## 1. INTRODUCTION

The sharing economy refers to a new means of distributing goods and services in a way that differs from the traditional model of corporations hiring employees and selling products to consumers. In the sharing economy, individuals are said to rent or “share” things such as their cars, homes and personal time with other individuals in a peer-to-peer (P2P) fashion (Hamari *et al*., 2015). Information and communications technologies have enabled the rise of what has been termed “Collaborative Consumption,” i.e., the P2P-based activities of obtaining, giving, and/or sharing access to goods and services, coordinated through community‐based online services. In this context, the sharing economy has been widely recognized as a main growth sector of the current economy (Ma *et al*., 2019;Roma *et al*., 2019;Wang *et al*., 2019). It has disrupted many industries by providing consumers with costefficient and convenient access to products or services without the burden of ownership (Gong *et al*., 2020;Yuan and Shen, 2019). Thus, the issues related to this emerging industry have attracted extensive attention from the various industry fields. Especially, due to the fast development of the sharing economy industry and the importance of platforms for it, the platform operations in the sharing economy have attracted growing attention: fashion products- sharing (Choi and He, 2019), car-sharing (Bellos *et al*., 2017), and bike-sharing (Albiński *et al*., 2018).

Despite the advantages of the sharing economy, such as convenience and affordability, consumers’ perceived risk formed by possible physical injury from strangers or unexpected poor service quality disturbs their active participation in the sharing economy. Recently, Hong *et al*. (2019) conducted excellent research on optimal risk management policies for the sharing economy platform. More specifically, they studied the implications of the two types of risks that are perceived by consumers in the sharing economy: physical risk, stemming from safety concerns, and performance risk, related to low service quality. They considered one general monopolistic sharing platform provider that devises risk management to maximize its profit. Given the price of a traditional firm’s service, the optimal price of a sharing platform’s service was determined. However, it is more realistic to assume price competition in the sharing economy market between the sharing platform provider and the traditional firm. According to Wallsten (2015), traditional taxi companies have long faced imperfect competition - from public transportation systems such as buses and subways to car services that pick up passengers who request a ride (generally via telephone) as well as so-called “gypsy cabs.” However, Uber and other ride-sharing services appear to compete more directly with taxis if for no other reason than their increasing ubiquity as a convenient, on-demand means of transportation. As competition increases, consumers have new options and incumbents may be forced to respond. The sharing economy is unambiguously increasing competition. Zervas *et al*. (2017) studied the effects of Airbnb on the hotel industry in Texas, finding that when more rentals exist on Airbnb, lower hotel revenues and prices follow. Hence, the purpose of this note is to study the price competition between the service offered by a traditional firm and that offered by a sharing platform firm.

In this note, we assume three types of pricing game: i) parallel pricing, ii) sequential pricing, and iii) unified pricing. When pricing, we can consider three possible situations in the duopoly market. In the first situation, two firms (or players) in the market have the same market power in terms of pricing. In the second situation, one firm has stronger pricing power than the other. Finally, in the third situation, the two firms agree to cooperate with each other in pricing. The first, second, and third situations correspond to the parallel, sequential, and unified pricing games, respectively. For each pricing game, we prove that there exists a unique equilibrium and optimal price pair that maximizes each player’s profit. To the best our knowledge, there has been no work studying the price competition under the sharing economy so far. Our contribution is to prove the uniqueness of equilibrium and optimal solutions of the three pricing games introduced above and is to show that, in the unified pricing game, a traditional firm and a sharing platform firm can achieve the win-win scenario by cooperating with each other.

The remainder of this note is organized as follows. In section 2, we describe the basic mathematical model and assumptions used in the analysis. In section 3, we introduce three pricing games between the sharing platform and the traditional firm and then propose a solution procedure for each pricing game. Moreover, at the end of each subsection in section 3, we conduct a numerical analysis to show the price competition between the sharing platform and the traditional firm.

## 2. MODEL DESCRIPTION AND ASSUMPTIONS

First, we consider the level of demand of the sharing platform. The demand of a sharing platform is determined by the self-selection of consumers. The concept of consumer’s self-selection originated with Mussa and Rosen (1978), after which there were a great deal of applications to analytical model buildings, including Moorthy (1984) and Moorthy and Png (1992). According to this concept, it is assumed that each consumer voluntarily participates in the sharing economy. The basic model of a consumer self-selection utility can be defined as follows:

The right-hand side of Eq. (1) represents the consumer utility when the consumer uses a service offered by a traditional firm, such as a major hotel chain or a city taxi company. The left-hand side of Eq. (1) indicates the consumer utility when the consumer chooses a service offered by a sharing platform, such as Airbnb or Uber. Thus, when the utility from the sharing platform is greater than that from the traditional company, the consumer is willing to choose the service offered by the sharing platform. *θ _{s}* and

*θ*denote the customer heterogeneities indicating consumer’s willingness to pay for the service offered by the sharing platform and the traditional firm, respectively. Both

_{t}*θ*and

_{s}*θ*are assumed to be mutually independent and to follow a uniform distribution in the interval [0, 1]. Note that we consider a situation where both the sharing platform and the traditional firm provide a service at the same quality level.

_{t}*p*and

_{s}*p*denote the prices offered by the sharing platform and the traditional firm, respectively. The notations

_{t}*R*and

*r*(0 <

*r*,

*R*<1) correspondingly represent the physical risk and the performance risk when choosing the sharing platform service. The physical risk represents fatal issues related to the life of the customer and injuries suffered by them, whereas the performance risk mainly comes from an unstable or lower service quality. Thus, as the performance risk and physical risk increase, the consumer utility in the sharing platform decreases. These two risks may also exist in the traditional firm side, but the

*R*and

*r*can be thought of as relatively greater risks in the sharing platform than in the traditional firm. So, we neglect the those risks in the traditional firm.

Hong *et al*. (2019) suggested the modified model of Eq. (1), in which the utility in the sharing economy captures both word-of-mouth and the investment in safety. It was assumed that the sharing platform service benefits from the word-of-mouth effect, which is exogenously determined among consumers. Another assumption is that the sharing platform makes an additional effort to enhance the physical protection of the consumers, such as by developing a more precise identification system. The corresponding modification of Eq. (1) is expressed as shown below.

In Eq. (2), *w* is the exogenous word-of-mouth effect on the demand of the sharing platform service *D _{s}*, and

*s*is the amount of investments in safety improvement. Intuitively, as

*w*increases, the boosted demand

*wD*alleviates the performance risk

_{s}*r*. Regarding the physical risk

*R*, the

*s*investment in safety is assumed to reduce the physical risk. Summarizing this fact,

*w*and

*s*have a positive effect on increasing the utility of consumers who choose the sharing platform service.

Our study mainly focuses on the price competition between the sharing platform and the traditional firm. Therefore, we must consider the price elasticity of the demand and the cross-price sensitivity in Eq. (2). The utility of a consumer who chooses the sharing platform’s service has the following relationship:

where *β _{s}* is the price sensitivity of a consumer when considering a sharing platform service. In fact,

*β*models the effect of consumers, who can be attracted by a lower price. The term

_{s}*γ*is the cross-price sensitivity, which reflects the degree of cannibalization between the sharing platform and the traditional firm. In other words,

*γ*represents the leakage of the demand from one side to the other. Therefore, the term

*γ*is competition parameter between the sharing platform and the traditional firm. Throughout the paper, it is assumed that

*β*>

_{s}*γ*. The demand of the sharing platform

*D*is thus given by

_{s}

where *f _{X}* is the probability density function of a random variable

*X*. Eq. (4) is simplified as

Similarly, from Eq. (3), the utility of a consumer who chooses a traditional firm’s service has the following relationship:

where *β _{t}* is the price sensitivity of a consumer when considering a traditional firm’s service. It is also assumed that

*β*>

_{t}*γ*. Integrating Eq. (5) into Eq. (6), the demand of the traditional firm

*D*is given by

_{t}

Let Π_{s} and Π_{t} denote the profit functions of the sharing platform and the traditional firm, respectively. These are correspondingly expressed as follows:

As in Hong *et al*. (2019), we assume that *cs*^{2} is the cost function for the safety investment in the sharing platform. And, we also assume that *w _{Ds}* < 1 and

*s*<1 , implying that it is impossible completely to eliminate the two risks associated with the service offered by the sharing platform. The objective function in Eq. (8) is used to solve the pricing games in next section. Finally, it is useless to study pricing decisions with no positive profit in practice; therefore, the following assumption is made: there is at least one feasible point (

*p*,

_{s}*p*) that satisfies both Π

_{t}_{s}> 0 and Π

_{t}> 0 .

## 3. PRICING COMPETITION MODELS

This section shows the results of the three pricing game models. The results contain the explicit form of the equilibrium prices of the sharing platform and the traditional firm. At the end of each subsection, a numerical example is presented.

### 3.1 Parallel Pricing Game (Nash Game)

In a parallel pricing game, two competing players, the sharing platform and the traditional firm, should decide on their service price simultaneously. To obtain equilibrium prices, one can adapt the concept of game theory. Their strategy spaces are continuous and their payoff functions are expressed by Eq. (8). The outcome of this game is analyzed using the concept of the Nash equilibrium. In the Nash equilibrium, none of the players can benefit by changing only his/her strategy. Thus, each player’s strategy is optimal against that of the other. Under this concept, we propose the following:

**Proposition 1.** In the parallel pricing mechanism, there exists a unique Nash equilibrium under the sharing platform’s price, ${p}_{s}^{NE}$, and the traditional firm’s price, ${p}_{t}^{NE}$:

where *K* = 1 − 2*r* − 2*R*(1− *s*) .

**Proof.** Using Eq. (8), we consider the following Hessian matrix:

Notice that each player’s profit function is strictly concave with respect to the player’s own decision variable because $\frac{{\partial}^{2}{\Pi}_{s}}{\partial {p}_{s}^{2}}=-\frac{2{\beta}_{s}}{1-rw}<0$ and $\frac{{\partial}^{2}{\Pi}_{t}}{\partial {p}_{t}^{2}}=\frac{2\left({\beta}_{t}\left(1-rw\right)+rw\gamma \right)}{1-rw}<0$. The determinant for the Hessian matrix * H_{N}* is given by

Since we assume that *β _{s}*>

*γ*and

*β*>

_{t}*γ*, |

*| is strictly positive; i.e.,*

**H**_{N}

Hence, there exists a unique Nash equilibrium for this parallel pricing game. Consequently, given the competitor’s price, each player can find its own pricing strategy by solving the following first-order conditions:

By solving the equations system of Eq. (10) simultaneously, one can find the Nash equilibrium prices of the sharing platform and the traditional firm in Eq. (9). ☐

From Proposition 1, we outline the following corollary.

**Corollary 1.***Given*$\frac{2(1-rw)(2{\beta}_{s}-\gamma )}{(1-rw)({\beta}_{s}+2{\beta}_{t}-3\gamma )+{\beta}_{s}+\gamma}>K$, then ${p}_{s}^{NE}<{p}_{t}^{NE}$, implying that the sharing platform provider more affordably serves consumers than the traditional firm.

**Proof.** It follows from Eq. (9) that

where $L=(1-rw)({\beta}_{s}+2{\beta}_{t}-3\gamma )+{\beta}_{s}+\gamma >0$ because ${\beta}_{s}>\gamma \hspace{0.17em}\text{and}\hspace{0.17em}{\beta}_{t}>\gamma $. Additionally, the denominator of Eq. (11) is strictly positive. Therefore, if the numerator of Eq. (11) is strictly negative, then ${p}_{s}^{NE}-{p}_{t}^{NE}<0$. This completes the proof. ☐

The parameter *γ* indicates the competition intensity between the sharing platform and the traditional firm. We are interested in an investigation of the effects of competition intensity on the equilibrium (and optimal) price, the demand, and the profit of each player. To do this, we present the result of a numerical example at the end of each subsection. The main experimental parameters used in the analysis are as follows: *r* = 0.2 , *R* = 0.1 , *w* = 0.5 , *s* = 0.3, *c* = 0.05, *β _{s}* = 0.8, and

*β*= 0.7. These parameters are used in the next subsections as well. Changing the cross-price sensitivity (competition parameter)

_{t}*γ*to from 0 to 0.6 in steps of 0.1, we record the equilibrium price, the demand, and the profit of each player in the case of the parallel pricing game in Table 1. It can be observed that the greater the competition intensity is, the higher the prices become. Severe price competition between two players helps to raise the prices. Ironically, the increased price of the competitor tends to increase the demand of each player. Due to the increased price and the increased demand, the profit of each player also increases.

### 3.2 Sequential Pricing Game (Traditional Firm- Leading Stackelberg Game)

This subsection covers a sequential pricing game in which two competing players, the sharing platform and the traditional firm, have different levels of power in pricing. This type of pricing game is usually modeled as a duopoly Stackelberg game, where one player is a pricing leader and the other is a pricing follower. For convenience, we regard the traditional firm and the sharing platform as the pricing leader and the follower, respectively. In practice, it is more realistic and reasonable to assume that the traditional firm has more power with regard to pricing, as the traditional firm has been in business longer than the sharing platforms, and the traditional firm has a relatively high market share because it does business both online and offline. Therefore, the traditional firm is assumed to be the pricing leader in the sequential pricing game. In the game, the sharing platform maximizes its profit based on the price suggested by the traditional firm, while the traditional firm maximizes its profit based on the sharing platform’s best response function. Under this concept, we propose the following:

**Proposition 2.** In the sequential pricing mechanism, there exists a unique Stackelberg equilibrium under the sharing platform’s price, ${p}_{s}^{SE}$, and the traditional firm’s price, ${p}_{t}^{SE}$:

**Proof.** Because $\frac{{\partial}^{2}{\Pi}_{s}}{\partial {p}_{s}^{2}}=-\frac{2{\beta}_{s}}{1-rw}<0$, then the sharing platform provider’s profit function is strictly concave with respect to *p _{s}*. Thus, given the traditional firm’s price

*p*, the sharing platform provider’s optimal price ${p}_{s}^{SE}({p}_{t})$ can be found by solving the first-order condition $\frac{\partial {\Pi}_{s}}{\partial {p}_{s}}=0$. $\frac{\partial {\Pi}_{s}}{\partial {p}_{s}}=0$. First, we substitute ${p}_{s}^{SE}({p}_{t})$ into the traditional firms’s profit function Π

_{t}_{t}. Since $\frac{{\partial}^{2}{\Pi}_{t}}{\partial {p}_{t}^{2}}=-\frac{(1-rw)(2{\beta}_{s}{\beta}_{t}-{\gamma}^{2})+rw{\beta}_{s}\gamma}{{\beta}_{s}(1-rw)}<0$, given the sharing platform provider’s best response function, the traditional firm’s profit function is strictly concave with respect to

*p*. This guarantees the existence of a unique equilibrium for the sequential pricing game described above. To find the traditional firm’s optimal price of ${p}_{t}^{SE}$, the first-order condition $\frac{\partial {\Pi}_{t}}{\partial {p}_{t}}=0$ is solved. One may substitute ${p}_{t}^{SE}$ into Eq. (10) to obtain ${p}_{s}^{SE}$. ☐

_{t}From Proposition 2, we have the following corollary.

**Corollary 2.***Given*$\frac{4{\beta}_{s}\left(1-rw\right)\left(2{\beta}_{s}-\gamma \right)}{\left(1-rw\right)\left(4{\beta}_{s}{\beta}_{t}-{\gamma}^{2}\right)+2{\beta}_{s}\left(2-rw\right)\left({\beta}_{s}-\gamma \right)+3rw{\beta}_{s}\gamma}$ > *K*, then ${p}_{s}^{SE}<{p}_{t}^{SE}$, implying that the sharing platform provider more affordably serves consumers than the traditional firm.

**Proof.** The proof of Corollary 2 is quite similar to that of Corollary 1. So, we omit it here. ☐

In Table 2, as *γ* increases, we can observe the same trends with regard to the equilibrium price, the demand, and the profit as presented in Table 1. Therefore, we omit these details in Table 2. By the way, it should be noted that the profits in case of the sequential pricing game are greater than or equal to those in case of the parallel pricing game; i.e.,

This indicates that the sequential pricing may be more beneficial than the parallel pricing in terms of maximizing not only the profit of each player but also the profit of the total market.

### 3.3 Unified Pricing Game

In the unified pricing game, it is assumed that both substitutable services are sold by a monopolist and that she jointly determines *p _{s}* and

*p*to maximize her profit. This case is similar to a case in which the sharing platform and the traditional firm cooperate in terms of pricing each other, set their prices to maximize the total market profit, and then share the added benefit by some agreedupon mechanism. Note that the sharing platform and the traditional firm cooperate in pricing but still compete in selling. The unified pricing game is modeled as

_{t}

where the objective function Π_{m} indicates the total market profit. From definitions in Eqs. (4) and (7), each demand should be bounded in (0,1] . We introduce the final proposition.

**Proposition 3.** Assuming that $4(1-rw)({\beta}_{s}{\beta}_{t}-{\gamma}^{2})>{\left(rw({\beta}_{s}-\gamma )\right)}^{2}$, the optimal values of *p _{s}* and

*p*in the unified pricing game are then obtained as follows:

_{t}

**Proof.** The Hessian matrix of the objective function in Eq. (13) is

We define Δ_{k} as the leading principal minor of order *k* in * H_{U}*. From Eq. (15), we find that ${\Delta}_{1}=-\frac{2{\beta}_{s}}{1-rw}<0.$ According to the assumption in Proposition 3, ${\Delta}_{2}=\frac{4\left(1-rw\right)\left({\beta}_{s}{\beta}_{t}-{\gamma}^{2}\right)-{\left(rw\left({\beta}_{s}-\gamma \right)\right)}^{2}}{{\left(1-rw\right)}^{2}}$. Therefore,

*is negatively definite, implying that Π*

**H**_{U}_{m}is strictly concave in the feasible region and that the stationary point of Π

_{m}becomes the global maximizer of the unified pricing game. The first derivatives of Π

_{m}with regard to

*p*and

_{s}*p*are, respectively,

_{t}

By solving the equations system in Eq. (16) simultaneously, one can find the optimal prices of the sharing platform and the traditional firms in Eq. (14). ☐

From Proposition 3, we have the following corollary.

**Corollary 3.***Given*$\frac{(2-rw)({\beta}_{s}-\gamma )}{{\beta}_{s}+{\beta}_{t}-2\gamma}<K$, then ${p}_{s}^{UP}<{p}_{t}^{UP}$, implying that the sharing platform provider more affordably serves the consumers than the traditional firm.

**Proof.** The proof of Corollary 1 is quite similar to that of Corollary 1. Accordingly, we omit it here. ☐

Table 3 shows the equilibrium results when *γ* increases. Because the trends of the values in Table 3 are also identical to those in Table 1, a detailed description will be omitted. Instead, comparing Table 3 with Table 2, we find that the profits in the unified pricing game are greater than those in the sequential pricing game; i.e.,

This implies that the unified pricing may be more beneficial than the sequential pricing. Hence, summarizing all numerical examples in Table 1, 2, and 3, we can conclude that the unified pricing strategy is the most advantageous in terms of maximizing not only the total market profit but also the profits of both the sharing platform and the traditional firm.

## 4. CONCLUDING REMARKS

In this note, we dealt with the three pricing games in the duopoly market in which one traditional firm and one sharing platform firm compete in pricing. Our numerical result showed that the two firms can achieve the win-win scenario by cooperating each other and setting their prices to maximize the total market profit. There are also new areas of improvements of this work which can be analyzed in future research. We can compare the three pricing games in the three types of the oligopoly market: (1) one traditional firm and multiple sharing platforms, (2) multiple traditional firms and one sharing platform, and (3) multiple traditional firms and multiple sharing platforms. The difficulty of the analysis of these three oligopoly markets is not expected to be very high, however, it will not be easy to obtain closed-form results for them.