How Does the Reference Price Mechanism Affect Competition?

This paper proposes an operational strategy based on the theory of transaction utility, aiming to mitigate the negative impact of customers’ free-riding behaviour. This operational strategy is named the reference price mechanism (RPM), in which the retailer gives a reference price that may be different from the retail price, and the customer’s purchase decision is affected by the reference price. Based on the Hotelling model, we develop a duopoly game by considering the free-riding cost to systematically discuss the effectiveness of the RPM. The results show that the brick-and-mortar/online retailer can benefit from implementing the RPM separately, vice versa. If both retailers implement the RPM, only the retailer with the higher reference price can attain more profits. In addition, there is a threshold of the free-riding cost that can affect whether the retailers implement the RPM.

is equivalent to what economists call "consumer surplus". Transaction utility is the transaction quality that consumers perceive. The difference between the actual price paid and the "reference price" is one of the ways to describe transaction utility, and the reference price is the consumer's expected price. To give a simple example, suppose you buy a sandwich in a scenic spot. This sandwich is the same as what you usually eat at noon, but the price twice as high as usual. There is nothing wrong with the sandwich itself, but the transaction is very unpleasant. This can result in a negative transaction utility because you feel like you have been cheated.
In contrast, if the paid price is lower than the reference price, the transaction utility is positive; it feels very costeffective. In practice, the application of a reference price is very common. For example, before Double Eleventh Day in China, retailers may raise the price from the previous period, take the raised price as the reference price of the product, and then sell the product to customers using various promotional methods on that day. Thus, customers have a sense of greater transaction utility with smaller discounts. The method stimulates customers to buy more products on Double Eleventh Day.
In this paper, we develop a duopoly competition model to examine the price decision of retailers and the impact of the reference price mechanism (RPM) on retailers' revenues under the free-riding behaviour of strategic customers. The Hotelling model is used to describe the competition between a BM retailer and an online retailer. We consider four types of RPM strategies. The first and second strategies are the "BM retailer only" (R) strategy and the "online retailer only" (O) strategy. In these strategies, only the BM/online retailers implement the RPM. The third strategy is the "BM & online" (RO) strategy in which both retailers implement the RPM. The last strategy is defined as Strategy F, where neither retailer implements the RPM. By analysing and comparing these strategies, we attempt to answer the following questions: (i) How does the reference price affect the purchase decision of consumers and the retailer's profit by considering the free-riding cost and transaction utility? (ii) Can the BM retailer and the online retailer increase profits by implementing the RPM? (iii) For the two retailers, which is the optimal strategy among the four strategies in the face of free-riding behaviour?
This study contributes to the literature on online and offline retailing and operational strategies of customers' strategic behaviour. First, many studies have focused on the impact of customers' free-riding behaviour on the competition between online and offline retailers [16][17]. However, none of these studies consider the cost of freeriding to study the impact of free-riding behaviour. In fact, customers need to spend more time and energy to learn about the product when they consume the service of the BM retailer for free, so there will be a free-riding cost. Therefore, this paper considers the free-riding cost based on Chen and Chen [2], which is more realistic. In addition, the present literature on operational strategies to address the adverse effects of strategic customer behaviour mainly focuses on inventory control and pricing. This paper aims to consider the impact of transaction utility in price decision-making and use the difference between the reference price and the actual paid price to describe the transaction utility. Last, we explore the equilibrium pricing strategy of the two retailers when customers have free-riding costs and transaction utility.
Our analysis leads to several interesting main results. First, different from other studies, considering the cost of free riding, free riding behaviour has a negative impact on not only the BM retailer but also the online retailer. At the same time, it is surprising that free riding can benefit the BM retailer when the cost of free-riding is high. Second, by considering the transaction utility, the RPM can effectively alleviate the negative impact of free-riding behaviour, but it cannot benefit both retailers. If the BM (online) retailer implements the RPM separately, the profit of the BM (online) retailer may increase, but the online (BM) retailer's profit will certainly decrease. Even if both retailers adopt the RPM, only the retailer with the higher reference price can increase its profits. Our results also show that BM retailers are more likely to achieve profit gains if they use a higher reference price. Therefore, from the perspective of the whole market, it is optimal for BM retailers to implement RPM separately. Furthermore, the four strategies are likely to become the equilibrium strategy of the RPM, which should be determined according to the threshold of the free-riding cost. The equilibrium strategy of the RPM varies with the threshold of the free-riding cost.
The rest of the paper is organized as follows. We first review the related literature in Section 2. Section 3 introduces the model and its related assumptions. In Section 4, we analyse the optimal price, demand and profit of retailers under the operational strategy of the RPM. We explore the effectiveness of the RPM and obtain the equilibrium strategy of the RPM in Section 5. Section 6 provides concluding remarks and further research directions.

II. SURVEY OF EXISTING LITERATURE
This paper is related to three research streams: the impact of free-riding behaviour on online and offline retailers, the operational strategies of strategic customer behaviour, and the reference price effect. levels and leads to a drop in demand. Antia et al. [8] demonstrate that since consumers take the services of BM retailers without paying the corresponding fees, free-riding behaviour would inhibit the enthusiasm of BM retailers to provide services, such as presale services and training of sales staff. However, some scholars believe that free-riding behaviour may have positive effects on service providers. Bernstein et al. [21] find that free-riding behaviour leads to further increases in product sales by spurring retailers to establish their own direct sales channels. Shin [1] shows that the differentiation of service provision under the free-riding behaviour of consumers will ease the price competition among retailers, thus leading to an increase in the profits of both competitors. Pun et al. [22] consider the differences in service capacity of different channels. They find that when customers exhibit free-riding behaviour, retailers bearing the free service costs are better compensated than the beneficiaries of free riding. Therefore, the impact of freeriding behaviour on retailers may depend on different scenarios, such as channel structure, purchase cost and different consumer preferences. In this paper, we consider that the strategic customers will incur a free-riding cost in the process of free riding on offline retailers' services. Under the influence of free-riding cost, whether the free-riding behaviour has a positive or negative impact on online and offline retailers needs to be reevaluated.

B. LITERATURE ON OPERATIONAL STRATEGIES OF STRATEGIC CUSTOMER BEHAVIOUR
Many scholars have proposed operational strategies to reduce the impact of strategic customer behaviour (including waiting and free-riding). The operational strategy is mainly presented from two perspectives: inventory control and pricing. In terms of inventory control, Levin et al. [23] find that the impact of strategic customer behaviour can be effectively reduced by controlling the initial inventory capacity. Su and Zhang [24] study the effect of inventory information on customer purchase behaviour. They demonstrate that the seller can improve his profits by combining inventory commitment and availability guarantees. Yin et al. [25] propose an operational strategy to control strategic behaviour via the inventory display format. They find a display format that conceals inventory information can be used to influence customers' perception of the risk of stockouts if they decide to wait. Aydinliyim et al. [26] describe an inventory level threshold within which inventory information should be disclosed, and it is optimal to ensure that the inventory exceeds this threshold. Cai et al. [27] propose a vendor-managed inventory decision model that considers strategic customer behaviour and, on this basis, introduce option contracts to achieve a Pareto improvement. In terms of pricing, some studies suggest price commitment to deter the strategic behaviour of customers. Perry et al. [28] research the operational strategy of a supplier's adoption of retail price maintenance to motivate retailers to improve the level of information service and reduce the negative impact of free riding. Li and Zhang [29] find that an effective way to attract strategic customers is to guarantee unchanged prices in the later period. Price matching is another operational strategy that can be used to discourage strategic customers. Png [30] and Lai et al. [31] find that price matching is profitable when the firm sells to two groups of customers with different valuations. Xing and Liu [32] design a contract with a price match and selective compensation rebate to achieve sales effort coordination for a supply chain with one manufacturer and two retail channels, where an online retailer offers a lower price and free rides a BM retailer's sales effort. Chen and Chen [2] analyse the effectiveness of price matching on free-riding behaviour. They show that the BM retailer can effectively prevent strategic customers from transferring online when customers' shopping costs are moderate. Liu et al. [33] propose a two-way revenue sharing mechanism to coordinate the decentralized supply chain to reduce the free-riding behaviour of strategic customers. Unlike the above studies, this paper considers the transaction utility of consumers and tries to address the free-riding behaviour of consumers by introducing the reference price.

C. LITERATURE ON THE REFERENCE PRICE EFFECT
The purchase decision of customers is affected by transaction utility. Many scholars use the difference between the reference price and the real purchase price to describe the transaction utility, which is the reference price effect. The initial research on the reference price effect focused on the influence of reference price on consumer behaviour. According to the perspectives of prospect theory [34] and adaptation level theory [35], if the actual price observed by consumers at the point of purchase is inconsistent with the reference price, it will affect consumers' purchasing behaviour. Much of the current literature focuses on the application of reference price mechanisms to increase profits. Chen et al. [36] explore a dynamic pricing model that describes the reference price effect as a combination of current and past prices over a limited period. Dye et al. [37] use the reference price effect to model the pricing strategy of deteriorating commodities and prove that the initial reference price greatly influences the pricing and replenishment strategy. Courty and Nasiry [38] use the reference dependence model to analyse and explain the unified pricing of entertainment products with quality differences. Wang [39] incorporates the reference price into the consumer choice model to study product classification and pricing strategy. Crettez et al. [40] discuss the existence of the optimal dynamic pricing strategy when the demand depends on the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. reference price, and the authors give the optimal pricing strategy under different initial values of the reference price. Using the reference price effect, Li [41] analyses the multiperiod pricing and inventory management issues of an omnichannel retailer and concludes that the convergence of pricing and ending inventory level to equilibrium depends on the relative position of the initial reference price relative to the unique equilibrium price. As the online sales environment matures, the reference price has an important impact on the price set by online retailers. Hardesty and Suter [42] use a two reference price environment (online, BM) × two external reference price (low, high) between-subject experimental design with a single control condition to show that consumers expect to pay less in online e-tail settings than in BM retail settings. Zheng [43] find that online customers' willingness to pay will rise as reference prices increase by designing two surveys with different reference prices. However, these papers do not discuss the attempt to prevent free-riding behaviour of strategic customers through the establishment of a reference price, nor do they determine the optimal strategy to determine whether retailers should provide a reference price, which we will discuss.

III. MODEL
The notations that will be used in this paper are shown in TABLE I. In practice, when selling products at different prices through BM retailers and online retailers, some customers prefer BM retailers (to experience products before purchasing), some prefer online retailers (for convenience), and others may take strategic action by experiencing products at BM retailers but purchasing them from online retailers at lower prices.
Consider the market where online and offline channels coexist and two channels sell the same product. Based on the Hotelling model framework [44][45][46][47][48], it is assumed that there are two different types of retailers in the market, where r represents the BM retailer located in the corner ( 0 ) of the Hotelling line and o represents the online retailer located in the corner (1 ). The two retailers are risk-neutral and pursue maximum profits by setting the prices (denoted by ). Customers are uniformly distributed on the Hotelling line, and each customer can buy at most one product. If a customer visits the BM retailer, the customer whose location is x will incur a transportation cost of tx , where t ( 01 t ) represents the transportation cost per unit distance. Since customers purchase goods from online retailers by "clicking", we assume that customers' online purchases will incur a uniform shopping cost ( s ), as described by Liu and Zhang [45].
The BM retailer educates and guides customers to learn about products in the store. When purchasing online, the customer cannot experience the service of the salesperson; touch, feel or see the product; or check it personally to determine whether it is appropriate, so the valuation will be discounted by the service differentiation factor  to distinguish between the BM retailer and the online retailer. Each customer's valuation of the product sold by the BM and online retailers is v and v  , respectively, as in Shum et al. [49], where (0,1]

 
. The smaller the service difference factor (  ) is, the higher the service level of the BM retailer is.
Customers experience the product at the BM retailer first and then purchase the product from the online retailer. This behaviour is called free-riding. A free-riding customer will visit the offline store to take advantage of the service provided by the BM retailer before purchasing the product online, so that there is no service difference between online and offline products, and the free-riding customer's valuation is v , as in Chen and Chen [2]. Compared with purchasing directly from BM retailers or online retailers, free-riding behaviour may cause customers to spend more time and energy to perceive, experience and understand the products and thus delay the delivery time. Accordingly, customers will have a free-riding cost, which is characterized by a parameter  ( 01  ). The greater the free-riding cost is, the less utility the product brings to the customer; for example, delaying receiving the product in the process of free-riding reduces the customer utility. Therefore, the higher the value of  is, the smaller freeriding cost, and the more likely that customers will free ride.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Considering the free-riding behaviour of strategic customers, this paper focuses on the impact of the reference price mechanism (RPM) on retailer's decision-making and profits. The RPM specifically refers to the reference price of a product given by the retailer to the customer, which is not necessarily the price paid by the customer for the product. The RPM is widely used in daily life, such as with the historical transaction price displayed by online shopping platform, the suggested retail price on products, and the tag price of clothes. The final transaction price is not necessarily equal to these reference prices, but the reference price will have a psychological impact on consumers (which may make consumers feel that their spending has been cost-effective or that they have been cheated). According to the transaction utility theory [15,[50][51][52], these reference prices ultimately affect customer demand. When the actual purchase price is lower than the reference price, the transaction utility is positive, meaning that customers think the transaction is a great deal, thus increasing their desire to buy products. For example, almost everyone has unworn clothes in their closets, but the sense that a purchase is cost-effective leads them to buy additional clothes. The transaction utility may also be negative. When the actual price paid is higher than the reference price, it will reduce the utility of consumers to buy the product, and even cause some customers not to buy it. This paper uses the reference price sensitivity coefficient (  ) and the difference between the reference price ( j P ) and the retail price ( j p ) to characterize the customer's transaction utility. Similar to Özer and Zheng [53], the customer's transaction utility is . Customers have greater reference price sensitivity (  ) to products for which quality is difficult to evaluate, such as carpets and mattresses [51]. If the quality of the product is difficult to evaluate, then the reference price will be used as an important indicator for customers to evaluate the quality of the product. At the same time, customers are more convinced of the authenticity of the reference price and thus have greater reference price sensitivity. The larger the  , the more sensitive the customer is to the reference price. The size of  depends on the difficulty of product quality assessment, and the present paper assumes that retailers sell the same products, so  for the BM retailer and the online retailer are the same. To facilitate the discussion, we also assume that all customers have the same  .
We discuss four strategies related to the RPM. With Strategy R, only the BM retailer offers a reference price to strategic customers. The strategy in which the online retailer unilaterally provides a reference price is called Strategy O. We define the strategy in which both the BM retailer and the online retailer provide reference prices to strategic customers as Strategy RO. The strategy in which neither of the two retailers implements the RPM is denoted Strategy F. We denote these strategies as W , where Under Strategy R, since only the BM retailer implements the RPM, customers' utility to the BM retailer will be added, with () R rr Pp   . The utility of customers who buy products directly from the BM retailer is: 1) Free-riding customers will eventually buy products from the online retailer. Therefore, the BM retailer's reference price does not affect the transaction utility of customers who purchase products directly from the online retailer or free-riding customers. The utility of customers who buy products directly from online retailers is: 2) The utility of free-riding customers is: 3) Similar to Strategy R, when the online retailer provides a reference price to the customer, the customer who finally purchases from the online retailer will obtain an additional Under Strategy RO, the utility functions become: Under Strategy F, the utility functions are:

IV. REFERENCE PRICING MECHANISM (RPM)
According to the transaction utility theory, the reference price may effectively enhance the transaction utility of customers. Therefore, this paper discusses the impact of reference price on the free-riding behaviour of strategic customers. The RPM is introduced to respond to free-riding behaviour by considering the reference price effect of customers. In this section, we analyse the influence of the RPM on strategic customer free-riding behaviour from the perspective of the BM retailer and the online retailer.

A. STRATEGY R
Considering the free-riding behaviour, the customers in the market can be divided into three categories: purchase This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
customer who is x units away from the BM retailer prefers to purchase from the BM retailer rather than from the online retailer if R xx  ; otherwise, the customer prefers the online retailer. Therefore, the demands for the two retailers are: Correspondingly, the profits for the two retailers are: With the demands in Equation (4.1) and profits in Equation (4.2) and Equation (4.3), the pricing decisions can be summarized in Proposition 1.
BM and online retailers have unique optimal pricing decisions **

( , )
RR ro pp, which are given by: The demands for the two retailers are: The profits of the two retailers are: The proof of Proposition 1 is given in the appendix. With Proposition 1, we have the following results: The proof of Corollary 1 is given in the appendix. Corollary 1 shows that under Strategy R, if the customer is more concerned about the reference price, the online retailer should lower the retail price, and the retail price of the BM retailer will be decreased or increased according to the reference price. If the reference price is low, the BM retailer should decrease the retail price; otherwise, it is the same as that of the online retailer. Corollary 2. The proof of Corollary 2 is given in the appendix. Corollary 2 indicates that under Strategy R, as the customer's price sensitivity to the reference price increases, the RPM can only increase the profit of the BM retailer if the reference price is high enough. The reason is that, when the reference price is sufficiently high, the BM retailer has enough range for a premium to increase its profit performance.

B. STRATEGY O
This subsection considers the case where only the online retailer implements the RPM. The reference price provided by the online retailer will affect the transaction utility of customers who purchase products online.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
We define With the demands in Equation (4.7) and the profits in Equation (4.8) and Equation (4.9), the pricing decisions of the two retailers can be summarized in Proposition 2 as follows. Proposition 2.
both of the two retailers have unique optimal pricing decisions ** ( , ) OO ro pp, which are given by: The demands for the two retailers are: The profits of the two retailers are: With Proposition 2, we have the following results: , the prices of the two  The proofs of Proposition 2, Corollary 3, Corollary 4 and some of the following propositions and corollaries are similar to those of Proposition 1, Corollary 1 and Corollary 2, which will not be repeated in this paper.
Corollary 3 and Corollary 4 demonstrate again that under Strategy O, as customers' price sensitivity to the reference price increases, the RPM can only increase the profits of the mechanism implementer when the reference price is high enough.

C. STRATEGY RO
We discuss the strategy in which both retailers choose the option of reference price (Strategy RO) in this section. Similar to the previous form, we determine the two retailers' demand ( RO With the demands in Equation (4.13) and profits in Equation (4.14) and Equation ( The demands for two retailers are: 12 12 (1 )( The profits of the two retailers are: the profits of the two retailers ( RO decreases. Corollary 5 and Corollary 6 show that under Strategy RO, as the customer's price sensitivity to the reference price increases, the RPM can only increase the profits of the retailer that sets the higher reference price when there is a large difference between the reference prices set by the two retailers.

D. STRATEGY F
Under Strategy F, the demands for the two retailers are obtained from Equation (3.6): (1 ) , The profit ( According to Equation (4.22) and Equation (4.19), the demands of these two retailers are: (1 )( With prices in Equation (4.22) and demands in Equation (4.23), the profits of these two retailers are: Proposition 4 indicates that in the case of strategic customers with free-riding behaviour, BM retailers and online retailers will attract more customers through fierce price competition to achieve the purpose of maximizing their profits. We can see from Equation (4.22) and Equation  The practical explanation is that with the reduction in the free-riding cost, all retailers in the market will reduce their retail prices due to fierce price competition.

V. COMPARATIVE ANALYSIS
In this section, the impact of free-riding behaviour is analysed by comparing the profit of the retailers in the two cases of strategic customers and non-strategic customers under Strategy F. Based on the situation where the RPM is not implemented, i.e., Strategy F, the profit changes of the three implementation situations of Strategy R, Strategy O and Strategy RO are analysed separately to explore the effectiveness of the RPM in response to free-riding behaviour. According to the profit comparison under the four strategies, the equilibrium strategy of the RPM is obtained.

A. IMPACT OF FREE-RIDING BEHAVIOUR
We have: Proposition 5.
 represent the profit of the BM retailer and the profit of the online retailer without free-riding behaviour, respectively.
The proof of Proposition 5 is given in the appendix. We attempt to use a numerical example to determine the conclusion. We set 6 v  , 5 t  , 0    Figure 1 shows that the free-riding behaviour will reduce the total profit of the market, but, surprisingly, it may increase the profit of the BM retailer. If the free-riding cost is large (smaller  ), the profit of the BM retailer will increase under Strategy F ( 0 F r   ). However, the profit of the online retailer and the whole market will always decrease under Strategy F ( 0 . This result is different from the conclusion of Chen and Chen (2019) that the free-riding behaviour of strategic customers will adversely impact BM retailers. The reason for this difference is that the free-riding cost is considered in this paper. It indicates that if customers care too much about experiencing the products and spend more time and energy to understand the products, customers' free-riding behaviour will positively impact BM retailers. In contrast, if the customers have a good service experience without spending too much time and energy, the free-riding behaviour of strategic customers has a negative impact on BM retailers. Therefore, many BM retailers have established complicated processes of consumer experience, such as paying a deposit, the purpose of which is to increase the cost of free-riding.  II shows that as the cost of free-riding decreases ( increases), the profits of both the BM retailer and the online retailer decrease. When the cost of free riding is high, the emergence of strategic customers may have a positive impact on the BM retailer. With the reduction in the cost of free riding, customers have more incentive to free ride on the BM retailer's service and to switch to the online retailer to buy the product at a lower price, resulting in fierce price competition among retailers.
A large number of studies show that online shopping customers are more strategic than offline customers. Therefore, most online shopping customers are more inclined to engage in free riding. However, many strategic customers in the process of free riding on the BM retailer's service may end up buying a product from the BM retailer since BM retailers try their best to provide customers with satisfactory service. In this way, customers' free-riding behaviour converts some online shoppers into customers of the BM retailer. Therefore, in the market with a high freeriding cost, the free-riding behaviour of customers increases the profit of the BM retailer. Take Double Eleventh Day in China, for example. Before that day, many online shopping customers will first go to physical stores to find and experience their favourite products. However, while visiting physical stores, many strategic customers will buy offline because the sense of acquisition from buying in physical stores is higher than that of free riding, thus increasing the demand of the BM retailer. In recent years, before Double Eleventh Day, BM retailers have seized the opportunity of such traffic and carried out promotional activities, enabling more customers who want to experience the service of the BM retailer for free to purchase products directly from the BM retailer. With the reduction in the cost of free riding, increasingly, customers prefer to buy their favourite products by taking advantage of the service of the BM retailer for free. To maximize its profits, the BM retailer should reduce the price to retain customers.
For online retailers, when there is free-riding behaviour in the market, the demand for online retailers decreases. In addition, in the face of the price reduction of the BM retailer, the online retailer has to reduce its retail price to ensure its profit maximization. Therefore, the free-riding behaviour of customers has an adverse effect on the online retailer.
In a market with low free-riding cost, the BM retailer and the online retailer tend to compete on price in pursuit of profit maximization. However, the fierce price competition will depress the whole market to a certain extent. This paper proposes the RPM to address the adverse effects of free-riding behaviour. The effectiveness of this mechanism is studied through comparative analysis. The specific research process is presented in the next section.

1) STRATEGY R
We obtain Proposition 5 by comparing R j  and F j  and then discuss the impact of free-riding behaviour on the profits of the two retailers under Strategy R. We define (max(0, , ), min( , )), 2 , (min(max(0, ), ), ), 2 , (max(0, ), min( , )). 2 The proof of Proposition 6 is given in the appendix. Proposition 6 illustrates the results in which under Strategy R, the RPM is effective for the BM retailer when 1 1 2 2 (max(0, , ), min( , )) 2 , which can alleviate the negative impact of the free-riding behaviour of strategic customers on the BM retailer. However, when the cost of free riding is high (smaller  ), the retailer can set a higher retail price, and the reference price is not enough to bring premium space to the BM retailer. Therefore, the RPM is ineffective for the BM retailer when the free-riding cost is small. Whether the RPM will eventually become invalid to BM retailers depends on the reference price of the BM retailer ( r P ). If the reference price of the BM retailer is high enough, the RPM is always effective for the BM retailer under Strategy R. However, the RPM will aggravate the negative impact of free-riding behaviour on the online retailer under Strategy R.
Considering that the reference price is generally set around the value of the goods in the market, we take the reference price as the customer's valuation in a numerical example. We set 6  We observe from Figure 2 that the unilateral implementation of the RPM by the BM retailer increases the profit of the BM retailer ( R r  ) and decreases the profit of the online retailer ( R o  ). For the whole market, it is only in the high range of free-riding cost that the RPM can increase the gross profit. If customers are more likely to free ride and are more sensitive to the reference price, the RPM is more effective for the BM retailer under Strategy R.  Under Strategy R, when the cost of free riding is low, the RPM can effectively alleviate the market downturn caused by the free-riding behaviour of strategic customers.

2) STRATEGY O
In this section, we explore the impact of the unilateral implementation of the RPM by the online retailer on freeriding behaviour, and Proposition 7 is obtained. We define (2 (1 ))(2 ) The proof of Proposition 7 is given in the appendix.
with  and  are illustrated in Figure 3.    IV shows that as the free-riding cost decreases ( increases), the demand of the online retailer increases, and other indicators decrease. This result indicates that under Strategy O, with the reduction in the free-riding cost, the online retailer can gain more market share to mitigate the adverse impact of free riding. However, Strategy O cannot solve the problem that free-riding behaviour intensifies the price competition between the two retailers. With the reduction in free-riding cost, the trend of retailers' retail price and profit reduction remains unchanged.
Similar to Strategy R, the online retailer can increase its profit by implementing the RPM because the reference price is provided to satisfy the customer's utility. The difference between Strategy R and Strategy O is the change in the whole market's profits. Compared to Strategy F, the profits of the market will decrease in Strategy O. We obtain two interesting conclusions:  Unilateral implementation of the RPM can increase the profit of implementers.
 From the perspective of the whole market, Strategy R is better than Strategy O.

3) STRATEGY RO
How does the simultaneous implementation of the RPM by the BM retailer and the online retailer affect the competition? Are the profits of both sides increased, or will this strategy lead to price competition? This problem is discussed below.
As it is challenging to measure the profit difference under Strategy RO analytically, we conduct a comprehensive numerical study to derive additional management insights. We define The changes in Figure 4.   Figure 4 and Figure 5 show that the RPM may increase the profit of a retailer who sets a reference price sufficiently higher than the competitor's price. In any case, when one retailer's profit increases, the other retailer's profit will decrease. Each retailer must set a higher reference price to maximize its profits. Due to the existence of free-riding cost (  ), the influence of the reference price of the BM This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  . Since the reference price has a greater impact on BM retailers, the most likely way to increase the market's profits is for the reference price set by the BM retailer to be higher than that of the online retailer ( ro PP  ). We can see from Figures 4(c) and 5(c) that when ro PP  , the overall market profit is most likely to increase. The reason is that the higher reference price of BM retailers prevents free-riding customers from switching to online purchases. The RPM effectively restrains the free-riding behaviour of strategic customers and alleviates the retail price competition between the BM retailer and the online retailer when r P is sufficiently greater than o P . We can observe the failure of the RPM from Figure 4(b). When the cost of free riding is low and the reference price of the BM retailer is low, again, as shown in Figure 4(b), the RPM will reduce the profits of both retailers. This effect occurs because the RPM not only reduces the price of the BM retailer but also reduces the price and demand of the online retailer. As the reference price-sensitive coefficient (  ) increases, the RPM is more effective.
By exploring the above three strategies of the RPM (Strategy R, Strategy O and Strategy RO) and performing a comparative analysis with the non-implementation of the RPM (Strategy F), we obtain the following conclusions:  The RPM may have a positive impact on a retailer that sets the reference price much higher than the competitor.  The introduction of the RPM will enable retailers to enter into a new round of price competition, bidding up their reference price.
 From the whole market perspective, the BM retailer should provide a higher reference price.
Although in our analysis, Strategy R is the best strategy for the whole market, retailers are profit-oriented and will not change their strategies to benefit the profit of the whole market. They will do their best to improve their own profits or reduce their own losses. In such a situation, what is the equilibrium strategy for implementing the RPM for the BM and the online retailer? We discuss this issue in the next section.

C. THE EQUILIBRIUM STRATEGY OF THE RPM
In this section, we derive the equilibrium strategy of implementing the RPM for the BM retailer and the online retailer by discussing the four situations where the online retailer and the BM retailer either implement the RPM or not. We define (2 (1 ))( 2 )

Interval
Range of interval Whether these two retailers should implement the RPM is discussed in the range of the value of  .
U   is assumed to ensure that the market is fully covered and that the two retailers are not going out of business.
According to the comparative study on the profit of retailers, we have the following proposition: Proposition 8. The equilibrium strategy of implementing RPM is:   Figure 6 shows , , , (4.25) According to Equation (4.25), we can obtain a game matrix between the BM retailer and the online retailer, as shown in Figure 7.
The BM retailer The online retailer The game matrix in Figure 7 shows that the equilibrium strategy is Strategy RO (i.e., both retailers implement the RPM) when In this situation, the BM retailer will implement the RPM regardless of whether the online retailer implements the RPM. Moreover, the online retailer will also implement RPM if the BM retailer implements RPM. Therefore, the implementation of the RPM, as a tacit agreement and collusion between retailers, is an equilibrium strategy for the two retailers. According to Proposition 8, other strategies may also be used as an equilibrium strategy of the two retailers in other thresholds of free-riding cost. Take , , , as an example. It is advantageous for the BM retailer to implement the RPM regardless of whether the online retailer implements the RPM. Given that the BM retailer implements the RPM, it is optimal for the online retailer not to implement the RPM, so Strategy R is the equilibrium strategy in this case. Similarly, we can also identify cases in which Strategy O or Strategy R is the equilibrium strategy. Therefore, the equilibrium strategy in which the two retailers implement the RPM will be different with different free-riding costs.

VI. CONCLUSION
Based on the Hotelling model and transaction utility theory, this paper develops a duopoly model that describes freeriding behaviour and the reference price effect. This research provides management with new insights for addressing the free-riding behaviour of strategic customers in the highly competitive retail environment of online and offline retailers.
Our research provides (i) suggestions on how to adjust prices for BM retailers and online retailers according to different strategies, (ii) a better understanding of whether BM retailers and online retailers should implement reference pricing strategies to respond to strategic customers, and (iii) a method to mitigate the adverse effects of strategic customers with free-riding behaviour.
According to the research of this paper, we obtain the following main conclusions: (i) The impact of customers' free-riding behaviour on the BM retailer depends on the free-riding cost, but it has a negative impact on the online retailer and the overall market. We show that when the free-riding cost is low, the existence of strategic customers leads to fierce competition, which decreases the profit of the BM retailer. However, when the free-riding cost is high, there is no need for fierce price competition between the BM retailer and the online retailer to fight for market share. Therefore, the increase in demand from the BM retailer leads to an increase in its own profit. The free-riding behaviour of strategic customers leads to price competition between the two retailers, which reduces the retail prices of both retailers and the demand of the online retailer, resulting in the reduction of profits of the online retailer and the whole market.
(ii) The RPM may have a positive impact on one retailer in the market only. The retailer who unilaterally implements the RPM will increase its profit and reduce the profit of the competitor when the reference price is high enough. Even if the two retailers implement the RPM simultaneously, it is only possible to increase the profit of the retailer who sets higher reference prices.
(iii) The equilibrium strategy for implementing the RPM of the two retailers varies with free-riding cost. Strategy R, Strategy O, Strategy RO, and Strategy F are all likely to become the equilibrium strategy of the RPM under different free-riding costs.
The reference price is widely used in practice. For BM retailers, examples include tag prices in many brand-name clothing stores and official guidance prices in the automobile industry. The marking of these reference prices can indeed bring product awareness to customers. Moreover, if the retail price is lower than the reference price, it can enhance the transaction utility of customers and reduce the possibility of free riding by customers. Online retailers can also use the reference price to improve the transaction utility of customers and their competitiveness in the market to cope with the free-This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. However, the effectiveness of this mechanism must be based on the retailer's credibility. That is, the customer must believe that the reference price reflects the real value of the product. Therefore, it can be seen in some small commodity markets that the reference price often induces customers to engage in a greater degree of bargaining, which may strengthen customers' determination to free ride. This line of research can be extended in several directions. First, we assume that the customer thinks that any reference price set by the retailer is real and that the customer can obtain transaction utility from it. However, customers may not necessarily trust the reference prices given by retailers, or they may have different levels of trust in different reference prices. It would be interesting to explore the effectiveness of reference price strategies in depth by introducing a variable to characterize customers' belief in different reference prices and thus obtain different levels of transaction utility. In addition, we believe that determining the reference price is of general interest and can deepen the insights provided here. Our initial attempt suggests that determining a reference price will often lead to considerable analytical challenges, and therefore, this issue also indicates future research directions. Finally, the reference price can also be regarded as the promotion of discount information. For example, retailers clearly tell the customer how much discount they are receiving instead of directly giving the sales price. The greater the discount of the product, the higher the transaction utility the customer will obtain but the lower the customer's trust in the price before the discount may be. Determining the appropriate discount intensity is also a future research direction.

A. THE PROOF OF PROPOSITION 1
This method applies to Corollary 3-Corollary 6.

C. THE PROOF OF PROPOSITION 5
In the case of non-strategic customers, only two types of customers are considered: those who purchase products directly from BM retailers or from online retailers.