Strategic information sharing under revenue-sharing contract: Explicit vs. tacit collusion in retailers

Computers & Industrial Engineering 131 (2019) 99–114

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Strategic information sharing under revenue-sharing contract: Explicit vs. tacit collusion in retailers

T

Daozhi Zhaoa, Mingyang Chena, , Yeming Gongb ⁎

a b

College of Management and Economics, Tianjin University, 300072 Tianjin, China EMLYON Business School, 23 Avenue Guy de Collogue, 69134 Ecully Cedex, France

ARTICLE INFO

ABSTRACT

Keywords: Supply chain management Information leakage Collusion Revenue-sharing contract

This work explores collusion and information sharing in a supply chain consisting of two downstream retailers and a mutual upstream manufacturer with the coordination by a revenue-sharing contract. We first analyze the equilibrium strategies of members in different types of collusion and the effect of collusion on information sharing. Considering the consumer surplus, downstream competition, collusion preference, and sharing with payment, we study the relationship among these factors and information sharing. We find that, under the revenue-sharing contract, explicit collusion will completely inhibit information sharing and tacit collusion can partially discourage the retailers from sharing information only if the wholesale price cannot accurately deliver the manufacturer’s information to retailers, as opposed to that under a wholesale-price contract. Moreover, we show that the downstream market will attend to use the explicit collusion with the increase of the accurate signal and the weaker quantity competition between retailers will contribute to collusion. However it will not change the ways of collusion and information sharing. Finally, we demonstrate that information sharing can be achieved through side payment if the manufacturer gives retailers subsidies and tacit collusion is not always better than explicit collusion in terms of consumer surplus. Our research provides new interesting insights and makes difference with the existing studies which show that tacit collusion can result in lower consumer surplus.

1. Introduction Our research is partially motivated by the evidence that some firms break the law and collude in a supply chain for obtaining higher profit margins, especially some firms that produce luxuries and single usage products. In 2014, the antitrust authorities (ACCC and EC) adjudicated two cases about the collusion: One is petrol retailers in Australia (ACCC, 2014), the other is German luxury automakers (Schmitt, 2017). The reason is that these retailers and manufacturers reach the vertical information sharing agreement with a mutual partner in privacy to communicate with each other about their prices and get a proportion of profits. In recent years, many organizations have started using the flexible supply contracts because they can share the risks among the partners, such as buy-back and revenue-sharing contracts (Hou, Zeng, & Zhao, 2009; Zhang, Donohue, & Cui, 2015). The number of organizations to use the flexible contracts will continue to grow (Doshi, 2010). The types of collusion which have been long stipulated by antitrust and competition law in the above cases include explicit collusion, such as vertical and horizontal information sharing in petrol retailers in Australia and only horizontal information sharing

⁎

in Colgate (ACCC, 2016) and 1–800 Contacts (FTC, 2018), and tacit collusion, such as vertical and horizontal information sharing in German luxury automakers and only horizontal information sharing in supermarkets in UK (OFT, 2011). The former, in particular, is illegal in most jurisdictions because a cartel is built among members to publicly exchange prices (Modak, Panda, & Sana, 2016). Thereby, we study these phenomena of collusion and the problems of sharing information between the collusive retailers and their mutual manufacturer under revenue-sharing contract. While scholars have analyzed information sharing and collusion, there are some gaps overlooked in academic research. For example, previous research mainly considers wholesale-price contract and ignores the impact of other contracts on information sharing and collusion. Revenue-sharing contract, as a common contract in the supply chain, is not only widely used in several industries, such as entertainment, sports leagues, software, and online retailing (Kong, Rajagopalan, & Zhang, 2013), but also prevalently applied by the aforementioned collusive organizations, such as automobile manufacturers (Guo & Hou, 2012) and other single usage producers (Palsule-Desai, 2013), since it can help coordinate the supply chain and promote information sharing.

Corresponding author at: College of Management and Economics, Tianjin University, Weijin Road No. 92, Tianjin 300072, China. E-mail addresses: [email protected] (D. Zhao), [email protected] (M. Chen), [email protected] (Y. Gong).

https://doi.org/10.1016/j.cie.2019.03.035 Received 21 October 2017; Received in revised form 1 February 2019; Accepted 17 March 2019 Available online 20 March 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

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Hence, we consider the combination of collusion with revenue-sharing contract in supply chains. Moreover, many other interesting topics are not fully considered in previous research, such as downstream competition, retailers’ preferences, and payment for sharing. To bridge these gaps, we consider what will happen to information sharing among the downstream partners if the upstream manufacturer uses the revenue-sharing contract to coordinate the supply chain. As far as we know, few scholars have studied the problem that retailers share the demand information with the manufacturer to establish a cartel under revenue-sharing contracts. In this paper, we will answer the following research questions:

2. Literature review Our research is relevant to the following research streams: (1) information leakage effect, (2) information sharing and collusion, and (3) revenue-sharing contract. 2.1. Information leakage effect By developing a supply chain model with one supplier and multiple retailers who are engaged in Cournot competition, Li (2002) studies the incentive of vertical information sharing and shows that when the supplier sets the wholesale price, those retailers who choose not to share information can infer the information of those who do it from the wholesale price. He calls this information leakage. If the partial substitutes and complements of retailers’ products are considered into the information sharing models, information sharing will not occur voluntarily unless the manufacturer offers subsidies (Zhang, 2002). Li and Zhang (2008) study the different types of confidentiality agreements and information sharing strategy subsequently. Anand and Goyal (2009) establish a supply chain system which consists of one upstream supplier and two downstream retailers, and study the problems of strategic information management under information leakage. If the revenue-sharing contract is used to coordinate the supply chain, it can better align the incentives of the members and prevent the information leakage (Kong et al., 2013). Shamir (2016) and Wang, Tang, and Zhao (2018) respectively analyze the strategic information leakage in cartel formation and information leakage in the presence of a dominant retailer. Our work is different from the above literature because we focus on the revenue-sharing contract. Although Kong et al. (2013) survey the information sharing in supply chain under revenue-sharing contract, they consider the information sharing with incumbent retailer and entrant retailer. Hence, this paper focuses on the impact of revenuesharing contract on retailers who have already existed in the market.

• What are the impact of different types of information sharing and • •

collusion on the optimal pricing strategies, the manufacturer’s profit, and retailers expected discounted profits under revenuesharing contract? What are the impact of different types of collusion on information sharing under revenue-sharing contract? Compared with wholesale-price contract, what are the differences in information sharing under revenue-sharing contract considering collusion?

To answer research questions, we build a two-echelon supply chain model consisting of two retailers (he) and a common manufacturer (she), with the purpose to analyze the effect of collusion among retailers on information sharing under revenue-sharing contract. We consider two types of collusion and two ways of information sharing in a supply chain: Model BA considering explicit collusion between two retailers and both sharing private information with the manufacturer. Model ND considering explicit collusion among retailers, but without information exchange in the supply chain. Model OM considering tacit collusion between two retailers, with both only sharing information with the manufacturer. Model CND considering tacit collusion among retailers, but with no information exchange in the supply chain. Models of BA and OM respectively correspond to the examples of petrol retailers and luxury automakers. Models of ND and CND respectively correspond to the examples of supermarkets in UK, Colgate and 1–800 Contacts. This work provides several theoretical contributions and practical implications. First, we contribute to the information sharing literature by combining an existing information sharing model with collusion and revenue-sharing contract since few previous studies have considered. Second, this paper finds some interesting results that have not been referred to in prior studies on the collusion and information sharing with the wholesale-price contract. Moreover, this work also considers a conditional probability about the market condition for the first time. The probability of high demand signals observed by retailers in high demand state is different from that of low demand signals in the low demand state. Finally, this paper offers valuable insights to antitrust enforcement policy and collusive downstream market: It is not always a good strategy for a retailer to share the information with a manufacturer in both types of collusion unless the manufacturer gives him subsidies. For antitrust authorities, they should not only pay attention to explicit collusion, but also tacit collusion, since tacit collusion can sometimes lead to greater harm under revenue-sharing contract. Considering that tacit collusion may not be illegal, antitrust agencies need to introduce policies to intervene in tacit collusion. The rest of this paper is organized as follows. The next section provides a literature review. Section 3 sets up the model. Section 4 presents a model with explicit collusion and provides an equilibrium analysis. Section 5 analyzes the information sharing with tacit collusion and provides an equilibrium analysis. Section 6 extends the base model to consider downstream competition, retailers’ preferences, consumer surplus, and sharing with payment. Section 7 concludes the paper. All proofs are provided in Appendix.

2.2. Information sharing and collusion We review two aspects of collusion and information sharing in supply chains. One is explicit collusion: In a two-echelon supply chain, the manufacturer will be hurt if there is sharing information in supply chain or explicit collusion between suppliers, whereas the suppliers can benefit from the cooperation and information sharing (Shi, Zhou, Wang, Xu, & Xiong, 2014). Gomez-Martinez, Onderstal, and Sonnemans (2016) study the impact of information with competitors’ actions on cartel stability and the formation of cartel incentive mechanism in the Cournot markets. Asker, Fershtman, Jeon, and Pakes (2016) analyze the competition effects of information sharing involving the influence of collusion information on stocks and bidding behaviors. Roy (2017) considers the action correction, information and collusion in an oligopoly market. The other is tacit collusion between the upstream and downstream. Bertomeu, Evans, Feng, and Tseng (2015) study how the automotive industry peers share information when they are engaged in tacit collusion. Wang, Zhou, Min, and Zhong (2011) analyze the problem of cooperative advertising in the monopoly market and discuss some collusion structures. Different from previous research on collusion in supply chains, we combine the supply chain contract with information sharing and collusion and focus on the revenue-sharing contract. To our knowledge, there are few researches in this field at present. Furthermore, our work is also different from other literature. First, we extend the research of wholesale-price contract to that of revenue-sharing contract. Second, we analyze the cases which have not been considered in the previous works, such as sharing with payment and retailers’ preferences for collusion. 100

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2.3. Revenue-sharing contract

Table 1 Summary of notations.

Revenue-sharing contract has been shown to be an effective mechanism that aligns firms’ incentives by inducing appropriate behavior in a supply chain in literature (Kong et al., 2013) and it is also widely used in other areas like low-carbon supply chain (Bai, Gong, Jin, & Xu, 2019) and logistics service (Liu, Xu, & Kouhpaenejad, 2013). In a supply chain model consisting of one supplier and one retailer, the revenue-sharing contract can achieve coordination of supply chain under voluntary compliance (Cachon, 2003). Cachon and Lariviere (2005) study revenue-sharing contract under a general model of supply chain and compare revenue sharing to a number of other supply chain contracts. Zhang et al. (2015) analyze the contract preference and supplier risk aversion of buy-back contract and revenue-sharing contract. Hou, Wei, Li, Huang, and Ashley (2017) consider the coordination problem of decentralized supply chain with simultaneous game or Stackelberg game based on revenue-sharing contract. Our work is also different from the previous studies since we combine revenue-sharing contract, information sharing, and collusion.

Notation

Definition and Comments

p A A

Market clearing price Potential market size Expectation of the uncertain market potential

(0, 1) Revenue-sharing ratio and Wholesale price Expected discounted total profit of the supply chain Expected discounted profit of the manufacturer Expected discounted profit of the retailer i Demand signals observed by retailers Collusive profit of the retailer i in the c model

w V VM Vri c ri ri (w ,

Common discount factor Deviation profit of retailer i

pi , p ic ) pc

Retail price in collusion Deviation price of retailer The competition factor The probability that retailers can observe the signals h/l Degree of uncertain demand Prior probability of high demand Probability of observing the signal = h given high demand state

p h/ l

r µ

3. The model We consider a supply chain consisting of one mutual upstream manufacturer (she) and two downstream retailers (he) with two types of collusion among retailers. In the explicit collusion, two retailers share information and determine the retail price through direct communication. In the tacit collusion, retailers do not share information and just infer the signals observed from others by the manufacturer’s wholesale price. All firms are risk neutral with objectives to maximize expected profits. We use a linear market demand function as shown in Eq. (1), which demand function forms can be found in existing literature (e.g., Li & Zhang, 2015).

Q = (A

where µ indicates the probability of observing the h signal under the high market demand. Eq. (2) shows that l signal can be accurately observed by downstream when the market size is low and h signal can only be observed with a probability of µ when the market size is high. The probabilities shown in Eq. (2) are different from those described in some literature which assumes Pr (h H ) = Pr (l L) (e.g., Li & Zhang, 2015). However, for some products with larger price elasticity of demand (e.g., luxuries) or single usage (e.g., glasses), retailers can observe perfect signals about market conditions when the demand is low, i.e., Pr (l L) . Thus, comPr ( = l A = L) = 1, and we can get Pr (h H ) bining with other forms of probability, and supposing the products are luxuries and non-essential and single-usage products (e.g., sunglasses), we introduce the semi-separating conditional probability which is shown in Eq. (2) for the first time. So we can get the probabilities that retailers can respectively observe the signals h and l are as follows:

(1)

p)+,

= max (A p , 0), A denotes the potential market size where (A and can take one of two values: a high value H with probability r (0, 1) and a low value L with probability 1 r . H and L respectively correspond to high and low demand states, with H > L > 0 . The prior probability of market states is a common knowledge to all parties. The expectation of the uncertain market potential is A and A = rH + (1 r ) L . p is the market clearing price. Q represents the quantity sold in the market. For the convenience of formulation, we assume that there is no competition between retailers in the market. We can also consider this practice as a full collusion instead of semi-collusion because of higher profits. The first incentives for collusion in the downstream market is to earn supra-competitive profits by coordinating the market conduct (Andreoli & Franck, 2015; Fonseca & Normann, 2012). Another incentive is to avoid a price war and zero profits in the future periods, such as the noncooperative behavior of the Joint Executive Committee railroad cartel in the 1880s (Porter, 1983). The similar setting has been used in literature (e.g., van den Berg & Bos, 2017). This assumption will be relaxed in Section 6.2 and we consider the downstream competition. Table 1 summarizes the main notations used in the paper. Before the beginning of each selling period, each retailer has the {h, l} about the random opportunity to observe the demand signal market condition. Let h and l respectively denote the signal of H and L demand. The signal can partially reflect the demand state of the market. The conditional probability is shown as Eq. (2): p )+

Pr

=h =

µ, A = H , 0, A = L,

Pr

=l =

1 µ, A = H , 1, A = L,

h

= Pr (h) = Pr (H ) Pr (h H ) + Pr (L) Pr (h L) = µr ,

l

= Pr (l) = Pr (H ) Pr (l H ) + Pr (L) Pr (l L) = (1

µ) r + (1

r ).

Clearly, h + l = 1. Therefore, the Bayesian updated probabilities of the demand states conditional on the signal are shown in Eq. (3) which means that each retailer will update the probability after the observation of signals:

Pr (H h) = 1, Pr H l =

(1

(1 µ) r µ) r + (1

r)

1 r µ) r + (1

r)

,

Pr (L h) = 0, Pr L l =

(1

.

(3)

The expectation of A conditional on the demand signal is:

H = E [A h] = H , L = E A l = H (1

(1 µ) r µ) r + (1

r)

+ L (1

1 r µ) r + (1

r)

,

and L < A < H . (4)

Eq. (4) shows that the manufacturer will believe that market demand is high if at least one retailer needs to provide the h signal, and the manufacturer will believe that market demand is low if both retailers offer l signals. The sequence of events and decisions under

(2) 101

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The retailers decide whether to disclose their demand signals to the manufacturer.

Upon receiving the wholesale price w , the collusive retailers set the retail price equal to the market clearing price p .

The retailers observe the demand signals . If two parties can reach a sharing agreement, the manufacturer will get the signals from retailers and set the wholesale price w according to the information received.

Based on the market clearing price p , the quantity sold in the market Q will be achieved and retailers share the market equally.

The profits of firms in the supply chain will be achieved.

Fig. 1. Sequence of events.

revenue-sharing contract is shown in Fig. 1: The purpose of observing the demand signal which is shared with the manufacturer is to help the stakeholders make better decisions and obtain the optimal profits. So the retailers will have no incentive to send the signal if there is no information sharing in the supply chain. Thereby, the retailers first decide whether to disclose the information and then obtain the signal. A similar setup has been used in literature (Li & Zhang, 2015; Zhang, 2002). We also assume that the collusive retailers obtain the equal demand since they set the same retail price and share the market equally, which assumption can be found in exiting literature (e.g., Shamir, 2016). If the retailer i sets a price pi which is higher than pc , the market share of retailer i will be zero. If the price pi is lower than pc , the collusion will not be sustained. Hence, retailers will share the market equally if they design a single cartel price, which leads to the equal demand. We consider as the market discount factor indicating the weight attached by firms to the future in the infinitely repeated games. To solve the maximal profit problem with collusion in the downstream market at a given discount rate , we use the one-stage deviation principle for infinite-horizon games (Benjamin, 2016; van den Berg & Bos, 2017). In an infinite-horizon multi-stage game with observed actions that is continuous at infinity, profile S is subgame perfect if and only if there is no player i and strategy st that agrees with st , except at a single t and ht , such that st is a better response to s t than st conditional on history ht being reached (Fudenberg & Tirole, 1991). Based on the definitions in Fudenberg and Tirole (1991), we can obtain Eq. (5): t

Vri =

Fudenberg and Tirole (1991) and Friedman (1990) (as shown in Lemma 1). Lemma 1. Collusion among retailers can be sustained when ri (w ,

Vri =

c ri

Vri =

1

.

c ri (w ,

q f d f

q f

f (q ) N f

,

f

0, 1 ,

,1 .

pc ) + Vri

ri (w ,

(6)

pi , p ic ).

pc ) and ri (w, pi , p ic ) respectively denote the collusive where profit per period and the deviation profit per period. pc is retail price with collusion. pi is deviation price of retailer i. Constraint (6) indicates that the retailer i’s profit that he deviates from collusion in the current period will be less than the profit that he colludes. Therefore, Constraint (6) prevents retailers from deviating collusion. Since any deviation will trigger a price war and lead to zero profits in all future periods, the collusive profit of retailer i is equal to ri (w, pi , p ic ) . 4. Explicit collusion and information sharing in a supply chain The structures of information sharing in supply chain with the explicit collusion are shown in Fig. 2: (1) Model BA (both disclosure to all) indicates that there exists the explicit collusion among retailers and information sharing between the manufacturer and retailers; (2) Model ND (no disclosure) shows that there exists the explicit collusion among retailers and no information sharing between the manufacturer and retailers.

(5)

d f

,

c ri (w ,

where Vri denotes the expected discounted profit of retailer i in the current period and ric is the profit of the retailer i considering collusion in the c model. Although one of the retailers deviates from the strategy at some stage, there is no punishment in an infinitely repeated game with nocollusive setting. However, the deviation at any time can incur punishment and lead to trigger strategy for collusive retailers. Friedman (1990) demonstrates that the trigger strategy is not only a Nash equilibrium, but also a subgame perfect Nash equilibrium—a refinement of a Nash equilibrium when the repeated game satisfies a certain (F , X , ) denote an infinitely repeated game condition. Let G consisting of a repetition of a simultaneous-move stage game and q N = (q1N , …, qFN ) X be a Nash equilibrium of G. (q , q N ) X × X represents a grim-trigger strategy combination. Then (q , q N ) is a subgame-perfect equilibrium if and only if

pc )

Lemma 1 shows that collusion can sustain when the market discount factor is higher than . Otherwise, collusion will not be sustainable. According to Colombo (2013), collusion is not only sustainable but also a subgame perfect Nash equilibrium when the discounted profit with collusion is higher than the discounted profit that players deviate from the cartel agreement, so the profits of collusion among retailers must satisfy the Constraint (6) in equilibrium:

c ri,

+ Vri

c ri (w ,

Vri

t=0 c ri

pi , p ic )

4.1. Model BA (both disclosure to all) In Model BA, two retailers share information with each other and with their common manufacturer, which lead to the explicit collusion in the downstream market and the formation of a cartel. As a result, the

M

F,

Ri

where fd ( q f ) is firm f’s payoff from its best response to other firms’ action q f . Thus, the conditions for sustainable collusion can be formulated in an infinitely repeated game based on the theorems in

M

Rj

Ri

Rj

Fig. 2. Supply chain structures with explicit collusion. 102

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retailer will set the retail price equal to the monopoly price p BA and the conditions that collusion can be sustained are as follows: c ri (H ,

EA [

whBA, phBA ) + Vri

c ri (A ,

wlBA,

ri (H ,

BA

pl ) l] + Vri

whBA, p ), p

EA [

ri (A ,

wlBA,

propositions need to satisfy constraints.

phBA , p ) l], p

Proposition 1. When

plBA ,

c BA ri (w

w BA , p BA

, p BA )

max VriBA p

c ri (H ,

whBA, phBA ) + Vri

c ri (A ,

EA [

ri (H ,

wlBA, plBA ) l] + Vri

whBA, p ),

EA [

ri (A ,

wlBA, p ) l],

and p

plBA . Based on the above constraints, the retail price is:

p BA =

A (3 2(2

(a) Sharing information is always beneficial to the supply chain, i.e., V BA > V ND . (b) The effects of revenue-sharing ratio on the gaps of optimal profits are as follows:

The manufacturer maximizes the profit by optimally choosing the wholesale price:

max VMBA. w

d (VMBA

The wholesale price can be derived by:

w BA =

A (1 2

VriND ) d

> 0,

d (V BA

V ND ) d

> 0.

d (V BA

)2

V ND )

M M > 0 suggests the V BA V ND are all increasing in . d manufacturer can set a revenue-sharing ratio closer to the optimal revenue-sharing ratio when information sharing exists in supply chain. The reason is that the manufacturer is better off with more information about the downstream market demand and she will get more profits,

d (V BA

V ND )

ri ri > 0 shows that with the inespecially when is larger. d crease of , the difference between retailers’ profits with information sharing and without that is diminishing because retailers have no incentive to share their demand information with the manufacturer and

The manufacturer’s optimal wholesale price is:

.

they will get larger profit margins when

Given the manufacturer’s wholesale price, the retail price is:

E [p ND ] =

d (VriBA

Corollary 1(a) shows that the total profit of supply chain with information sharing is higher because sharing can mitigate the double marginal effects in supply chain, thus V BA > V ND . The above illustrations are also consistent with practices: Costco and 7-Eleven share the information with the supplier offering the advice to the retailers VMND , VriBA VriND and (Sillitoe, 2015). Corollary 1(b) confirms VMBA

w+A( 1+ ) . 2( 1 + )

A (1 2

> 0,

.

In Model ND, there is no information sharing in a supply chain and explicit collusion exists among the retailers. The requirement of Constraint (6) must be satisfied if retailers want to maintain the collusive price. When the manufacturer sets the wholesale price without knowing the demand signals of retailers, her expectation of the quantity is given by:

w ND =

VMND ) d

4.2. Model ND (no disclosure)

E [q ND ] =

, 1 , there exists the explicit collusion

Corollary 1. Under the explicit collusion model, we can get:

2 ) . )

)2

and off-schedule

(For proof, see Appendix.) Proposition 1(a) indicates that retailers have no incentive to share their demand information with the manufacturer. However, the manufacturer is better off with sharing because she can gain more profits by adjusting the wholesale price and revenue-sharing ratio based on the information, so VMBA > VMND . In Proposition 1(b), retailers do not need to infer rival’s signal from the wholesale price w when market demand is high and information sharing exists in supply chain. Although sharing allows retailers to set a higher retail price p BA > p ND , it can incur a higher wholesale price at the same time. Thus, compared the case without information sharing, retailers will get a lower profit in the case with information sharing, so VriBA < VriND . Conversely, retailers will not only receive a lower wholesale price wlBA and higher revenue-sharing ratio 1 by sharing information with the manufacturer under low market demand, but also obtain more profits with no information sharing than with information sharing by setting a higher retail price plND > plBA . Thereby, VriND > VriBA . Proposition 1 is consistent with the existing studies in nocollusive settings (e.g., Zhang, 2002; Li & Zhang, 2015).

According to the description in Section 1, the retailer i’s profit is given by

s. t .

r)

c ri (w, pc )

Vri

(a) Information sharing is always conducive to the manufacturer, i.e., VMBA > VMND . (b) Sharing information always hurts retailers, i.e., VriND > VriBA .

1 . 2

1

1 1 + µ (1

ri (w, pi , p ic )

among retailers and no information will be shared in supply chain.

where p is a retail price which is different from the monopoly price p BA . The above two constraints are the off-schedule constraints which mean that the retailer i must set an appropriate collusive price based on the signal when collusion exists in the downstream market. Otherwise, any deviation from collusive price will lead to a price war and eventually result in zero profits (Rao, Bergen, & Davis, 2000). The symmetric perfect public equilibrium (SPPE) is used as the solution concept and collusion members will yield the highest equilibrium payoff under SPPE. To satisfy the off-schedule constraints under the optimal SPPE, we give the following equation: c ri

=

is smaller.

d (V BA

V ND )

d

>0

VMND is higher than that of indicates that the growth rate of VMBA VriND VriBA , and the total profit of supply chain is increasing.

A( 2 + ) + A ( 1 + ) . 2( 2 + )

5. Tacit collusion and information sharing in a supply chain

4.3. Equilibrium analysis with explicit collusion

We now analyze the case when there is the tacit collusion among downstream retailers. In general, the common form of tacit collusion is that members communicate indirectly by price signals. When one raises

In this subsection, we compare the manufacturer’s profit, retailers’ profits, and total profit of supply chain in two models. All these 103

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M

max VM

s. t .

Ri

Rj

Ri

(8)

w

M

Rj

OM SM (H , pSh , wSh ) OM EA [ SM (A , pSl , wSl ) OM SM (H , pSh , wSh ) OM EA [ SM (A , pSl , wSl )

OM SM (H ,

l]

p (w ), w ), OM SM (A,

EA [

OM SM (H ,

l]

p (w ), w ) l],

pSl , wSl ),

EA [

OM SM (A,

and w

wSh/ wSl.

pSh , wSh ) l],

(9) The first two constraints in Eq. (9) show that the manufacturer has no incentive to produce off-schedule and the latter two constraints restrict the manufacturer’s on-schedule. For example, the manufacturer will set wSl when she receives the signal of l. Moreover, the objective of retailer i is to choose the price to maximize his expected discounted OM profit V Sri subject to the set of incentive constraints and Constraint (6):

Fig. 3. Supply chain structures with tacit collusion.

the price, he will expect that other members can understand his behavior by means of collusion and respond to him by raising the price (Shi et al., 2014). The structures of information sharing in supply chain with the tacit collusion are shown in Fig. 3: (1) Model OM (one disclosure to manufacturer only) indicates that there exists the tacit collusion among retailers without communication directly and they can only infer their rival’s signal through the wholesale price; (2) Model CND (collusion but not disclosure) is similar to Model ND and the difference is that the tacit collusion exists in the downstream market in CND model. For analytical tractability, we assume that is large enH = , where the parameter ough in equilibrium. We define is a L proxy for demand uncertainty given the set of observed signals, as it measures the ratio between the demand intercept during high demand periods and the expected demand intercept during the periods in which all retailers observe the low signals. A similar measure of demand uncertainty is used by Anand and Goyal (2009).

OM max V Sri p

s. t .

OM PM (H , p , w ) OM EA [ PM (A, p , w OM ( H , p, w ) PM OM EA [ PM (A, p , w

Fig. 3 shows that both retailers will disclose their private information vertically and no direct information sharing happens in the downstream market. Retailers determine the optimal retail price based on the wholesale price and private information in tacit collusion. Although there is no direct information exchange between duopoly retailers, they can infer the rival’s signal from w because of the information leakage effect (Zhang, 2002). To illustrate this further, we take the retailer i as an example. When he observes the signal l and the wholesale price is wSh , he can derive that the signal observed by retailer j must be h. When he observes the signal l and the wholesale price given by the manufacturer is wSl , he knows that the signal observed by retailer j will also be l. In Model OM, there exist two kinds of pure-strategy PBNE including separating and pooling equilibriums. In a pooling equilibrium, retailers will not conjecture the signals received by the manufacturer. Conversely, in a separating equilibrium, retailers can infer which types of signals the manufacturer has received according to the wholesale price and the probability of conjecture is shown in Eq. (7): (1

(1 µ) r µ) r + (1

r)

1, ws > ws .

, ws

EA [

whOM , phOM ) + Vri

c Sri (A ,

wlOM ,

plOM )

Sri (H ,

l] + Vri

whOM , p ),

EA [

Sri (A ,

. wlOM , p ) l]

Then, we analyze the behavior of both sides in a pooling equilibrium. In this case, retailers will not infer the rival’s signal based on the wholesale price wP . As a result, the manufacturer may deceive the retailers in the manner of high market demand by setting a wholesale price wPh when she receives low-demand signals, which leads retailers to increase the quantities. Generally speaking, the manufacturer who receives high demand signals will not imitate the strategy of low demand. Thus, the manufacturer receives high and low demand signals that are prone to pooling when the following constraints are satisfied:

5.1. Model OM (Both Disclosure to Manufacturer only)

Pr A = H =

c Sri (H ,

OM PM (H ,

) l]

OM PM (H ,

) l]

p , w ),

EA [

OM PM (A ,

p , w ) l],

and w

pPh , wPh),

EA [

OM PM (A ,

w.

pPl , wPl ) l],

(10)

Similar to the separating equilibrium, the first two constraints in Eq. (10) show that the manufacturer must design a uniform wholesale price w and the latter two constraints show that the uniform pricing is better than the separating pricing. The retailer’s objective is to choose the OM price to maximize his expected discounted profit V Pri subject to Incentive Constraint (6) and the following constraint: OM max V Pri p

s. t .

c Pri (H ,

EA [

whOM , phOM ) + Vri

c Pri (A ,

wlOM ,

plOM )

Pri (H ,

l] + Vri

whOM , p ),

EA [

Pri (A ,

wlOM , p ) l].

5.2. Model CND (collusion but not disclosure)

ws , (7)

In Model CND, there is no information sharing between the upstream and downstream markets, and the tacit collusion exists in the downstream market. Different from Model OM, retailers cannot infer their rival’s demand signal through the wholesale price in Model CND. The possible signals’ set received by the manufacturer is {(h, h), (l, h), (h , l), (l, l)} . When the retailer i observes signal h and the retailer j observes signal h simultaneously, the manufacturer believes that the market demand is H. So we can get that ri = rj = H . When the retailer i observes signal h and the retailer j observes signal l, she will also think that the market demand is H. Since retail price pjl is less than pih , we can derive rj = H > H . When the retailer i observes signal l and the retailer j also observes signal l, the manufacturer thinks that the market demand will be L and we can get ri = rj = L . Thus, the payoff matrix of retailers can be shown in Table 2. To avoid a price war after the collapse of tacit collusion, both

We first consider a separating equilibrium. To maximize profits, the manufacturer needs to choose the pricing strategy based on signals. Moreover, it can also produce two types of deviations, off-schedule and on-schedule. In off-schedule, the manufacturer may choose a wholesale {wh , wl } to gain more profits. In on-schedule, the manuprice w facturer will respectively set a wholesale price wSl and wSh when she receives h and l signals. Here, we consider on-schedule where the manufacturer receives signal l and sets wSh to illustrate how the manufacturer benefits from this deviation. To entice retailers to increase their orders and gain a higher profit, the manufacturer has the incentive to set wSh when she receives low-demand signals shared by retailers. The manufacturer’s objective is to choose the wholesale price to maximize her expected discounted profit VM subject to the following constraints in a separating equilibrium: 104

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Table 2 Payoff matrix of retailers in model CND. h h

H,

l

H,

2

0,

H

L,

0

1)2L

(

l H

1

L

, when 2

H

3 (H

L+ 1

ri (pl

1, 3 ,

p

p ((1

)p

pPlOM =

w ).

w

(3 2 ) A . 2(2 )

p

p+w .

The manufacturer’s optimal wholesale price is:

wCND =

)2A

(1 2

.

5.3. Equilibrium analysis with tacit collusion In this subsection, we compare Model OM and Model CND. All propositions must satisfy Constraint (11) and ri (w, p , p

c (w , p )

)

ic c ri = . Otherwise, the manufacturer or retailers Vri can distort the signals and tacit collusion will not be sustainable. The subscript S and P respectively represent separating equilibrium and pooling equilibrium.

Proposition 2. If the tacit collusion exists in the downstream retailers, we can get: (a) The optimal wholesale price w and retail price p in a separating equilibrium and pooling equilibrium are as follows: OM pSh =

(3 2 ) H ( 1)2H OM , wSh = , 2(2 ) 2

pSlOM (3 2 ) L , 2(2 ) 2

3

when

H+ 1

L+ 2(2

=

0,

3 2

0,

3 2

,

3

1, 3 ,

(H

L )((3

) H + (1 1

)

2

,1 ,

)

OM , wPl =

Proposition

3.

whOM = whBA

and wlOM = wlBA . If

wlOM

)L)

, when

A w + 2 2(1

(1

)((2 2

)L

A)

+

.

(For proof, see Appendix.) Proposition 2(a) summarizes the pure-strategy equilibrium outcomes with the tacit collusion. These equilibrium prices correspond to the examples of a single cartel price which is set by the collusive retailers in the introduction. As described in Proposition 2(a), the manufacturer and retailers design the optimal prices which depend on the condition of the market demand and signals. In a separating equilibrium, the manufacturer will think the market demand is high and set a OM wholesale price wSh when there is at least one retailer who observes OM the signal h. Then, retailers choose a retail price pSh equal to the market clearing price. When both retailers observe the signals l, the manufacturer will believe that the market demand is low and design a wholesale price w = wSlOM , and retailers choose the price pSlOM subseOM when quently. The manufacturer has no incentive to deviate from wSh receiving high demand signals, but she may deviate from wSlOM and OM choose wSh when the low demand signals are received. Hence, there is only one optimal strategy for the manufacturer or retailers in the high market demand. When the market demand is low, the optimal strategies can be obtained in two cases according to whether the manufacturer can set a wholesale price easily for retailers to infer or not. Proposition 2(b) provides a filter of Pareto-dominance in obtaining a pooling equilibrium in closed form and states that in some cases, the existence of pooling equilibrium wherein the retailers’ conjectures are “reasonable” is precluded. In a pooling equilibrium, the manufacturer OM will set a uniform wholesale price wPl no matter which types of signals are received and retailers only choose the price pPlOM . Generally, a pooling equilibrium does not exist in a fairly general situation because the manufacturer usually has no motivation to set a uniform price when she receives different signals. Moreover, we can also prove that there is no information sharing in the pooling equilibrium and information sharing is disadvantageous to the manufacturer since 1 OM V Pl V CND = 4 r 2µ2 ( 2)( 1)2 < 0 . wOM = Naturally, wSOM , pOM = pSOM , OM = SOM . Similar assumptions are made in the literature of information sharing (e.g., Kong et al., 2013).

The ex-ante expected profit for the manufacturer is given by:

max A

, when

h (b) When wP < wPmin , there can be no pooling equilibrium. When (1, ( , r , µ )], a Pareto-dominant pooling equilibrium exists.

The optimal retail price is:

pCND =

)L)

(11)

= h).

Under Constrain (11), the retailer will respectively set ph and pl when observing the signals of high demand and low demand. Furthermore, Model CND can be implemented when Constraint (11) is satisfied, which is different from Model ND. Ex-ante expected profit for retailers is given by:

max A

) H + (1

2

retailers will choose the optimal strategy { L, L} . Moreover, a retailer observing the high demand signal has the motivation to mimic the low demand signal and sets a lower retail price pl . To prevent retailers from distorting the signals, Constraint (11) is needed:

= h)

L )((3

,1 ,

2

1

wSlOM =

ri (ph

3

1, 3 ,

wlBA .

If

3

(1, 3),

[

3

(1, 3),

2

, 1) ,

(0,

we 3 2

can

get

) , we have

(For proof, see Appendix.) As described in Proposition 2, downstream retailers can correctly infer whether or not the manufacturer has received useful signals when 3, so the manufacturer has no incentive to imitate the wholesale price whOM in the setting with high demand with wlOM in the setting with low demand. Hence, she must set a wholesale price equal to w BA to encourage retailers to share private information. When

1, 3 ,

,

105

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(a) Information sharing always makes the supply chain better off:

3

(1, 3), (0, 2 ) , retailers cannot distinguish directly whether the manufacturer has received useful signals, but they can infer whether or not the manufacturer intentionally conceals the information of signals based on the revenue-sharing ratio . When the revenue-sharing 3 [ 2 , 1) , retailers will derive that the manuratio is larger, i.e., facturer has received the information of high demand. The reason is that retailers’ profits will be higher in the high demand and the manufacturer will surely set a larger sharing ratio to gain more profits. In this case, the manufacturer is likely to replace the wholesale price wlOM with whOM when receiving the low demand signals. Evidently, retailers can also predict this distortion and may be reluctant to share information with the manufacturer when the market demand cannot be differentiated and the ratio is smaller. To stimulate retailers to share information, the manufacturer must design a wholesale price wlOM wlBA , which can convey the true demand state to retailers. Proposition 3 can be verified with the corollary in Gal-Or, Geylani, and Dukes (2008), and it also further complements it.

V OM > V CND , when

V OM > V CND , when

OM d (V M

3

1, 3 ,

2

CND VM )

d

OM d (V M

CND VM )

d

d (VriOM

(a) Information sharing is always beneficial to the manufacturer:

3

VriCND ) d

VriCND ) d

1, 3 ,

0,

3

(b) Information sharing is not always advantageous to the collusive retailers:

VriOM < VriCND , when

3

1, 3 ,

3

1, 3 ,

0,

0,

3

,1 ,

.

2

> 0, when

> 0, when

< 0, when

3

3

1, 3 ,

1, 3 ,

3

0,

2

3

,

2

3

1, 3 ,

,1 ,

2

,1 ,

< 0, when

1, 3 ,

0,

3

,

2

.

2

d (V OM

VriOM > VriCND , when

1, 3 ,

2

,1 ,

d (VriOM OM CND VM > VM , when

3

1, 3 ,

(b) The effects of revenue-sharing ratio on the gaps of optimal profits are as follows:

Proposition 4. When the tacit collusion happens in the downstream market, there will be information sharing on condition that retailers can infer whether the manufacturer has received useful information from the wholesale price. Otherwise, no information sharing is the unique equilibrium.

OM CND VM > VM , when

3

2

3 2

V CND ) d

d (V OM

,1 ,

V CND ) d

< 0,

3

> 0, when

1, 3 , when

3 2

,1 ,

1.

Corollary 2(a) shows that, in the tacit collusion, the total profit of supply chain with information sharing will be higher than that without information sharing because sharing alleviates the problem of double marginalization. This is consistent with Corollary 1 and the example of the German luxury automakers who share the information with the upstream market to improve the performance in cost savings. When 3 3 (1, 3), [ 2 , 1) , we have V OM > V CND . Let r = µ =

.

(For proof, see Appendix.) Proposition 4 emphasizes that the manufacturer is better off with more information about the downstream market demand when 3 3 (1, 3), (0, 2 ) because she 3 (1, 3), [ 2 , 1) and can seek more economic rents and information rents with better information through the direct effects of information sharing (Li, 2002; 3 3 (1, 3), [ 2 , 1) , information Zhang, 2002). When sharing will be possible because it is helpful to the formation of tacit collusion and strengthens stability of the collusion, which is beneficial to retailers (Lee & Whang, 2000). Thus we have VriOM > VriCND . When 3 (1, 3), (0, 2 ) , retailers will not be able to speculate on whether the manufacturer has received h signal from the wholesale price. Then there is no information sharing between members of upstream and downstream markets because the manufacturer can improve the usefulness of information and gain more profits by pooling (Zhang, 2002). Furthermore, information sharing is harmful to retailers in this case and we have VriOM < VriCND . Proposition 4 (b) is consistent with the evidence that Walmart would no longer share its sales data with outside companies like Information Resources and ACNielsen (Hays, 2004). By comparing the total profit of supply chain in two models, we can get the following corollary:

3

(1, 3), (0, 2 ) (Jiang, Tian, Xu, & Zhang, 2016), then when we can get V OM > V CND . Therefore, information sharing is profitable for the whole supply chain. The conditions of 3 3 (1, 3), (0, 2 ) mean the 3 (1, 3), [ 2 , 1) and signal of signaling problem because of asymmetric information in contracting problems. The retailers know whether the manufacturer receives the useful signals based on these parameters. A relevant example is the Spence’s model of education (Spence, 1973) where he uses the education as a signal to distinguish high- and low-productivity workers. In Corollary 2(b),

OM d (V M

CND ) VM

d

indicates that the difference be-

OM CND tween V M and V M is increasing in revenue-sharing ratio . The reason is that as increases, the manufacturer can design a ratio is after she receives information, so the closer to the optimal ratio payoff pq will increase. Different from Corollary 1, Corollary 2 pre-

d (V OM

V CND )

ri ri < 0 since retailers cannot directly transmit signals sents d in the tacit collusion, which reduces the stability of collusion among retailers. Accordingly, the loss of retailers’ profits cannot be compenOM V CND ) sated when is larger. Another difference is d (V . We can find

Corollary 2. In the tacit collusion model, we can find:

d

106

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D. Zhao, et al.

when

3

1, 3 ,

3 2

,1 ,

d (V OM

V CND ) d

some extent, Proposition 5 can be explained by the communication experiments in which both tacit collusion and talking explicitly are regarded as an option (Fonseca & Normann, 2012). If the wholesale price cannot transmit useful signals to retailers, the comparison result is shown in Fig. 4. By Fig. 4, we can find that with the increase of signal accuracy and the decrease of market demand uncertainty, the types of collusion will (1, 3) and change from tacit collusion to explicit collusion. When 3 (0, 2 ) , the retailer will tend to adopt the explicit collusion if the signal accuracy is higher because he cannot judge the rival’s signal based on the wholesale price. When the signal accuracy is lower and market demand uncertainty is higher, as stated in Proposition 3, tacit collusion will allow the manufacturer to set a lower wholesale price so that retailers can indirectly observe the rival’s signal.

depends on the

retailers’ profits because their changes directly influence the signal re(1, 3) ceived by the manufacturer. When and

0,

3 2

,

d (V OM

V CND ) d

relies on the manufacturer’s profit since

she cannot accurately observe the signal no matter how the retailers’ profits change. Thereby, we have

d (V OM

V CND )

d

> 0.

6. Extensions and discussions In this section, we extend the base models considering the following settings:

• In view of the concealment of tacit collusion, we consider the re• • •

tailers’ preferences for collusion and analyze which model is better for the retailers in the same conditions. We analyze the impact of downstream competition on the equilibrium strategies in four base models and stability of the collusion. Since collusion may hurt the consumers, we consider the consumer surplus and study which type of collusion is more disadvantageous to consumers. When there is no information sharing in a supply chain, we analyze the manufacturer whether to give retailers subsidies to encourage them to share the information.

6.2. Downstream competition In this subsection, we analyze the competition between the downstream retailers (Palsule-Desai, 2013; Raweewan & Ferrell, 2018) and the purpose is to analyze the equilibrium strategies of members generally considering the downstream competition and two types of collusion. Now, the market demand function becomes:

p=A

qi

q i,

[0, 1) represents the competition factor (Guo, Li, & Zhang, where 2014). According to the solutions of four base models presented in Section 4 and 5, we can derive the optimal equilibrium solutions in the competition setting, as shown in Table 3: The expressions of B1, B2, B3 , and B4 are provided in the proof of Proposition 6.

6.1. Retailers’ preferences for collusion We will analyze the preferences of retailers’ collusion and answer the research question: Whether retailers’ original types of collusion will be changed in a certain moment in the infinitely repeated games. Hence, we consider that two retailers will simultaneously change the types of collusion and extend the research to the analysis of collusion preferences.

Proposition 6. When there is the explicit collusion among retailers or tacit collusion in Region 1, weaker quantity competition among retailers (i.e., the [0, 2 1) ) can promote the formation of collusion, competition factor and the competition will not change the ways of collusion and information sharing.

Proposition 5. When the signal accuracy is higher and market demand uncertainty is lower, retailers tend to adopt the explicit collusion. Otherwise, the tacit collusion is used.

(For proof, see Appendix.) Proposition 6 shows that when the product substitutability is weaker, the stability of collusion will be increased. Collusion is likely to exist in a weaker quantity competitive market under revenue-sharing contract because of a higher product differentiation (Ross, 1992).

(For proof, see Appendix.) Proposition 5 shows that if retailers can distinguish which signals the manufacturer has received based on the wholesale price, profits with explicit collusion will be higher than those with tacit collusion. Therefore, in this case, the types of collusion will not be changed. The main reason is that retailers have more information advantages in Model ND than those in Model OM when the market uncertainty is lower, so they are likely to take this advantage to gain more profits. To

Table 3 Equilibrium solutions under downstream competition. Case

BA

ND

OM

CND

Fig. 4. Region of dominant strategy ( = 0.5 ). 107

w

Condition

p

H (2 + 2 )(1 2(2 + )

)

H (3 + 2(2 +

2 ) )

=h

L (2 + 2 )(1 2(2 + )

)

L (3 + 2(2 +

2 ) )

=l

A (2 + 2 )(1 2(2 + )

)

A (1 + )(2 + 2 ) + 2H (2 + 2(2 + )(2 + )

)

=h

A (2 + 2 )(1 2(2 + )

)

A (1 + )(2 + 2 ) + 2L (2 + 2(2 + )(2 + )

)

=l

H (2 + 2 )(1 2(2 + )

)

H (3 + 2(2 +

2 ) )

=h

L (2 + 2 )(1 2(2 + )

)

L (3 + 2(2 +

2 ) )

B1

B4

B3

B2

A (2 + 2 )(1 2(2 + )

)

A (3 + 2(2 +

2 ) )

=h

A (2 + 2 )(1 2(2 + )

)

A (3 + 2(2 +

2 ) )

=l

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D. Zhao, et al.

6.3. Consumer surplus In practice, explicit collusion is rarely observed because it is hard to be sustained and illegal. Compared with explicit collusion, it is more common for firms to engage in tacit collusion, such as cases in oil industry (e.g., CNPC, Sinopec, and CNOOC) and telecommunications industry (e.g., China Mobile, China Unicom, and China Telecom). Firms will not choose the explicit collusion for the reason of illegality. However, for consumers, does the choice of tacit collusion lead to more consumer surplus than explicit collusion? Based on the consideration above, it is necessary to analyze the consumer surplus in different collusion models. Proposition

7.

When

3

(1, 3), 3

[

3 2

, 1) ,

we

have

(1, 3), (0, 2 ) , we can find CSND CS OM . CSND CS OM . When where CS means consumer surplus. Proposition 7 states that when 3 3 (1, 3), [ 2 , 1) , explicit collusion will bring more con-

Fig. 5.

3

=

(µ) for µ

(0, 1) .

(1, 3), (0, 2 ) , the sumer surplus. In a similar vein, when consumer surplus in tacit collusion will be higher than that in explicit collusion. Thus, the choice of tacit collusion for downstream enterprises is not always beneficial to consumers since information sharing with the manufacturer can allow her to design a wholesale price which is good for herself and lead to decreasing in consumer surplus in the first case. In the second case, compared with tacit collusion, explicit collusion can form a cartel and set a monopoly and stable retail price to reduce the consumer surplus. This provides different insights from Shamir (2016) who presents that tacit collusion will result in a lower consumer surplus under the wholesale-price contract.

OM VM

6.4. Sharing with payment

7. Conclusions

So far, we assume that information sharing will not be happened if retailers do not actively share the information. However, according to the propositions in Zhang (2002) and Li and Zhang (2015), the manufacturer can compensate retailers for the loss of sharing by paying a certain fee to encourage them to share their information when information sharing is always beneficial to her. As a result, under revenue-sharing contract, will the manufacturer in the same way be able to make downstream retailers in the tacit collusion share the information with her? Suppose that if retailers do not choose to share in the first stage of the game, the manufacturer can obtain their information by purchase or signing an agreement. The condition of the manufacturer’s purchase behavior is possible only if the total profit of supply chain with information sharing is higher than that without it.

In this paper, we explore the relationship between the collusion of retailers and information sharing in a two-echelon supply chain consisting of two downstream retailers and a common upstream manufacturer under revenue-sharing contact. Retailers will choose to share or not to share their demand information on the premise of maximizing their own profits in each type of collusion. We demonstrate what the optimal strategies of information sharing and optimal equilibrium are in different collusion structures. Moreover, we also investigate retailers’ choices of collusion preferences, the impact of downstream competition on the equilibrium strategies, consumer surplus in different types of collusion, and manufacturer’s payment for sharing. We show that when there is the explicit collusion among retailers, the information sharing in the supply chain will not be happened. Although information sharing is beneficial to the whole supply chain and the manufacturer, it hurts the retailers. The gaps of profits between members with information sharing and without that are decreasing in revenue-sharing ratio. Moreover, we find that if the retailers can infer which type of signals that the manufacturer has received from the wholesale price, the manufacturer will set the same wholesale price with that of perfect information and retailers will share information. If the retailers cannot infer the information from the wholesale price, the manufacturer will design a wholesale price which is lower than the price with the perfect information to stimulate them to share information. In this case, information sharing is beneficial to the supply chain and manufacturer, but it is not conducive to retailers, so there will be no information sharing. After analyzing the choices of retailers’ preferences for collusion, downstream competition, consumer surplus, and information sharing with payment, we find that when the signal accuracy is higher and the uncertainty of market demand is lower, retailers tend to choose explicit collusion. Otherwise, retailers’ preferences will become tacit collusion. When there is the explicit collusion or tacit collusion which satisfies some conditions in the market, weaker quantity competition among retailers will foster the formation of collusion because of a higher

Different from common belief, Proposition 8 shows that in the case of collusion under the revenue-sharing contract, the information sharing can be enforced through side payment. According to the equilibrium analysis in Section 4 and 5, there are two cases without information sharing: 3

(1, 3), (0, 2 ) , information sharing hurts the re(1) When tailers, so there is no information sharing in supply chain. (2) When there is the explicit collusion in the downstream market, information sharing is also disadvantageous to retailers. It is clear that the total profit of supply chain with sharing information is higher than that without it (see Corollaries 1 and 2), so the manufacturer can pay retailers T to prompt them to share information. The payment T = (T1, T2 ) needs to satisfy the following constraints:

VMND

T1

VriND

T2

VriCND

VriOM .

Here, we take the explicit collusion as an example, and an implicit VMND (VriND VriBA ) = 0 , is = (µ ) by setting VMBA function shown in Fig. 5. > (µ ), VMBA VMND (VriND VriBA ) > 0 holds. By When Fig. 5, we can find that > (µ ) when is larger or µ is larger. A larger µ shows that the signal is less accurate and a larger is conducive to retailers. So we can find that the manufacturer can always receive information from the downstream market by giving subsidies (Vives, 1984). Moreover, Model BA and Model OM exist when there is a 3 (1, 3), (0, 2 ) . manufacturer’s subsidy behavior and

Proposition 8. There exists a transfer payment T = (T1, T2 ) that allows the manufacturer to pay retailers T to achieve information sharing. Moreover, both the manufacturer and retailers can benefit from sharing.

VMBA

CND VM

VriBA , 108

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D. Zhao, et al.

product differentiation and the competition will not change the ways of collusion and information sharing. The consumer surplus in different models is closely related to whether the wholesale price can bring useful information to retailers. Furthermore, the manufacturer can achieve information sharing by giving subsidies to retailers and both sides will benefit from the sharing. For future research, one extension is to consider the nonlinear demand. Moreover, it is interesting to extend our work to analyze whether the effect of collusion on information sharing will be different under other contracts, such as upstream protective contracts and downstream protective contracts. Finally, we consider a two-echelon supply chain, so another future research problem is to consider a multi-echelon

supply chain and study how manufacturers participating in the collusion of downstream market will affect information sharing in a multiechelon supply chain. Acknowledgements The authors are grateful to the Editor-in-Chief, Associate Editor, Area Editors, and three anonymous referees for their instructive comments and suggestions, which have improved this paper. This work was supported by the National Natural Science Foundation of China [Grant 71472134]. This research was partially supported by NSFC (No. 71620107002), NSSFC (No. 16ZDA013), and Chutian Scholarship.

Appendix A. Proofs A.1. Proof of Proposition 1 We assume that the retailer i will not make the price deviate too much, that is pi = pc constraints, it can be deduced that when

c ri (w, pc )

ri (w, pi , p i )

=

Vri

and

, collusion can sustain. Clearly, when

0 . From Corollary 1 and off-schedule c ri

w, pc

c ri (w, pc ) µ (1

1

r)

, the collusive

retailers will not deviate. Therefore, the minimal discount factor that supports setting BA and ND is given by: 1 1 + µ (1 1 2

r)

, µr

,

=h

1 1 + µ (1

=l

1 r)

2

=

µr

(2

1 µ µr )(1 + µ

µr )

0.

1

Then is obtained. 1 + µ (1 r ) Based on the optimal retail price p and wholesale price w which are obtained in Model BA and Model ND, it is concluded that the optimal expected profits for the manufacturer M are given by: BA M

=

h

H2 8 4

+

l

L

2

8

,

4

ND M

=

A2 . 8 4

After comparing the optimal expected profits of the manufacturer in BA and ND models, we can derive: BA M

ND M

=

( 1 + r )2rµ (H L)2 > 0. 8( 1 + rµ)( 2 + )

The profits of retailers BA ri

=

h

ND ri

=

h

H 2 (1 8(2

) + )2

ri

are as follows:

2

l

L (1 8(2

) , )2

)(A (1 ) + H ( 2 + ))2 + 8( 2 + )2

(1

l

)(A (1 ) + L ( 2 + ))2 . 8( 2 + ) 2

(1

By comparing the profits in two models, we can derive: BA ri

ND ri

=

( 1 + r ) 2rµ ( 3 + )( 1 + )2 (H 8( 1 + rµ )( 2 + ) 2

The profits of total supply chain BA

=

h

8

H2 4

+

l

L

4

,

ND

> 0.

are given by:

2

8

L)2

=

8

A2 4

By comparing them, we can get: BA

ND

=

( 1 + r )2rµ (1 + 5 5 2 + 3)(H 8( 1 + rµ)( 2 + )2

L)2

> 0.

A.2. Proof of Proposition 2 Taking the first order derivative with respect to the retail price p based on Eq. (8), and deriving the optimal retail price under the high and low market demand, we obtain: OM pSh (w ) =

H w+H , pSlOM (w ) = 2( 1 + )

L w+L . 2( 1 + )

Eq. (9) is the IC constraint (Incentive Compatibility) for a separating equilibrium. The first two constraints in Eq. (9) make it impossible for the OM manufacturer to choose other wholesale prices w , and she only sets the optimal wholesale price wSh and wSlOM . With the latter two constraints, the manufacturer has no motivation to mimic others after receiving the corresponding signal. Generally, when the discount factor is high enough, the off109

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D. Zhao, et al.

schedule deviation will not happen. In addition, the manufacturer who receives the high demand signal has no incentive to imitate the low market demand. Meanwhile, retailers can immediately detect and stop sharing information when the manufacturer designs a wholesale price OM w {wSh , wSlOM } . Thus, the optimization program for the low type can be simplified to:

max VM w

(w + 2( 1 + ) H

s. t .

H2 . 8 4

( 1 + ) L )(w ( 2 + ) + ( 1 + ) L ) 4( 1 + ) 2

The Lagrangian function for the above formulation is:

L (w, µ)= max

(w + ( 1 + ) L)(w ( 2 + ) + ( 1 + ) L) 4( 1 + )2

w

(w + 2( 1 + ) H

µ

( 1 + L))(w ( 2 + ) + ( 1 + ) L) 4( 1 + )2

8

H2 4

.

The first-order Karush-Kuhn-Tucker (KKT) conditions for the Lagrangian are: (1) µ = 0, wSl =

( 1 + )2L L ( 2( 2 + ) H + ( 3 + 2 ) L ) , 2 4( 2 + )

1

(2) µ

2

H

(H

L+ 1

L )((3

)H

(3 2 ) L 2(2 )

(3 2 ) H OM , pSl = 2(2 )

)L )

, 3L > H .

2

= . Solving the above system, we can get:

H L

,

3

H+ 1

1)2L

( 2

1)2H 2

, wSlOM =

1

,

(H

L+

L )((3

) H + (1

)L )

,

)

3

1, 3 ,

H

,1 ,

2

1

3

2

3

1, 3 ,

2(2

(

H;

, 3L (1

2

For analytical tractability, we define

OM wSh =

H2 4

1

0, wSl =

OM pSh =

8

(H

L+ 1

0,

3

,

2

,1 ,

2 L )((3

1, 3 ,

) H + (1

)L )

1

,

2

1, 3 ,

0,

3 2

.

In a pooling equilibrium, the collusive profit of retailers is given by:

max p

Pr

=r H

p

1

p

w + 1

r

L

p

1

p

w .

Thus, we have

pP =

A w + 2 2(1

)

, wPh = 2ph

A

1

, wPl = 2pl

A

1

.

) . Hence, we show that a pooling equilibrium can be sustained if and only if We can verify that min(wPh, wPl ) = wPl = (2pl A )(1 wP = [(2pl A )(1 )]+ > wPl and the optimal wholesale price w = wP . Then we analyze Proposition 2(b). First, we analyze a case that the manufacturer receives the high demand signal. Based on Eq. (11), we can derive:

H

A 2

w 2(1

)

A w + 2 2(1

w+

)

8

H2 . 4

Accordingly, the minimal pooling wholesale price for the manufacturer receiving the high demand signal is given by: h wPmin =

(1

)((2

)H

A

(1

)(A

H )(A (1

)

(3

)H) )

2

.

From Eq. (10), we can get the condition that the manufacturer needs to satisfy when she receives the low demand signals:

L

A 2

w 2(1

)

w+

A w + 2 2(1

2(2 )

) LH 4(2

(3 )

2 )H2

.

Similarly, we can get the minimal pooling wholesale price for the manufacturer who receives the low demand signals:

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D. Zhao, et al.

1

2

L

A +

2 ) H 2 + (A (1

(3

l wPmin =

(2

)L )

2

2(2

) HL ,

2

h Let wPmin

(1

=

)

2(2 ) . 3 2

. After comparison, we have:

l wPmin =

)((2

1,

)(

1)

(1

)(1

rµ)(

1)((1 + rµ (

1))(1

)

2(2

)

(3

) ))

2 (1

+

) (3

2

2 )

+ ((1 + (

1) rµ )(1

)

) )2

(2

< 0.

2

h Thus, there will be no pooling equilibria when wP < wPmin and the pooling equilibrium exists when wP > wPl . By Proposition 2, it will be:

(1

wP =

)((2 2

)L

(1

)((2

h wPl = wPmin =

+

A)

,

)H

A

(1

)(A

H )(A (1

)

(3

)H) )

H )(A (1

)

2

.

So the condition of the pooling must be satisfied:

(1

)((2 2

)L

A)

>

(1

)((2

)H

A

(1

)(A

(3

)H) )

2

After solving Eq. (12), we obtain

.

(12)

(1, ( , µ, r )], where:

, µ, r =

3 2rµ ( 1 + )2 + r 2µ2 ( 1 + )2 + 2 . 1 + r 2µ2 ( 1 + )2 2rµ (2 3 + 2 )

When

(1, ( , µ, r )], the pooling equilibrium is reached.

A.3. Proof of Proposition 3 When

wlOM =

(1, 3),

(1

)((2

(0, )H

3 2

) , we show:

L

(H

L )((1

)L + (

3) H ) )

L

L )((1

2 1)2L

wlBA =

(

wlBA

wlOM =

2

=

.

in Model BA. Hence, we show:

1)2L

(

(1

)((2

)H

2

(H

)L + (

3) H ) )

2

( 1 + )( ( 2 + ) L

2H + H +

(H

L )((1

)L + (

2+

3) H ) )

.

We can simplify Eq. (13) as follows and show it in Fig. 6:

= (7

5 +

2)

2

2(6

5 +

2)

+ (5

5 +

2)

0.

Fig. 6. Relationship between wlBA 111

wlOM and , .

(13)

Computers & Industrial Engineering 131 (2019) 99–114

D. Zhao, et al.

A.4. Proof of Proposition 4 According to the optimal wholesale price w and retail price p when tacit collusion exists in the downstream market, the profits of manufacturer and retailers in Model OM and Model CND can be solved respectively: 8

M

OM M

=

H2 4 L

=

8

, 2

,

4

3

(1 + b ( 1 + ) + ( 1 + )

) H2

(1

)2

8(2

ri

OM ri

=

(1

=

,1 ,

2

2 )( 1 + a ( 1 + ) + 4( 2 + )

+2

CND M )

2

L ,

2

)2

,

3

3

1, 3 ,

2

( 1 + )(3 + b ( 1 + ) + ( 1 + )

(0, 1), r

(0, 1), µ

3

OM M

CND M

> 0,

OM M

CND M

> 0,

OM ri

CND ri

> 0,

OM ri

CND ri

< 0,

2 )((1 + a)( 1 + )

b

3

8

A2 4

,

,

2

CND ri ( 2+ ) )

3

1, 3 ,

1, 3 ,

3

0,

2

3

0 . We show a

1, 3 ,

0,

3 2

8(2

) )2

.

,

0,

,1 ,

,

2

3

1, 3 ,

1, 3 ,

2

L ,

A 2 (1

=

(0, 1) , we can get:

2

3

,1 ,

.

2

For the convenience of calculation, we have can get a

0,

,1 ,

8( 2 + )2

Since

1, 3 ,

=

,

)L

8(2

3

1, 3 ,

( 1 + )( 1 + + ( 3 + ) ) 1+

b in Fig. 7.

= a and

( 1 + )(1 + ( 1 + ) 1+

3 )

= b . When

and

are not large enough, we

A.5. Proof of Proposition 5 According to the profits of the manufacturer and retailers in four base models (BA, ND, OM and CND), we can find that when 3 3 (1, 3), (0, 2 ) , Model CND and Model ND exist. By 3 (1, 3), [ 2 , 1) , Model OM and Model ND exist in supply chain. When comparing retailers’ profits in different models, we analyze their collusion preferences. Here we ignore the illegality of explicit collusion. 3 3 (1, 3), [ 2 , 1) , we will have: If

VriND > VriOM . To keep the analysis tractable and make the results more comparable, let r = µ =

VriCND

VriND =

2( 1 +

+ 2 ( 1 + )( 1 + ) + 2

)

2

( 1 + )2 (1 + 2 ( 1 + )( 1 + ))

4( 2 + )2 (1 + ( ( 1 + ) 2 (1 + 2 ( 1 + ))

8( 2 + )2 (1 + ( ( 1 + )

))2

if

2

))2

,

Fig. 7. Relationship between a 112

b and .

(1, 3),

(0,

3 2

) , the result is shown in Fig. 4:

Computers & Industrial Engineering 131 (2019) 99–114

D. Zhao, et al.

where

denotes the signal precision (Li & Zhang, 2015).

A.6. Proof of Proposition 6 According to the description of Section 4 and 5, the optimal profits of retailers and the manufacturer can be calculated when the competition exists in the downstream market. To conserve space, we only give collusive profits of the retailer i in four models here: BA ri

=

h

ND ri

=

h

OM ri

=

h

CND ri

=

H 2 (1 4(2 + (1

2

) )2

+

l

L (1 4(2 +

) )2

, ))2

)(A (2 + 2 ) 2H (2 + 4(2 + )2 (2 + )2

H 2 (1 )(1 + ) + 4(2 + )2

A 2 (1 4(2 +

+

l

(1

)(A (2 + 2 ) 2L (2 + 4(2 + )2 (2 + )2

))

2

,

2

l

L (1 )(1 + ) , Region 1, 4(2 + )2

) . )2

Thus, we can get the optimal equilibrium solutions which are shown in Table 3, where B1 denotes the regions:

1,

4 3

4 3 2

&

4 3 2

0,

,1 &

&

0, 1

>

0,

(2

4 3

&

)(

1)

2

0, 1 &

0, 1

.

B2 denotes the regions: 1,

B3 =

B4 =

4 & 3

(1

(2 +

0,

4 2

3

(2

&

)(2 + )(2 + ) H (1 2(1 + )(2 +

+

) L (2 + 2(2 + )(2 +

)(

1)

2

)(2 +

,1 .

+

)L

2+

)2

(1

(6 + 2 + (5

5 + 2) H2 2

)

)(2L + (2 + ) H ) + )

2 )

2(1 + )(2 + 1

(6 + 2 + (5

2 )

5 + 2) H2 2

2(2 +

2(2 +

)(2

2(2 +

)(2

) HL + (2

3 + 2

.

) ) HL + (2

3 + 2

)

2 3 + 2) L

2 3 + 2) L

.

(1) When there is explicit collusion among the downstream retailers, the results are shown as follows:

VMBA > VMND , VriBA < VriND . (2) When there is tacit collusion among retailers within the Region 1, we can get: OM CND VM > VM , VriOM < VriCND .

By comparison, we find that the results will be the same when there is explicit collusion or tacit collusion in Region 1 if competition between retailers is taken into consideration. Therefore, the competition between retailers may not change the types of collusion in supply chain and the ways of information sharing. Next, we discuss the impact of the competition on the critical discount factor. In accordance with the constraint of the discount factor in explicit collusion, we can derive the following equations: ri (H ,

whBA, p )

c ri (H ,

whBA, phBA )

Vri

.

Compared with the case where there is no competition, the numerator of the critical discount in the competition setting is decreasing in and the [0, 2 1) ( 2 1, 2( 2 1))& (0, 2 (1 + 2 )) . Thus, the discount factor with competition denominator is increasing in when will be lower than without. Weaker quantity competition among retailers can contribute to the formation of collusion. A.7. Proof of Proposition 7 The consumer surplus is CS = 2 Q 2 because the demand function is linear in this paper. Accordingly, the consumer surplus in different models can be expressed as follows: 1

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CSND =

1 2

CS CND =

H

h

1 2

H

h

1 2

h

CS OM = 1 2

h

(H

(H

H( 2 + ) + A( 1 + ) 2( 2 + ) 2

(3 2 ) A 2(2 )

)

(3 2 ) H 2 2(2 )

)

(3 2 ) H 2 2(2 )

+

+

l

l

(L

l

+

l

(3 2 ) L 2(2 )

)

2

,

2

,

2

H+ 1

L

L( 2 + ) + A( 1 + ) 2( 2 + )

L

(3 2 ) A 2(2 )

L

2

+

2

L+ 2(2

(H

L )((3

) H + (1 1

)L )

2

.

)

After comparison, we find: 3 3 (1, 3), [ 2 , 1) , we have: If

CSND If

CSND

CS OM . (1, 3),

(0,

3 2

) , we can derive:

CS CND.

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