# Strategic Third-Party Product Entry and Mode Choice under Self-Operating Channels and Marketplace Competition: A Game-Theoretical Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Provider-Side Network Effects in the Context of a Two-Sided Market

#### 2.2. Platform Operations under Different Business Models

#### 2.3. Product Entry in Platform-Based Markets

## 3. The Model

_{1}; we refer to this self-operating mode as the reselling self-operating mode). Under the revenue-sharing mode, the incumbent supplier directly sells their products to their end consumers by sharing a proportion (λ) of its revenue to the platform, where λ∈(0,1) represents the revenue-sharing rate (we refer to this self-operating mode as the revenue-sharing self-operating mode). On the other hand, the platform also serves as a marketplace, and a third-party supplier chooses whether to join and engage in direct selling by paying a commission at a commission rate (ρ∈(0, 1)) as a fee to access consumers. The product retail price set by the entrant supplier was denoted as P

_{2}. We regarded the retail price, P

_{1}or P

_{2}, as a total price of the bundle of the product and associated logistics service. The products offered by the two suppliers were completely substituted, and the marginal production costs were constant across the two suppliers, which were assumed to be zero without loss of generality. Both suppliers have no consumer base and can only reach their end consumers through the platform.

_{1}and s

_{2}denote the platform’s logistics service level and the third-party supplier’s logistics service level, respectively. On the consumer side, each consumer was assumed to have one unit of product demand and aimed to maximize their own utility. Based on the Hotelling model, we assumed that consumers’ product preferences are distributed uniformly on [0, 1], with the incumbent supplier’s product located at 0 and the entrant supplier’s product located at 1. If a consumer purchases a product that is distinct from their ideal product, they incur a misfit cost, which increases with the distance between their location and the given product. We let v denote the quality of the product sold by either supplier and t represent the disutility cost per unit distance; then, a consumer located at x derives the utility of v + s

_{1}− tx when buying the product from the incumbent supplier and the utility of v + s

_{2}− t (1 − x) when buying the product from the entrant supplier. We assumed that the consumer utility increases along with the logistics service level as a high-quality logistics service can enhance the customer experience and thus derive a higher consumer utility [39,40,41].

_{i}+ N), where N is the supplier number and α denotes the strength of the indirect network effects, which measures how much consumers care about supplier-side variety. The approach to model the network effects as a function of product quality and the network size has been widely used in the existing literature (e.g., Baake and Boom, 2001 [43], Jing, 2007 [44], and Zhang et al., 2016 [45]). The third-party supplier has the option to join the marketplace to sell its products and needs to determine whether to do so. If the third-party supplier decides to enter, it bears an opportunity cost (c). We interpreted the opportunity cost as some of the other outputs the entrant supplier must sacrifice in operating an online retail. This is rational for those suppliers with a limited budget. In our analysis, we assumed that c is not so high (i.e., c < min{$\frac{{\left(3\mathrm{t}-\left(1+\mathsf{\alpha}\right)\left({\mathrm{s}}_{1}-{\mathrm{s}}_{2}\right)\right)}^{2}}{18\mathrm{t}}$, $\frac{{\left(9\mathrm{t}-\left(1+\mathsf{\alpha}\right)\left({\mathrm{s}}_{1}-{\mathrm{s}}_{2}\right)\right)}^{2}}{72\mathrm{t}}$}); otherwise, the third-party supplier would have no incentive to join the marketplace even if the charged commission was 0.

^{RC}), and then the platform and entrant supplier simultaneously determine the retail prices P

_{1}

^{RC}and P

_{2}

^{RC}, respectively. In scenario RM, the incumbent supplier first determines the wholesale price (w

^{RM}), and then the platform determines the retail price (P

_{1}

^{RM}). In scenario SC, the platform and the entrant supplier simultaneously determine the retail prices P

_{1}

^{SC}and P

_{2}

^{SC}, respectively. In scenario SM, the incumbent supplier sets the retail price (P

_{1}

^{SM}). In Stage 4, customers observe the product prices and decide whether to buy a product and from which supplier. The time sequence of this game is illustrated in Figure 1.

_{1}

^{RC}and P

_{2}

^{RC}indicate the retail prices in scenario RC. Table 1 summarizes the key notations that were used in this study.

## 4. Pricing Analysis

#### 4.1. Pricing Strategies without an Entrant Supplier

_{1}+α(v+s

_{1}+1) − P

_{1}− tx. In Stage 4, consumers will buy the product as long as they derive non-negative utilities (i.e., U ≥ 0). By setting the surplus function as U = 0, we can obtain the location point of the marginal customer (${x}_{0}=\frac{v+{s}_{1}+\alpha \left(v+{s}_{1}+1\right)-{P}_{1}}{t}$). Among all the consumers in the market, those located in (0, x

_{0}) buy the product, and the others do not. As a result, we obtain the demand (${D}_{1}={x}_{0}=\frac{v+{s}_{1}+\alpha \left(v+{s}_{1}+1\right)-{P}_{1}}{t}$). Figure 2 illustrates the market segmentation in this case.

^{RM}). We then derived the incumbent supplier’s optimal wholesale price based on the platform’s optimal response. Finally, we were able to obtain the optimal profits by inserting the optimal prices into the profit functions in Equations (1) and (2). The results are summarized in the following lemma (All proofs are provided in Appendix A, unless indicated otherwise).

**Lemma**

**1.**

_{1}) would lead to an increase in the optimal wholesale price, retail price, demand, and the platform’s and incumbent supplier’s profits. By contrast, the disutility cost (t) exerts negative impacts on the demand and the platform’s and incumbent supplier’s profits.

**Lemma**

**2.**

#### 4.2. Pricing Strategies with an Entrant Supplier

_{1}= v + s

_{1}+ α(v + s

_{1}+ 2) − P

_{1}− tx and that of buying the entrant supplier’s product is U

_{2}= v + s

_{2}+ α(v + s

_{2}+ 2) − P

_{2}− t(1 − x). Out of these two options, consumers choose the one that gives them the maximum surplus under Stage 4. By setting the surplus function as U

_{1}= U

_{2}, we can obtain the location point of the marginal customer (${x}_{1}=\frac{t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)+{P}_{2}-{P}_{1}}{2t}$). All consumers in the market are divided into two segments. Consumers located in (0, x

_{1}) buy from the incumbent supplier, as they derive the higher surplus from purchasing the incumbent supplier’ product than from purchasing the entrant supplier’ product. In contrast, consumers located in (x

_{1},1) choose to buy from the entrant supplier. As a result, we found that ${D}_{1}={x}_{1}=\frac{t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)+{P}_{2}-{P}_{1}}{2t}$ and ${D}_{2}={1-x}_{1}=\frac{t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)-{P}_{2}+{P}_{1}}{2t}$. Figure 3 illustrates the market segmentation in this case.

**Lemma**

**3.**

_{1}and s

_{2}), the strength of the indirect network effects (α), the commission rate (ρ), and the revenue-sharing rate (λ). By examining the equilibrium outcome, we obtained the following corollary:

**Corollary**

**1.**

- (1)
- $\frac{\partial {P}_{i}^{SC}}{\partial {s}_{i}}>0$,$\frac{\partial {P}_{i}^{SC}}{\partial {s}_{j}}0$,$\frac{\partial {D}_{i}^{SC}}{\partial {s}_{i}}0$,$\frac{\partial {D}_{i}^{SC}}{\partial {s}_{j}}0$, $\frac{\partial {\pi}_{Si}^{RC}}{\partial {s}_{i}}>0$,$\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial {\pi}_{Si}^{RC}}{\partial {s}_{j}}0$;
- (2)
- If s
_{i}> s_{j}, $\frac{\partial {P}_{i}^{SC}}{\partial \alpha}>0$, $\frac{\partial {P}_{j}^{SC}}{\partial \alpha}<0$,$\frac{\partial {D}_{i}^{SC}}{\partial \alpha}0$,$\frac{\partial {D}_{j}^{SC}}{\partial \alpha}0$, $\frac{\partial {\pi}_{Si}^{RC}}{\partial \alpha}>0$, and $\frac{\partial {\pi}_{Sj}^{RC}}{\partial \alpha}<0$; - (3)
- If $\alpha >\frac{3t\left(\lambda -\rho \right)+\left(\lambda +\rho \right)\left({s}_{2}-{s}_{1}\right)}{\left(\lambda +\rho \right)\left({s}_{1}-{s}_{2}\right)}$, $\frac{\partial {\pi}_{P}^{SC}}{\partial \alpha}>0$; otherwise, $\frac{\partial {\pi}_{P}^{SC}}{\partial \alpha}\le 0$;
- (4)
- $\frac{\partial {\pi}_{S1}^{RC}}{\partial \lambda}<0$,$\frac{\partial {\pi}_{S2}^{RC}}{\partial \rho}0$,$\frac{\partial {\pi}_{P}^{SC}}{\partial \lambda}0$,$\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial {\pi}_{P}^{SC}}{\partial \rho}0$.

_{i}) increases or as its rival’s logistics service level (s

_{j}) decreases under scenario SC. This is intuitive, as a high-quality logistics service can enhance consumers’ utilities directly and thus strengthens the supplier’s competitiveness. Corollary 1(2) and corollary 1(3) examined the impact of indirect network effects on the equilibrium outcomes. The result revealed that the supplier with a logistics service quality advantage can benefit from an increasing indirect network effect intensity. For the platform, only if the network effect intensity is sufficiently large can an increasing indirect network effect enhance its profits. Corollary 1(4) identifies the varying profits via the commission rate or the revenue-sharing rate. We can find that the incumbent (entrant) supplier suffers from a higher revenue-sharing (commission) rate, whereas the platform always benefits from charging a higher revenue-sharing (commission) rate. Under scenario SC, both suppliers directly determine their retail prices, and the platform does not involve the price competition. As a result, the platform’s profits would increase as the revenue-sharing (commission) rate increases.

^{RC}). We then derived the incumbent supplier’s wholesale price based on the optimal response. Finally, we derived the optimal profits by inserting the optimal prices into the profit functions in Equations (5)–(7). The following lemma solves these above profit maximization problems and characterizes the equilibrium outcome under this scenario:

**Lemma**

**4.**

_{1}and s

_{2}) and the indirect network effect intensity (α) have similar impacts on the equilibrium outcomes under scenario RC with what they carry out under scenario SC. However, one can observe that the commission rate (ρ) has more complicated impacts on the equilibrium outcomes under scenario RC. Thus, we focused on examining the impact of the commission rate under this scenario and provided the following corollary:

**Corollary**

**2**.

- (1)
- $\frac{\partial {w}^{RC}}{\partial \rho}<0$,$\frac{\partial {P}_{1}^{RC}}{\partial \rho}0$,$\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial {P}_{2}^{RC}}{\partial \rho}0$;
- (2)
- $\frac{\partial {D}_{1}^{RC}}{\partial \rho}<0$,$\frac{\partial {D}_{2}^{RC}}{\partial \rho}0$;
- (3)
- $\frac{\partial {\pi}_{S1}^{RC}}{\partial \rho}<0$,$\frac{\partial {\pi}_{S2}^{RC}}{\partial \rho}0$,$\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial {\pi}_{P}^{RC}}{\partial \rho}0$.

## 5. Equilibrium Analysis

#### 5.1. Impact of a Third-Party Supplier’s Entry

**Proposition**

**1.**

- (a)
- Under the reselling self-operating mode, the incumbent supplier charges a higher wholesale price (i.e., ${w}^{RC}>{w}^{RM}$) if v < v
_{1}and gains a higher profit (i.e., ${\pi}_{S1}^{RC}>{\pi}_{S1}^{RM}$) if v < v_{3}, and the platform charges a higher retail price (i.e., ${P}_{1}^{RC}>{P}_{1}^{RM}$) if v < v_{2}and gains a higher profit (i.e., ${\pi}_{P}^{RC}>{\pi}_{P}^{RM}$) if v < v_{4}; - (b)
- Under the revenue-sharing self-operating mode, the incumbent supplier charges a higher retail price (i.e., ${P}_{1}^{SC}>{P}_{1}^{SM}$) if v < v
_{5}and gains a higher profit (i.e., ${\pi}_{S1}^{SC}>{\pi}_{S1}^{SM}$) if v < v_{6}, and the platform gains a higher profit (i.e., ${\pi}_{P}^{SC}>{\pi}_{P}^{SM}$) if v < v_{7}, where v_{i}(i=1,…,7) is defined in Appendix A.

#### 5.2. Equilibrium Partnership

**Lemma**

**5.**

- (a)
- If v < v
_{4}, the platform would open its marketplace under both self-operating modes; - (b)
- If v
_{4}< v < v_{7}, the platform would open its marketplace under the revenue-sharing self-operating mode; - (c)
- If v > v
_{7}, the platform would not open its marketplace, where v_{4}and v_{7}are given in Proposition 1.

**Lemma**

**6.**

- (a)
- If ρ < ρ
_{1}, the third-party supplier would join the marketplace under both self-operating modes; - (b)
- If ρ
_{1}< ρ < ρ_{2}, the third-party supplier would join the marketplace under both self-operating modes; - (c)
- If ρ > ρ
_{2}, the third-party supplier would not join the marketplace, where ρ_{1}and ρ_{2}are defined in Appendix A.

**Proposition**

**2.**

_{4}and ρ < ρ

_{1}}, R2 = {(v, ρ)| v < v

_{4}and ρ

_{1}< ρ < ρ

_{2}}, and R3 = {(v, ρ)| v

_{4}< v < v

_{7}and ρ < ρ

_{1}}, such that: (i) the platform opens its marketplace and the third-party supplier joins the marketplace under both of the self-operating modes if (v, ρ)∈ R1; (ii) the platform opens its marketplace and the third-party supplier only joins the marketplace under the reselling self-operating mode if (v, ρ)∈ R2; and (iii) the platform opens its marketplace and the third-party supplier only joins the marketplace under the revenue-sharing self-operating mode if (v, ρ)∈ R3; otherwise, the partnership between the platform and the entrant supplier cannot emerge as an equilibrium.

#### 5.3. Self-Operating Mode Choice

**Proposition**

**3**.

_{1}, λ

_{2}, and λ

_{3}are defined in Appendix A):

- (a)
- When (v, ρ) ∈ R1:(a1) If λ > λ
_{1}, the incumbent supplier chooses the reselling mode and scenario RC is under an equilibrium;(a2) If λ < λ_{1}, the incumbent supplier chooses the revenue-sharing mode and scenario SC is under an equilibrium; - (b)
- When (v, ρ) ∈ R2:(b1) If λ > λ
_{2}, the incumbent supplier chooses the reselling mode and scenario RC is under an equilibrium;(b2) If λ < λ_{2}, the incumbent supplier chooses the revenue-sharing mode and scenario SM is under an equilibrium; - (c)
- When (v, ρ) ∈ R3:(c1) If λ > λ
_{3}, the incumbent supplier chooses the reselling mode and scenario RM is under an equilibrium;(c2) If λ < λ_{3}, the incumbent supplier chooses the revenue-sharing mode and scenario SC is under an equilibrium; - (d)
- Otherwise, the third-party supplier does not appear on the platform’s marketplace and:(d1) If λ > 1/2, the incumbent supplier chooses the reselling mode and scenario RM is under an equilibrium;(d2) If λ < 1/2, the incumbent supplier chooses the reselling mode and scenario SM is under an equilibrium.

_{1}, ${\pi}_{S1}^{RC}>{\pi}_{S1}^{SC}$, and the incumbent supplier would choose the reselling mode; if λ < λ

_{1}, ${\pi}_{S1}^{RC}<{\pi}_{S1}^{SC},$ and the incumbent supplier would choose the revenue-sharing mode. When (v, ρ) ∈ R2, the platform and the entrant supplier only form a partnership if the reselling mode is chosen; otherwise, their partnership cannot be formed. In this case, the incumbent supplier should compare its profit under scenario RC (i.e., ${\pi}_{S1}^{RC}$) with that under scenario SM (i.e., ${\pi}_{S1}^{SM}$). If λ > λ

_{2}, ${\pi}_{S1}^{RC}>{\pi}_{S1}^{SM}$, and the incumbent supplier would choose the reselling mode; if λ < λ

_{2}, ${\pi}_{S1}^{RC}<{\pi}_{S1}^{SM}$, and the incumbent supplier would choose the revenue-sharing mode. When (v, ρ) ∈ R3, the platform and the entrant supplier only form a partnership if the revenue-sharing self-operating mode is chosen; otherwise, their partnership cannot be formed. In this case, the incumbent supplier should compare its profit under scenario SC (i.e., ${\pi}_{S1}^{SC}$) with that under scenario RM (i.e., ${\pi}_{S1}^{RM}$). If λ > λ

_{2}, ${\pi}_{S1}^{RM}>{\pi}_{S1}^{SC}$, and the incumbent supplier would choose the reselling mode; if λ < λ

_{2}, ${\pi}_{S1}^{RM}<{\pi}_{S1}^{SC}$, and the incumbent supplier would choose the revenue-sharing mode. When (v, ρ) ∉ R1∪R2∪R3, the partnership between the platform and the entrant supplier cannot emerge as an equilibrium. In this case, the incumbent supplier should compare its profit under scenario RM (i.e., ${\pi}_{S1}^{RM}$) with that under scenario SM (i.e., ${\pi}_{S1}^{SM}$). If λ > 1/2, ${\pi}_{S1}^{RM}>{\pi}_{S1}^{SM}$, and the incumbent supplier would choose the reselling mode; if λ < 1/2, ${\pi}_{S1}^{RM}<{\pi}_{S1}^{SM}$, and the incumbent supplier would choose the revenue-sharing mode.

_{1}= s

_{2}= 0.1 to illustrate the equilibrium outcomes. Figure 4 depicts a spectrum of equilibrium regions depending on the revenue-sharing rate (vertical axis) and the commission rate (horizontal axis). As illustrated in Figure 4, the entire parameter space ((ρ, λ)∈[0, 1]

^{2}) is numerically divided into six regions. Since v < v

_{4}was always satisfied in this numerical example, R1 and R2 were reduced to (0, 0.4) and (0.4, 0.78), respectively. First, moving horizontally from left to right, the equilibrium outcomes switch from competitive scenarios to monopolistic scenarios. Intuitively, this switching behavior was due to the impact of an increasing commission rate on the entrant supplier. As indicated in Lemma 6, an increasing commission rate lowers the third-party supplier’s willingness to join the marketplace. Second, moving vertically from low to high, the optimal self-operating mode shift from the revenue-sharing mode to the reselling mode. For instance, the equilibrium outcomes switch from scenario SC to scenario RC in R1, from scenario SM to scenario RC in R2, and from scenario SM to scenario RM in the other region. A larger revenue-sharing rate undermines the incumbent supplier’s profit, which leads to its choice of the reselling mode.

## 6. Conclusions and Discussion

#### 6.1. Theoretical Implications

#### 6.2. Managerial Implications

#### 6.3. Limitations and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Corollary**

**1.**

- (1)
- $\frac{\partial {P}_{1}^{SC}}{\partial {s}_{1}}=\frac{1+\alpha}{3}>0$, $\frac{\partial {P}_{1}^{SC}}{\partial {s}_{2}}=-\frac{1+\alpha}{3}<0$,$\frac{\partial {P}_{2}^{SC}}{\partial {s}_{2}}=\frac{1+\alpha}{2}0$, $\frac{\partial {P}_{2}^{SC}}{\partial {s}_{1}}=-\frac{1+\alpha}{2}<0$, $\frac{\partial {D}_{i}^{SC}}{\partial {s}_{i}}=\frac{1+\alpha}{6t}>0$,$\frac{\partial {D}_{i}^{SC}}{\partial {s}_{j}}=-\frac{1+\alpha}{6t}0$, $\frac{\partial {\pi}_{S1}^{RC}}{\partial {s}_{1}}=\frac{2\left(1-\lambda \right)\left(1+\alpha \right)\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{18t}>0$, $\frac{\partial {\pi}_{S1}^{RC}}{\partial {s}_{2}}=-\frac{2\left(1-\lambda \right)\left(1+\alpha \right)\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{18t}<0$, $\frac{\partial {\pi}_{S2}^{RC}}{\partial {s}_{2}}=\frac{2\left(1-\rho \right)\left(1+\alpha \right)\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{18t}>0$, and $\frac{\partial {\pi}_{S2}^{RC}}{\partial {s}_{1}}=-\frac{2\left(1-\rho \right)\left(1+\alpha \right)\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{18t}0$;
- (2)
- If s
_{1}> s_{2}holds, $\frac{\partial {P}_{1}^{SC}}{\partial \alpha}=\frac{1+\alpha}{3}>0$, $\frac{\partial {P}_{2}^{SC}}{\partial \alpha}=-\frac{1+\alpha}{2}<0$,$\frac{\partial {D}_{1}^{SC}}{\partial \alpha}=\frac{1+\alpha}{6t}0$,$\frac{\partial {D}_{2}^{SC}}{\partial \alpha}=-\frac{1+\alpha}{6t}0$, $\frac{\partial {\pi}_{S1}^{RC}}{\partial \alpha}=\frac{\left(1-\lambda \right)\left(1+\alpha \right)\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{9t}>0$, and $\frac{\partial {\pi}_{S2}^{RC}}{\partial \alpha}=-\frac{\left(1-\rho \right)\left(1+\alpha \right)\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{2t}<0$; if s_{1}< s_{2}holds, the following results can be derived similarly; - (3)
- $\frac{\partial {\pi}_{P}^{SC}}{\partial \alpha}=\frac{2\left(\lambda +\rho \right){\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}^{2}+6t\left(\lambda -\rho \right)\left({s}_{1}-{s}_{2}\right)}{8t{\left(3-\rho \right)}^{2}}$ > 0 is equivalent to $\alpha >\frac{3t\left(\lambda -\rho \right)+\left(\lambda +\rho \right)\left({s}_{2}-{s}_{1}\right)}{\left(\lambda +\rho \right)\left({s}_{1}-{s}_{2}\right)}$;
- (4)
- $\frac{\partial {\pi}_{S1}^{RC}}{\partial \lambda}=-\frac{{\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}{18t}<0$,$\frac{\partial {\pi}_{S2}^{RC}}{\partial \rho}=-\frac{{\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}{18t}0$,$\frac{\partial {\pi}_{P}^{SM}}{\partial \lambda}=\frac{{\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}{18t}0$,$\mathrm{a}\mathrm{n}\mathrm{d}\frac{\partial {\pi}_{P}^{SM}}{\partial \rho}=\frac{{\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}{18t}0$. □

**Proof**

**of**

**Lemma**

**4**.

^{RC}), we first solve the profit maximization problems of the platform and the entrant supplier in (8) and (10). By solving the first-order conditions $\frac{\partial {\pi}_{P}^{RC}}{\partial {P}_{1}^{RC}}=0$ and $\frac{\partial {\pi}_{S2}^{RC}}{\partial {P}_{2}^{RC}}=0$ simultaneously, we obtain ${P}_{1}^{RC}=\frac{\left(3+\rho \right)t+2{w}^{RC}+\left(1-\rho \right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{3-\rho}$ and ${P}_{2}^{RC}=\frac{3t+{w}^{RC}-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{2}$. Inserting the above retail prices into (9), we obtain the profit maximization problems of the incumbent supplier as follows:

**Proof**

**of**

**Corollary**

**2.**

- (1)
- $\frac{\partial {w}^{RC}}{\partial \rho}=-t<0$,$\frac{\partial {P}_{1}^{RC}}{\partial \rho}=\frac{3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{{\left(3-\rho \right)}^{2}}0$, and $\frac{\partial {P}_{2}^{RC}}{\partial \rho}=\frac{3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{2{\left(3-\rho \right)}^{2}}>0$;
- (2)
- $\frac{\partial {D}_{1}^{RC}}{\partial \rho}=-\frac{3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{4t{\left(3-\rho \right)}^{2}}<0$, and $\frac{\partial {D}_{2}^{RC}}{\partial \rho}=\frac{3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)}{4t{\left(3-\rho \right)}^{2}}>0$;
- (3)
- $\frac{\partial {\pi}_{S1}^{RC}}{\partial \rho}=-\frac{\left(\left(9-2\rho \right)t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)\left(\left(3-2\rho \right)t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{8t{\left(3-\rho \right)}^{2}}<0$;
- (4)
- $\frac{\partial {\pi}_{S2}^{RC}}{\partial \rho}=-\frac{(3-\rho )\left(\left(9-2\rho \right)t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)\left(\left(21+2{\rho}^{2}-9\rho \right)t-\left(1+\rho \right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{8t{\left(3-\rho \right)}^{4}}<0$,$$\frac{\partial {\pi}_{P}^{RC}}{\partial \rho}=\frac{(3-\rho )\left(\left(306-144\rho -40{\rho}^{2}-8{\rho}^{3}\right){t}^{2}-\left(33-9\rho \right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)t+2{\left(1+\alpha \right)}^{2}{\left({s}_{1}-{s}_{2}\right)}^{2}\right)}{8t{\left(3-\rho \right)}^{4}}>0.$$

**Proof**

**of**

**Proposition**

**1.**

- (a)
- Denote ${v}_{1}=\frac{\left(3-2\rho \right)t-\left(1+\alpha \right){s}_{2}-\alpha}{1+\alpha}$, ${v}_{2}=\frac{4\left(6-\rho \right)t-\left(1+\rho \right)\left(1+\alpha \right){s}_{1}-4\left(2-\rho \right)\left(1+\alpha \right){s}_{2}-3\left(3-\rho \right)\alpha}{3\left(3-2\rho \right)\left(1+\alpha \right)}$, ${v}_{3}=\frac{\left(3-2\rho \right)t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)-\sqrt{3-\rho}\left(\left(1+\alpha \right){s}_{1}+\alpha \right)}{\left(1+\alpha \right)\sqrt{3-\rho}}$, and ${v}_{4}=\frac{\sqrt{2\left\{\begin{array}{c}t\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\left(6-16\rho +4{\rho}^{2}\right)+\\ \left(9+96\rho -56{\rho}^{2}+8{\rho}^{3}\right){t}^{2}+\\ {{\left(1+\alpha \right)}^{2}\left({s}_{1}-{s}_{2}\right)}^{2}\end{array}\right\}}-\left(3-\rho \right)\left(\left(1+\alpha \right){s}_{1}+\alpha \right)}{\left(3-\rho \right)\left(1+\alpha \right)}$. For the wholesale price, we can show that ${w}^{RM}<{w}^{RC}$ is equivalent to v < v
_{1}. For the incumbent supplier’s profit, ${\pi}_{S1}^{RC}>{\pi}_{S1}^{RM}$ holds if v < v_{3}is satisfied. For the retail price, ${P}_{1}^{RC}>{P}_{1}^{RM}$ is equivalent to v < v_{2}. For the platform’s profit, ${\pi}_{P}^{RC}>{\pi}_{P}^{RM}$ holds if v < v_{4}is satisfied; - (b)
- Denote ${v}_{5}=\frac{6t-\left(1+\alpha \right)\left({s}_{1}+2{s}_{2}\right)-3\alpha}{3\left(1+\alpha \right)}$, ${v}_{6}=\frac{3\sqrt{2}t-\left(3-\sqrt{2}\right)\left(1+\alpha \right){s}_{1}-\sqrt{2}\left(1+\alpha \right){s}_{2}-3\alpha}{3\left(1+\alpha \right)}$, and ${v}_{7}=\frac{\sqrt{2\left\{\left(\lambda +\rho \right)\left({9t}^{2}+{{\left(1+\alpha \right)}^{2}\left({s}_{1}-{s}_{2}\right)}^{2}\right)+6t\left(\lambda -\rho \right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right\}}-3\lambda \left(\left(1+\alpha \right){s}_{1}+\alpha \right)}{3\lambda \left(1+\alpha \right)}$. For the retail price, we can show that ${P}_{1}^{SC}>{P}_{1}^{SM}$ is equivalent to v < v
_{5}. For the incumbent supplier’s profit, ${\pi}_{S1}^{SC}>{\pi}_{S1}^{SM}$ holds if v < v_{6}is satisfied. For the platform’s profit, ${\pi}_{P}^{SC}>{\pi}_{P}^{SM}$ holds if v < v_{7}is satisfied. □

**Proof**

**of**

**Lemma**

**5.**

_{4}, the platform gains a higher profit by opening its marketplace (i.e., ${\pi}_{P}^{RC}>{\pi}_{P}^{RM}$) under the reselling self-operating channel mode. If v < v

_{7}, the platform gains a higher profit by opening its marketplace (i.e., ${\pi}_{P}^{SC}>{\pi}_{P}^{SM}$) under the revenue-sharing self-operating channel mode.

_{4}< v

_{7}. It was easy to check that v

_{7}(λ) decreases via λ. Thus, we only needed to prove that min v

_{7}(λ) > v

_{7}(λ = 1) > v

_{4}, which is equivalent to $\frac{\sqrt{\left\{\begin{array}{c}\left(1+\rho \right)\left({9t}^{2}+{{\left(1+\alpha \right)}^{2}\left({s}_{1}-{s}_{2}\right)}^{2}\right)+\\ 6t\left(1-\rho \right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\end{array}\right\}}}{3}>\frac{\sqrt{\left\{\begin{array}{c}t\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\left(6-16\rho +4{\rho}^{2}\right)+\\ \left(9+96\rho -56{\rho}^{2}+8{\rho}^{3}\right){t}^{2}+\\ {{\left(1+\alpha \right)}^{2}\left({s}_{1}-{s}_{2}\right)}^{2}\end{array}\right\}}}{\left(3-\rho \right)}$. It was easy to show that ${\pi}_{P}^{SC}\left(\lambda =1\right)>{\pi}_{P}^{RC}$, which is equivalent to the above inequation. As a result, we can conclude as follows: when v < v

_{4}, the platform would open its marketplace under both self-operating channel modes; when v

_{4}< v < v

_{7}, the platform would open its marketplace under the revenue-sharing self-operating channel modes; when v > v

_{7}, the platform would not open its marketplace. □

**Proof**

**of**

**Lemma**

**6.**

_{1}= $\frac{{\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}-18tc}{{\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}$ and ρ

_{2}as the unique root of ${(1-\rho )\left(\left(9-2\rho \right)t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}-8tc{\left(3-\rho \right)}^{2}=0$ in ρ∈[0, 1]; thus, we found that ${\pi}_{S2}^{SC}\left({\rho}_{1}\right)=0$ and ${\pi}_{S2}^{RC}\left({\rho}_{2}\right)=0$.

_{1}< ρ

_{2}. Since ${\pi}_{S2}^{RC}>{\pi}_{S2}^{SC}$ was equivalent to $3t\left(18-12\rho +2{\rho}^{2}\right)+\left(7-3\rho \right)\left(3t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)>0$, which was always satisfied, we found that ${\pi}_{S2}^{RC}>{\pi}_{S2}^{SC}$ always holds. As ρ

_{1}and ρ

_{2}denote the roots of ${\pi}_{S2}^{RC}\left(\rho \right)$ = 0 and ${\pi}_{S2}^{SC}\left(\rho \right)=0$, respectively, it was easy to show that ρ

_{1}< ρ

_{2}.

_{1}, the third-party supplier would join the marketplace under both self-operating channel modes; if ρ

_{1}< ρ < ρ

_{2}, the platform would join the marketplace under the reselling self-operating channel mode; and if ρ > ρ

_{2}, the platform would not join the marketplace. □

**Proof**

**of**

**Proposition**

**2.**

_{4}and ρ < ρ

_{1}, both the platform and the third-party supplier have incentives to cooperate under the two self-operating channel modes. If v < v

_{4}and ρ

_{1}< ρ < ρ

_{2}, the platform is willing to open the marketplace under both self-operating channel modes, but the third-party supplier only has a willingness to join under the reselling self-operating channel mode. Thus, the partnership can only emerge as an equilibrium under the reselling self-operating channel mode. If v

_{4}< v < v

_{7}and ρ < ρ

_{1}, the third-party supplier is willing to join the marketplace under both self-operating channel modes, but the platform only has an incentive to open its marketplace under the revenue-sharing self-operating channel mode. Thus, the partnership can only emerge as an equilibrium under the revenue-sharing self-operating channel mode. In other cases, the incentives of the platform and the entrant supplier cannot be aligned in either self-operating channel mode, and the partnership cannot be formed in equilibrium. □

**Proof**

**of**

**Proposition**

**3.**

_{1}= $\frac{\left(3t\left(2\sqrt{3-\rho}+3-2\rho \right)+\left(2\sqrt{3-\rho}+3\right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)\left(3t\left(2\sqrt{3-\rho}-3+2\rho \right)+\left(2\sqrt{3-\rho}-3\right)\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{4\left(3-\rho \right){\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}$. If λ > λ

_{1}, ${\pi}_{S1}^{SC}<{\pi}_{S1}^{RC}$, and the incumbent supplier chooses the reselling mode. If λ < λ

_{1}, ${\pi}_{S1}^{SC}>{\pi}_{S1}^{RC},$ and the incumbent supplier chooses the revenue-sharing mode.

_{2}= $\frac{\left(\sqrt{2\left(3-\rho \right)}\left(\left(v+{s}_{1}\right)\left(1+\alpha \right)+\alpha \right)+\left(3-2\rho \right)t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)\left(\sqrt{2\left(3-\rho \right)}\left(\left(v+{s}_{1}\right)\left(1+\alpha \right)+\alpha \right)-\left(3-2\rho \right)t-\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}{2\left(3-\rho \right){\left(\left(v+{s}_{1}\right)\left(1+\alpha \right)+\alpha \right)}^{2}}$. If λ > λ

_{2}, ${\pi}_{S1}^{SC}<{\pi}_{S1}^{RC},$ and the incumbent supplier chooses the reselling mode. If λ < λ

_{2}, ${\pi}_{S1}^{SC}>{\pi}_{S1}^{RC},$ and the incumbent supplier chooses the revenue-sharing mode.

_{3}= $\frac{\left(6t+3\alpha +\left(1+\alpha \right)\left(3v+5{s}_{1}-2{s}_{2}\right)\right)\left(6t+3\alpha -\left(1+\alpha \right)\left(3v+{s}_{1}+2{s}_{2}\right)\right)}{4{\left(3t+\left(1+\alpha \right)\left({s}_{1}-{s}_{2}\right)\right)}^{2}}$. If λ > λ

_{3}, ${\pi}_{S1}^{SC}<{\pi}_{S1}^{RM},$ and the incumbent supplier chooses the reselling mode. If λ < λ

_{3}, ${\pi}_{S1}^{SC}>{\pi}_{S1}^{RM},$ and the incumbent supplier chooses the revenue-sharing mode.

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Notation | Description |
---|---|

Decision variables | |

${P}_{1}^{i}$ | The retail price set by the platform or the incumbent supplier under scenario i, i ∈{RM, RC, SM, SC} |

${P}_{2}^{j}$ | The retail price set by the entrant supplier under scenario j, j∈{RC, SC} |

${w}^{m}$ | The wholesale price set by the incumbent supplier under scenario m, m∈{RM, RC} |

Other variables | |

${\pi}_{S1}^{i}$ | The incumbent supplier’s profit under scenario i |

${\pi}_{S2}^{j}$ | The entrant supplier’s profit under scenario j |

${\pi}_{P}^{i}$ ${D}_{1}^{i}$ ${D}_{2}^{j}$ | The platform’s profit under scenario i The incumbent supplier’s demand under scenario i The entrant supplier’s demand under scenario j |

Parameters | |

s_{1}/s_{2} | The logistics service level of the platform/entrant supplier |

v | Product quality of the competitive goods |

c | The entrant supplier’s opportunity cost incurred from joining the marketplace |

λ | The revenue-sharing rate under the revenue-sharing self-operating channel mode and λ∈(0, 1) |

ρ | The commission rate and ρ∈(0, 1) |

t | The disutility cost per unit distance |

α | The strength of the indirect network effects |

N | The supplier number and N∈{1, 2} |

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, B.; Huang, J.; Zhang, X.; Alejandro, T.B.
Strategic Third-Party Product Entry and Mode Choice under Self-Operating Channels and Marketplace Competition: A Game-Theoretical Analysis. *J. Theor. Appl. Electron. Commer. Res.* **2024**, *19*, 73-94.
https://doi.org/10.3390/jtaer19010005

**AMA Style**

Xu B, Huang J, Zhang X, Alejandro TB.
Strategic Third-Party Product Entry and Mode Choice under Self-Operating Channels and Marketplace Competition: A Game-Theoretical Analysis. *Journal of Theoretical and Applied Electronic Commerce Research*. 2024; 19(1):73-94.
https://doi.org/10.3390/jtaer19010005

**Chicago/Turabian Style**

Xu, Biao, Jinting Huang, Xiaodan Zhang, and Thomas Brashear Alejandro.
2024. "Strategic Third-Party Product Entry and Mode Choice under Self-Operating Channels and Marketplace Competition: A Game-Theoretical Analysis" *Journal of Theoretical and Applied Electronic Commerce Research* 19, no. 1: 73-94.
https://doi.org/10.3390/jtaer19010005