & Planning H.Q (2017), Hanwha also singed wholesale price contracts with GE and Rolls-Royce to provide large-scale aircraft engine parts to the latter; In China, Procter & Gamble (P&G) offers a lower wholesale price to the distributors compared with small retailers and the distributors need to share part of revenue with P&G at the end of the year, which is pointed out by Yang Huabin, a
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professional cosmetic consultant in Shenzhen (Wu and Li 2010); In the household appliance industry, Haier sets up a partnership with a big distributor in Shanghai and offers procurement preferences (Spring and Anantharaman 2016). Other examples include Gree Electric Appliances, Inc. and Kweichow Moutai Co., Ltd., who have different collaboration schemes to maintain close relationships with big retailers and small ones. In general, considering factors such as financial returns and ease of implementation of
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contracts, suppliers often have the flexibility to select a contract type with certain retailers and may choose an asymmetric contract scheme for different types of retailers.
Our goal in this paper is to investigate information leakage under different contracts (WPC or RSC) with different retailers. We consider a stylized supply chain including a supplier, an incumbent retailer, and an entrant retailer. The supplier may sign different contracts with the two retailers in one of the following four scenarios:
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Table 1 Different Contract Scenarios
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Contract Type
WPC
RSC
RSC
Scenario 1 Scenario 4
WPC
Scenario 3 Scenario 2
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Incumbent retailer
Entrant retailer
We assume that the retailers compete by selling completely substitutable products with short life cycles in
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a common market with uncertain demand. That is, the incumbent and the entrant engage in Cournot competition and the product price is a decreasing function of the overall order quantity available in the
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market. In this paper, we consider wholesale price as an endogenous variable, i.e., the supplier is allowed to optimally choose a wholesale price under any given revenue-sharing rate for each retailer independently. Note that this assumption may result in different wholesale prices for retailers, which is different from the assumption in Anand and Goyal (2009) and Kong et al. (2013). In general, the supplier can be informed of consumer preferences and market demand only through experience (Blattberg and Fox 1995). The entrant retailer lacks the necessary channel and expertise and has to depend on upstream suppliers for information (Agency Sales 1992, 2003). Therefore, we employ a two-state demand distribution and assume that the incumbent retailer has private and perfect information 4
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about the market demand while the supplier and the entrant have no channel to acquire the actual demand state. However, the supplier may leak the incumbent order information to the entrant in order to lock more profits from the latter. We assume that the entrant plays a passive role in the supply chain and has no ability to distinguish the information being leaked by the supplier. In other words, if the supplier leaks the incumbent’s order information to the entrant, the entrant would accept the information and make order decisions accordingly.
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We first investigate the information leakage problem under Scenario 1. By analyzing the sustainability of non-leakage equilibrium (NLE) in a supply chain, we find that the supplier would not leak information and the incumbent would not deviate from NLE under certain parameter regions. More specifically, NLE exists more likely when the revenue-sharing rate is relatively high and the probability of high demand state is relatively low.
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Following the discussions under Scenario 1, we further study information leakage under other three contract scenarios. We find that the supplier would leak information or the incumbent may deviate from NLE under high demand variation and NLE doesn’t exist under Scenarios 2 and 3. That is, as long as the supplier signs WPC with the incumbent, the former would always leak information to the entrant. Similar to Kong et al. (2013) when the supplier has an identical RSC with the two retailers (Scenario 4), information leakage can be prevented under some appropriate regions and the result is robust when the
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wholesale price is considered as an endogenous variable. In summary, in case of high demand variations,
2 Literature Review
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NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract.
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Our work is related to a stream of the literature that investigates vertical information sharing between two partners in a supply chain. Lee et al. (2000) show information sharing can provide significant benefit
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to the supplier and the retailer, such as inventory reduction and cost savings. Özer and Wei (2006) show how the distortion of demand forecast information occurs in a wholesale-price contract and study contracts to ensure information shared credibly between a manufacturer and a supplier. Yue and
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Raghunathan (2007) research the effect of a returns policy and information sharing on the performance of the supplier and the retailer, respectively. Zhang and Chen (2013) investigate coordination of information sharing in a supply chain consisting of one supplier and one retailer and show coordinative contact can ensure that they share their information better. Oh and Özer (2013) study how a supplier can get credible information from a manufacturer and how a supplier makes decisions to obtain more demand information under asymmetric information. These papers analyze vertical information sharing in a supply chain, but they do not consider information leakage and horizontal competition, which is our concern.
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Another stream of research considers vertical information sharing between one supplier and its retailer under two supply chains. Ha and Tong (2008) study vertical information sharing under two competing supply chains and show how important the contact type is to information sharing. Ha et al. (2011) consider two competing supply chains, each consisting of one supplier and one retailer and show how information sharing affects the performance of a supply chain under Cournot or Bertrand competition, respectively. Guo et al. (2014) show how retailers in two competing supply chains can strategically
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disclose demand information to affect the suppliers’ pricing decision. Shamir and Shin (2015) investigate when and how forecast information can be credibly shared in two competing supply chains under a wholesale price contract by making the forecast information publicly available. These papers focus on vertical information sharing under two competing supply chains and don’t consider the impact of information leakage on player’s decision-making.
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There is a stream of the literature researching vertical information sharing when horizontal competition exists in a supply chain where information leakage may occur. Li (2002) investigates the incentives of information sharing in a supply chain consisting of a supplier and n symmetric competing retailers. He refers to the phenomenon as “information leakage” and shows no information sharing is an equilibrium, which is undesirable for the supplier. Zhang (2002) shows a similar research to Li (2002) and further studies the incentives of information sharing under Bertrand competition between the retailers. Li and
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Zhang (2008) consider information sharing in a supply chain including a manufacturer and multiple retailers and study the effects of information leakage. They assume that each retailer has some private
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information about uncertain demand. Anand and Goyal (2009) study the behavior of information leakage in a one-supplier-two-retailers supply chain under a wholesale-price contract and find the supplier will always leak information. Kong et al. (2013) continue the research of Anand and Goyal (2009) and
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indicate that RSC can prevent the supplier from leaking information. Shamir (2016) shows retailers’ ability to share information with a mutual supplier and demonstrates how the retailers take advantage of
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information leakage to form a cartel. These works emphasize the effect of information leakage on the incentives of information sharing in a supply chain. However, we focus on information leakage under
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different contract configurations. The final stream of research relates to contract mechanism. Cachon and Lariviere (2001) show how a
downstream manufacturer uses contracts to share credible demand information with a supplier. Cachon (2003) provides the literature review about contract mechanism in supply chains under asymmetric information. Cachon and Lariviere (2005) show that RSC coordinates well a supply chain with one manufacturer and multiple competing retailers. Li and Wang (2007) review coordination mechanisms of supply chain systems. Yao et al. (2008) demonstrate that RSC is better than WPC for coordinating a supply chain. Pan et al. (2010) discuss how the manufacturers or retailers choose an optimal contract 6
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under different channel power structures, such as RSC or WPC. Zhang et al. (2012) investigate how to use RSC to coordinate a supply chain consisting of one manufacturer and two retailers with demand disruptions. These papers focus on how to use a contract to coordinate supply chains, but they do not consider the effect of information leakage in a supply chain, which is our focus. The remainder of this paper is organized as follows. Section 3 presents the model framework in detail. Section 4 studies the existence of NLE under Scenario 1. Section 5 investigates the impact of different
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contract configurations on NLE. Section 6 offers conclusions and future research. Proofs of main results are given in Appendices.
3 Model Framework
We consider a supply chain including a supplier and two retailers. The incumbent retailer and the entrant retailer sell completely substitutable products in a common market and engage in a Cournot
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competition on quantity. We assume that all players are risk neutral and develop a dynamic game among the three players to maximize their own expected profits under incomplete demand information. We further assume that the incumbent has been in a market for a long time and therefore has enough ability to acquire the actual demand state. On the other hand, the supplier and the entrant have no channel to access the accurate market demand state. Therefore, the actual market demand state is private information for the
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incumbent. We shall investigate the behavior of information leakage between the supplier and the entrant retailer under different contract scenarios in Table 1 where the supplier can optimally choose a wholesale
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price for a given revenue-sharing rate. The supplier, the incumbent retailer, and the entrant retailer are denoted by s, i and e, respectively.
We assume that the demand function is linear of the retail price P with intercept A , i.e., the inverse
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demand curve is given by P(Q) A Q , where Q qi qe is the total quantity in the market and qi / qe is the order quantity of the incumbent/entrant, respectively. The intercept A is a random variable which
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may take a high value H with probability p (0,1) or a low value L with probability 1 p and
H L 0 . The mean demand intercept is given by pH (1 p) L . These priors are common
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knowledge to all supply chain players.
3.1 Sequence of Events Suppose that the retailers bring their entire order quantities to the market. Let ww and wr denote the
wholesale price offered by the supplier under WPC and RSC, respectively. The following events take place in sequence.
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1) The supplier offers the retailers a WPC or an RSC, consisting of wholesale price and/or revenue-sharing rate (0,1) ; 2) The incumbent can acquire the actual market demand state A , which takes value H (high demand state) or L (low demand state); 3) Considering the market demand state, the incumbent places an order qiA to the supplier;
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4) The supplier decides whether to leak the incumbent’s order quantity to the entrant; 5) The entrant places an order qeA to the supplier according to the information available to him; 6) The demand state A is revealed to all parties and the three players realize their profits. 3.2 Supply Chain Player’s Incentives and Decisions
Given order quantities of the two retailers qiA and qeA in face of the demand state A , Table 2 presents
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the three players’ profits under the four contract scenarios in Table 1.
Table 2 Players’ Profits under Different Contract Scenarios Contract
Scenario 1
Configuration
Incumbent under RSC; entrant under WPC
Scenario 2
Incumbent under WPC; entrant under RSC
iA 1 qiA A qiA qeA wr qiA
iA qiA A qiA qeA wwqiA
Entrant’s Profit
eA qeA A qiA qeA wwqeA
eA 1 qeA A qiA qeA wr qeA
Supplier’s Profit
sA wr qiA wwqeA qiA A qiA qeA
sA ww qiA wr qeA qeA A qiA qeA
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Incumbent’s Profit
Scenario 4
Incumbent and entrant under WPC
Incumbent and entrant under RSC
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Scenario 3
iA qiA A qiA qeA wwqiA
iA 1 qiA A qiA qeA wr qiA
Entrant’s Profit
eA qeA A qiA qeA wwqeA
eA 1 qeA A qiA qeA wr qeA
Supplier’s Profit
sA ww qiA qeA
sA wr qiA qeA qiA qeA A qiA qeA
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Incumbent’s Profit
Supplier Given a particular type of contract with a retailer, the supplier optimally chooses a wholesale
price and decides whether or not to leak the incumbent’s order quantity to the entrant. In this paper, we assume that the supplier can’t distort the incumbent’s order quantity when she leaks it. We shall investigate whether there exists an equilibrium of information non-leakage if the supplier cooperates with the retailers under different contract schemes.
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Incumbent Considering that the supplier may leak information to the entrant, the incumbent decides an optimal order quantity to maximize his profit. The incumbent can acquire the actual market demand state to place an order, implying the order quantity reflects the demand information. If the supplier has an incentive to leak the incumbent’s order to the entrant, the entrant makes order decisions depending on the information. Therefore, the incumbent would place a strategic order to reduce the effect of information leakage.
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Entrant The entrant’s decision is his order quantity to maximize his expected profit. Comparing to the incumbent and the supplier, the entrant plays a passive role in the supply chain. If the supplier leaks the incumbent’s order information to the entrant, we assume that the entrant always accepts the information and makes his optimal order decision correspondingly. The two retailers play a Stackelberg game where the incumbent is the leader and the entrant is the follower. If the supplier wouldn’t leak the information,
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the retailers play a simultaneous game where the entrant can’t acquire the actual state of market demand and the two retailers simultaneously place the optimal orders.
4 Game Theoretic Analysis of Scenario 1
In this section, we analyze the retailers’ order quantities and profits when the supplier leaks information
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or never leaks information in contract Scenario 1, where the incumbent has an RSC and the entrant has a WPC with the supplier respectively. Sections 4.1 and 4.2 discuss the effect of information leakage and
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non-leakage on the retailers’ order decisions, respectively. The benchmark analysis is helpful in deriving and understanding the NLE under high demand variation in Section 4.3. Section 4.4 gives a brief discussion regarding NLE under low demand variation.
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The notations used for order quantity, profits and wholesale price are as follows. To compare the E E results under leakage (S) and non-leakage (N) equilibriums (or games), let qmA and mA denote the order
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quantity and profit of player m (m=i, e, or s) in equilibrium E (S or N) under demand state A (H or L), respectively. For example, qiHS denotes the order quantity of the incumbent in a separating equilibrium
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under the high demand state when the supplier leaks information. Besides, qeN and eN denote the entrant’s order quantity and profit in the NLE. Let wwE and wrE denote the wholesale price that the retailers should pay for each product under WPC and RSC in E equilibrium, respectively. Additional notations will be introduced later.
4.1 Benchmark Analysis under Information Leakage
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We first discuss the case where the supplier leaks the incumbent’s order information. In this case, we assume that the entrant always accepts the information leaked by the supplier and places his order accordingly. The incumbent makes order decision after learning the entrant’s belief of the demand state and decisions. Therefore, the game between the incumbent and the entrant is a Stackelberg game. We may consider two pure strategic perfect Bayesian equilibriums – pooling or separating equilibrium. In a pooling equilibrium, the incumbent places the same order quantity whether the demand state is
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high or low. Thus, although the supplier may leak the order information to the entrant, the entrant can’t distinguish the actual demand state. However, pooling equilibrium is different from NLE, under which the entrant has to guess the demand state to maximize his expected profit. Nevertheless, in pooling equilibrium, the retailers play a sequential-move game and the entrant places an optimal order when he observes the incumbent’s order quantity leaked by the supplier. When the supplier has identical WPC
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with the two retailers, Tian and Jiang (2016) illustrate that all pooling outcomes can be eliminated by the intuitive criterion of Cho and Kreps (1987). Kong et al. (2013) have shown that, under identical RSC with the two retailers, pooling equilibrium may not exist when demand variation is high and even when it exists the incumbent may prefer NLE. Similar observations can be made when the two retailers have different contracts with the supplier, i.e., there does not exist pooling equilibrium when demand variation is relatively high and it is feasible that the NLE replaces the pooling equilibrium in most scenarios. The
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detailed proofs are presented in Appendix D. Therefore, in this paper, we shall focus on separating equilibrium and analyze the existence of NLE.
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Under a separating equilibrium, the incumbent places different order quantity based on demand state. The entrant can infer the demand state from the incumbent’s order quantity leaked by the supplier. If the incumbent’s order is low enough, the entrant believes that the demand state is low. Otherwise, he believes
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that the demand state is high. As shown in Anand and Goyal (2009), the entrant’s belief is
0 if the supplier leaks and qi qiLS , Pr A H S e 1 if the supplier leaks and qi qiL .
Therefore, the high-type incumbent has an incentive to mimic a low-type order to induce the entrant to
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order less, which better benefits the incumbent. Thus, the separating equilibrium requires that (i) it is too costly for the high-type incumbent to pretend to be a low type and (ii) it is valuable for the low-type incumbent to place a low-type order. Similar as Kong et al. (2013), we use r H / L to measure the relative variation in demand. It can be easily shown that the entrant would not participate if
r 1 1 / p
(1)
does not hold. Therefore, we shall assume this condition in the rest of this section.
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The low-type incumbent’s order can be divided into two types depending on demand variations. Define
r 1 2 / [1 2 p 11 ] as a threshold to classify situations of high and low demand variations. More specifically, when r r , i.e., under the high demand variation, the high- and low-demand states are far apart and there is no cost for the low-type incumbent to separate out. If Equation (1) holds, it can be shown that 1/ 2 p 1 . When r r , i.e., under the low demand variation, considering the entrant’s
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belief, the incumbent has to place a small enough order to prevent the high-type incumbent from mimicking a low type. Lemma 1 below presents the optimal price and order quantities of all players under the high demand variation when r r . Note that the incumbent always places a positive order as long as the entrant participates in the leakage case. The proof is given in Appendix A.
Lemma 1 Suppose the supplier always leaks the incumbent’s order information to the entrant.
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1) For any given value of p , if 1/ 2 p 1 , we have r 1 1 / p . 2) If r r ,1 1 / p , the supplier’s optimal wholesale prices are
wrS 1 4 3 / 8 4
and
wwS / 2
for the incumbent and the entrant, respectively. (i)
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3) The order quantities of the incumbent and the entrant in the separating equilibrium are as follows: If the demand state is high, the incumbent orders
otherwise
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qiHS H H L 1 p 1 / 4 2 ,
If Pr A H 1 , the entrant orders e
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(ii)
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qiLS L H L 1 p / 4 2 .
S qeH H L 1 p H 1 / 8 4 ,
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otherwise
qeLS L 1 H L p / 8 4 .
The condition 1/ 2 p 1 ensures that the threshold r is always lower than the right-hand side of
condition (1) so that an analytical result can be obtained for the separating game. In practice, considering long-term cooperation and bargaining between the supplier and retailer, the revenue-sharing rate is often not too high. More specifically, when 1/ 3 , 1/ 2 p 1 always holds for any value of p .
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When r r , i.e., under the case of low demand variation, a closed form of the optimal wholesale price cannot be obtained. In practice, demand variation is often relatively high for products of short-life cycles, such as fashion apparel, electronics, etc. Thus, in this paper, we focus on the existence of NLE under high demand variation. With respect to low demand variation, we present a brief discussion in Section 4.4.
4.2. Benchmark Analysis under Information Non-leakage
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We now analyze the situation where the supplier never leaks incumbent’s order quantity to the entrant. Thus, the entrant can’t observe the incumbent’s order decision and the game between the incumbent and the entrant is a simultaneous-move game. In non-leakage case, because the entrant’s order quantity is the same in both demand states and the incumbent places a relatively small order in the low demand state, it is more difficult to guarantee the low-type incumbent’s participation. The incumbent would not participate in low demand state if
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r 1 1/ [2 p 1 ]
(2)
does not hold. Therefore, we shall assume the condition in the rest of this section. Under the non-leakage game, the incumbent will make order decision depending on the actual market demand state, implying that the incumbent’s order quantity truthfully reveals the demand state to the
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supplier. Because the entrant can’t acquire any information about market demand state, he has to decide order quantity depending on the mean demand. Lemma 2 presents the optimal prices and order quantities
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of the two retailers.
Lemma 2 Suppose the supplier never leaks the incumbent order quantity. The supplier’s optimal
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wholesale prices are
wrN 3 1 / 6 4 2
and
wwN / 2
The high-type incumbent will order
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(i)
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for the incumbent and the entrant, respectively. qiHN H 2 1 1 p H L / 6 4 ,
while the low-type incumbent will order
(ii)
qiLN L 2 p 1 H L / 6 4 ;
The entrant will order
qeN 1 / 6 4 .
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We now examine the dominance of non-leakage game over the separating game when r r from the supplier’s and the incumbent’s incentives with information non-leakage in the case of high demand variations. We shall first discuss the existence of the two games. When both games exist, we then consider the incumbent’s incentive with information non-leakage assuming leakage choice as an endogenous decision for the supplier. For the case of low demand variations, it will be discussed in Section 4.4.
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Equations (1) and (2) provide sufficient and necessary conditions for all players to participate in the separating and non-leakage games, respectively. Figure 1 illustrates four parameter regions for the existence of the separating and non-leakage games when p = 0.1 and 0.5, i.e., OS (only separating game exists), ON (only non-leakage game exists), BN (both games don’t exist), and B (both games exist). It is easy to see that when p increases, the participation regions for both games will get smaller. Figure 1 also
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depicts the area for low demand variations (LD).
BN OS
BN
ON
ON
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B
LD
OS
B
LD
In the parameter region OS, 0.2929 and r UBPN ,UBPS , where UBPS 1 1 / p and
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1)
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Figure 1 Participation Regions of Separating and Non-Leakage Games
UBPN 1 1/ [2 p 1 ] respectively correspond to the participation constraint under the separating
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game and the non-leakage game in Equations (1) and (2). When is small, the supplier cannot offer a very low wholesale price to encourage the low-type incumbent to increase the order quantity. In low demand state, the entrant would place a larger order depending on her own estimation for the average demand under non-leakage, compared with that under the separating game. Therefore the entrant would push the low-type incumbent out of business under the non-leakage game. Hence, the low-type incumbent has no incentive to participate in the non-leakage game, i.e., there only exists the separating game, 13
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2)
In the parameter region ON, 0.2929 and r max r ,UBPS ,UBPN . When is relatively high, the incumbent would place a big order due to the low wholesale price offered by the supplier, which results in that the entrant withdraws from the market under the separating game in low demand state. Hence, the low-type entrant wouldn’t participate the separating game, i.e., there only exists the nonleakage game In the parameter region BN, r max UBPS ,UBPN . Given the low demand L , when r is very high,
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3)
the mean demand is high too. Under the separating game, faced with high potential mean market demand, the retailers would order more and the supplier would raise the wholesale prices for the retailers. Thus, in low demand state, for any given , the incumbent would place a large enough order because of the relatively low wholesale price compared with the entrant and therefore push the
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low-type entrant out of business. On the other hand, under the non-leakage game, the entrant would place a big order based on the high average market demand, which results in that the low-type incumbent cannot make a profit and is out of the market. Hence when r is very high, the low-type incumbent and the low-type entrant wouldn’t participate the non-leakage game and the separating game, respectively, and the two games do not exist.
In the parameter region B, 1/ 2 p 1 and r r min UBPS ,UBPN . We shall focus on the
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4)
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parameter regions where the non-leakage game dominates over the separating game.
The following sufficient conditions guarantee that the NLE dominates the separating game in the parameter region B:
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a) The supplier would not leak information in both demand states, i.e., S N and sLS sLN . sH sH
(3)
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b) The incumbent would order qiHN and qiLN in corresponding demand states under NLE, i.e.,
iHS iHN and iLS iLN .
(4)
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c) Guarantee that the incumbent and the entrant participate in the market under NLE, i.e., qiHN 0 , qiLN 0 and qeN 0 .
Note that if iLS iLN , the incumbent may place an order deviating from the non-leakage game qiLN in
low demand state. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information, which benefits both incumbent and supplier. Because the supplier informs the entrant about the low-type incumbent’s order information, the entrant would reduce the order quantity compared with 14
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the non-leakage game one, resulting in that the market price rises and the profits of incumbent and supplier increase. Hence, the separating game dominates the non-leakage game and there does not exist NLE. In this paper, for ease of exposition, we present the sufficient condition b) to guarantee that the incumbent would not deviate under NLE. Appendices B and C present parameter regions of , p and r for the above conditions (3) and (4), respectively. After analyzing these conditions and considering the supplier’s and incumbent’s incentives
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with non-leakage, we have the following results.
Proposition 1 In parameter region B, NLE dominates the separating game only if the parameters of , p and r lie in the regions defined in Table 3. Otherwise, the separating dominates the NLE. Each subregion is determined by the upper and lower bounds on p and r given in Table 4.
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Table 3 Parameter Regions of NLE in Parameter Region B
0.6112 0.7469
pB
0.7469 0.7808
p B1p
p
1 p
max r , LB , LB 1 r
2 r
r
r min LB ,UB 3 r
S P
max r , LBr2 r UBPS
1 2 2 3 2 2 3 3 2 6 2 p 5 5 2 p 8 11 2 4 3 1 p
6 7 2 2 2 p 3 2 6 2 2 3 p 2 5 5 2 2 3
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3 2 2 1 p 5 5 2 2 3
LBr3 :
7 4 2 6 7 2 2 p 23 43 24 2 4 3 p 1 23 20 4 2
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LBr2 :
B1p :
ED
: 9 18 9 2 3 4 LBr1 :
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Table 4 Definitions of the Bounds on p and r
From Proposition 1, it is straightforward to prove the following corollary.
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Corollary 1 If p 0.32 , NLE does not exist.
When p is in the medium range, due to the large demand uncertainty, the supplier always has an
incentive to inform the entrant about the true demand state because she can benefit more by leaking information to induce that the entrant makes a correct order decision. When p is high, demand uncertainty is small but the high demand state is more likely. The incumbent always induces the supplier to leak
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information about the actual demand state to the entrant, which can avoid causing a loss to the incumbent or even the supply chain due to the entrant’s over order in low demand state. For illustration, Figure 2 shows the existence region of NLE in parameter region B when p 0.1 , where N and S denote the cases where the non-leakage game dominates the separating game/the
OS
BN ON S
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N
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separating game dominates the non-leakage game, respectively.
LD
Figure 2 Dominance Region of NLE for p 0.1
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In parameter region N, where 0.6112 0.7808 and r LBr2 , min LBr3 ,UBPS , NLE dominates the
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separating game. When is relatively high, the supplier’s profit becomes more consistent with the whole supply chain and hence the supplier would control the total order quantity in the supply chain to maximize her own profit. The supplier makes leakage or non-leakage decision based on the trade-off
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between total order quantity and market price. Compared with the total order quantity under information non-leakage, the retailers may together place a bigger order in high demand state and smaller order in low
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demand state under information leakage. If the supplier would not leak the incumbent’s order information to the entrant, the entrant has to place an intermediate order depending on own estimation for the market demand, which is beneficial to the supplier in both demand states under certain regions. Meanwhile, the
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incumbent would not deviate from NLE in both demand states because he can benefit more from information non-leakage by reducing sales volume and raising market price, compared with information leakage.
In parameter region S, the separating game dominates the NLE because the supplier has an incentive to
leak information or the incumbent may deviate from NLE. For example, when 0.2929 and r r , UBPN , the incumbent would have an incentive to deviate from the non-leakage game in low
demand state. The supplier acquiesces in the incumbent’s action and would leak information to the entrant. 16
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In fact, when is relatively low, the incumbent can reserve most of the sales revenue and hence would like to increase the order quantity to lift his own profit. Compared with the order under information nonleakage, the entrant will place a smaller order when he is informed that the actual demand state is low, which is beneficial to the low-type incumbent.
4.4 Dominance of NLE under Low Demand Variations
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Under low demand variations, i.e., r r , we now use some numerical examples to investigate the dominance of non-leakage game over the separating game. Assume that H 700 , L 500 , p 0.1 and
r 1.4 , the mean demand is pH 1 p L 520 . For any 0,1 , r r , i.e., the demand variation is low. To investigate the supplier’s and incumbent’s incentives of information non-leakage, we list the optimal wholesale price offered by the supplier and all parties’ profits under various in Table 5.
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Note that “N” and “S” denote the non-leakage and the separating game, respectively.
Table 5 Profits Under Various Values of for H 700 , L 500 , p 0.1 and r 1.4
0.1
0.3
Game
S
N
wrN
wwN
wrS
wwS
wrN
225.6
260
242.2
260
159.3
iH
30,093
20,013
eH
14,506
13,123
sH
70,917
88,761
iL
6,179
eL
6,148
260
N
S
N
S
wrS
wwS
wrN
wwN
wrS
wwS
wrN
wwN
wrS
wwS
171
260
69.3
260
75.2
260
6.5
260
8.5
260
19,446
21,986
17,053
9,404
7,801
12,576
10,433
8,538
5,476
2,419
506
76,638
92,473
88,638
99,715
110,207
109,636
5,170
6,769
5,913
7,230
6,616
4,271
3,868
6,204
4,992
5,136
2,760
2,999
253
469
44,988
44,782
49,171
48,690
57,507
57,160
27,535
ED
PT
CE 43,189
wwN
0.9
43,246
AC
sL
S
M
N
0.6
In this particular example, it is found that NLE dominates the separating game only if the revenue-
sharing rate is relatively high. When p is relatively low, the low demand state is more likely. On the other hand, when r is very low, the two demand states are very close, implying that the entrant is not easy to distinguish them. Therefore, in high demand state, in most cases where 0.1,0.3 and 0.6 , the supplier would always inform the entrant about the actual demand state, which can induce that the entrant orders more products. However, in case 0.9 , where is relatively high, the supplier’s profit is more in line
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with the whole supply chain and hence she would like to control the total order quantity in the supply chain to lift own profit. In this case, compared with information leakage, both the supplier and the incumbent can benefit more from the non-leakage game by reducing order and raising price. Besides, we demonstrate another two examples when p 0.5 and p 0.9 . We find that the supplier would always leak order information to the entrant and the separating game dominates the non-leakage
and p is relatively low, NLE may exist under low demand variations.
5 Impact of Different Contract Scenarios
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game. In short, similar as the results in the situation of high demand variations, only if is relatively high
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To further analyze the effect of different contract configurations on information leakage, we investigate the existence of NLE under Scenarios 2 and 3 and summarize the results under Scenario 4 studied in Kong et al. (2013).
Analogous to the analysis under Scenario 1, we define r 1 4 1 / 4 4 p 3 2 p and
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r 1 1/ 1 p for Scenarios 2 and 3, respectively, as the threshold to classify situations of high and low demand variations. We primarily analyze the incentive of supporting non-leakage from the perspective of supplier and incumbent when r r and further investigate the sustainability and existence
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of NLE.
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Proposition 2 For Scenarios 2 and 3, the supplier always leaks information in high demand state and the incumbent deviates from NLE in low demand state when r r in the corresponding scenario. Therefore
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NLE doesn’t exist when r r .
In Scenario 2, faced with the high demand state, the supplier always has an incentive to leak the high-
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demand signal to induce that the entrant increases the order quantity. When the supplier cooperates with the entrant under an RSC, the wholesale price offered by the supplier is relatively low and the risk from market uncertainty for the entrant diminishes. Hence the entrant is more likely to place a relatively big order under information non-leakage, resulting in that the incumbent’s profit reduces, especially in low demand state. Thus the low-type incumbent has an incentive to deviate from the non-leakage game. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information to inform the entrant of the actual demand state, inducing that the entrant’s order quantity decreases and the market price rises. 18
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Anand and Goyal (2009) have shown that the supplier always leaks information under exogenously fixed wholesale price in Scenario 3. They used only a numerical example to illustrate the impact of endogenous wholesale price on information and material flows and analyze how to set wholesale price when the supplier takes the incumbent’s informational imperative and double marginalization into account. Here we show that NLE doesn’t exist when the supplier chooses the optimal wholesale price. Kong et al. (2013) have demonstrated the existence of NLE in Scenario 4 when the wholesale price and
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revenue-sharing rate are appropriately given. In addition, they found that NLE remains existence when the supplier is allowed to choose an optimal wholesale price for any given value of . However, they only proved that NLE is a robust outcome by comparing the expected profit of the supplier and incumbent under the separating game and the non-leakage game, respectively. In this study, we shall investigate the region where NLE dominates the separating game under both demand states in the case of high demand
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variations. Similar as Scenario 1, we define r 1 4 / 4 4 p 2 p as the threshold to classify situations of high and low demand variations in Scenario 4. Figure 3 presents regions of BN, OS, and B in the case of high demand variations. The regions are found different from those in Figure 1. We shall discuss the dominance of the non-leakage game over the separating game in region B, which is shown in
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BN
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Proposition 3.
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OS
OS B
LD
LD
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B
BN
AC
Figure 3 Participation Regions of Separating and Non-Leakage Games under Scenario 4 Proposition 3 For Scenario 4, in parameter region B, where r r 1 1/ p 2 , both the separating game and the non-leakage game exist. NLE dominates the separating game only if the parameters of , p and r lie in the regions defined in Table 6. Otherwise, the separating dominates the NLE. Each sub-region is determined by the upper and lower bounds on p and r given in Table 7.
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Table 6 Parameter Regions of NLE in Parameter Region B
0.9775
p
pB
max r , LB
2 p
r
4 r
r LB
5 r
Table 7 Definitions of the Bounds on p and r 43 2
22
LBr5 :
2 1
LB : 1
2
4 r
6 6 12 7 2 3 4 5 2
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B p2 :
p 20 12 3 2 3
14 4 2 12 7 2 p 46 43 12 2 3 p 2 23 10 2
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Similar as Scenario 1, we can easily obtain the following corollary from Proposition 3.
BN
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Corollary 2 For scenario 4, if p 0.086 , NLE does not exist.
S LD
N
AC
CE
PT
ED
OS
Figure 6 Dominance Region of NLE for p 0.05 under Scenario 4
From Proposition 3 and Corollary 2, we can conclude that the separating game dominates the non-
leakage game in most cases. Only if is very high and p is very low, may there exist NLE. We illustrate an example in Figure 6 when p 0.05 . In parameter region N, where 0.9775 and max r , LBr4 r LBr5 , non-leakage game dominates over the separating game. Similar as the analysis in
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Scenario 1, when is very large, the supplier would control the total order quantity in the supply chain to lift profit. In this case, both the supplier and the incumbent obtain more profits under information nonleakage by reducing order quantity and increase market price. Finally, we have the following summary of the four scenarios.
Proposition 4 In presence of high demand variations, NLE may exist only if the supplier signs RSC with
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the incumbent (Scenarios 1 and 4). The dominance region of NLE shrinks when the supplier signs RSC with the entrant, compared with WPC.
Proposition 4 is interesting since the supplier’s choice of cooperation with the incumbent is the key to prevent information leakage. Under WPC, it is difficult to balance the profits of supplier and incumbent.
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Thus the supplier always has an incentive to leak information or the incumbent would deviate from NLE for her/his own sake, resulting in that NLE does not exist in a supply chain. If the supplier signs RSC with the incumbent, she would offer a relatively low wholesale price for each unit, compared with WPC. The incumbent thus places a higher order and needs to share a part of revenue with the supplier. The revenue is felicitously allocated between the supplier and incumbent, which prevents information leakage under an appropriately given revenue-sharing rate.
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Although the entrant plays a passive role in the supply chain, the contract type of the entrant has an important impact on the dominance region of NLE. Compared with the dominance region of NLE in
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Scenario 1, under Scenario 4, the entrant can purchase more products from the supplier under RSC, resulting in a decrease of the incumbent’s market share and profit. Therefore, the incumbent would induce the supplier to turn the non-leakage game into the separating game because the incumbent can better
PT
utilize the information and first-mover advantage under the separating game to improve the market share. As a result, due to RSC with the entrant, the dominance region of NLE diminishes, compared with WPC.
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As an extension, we also investigate the above comparison when both wholesale price and revenuesharing rate are endogenous. In this case, we can find that the supplier would offer a unique optimal
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revenue-sharing contract with 1 and w 0 . The specific result does not provide any valuable managerial insights because it shows that the supplier is powerful enough to effectively control the entire channel, which does not appear in practice.
6 Concluding Remarks In this paper, we investigate information leakage problem under different contract configurations where the supplier is allowed to optimally choose a wholesale price for given revenue-sharing rate. When the 21
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supplier signs RSC and WPC respectively with the incumbent and entrant retailers (Scenario 1), NLE regions are found analytically, where the supplier would not leak information and the incumbent retailer would not deviate from NLE in the situations of high demand variation. It is found that NLE exists more likely when the revenue-sharing rate is relatively high and the probability of high demand state is relatively low. By further investigating the existence of NLE when the above asymmetric contract types are reversed and following the results in Anand and Goyal (2009) and Kong et al. (2013), we show that
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NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract. Otherwise, the supplier would always leak information to the entrant and NLE doesn’t exist. In other words, the supplier’s coordination with the incumbent is the key to prevent information leakage. When the supplier has a revenue-sharing contract with the incumbent, the revenue is felicitously allocated between the supplier and the incumbent under an appropriately given revenue-sharing rate, which
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prevents information from being leaked.
There are several directions for future research. It would be interesting to investigate information leakage problems under different contracts when the products are imperfect substitutable or complementary. We can also investigate the effect of player’s risk preference on information leakage or non-leakage. Another interesting extension is to study how the collusion between the supplier and the
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Acknowledgments
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incumbent affects information leakage and the entrant’s order decision.
The authors thank the area editor, the associate editor and three anonymous referees for their
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constructive suggestions and comments that improved the paper. Feng’s research was partially supported by National Natural Science Foundation of China [Grant 71572145]. Jiang’s research
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was partially supported by National Natural Science Foundation of China [Grants 71831006,
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71531010 and 71325003].
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Appendices Appendix A Proof of Lemmas 1 and 2 Proof of Lemma 1: To obtain a pure separating equilibrium, the incumbent has to satisfy the following constrained optimization program: iLS max 1 qiLS ( L qiLS qeLS (qiLS )) wrS qiLS
(A-1)
S S S iHS max 1 qiHS ( H qiHS qeH (qiH )) wrS qiH
(A-2)
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qiL
qiH
Subject to：
(A-3)
1 qiLS (H qiLS qeLS (qiLS )) wrS qiLS 1 qiHS (H qiHS qeHS (qiHS )) wrS qiHS
(A-4)
S S S qeLS (qiLS ) argmax( L qiLS qeL )qeL wwS qeL ( L qiLS wwS ) / 2
(A-5)
S S S S S S S qeH (qiH ) argmax( H qiH qeH )qeH wwS qeH ( H qiH wwS ) / 2
(A-6)
where
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1 qiHS (L qiHS qeHS (qiHS )) wrS qiHS 1 qiLS (L qiLS qeLS (qiLS )) wrS qiLS
S qeL
S qeH
Inequalities (A-3) and (A-4) are the incentive compatibility constraints which ensure that each
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type of incumbent has no incentive to mimic the other. Formula (A-5) and (A-6) are the entrant’s optimal quantity when the entrant accepts the information being leaked by the supplier. i) In low demand state, in order to prevent the high-type incumbent from mimicking low-type
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incumbent, the optimization program for the low-type incumbent reduces to: iLS max(1 )qiLS ( L qiSL wwS 2wrS / (1 )) / 2 S qiL
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s.t. (1 )qiLS ( H qiLS ( L qiLS wwS ) / 2 wrS / (1 )) ((1 )( H wwS ) 2wrS )2 / (8(1 )) (A-7) The Lagrangian for the above formulation is
AC
CE
(1 )qiLS ( L qiLS wwS 2wrS / (1 )) / 2 L(qiLS , ) max S S S S S S S S 2 qiL ((1 )qiL ( H qiL ( L qiL ww ) / 2 wr / (1 )) ((1 )( H ww ) 2wr ) / (8(1 )))
The first-order Karush-Kuhn-Tucker (KKT) conditions for the Lagrangian are:
1a) :
L L(qiLS , ) wwS wwS wrS wrS L 0 1 q 1 H q v1 0 iL iL qiSL 2 1 2 2 1 2
1b) :
L(qiLS , ) S qiL 0 qiLS
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2a ) :
L(qiLS , ) ((1 )( H wwS ) 2wrS ) 2 wS L q S wS 0 (1 )qiLS ( H qiSL ( iL w ) r ) v2 2 2 2 1 8(1 )
2b) :
L(qiLS , ) 0
Solve the above system and we get:
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1) For 0 (unconstrained optimal), qiLS L 2wrS / 1 wwS / 2 . Thus, by (A-5), we get qeLS L 2wrS / 1 3wwS / 4 . In the following part, we prove that the high-type incumbent has
no incentive to mimic the low-type incumbent. Thus, the order quantities of incumbent and
entrant in the high demand state are as follows: qiHS H 2wrS / 1 wwS / 2 ,
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S qeH H 2wrS / 1 3wwS / 4 .
We have the supplier’s profit in high and low demand state as follows: S S sH qiHS H qiHS qeH wrS qiHS wwS qeHS ,
sLS qiLS L qiLS qeLS wrS qiLS wwS qeLS .
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Then, we have the supplier’s expected profit as follows:
S E sS p sH 1 p sLS .
E sS 0 wrS
E sS 0 , and get the optimal wholesale price offered by the supplier: wwS
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and
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We solve the following first-order conditions under a given simultaneously, i.e.,
wrS Hp L 1 p 1 4 3 / 8 4 , wwS Hp L 1 p / 2 .
(A-8)
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By the KKT conditions, we get v2 0 H L 1 H 3L 2wwS 4wrS 0 . Substitute (A-8)
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into the above inequality, we get r 1 2 / 2 2 p 1 . Thus, the incumbent and entrant order qiLS L H L 1 p / 4 2 and qeLS L 1 H L p / 8 4 in low demand state when r 1 2 / 2 2 p 1 , respectively. To guarantee the low-type entrant’s participation, r 1 1 / p must hold.
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2) For 0 , we have qiLS 2H L 2wrS / 1 wwS H L 3H L 2wwS 4wrS / 1 / 2 . Denote the lower root qiLS 1 and the upper root qiLS 2 . For the constraint (A-7), when qiL 2H L 2wrS / 1 wwS / 2 , LHS of constraint (7) get the maxima which is bigger than
RHS of constraint (A-7). Because qiLS1 2H L 2wrS / 1 wwS / 2 qiSL 2 , we know that the
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LHS of constraint (A-7) is increasing for any qiLS qiLS1 , 2H L 2wrS / 1 wwS / 2 . Therefore, in order to prevent the high-type incumbent from mimicking the low type, the incumbent has to place an order qiLS 1 to realize the pure separating equilibrium when r 1 2 / 2 2 p 1 . However, we can’t get the closed form of the optimal wholesale
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price offered by the supplier in this case and still use wrS and wwS to denote the wholesale price under RSC and WPC, respectively. Nonetheless, we give the complete proof to obtain the pure separating equilibrium.
ii) Then we need to check whether the low-type incumbent places an order which deviates from the low-type order in the separating equilibrium. When the demand state is low, if the
belief.
Thus,
the
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incumbent places an order qiL* qiLS , the entrant thinks that the demand state is high based on own entrant’s
order
quantity
is:
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* qeL (qiL ) argmax( H qiL qeL )qeL wwS qeL ( H qiL wwS ) / 2 . The incumbent’s order quantity is: qeL
* qiL qiL
(1)
When
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* qiL* argmax(1 )( L qiL qeL (qiL ) wrS / (1 ))qiL [2 L H wwS 2wrS / (1 )] / 2 .
r 1 2 / 2 2 p 1 ,
substitute
(A-8)
into
qiL*
and
we
get
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qiL* L 3 p 1 H 2 p 1 / 4 2 . However, qiL* 0 always holds in this
case. Thus, the low-type incumbent has no incentive to place the high-type order.
AC
(2) When r 1 2 / 2 2 p 1 , for all wwS and wrS , we have the following proof: When the low-type order of incumbent doesn’t deviate, the low-type incumbent’s order
quantity is qiLS 2H L 2wrS / 1 wwS H L 3H L 2wwS 4wrS / 1 / 2 , which is
positive if L 2wrS / 1 wwS . Define 1 H 2wrS / 1 wwS / L 2wrS / 1 wwS .
Only when 1 2 , qiL* 0 and the incumbent’s profit iL is 1 H 2L wwS 2wrS
2
/ 8 8 . 28
ACCEPTED MANUSCRIPT
Thus, compare iLS iL when 1 2 and we have following inequality:
1 1 2 21 1 1 31 1 0
Since 1 1 , it is sufficient to verify that 2 21 1 1 31 1 0 . The inequality holds for all 1 2 . Therefore, the low type always separates out. sum
up,
the
incumbent’s
order
quantity
in
high
demand
state
CR IP T
To
is
qiHS H 2wrS / 1 wwS / 2 for all r because the low type separates out. In low demand
state, the incumbent orders qiLS L 2wrS / 1 wwS / 2 when r 1 2 / 2 2 p 1 . the
qiLS 2H L 2wrS / 1 wwS
incumbent
orders
H L 3H L 2wwS 4wrS / 1 / 2 .
AN US
Otherwise,
Proof of Lemma 2: If the supplier does not leak information, the game between the incumbent and the entrant is a simultaneous move game. We have the following profit function of the retailers: qiH
M
iHN max(1 )qiHN ( H qiHN qeN ) wrN qiHN ,
ED
iLN max(1 )qiLN ( L qiLN qeN ) wrN qiLN , qiL
eN max( p( H qiHN qeN )qeN (1 p)( L qiLN qeN )qeN ) wwN qeN qe
PT
Solve the first-order conditions of the three equations simultaneously, and we get: The order quantities of the high-type incumbent, the low-type incumbent and the entrant
CE
respectively are:
qiHN H 3 p L 1 p 4wrN / 1 2wwN / 6 , qiLN L 2 p Hp 4wrN / 1 2wwN / 6 ,
AC
qeN Hp L 1 p wrN / 1 2wwN / 3 .
We have the supplier’s profit in high and low demand state as follows: N sH qiHN H qiHN qeN wrN qiHN wwN qeN ,
sLN qiLN L qiLN qeN wrN qiLN wwN qeN .
Then, we have the supplier’s expected profit as follows:
29
ACCEPTED MANUSCRIPT
N E sN p sH 1 p sLN .
We solve the following first-order conditions under a given simultaneously, i.e., and
E sN 0 wrN
E sN 0 , and get the optimal wholesale price offered by the supplier: wwN
wrN 3 Hp L 1 p 1 / 6 4 , wwN Hp L 1 p / 2 .
CR IP T
2
Consider the participation constraint ( qiHN 0 , qiLN 0 and qeN 0 ) and we get:
These three inequalities simultaneously hold when r 1 1/ 2 p 2 p , which guarantees the
AC
CE
PT
ED
M
AN US
incumbent’s and entrant’s participation.
30
ACCEPTED MANUSCRIPT
Appendix B Supplier’s Incentives of Non-leakage S N The supplier would not leak information in both demand states if sH and sLS sLN holds. sH
Thus, the non-leakage is efficacious and the entrant can’t observe the incumbent’s order information to aid decision making. We summarize the results and show the non-leakage region
CR IP T
supported by the supplier in both demand states under Scenario 1.
Proposition B In parameter region B, the supplier would not leak information in both demand states if the parameters of , p and r lie in the region defined in Table B1. Additional notations on p and r are shown in Table B2.
Table B1 Non-leakage Region Supported by the Supplier in Parameter Region B 0.2929
p
Subregions i
r LBr2 r UBPN
B3p p Bp7
LBr1 r UBPN
p B3p
LBr2 r UBPS
B3p p Bp4
LBr1 r UBPS
p B5p
LBr2 r UBPS
ii
B5p p Bp6
r r UBPS
iii
Bp6 p Bp4
LBr1 r UBPS
i
p B5p
LBr2 r UBPS
ii
B5p p B1p
r r UBPS
AN US p B3p
ii 0.2929 0.7325
i ii
0.7325 0.7469
M
i
PT
ED
0.7469 0.7808
Table B2 Addition Notations on p and r
1 B : 1 3 9 9 3
CE
3 p
5 2 2 3 2 4 2 2
AC
B p6 :
B9p :
/ 2
Bp4 :
Bp7 :
16 9 2 6 2 10
6 r
LB :
3 4 2 2 3 4 4 3 2 2 3
B10 p :
4 4
1 p 1 5
6 7 2
2 2
2 3 2 3
B8p :
1 2 4 2 2
43 2 2
4 4
1
2 2 6 7 2 2 p 23 43 24 2 4 3 2
2 1
2 1 2
p 1 23 20 4 2
B5p :
2
2 p 30 77 71
2
28 4 4 3
31
ACCEPTED MANUSCRIPT
From Proposition B, we have the following conclusions regarding non-leakage regions supported by the supplier: 1) If 0.2929 , the supplier would not leak information under the two following subregions: p B3p and LBr2 r UBPN ; B3p p Bp7 and LBr1 r UBPN .
2) If 0.2929 0.7808 , the supplier would not leak information only when the parameters
CR IP T
of p and r lie in the subregions shown in Table B1. Proof:
1) We have the following supplier’s profit under information leakage or non-leakage in high demand state in parameter region B:
S S sH qiHS H qiHS qeH wrS qiHS wwS qeHS , sHN qiHN H qiHN qeN wrN qiHN wwN qeN .
AN US
S N By the Lemma 1 and 2, sH can be rewritten as sH
3 3 H 2 p L 1 p Hp 1 p L H 2 / 8 2 H 3 2 p 3 7 4 H p L 1 p 1 3 2 / 12 8 2
2
2
2
2
2
2
2
2
HL 1 p 1 3 4 p 4 2
M
Simplify it by r H / L and we can obtain
3 3 r 2 p 1 p rp 1 p r 2 / 8 2 r 3 2 p 3 7 4 r p 1 p 1 3 2 / 12 8
ED
2
2
2
r 1 p 1 3 4 p 4 2
2
2
2
2
2
(B-1)
PT
From Lemma 1, we can express the parameter region B by using , p and r :
CE
p min 1 / 2 ,1 and r r min UBPS ,UBPN for any given . Then we further compare the
participation constraint under non-leakage with that under leakage and get: i) When 0.2929 , UBPN UBPS always holds for any p . We further verify that r UBPN always
AC
holds for any and p . If p min 1 / 2 ,1 and r r ,UBPN , we solve the inequality (B1).
ii) When 0.2929 , UBPN UBPS always holds for any p . If p min 1 / 2 ,1 and
r r ,UBPS , we solve the inequality (B-1).
32
ACCEPTED MANUSCRIPT
Combine i) and ii), and we can obtain the non-leakage region supported by the supplier in high
CR IP T
demand state in parameter region B, shown in Table B3.
Table B3 Non-leakage Region Supported by the Supplier in High Demand State in Parameter Region B 0.2929
p
Subregions
0.6052 0.7469
i ii
0.7469
r LBr1 r UBPN
p Bp4
LBr1 r UBPS
p Bp6
r r UBPS
Bp6 p Bp4
LBr1 r UBPS
p B1p
r r UBPS
AN US
0.2929 0.6052
p Bp7
2) We have the following supplier’s profit under information leakage or non-leakage in low demand state in parameter region B:
M
sLS qiLS L qiLS qeLS wrS qiLS wwS qeLS , sLN qiLN L qiLN qeN wrN qiLN wwN qeN .
ED
By the Lemma 1 and 2, sLS sLN can be rewritten as
3 L 3 3 p H L 2
2
PT
8 2
2
2
L2 2 p p H 2 p 2 L2 p 2 1 3 2 2 HLp 1 1 p 2 4
12 8
CE
Simplify it by r H / L and we can obtain
AC
3 3 3 p r 1 2
2
2
8 2
2 p p r 2 p 2 p 2 1 3 2 2 rp 1 1 p 2 4
12 8
(B-2)
As the similar analysis in high demand state, we can obtain the non-leakage region supported
by the supplier in low demand state in parameter region B, shown in Table B4.
Table B4 Non-leakage Region Supported by the Supplier in Low Demand State in Parameter Region B
Subregions
p
r 33
ACCEPTED MANUSCRIPT
0.2929 1/ 3 1/ 3 0.7808
i
p B5p
LBr2 r UBPN
ii
p B5p
r r UBPN
i
p B5p
LBr2 r UBPS
ii
p B5p
r r UBPS
i
p B5p
LBr2 r UBPS
ii
B5p p B1p
r r UBPS
CR IP T
0.2929
Combine Table B3 and Table B4, and we can obtain the non-leakage region supported by the
AC
CE
PT
ED
M
AN US
supplier in both demand states in parameter region B.
34
ACCEPTED MANUSCRIPT
Appendix C Incumbent’s Incentives of Non-leakage The incumbent would order qiHN and qiLN in corresponding demand states under NLE if iHS iHN and iLS iLN holds. Otherwise, the incumbent may place an order deviating from NLE. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information, which is beneficial for the incumbent and supplier. Hence, the separating game dominates the
CR IP T
non-leakage game and there does not exist NLE. Considering the sufficient condition, we have the following results to ensure that the incumbent would not deviate from NLE in both demand states under Scenario 1.
Proposition C In parameter region B, the incumbent would not deviate from NLE in both
AN US
demand states if the parameters of , p and r lie in the region defined in Table C1.
Table C1 Non-leakage Region Supported by the Incumbent in Parameter Region B 0.2929 0.6739
p
8 p
r r r LBr3
p B1p
r r UBPS
pB
M
0.6739
From Proposition C, we have the following conclusions regarding non-leakage regions supported
ED
by the incumbent:
1) If 0.2929 , the incumbent would place an order deviating from NLE in low demand state. The supplier would acquiesce in the incumbent’s action and leak information. Thus,
PT
information non-leakage does not work in both demand states. 2) If 0.2929 , the incumbent would not deviate from NLE only when the parameters of
Proof:
CE
, p and r lie in the subregions shown in Table C1.
AC
1) We have the following incumbent’s profit under information leakage or non-leakage in high
demand state in parameter region B: S iHS 1 H qiHS qeH qiHS wrS qiHS , iHN 1 H qiHN qeN qiHN wrN qiHN .
By the Lemma 1 and 2, iHS iHN can be rewritten as
35
ACCEPTED MANUSCRIPT
1 H 2 p p L 1 p 1 2 8 2
1 H 3 2 p 2 p 2 2L 1 p 1 2 4 3 2
2
2
Simplify it by r H / L and we can obtain
r 2 p p 1 p 1 2
2
2
2 r 3 2 p 2 p 2 2 1 p 1
3 2
2
2
(C-1)
CR IP T
As the similar analysis in Appendix B, we can obtain the non-leakage region supported by the incumbent in high demand state in parameter region B, shown in Table C2. Addition notation on p and r are shown in Table B2.
Table C2 Non-leakage Region Supported by the Incumbent in High Demand State in Parameter Region B
p
Subregions i
pB
ii
B p B10 p
LBr6 r UBPN
p 0,1
r r UBPS
p B1p
r r UBPS
r r r UBPN
AN US
0.2929
9 p
9 p
0.2929 1/ 3
M
1/ 3
2) We have the following incumbent’s profit under information leakage or non-leakage in low demand state in parameter region B:
ED
iLS 1 L qiLS qeLS qiLS wrS qiLS , iLN 1 L qiLN qeN qiLN wrN qiLN .
PT
By the Lemma 1 and 2, iLS iLN can be rewritten as
CE
1 Hp 1 L 1 p p 2 8 2
2
1 2Hp 1 L 1 2 p 1 2 4 3 2
2
AC
Simplify it by r H / L and we can obtain
rp 1 1 p p 2
2
2
2 2rp 1 1 2 p 1
3 2
2
2
(C-2)
As the similar analysis in Appendix B, we can obtain the non-leakage region supported by the
incumbent in low demand state in parameter region B, shown in Table C3. Table C3 Non-leakage Region Supported by the Incumbent in Low Demand State in Parameter Region B
p
r 36
ACCEPTED MANUSCRIPT
0.2929 0.6739
p B8p
r r LBr3
0.6739
p B1p
r r UBPS
Combine Table C2 and Table C3, and we can obtain the non-leakage region supported by the incumbent in both demand states in parameter region B. Combine Table B1 and C1, and we can
CR IP T
obtain the region of NLE in parameter region B under Scenario 1. As similar argument above,
AC
CE
PT
ED
M
AN US
we can prove Proposition 2 and 3.
37
ACCEPTED MANUSCRIPT
Appendix D Pooling Equilibrium In this Appendix, we summarize the results of the pooling equilibrium under Scenario 1 and show that the pooling equilibrium does not exist or are dominated by other equilibrium in most cases. Therefore, we omit the pooling equilibrium from our main analysis. Firstly, we analyze
Benchmark: Pooling Case where Supplier Always Leak Proposition
D
Suppose
that
the
supplier
always
r r P 20 p 2 15 8
CR IP T
the benchmark case of pooling equilibrium where the supplier always leaks.
leaks
information.
When
AN US
p 2 2 11 2 1 p 32 20 3 2 p 8 8 3 2 / 8 p 8 16 4 p 2 2 11 , a
Pareto-dominant pooling equilibrium exists and is as follows: (i) The supplier’s optimal wholesale prices are
wrP 1 4L 3 / 8 4 and wwP / 2
For the incumbent and the entrant, respectively.
M
(ii) The incumbent orders qiP 2L 1 1 2 / 4 2 .
(iii) The entrant orders qeP 1 3 2L / 8 4 , consistent with his beliefs that:
H 1 3H 2wwP 4wrP / 1 2wrP / 1
CE
q P 2H
PT
ED
0, if the supplier leaks and qiP q P , Pre ( A H ) p, if the supplier leaks and q P qiP qiP and q P qPiL min qiP , where 1, if the supplier leaks and qiP q P
AC
2 1 L wwP 2wrP 1 P P 1 L ww 2wr 2 1 4 1 qiP 1 2 L qiP wwP 2wrP
/ 2 and q
P
iL min
q P i
.
Proof:
The proof is through a series of Lemmas. qiP denotes a candidate pooling quantity for the incumbent.
Lemma D1 The maximum quantity where a pooling equilibrium can be sustained is 38
ACCEPTED MANUSCRIPT
qiP L 1 p Hp wwP / 2 wrP / 1
Proof. The entrant orders qeP where
qeP qiP arg max p H qiP qeP wwP qeP 1 p L qiP qeP wwP qeP qiP wwP / 2 (D-1) P qe
The high-type incumbent maximizes
qiP
The low-type incumbent maximizes
CR IP T
iHP max 1 H qiP qeP qiP qiP wrP qiP
iLP max 1 L qiP qeP qiP qiP wrP qiP qiP
(D-2)
(D-3)
From (D-1), (D-2) and (D-3), we obtain the optimal quantity for the high-type and the low-type incumbent as follows:
AN US
qiPH H 2 p L 1 p wwP / 2 wrP / 1 ; qiLP L 1 p Hp wwP / 2 wrP / 1
We now show that any candidate quantity qiP for a pooling equilibrium must satisfy:
M
qiP min qiLP , qiHP qiLP L 1 p Hp wwP / 2 wrP / 1 qiP .
The high-type incumbent prefers to pool as long as his profits under pooling are higher than the profits he would obtain by ordering a high enough quantity to reveal his type. That is,
ED
1 H qiP qeP qiP qiP wrP qiP qmax 1 H qiH qeH qiH qiH wrP qiH , q iH
(D-4)
P i
qeH qiH arg max H qiH qeH wwP qeH H qiH wwP / 2 .
PT
where
qeH
Simplify inequality (D-4):
CE
1 H qiP wwP / 2 qiP wrP qiP 1 H wwP 2wrP / 8 8 2
Solve it and we get,
H 1 3H 2wwP 4wrP / 1 2wrP / 1
AC
qiP 2 H
2H
H 1 3H 2wwP 4wrP / 1 2wrP / 1
/ 2,
/ 2
Thus, the lowest pooling quantity for the high-type incumbent is
qPiH min 2 H
H 1 3H 2wwP 4wrP / 1 2wrP / 1
/2 .
Similarly, for the low-type incumbent, we verify 39
ACCEPTED MANUSCRIPT
1 L qiP qeP qiP qiP wrP qiP qmax 1 L qiL qeL qiL qiL wrP qiL q iL
P i
(D-5)
qeL qiL arg max H qiL qeL wwP qeL H qiL wwP / 2 .
where
qeL
Simplify inequality (D-5):
2wP 1 H 2 L wP 2 r w wwP wrP H q w P if L 0 qiL 0 P P 1 L qi wr qi 8 1 2 1 2 2 0 otherwise P P w w H Thus, we only need to check for L r w 0 . Solve it and we obtain 2 1 2
2L
H 1 4 L H 2wwP 4wrP / 1 2wrP / 1 H 1 4 L H 2wwP 4wrP / 1 2wrP / 1
AN US
qiP 2 L
P w
CR IP T
P i
/ 2,
/ 2
Thus, the lowest pooling quantity for the low-type incumbent is
qPiL min 2 L
H 1 4L H 2wwP 4wrP / 1 2wrP / 1
/ 2.
M
Therefore, we can obtain the following corollary from the above analysis.
ED
Corollary D1 There is no pooling equilibria below max qPiH min , qPiL min qPiH min q P .
In order to hold the pooling equilibrium, we also need to guarantee that the incumbent has no
PT
incentive to profitably deviate to a low enough order quantity to be considered as the low type. Firstly, the profit of high-type incumbent under pooling should be higher than its profit from
CE
ordering a low enough quantity to reveal the low type:
1 H qiP wwP / 2 qiP wrP qiP qmax 1 H qiH L qiH wwP / 2 qiH wrP qiH (D-6) q P
AC
iH
Secondly, the profit of low-type incumbent under pooling should be higher than its profit from ordering a low enough quantity to reveal the low type:
1 L qiP wwP / 2 qiP wrP qiP qmax 1 L qiL L qiL wwP / 2 qiL wrP qiL q P
(D-7)
iL
Solve constraint (D-6) step by step: the constraint (D-6) is equivalent to solve:
40
ACCEPTED MANUSCRIPT
1 2H wwP qiP qiP 2wrP qiP qmax 1 2H L wwP qiH qiH 2wrP qiH q
(D-8)
P
iH
We examine the two cases for q P . Suppose q P 2H L 2wrP / 1 wwP / 2 . Then,
maxP 1 2 H L wwP qiH qiH 2wrP qiH 1 2 H L wwP 2wrP
qiH q
1 2H w 2w P w
P r
2
2
/ 4 4
/ 4 4 1 2H wwP qiP qiP 2wrP qiP , which contradicts (D-
CR IP T
8). Thus, we must have q P 2H L 2wrP / 1 wwP / 2 . Given q P 2H L 2wrP / 1 wwP / 2 ,
max 1 2H L wwP qiH qiH 2wrP qiH 1 2H L wwP q P q P 2wrP q P . That is, we need
qiH q P
AN US
to solve 1 2H wwP qiP qiP 2wrP qiP 1 2H L wwP q P q P 2wrP q P . Solve the inequality and we obtain
2 1 2 H L wwP 2wrP 1 P P q 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
P
2 1 2 H L wwP 2wrP 1 P P q 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
M
ED
P
or
.
Combining with the condition q P 2H L 2wrP / 1 wwP / 2 , we get q P qPiH min qiP 2 1 2 H L wwP 2wrP 1 P P 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
, which is
CE
PT
smaller than 2H L 2wrP / 1 wwP / 2 .
AC
Next we solve constraint (D-7), which is equivalent to solve:
1 2L wwP qiP qiP 2wrP qiP qmax 1 L wwP qiL qiL 2wrP qiL q P
(D-9)
iL
We examine the two cases for q P . Suppose q P L 2wrP / 1 wwP / 2 . Then,
maxP 1 L wwP qiL qiL 2wrP qiL 1 L wwP 2wrP
qiL q
2
/ 4 4
41
ACCEPTED MANUSCRIPT
1 2L w 2w P w
P r
2
/ 4 4 1 2L wwP qiP qiP 2wrP qiP , which contradicts (D-
9). Thus, we must have q P L 2wrP / 1 wwP / 2 . Given q P L 2wrP / 1 wwP / 2 ,
max 1 L wwP qiL qiL 2wrP qiL 1 L wwP q P q P 2wrP q P . That is, we need to solve
qiL q P
CR IP T
1 2L wwP qiP qiP 2wrP qiP 1 L wwP q P q P 2wrP q P . Solve this inequality and we obtain qP
1 P P 1 L ww 2wr 2 1
qP
1 P P 1 L ww 2wr 2 1
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP or
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP .
AN US
Combining with the condition q P L 2wrP / 1 wwP / 2 , we get q P qPiL min qiP 1 P P 1 L ww 2wr 2 1
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP ,
M
which is smaller than L 2wrP / 1 wwP / 2 . Lemma D2 qPiL min qiP qPiH min qiP qiP .
ED
Proof. First, we show qPiL min qiP qPiH min qiP . Substituting the expressions for qPiL min qiP and
1 2H L w 2w P w
P r
CE
G1
PT
qPiH min qiP , we obtain qPiL min qiP qPiH min qiP
G2 2 1 H L
2
G1 G2 , where 2 1
4 1 qiP 1 2 H qiP wwP 2wrP
1 L w 2w P w
P r
2
and
4 1 qiP 1 2L qiP wwP 2wrP .
AC
Note that G12 G22 4 1 H L g1 g2 , where g1 1 L wwP 2wrP 2 1 qiP and g2
1 L w 2w P w
P r
2
4 1 qiP 1 2 L qiP wwP 2wrP . Note that
g12 g22 4 1 qiP L 0 . Since g 2 0 , we obtain g1 g 2 , implying G1 G2 . Thus, we 2
obtain qPiL min qiP qPiH min qiP .
42
ACCEPTED MANUSCRIPT
Next we show qPiH min qiP qiP . Substituting the expression for qPiH min qiP , we obtain qPiH min qiP qiP
F2
F1 F2 , where F1 1 2H L wwP 2wrP 2 1 qiP and 2 1
1 2H L w 2w P w
P r
2
4 1 qiP 1 2 H qiP wwP 2wrP . Note that
F12 F22 4 1 qiP L 0 . Since F2 0 , we obtain F1 F2 , implying qPiH min qiP qiP .
CR IP T
2
Therefore, we obtain the following corollary from the above analysis.
Corollary D2 For a pooling-equilibrium quantity, the low threshold in the entrant’s belief system must satisfy q P qPiL min qiP .
AN US
Lemma D3 When r r P , the pooling equilibrium exists.
Proof. From Lemma D1, in a Pareto-dominant pooling equilibrium, the incumbent and entrant respectively orders
qiP L 1 p Hp wwP / 2 wrP / 1 , qeP L 3 p H L 3wwP / 4 wrP / 2 2 . (D-10)
M
We have the supplier’s profit in high and low demand state as follows: P sH qiP H qiP qeP wrP qiP wwP qeP ,
ED
sLP qiP L qiP qeP wrP qiP wwP qeP .
PT
Then, we have the supplier’s expected profit as follows: P E sP p sH 1 p sLP .
E sP 0 wrP
E sP 0 , and get the optimal wholesale price offered by the supplier: wwP
AC
and
CE
We solve the following first-order conditions under a given simultaneously, i.e.,
wrP 1 4L 3 / 8 4 , wwP / 2 .
(D-11)
Substituting (D-11) into (D-10), we obtain qiP 2L 1 1 2 / 4 2 , qeP 1 3 2L / 8 4 .
From Corollary D1, the pooling equilibrium exists if q P qiP . Substituting (D-11) into q P , we solve the following inequality q P qiP and obtain r r P .
43
ACCEPTED MANUSCRIPT
Hence, from Lemma D1-D3 and Corollary D1-D2, we can obtain the Proposition D.
Justification for Ignoring Pooling We further discuss the existence of the pooling equilibrium and find that it is unnecessary to consider the pooling equilibrium in most scenarios, for the following reasons:
CR IP T
Firstly, from the Proposition D, a pooling equilibrium does not exist if r r P because it is costly for the high-type incumbent to pool with the low type. The condition r r P includes the many interesting cases.
Secondly, if a pooling equilibrium is possible (when r r P ), we compare the supplier’s
preference for the pooling and non-leakage cases. We find that the supplier would support the
rr P s
AN US
pooling equilibrium when
5 21 18 2 4 3 p 1 6 2 4 3 2 6 13 9 2 2 3 p 2 3 p 1 6 2 4 3 2 6 7 2 2
.
From the above analysis, the pooling case may exist if r min r P , rsP . It’s clear from Figure
M
D1 that the region of pooling case is very small. Therefore, although the pooling equilibrium strictly shrinks the region of NLE, we believe that we cover the majority of practical situations to
ED
analyze the existence of NLE in the paper. Note that if we further investigate preference of the high- and low-type incumbent for the pooling and non-leakage cases, the region of pooling case will become smaller. In addition, if necessary, pooling can be prevented by adding a constraint
PT
qiP q P in Proposition 1. Hence, in this paper, we ignore the pooling equilibrium and focus on
CE
the separating equilibrium when there exists information leakage.
Region where
Pooling cannot
be ignored
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Figure D1 The
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Information Leakage and Supply Chain Contracts Hao Liu , Wei Jiang , Gengzhong Feng , Kwai-Sang Chin PII: DOI: Reference:
S0305-0483(17)30659-X https://doi.org/10.1016/j.omega.2018.11.003 OME 1994
To appear in:
Omega
Received date: Accepted date:
8 July 2017 2 November 2018
Please cite this article as: Hao Liu , Wei Jiang , Gengzhong Feng , Kwai-Sang Chin , Information Leakage and Supply Chain Contracts, Omega (2018), doi: https://doi.org/10.1016/j.omega.2018.11.003
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Highlights We investigate information leakage under different contracts (WPC or RSC) with different retailers.
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When the supplier signs RSC and WPC respectively with the incumbent and entrant, there exists the non-leakage equilibrium (NLE) under certain parameter regions. When the supplier signs WPC and RSC respectively with the incumbent and entrant, NLE does not exist in high demand variation. Following the results in Anand and Goyal (2009) and Kong et al. (2013), we summarize that in case of high demand variations, NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract.
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Information Leakage and Supply Chain Contracts Hao Liu1,3, Wei Jiang2, Gengzhong Feng1, Kwai-Sang Chin3 1 2
Xi’an Jiaotong University
Dongbei University of Finance and Economics City University of Hong Kong
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Abstract: This paper investigates information leakage under different contract configurations in a supply chain. We consider two competing retailers, one of whom (incumbent) has private information about market demand while the other (entrant) has no access to acquire any market demand information. We assume that the supplier is the leader who may choose a wholesale price contract (WPC) or a
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revenue-sharing contract (RSC) with each retailer independently. We explore the effect of the supplier’s and incumbent’s incentives on non-leakage equilibrium and find out that there exists a non-leakage equilibrium only when the incumbent signs an RSC with the supplier under an appropriately given revenue-sharing rate in situations of high demand variation.
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Key words: information leakage; contract mechanisms; game theory; supply chain management
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1 Introduction Contemporary supply chains face uncertain demand due to long lead time, short product life cycle, and proliferation of products and categories, such as fashion apparel, popular toys, electronics, etc. (Burnetas et al. 2007). Demand uncertainty leads to a series of problems, e.g., the mismatch between suppliers and retailers, shortages of products, and reducing customer service levels. Information sharing has become a critical driver for improving supply chain performance and weakening the negative effect of asymmetric
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demand information between suppliers and retailers. Advanced information sharing technology promotes the collaboration between the enterprises in a supply chain, which results in various initiatives such as Collaborative Planning, Forecasting, and Replenishment (CPFR), VMI (Vendor Management Inventory), Efficient Consumer Response (ECR), and Forecast Information Sharing (FIS) (Ali et al. 2017).
However, these coordination initiatives have increased the likelihood of information leakage to
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competitors or third parties due to deliberate efforts to lift supplier’s own profits. In the UK clothing industry, many retailers realize that suppliers may divulge important information to other companies and they are reluctant to share information with those suppliers (Adewole 2005). A survey made by supplychainaccess.com shows that common suppliers sharing product/demand information with competing retailers have severe threats to the security of online supply chain activities (Zhang and Li 2006). In general, information leakage is pervasive across all sectors of industry and consequently
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becomes a major obstacle to information sharing between suppliers and retailers. There are multifarious types of contract mechanisms for suppliers to cooperate with retailers, e.g.,
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wholesale price contract (WPC), revenue-sharing contract (RSC), buy-back contract (BBC), quantity discount contract (QDC), quantity–flexibility contract (QFC), etc. (Cachon 2003). Information leakage
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may occur under contract mechanisms when horizontal competition exists between retailers. For example, when a supplier signs WPC with two retailers, Anand and Goyal (2009) develop a game theoretic model and show that there always exists information leakage. When a supplier signs RSC with two retailers,
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however, Kong et al. (2013) indicate that private information may be protected by an appropriately designed revenue-sharing rate.
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While WPC can be considered a special case of RSC with zero revenue-sharing rate, a supplier must be able to ex post verify a retailer’s revenue under an RSC and needs to balance the costs of running RSC with the profit sacrificed by using WPC (Cachon and Lariviere 2005). Since small retailers’ contribution to the supplier’s profit is relatively low, the supplier is more willing to choose a WPC with small retailers considering regulatory costs. In practice, suppliers may choose different types of contract with different retailers depending on the market, negotiation power of the retailers, and regulatory capability of the supplier (Pan et al. 2010). For example, Hanwha Techwin Co., Ltd signed a revenue-sharing contract with Pratt & Whitney, a subsidiary of United Technologies International Corporation-Asia Private Limited 3
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(Watson and Tan 2016). Hanwha can enhance delivery of critical parts more efficiently to Pratt & Whitney so as to meet Pratt & Whitney’s increased demand for orders. However, according to Hanwha Group Management & Planning H.Q (2017), Hanwha also singed wholesale price contracts with GE and Rolls-Royce to provide large-scale aircraft engine parts to the latter; In China, Procter & Gamble (P&G) offers a lower wholesale price to the distributors compared with small retailers and the distributors need to share part of revenue with P&G at the end of the year, which is pointed out by Yang Huabin, a
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professional cosmetic consultant in Shenzhen (Wu and Li 2010); In the household appliance industry, Haier sets up a partnership with a big distributor in Shanghai and offers procurement preferences (Spring and Anantharaman 2016). Other examples include Gree Electric Appliances, Inc. and Kweichow Moutai Co., Ltd., who have different collaboration schemes to maintain close relationships with big retailers and small ones. In general, considering factors such as financial returns and ease of implementation of
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contracts, suppliers often have the flexibility to select a contract type with certain retailers and may choose an asymmetric contract scheme for different types of retailers.
Our goal in this paper is to investigate information leakage under different contracts (WPC or RSC) with different retailers. We consider a stylized supply chain including a supplier, an incumbent retailer, and an entrant retailer. The supplier may sign different contracts with the two retailers in one of the following four scenarios:
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Table 1 Different Contract Scenarios
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Contract Type
WPC
RSC
RSC
Scenario 1 Scenario 4
WPC
Scenario 3 Scenario 2
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Incumbent retailer
Entrant retailer
We assume that the retailers compete by selling completely substitutable products with short life cycles in
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a common market with uncertain demand. That is, the incumbent and the entrant engage in Cournot competition and the product price is a decreasing function of the overall order quantity available in the
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market. In this paper, we consider wholesale price as an endogenous variable, i.e., the supplier is allowed to optimally choose a wholesale price under any given revenue-sharing rate for each retailer independently. Note that this assumption may result in different wholesale prices for retailers, which is different from the assumption in Anand and Goyal (2009) and Kong et al. (2013). In general, the supplier can be informed of consumer preferences and market demand only through experience (Blattberg and Fox 1995). The entrant retailer lacks the necessary channel and expertise and has to depend on upstream suppliers for information (Agency Sales 1992, 2003). Therefore, we employ a two-state demand distribution and assume that the incumbent retailer has private and perfect information 4
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about the market demand while the supplier and the entrant have no channel to acquire the actual demand state. However, the supplier may leak the incumbent order information to the entrant in order to lock more profits from the latter. We assume that the entrant plays a passive role in the supply chain and has no ability to distinguish the information being leaked by the supplier. In other words, if the supplier leaks the incumbent’s order information to the entrant, the entrant would accept the information and make order decisions accordingly.
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We first investigate the information leakage problem under Scenario 1. By analyzing the sustainability of non-leakage equilibrium (NLE) in a supply chain, we find that the supplier would not leak information and the incumbent would not deviate from NLE under certain parameter regions. More specifically, NLE exists more likely when the revenue-sharing rate is relatively high and the probability of high demand state is relatively low.
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Following the discussions under Scenario 1, we further study information leakage under other three contract scenarios. We find that the supplier would leak information or the incumbent may deviate from NLE under high demand variation and NLE doesn’t exist under Scenarios 2 and 3. That is, as long as the supplier signs WPC with the incumbent, the former would always leak information to the entrant. Similar to Kong et al. (2013) when the supplier has an identical RSC with the two retailers (Scenario 4), information leakage can be prevented under some appropriate regions and the result is robust when the
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wholesale price is considered as an endogenous variable. In summary, in case of high demand variations,
2 Literature Review
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NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract.
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Our work is related to a stream of the literature that investigates vertical information sharing between two partners in a supply chain. Lee et al. (2000) show information sharing can provide significant benefit
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to the supplier and the retailer, such as inventory reduction and cost savings. Özer and Wei (2006) show how the distortion of demand forecast information occurs in a wholesale-price contract and study contracts to ensure information shared credibly between a manufacturer and a supplier. Yue and
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Raghunathan (2007) research the effect of a returns policy and information sharing on the performance of the supplier and the retailer, respectively. Zhang and Chen (2013) investigate coordination of information sharing in a supply chain consisting of one supplier and one retailer and show coordinative contact can ensure that they share their information better. Oh and Özer (2013) study how a supplier can get credible information from a manufacturer and how a supplier makes decisions to obtain more demand information under asymmetric information. These papers analyze vertical information sharing in a supply chain, but they do not consider information leakage and horizontal competition, which is our concern.
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Another stream of research considers vertical information sharing between one supplier and its retailer under two supply chains. Ha and Tong (2008) study vertical information sharing under two competing supply chains and show how important the contact type is to information sharing. Ha et al. (2011) consider two competing supply chains, each consisting of one supplier and one retailer and show how information sharing affects the performance of a supply chain under Cournot or Bertrand competition, respectively. Guo et al. (2014) show how retailers in two competing supply chains can strategically
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disclose demand information to affect the suppliers’ pricing decision. Shamir and Shin (2015) investigate when and how forecast information can be credibly shared in two competing supply chains under a wholesale price contract by making the forecast information publicly available. These papers focus on vertical information sharing under two competing supply chains and don’t consider the impact of information leakage on player’s decision-making.
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There is a stream of the literature researching vertical information sharing when horizontal competition exists in a supply chain where information leakage may occur. Li (2002) investigates the incentives of information sharing in a supply chain consisting of a supplier and n symmetric competing retailers. He refers to the phenomenon as “information leakage” and shows no information sharing is an equilibrium, which is undesirable for the supplier. Zhang (2002) shows a similar research to Li (2002) and further studies the incentives of information sharing under Bertrand competition between the retailers. Li and
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Zhang (2008) consider information sharing in a supply chain including a manufacturer and multiple retailers and study the effects of information leakage. They assume that each retailer has some private
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information about uncertain demand. Anand and Goyal (2009) study the behavior of information leakage in a one-supplier-two-retailers supply chain under a wholesale-price contract and find the supplier will always leak information. Kong et al. (2013) continue the research of Anand and Goyal (2009) and
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indicate that RSC can prevent the supplier from leaking information. Shamir (2016) shows retailers’ ability to share information with a mutual supplier and demonstrates how the retailers take advantage of
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information leakage to form a cartel. These works emphasize the effect of information leakage on the incentives of information sharing in a supply chain. However, we focus on information leakage under
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different contract configurations. The final stream of research relates to contract mechanism. Cachon and Lariviere (2001) show how a
downstream manufacturer uses contracts to share credible demand information with a supplier. Cachon (2003) provides the literature review about contract mechanism in supply chains under asymmetric information. Cachon and Lariviere (2005) show that RSC coordinates well a supply chain with one manufacturer and multiple competing retailers. Li and Wang (2007) review coordination mechanisms of supply chain systems. Yao et al. (2008) demonstrate that RSC is better than WPC for coordinating a supply chain. Pan et al. (2010) discuss how the manufacturers or retailers choose an optimal contract 6
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under different channel power structures, such as RSC or WPC. Zhang et al. (2012) investigate how to use RSC to coordinate a supply chain consisting of one manufacturer and two retailers with demand disruptions. These papers focus on how to use a contract to coordinate supply chains, but they do not consider the effect of information leakage in a supply chain, which is our focus. The remainder of this paper is organized as follows. Section 3 presents the model framework in detail. Section 4 studies the existence of NLE under Scenario 1. Section 5 investigates the impact of different
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contract configurations on NLE. Section 6 offers conclusions and future research. Proofs of main results are given in Appendices.
3 Model Framework
We consider a supply chain including a supplier and two retailers. The incumbent retailer and the entrant retailer sell completely substitutable products in a common market and engage in a Cournot
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competition on quantity. We assume that all players are risk neutral and develop a dynamic game among the three players to maximize their own expected profits under incomplete demand information. We further assume that the incumbent has been in a market for a long time and therefore has enough ability to acquire the actual demand state. On the other hand, the supplier and the entrant have no channel to access the accurate market demand state. Therefore, the actual market demand state is private information for the
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incumbent. We shall investigate the behavior of information leakage between the supplier and the entrant retailer under different contract scenarios in Table 1 where the supplier can optimally choose a wholesale
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price for a given revenue-sharing rate. The supplier, the incumbent retailer, and the entrant retailer are denoted by s, i and e, respectively.
We assume that the demand function is linear of the retail price P with intercept A , i.e., the inverse
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demand curve is given by P(Q) A Q , where Q qi qe is the total quantity in the market and qi / qe is the order quantity of the incumbent/entrant, respectively. The intercept A is a random variable which
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may take a high value H with probability p (0,1) or a low value L with probability 1 p and
H L 0 . The mean demand intercept is given by pH (1 p) L . These priors are common
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knowledge to all supply chain players.
3.1 Sequence of Events Suppose that the retailers bring their entire order quantities to the market. Let ww and wr denote the
wholesale price offered by the supplier under WPC and RSC, respectively. The following events take place in sequence.
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1) The supplier offers the retailers a WPC or an RSC, consisting of wholesale price and/or revenue-sharing rate (0,1) ; 2) The incumbent can acquire the actual market demand state A , which takes value H (high demand state) or L (low demand state); 3) Considering the market demand state, the incumbent places an order qiA to the supplier;
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4) The supplier decides whether to leak the incumbent’s order quantity to the entrant; 5) The entrant places an order qeA to the supplier according to the information available to him; 6) The demand state A is revealed to all parties and the three players realize their profits. 3.2 Supply Chain Player’s Incentives and Decisions
Given order quantities of the two retailers qiA and qeA in face of the demand state A , Table 2 presents
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the three players’ profits under the four contract scenarios in Table 1.
Table 2 Players’ Profits under Different Contract Scenarios Contract
Scenario 1
Configuration
Incumbent under RSC; entrant under WPC
Scenario 2
Incumbent under WPC; entrant under RSC
iA 1 qiA A qiA qeA wr qiA
iA qiA A qiA qeA wwqiA
Entrant’s Profit
eA qeA A qiA qeA wwqeA
eA 1 qeA A qiA qeA wr qeA
Supplier’s Profit
sA wr qiA wwqeA qiA A qiA qeA
sA ww qiA wr qeA qeA A qiA qeA
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Incumbent’s Profit
Scenario 4
Incumbent and entrant under WPC
Incumbent and entrant under RSC
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Scenario 3
iA qiA A qiA qeA wwqiA
iA 1 qiA A qiA qeA wr qiA
Entrant’s Profit
eA qeA A qiA qeA wwqeA
eA 1 qeA A qiA qeA wr qeA
Supplier’s Profit
sA ww qiA qeA
sA wr qiA qeA qiA qeA A qiA qeA
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Incumbent’s Profit
Supplier Given a particular type of contract with a retailer, the supplier optimally chooses a wholesale
price and decides whether or not to leak the incumbent’s order quantity to the entrant. In this paper, we assume that the supplier can’t distort the incumbent’s order quantity when she leaks it. We shall investigate whether there exists an equilibrium of information non-leakage if the supplier cooperates with the retailers under different contract schemes.
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Incumbent Considering that the supplier may leak information to the entrant, the incumbent decides an optimal order quantity to maximize his profit. The incumbent can acquire the actual market demand state to place an order, implying the order quantity reflects the demand information. If the supplier has an incentive to leak the incumbent’s order to the entrant, the entrant makes order decisions depending on the information. Therefore, the incumbent would place a strategic order to reduce the effect of information leakage.
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Entrant The entrant’s decision is his order quantity to maximize his expected profit. Comparing to the incumbent and the supplier, the entrant plays a passive role in the supply chain. If the supplier leaks the incumbent’s order information to the entrant, we assume that the entrant always accepts the information and makes his optimal order decision correspondingly. The two retailers play a Stackelberg game where the incumbent is the leader and the entrant is the follower. If the supplier wouldn’t leak the information,
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the retailers play a simultaneous game where the entrant can’t acquire the actual state of market demand and the two retailers simultaneously place the optimal orders.
4 Game Theoretic Analysis of Scenario 1
In this section, we analyze the retailers’ order quantities and profits when the supplier leaks information
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or never leaks information in contract Scenario 1, where the incumbent has an RSC and the entrant has a WPC with the supplier respectively. Sections 4.1 and 4.2 discuss the effect of information leakage and
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non-leakage on the retailers’ order decisions, respectively. The benchmark analysis is helpful in deriving and understanding the NLE under high demand variation in Section 4.3. Section 4.4 gives a brief discussion regarding NLE under low demand variation.
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The notations used for order quantity, profits and wholesale price are as follows. To compare the E E results under leakage (S) and non-leakage (N) equilibriums (or games), let qmA and mA denote the order
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quantity and profit of player m (m=i, e, or s) in equilibrium E (S or N) under demand state A (H or L), respectively. For example, qiHS denotes the order quantity of the incumbent in a separating equilibrium
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under the high demand state when the supplier leaks information. Besides, qeN and eN denote the entrant’s order quantity and profit in the NLE. Let wwE and wrE denote the wholesale price that the retailers should pay for each product under WPC and RSC in E equilibrium, respectively. Additional notations will be introduced later.
4.1 Benchmark Analysis under Information Leakage
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We first discuss the case where the supplier leaks the incumbent’s order information. In this case, we assume that the entrant always accepts the information leaked by the supplier and places his order accordingly. The incumbent makes order decision after learning the entrant’s belief of the demand state and decisions. Therefore, the game between the incumbent and the entrant is a Stackelberg game. We may consider two pure strategic perfect Bayesian equilibriums – pooling or separating equilibrium. In a pooling equilibrium, the incumbent places the same order quantity whether the demand state is
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high or low. Thus, although the supplier may leak the order information to the entrant, the entrant can’t distinguish the actual demand state. However, pooling equilibrium is different from NLE, under which the entrant has to guess the demand state to maximize his expected profit. Nevertheless, in pooling equilibrium, the retailers play a sequential-move game and the entrant places an optimal order when he observes the incumbent’s order quantity leaked by the supplier. When the supplier has identical WPC
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with the two retailers, Tian and Jiang (2016) illustrate that all pooling outcomes can be eliminated by the intuitive criterion of Cho and Kreps (1987). Kong et al. (2013) have shown that, under identical RSC with the two retailers, pooling equilibrium may not exist when demand variation is high and even when it exists the incumbent may prefer NLE. Similar observations can be made when the two retailers have different contracts with the supplier, i.e., there does not exist pooling equilibrium when demand variation is relatively high and it is feasible that the NLE replaces the pooling equilibrium in most scenarios. The
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detailed proofs are presented in Appendix D. Therefore, in this paper, we shall focus on separating equilibrium and analyze the existence of NLE.
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Under a separating equilibrium, the incumbent places different order quantity based on demand state. The entrant can infer the demand state from the incumbent’s order quantity leaked by the supplier. If the incumbent’s order is low enough, the entrant believes that the demand state is low. Otherwise, he believes
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that the demand state is high. As shown in Anand and Goyal (2009), the entrant’s belief is
0 if the supplier leaks and qi qiLS , Pr A H S e 1 if the supplier leaks and qi qiL .
Therefore, the high-type incumbent has an incentive to mimic a low-type order to induce the entrant to
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order less, which better benefits the incumbent. Thus, the separating equilibrium requires that (i) it is too costly for the high-type incumbent to pretend to be a low type and (ii) it is valuable for the low-type incumbent to place a low-type order. Similar as Kong et al. (2013), we use r H / L to measure the relative variation in demand. It can be easily shown that the entrant would not participate if
r 1 1 / p
(1)
does not hold. Therefore, we shall assume this condition in the rest of this section.
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The low-type incumbent’s order can be divided into two types depending on demand variations. Define
r 1 2 / [1 2 p 11 ] as a threshold to classify situations of high and low demand variations. More specifically, when r r , i.e., under the high demand variation, the high- and low-demand states are far apart and there is no cost for the low-type incumbent to separate out. If Equation (1) holds, it can be shown that 1/ 2 p 1 . When r r , i.e., under the low demand variation, considering the entrant’s
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belief, the incumbent has to place a small enough order to prevent the high-type incumbent from mimicking a low type. Lemma 1 below presents the optimal price and order quantities of all players under the high demand variation when r r . Note that the incumbent always places a positive order as long as the entrant participates in the leakage case. The proof is given in Appendix A.
Lemma 1 Suppose the supplier always leaks the incumbent’s order information to the entrant.
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1) For any given value of p , if 1/ 2 p 1 , we have r 1 1 / p . 2) If r r ,1 1 / p , the supplier’s optimal wholesale prices are
wrS 1 4 3 / 8 4
and
wwS / 2
for the incumbent and the entrant, respectively. (i)
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3) The order quantities of the incumbent and the entrant in the separating equilibrium are as follows: If the demand state is high, the incumbent orders
otherwise
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qiHS H H L 1 p 1 / 4 2 ,
If Pr A H 1 , the entrant orders e
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(ii)
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qiLS L H L 1 p / 4 2 .
S qeH H L 1 p H 1 / 8 4 ,
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otherwise
qeLS L 1 H L p / 8 4 .
The condition 1/ 2 p 1 ensures that the threshold r is always lower than the right-hand side of
condition (1) so that an analytical result can be obtained for the separating game. In practice, considering long-term cooperation and bargaining between the supplier and retailer, the revenue-sharing rate is often not too high. More specifically, when 1/ 3 , 1/ 2 p 1 always holds for any value of p .
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When r r , i.e., under the case of low demand variation, a closed form of the optimal wholesale price cannot be obtained. In practice, demand variation is often relatively high for products of short-life cycles, such as fashion apparel, electronics, etc. Thus, in this paper, we focus on the existence of NLE under high demand variation. With respect to low demand variation, we present a brief discussion in Section 4.4.
4.2. Benchmark Analysis under Information Non-leakage
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We now analyze the situation where the supplier never leaks incumbent’s order quantity to the entrant. Thus, the entrant can’t observe the incumbent’s order decision and the game between the incumbent and the entrant is a simultaneous-move game. In non-leakage case, because the entrant’s order quantity is the same in both demand states and the incumbent places a relatively small order in the low demand state, it is more difficult to guarantee the low-type incumbent’s participation. The incumbent would not participate in low demand state if
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r 1 1/ [2 p 1 ]
(2)
does not hold. Therefore, we shall assume the condition in the rest of this section. Under the non-leakage game, the incumbent will make order decision depending on the actual market demand state, implying that the incumbent’s order quantity truthfully reveals the demand state to the
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supplier. Because the entrant can’t acquire any information about market demand state, he has to decide order quantity depending on the mean demand. Lemma 2 presents the optimal prices and order quantities
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of the two retailers.
Lemma 2 Suppose the supplier never leaks the incumbent order quantity. The supplier’s optimal
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wholesale prices are
wrN 3 1 / 6 4 2
and
wwN / 2
The high-type incumbent will order
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(i)
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for the incumbent and the entrant, respectively. qiHN H 2 1 1 p H L / 6 4 ,
while the low-type incumbent will order
(ii)
qiLN L 2 p 1 H L / 6 4 ;
The entrant will order
qeN 1 / 6 4 .
4.3 Dominance of NLE under High Demand Variations 12
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We now examine the dominance of non-leakage game over the separating game when r r from the supplier’s and the incumbent’s incentives with information non-leakage in the case of high demand variations. We shall first discuss the existence of the two games. When both games exist, we then consider the incumbent’s incentive with information non-leakage assuming leakage choice as an endogenous decision for the supplier. For the case of low demand variations, it will be discussed in Section 4.4.
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Equations (1) and (2) provide sufficient and necessary conditions for all players to participate in the separating and non-leakage games, respectively. Figure 1 illustrates four parameter regions for the existence of the separating and non-leakage games when p = 0.1 and 0.5, i.e., OS (only separating game exists), ON (only non-leakage game exists), BN (both games don’t exist), and B (both games exist). It is easy to see that when p increases, the participation regions for both games will get smaller. Figure 1 also
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depicts the area for low demand variations (LD).
BN OS
BN
ON
ON
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B
LD
OS
B
LD
In the parameter region OS, 0.2929 and r UBPN ,UBPS , where UBPS 1 1 / p and
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1)
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Figure 1 Participation Regions of Separating and Non-Leakage Games
UBPN 1 1/ [2 p 1 ] respectively correspond to the participation constraint under the separating
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game and the non-leakage game in Equations (1) and (2). When is small, the supplier cannot offer a very low wholesale price to encourage the low-type incumbent to increase the order quantity. In low demand state, the entrant would place a larger order depending on her own estimation for the average demand under non-leakage, compared with that under the separating game. Therefore the entrant would push the low-type incumbent out of business under the non-leakage game. Hence, the low-type incumbent has no incentive to participate in the non-leakage game, i.e., there only exists the separating game, 13
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2)
In the parameter region ON, 0.2929 and r max r ,UBPS ,UBPN . When is relatively high, the incumbent would place a big order due to the low wholesale price offered by the supplier, which results in that the entrant withdraws from the market under the separating game in low demand state. Hence, the low-type entrant wouldn’t participate the separating game, i.e., there only exists the nonleakage game In the parameter region BN, r max UBPS ,UBPN . Given the low demand L , when r is very high,
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3)
the mean demand is high too. Under the separating game, faced with high potential mean market demand, the retailers would order more and the supplier would raise the wholesale prices for the retailers. Thus, in low demand state, for any given , the incumbent would place a large enough order because of the relatively low wholesale price compared with the entrant and therefore push the
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low-type entrant out of business. On the other hand, under the non-leakage game, the entrant would place a big order based on the high average market demand, which results in that the low-type incumbent cannot make a profit and is out of the market. Hence when r is very high, the low-type incumbent and the low-type entrant wouldn’t participate the non-leakage game and the separating game, respectively, and the two games do not exist.
In the parameter region B, 1/ 2 p 1 and r r min UBPS ,UBPN . We shall focus on the
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4)
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parameter regions where the non-leakage game dominates over the separating game.
The following sufficient conditions guarantee that the NLE dominates the separating game in the parameter region B:
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a) The supplier would not leak information in both demand states, i.e., S N and sLS sLN . sH sH
(3)
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b) The incumbent would order qiHN and qiLN in corresponding demand states under NLE, i.e.,
iHS iHN and iLS iLN .
(4)
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c) Guarantee that the incumbent and the entrant participate in the market under NLE, i.e., qiHN 0 , qiLN 0 and qeN 0 .
Note that if iLS iLN , the incumbent may place an order deviating from the non-leakage game qiLN in
low demand state. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information, which benefits both incumbent and supplier. Because the supplier informs the entrant about the low-type incumbent’s order information, the entrant would reduce the order quantity compared with 14
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the non-leakage game one, resulting in that the market price rises and the profits of incumbent and supplier increase. Hence, the separating game dominates the non-leakage game and there does not exist NLE. In this paper, for ease of exposition, we present the sufficient condition b) to guarantee that the incumbent would not deviate under NLE. Appendices B and C present parameter regions of , p and r for the above conditions (3) and (4), respectively. After analyzing these conditions and considering the supplier’s and incumbent’s incentives
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with non-leakage, we have the following results.
Proposition 1 In parameter region B, NLE dominates the separating game only if the parameters of , p and r lie in the regions defined in Table 3. Otherwise, the separating dominates the NLE. Each subregion is determined by the upper and lower bounds on p and r given in Table 4.
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Table 3 Parameter Regions of NLE in Parameter Region B
0.6112 0.7469
pB
0.7469 0.7808
p B1p
p
1 p
max r , LB , LB 1 r
2 r
r
r min LB ,UB 3 r
S P
max r , LBr2 r UBPS
1 2 2 3 2 2 3 3 2 6 2 p 5 5 2 p 8 11 2 4 3 1 p
6 7 2 2 2 p 3 2 6 2 2 3 p 2 5 5 2 2 3
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3 2 2 1 p 5 5 2 2 3
LBr3 :
7 4 2 6 7 2 2 p 23 43 24 2 4 3 p 1 23 20 4 2
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LBr2 :
B1p :
ED
: 9 18 9 2 3 4 LBr1 :
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Table 4 Definitions of the Bounds on p and r
From Proposition 1, it is straightforward to prove the following corollary.
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Corollary 1 If p 0.32 , NLE does not exist.
When p is in the medium range, due to the large demand uncertainty, the supplier always has an
incentive to inform the entrant about the true demand state because she can benefit more by leaking information to induce that the entrant makes a correct order decision. When p is high, demand uncertainty is small but the high demand state is more likely. The incumbent always induces the supplier to leak
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information about the actual demand state to the entrant, which can avoid causing a loss to the incumbent or even the supply chain due to the entrant’s over order in low demand state. For illustration, Figure 2 shows the existence region of NLE in parameter region B when p 0.1 , where N and S denote the cases where the non-leakage game dominates the separating game/the
OS
BN ON S
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N
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separating game dominates the non-leakage game, respectively.
LD
Figure 2 Dominance Region of NLE for p 0.1
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In parameter region N, where 0.6112 0.7808 and r LBr2 , min LBr3 ,UBPS , NLE dominates the
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separating game. When is relatively high, the supplier’s profit becomes more consistent with the whole supply chain and hence the supplier would control the total order quantity in the supply chain to maximize her own profit. The supplier makes leakage or non-leakage decision based on the trade-off
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between total order quantity and market price. Compared with the total order quantity under information non-leakage, the retailers may together place a bigger order in high demand state and smaller order in low
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demand state under information leakage. If the supplier would not leak the incumbent’s order information to the entrant, the entrant has to place an intermediate order depending on own estimation for the market demand, which is beneficial to the supplier in both demand states under certain regions. Meanwhile, the
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incumbent would not deviate from NLE in both demand states because he can benefit more from information non-leakage by reducing sales volume and raising market price, compared with information leakage.
In parameter region S, the separating game dominates the NLE because the supplier has an incentive to
leak information or the incumbent may deviate from NLE. For example, when 0.2929 and r r , UBPN , the incumbent would have an incentive to deviate from the non-leakage game in low
demand state. The supplier acquiesces in the incumbent’s action and would leak information to the entrant. 16
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In fact, when is relatively low, the incumbent can reserve most of the sales revenue and hence would like to increase the order quantity to lift his own profit. Compared with the order under information nonleakage, the entrant will place a smaller order when he is informed that the actual demand state is low, which is beneficial to the low-type incumbent.
4.4 Dominance of NLE under Low Demand Variations
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Under low demand variations, i.e., r r , we now use some numerical examples to investigate the dominance of non-leakage game over the separating game. Assume that H 700 , L 500 , p 0.1 and
r 1.4 , the mean demand is pH 1 p L 520 . For any 0,1 , r r , i.e., the demand variation is low. To investigate the supplier’s and incumbent’s incentives of information non-leakage, we list the optimal wholesale price offered by the supplier and all parties’ profits under various in Table 5.
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Note that “N” and “S” denote the non-leakage and the separating game, respectively.
Table 5 Profits Under Various Values of for H 700 , L 500 , p 0.1 and r 1.4
0.1
0.3
Game
S
N
wrN
wwN
wrS
wwS
wrN
225.6
260
242.2
260
159.3
iH
30,093
20,013
eH
14,506
13,123
sH
70,917
88,761
iL
6,179
eL
6,148
260
N
S
N
S
wrS
wwS
wrN
wwN
wrS
wwS
wrN
wwN
wrS
wwS
171
260
69.3
260
75.2
260
6.5
260
8.5
260
19,446
21,986
17,053
9,404
7,801
12,576
10,433
8,538
5,476
2,419
506
76,638
92,473
88,638
99,715
110,207
109,636
5,170
6,769
5,913
7,230
6,616
4,271
3,868
6,204
4,992
5,136
2,760
2,999
253
469
44,988
44,782
49,171
48,690
57,507
57,160
27,535
ED
PT
CE 43,189
wwN
0.9
43,246
AC
sL
S
M
N
0.6
In this particular example, it is found that NLE dominates the separating game only if the revenue-
sharing rate is relatively high. When p is relatively low, the low demand state is more likely. On the other hand, when r is very low, the two demand states are very close, implying that the entrant is not easy to distinguish them. Therefore, in high demand state, in most cases where 0.1,0.3 and 0.6 , the supplier would always inform the entrant about the actual demand state, which can induce that the entrant orders more products. However, in case 0.9 , where is relatively high, the supplier’s profit is more in line
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with the whole supply chain and hence she would like to control the total order quantity in the supply chain to lift own profit. In this case, compared with information leakage, both the supplier and the incumbent can benefit more from the non-leakage game by reducing order and raising price. Besides, we demonstrate another two examples when p 0.5 and p 0.9 . We find that the supplier would always leak order information to the entrant and the separating game dominates the non-leakage
and p is relatively low, NLE may exist under low demand variations.
5 Impact of Different Contract Scenarios
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game. In short, similar as the results in the situation of high demand variations, only if is relatively high
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To further analyze the effect of different contract configurations on information leakage, we investigate the existence of NLE under Scenarios 2 and 3 and summarize the results under Scenario 4 studied in Kong et al. (2013).
Analogous to the analysis under Scenario 1, we define r 1 4 1 / 4 4 p 3 2 p and
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r 1 1/ 1 p for Scenarios 2 and 3, respectively, as the threshold to classify situations of high and low demand variations. We primarily analyze the incentive of supporting non-leakage from the perspective of supplier and incumbent when r r and further investigate the sustainability and existence
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of NLE.
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Proposition 2 For Scenarios 2 and 3, the supplier always leaks information in high demand state and the incumbent deviates from NLE in low demand state when r r in the corresponding scenario. Therefore
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NLE doesn’t exist when r r .
In Scenario 2, faced with the high demand state, the supplier always has an incentive to leak the high-
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demand signal to induce that the entrant increases the order quantity. When the supplier cooperates with the entrant under an RSC, the wholesale price offered by the supplier is relatively low and the risk from market uncertainty for the entrant diminishes. Hence the entrant is more likely to place a relatively big order under information non-leakage, resulting in that the incumbent’s profit reduces, especially in low demand state. Thus the low-type incumbent has an incentive to deviate from the non-leakage game. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information to inform the entrant of the actual demand state, inducing that the entrant’s order quantity decreases and the market price rises. 18
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Anand and Goyal (2009) have shown that the supplier always leaks information under exogenously fixed wholesale price in Scenario 3. They used only a numerical example to illustrate the impact of endogenous wholesale price on information and material flows and analyze how to set wholesale price when the supplier takes the incumbent’s informational imperative and double marginalization into account. Here we show that NLE doesn’t exist when the supplier chooses the optimal wholesale price. Kong et al. (2013) have demonstrated the existence of NLE in Scenario 4 when the wholesale price and
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revenue-sharing rate are appropriately given. In addition, they found that NLE remains existence when the supplier is allowed to choose an optimal wholesale price for any given value of . However, they only proved that NLE is a robust outcome by comparing the expected profit of the supplier and incumbent under the separating game and the non-leakage game, respectively. In this study, we shall investigate the region where NLE dominates the separating game under both demand states in the case of high demand
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variations. Similar as Scenario 1, we define r 1 4 / 4 4 p 2 p as the threshold to classify situations of high and low demand variations in Scenario 4. Figure 3 presents regions of BN, OS, and B in the case of high demand variations. The regions are found different from those in Figure 1. We shall discuss the dominance of the non-leakage game over the separating game in region B, which is shown in
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BN
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Proposition 3.
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OS
OS B
LD
LD
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B
BN
AC
Figure 3 Participation Regions of Separating and Non-Leakage Games under Scenario 4 Proposition 3 For Scenario 4, in parameter region B, where r r 1 1/ p 2 , both the separating game and the non-leakage game exist. NLE dominates the separating game only if the parameters of , p and r lie in the regions defined in Table 6. Otherwise, the separating dominates the NLE. Each sub-region is determined by the upper and lower bounds on p and r given in Table 7.
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Table 6 Parameter Regions of NLE in Parameter Region B
0.9775
p
pB
max r , LB
2 p
r
4 r
r LB
5 r
Table 7 Definitions of the Bounds on p and r 43 2
22
LBr5 :
2 1
LB : 1
2
4 r
6 6 12 7 2 3 4 5 2
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B p2 :
p 20 12 3 2 3
14 4 2 12 7 2 p 46 43 12 2 3 p 2 23 10 2
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Similar as Scenario 1, we can easily obtain the following corollary from Proposition 3.
BN
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Corollary 2 For scenario 4, if p 0.086 , NLE does not exist.
S LD
N
AC
CE
PT
ED
OS
Figure 6 Dominance Region of NLE for p 0.05 under Scenario 4
From Proposition 3 and Corollary 2, we can conclude that the separating game dominates the non-
leakage game in most cases. Only if is very high and p is very low, may there exist NLE. We illustrate an example in Figure 6 when p 0.05 . In parameter region N, where 0.9775 and max r , LBr4 r LBr5 , non-leakage game dominates over the separating game. Similar as the analysis in
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Scenario 1, when is very large, the supplier would control the total order quantity in the supply chain to lift profit. In this case, both the supplier and the incumbent obtain more profits under information nonleakage by reducing order quantity and increase market price. Finally, we have the following summary of the four scenarios.
Proposition 4 In presence of high demand variations, NLE may exist only if the supplier signs RSC with
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the incumbent (Scenarios 1 and 4). The dominance region of NLE shrinks when the supplier signs RSC with the entrant, compared with WPC.
Proposition 4 is interesting since the supplier’s choice of cooperation with the incumbent is the key to prevent information leakage. Under WPC, it is difficult to balance the profits of supplier and incumbent.
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Thus the supplier always has an incentive to leak information or the incumbent would deviate from NLE for her/his own sake, resulting in that NLE does not exist in a supply chain. If the supplier signs RSC with the incumbent, she would offer a relatively low wholesale price for each unit, compared with WPC. The incumbent thus places a higher order and needs to share a part of revenue with the supplier. The revenue is felicitously allocated between the supplier and incumbent, which prevents information leakage under an appropriately given revenue-sharing rate.
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Although the entrant plays a passive role in the supply chain, the contract type of the entrant has an important impact on the dominance region of NLE. Compared with the dominance region of NLE in
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Scenario 1, under Scenario 4, the entrant can purchase more products from the supplier under RSC, resulting in a decrease of the incumbent’s market share and profit. Therefore, the incumbent would induce the supplier to turn the non-leakage game into the separating game because the incumbent can better
PT
utilize the information and first-mover advantage under the separating game to improve the market share. As a result, due to RSC with the entrant, the dominance region of NLE diminishes, compared with WPC.
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As an extension, we also investigate the above comparison when both wholesale price and revenuesharing rate are endogenous. In this case, we can find that the supplier would offer a unique optimal
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revenue-sharing contract with 1 and w 0 . The specific result does not provide any valuable managerial insights because it shows that the supplier is powerful enough to effectively control the entire channel, which does not appear in practice.
6 Concluding Remarks In this paper, we investigate information leakage problem under different contract configurations where the supplier is allowed to optimally choose a wholesale price for given revenue-sharing rate. When the 21
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supplier signs RSC and WPC respectively with the incumbent and entrant retailers (Scenario 1), NLE regions are found analytically, where the supplier would not leak information and the incumbent retailer would not deviate from NLE in the situations of high demand variation. It is found that NLE exists more likely when the revenue-sharing rate is relatively high and the probability of high demand state is relatively low. By further investigating the existence of NLE when the above asymmetric contract types are reversed and following the results in Anand and Goyal (2009) and Kong et al. (2013), we show that
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NLE exists only when the supplier cooperates with the incumbent under a revenue-sharing contract. Otherwise, the supplier would always leak information to the entrant and NLE doesn’t exist. In other words, the supplier’s coordination with the incumbent is the key to prevent information leakage. When the supplier has a revenue-sharing contract with the incumbent, the revenue is felicitously allocated between the supplier and the incumbent under an appropriately given revenue-sharing rate, which
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prevents information from being leaked.
There are several directions for future research. It would be interesting to investigate information leakage problems under different contracts when the products are imperfect substitutable or complementary. We can also investigate the effect of player’s risk preference on information leakage or non-leakage. Another interesting extension is to study how the collusion between the supplier and the
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Acknowledgments
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incumbent affects information leakage and the entrant’s order decision.
The authors thank the area editor, the associate editor and three anonymous referees for their
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constructive suggestions and comments that improved the paper. Feng’s research was partially supported by National Natural Science Foundation of China [Grant 71572145]. Jiang’s research
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was partially supported by National Natural Science Foundation of China [Grants 71831006,
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71531010 and 71325003].
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Appendices Appendix A Proof of Lemmas 1 and 2 Proof of Lemma 1: To obtain a pure separating equilibrium, the incumbent has to satisfy the following constrained optimization program: iLS max 1 qiLS ( L qiLS qeLS (qiLS )) wrS qiLS
(A-1)
S S S iHS max 1 qiHS ( H qiHS qeH (qiH )) wrS qiH
(A-2)
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qiL
qiH
Subject to：
(A-3)
1 qiLS (H qiLS qeLS (qiLS )) wrS qiLS 1 qiHS (H qiHS qeHS (qiHS )) wrS qiHS
(A-4)
S S S qeLS (qiLS ) argmax( L qiLS qeL )qeL wwS qeL ( L qiLS wwS ) / 2
(A-5)
S S S S S S S qeH (qiH ) argmax( H qiH qeH )qeH wwS qeH ( H qiH wwS ) / 2
(A-6)
where
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1 qiHS (L qiHS qeHS (qiHS )) wrS qiHS 1 qiLS (L qiLS qeLS (qiLS )) wrS qiLS
S qeL
S qeH
Inequalities (A-3) and (A-4) are the incentive compatibility constraints which ensure that each
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type of incumbent has no incentive to mimic the other. Formula (A-5) and (A-6) are the entrant’s optimal quantity when the entrant accepts the information being leaked by the supplier. i) In low demand state, in order to prevent the high-type incumbent from mimicking low-type
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incumbent, the optimization program for the low-type incumbent reduces to: iLS max(1 )qiLS ( L qiSL wwS 2wrS / (1 )) / 2 S qiL
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s.t. (1 )qiLS ( H qiLS ( L qiLS wwS ) / 2 wrS / (1 )) ((1 )( H wwS ) 2wrS )2 / (8(1 )) (A-7) The Lagrangian for the above formulation is
AC
CE
(1 )qiLS ( L qiLS wwS 2wrS / (1 )) / 2 L(qiLS , ) max S S S S S S S S 2 qiL ((1 )qiL ( H qiL ( L qiL ww ) / 2 wr / (1 )) ((1 )( H ww ) 2wr ) / (8(1 )))
The first-order Karush-Kuhn-Tucker (KKT) conditions for the Lagrangian are:
1a) :
L L(qiLS , ) wwS wwS wrS wrS L 0 1 q 1 H q v1 0 iL iL qiSL 2 1 2 2 1 2
1b) :
L(qiLS , ) S qiL 0 qiLS
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2a ) :
L(qiLS , ) ((1 )( H wwS ) 2wrS ) 2 wS L q S wS 0 (1 )qiLS ( H qiSL ( iL w ) r ) v2 2 2 2 1 8(1 )
2b) :
L(qiLS , ) 0
Solve the above system and we get:
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1) For 0 (unconstrained optimal), qiLS L 2wrS / 1 wwS / 2 . Thus, by (A-5), we get qeLS L 2wrS / 1 3wwS / 4 . In the following part, we prove that the high-type incumbent has
no incentive to mimic the low-type incumbent. Thus, the order quantities of incumbent and
entrant in the high demand state are as follows: qiHS H 2wrS / 1 wwS / 2 ,
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S qeH H 2wrS / 1 3wwS / 4 .
We have the supplier’s profit in high and low demand state as follows: S S sH qiHS H qiHS qeH wrS qiHS wwS qeHS ,
sLS qiLS L qiLS qeLS wrS qiLS wwS qeLS .
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Then, we have the supplier’s expected profit as follows:
S E sS p sH 1 p sLS .
E sS 0 wrS
E sS 0 , and get the optimal wholesale price offered by the supplier: wwS
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and
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We solve the following first-order conditions under a given simultaneously, i.e.,
wrS Hp L 1 p 1 4 3 / 8 4 , wwS Hp L 1 p / 2 .
(A-8)
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By the KKT conditions, we get v2 0 H L 1 H 3L 2wwS 4wrS 0 . Substitute (A-8)
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into the above inequality, we get r 1 2 / 2 2 p 1 . Thus, the incumbent and entrant order qiLS L H L 1 p / 4 2 and qeLS L 1 H L p / 8 4 in low demand state when r 1 2 / 2 2 p 1 , respectively. To guarantee the low-type entrant’s participation, r 1 1 / p must hold.
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2) For 0 , we have qiLS 2H L 2wrS / 1 wwS H L 3H L 2wwS 4wrS / 1 / 2 . Denote the lower root qiLS 1 and the upper root qiLS 2 . For the constraint (A-7), when qiL 2H L 2wrS / 1 wwS / 2 , LHS of constraint (7) get the maxima which is bigger than
RHS of constraint (A-7). Because qiLS1 2H L 2wrS / 1 wwS / 2 qiSL 2 , we know that the
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LHS of constraint (A-7) is increasing for any qiLS qiLS1 , 2H L 2wrS / 1 wwS / 2 . Therefore, in order to prevent the high-type incumbent from mimicking the low type, the incumbent has to place an order qiLS 1 to realize the pure separating equilibrium when r 1 2 / 2 2 p 1 . However, we can’t get the closed form of the optimal wholesale
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price offered by the supplier in this case and still use wrS and wwS to denote the wholesale price under RSC and WPC, respectively. Nonetheless, we give the complete proof to obtain the pure separating equilibrium.
ii) Then we need to check whether the low-type incumbent places an order which deviates from the low-type order in the separating equilibrium. When the demand state is low, if the
belief.
Thus,
the
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incumbent places an order qiL* qiLS , the entrant thinks that the demand state is high based on own entrant’s
order
quantity
is:
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* qeL (qiL ) argmax( H qiL qeL )qeL wwS qeL ( H qiL wwS ) / 2 . The incumbent’s order quantity is: qeL
* qiL qiL
(1)
When
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* qiL* argmax(1 )( L qiL qeL (qiL ) wrS / (1 ))qiL [2 L H wwS 2wrS / (1 )] / 2 .
r 1 2 / 2 2 p 1 ,
substitute
(A-8)
into
qiL*
and
we
get
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qiL* L 3 p 1 H 2 p 1 / 4 2 . However, qiL* 0 always holds in this
case. Thus, the low-type incumbent has no incentive to place the high-type order.
AC
(2) When r 1 2 / 2 2 p 1 , for all wwS and wrS , we have the following proof: When the low-type order of incumbent doesn’t deviate, the low-type incumbent’s order
quantity is qiLS 2H L 2wrS / 1 wwS H L 3H L 2wwS 4wrS / 1 / 2 , which is
positive if L 2wrS / 1 wwS . Define 1 H 2wrS / 1 wwS / L 2wrS / 1 wwS .
Only when 1 2 , qiL* 0 and the incumbent’s profit iL is 1 H 2L wwS 2wrS
2
/ 8 8 . 28
ACCEPTED MANUSCRIPT
Thus, compare iLS iL when 1 2 and we have following inequality:
1 1 2 21 1 1 31 1 0
Since 1 1 , it is sufficient to verify that 2 21 1 1 31 1 0 . The inequality holds for all 1 2 . Therefore, the low type always separates out. sum
up,
the
incumbent’s
order
quantity
in
high
demand
state
CR IP T
To
is
qiHS H 2wrS / 1 wwS / 2 for all r because the low type separates out. In low demand
state, the incumbent orders qiLS L 2wrS / 1 wwS / 2 when r 1 2 / 2 2 p 1 . the
qiLS 2H L 2wrS / 1 wwS
incumbent
orders
H L 3H L 2wwS 4wrS / 1 / 2 .
AN US
Otherwise,
Proof of Lemma 2: If the supplier does not leak information, the game between the incumbent and the entrant is a simultaneous move game. We have the following profit function of the retailers: qiH
M
iHN max(1 )qiHN ( H qiHN qeN ) wrN qiHN ,
ED
iLN max(1 )qiLN ( L qiLN qeN ) wrN qiLN , qiL
eN max( p( H qiHN qeN )qeN (1 p)( L qiLN qeN )qeN ) wwN qeN qe
PT
Solve the first-order conditions of the three equations simultaneously, and we get: The order quantities of the high-type incumbent, the low-type incumbent and the entrant
CE
respectively are:
qiHN H 3 p L 1 p 4wrN / 1 2wwN / 6 , qiLN L 2 p Hp 4wrN / 1 2wwN / 6 ,
AC
qeN Hp L 1 p wrN / 1 2wwN / 3 .
We have the supplier’s profit in high and low demand state as follows: N sH qiHN H qiHN qeN wrN qiHN wwN qeN ,
sLN qiLN L qiLN qeN wrN qiLN wwN qeN .
Then, we have the supplier’s expected profit as follows:
29
ACCEPTED MANUSCRIPT
N E sN p sH 1 p sLN .
We solve the following first-order conditions under a given simultaneously, i.e., and
E sN 0 wrN
E sN 0 , and get the optimal wholesale price offered by the supplier: wwN
wrN 3 Hp L 1 p 1 / 6 4 , wwN Hp L 1 p / 2 .
CR IP T
2
Consider the participation constraint ( qiHN 0 , qiLN 0 and qeN 0 ) and we get:
These three inequalities simultaneously hold when r 1 1/ 2 p 2 p , which guarantees the
AC
CE
PT
ED
M
AN US
incumbent’s and entrant’s participation.
30
ACCEPTED MANUSCRIPT
Appendix B Supplier’s Incentives of Non-leakage S N The supplier would not leak information in both demand states if sH and sLS sLN holds. sH
Thus, the non-leakage is efficacious and the entrant can’t observe the incumbent’s order information to aid decision making. We summarize the results and show the non-leakage region
CR IP T
supported by the supplier in both demand states under Scenario 1.
Proposition B In parameter region B, the supplier would not leak information in both demand states if the parameters of , p and r lie in the region defined in Table B1. Additional notations on p and r are shown in Table B2.
Table B1 Non-leakage Region Supported by the Supplier in Parameter Region B 0.2929
p
Subregions i
r LBr2 r UBPN
B3p p Bp7
LBr1 r UBPN
p B3p
LBr2 r UBPS
B3p p Bp4
LBr1 r UBPS
p B5p
LBr2 r UBPS
ii
B5p p Bp6
r r UBPS
iii
Bp6 p Bp4
LBr1 r UBPS
i
p B5p
LBr2 r UBPS
ii
B5p p B1p
r r UBPS
AN US p B3p
ii 0.2929 0.7325
i ii
0.7325 0.7469
M
i
PT
ED
0.7469 0.7808
Table B2 Addition Notations on p and r
1 B : 1 3 9 9 3
CE
3 p
5 2 2 3 2 4 2 2
AC
B p6 :
B9p :
/ 2
Bp4 :
Bp7 :
16 9 2 6 2 10
6 r
LB :
3 4 2 2 3 4 4 3 2 2 3
B10 p :
4 4
1 p 1 5
6 7 2
2 2
2 3 2 3
B8p :
1 2 4 2 2
43 2 2
4 4
1
2 2 6 7 2 2 p 23 43 24 2 4 3 2
2 1
2 1 2
p 1 23 20 4 2
B5p :
2
2 p 30 77 71
2
28 4 4 3
31
ACCEPTED MANUSCRIPT
From Proposition B, we have the following conclusions regarding non-leakage regions supported by the supplier: 1) If 0.2929 , the supplier would not leak information under the two following subregions: p B3p and LBr2 r UBPN ; B3p p Bp7 and LBr1 r UBPN .
2) If 0.2929 0.7808 , the supplier would not leak information only when the parameters
CR IP T
of p and r lie in the subregions shown in Table B1. Proof:
1) We have the following supplier’s profit under information leakage or non-leakage in high demand state in parameter region B:
S S sH qiHS H qiHS qeH wrS qiHS wwS qeHS , sHN qiHN H qiHN qeN wrN qiHN wwN qeN .
AN US
S N By the Lemma 1 and 2, sH can be rewritten as sH
3 3 H 2 p L 1 p Hp 1 p L H 2 / 8 2 H 3 2 p 3 7 4 H p L 1 p 1 3 2 / 12 8 2
2
2
2
2
2
2
2
2
HL 1 p 1 3 4 p 4 2
M
Simplify it by r H / L and we can obtain
3 3 r 2 p 1 p rp 1 p r 2 / 8 2 r 3 2 p 3 7 4 r p 1 p 1 3 2 / 12 8
ED
2
2
2
r 1 p 1 3 4 p 4 2
2
2
2
2
2
(B-1)
PT
From Lemma 1, we can express the parameter region B by using , p and r :
CE
p min 1 / 2 ,1 and r r min UBPS ,UBPN for any given . Then we further compare the
participation constraint under non-leakage with that under leakage and get: i) When 0.2929 , UBPN UBPS always holds for any p . We further verify that r UBPN always
AC
holds for any and p . If p min 1 / 2 ,1 and r r ,UBPN , we solve the inequality (B1).
ii) When 0.2929 , UBPN UBPS always holds for any p . If p min 1 / 2 ,1 and
r r ,UBPS , we solve the inequality (B-1).
32
ACCEPTED MANUSCRIPT
Combine i) and ii), and we can obtain the non-leakage region supported by the supplier in high
CR IP T
demand state in parameter region B, shown in Table B3.
Table B3 Non-leakage Region Supported by the Supplier in High Demand State in Parameter Region B 0.2929
p
Subregions
0.6052 0.7469
i ii
0.7469
r LBr1 r UBPN
p Bp4
LBr1 r UBPS
p Bp6
r r UBPS
Bp6 p Bp4
LBr1 r UBPS
p B1p
r r UBPS
AN US
0.2929 0.6052
p Bp7
2) We have the following supplier’s profit under information leakage or non-leakage in low demand state in parameter region B:
M
sLS qiLS L qiLS qeLS wrS qiLS wwS qeLS , sLN qiLN L qiLN qeN wrN qiLN wwN qeN .
ED
By the Lemma 1 and 2, sLS sLN can be rewritten as
3 L 3 3 p H L 2
2
PT
8 2
2
2
L2 2 p p H 2 p 2 L2 p 2 1 3 2 2 HLp 1 1 p 2 4
12 8
CE
Simplify it by r H / L and we can obtain
AC
3 3 3 p r 1 2
2
2
8 2
2 p p r 2 p 2 p 2 1 3 2 2 rp 1 1 p 2 4
12 8
(B-2)
As the similar analysis in high demand state, we can obtain the non-leakage region supported
by the supplier in low demand state in parameter region B, shown in Table B4.
Table B4 Non-leakage Region Supported by the Supplier in Low Demand State in Parameter Region B
Subregions
p
r 33
ACCEPTED MANUSCRIPT
0.2929 1/ 3 1/ 3 0.7808
i
p B5p
LBr2 r UBPN
ii
p B5p
r r UBPN
i
p B5p
LBr2 r UBPS
ii
p B5p
r r UBPS
i
p B5p
LBr2 r UBPS
ii
B5p p B1p
r r UBPS
CR IP T
0.2929
Combine Table B3 and Table B4, and we can obtain the non-leakage region supported by the
AC
CE
PT
ED
M
AN US
supplier in both demand states in parameter region B.
34
ACCEPTED MANUSCRIPT
Appendix C Incumbent’s Incentives of Non-leakage The incumbent would order qiHN and qiLN in corresponding demand states under NLE if iHS iHN and iLS iLN holds. Otherwise, the incumbent may place an order deviating from NLE. Meanwhile, the supplier would acquiesce in the incumbent’s order decision and leak information, which is beneficial for the incumbent and supplier. Hence, the separating game dominates the
CR IP T
non-leakage game and there does not exist NLE. Considering the sufficient condition, we have the following results to ensure that the incumbent would not deviate from NLE in both demand states under Scenario 1.
Proposition C In parameter region B, the incumbent would not deviate from NLE in both
AN US
demand states if the parameters of , p and r lie in the region defined in Table C1.
Table C1 Non-leakage Region Supported by the Incumbent in Parameter Region B 0.2929 0.6739
p
8 p
r r r LBr3
p B1p
r r UBPS
pB
M
0.6739
From Proposition C, we have the following conclusions regarding non-leakage regions supported
ED
by the incumbent:
1) If 0.2929 , the incumbent would place an order deviating from NLE in low demand state. The supplier would acquiesce in the incumbent’s action and leak information. Thus,
PT
information non-leakage does not work in both demand states. 2) If 0.2929 , the incumbent would not deviate from NLE only when the parameters of
Proof:
CE
, p and r lie in the subregions shown in Table C1.
AC
1) We have the following incumbent’s profit under information leakage or non-leakage in high
demand state in parameter region B: S iHS 1 H qiHS qeH qiHS wrS qiHS , iHN 1 H qiHN qeN qiHN wrN qiHN .
By the Lemma 1 and 2, iHS iHN can be rewritten as
35
ACCEPTED MANUSCRIPT
1 H 2 p p L 1 p 1 2 8 2
1 H 3 2 p 2 p 2 2L 1 p 1 2 4 3 2
2
2
Simplify it by r H / L and we can obtain
r 2 p p 1 p 1 2
2
2
2 r 3 2 p 2 p 2 2 1 p 1
3 2
2
2
(C-1)
CR IP T
As the similar analysis in Appendix B, we can obtain the non-leakage region supported by the incumbent in high demand state in parameter region B, shown in Table C2. Addition notation on p and r are shown in Table B2.
Table C2 Non-leakage Region Supported by the Incumbent in High Demand State in Parameter Region B
p
Subregions i
pB
ii
B p B10 p
LBr6 r UBPN
p 0,1
r r UBPS
p B1p
r r UBPS
r r r UBPN
AN US
0.2929
9 p
9 p
0.2929 1/ 3
M
1/ 3
2) We have the following incumbent’s profit under information leakage or non-leakage in low demand state in parameter region B:
ED
iLS 1 L qiLS qeLS qiLS wrS qiLS , iLN 1 L qiLN qeN qiLN wrN qiLN .
PT
By the Lemma 1 and 2, iLS iLN can be rewritten as
CE
1 Hp 1 L 1 p p 2 8 2
2
1 2Hp 1 L 1 2 p 1 2 4 3 2
2
AC
Simplify it by r H / L and we can obtain
rp 1 1 p p 2
2
2
2 2rp 1 1 2 p 1
3 2
2
2
(C-2)
As the similar analysis in Appendix B, we can obtain the non-leakage region supported by the
incumbent in low demand state in parameter region B, shown in Table C3. Table C3 Non-leakage Region Supported by the Incumbent in Low Demand State in Parameter Region B
p
r 36
ACCEPTED MANUSCRIPT
0.2929 0.6739
p B8p
r r LBr3
0.6739
p B1p
r r UBPS
Combine Table C2 and Table C3, and we can obtain the non-leakage region supported by the incumbent in both demand states in parameter region B. Combine Table B1 and C1, and we can
CR IP T
obtain the region of NLE in parameter region B under Scenario 1. As similar argument above,
AC
CE
PT
ED
M
AN US
we can prove Proposition 2 and 3.
37
ACCEPTED MANUSCRIPT
Appendix D Pooling Equilibrium In this Appendix, we summarize the results of the pooling equilibrium under Scenario 1 and show that the pooling equilibrium does not exist or are dominated by other equilibrium in most cases. Therefore, we omit the pooling equilibrium from our main analysis. Firstly, we analyze
Benchmark: Pooling Case where Supplier Always Leak Proposition
D
Suppose
that
the
supplier
always
r r P 20 p 2 15 8
CR IP T
the benchmark case of pooling equilibrium where the supplier always leaks.
leaks
information.
When
AN US
p 2 2 11 2 1 p 32 20 3 2 p 8 8 3 2 / 8 p 8 16 4 p 2 2 11 , a
Pareto-dominant pooling equilibrium exists and is as follows: (i) The supplier’s optimal wholesale prices are
wrP 1 4L 3 / 8 4 and wwP / 2
For the incumbent and the entrant, respectively.
M
(ii) The incumbent orders qiP 2L 1 1 2 / 4 2 .
(iii) The entrant orders qeP 1 3 2L / 8 4 , consistent with his beliefs that:
H 1 3H 2wwP 4wrP / 1 2wrP / 1
CE
q P 2H
PT
ED
0, if the supplier leaks and qiP q P , Pre ( A H ) p, if the supplier leaks and q P qiP qiP and q P qPiL min qiP , where 1, if the supplier leaks and qiP q P
AC
2 1 L wwP 2wrP 1 P P 1 L ww 2wr 2 1 4 1 qiP 1 2 L qiP wwP 2wrP
/ 2 and q
P
iL min
q P i
.
Proof:
The proof is through a series of Lemmas. qiP denotes a candidate pooling quantity for the incumbent.
Lemma D1 The maximum quantity where a pooling equilibrium can be sustained is 38
ACCEPTED MANUSCRIPT
qiP L 1 p Hp wwP / 2 wrP / 1
Proof. The entrant orders qeP where
qeP qiP arg max p H qiP qeP wwP qeP 1 p L qiP qeP wwP qeP qiP wwP / 2 (D-1) P qe
The high-type incumbent maximizes
qiP
The low-type incumbent maximizes
CR IP T
iHP max 1 H qiP qeP qiP qiP wrP qiP
iLP max 1 L qiP qeP qiP qiP wrP qiP qiP
(D-2)
(D-3)
From (D-1), (D-2) and (D-3), we obtain the optimal quantity for the high-type and the low-type incumbent as follows:
AN US
qiPH H 2 p L 1 p wwP / 2 wrP / 1 ; qiLP L 1 p Hp wwP / 2 wrP / 1
We now show that any candidate quantity qiP for a pooling equilibrium must satisfy:
M
qiP min qiLP , qiHP qiLP L 1 p Hp wwP / 2 wrP / 1 qiP .
The high-type incumbent prefers to pool as long as his profits under pooling are higher than the profits he would obtain by ordering a high enough quantity to reveal his type. That is,
ED
1 H qiP qeP qiP qiP wrP qiP qmax 1 H qiH qeH qiH qiH wrP qiH , q iH
(D-4)
P i
qeH qiH arg max H qiH qeH wwP qeH H qiH wwP / 2 .
PT
where
qeH
Simplify inequality (D-4):
CE
1 H qiP wwP / 2 qiP wrP qiP 1 H wwP 2wrP / 8 8 2
Solve it and we get,
H 1 3H 2wwP 4wrP / 1 2wrP / 1
AC
qiP 2 H
2H
H 1 3H 2wwP 4wrP / 1 2wrP / 1
/ 2,
/ 2
Thus, the lowest pooling quantity for the high-type incumbent is
qPiH min 2 H
H 1 3H 2wwP 4wrP / 1 2wrP / 1
/2 .
Similarly, for the low-type incumbent, we verify 39
ACCEPTED MANUSCRIPT
1 L qiP qeP qiP qiP wrP qiP qmax 1 L qiL qeL qiL qiL wrP qiL q iL
P i
(D-5)
qeL qiL arg max H qiL qeL wwP qeL H qiL wwP / 2 .
where
qeL
Simplify inequality (D-5):
2wP 1 H 2 L wP 2 r w wwP wrP H q w P if L 0 qiL 0 P P 1 L qi wr qi 8 1 2 1 2 2 0 otherwise P P w w H Thus, we only need to check for L r w 0 . Solve it and we obtain 2 1 2
2L
H 1 4 L H 2wwP 4wrP / 1 2wrP / 1 H 1 4 L H 2wwP 4wrP / 1 2wrP / 1
AN US
qiP 2 L
P w
CR IP T
P i
/ 2,
/ 2
Thus, the lowest pooling quantity for the low-type incumbent is
qPiL min 2 L
H 1 4L H 2wwP 4wrP / 1 2wrP / 1
/ 2.
M
Therefore, we can obtain the following corollary from the above analysis.
ED
Corollary D1 There is no pooling equilibria below max qPiH min , qPiL min qPiH min q P .
In order to hold the pooling equilibrium, we also need to guarantee that the incumbent has no
PT
incentive to profitably deviate to a low enough order quantity to be considered as the low type. Firstly, the profit of high-type incumbent under pooling should be higher than its profit from
CE
ordering a low enough quantity to reveal the low type:
1 H qiP wwP / 2 qiP wrP qiP qmax 1 H qiH L qiH wwP / 2 qiH wrP qiH (D-6) q P
AC
iH
Secondly, the profit of low-type incumbent under pooling should be higher than its profit from ordering a low enough quantity to reveal the low type:
1 L qiP wwP / 2 qiP wrP qiP qmax 1 L qiL L qiL wwP / 2 qiL wrP qiL q P
(D-7)
iL
Solve constraint (D-6) step by step: the constraint (D-6) is equivalent to solve:
40
ACCEPTED MANUSCRIPT
1 2H wwP qiP qiP 2wrP qiP qmax 1 2H L wwP qiH qiH 2wrP qiH q
(D-8)
P
iH
We examine the two cases for q P . Suppose q P 2H L 2wrP / 1 wwP / 2 . Then,
maxP 1 2 H L wwP qiH qiH 2wrP qiH 1 2 H L wwP 2wrP
qiH q
1 2H w 2w P w
P r
2
2
/ 4 4
/ 4 4 1 2H wwP qiP qiP 2wrP qiP , which contradicts (D-
CR IP T
8). Thus, we must have q P 2H L 2wrP / 1 wwP / 2 . Given q P 2H L 2wrP / 1 wwP / 2 ,
max 1 2H L wwP qiH qiH 2wrP qiH 1 2H L wwP q P q P 2wrP q P . That is, we need
qiH q P
AN US
to solve 1 2H wwP qiP qiP 2wrP qiP 1 2H L wwP q P q P 2wrP q P . Solve the inequality and we obtain
2 1 2 H L wwP 2wrP 1 P P q 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
P
2 1 2 H L wwP 2wrP 1 P P q 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
M
ED
P
or
.
Combining with the condition q P 2H L 2wrP / 1 wwP / 2 , we get q P qPiH min qiP 2 1 2 H L wwP 2wrP 1 P P 1 2 H L ww 2wr 2 1 4 1 qiP 1 2 H qiP wwP 2wrP
, which is
CE
PT
smaller than 2H L 2wrP / 1 wwP / 2 .
AC
Next we solve constraint (D-7), which is equivalent to solve:
1 2L wwP qiP qiP 2wrP qiP qmax 1 L wwP qiL qiL 2wrP qiL q P
(D-9)
iL
We examine the two cases for q P . Suppose q P L 2wrP / 1 wwP / 2 . Then,
maxP 1 L wwP qiL qiL 2wrP qiL 1 L wwP 2wrP
qiL q
2
/ 4 4
41
ACCEPTED MANUSCRIPT
1 2L w 2w P w
P r
2
/ 4 4 1 2L wwP qiP qiP 2wrP qiP , which contradicts (D-
9). Thus, we must have q P L 2wrP / 1 wwP / 2 . Given q P L 2wrP / 1 wwP / 2 ,
max 1 L wwP qiL qiL 2wrP qiL 1 L wwP q P q P 2wrP q P . That is, we need to solve
qiL q P
CR IP T
1 2L wwP qiP qiP 2wrP qiP 1 L wwP q P q P 2wrP q P . Solve this inequality and we obtain qP
1 P P 1 L ww 2wr 2 1
qP
1 P P 1 L ww 2wr 2 1
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP or
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP .
AN US
Combining with the condition q P L 2wrP / 1 wwP / 2 , we get q P qPiL min qiP 1 P P 1 L ww 2wr 2 1
1 L w 2w P w
P r
2
4 1 qiP 1 2L qiP wwP 2wrP ,
M
which is smaller than L 2wrP / 1 wwP / 2 . Lemma D2 qPiL min qiP qPiH min qiP qiP .
ED
Proof. First, we show qPiL min qiP qPiH min qiP . Substituting the expressions for qPiL min qiP and
1 2H L w 2w P w
P r
CE
G1
PT
qPiH min qiP , we obtain qPiL min qiP qPiH min qiP
G2 2 1 H L
2
G1 G2 , where 2 1
4 1 qiP 1 2 H qiP wwP 2wrP
1 L w 2w P w
P r
2
and
4 1 qiP 1 2L qiP wwP 2wrP .
AC
Note that G12 G22 4 1 H L g1 g2 , where g1 1 L wwP 2wrP 2 1 qiP and g2
1 L w 2w P w
P r
2
4 1 qiP 1 2 L qiP wwP 2wrP . Note that
g12 g22 4 1 qiP L 0 . Since g 2 0 , we obtain g1 g 2 , implying G1 G2 . Thus, we 2
obtain qPiL min qiP qPiH min qiP .
42
ACCEPTED MANUSCRIPT
Next we show qPiH min qiP qiP . Substituting the expression for qPiH min qiP , we obtain qPiH min qiP qiP
F2
F1 F2 , where F1 1 2H L wwP 2wrP 2 1 qiP and 2 1
1 2H L w 2w P w
P r
2
4 1 qiP 1 2 H qiP wwP 2wrP . Note that
F12 F22 4 1 qiP L 0 . Since F2 0 , we obtain F1 F2 , implying qPiH min qiP qiP .
CR IP T
2
Therefore, we obtain the following corollary from the above analysis.
Corollary D2 For a pooling-equilibrium quantity, the low threshold in the entrant’s belief system must satisfy q P qPiL min qiP .
AN US
Lemma D3 When r r P , the pooling equilibrium exists.
Proof. From Lemma D1, in a Pareto-dominant pooling equilibrium, the incumbent and entrant respectively orders
qiP L 1 p Hp wwP / 2 wrP / 1 , qeP L 3 p H L 3wwP / 4 wrP / 2 2 . (D-10)
M
We have the supplier’s profit in high and low demand state as follows: P sH qiP H qiP qeP wrP qiP wwP qeP ,
ED
sLP qiP L qiP qeP wrP qiP wwP qeP .
PT
Then, we have the supplier’s expected profit as follows: P E sP p sH 1 p sLP .
E sP 0 wrP
E sP 0 , and get the optimal wholesale price offered by the supplier: wwP
AC
and
CE
We solve the following first-order conditions under a given simultaneously, i.e.,
wrP 1 4L 3 / 8 4 , wwP / 2 .
(D-11)
Substituting (D-11) into (D-10), we obtain qiP 2L 1 1 2 / 4 2 , qeP 1 3 2L / 8 4 .
From Corollary D1, the pooling equilibrium exists if q P qiP . Substituting (D-11) into q P , we solve the following inequality q P qiP and obtain r r P .
43
ACCEPTED MANUSCRIPT
Hence, from Lemma D1-D3 and Corollary D1-D2, we can obtain the Proposition D.
Justification for Ignoring Pooling We further discuss the existence of the pooling equilibrium and find that it is unnecessary to consider the pooling equilibrium in most scenarios, for the following reasons:
CR IP T
Firstly, from the Proposition D, a pooling equilibrium does not exist if r r P because it is costly for the high-type incumbent to pool with the low type. The condition r r P includes the many interesting cases.
Secondly, if a pooling equilibrium is possible (when r r P ), we compare the supplier’s
preference for the pooling and non-leakage cases. We find that the supplier would support the
rr P s
AN US
pooling equilibrium when
5 21 18 2 4 3 p 1 6 2 4 3 2 6 13 9 2 2 3 p 2 3 p 1 6 2 4 3 2 6 7 2 2
.
From the above analysis, the pooling case may exist if r min r P , rsP . It’s clear from Figure
M
D1 that the region of pooling case is very small. Therefore, although the pooling equilibrium strictly shrinks the region of NLE, we believe that we cover the majority of practical situations to
ED
analyze the existence of NLE in the paper. Note that if we further investigate preference of the high- and low-type incumbent for the pooling and non-leakage cases, the region of pooling case will become smaller. In addition, if necessary, pooling can be prevented by adding a constraint
PT
qiP q P in Proposition 1. Hence, in this paper, we ignore the pooling equilibrium and focus on
CE
the separating equilibrium when there exists information leakage.
Region where
Pooling cannot
be ignored
AC
Figure D1 The
44