# Optimal Pricing and Service Investment of Home-Sharing Platforms in a Duopoly Market

Authors: Yingna Li; Pengfei Ma
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we construct a fully covered duopoly market model. In this market, two home-sharing platforms provide differentiated rental services to consumers, respectively, and each platform has two strategies: short-term rental strategy and long-term rental strategy. This paper studies the pricing decisions and service investment of home-sharing platforms in a competitive market. The results show that, in the market equilibrium, how the platform chooses the strategy largely depends on the service quality of competitors. Specifically, when the difference in service quality is small, it is better for the two platforms to adopt the short-term rental strategy; otherwise, the two platforms are more inclined to adopt the long-term rental strategy. We also find that the commission rate and service cost will also affect the profitability of the platform. Finally, we extend the model to the uncovered market.


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## Body

## 1. Introduction

In recent years, some online platforms have emerged in the accommodation industry to promote home-sharing, such asAirbnb, Homeway, Booking.com, and VRBO [1]. In these home-sharing platforms, service provides rent idle homes to consumers in need and obtain income [2]. This satisfies the needs of consumers and improves the overall utilization efficiency of the house [3, 4]. The most popular home-sharing platform is Airbnb. Airbnb is valued at over $31 billion [5]. Airbnb has more than 6 million unique listings in more than 99800 cities in more than 191 countries and regions. As of May 2019, hosts on Airbnb have generated over $65 billion in revenue [6].The latest development in the field of home-sharing is several commercial operators to home-sharing platforms, such asPillow.com, Marriott Bonvoy, and Tujia. These commercial operators collect the idle homes for unified operation and management and then rent them through the home-sharing platform. Different from the Consumer-to-Consumer (C2C) home-sharing platform based on market pricing, these commercial operators have pricing power. They can set the prices on behalf of the hosts to maximize their profits and share the profits with the owners according to the predetermined proportion of income sharing [7].For example, while most hosts on Airbnb are individuals, commercial operators are becoming increasingly influential [8]. In New York, 12 percent of Airbnb hosts are commercial operators, which account for 28 percent of total rental income, according to the data [9]. Meanwhile, more than 20 percent of listings are from hosts who own at least ten properties, indicating that the platform controlling all idle homes has become a phenomenon worthy of attention in the field of home-sharing.The services provided by the platform often play a crucial role in influencing the utility of consumers [10]. For example, the platform can provide consumers with infrastructure, cleaning services, and a feedback mechanism for consumer evaluation. These services are important factors that influence consumers’ purchasing decisions. These services play an important role in stimulating demand. In addition, the pricing decision of the platform is very important, which plays a decisive role in the profitability of the platform [11, 12]. Therefore, we pay special attention to home-sharing platforms with pricing power.Generally, each platform has two strategies to choose from short-term rental strategy and long-term rental strategy. Not all home-sharing platforms choose the short-term rental strategy. If the platform chooses the short-term rental strategy, it will have more customers sources and wider geographical scope, but it also has the risk of a low occupancy rate. If the platform chooses the long-term leasing strategy, once the transaction is completed, it will have fixed consumers who come from the region. Although both the short-term rental strategy and the long-term rental strategy can be observed in the home-sharing market, it is difficult to judge which strategy should be chosen under what conditions when the platform controls all listed homes. Therefore, we focus on the following key issues:(1)
In the duopoly market, under what conditions will the platform adopt the short-term rental strategy better than the long-term rental strategy?(2)
How does the platform determine the optimal service price and service level?(3)
How do the commission rate and service cost affect the strategy selection of the platform?The purpose of this paper is to provide reference and theoretical support for the operational decision-making and strategy selection of competitive platforms. Based on the practice of home-sharing platform, we consider that there are two competing platforms in the market and provide differentiated housing rental services, respectively. We analyze four cases: Case SS, both platforms choose short-term rental strategy; Case SL, platform 1 chooses the short-term rental strategy, and platform 2 chooses the long-term rental strategy; Case LS, platform 1 chooses the long-term rental strategy, and platform 2 chooses the short-term rental strategy; Case LL, both platforms choose the long-term rental strategy. Firstly, based on the Hotelling model, this paper constructs the profit maximization model of the platform in four different cases. Furthermore, it gives the optimal service price and service level in each case and analyzes the influence of service cost and commission rate on the optimal service price, service level, and profit. Finally, by comparing the platform’s profits in different cases, we discuss the conditions under which the platforms adopt different strategies and under what conditions to achieve equilibrium.The main contributions of this paper are as follows. Firstly, this paper focuses on the optimal pricing decisions and service investment of the home-sharing platform under the competitive environment and examines the impact of commission rate and service cost on the platform’s profit. Existing studies have analyzed pricing models in the market through the demand response management framework of contract theory [13]. Our research can fill the gap of the existing research. Secondly, through theoretical analysis, we can gain some new conclusions. For example, under some conditions, there may be two equilibria; that is, under different circumstances, both platforms can get the optimal profits. Therefore, we also give the condition of dominant equilibrium.The rest of this paper is as follows. Section2 reviews the relevant literature. Section 3 gives the problem description. Section 4 gives the equilibrium price, service level, and profit of the two platforms under four cases. In Section 5, we analyze the market equilibrium and discuss the impact of the service cost and commission rate on the profits of the two platforms. In Section 6, we extend our analysis to markets that are not fully covered and give the equilibrium results of the two platforms. Section 7 summarizes the main conclusions and management implications of this paper and gives future research. All the proofs are shown in Appendix.


## 2. Literature Review

This study is related to two research streams: the business model and the pricing decisions of the home-sharing platform. Next, we review the existing research on these two streams and discuss the interaction between our research and the existing research.With the application of Internet and mobile technology [14, 15], home-sharing platforms have developed rapidly. There is a large amount of literature that studies the business models of housing sharing platforms. For example, Zhu et al. point out that the current home-sharing platform has two business models: the business model priced by the market and the business model priced by the platform [7]. In the study of home-sharing based on market pricing, Zhou et al. find that the price is naturally determined by the supply and demand in the market; the purpose is to maintain the balance between supply and demand [16]. Moreover, Gibbs et al. point out that the price of listings on Airbnb is determined by the host based on the market price, and the platform only collects commissions and does not enforce pricing [17]. Zhu et al. point out that, compared with the home-sharing platform priced by the market, in the home-sharing platform priced by the platform, although it is still owned by the individual owner voluntarily, the price is determined by the platform, and the goal of the platform is often to maximize its benefit [7]. Different from the above research, we focus on the optimal pricing decision and service level investment of home-sharing platforms under a competitive environment. We analyze different situations: both platforms adopt the short-term rental strategy, one platform adopts the short-term rental strategy, and the other adopts the long-term rental strategy; both platforms adopt the long-term rental strategy. Furthermore, the conditions of different strategies are further analyzed.On the pricing decision of the home-sharing platform, we can see that scholars have carried out some research work. For example, Gal-Or studies the competition between the home-sharing platform and traditional hotels, and the results show that if the service fee of the platform is low enough, the consumer surplus will increase [18]. Gal-Or studies the impact of competition between home-sharing platforms on prices. They find that the competition can reduce the fees that platforms can charge consumers [19]. Zhou et al. study the competition between home-sharing platforms. They point out that, under certain conditions, two platforms will adopt the same commission rate contract [16]. Li et al. conduct an empirical study on the pricing behavior of experienced and inexperienced hosts on Airbnb. They find that experienced hosts are more likely to earn higher incomes because they often adjust prices and respond to high demand [20]. The platform can analyze consumers’ preference for service quality by collecting data about transactions [21–23]. Hong and Yoo study the relationship between service price and pricing, and their findings deepen the understanding of hosts’ pricing decisions [24]. Wang and Nicolau point out the factors that affect the rental price of Airbnb and also analyze the impact of the pricing of Airbnb on hotel channel performance [10]. Zervas et al. find that Airbnb hinders the development of low-end hotels and hotels that do not cater to business tourists. They also believe that, compared with hotels, the home-sharing platform has the advantages of low marginal costs and a wide range of services [25]. Barron et al. analyze the impact of home-sharing on housing sales and rents through Airbnb’s transaction data [26]. Different from the above research, we consider both the optimal pricing decision and service level investment for home-sharing platforms. We find that the service price and service level are related to factors such as the intrinsic value and commission rate of the services of its competitors.


## 3. Problem Description

In this paper, we consider that there are two home-sharing platforms (hereinafter referred to as “the platform”) to collect individual hosts’ homes, respectively, carry out unified decoration and management, and then for rental. Each platform has two strategies to choose from the short-term rental strategy and the long-term rental strategy. Each consumer who chooses the short-term rental service can only rent one day at most, and each consumer who chooses the long-term rental service can only rent one period at most. The rental services of the two platforms are characterized by quality attributes (i.e., vertical differentiation) and suitability attributes (i.e., horizontal differentiation). All consumers prefer high-quality service, but different consumers have different preferences. Quality determines the intrinsic value obtained by consumers from products, and the suitability attribute describes the negative utility caused by the inappropriate relationship between consumers’ ideal service and their expected service.For the convenience of analysis, the definition and description of related symbols and variables involved in this paper are shown in Table1, where i=1,2 and k∈SS,SL,LS,LL.Table 1
Summary of notations.
NotationsDescriptionspikService price (decision variable).xikService level (decision variables).NThe number of days in a period.vSiThe intrinsic value of short-term rental services.vLiThe intrinsic value of long-term rental services.xikUtility per unit of service level.λ/βλThe unsuitable cost for consumers to choose the short-term rental service/ the long-term rental service, whereβ>1.cUnit cost of services provided by the platform.riCommission rate.


## 4. Model

In this paper, we build the model based on the Hotelling model and assume that the market size isN. Each consumer only needs one service from two platforms at most, and the preferences of consumers are evenly divided in 0,1, where platform 1 is located at 0 and platform 2 is located at 1. A consumer’s location represents his ideal service. If a service does not exactly match the consumer’s preferences, then the consumer incurs inappropriate costs, which increase with the distance between his location and the given platform. Assume that the inappropriate cost for consumers to choose short-term rental service is λ. Since the rental time of long-term rental service is long, the inappropriate cost of long-term rental service is assumed to be βλ, where β>1. The utility functions of consumers are as follows:(1)US1=vS1−p1k+αx1k−λt,(2)UL1=vL1−p1kN+αx1k−βλt,(3)US2=vS2−p2k+αx2k−λ1−t,(4)UL2=vL2−p2kN+αx2k−βλ1−t.Next, we will analyze the different strategies adopted by each platform.


### 4.1. Case SS: Both Platforms Choose the Short-Term Rental Strategy

When both platforms choose the short-term rental strategy, the utility of consumers choosing platforms 1 and 2 is equations (1) and (3), respectively, as shown in Figure 1.Figure 1
Market segmentation under Case SS.The goal of each consumer is to maximize their utility. Therefore, the whole market is divided into0,t0 and t0,1. Consumers in 0,t0 prefer to choose the service of platform 1 rather than that of platform 2. On the contrary, consumers in t0,1 prefer the service of platform 2 rather than that of platform 1. The marginal consumers at t0 have no difference in choosing the service of platform 1 and platform 2.ByUS1=US2, we can obtain t0=λ−p1SS+p2SS+vS1−vS2+αx1SS−αx2SS/2λ; then the demand functions of platform 1 and platform 2 are DSS1=t0 and DSS2=1−t0, respectively.Under Case SS, the profit functions of the platforms are, respectively, in equations (5) and (6):(5)Π1SS=r1p1SSD1SSN−cx1SS2,(6)Π2SS=r2p2SSD2SSN−cx2SS2.Each platform determines the optimal short-term rental service pricepiSS∗ and the service level xiSS∗ to maximize its profit. The optimal solutions can be obtained by substituting t0 into equations (5) and (6), respectively. In this paper, we assume that the second-order conditions are positive (i.e., 8cλ−α2r1N>0 and 8cλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 1.
The optimal short-rental price and the service level of platform 1 and platform 2, are, respectively,(7)p1SS∗=2λ6cλ−α2r2N+2cvS1−2cvS212cλ−α2r1N−α2r2N,x1SS∗=αr1N6cλ−α2r2N+2cvS1−2cvS22c12cλ−α2r1N−α2r2N,p2SS∗=2λ6cλ−α2r1N−2cvS1+2cvS212cλ−α2r1N−α2r2N,x2SS∗=αr2N6cλ−α2r1N−2cvS1+2cvS22c12cλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 can be obtained as(8)D1SS∗=6cλ−α2r2N+2cvS1−2cvS212cλ−α2r1N−α2r2N,Π1SS∗=r1N8cλ−α2r1N6cλ−α2r2N+2cvS1−2cvS224c12cλ−α2r1N−α2r2N2.
The demand and the profit of platform 2 can be obtained as(9)D2SS∗=6cλ−α2r1N−2cvS1+2cvS212cλ−α2r1N−α2r2N,Π2SS∗=r2N8cλ−α2r2N6cλ−α2r1N−2cvS1+2cvS224c12cλ−α2r1N−α2r2N2.


### 4.2. Case SL: Platform 1 Chooses the Short-Term Rental Strategy, and Platform 2 Chooses the Long-Term Rental Strategy

Under Case SL, we analyze the operational decisions of the two platforms. The utility obtained by consumers using the services of platform 1 and platform 2 is equations (1) and (4), respectively, as shown in Figure 2.Figure 2
Market segmentation under Case SL.Similar to Case SS, the whole market is divided in0,t1 and t1,1. The consumers in 0,t1 prefer to choose the service of platform 1 rather than that of platform 2. On the contrary, the consumers in t1,1 prefer the service of platform 2 rather than that of platform 1. The marginal consumer at t1 has no difference in choosing the service of platform 1 and platform 2.ByUS1=UL2, we can obtain t1=NvS1−p1SL+βλ+αx1SL−αx2SL+p2SL−vL2/Nλβ+1; then the demand functions of platform 1 and platform 2 are D1SL=t1 and DSS2=1−t0, respectively.Under Case SL, the profit functions of the platforms are, respectively, in equations (10) and (11):(10)Π1SL=r1p1SLD1SLN−cx1SL2,(11)Π2SL=r2p2SLD2SL−cx2SL2.Each platform determines the optimal short-term rental service pricepiSL∗ and the service level xiSL∗ to maximize its profit. The optimal solutions can be obtained by substituting t1 into equations (10) and (11), respectively. We assume that the second-order conditions are positive (i.e., 4cλ+4βcλ−α2r1N>0 and 4cλ+4βcλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 2.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(12)p1SL∗=λβ+12cλN+4βcλN−α2r2N2+2cNvS1−2cvL2N6cλ+6βcλ−α2r1N−α2r2N,x1SL∗=αr12cλN+4βcλN−α2r2N2+2cNvS1−2cvL22c6cλ+6βcλ−α2r1N−α2r2N,p2SL∗=λβ+12βcλN+4cλN−α2r1N2−2cNvS1+2cvL26cλ+6βcλ−α2r1N−α2r2N,x2SL∗=αr22βcλN+4cλN−α2r1N2−2cNvS1+2cvL22c6cλ+6βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(13)D1SL∗=2cλN+4βcλN−α2r2N2+2cNvS1−2cvL2N6cλ+6βcλ−α2r1N−α2r2N,Π1SL∗=r14cλ+4βcλ−α2r1N2cλN+4βcλN−α2r2N2+2cNvS1−2cvL224cN6cλ+6βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(14)D2SL∗=2βcλN+4cλN−α2r1N2−2cNvS1+2cvL2N6cλ+6βcλ−α2r1N−α2r2N,Π2SL∗=r24cλ+4βcλ−α2r2N2βcλN+4cλN−α2r1N2−2cNvS1+2cvL224cN6cλ+6βcλ−α2r1N−α2r2N2.


### 4.3. Case LS: Platform 1 Chooses the Long-Term Rental Strategy, and the Platform 2 Chooses the Short-Term Rental Strategy

The results of Case LS can be solved in the same way as Case SL. Under Case LS, the utility obtained by consumers using the services of platform 1 and platform 2, respectively, is equations (2) and (3), as shown in Figure 3.Figure 3
Market segmentation under Case LS.Similar to Case SL, by solvingUL1=US2, we can obtain t2=Nλ+p2LS−vS2+αx1LS−αx2LS−p1LS+vL1/Nλβ+1; then the demand functions of platform 1 and platform 2 are D1LS=t2 and D2LS=1−t2.Under Case LS, the profit functions of the platforms are, respectively, in equations (15) and (16):(15)Π1LS=r1p1LSD1LS−cx1LS2,(16)Π2LS=r2p2LSD2LSN−cx2LS2.Each platform determines the optimal service pricepiLS∗ and the service level xiLS∗ to maximize its profit. The optimal solutions can be obtained by substituting t2 into equations (15) and (16), respectively. We assume that the second-order conditions are positive (i.e., 4cλ+4βcλ−α2r1N>0 and 4cλ+4βcλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 3.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(17)p1LS∗=λβ+12βcλN+4cλN−α2r2N2+2cvL1−2cNvS26cλ+6βcλ−α2r1N−α2r2N,x1LS∗=αr12βcλN+4cλN−α2r2N2+2cvL1−2cNvS22c6cλ+6βcλ−α2r1N−α2r2N,p2LS∗=λβ+12cλN+4βcλN−α2r1N2−2cvL1+2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,x2LS∗=αr22cλN+4βcλN−α2r1N2−2cvL1+2cNvS22c6cλ+6βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(18)D1LS∗=2βcλN+4cλN−α2r2N2+2cvL1−2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,Π1LS∗=r14cλ+4βcλ−α2r1N2βcλN+4cλN−α2r2N2+2cvL1−2cNvS224cN6cλ+6βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(19)D2LS∗=2cλN+4βcλN−α2r1N2−2cvL1+2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,Π2LS∗=r24cλ+4βcλ−α2r2N2cλN+4βcλN−α2r1N2−2cvL1+2cNvS224cN6cλ+6βcλ−α2r1N−α2r2N2.


### 4.4. Case LL: Both Platforms Choose the Long-Term Rental Strategy

Under Case LL, the utility obtained by the consumers using the services of platform 1 and platform 2, respectively, is equations (2) and (4), as shown in Figure 4.Figure 4
Market segmentation under Case LL.Similar to Case LS, by solvingUL1=UL2, we can obtain t3=Nβλ+αx1LL−αx2LL−p1LL+p2LL+vL1−vL2/2βλN; then the demand functions of platform 1 and platform 2 are D1LL=t3 and D2LL=1−t3.Under Case LL, the profit functions of the platforms are, respectively, in equations (20) and (21):(20)Π1LL=r1p1LLD1LL−cx1LL2,(21)Π2LL=r2p2LLD2LL−cx2LL2.Each platform determines the optimal service pricepiLL∗ and the service level xiLL∗ to maximize its profit. The optimal solutions can be obtained by substituting t3 into equations (20) and (21), respectively. We assume that the second-order conditions are positive (i.e., 8βcλ−α2r1N>0 and 8βcλ−α2r1N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 4.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(22)p1LL∗=2βλ6βcλN−α2r2N+2cvL1−2cvL212βcλ−α2r1N−α2r2N,x1LL∗=αr16βcλN−α2r2N2+2cvL1−2cvL22c12βcλ−α2r1N−α2r2N,p2LL∗=2βλ6βcλN−α2r1N2−2cvL1+2cvL212βcλ−α2r1N−α2r2N,x2LL∗=αr26βcλN−α2r1N2−2cvL1+2cvL22c12βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(23)D1LL∗=6βcλN−α2r2N+2cvL1−2cvL2N12βcλ−α2r1N−α2r2N,Π1LL∗=r18βcλ−α2r1N6βcλN−α2r2N2+2cvL1−2cvL224cN12βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(24)D2LL∗=6βcλN−α2r1N2−2cvL1+2cvL2N12βcλ−α2r1N−α2r2N,Π2LL∗=r28βcλ−α2r2N6βcλN−α2r1N2−2cvL1+2cvL224cN12βcλ−α2r1N−α2r2N2.


## 4.1. Case SS: Both Platforms Choose the Short-Term Rental Strategy

When both platforms choose the short-term rental strategy, the utility of consumers choosing platforms 1 and 2 is equations (1) and (3), respectively, as shown in Figure 1.Figure 1
Market segmentation under Case SS.The goal of each consumer is to maximize their utility. Therefore, the whole market is divided into0,t0 and t0,1. Consumers in 0,t0 prefer to choose the service of platform 1 rather than that of platform 2. On the contrary, consumers in t0,1 prefer the service of platform 2 rather than that of platform 1. The marginal consumers at t0 have no difference in choosing the service of platform 1 and platform 2.ByUS1=US2, we can obtain t0=λ−p1SS+p2SS+vS1−vS2+αx1SS−αx2SS/2λ; then the demand functions of platform 1 and platform 2 are DSS1=t0 and DSS2=1−t0, respectively.Under Case SS, the profit functions of the platforms are, respectively, in equations (5) and (6):(5)Π1SS=r1p1SSD1SSN−cx1SS2,(6)Π2SS=r2p2SSD2SSN−cx2SS2.Each platform determines the optimal short-term rental service pricepiSS∗ and the service level xiSS∗ to maximize its profit. The optimal solutions can be obtained by substituting t0 into equations (5) and (6), respectively. In this paper, we assume that the second-order conditions are positive (i.e., 8cλ−α2r1N>0 and 8cλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 1.
The optimal short-rental price and the service level of platform 1 and platform 2, are, respectively,(7)p1SS∗=2λ6cλ−α2r2N+2cvS1−2cvS212cλ−α2r1N−α2r2N,x1SS∗=αr1N6cλ−α2r2N+2cvS1−2cvS22c12cλ−α2r1N−α2r2N,p2SS∗=2λ6cλ−α2r1N−2cvS1+2cvS212cλ−α2r1N−α2r2N,x2SS∗=αr2N6cλ−α2r1N−2cvS1+2cvS22c12cλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 can be obtained as(8)D1SS∗=6cλ−α2r2N+2cvS1−2cvS212cλ−α2r1N−α2r2N,Π1SS∗=r1N8cλ−α2r1N6cλ−α2r2N+2cvS1−2cvS224c12cλ−α2r1N−α2r2N2.
The demand and the profit of platform 2 can be obtained as(9)D2SS∗=6cλ−α2r1N−2cvS1+2cvS212cλ−α2r1N−α2r2N,Π2SS∗=r2N8cλ−α2r2N6cλ−α2r1N−2cvS1+2cvS224c12cλ−α2r1N−α2r2N2.


## 4.2. Case SL: Platform 1 Chooses the Short-Term Rental Strategy, and Platform 2 Chooses the Long-Term Rental Strategy

Under Case SL, we analyze the operational decisions of the two platforms. The utility obtained by consumers using the services of platform 1 and platform 2 is equations (1) and (4), respectively, as shown in Figure 2.Figure 2
Market segmentation under Case SL.Similar to Case SS, the whole market is divided in0,t1 and t1,1. The consumers in 0,t1 prefer to choose the service of platform 1 rather than that of platform 2. On the contrary, the consumers in t1,1 prefer the service of platform 2 rather than that of platform 1. The marginal consumer at t1 has no difference in choosing the service of platform 1 and platform 2.ByUS1=UL2, we can obtain t1=NvS1−p1SL+βλ+αx1SL−αx2SL+p2SL−vL2/Nλβ+1; then the demand functions of platform 1 and platform 2 are D1SL=t1 and DSS2=1−t0, respectively.Under Case SL, the profit functions of the platforms are, respectively, in equations (10) and (11):(10)Π1SL=r1p1SLD1SLN−cx1SL2,(11)Π2SL=r2p2SLD2SL−cx2SL2.Each platform determines the optimal short-term rental service pricepiSL∗ and the service level xiSL∗ to maximize its profit. The optimal solutions can be obtained by substituting t1 into equations (10) and (11), respectively. We assume that the second-order conditions are positive (i.e., 4cλ+4βcλ−α2r1N>0 and 4cλ+4βcλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 2.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(12)p1SL∗=λβ+12cλN+4βcλN−α2r2N2+2cNvS1−2cvL2N6cλ+6βcλ−α2r1N−α2r2N,x1SL∗=αr12cλN+4βcλN−α2r2N2+2cNvS1−2cvL22c6cλ+6βcλ−α2r1N−α2r2N,p2SL∗=λβ+12βcλN+4cλN−α2r1N2−2cNvS1+2cvL26cλ+6βcλ−α2r1N−α2r2N,x2SL∗=αr22βcλN+4cλN−α2r1N2−2cNvS1+2cvL22c6cλ+6βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(13)D1SL∗=2cλN+4βcλN−α2r2N2+2cNvS1−2cvL2N6cλ+6βcλ−α2r1N−α2r2N,Π1SL∗=r14cλ+4βcλ−α2r1N2cλN+4βcλN−α2r2N2+2cNvS1−2cvL224cN6cλ+6βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(14)D2SL∗=2βcλN+4cλN−α2r1N2−2cNvS1+2cvL2N6cλ+6βcλ−α2r1N−α2r2N,Π2SL∗=r24cλ+4βcλ−α2r2N2βcλN+4cλN−α2r1N2−2cNvS1+2cvL224cN6cλ+6βcλ−α2r1N−α2r2N2.


## 4.3. Case LS: Platform 1 Chooses the Long-Term Rental Strategy, and the Platform 2 Chooses the Short-Term Rental Strategy

The results of Case LS can be solved in the same way as Case SL. Under Case LS, the utility obtained by consumers using the services of platform 1 and platform 2, respectively, is equations (2) and (3), as shown in Figure 3.Figure 3
Market segmentation under Case LS.Similar to Case SL, by solvingUL1=US2, we can obtain t2=Nλ+p2LS−vS2+αx1LS−αx2LS−p1LS+vL1/Nλβ+1; then the demand functions of platform 1 and platform 2 are D1LS=t2 and D2LS=1−t2.Under Case LS, the profit functions of the platforms are, respectively, in equations (15) and (16):(15)Π1LS=r1p1LSD1LS−cx1LS2,(16)Π2LS=r2p2LSD2LSN−cx2LS2.Each platform determines the optimal service pricepiLS∗ and the service level xiLS∗ to maximize its profit. The optimal solutions can be obtained by substituting t2 into equations (15) and (16), respectively. We assume that the second-order conditions are positive (i.e., 4cλ+4βcλ−α2r1N>0 and 4cλ+4βcλ−α2r2N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 3.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(17)p1LS∗=λβ+12βcλN+4cλN−α2r2N2+2cvL1−2cNvS26cλ+6βcλ−α2r1N−α2r2N,x1LS∗=αr12βcλN+4cλN−α2r2N2+2cvL1−2cNvS22c6cλ+6βcλ−α2r1N−α2r2N,p2LS∗=λβ+12cλN+4βcλN−α2r1N2−2cvL1+2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,x2LS∗=αr22cλN+4βcλN−α2r1N2−2cvL1+2cNvS22c6cλ+6βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(18)D1LS∗=2βcλN+4cλN−α2r2N2+2cvL1−2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,Π1LS∗=r14cλ+4βcλ−α2r1N2βcλN+4cλN−α2r2N2+2cvL1−2cNvS224cN6cλ+6βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(19)D2LS∗=2cλN+4βcλN−α2r1N2−2cvL1+2cNvS2N6cλ+6βcλ−α2r1N−α2r2N,Π2LS∗=r24cλ+4βcλ−α2r2N2cλN+4βcλN−α2r1N2−2cvL1+2cNvS224cN6cλ+6βcλ−α2r1N−α2r2N2.


## 4.4. Case LL: Both Platforms Choose the Long-Term Rental Strategy

Under Case LL, the utility obtained by the consumers using the services of platform 1 and platform 2, respectively, is equations (2) and (4), as shown in Figure 4.Figure 4
Market segmentation under Case LL.Similar to Case LS, by solvingUL1=UL2, we can obtain t3=Nβλ+αx1LL−αx2LL−p1LL+p2LL+vL1−vL2/2βλN; then the demand functions of platform 1 and platform 2 are D1LL=t3 and D2LL=1−t3.Under Case LL, the profit functions of the platforms are, respectively, in equations (20) and (21):(20)Π1LL=r1p1LLD1LL−cx1LL2,(21)Π2LL=r2p2LLD2LL−cx2LL2.Each platform determines the optimal service pricepiLL∗ and the service level xiLL∗ to maximize its profit. The optimal solutions can be obtained by substituting t3 into equations (20) and (21), respectively. We assume that the second-order conditions are positive (i.e., 8βcλ−α2r1N>0 and 8βcλ−α2r1N>0), which can ensure that the demand functions of each platform are positive.By solving the profit functions of platform 1 and platform 2, the following lemma can be obtained.Lemma 4.
The optimal short-rental price and the service level of platform 1 and platform 2 are, respectively,(22)p1LL∗=2βλ6βcλN−α2r2N+2cvL1−2cvL212βcλ−α2r1N−α2r2N,x1LL∗=αr16βcλN−α2r2N2+2cvL1−2cvL22c12βcλ−α2r1N−α2r2N,p2LL∗=2βλ6βcλN−α2r1N2−2cvL1+2cvL212βcλ−α2r1N−α2r2N,x2LL∗=αr26βcλN−α2r1N2−2cvL1+2cvL22c12βcλ−α2r1N−α2r2N.
Furthermore, the demand and profit of platform 1 are as follows:(23)D1LL∗=6βcλN−α2r2N+2cvL1−2cvL2N12βcλ−α2r1N−α2r2N,Π1LL∗=r18βcλ−α2r1N6βcλN−α2r2N2+2cvL1−2cvL224cN12βcλ−α2r1N−α2r2N2.
The demand and profit of platform 2 are as follows:(24)D2LL∗=6βcλN−α2r1N2−2cvL1+2cvL2N12βcλ−α2r1N−α2r2N,Π2LL∗=r28βcλ−α2r2N6βcλN−α2r1N2−2cvL1+2cvL224cN12βcλ−α2r1N−α2r2N2.


## 5. Market Equilibrium

The game is divided into three stages. First, platform 1 and platform 2 simultaneously decide whether to adopt the short-term or long-term rental strategy. Next, the two platforms determine the service price and the service level at the same time. Finally, consumers make purchase decisions after obtaining market information (e.g., service price, the rental strategy of the platform). In addition, due to the complexity of the model of duopoly market, we also give some numerical analysis to visually display the results to obtain more management insights, in which the parameter is set tovL1=309, vL2=306, vs1=10, vs2=9.8, β=1.2, λ=13, α=7, N=30, r1=0.56, and r2=0.55.In this paper, the equilibrium results under four different cases can be obtained by backward induction. In each subgame, we can obtain the optimal service price, service level, and profit of the two platforms. Table2 summarizes the profits of the two platforms. By solving the payoff matrix in Table 2, the following proposition is obtained.Table 2
Equilibrium profits.
Platform 2Short-term rental strategyLong-term rental strategyPlatform 1Short-term rental strategyΠ1SS∗,Π2SS∗Π1SL∗,Π2SL∗Long-term rental strategyΠ1LS∗,Π2LS∗Π1LL∗,Π2LL∗Proposition 1.
The market equilibria under different conditions are as follows:(i)
ifΔV1<T1, and ΔV2<T2, Case SS is an equilibrium;(ii)
ifΔV1<T3, and ΔV2>T2, Case SL is an equilibrium;(iii)
ifΔV1>T1, and ΔV2<T4, Case LS is an equilibrium;(iv)
ifΔV1>T3, and ΔV2>T4, Case LL is an equilibrium;where ΔV=vS1−vS2, ΔV1=vL1−NvS1ΔV2=vL2−NvS2,(25)T1=α2r2N2−2cNΔV−2βcλN−4cλN+A12c,T2=α2r1N2+2cNΔV−2βcλN−4cλN+A22c,T3=α2r2N2−2cNΔV−6βcλN+2cΔV2+A32c,T4=2cΔV1+2cNΔV+α2r1N2−6βcλN+A42c,A1=N6cλ+6βcλ−α2r1N−α2r2N6cλ−α2r2N+2cΔV8cλ−α2r1N12cλ−α2r1N−α2r2N4cλ+4βcλ−α2r1N,A2=N6cλ+6βcλ−α2r1N−α2r2N6cλ−α2r1N−2cΔV8cλ−α2r2N12cλ−α2r1N−α2r2N4cλ+4βcλ−α2r2N,A3=12βcλ−α2r1N−α2r2N2cλN+4βcλN−α2r2N2+2cNΔV−2cΔV24cλ+4βcλ−α2rN6cλ+6βcλ−α2r1N−α2r2N8βcλ−α2r1N,A4=12βcλ−α2r1N−α2r2N2cλN+4βcλN−α2r1N2+2cNΔV−2cΔV14cλ+4βcλ−α2r2N6cλ+6βcλ−α2r1N−α2r2N8βcλ−α2r2N.Proposition1 describes four possible conditions of market equilibrium. Specifically, if the difference in the intrinsic values of the service (ΔV1 and ΔV2) is small, then both platforms choose to adopt the short-term rental strategy in equilibrium. If the difference in the intrinsic values of the service (i.e., ΔV1 and ΔV2) is high enough, then the two platforms adopt the long-term rental strategy. When the intrinsic value of the service of platform 1 is high and and the intrinsic value of the service of platform 2 is small, then the platform 1 adopts the short-term rental strategy, while platform 2 adopts the long-term rental strategy.When choosing the rental strategy, the platform must consider the intrinsic value of the services provided by competitors. If there is a significant difference in the intrinsic value difference between the services of the two platforms, the platform with the high intrinsic value difference chooses to adopt a long-term rental strategy in equilibrium, while the competitor chooses to adopt the short-term rental strategy. When faced with competitors with higher quality services, services can attract more consumers and expand the consumer base. Therefore, it is more profitable to choose the long-term rental strategy for a platform with high service quality. If the difference in the intrinsic value of services of each platform is significant, there are two situations: if the difference in the intrinsic value of services is small, the two platforms adopt the short-term rental strategy when in equilibrium; if there is a large difference in the intrinsic value of services, the two platforms will adopt the long-term rental strategy when they are in equilibrium.However, when the quality difference between the long-term rental service and the short-term rental service is moderate, there are two equilibria in both cases, as shown in Proposition2.Proposition 2.
Table3 summarizes the conditions under which the two equilibria exist and the corresponding dominant equilibria, as shown in Figure 5.In Table3, we can find that when the threshold of the difference in service quality satisfies T1<T3 and T2<T4, there may be two equilibria in some similar regions. As shown in Figure 5(a), the region A-B-C-D is marked as SL/LS. We can also find that when the threshold of the difference in service quality satisfies T3<T1 and T4<T2, there are two equilibria in some similar regions; that is, both platforms adopt the short-term rental strategy or the long-term rental strategy. As shown in Figure 5(b), the region E-F-G-H is marked as SS/LL. This is because a platform’s competitors choose the short-term rental strategy; there are two reasons for the platform to give up the long-term rental strategy. One is the quality of its service, and the other is to compete with it for more consumers. The more the competitor’s customer base, the greater the consumer’s demand for rental services. Therefore, the platform will lose market share and exit the market.Table 3
Two equilibria.
Two equilibriaDominant equilibriumParameter conditionSS and LLSST3<ΔV1<T1 and T4<ΔV2<T2SL and LS—T1<ΔV1<T3 and T2<ΔV2<T4Figure 5
Two equilibria.
(a)(b)Proposition 3.
(i)
If both platforms choose the short-term rental strategy in equilibrium, whenri<4cλ12cλ−α2rjN/α2N8cλ−α2rjN, ∂Πi/∂ri>0, and ∂Πi/∂rj>0.(ii)
If the platformi chooses the short-term rental strategy and the platform j chooses the long-term rental strategy in equilibrium, when ri<2cλ1+β6cλ1+β−α2rjN/α2N4cλ+4βcλ−α2rjN, ∂Πi/∂ri>0, and ∂Πi/∂rj>0.(iii)
If both platforms choose the log-term rental strategy in equilibrium, whenri>4βcλ12βcλ−α2rjN/α2N8βcλ−α2rjN, ∂Πi/∂ri>0, and ∂Πi/∂rj>0, where i,j∈1,2,i≠j.Proposition3 shows that when both platforms choose the short-term rental strategy or the long-term rental strategy, the profit of platform i increases with the increase of its commission rate (ri); no matter how ri changes, the profit of platform i increases with the increase of commission rate (rj) of the competitive platform. In other words, if any commission rate increases, the profit of the platform will increase. The difference is that when the two platforms choose different rental strategies, the profit of platform i increases with the increase of its commission rate (ri); no matter how ri changes, the profit of platform i decreases with the increase of commission rate (rj) of competitive platform.Proposition 4.
In equilibrium, there are∂Πi/∂c>0 and ∂Πj/∂c>0, where i,j∈1,2,i≠j.Proposition4 shows that regardless of whether platform i and platform j choose the short-term rental strategy or the long-term rental strategy, the profits of platform i and platform j increase with their service cost c increases. The platform can reduce service costs and make it more profitable under the premise of ensuring service quality.From Figures6–9, we can find that the results of numerical analysis are consistent with those of Proposition 4. It can be seen from Figures 6–8 that, in the equilibrium state, the profit of platform 2 is larger than that of platform 1 under Cases SS, SL, and LS. In Figure 9, the profit of platform 1 is larger than that of platform 2 under Case LL. Furthermore, the platform can actually estimate the service cost of the industry through sufficient market research.Figure 6
Impact ofc on Π∗ under Case SS.Figure 7
Impact ofc on Π∗ under Case SL.Figure 8
Impact ofc on Π∗ under Case LS.Figure 9
Impact ofc on Π∗ under Case LL.


## 6. Extension

In the previous sections, this paper assumes that the market is fully covered. In this section, we relax this assumption to explore the market strategy and operational decisions of home-sharing platforms as local monopoly platforms.


### 6.1. Case SS: Both Platforms Are Local Monopolies and Both Platforms Choose the Short-Term Rental Strategy

When the market is not fully covered, both platforms are local monopolies. The marginal consumers who choose the service of platform 1 are att1e=vS1−p1eSS+αx1eSS/λ, and the marginal consumers who choose the service of platform 2 are at t2e=λ+p2eSS−vS2−αx2eSS/λ. The consumers between 0,t1e choose the short-term rental service of platform 1, and the consumers between t2e,1 choose the short-term rental service of platform 2, where t1e<t2e. In this setting, these platforms do not directly compete with marginal consumers. Therefore, the profit functions of platform 1 and platform 2 are as follows: Π1eSS=r1p1eSSt1eN−cx1eSS2 and Π2eSS=r2p2eSS1−t2eN−cx2eSS2. In equilibrium, the platforms determine the prices and the service levels, respectively, to maximize the profits. According to the first-order conditions of the profit functions on price and service level, the price and service level of platform 1 under equilibrium can be obtained as p1eSS∗=2cλvS1/4cλ−α2r1N, p1eSS∗=2cλvS1/4cλ−α2r1N, and x1eSS∗=αr1vS1N/4cλ−α2r1N. Furthermore, the demand and profit of platform 1 can be obtained as D1eSS∗=2cvS1/4cλ−α2r1N and Π1eSS∗=cr1vS12N/4cλ−α2r1N. Similarly, we can get that the price and the service level of platform 2 are p2eSS∗=2cλvS2/4cλ−α2r2N and x2eSS∗=αr2vS2N/4cλ−α2r2N. Furthermore, the demand and profit of platform 2 are D2eSS∗=2cvS2/4cλ−α2r2N and Π2eSS∗=cr2vS22N/4cλ−α2r2N.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvS14cλ−α2r2N+vS24cλ−α2r1N<4cλ−α2r1N4cλ−α2r2N must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


### 6.2. Case SL: Both Platforms Are Local Monopolies, Platform 1 Chooses the Short-Term Rental Strategy, and Platform 2 Chooses the Long-Term Rental Strategy

When platform 1 chooses the short-term rental strategy and platform 2 chooses the long-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=vS1−p1eSL+αx1eSL/λ, and the marginal consumers who choose the service of platform 2 are at t2e=βλN−αx2eSLN+p2eSL−vL2/βλN. The consumers between 0,t1e choose the short-term rental service of platform 1, and the consumers between t2e,1 choose the long-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eSL=r1p1eSLt1eN−cx1eSL2 and Π2eSL=r2p2eSL1−t2e−cx2eSL2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eSL∗=2cλvS1/4cλ−α2r1N and x1eSL∗=αr1vS1N/4cλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eSL∗=2cvS1/4cλ−α2r1N and Π1eSL∗=cr1vS12N/4cλ−α2r1N. Similarly, the price and service level of platform 2 are p2eSL∗=2βcλvL2/4βcλ−α2r2N and x2eSL∗=αr2vL2/4βcλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eSL∗=2cvL2/4βcλ−α2r2NN and Π2eSL∗=cr2vL22/4βcλ−α2r2NN, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvS14βcλ−α2r2NN+vL24cλ−α2r1N<4cλ−α2r1N4βcλ−α2r2NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


### 6.3. Case LS: Both Platforms Are Local Monopolies, Platform 1 Chooses the Long-Term Rental Strategy, and Platform 2 Chooses the Short-Term Rental Strategy

When platform 1 chooses the long-term rental strategy and platform 2 chooses short-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=αx1eLSN−p1eLS+vL1/βλN, and the marginal consumers who choose the service of platform 2 are at t2e=λ+p2eLS−αx2eLS−vL2/λ. The consumers between 0,t1e choose the long-term rental service of platform 1, and the consumers between t2e,1 choose the short-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eLS=r1p1eLSt1e−cx1eLS2 and Π2eLS=r2p2eLS1−t2eN−cx2eLS2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eLS∗=2βcλvS1/4βcλ−α2r1N and x1eLS∗=αr1vL1/4βcλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eLS∗=2cvL1/4βcλ−α2r1NN and Π1eLS∗=cr1vL12/4βcλ−α2r1NN. Similarly, the price and service level of platform 2 are p2eLS∗=2cλvS2/4cλ−α2r2N and x2eLS∗=αr2vS2N/4cλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eLS∗=2cvS2/4cλ−α2r2N and Π2eLS∗=cr2vS22N/4cλ−α2r2N, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvL14cλ−α2r2N+vS24βcλ−α2r1NN<4cλ−α2r2N4βcλ−α2r1NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


### 6.4. Case LL: Both Platforms Are Local Monopolies and Both Platforms Choose Long-Term Rental Strategy

When platform 1 and platform 2 choose the long-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=αx1eLLN−p1eLL+vL1/βλN, and the marginal consumers who choose the service of platform 2 are at t2e=βλN−αx2eLLN+p2eLL−vL2/βλN. The consumers between 0,t1e choose the long-term rental service of platform 1, and the consumers between t2e,1 choose the long-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eLL=r1p1eLLt1e−cx1eLL2 and Π2eLL=r2p2eLL1−t2e−cx2eLL2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eLL∗=2βcλvL1/4βcλ−α2r1N and x1eLL∗=αr1vL1/4βcλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eLL∗=2cvL1/4βcλ−α2r1NN and Π1eLL∗=cr1vL12/4βcλ−α2r1NN. Similarly, the price and service level of platform 2 are p2eLL∗=2βcλvL2/4βcλ−α2r2N and x2eLL∗=αr2vL2/4βcλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eLL∗=2cvL2/4βcλ−α2r2NN and Π2eLL∗=cr2vL22/4βcλ−α2r2NN, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvL14βcλ−α2r2N+vL24βcλ−α2r1N<4βcλ−α2r1N4βcλ−α2r2NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


## 6.1. Case SS: Both Platforms Are Local Monopolies and Both Platforms Choose the Short-Term Rental Strategy

When the market is not fully covered, both platforms are local monopolies. The marginal consumers who choose the service of platform 1 are att1e=vS1−p1eSS+αx1eSS/λ, and the marginal consumers who choose the service of platform 2 are at t2e=λ+p2eSS−vS2−αx2eSS/λ. The consumers between 0,t1e choose the short-term rental service of platform 1, and the consumers between t2e,1 choose the short-term rental service of platform 2, where t1e<t2e. In this setting, these platforms do not directly compete with marginal consumers. Therefore, the profit functions of platform 1 and platform 2 are as follows: Π1eSS=r1p1eSSt1eN−cx1eSS2 and Π2eSS=r2p2eSS1−t2eN−cx2eSS2. In equilibrium, the platforms determine the prices and the service levels, respectively, to maximize the profits. According to the first-order conditions of the profit functions on price and service level, the price and service level of platform 1 under equilibrium can be obtained as p1eSS∗=2cλvS1/4cλ−α2r1N, p1eSS∗=2cλvS1/4cλ−α2r1N, and x1eSS∗=αr1vS1N/4cλ−α2r1N. Furthermore, the demand and profit of platform 1 can be obtained as D1eSS∗=2cvS1/4cλ−α2r1N and Π1eSS∗=cr1vS12N/4cλ−α2r1N. Similarly, we can get that the price and the service level of platform 2 are p2eSS∗=2cλvS2/4cλ−α2r2N and x2eSS∗=αr2vS2N/4cλ−α2r2N. Furthermore, the demand and profit of platform 2 are D2eSS∗=2cvS2/4cλ−α2r2N and Π2eSS∗=cr2vS22N/4cλ−α2r2N.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvS14cλ−α2r2N+vS24cλ−α2r1N<4cλ−α2r1N4cλ−α2r2N must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


## 6.2. Case SL: Both Platforms Are Local Monopolies, Platform 1 Chooses the Short-Term Rental Strategy, and Platform 2 Chooses the Long-Term Rental Strategy

When platform 1 chooses the short-term rental strategy and platform 2 chooses the long-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=vS1−p1eSL+αx1eSL/λ, and the marginal consumers who choose the service of platform 2 are at t2e=βλN−αx2eSLN+p2eSL−vL2/βλN. The consumers between 0,t1e choose the short-term rental service of platform 1, and the consumers between t2e,1 choose the long-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eSL=r1p1eSLt1eN−cx1eSL2 and Π2eSL=r2p2eSL1−t2e−cx2eSL2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eSL∗=2cλvS1/4cλ−α2r1N and x1eSL∗=αr1vS1N/4cλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eSL∗=2cvS1/4cλ−α2r1N and Π1eSL∗=cr1vS12N/4cλ−α2r1N. Similarly, the price and service level of platform 2 are p2eSL∗=2βcλvL2/4βcλ−α2r2N and x2eSL∗=αr2vL2/4βcλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eSL∗=2cvL2/4βcλ−α2r2NN and Π2eSL∗=cr2vL22/4βcλ−α2r2NN, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvS14βcλ−α2r2NN+vL24cλ−α2r1N<4cλ−α2r1N4βcλ−α2r2NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


## 6.3. Case LS: Both Platforms Are Local Monopolies, Platform 1 Chooses the Long-Term Rental Strategy, and Platform 2 Chooses the Short-Term Rental Strategy

When platform 1 chooses the long-term rental strategy and platform 2 chooses short-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=αx1eLSN−p1eLS+vL1/βλN, and the marginal consumers who choose the service of platform 2 are at t2e=λ+p2eLS−αx2eLS−vL2/λ. The consumers between 0,t1e choose the long-term rental service of platform 1, and the consumers between t2e,1 choose the short-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eLS=r1p1eLSt1e−cx1eLS2 and Π2eLS=r2p2eLS1−t2eN−cx2eLS2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eLS∗=2βcλvS1/4βcλ−α2r1N and x1eLS∗=αr1vL1/4βcλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eLS∗=2cvL1/4βcλ−α2r1NN and Π1eLS∗=cr1vL12/4βcλ−α2r1NN. Similarly, the price and service level of platform 2 are p2eLS∗=2cλvS2/4cλ−α2r2N and x2eLS∗=αr2vS2N/4cλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eLS∗=2cvS2/4cλ−α2r2N and Π2eLS∗=cr2vS22N/4cλ−α2r2N, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvL14cλ−α2r2N+vS24βcλ−α2r1NN<4cλ−α2r2N4βcλ−α2r1NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


## 6.4. Case LL: Both Platforms Are Local Monopolies and Both Platforms Choose Long-Term Rental Strategy

When platform 1 and platform 2 choose the long-term rental strategy, the marginal consumers who choose the service of platform 1 are att1e=αx1eLLN−p1eLL+vL1/βλN, and the marginal consumers who choose the service of platform 2 are at t2e=βλN−αx2eLLN+p2eLL−vL2/βλN. The consumers between 0,t1e choose the long-term rental service of platform 1, and the consumers between t2e,1 choose the long-term rental service of platform 2, where t1e<t2e. The profit functions of platform 1 and platform 2 are as follows: Π1eLL=r1p1eLLt1e−cx1eLL2 and Π2eLL=r2p2eLL1−t2e−cx2eLL2. According to the first-order conditions of platform 1’s profit function on price and service level, we can find that the price and service level of platform 1 are p1eLL∗=2βcλvL1/4βcλ−α2r1N and x1eLL∗=αr1vL1/4βcλ−α2r1N. Therefore, the demand and profit of platform 1 are D1eLL∗=2cvL1/4βcλ−α2r1NN and Π1eLL∗=cr1vL12/4βcλ−α2r1NN. Similarly, the price and service level of platform 2 are p2eLL∗=2βcλvL2/4βcλ−α2r2N and x2eLL∗=αr2vL2/4βcλ−α2r2N, respectively. Furthermore, the demand and profit of platform 2 are D2eLL∗=2cvL2/4βcλ−α2r2NN and Π2eLL∗=cr2vL22/4βcλ−α2r2NN, respectively.We need to ensure that0<t1e<t2e<1, so that both platforms can act as local monopolies. To ensure t1e<t2e, 2cvL14βcλ−α2r2N+vL24βcλ−α2r1N<4βcλ−α2r1N4βcλ−α2r2NN must be met. If this condition is not met, the two platforms will directly complete with marginal consumers.


## 7. Conclusion

In this paper, we examine a duopoly housing sharing platform market, in which two platforms provide rental services to consumers. Each platform can choose the short-term rental strategy or the long-term rental strategy. When two platforms make decisions, there are four cases (i.e., Case SS, Case SL, Case LS, and Case LL). The two platforms determine the optimal price and service level to maximize their own profits, based on the competitors’ observed decisions. In this paper, the equilibrium price and profit are given by using game theory, and the market equilibrium is analyzed to determine the applicable conditions of each strategy.The results show that whether the platform adopts the short-term rental strategy or the long-term rental strategy largely depends on the service quality of competitors. Specifically, when the difference of service value is small, it is better for the two platforms to adopt the short-term rental strategy, and when the difference of services value is large, the two platforms intend to adopt the long-term rental strategy. We also find that there may be two equilibria under certain conditions, and we also give the conditions of dominant equilibrium. In addition, we also find that the commission rate of the two platforms will affect the profits of the platforms, and when the service quality is fixed, the platform can benefit from the decreasing service cost. Finally, we analyze the uncovered market and get the equilibrium profits of the two platforms.Our results can provide some important management implications for the strategic choice of home-sharing platforms in the competitive home-sharing market. First, when a platform plans to choose the short-term rental strategy or the long-term rental strategy, it needs to compare the service quality with its competitors. Each platform should build its own user base to improve profits. Second, when the platform and the competitor have the same or almost the same level of service quality, the platform will choose the strategy according to the intrinsic value of the competitor’s service. Finally, the commission rate or service cost will also affect the profits of the platforms. Therefore, these management implications provide a reference for home-sharing platforms in different cases.Future work can also be carried out in two aspects. First, we focus on all listed homes on the commercial operator control platform in this paper. In some home-sharing platforms, commercial operators and individual hosts also coexist. Second, the home-sharing platform holds a large number of data, and the influence of data-driven on the operational decision of the home-sharing platform can be considered in the future.



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*Source: 3104478-2021-12-24.xml*