#### 4.1. Model 1: Optimal Subsidy with a Dominant Platform

Based on the above section, the platform determines her subsidies to the retailer, while the retailer decides the sales price and logistics service level. In the process of decision-making, the platform and the retailer play games with each other. The sequence and outcome of the game depends on the power structure between them. The dominant platform means she obtains strong bargaining power with huge user resources, while the weaker retailer can only accept the decision made by the platform. For instance, Tmall and JD.com, the famous e-commerce platforms in China, can easily dominate numerous small sellers on it.

To summarize, the timing of the Stackelberg game with a dominant platform is as follows. In the first stage, the platform determines her sales subsidy

m. In the second stage, the retailer determines his price

p and logistics service level

t simultaneously after observing the sales subsidy decision by the platform. This paper uses the superscript

ed to denote this mode and its decision problem can be determined by the following:

We solve the game backward and obtain its subgame-perfect Nash equilibrium as follows:

**Lemma** **1.** With the platform dominating the retailer, the optimal sales subsidy m^{ed*}, the optimal price p^{ed*}, and the optimal logistics service level t^{ed*} are as follows:where

${\Omega}_{1}=\{(T,b)|T\le {T}_{0}\}$,${\Omega}_{2}=\{(T,b)|T>{T}_{1},b\le \overline{b}\}$,${\Omega}_{3}=\{(T,b)|T>{T}_{1},b>\overline{b}\}$,${T}_{0}=\frac{\phi nv}{2-\phi {n}^{2}}$,${T}_{1}=\frac{\phi nv}{1-\phi {n}^{2}}$,$\overline{b}=\frac{2-{n}^{2}}{2-\phi {n}^{2}}$. This decision spaces with dominant platform can be clearly understood in Figure 1. Substituting (6)–(8) into Equation (4), it can be obtained that:

**Proposition** **1.** When and only when logistics constraints T and the altruistic preference coefficient b are both higher than a certain value, respectively, the platform will give the retailer a certain subsidy m^{ed}. At this time, $\frac{\partial {m}^{ed*}}{\partial b}>0,\frac{\partial {m}^{ed*}}{\partial n}>0$.

Proposition 1 indicates the conditions of giving sales subsidy. It can be understood the platform with higher altruistic preference is more likely to give the retailer sales subsidies, because it can lower the price and attract more buyers to improve the retailer’s profit and clove to her altruistic ideal. However, when the logistics service level is limited to a low value, the market demand is limited as well. At this time, the sales subsidies can only help the retailer increase few profits which cannot make up for the decrease of platform utility caused by subsidizing the retailer. That is the reason why the conditions of giving sales subsidy hold. When the platform has no altruistic preference (b = 0), she will not subsidize the retailer even there is no limitation on the logistics service level (i.e., T = +∞).

When the platform implements the sales subsidy program, m^{ed} is increasing in b and n. The higher n indicates the higher value of logistics service, and the platform is also willing to give more subsidies to the retailer to encourage him to improve his logistics service level and attract more customers. In reality, the platforms highly subsidize the retailers who sell badly needed products (higher n) like mask during COVID-19. For instance, JD.com gives 50% sales subsidies to the retailers who sell badly needed products, while 30% sales subsidies to the retailers who sell general products from 11 February 2020 to 31 March 2020.

**Proposition** **2.** (i) p^{ed} is increasing in n and φ, decreasing in b. (ii) t^{ed} is increasing in n, φ and b.

Proposition 2 can be proved by $\frac{\partial {p}^{ed*}}{\partial n}>0,\frac{\partial {p}^{ed*}}{\partial \phi}\ge 0,\frac{\partial {p}^{ed*}}{\partial b}\le 0$ and $\frac{\partial {t}^{ed*}}{\partial n}\ge 0,\frac{\partial {t}^{ed*}}{\partial \phi}\ge 0,\frac{\partial {t}^{ed*}}{\partial b}\ge 0$.

The platform with higher altruistic preference should improve the subsidies to the retailer, which stimulates him to increase market demand by charging lower price and providing higher logistics service level. From Equation (1), the higher n allows the retailer to charge higher price to keep the demand unchanged and obtain more profits. Furthermore, it also indicates that higher logistics service level can attract more consumers and obtain more profits. The higher φ means boosting the market demand can obtain more profits, so the retailer will provide higher logistics service level. However, he will not lower price for that, because it cannot compensate the profits obtained by charging a higher price when the higher φ holds. In China, some platforms lower the commission and advance altruistic preference to stabilize the price of badly needed product which is called by the government during early COVID-19.

**Proposition** **3.** (i) When $(T,b)\in {\Omega}_{1}\cup {\Omega}_{2}$, ${u}_{e}^{ed}$ is increasing in n, decreasing in φ. (ii) When $(T,b)\in {\Omega}_{3}$, ${u}_{e}^{ed}$ is increasing in n and φ.

Proposition 3 depicts the impacts of n and φ on the platform’s utility. Obviously, the market demand is increasing in n, and the platform and the retailer can both benefit from higher n. φ involves profit sharing between the platform and the retailer, and its impact on the platform’s utility is more complex. When T or b is low, the platform will not implement sales subsidy, and they show a competitive relationship. In such conditions ${u}_{o}^{ed}$ is decreasing in φ.

On the contrary, when T and b are all high enough, the platform should implement a sales subsidy, and they show a cooperative relationship. The higher the profit that the platform transfers to the retailer, the greater the utility it gets. It can be observed that many platforms lower their commission during COVID-19 to clove to their altruistic ideal. For instance, Tophatter, a cross border e-commerce platform in North America, claims that it remains a commission at 9% which should have risen after the Spring Festival on 20 February 2020 because of COVID-19. In China, MOGU, the most famous platform focusing on fashionable female consumers, announces that the cooperative brands will not need to pay the platform commission before the end of March 2020 within 10 million RMB on 4 February.

#### 4.2. Model 2: Optimal Subsidy with a Dominant Retailer

The preceding

Section 4.1 studies the optimal sales subsidy with a dominant platform who fully control the bargaining power. However, some dominant retailers and weaker platforms in reality. Dominant retailer means he can move first to decide his retail margin and the weaker platform has to accept the decision made by him. Some large brand retailers always take the role of dominant retailers because they have won massive loyal customers and can impact on the reputation of weaker platforms.

To summarize, the timing of the Stackelberg game with a dominant retailer is as follows. In the first stage, the retailer determines his retail margin

h = φ p + m and logistics service level

t simultaneously. In the second stage, the platform determines her sales subsidy

m after observing the decision by the retailer. This paper uses the superscript

rd to denote this mode and its decision problem can be determined by the following:

Solving the game backward and obtain its subgame-perfect Nash equilibrium as follows:

**Lemma** **2.** With the retailer dominating the platform, the optimal sales subsidy m^{rd*}, the optimal price p^{rd*}, and the optimal logistics service level t^{rd*} are as follows: where${\Omega}_{4}=\{(T,b)|T>{T}_{0},b\le \underset{\_}{b}\}$,${\Omega}_{5}=\{(T,b)|T\le {T}_{2},b>\underset{\_}{b}\}$,${\Omega}_{6}=\{(T,b)|T>{T}_{2},b>\underset{\_}{b}\}$,${T}_{2}=\frac{nv}{4-4b-{n}^{2}}$,$\underset{\_}{b}=1-\frac{1}{2\phi}$. This decision spaces with dominant retailer can be clearly understood in Figure 2. Substituting (11)–(13) into Equation (4), it can be obtained that

**Proposition** **4.** When and only when the altruistic preference coefficient b is higher enough to meet $b>1-\frac{1}{2\phi}$, the weaker platform will give the retailer a certain subsidy m^{rd*}. At this time, (i) m^{rd*} is increasing in n and b, but decreasing in φ; (ii) p^{rd*} is increasing in n and b.

Propositions 1 and 4 indicate the higher b can ensure the implementation of sales subsidy no matter what the power structure is. It is decreasing in φ because higher φ means the retailer can obtain more profits and the platform can give few subsidies to achieve the same incentives. Furthermore, an interesting conclusion is that the price is increasing in b, while decreasing in b with a dominant platform. The reason is the dominant retailer should charge a higher retail margin because he sees that the weaker platform can only accept it. The more altruistic preference, the higher retail margin she can accept, and the price goes up. That first move advantage can be described as “all lay loads on a willing house”.

**Proposition** **5.** If the weaker platform can select his altruistic preference b, he will find the best choice is m^{rd*} = 0 when $\phi \ge \frac{1}{2}$.

We define this selection as a weaker platform which will lower her altruistic preference b if it can improve his utility. The definition is based on the cost-benefit analysis that lowering her altruistic preference does not cost anything but can obtain more utility.

Proposition 5 is interesting that the weaker platform can get more utility by lowering her altruistic preference. It is contrary to our common sense that the platform’s utility is increasing in her altruistic preference from Equation (4). The reason is the dominant retailer should charge a higher h considering the weaker platform with higher b. The higher h would seriously damage her profits and utility, and the weaker platform would rather have a lower b. This proposition can explain the reality that most weaker platforms have low altruistic preference and give the dominant retailers few sales subsidies during an epidemic scenario considering most platforms’ commission is below 1/2.