Pricing of platform service supply chain with dual credit: Can you have the cake and eat it?

Abstract

Online shopping has become the main form of daily consumption, which contributes to the growing maturity of platform service supply chains (PSSCs). However, financial constraints of consumers and sellers have emerged as the key factors hindering their participation. To attract both sides on board and expand transaction volume, seller’s credit combined with buyer’s credit, or dual credit, has been launched by platforms. However, studies on whether this dual credit service is worth implementing remains to be an uncharted area. This paper aims to explore the e-commerce (EC) platform ecosystem with credit services, which is composed of a platform, a seller, and a group of consumers. By using game-theoretic approaches, our study finds that the dual credit drives down product price, which always profits consumers compared to the single buyer’s credit; platform charge, however, changes vaguely so that other agents are not always profited. For mid-to-high end products, platforms’ incentive in providing dual credit can be enhanced by raising seller’s capital or consumers’ dishonesty aversion coefficient, which also increases social welfare; while sellers engage in credit only when under severe shortage of capital. The effectiveness of raising consumers’ aversion coefficient in motivating the acceptance of financing services for sellers depends on the product type. Besides, for low-end products, the welfare of both the seller and the society always suffer, making the dual credit not worth promoting in this case.

Appendix

1.1 Appendix A

We first perform a detailed analysis of consumers’ demand. Given the product price $p_{s}$, consumers make product purchase decisions: purchase with buyer’s credit payment, purchase with personal income payment or no purchase. Obviously, the only source of return when consumers do not purchase products is the risk-free investment market, i.e., the reserved utility is $U_{i}^{r} = \left( {1 + r_{F} } \right)x_{i}$. When the consumer chooses buyer’s credit payment, consumer $i$ can use the credit payment to purchase a product at the beginning and repay the debt with the principal and interest gained from risk-free investment at the end of the period, thus the utility obtained by consumer $i$ from purchasing the product is

$$\begin{aligned} {\text{E}}U_{i}^{A} &= v + \left[ {\left( {1 + r_{F} } \right)x_{i} - p_{s} } \right]^{ + } - k_{A} \left[ {p_{s} - \left( {1 + r_{F} } \right)x_{i} } \right]^{ + } \\&= \left\{ {\begin{array}{*{20}l} {v + \left( {1 + r_{F} } \right)x_{i} - p_{s} ,} \hfill & {\quad x_{i} \ge \frac{{p_{s} }}{{1 + r_{F} }}} \hfill \\ {v + k_{A} \left( {1 + r_{F} } \right)x_{i} - k_{A} p_{s} ,} \hfill & {\quad x_{i} < \frac{{p_{s} }}{{1 + r_{F} }}} \hfill \\ \end{array} } \right..\end{aligned} $$

(A1)

And when consumers pay with their personal income, they can only invest their surplus in risk-free market, and thus the consumer’s utility function is

$$ EU_{i}^{N} = \left\{ {\begin{array}{*{20}l} {v + \left( {1 + r_{F} } \right)\left( {x_{i} - p_{s} } \right), } \hfill & {\quad x_{i} \ge p_{s} } \hfill \\ {0,} \hfill & {\quad x_{i} < p_{s} } \hfill \\ \end{array} } \right.. $$

(A2)

When $x_{i} \ge p_{s}$, ${\text{E}}U_{i}^{A} - EU_{i}^{N} = r_{F} p_{s} \ge 0$, ${\text{E}}U_{i}^{A} - U_{i}^{r} = v - p_{s}$, and thus consumer $i$’s product purchase decision is $\omega^{i} = \left\{ {\begin{array}{*{20}c} {1, v \ge p_{s} } \\ {0, v < p_{s} } \\ \end{array} } \right.$; when $\frac{{p_{s} }}{{1 + r_{F} }} \le x_{i} < p_{s}$, ${\text{E}}U_{i}^{A} - U_{i}^{r} = v - p_{s}$, consumer $i$’s product purchase decision is likewise $\omega^{i} = \left\{ {\begin{array}{*{20}c} {1, v \ge p_{s} } \\ {0, v < p_{s} } \\ \end{array} } \right.$; while $0 \le x_{i} < \frac{{p_{s} }}{{1 + r_{F} }}$, ${\text{E}}U_{i}^{A} - U_{i}^{r} = v - \left( {1 - k_{A} } \right)\left( {1 + r_{F} } \right)x_{i} - k_{A} p_{s}$, and consumer $i$’s product purchase decision is

$$\omega^{i} = \left\{ {\begin{array}{*{20}c} {1, v - k_{A} p_{s} \ge \left( {1 - k_{A} } \right)\left( {1 + r_{F} } \right)x_{i} } \\ {0, v - k_{A} p_{s} < \left( {1 - k_{A} } \right)\left( {1 + r_{F} } \right)x_{i} } \\ \end{array} } \right..$$

Thus, consumers all use credit payments and $\frac{{p_{s} }}{{1 + r_{F} }} \le \frac{{v - k_{A} p_{s} }}{{\left( {1 - k_{A} } \right)\left( {1 + r_{F} } \right)}}$ holds if $v \ge p_{s}$, the market demand is 1. If $v < p_{s}$, there is $\frac{{p_{s} }}{{1 + r_{F} }} > \frac{{v - k_{A} p_{s} }}{{\left( {1 - k_{A} } \right)\left( {1 + r_{F} } \right)}}$, and the market demand is $\frac{{v - k_{A} p_{s} }}{{a\left( {1 + r_{F} } \right)\left( {1 - k_{A} } \right)}}$. Therefore, consumers’ demand function is

$$ D = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\quad p_{s} \le v} \hfill \\ {\frac{{v - k_{A} p_{s} }}{{a\left( {1 + r_{F} } \right)\left( {1 - k_{A} } \right)}},} \hfill & {\quad v < p_{s} \le \frac{v}{{k_{A} }}} \hfill \\ \end{array} } \right.. $$

(A3)

□

1.2 Appendix B

1.2.1 Proof of Proposition 1

The maximized profit for the seller is

$$ \left\{ {\begin{array}{*{20}l} {\max\;\; \Pi_{s}^{Ns} \left( {p_{s}^{Ns} } \right) = \left( {p_{s}^{Ns} - p_{p}^{Ns} } \right)D + \left( {1 + r_{F} } \right)\left( {B - cD} \right)} \hfill \\ {{\text{s.t.}}\; \;cD \le B} \hfill \\ \end{array} } \right.. $$

(B1)

First, the financial constraint is not considered. According to the demand function, the seller’s profit is a piecewise function of product price. When $p_{s}^{Ns} \le v$, the seller’s pricing decision is $p_{s}^{Ns} = v$; when $v < p_{s}^{Ns} \le \frac{v}{{k_{A} }}$,

$$\begin{aligned}{p}_{s}^{Ns}\left({p}_{p}^{Ns}\right)&=\mathrm{max}\left[\mathrm{min}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{Ns}+\left(1+{r}_{F}\right)c}{2},\frac{v}{{k}_{A}}\right),v\right]\\&=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} ,& {p}_{p}^{Ns}>\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\\ \frac{v}{2{k}_{A}}+\frac{{p}_{p}^{Ns}+\left(1+{r}_{F}\right)c}{2},& \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\\ v , &{p}_{p}^{Ns}\le \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\end{array}\right.\end{aligned}.$$

Second, consider the financial constraint. When $p_{s}^{Ns} = v$, the seller with limited capital cannot afford to satisfy all consumers and has to give up this price; when $p_{s}^{Ns} > v$, the capital constraint is equivalent to $p_{s}^{Ns} \ge \frac{{cv - a\left( {1 + r_{F} } \right)\left( {1 - k_{A} } \right)B}}{{ck_{A} }}$.

Thus, the seller’s price response function after considering the capital constraint is as follows.

(i) When $B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\ge v$ holds. The seller’s price response function can be further expressed as

$$\begin{aligned}&{p}_{s}^{Ns}\left({p}_{p}^{Ns}\right)\\&=\left\{\begin{array}{ll}\frac{v}{{k}_{A}},& {p}_{p}^{Ns}>\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\\ \frac{v}{2{k}_{A}}+\frac{{p}_{p}^{Ns}+\left(1+{r}_{F}\right)c}{2},& \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c.\\ \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &{p}_{p}^{Ns}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\end{array}\right.\end{aligned}$$

(B2)

(ii) When $B>\frac{vc}{a\left(1+{r}_{F}\right)}$, $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}<v$. The seller’s price response function can be expressed as.

$${p}_{s}^{Ns}\left({p}_{p}^{Ns}\right)=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} ,& {p}_{p}^{Ns}>\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\\ \frac{v}{2{k}_{A}}+\frac{{p}_{p}^{Ns}+\left(1+{r}_{F}\right)c}{2},& \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\\ v ,& {p}_{p}^{Ns}\le \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\end{array}\right..$$

(B3)

The maximum profit of the platform is

$$\mathrm{max }{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)={p}_{p}^{Ns}D+{\int }_{i\in S}\left[\mathrm{min}\left[{p}_{s}^{Ns},\left(1+{r}_{F}\right){x}_{i}\right]-\left(1+{r}_{F}\right){p}_{s}^{Ns}\right]f\left({x}_{i}\right)d{x}_{i}.$$

(B4)

Since the seller’s response function is related not only to the platform charge but also to the seller’s capital, platform’s profit is likewise related to these two parameters. (1.1) $\le \frac{vc}{a\left(1+{r}_{F}\right)}$. We first determine the best choice of platform in different charging intervals. When${p}_{p}^{Ns}>\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, ${\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)=0$, and any charge that not lower than $\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$ has no effect on equilibrium strategies, which can be set to ${p}_{p}^{Ns}=\frac{v}{{k}_{A}}$, namely consumers’ reservation price, and the seller’s profit is $\left(1+{r}_{F}\right)B$. When ${p}_{p}^{Ns}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c$, $D=\frac{B}{c}$.

$$\begin{aligned}{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)&={p}_{p}^{Ns}D-\left(1+{r}_{F}\right){p}_{s}^{Ns}D+{\int }_{0}^{\frac{v-{k}_{A}{p}_{s}^{Ns}}{\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}}\left(1+{r}_{F}\right){x}_{i}f\left({x}_{i}\right)d{x}_{i}\\&=\frac{B(2c{k}_{A}{p}_{p}^{Ns}+2(1+{r}_{F})(aB\left(1+{r}_{F}\right)-cv)-aB{k}_{A}(1+3{r}_{F}+2{r}_{F}^{2}))}{2{c}^{2}{k}_{A}},\end{aligned}$$

and ${\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)$ is an increasing function of ${p}_{p}^{Ns}$, so we have ${p}_{p}^{Ns}=\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c$. When $\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, $D=\frac{v-{k}_{A}(\left(1+{r}_{F}\right)c+{p}_{p}^{Ns})}{2a(1-{k}_{A})(1+{r}_{F})}$,

$$\begin{aligned}&{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)=\frac{1}{8a{(1-{k}_{A})}^{2}(1+{r}_{F})}\\&\quad\left\{\left(v-{k}_{A}\left(\left(1+{r}_{F}\right)c+{p}_{p}^{Ns}\right)\right)\left[v-{k}_{A}\left(\left(1+{r}_{F}\right)c+{p}_{p}^{Ns}\right)+4(1-{k}_{A})\right.\right. \\ &\quad \left.\left.\left({p}_{p}^{Ns}-\frac{(1+{r}_{F})\left(v+{k}_{A}\left(\left(1+{r}_{F}\right)c+{p}_{p}^{Ns}\right)\right)}{2{k}_{A}}\right)\right]\right\},\end{aligned}$$

the first-order and second-order derivative of the platform profit with respect to the charge are

$$\begin{aligned}&\frac{\partial {\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {p}_{p}^{Ns}}\\&\quad=\frac{2v+{k}_{A}^{2}(c+3{p}_{p}^{Ns}-(c+2{p}_{p}^{Ns}){r}_{F}-2c{r}_{F}^{2})+{k}_{A}(-3v-2{p}_{p}^{Ns}+2(c+{p}_{p}^{Ns}){r}_{F}+2c{r}_{F}^{2})}{4a{(-1+{k}_{A})}^{2}(1+{r}_{F})},\end{aligned}$$

(B5)

$$\frac{{\partial }^{2}{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {\left({p}_{p}^{Ns}\right)}^{2}}=\frac{{k}_{A}^{2}\left(3-2{r}_{F}\right)-2{k}_{A}(1-{r}_{F})}{4a{(1-{k}_{A})}^{2}(1+{r}_{F})}.$$

(B6)

(1.1.1) When ${k}_{A}<\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{{\partial }^{2}{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {\left({p}_{p}^{Ns}\right)}^{2}}<0$, and at the point

$${p}_{p}^{Ns{\ddag} }=\frac{2v-c{k}_{A}^{2}(-1+{r}_{F}+2{r}_{F}^{2})+{k}_{A}(-3v+2c{r}_{F}(1+{r}_{F}))}{{k}_{A}(2\left(1-{r}_{F}\right)-{k}_{A}(3-2{r}_{F}))},$$

there is $\frac{\partial {\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {p}_{p}^{Ns}}=0$. Since

$${p}_{p}^{Ns{\ddag} }-\left(\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)=\frac{2\left(1-{k}_{A}\right)\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\left(3-2{r}_{F}\right)\right)}>0,$$

the optimal charge of the platform is $\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$. (1.1.2) when ${k}_{A}=\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{\partial {\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {p}_{p}^{Ns}}=\frac{2c+3v{r}_{F}-2(c+v){r}_{F}^{2}}{2a(1+{r}_{F})}>0$. In the interval of

$$\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c,$$

the optimal charge is likewise $\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, and

$${\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)={\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right)=0,$$

Thus, when ${k}_{A}\le \frac{2(1-{r}_{F})}{3-2{r}_{F}}$, the optimal charge for the platform over the entire defined domain is

$${p}_{p}^{Ns*}=\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right].$$

(1.1.3) When ${k}_{A}>\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{{\partial }^{2}{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {\left({p}_{p}^{Ns}\right)}^{2}}>0$, and thus the optimal charge is

$$\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right]$$

in the interval of $\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, the globally optimal decision is also

$$\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right].$$

Next, we provide further analysis of the optimal decision for the platform.

$$\begin{aligned}&{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\\&\quad=\frac{B}{2{c}^{2}{k}_{A}}\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})\right.,\end{aligned}$$

and denote that $G\left({k}_{A}\right)=-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)$. For $B\le \frac{2{c}^{2}}{a\left(3-2{r}_{F}\right)}$, $\frac{\partial G\left({k}_{A}\right)}{\partial {k}_{A}}\le 0$, thus $G\left({k}_{A}\right)\le G\left(0\right)=-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}<0$. The optimal charge of the platform over the entire definition domain is ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$. When $B>\frac{2{c}^{2}}{a\left(3-2{r}_{F}\right)}$, $\frac{\partial G\left({k}_{A}\right)}{\partial {k}_{A}}>0$ and $G\left(1\right)=\left(aB-2{c}^{2}\right)\left(1+{r}_{F}\right)-2vc{r}_{F}$. If $B\le \frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}$, $G\left({k}_{A}\right)<G\left(1\right)<0$ and ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, moreover, $\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}>\frac{2{c}^{2}}{a\left(3-2{r}_{F}\right)}$ obviously holds. If $B>\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}$, there exists a threshold $\frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}$, and $\frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\in \left(0,1\right)$. For ${k}_{A}\le \frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}$, $G\left({k}_{A}\right)\le 0$, and $G\left({k}_{A}\right)>0$ if ${k}_{A}>\frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}$. What’s more $\frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}$ is derived with respect to $B$ to obtain $\frac{2ac(-2c-3v{r}_{F}+2(c+v){r}_{F}^{2})}{(1+{r}_{F}){(-3aB+2{c}^{2}+2aB{r}_{F})}^{2}}$, thus $\frac{2aB+2cv{r}_{F}-2aB{r}_{F}^{2}}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}$ is a decreasing function of $B$.

(1.2)$B>\frac{vc}{a\left(1+{r}_{F}\right)}$. Similar to (1.1), the best choice of the platform on each interval is as follows: when ${p}_{p}^{Ns}>\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, ${\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)=0$ and set ${p}_{p}^{Ns}=\frac{v}{{k}_{A}}$. When ${p}_{p}^{Ns}\le \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, $D=\frac{v}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)=\frac{v(2{p}_{p}^{Ns}-v-2v{r}_{F})}{2a(1+{r}_{F})}$, ${\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)$ is an increasing function of ${p}_{p}^{Ns}$, and thus the platform charge is ${p}_{p}^{Ns}=\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, moreover ${\Pi }_{p}^{Ns}\left(\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)=\frac{v({k}_{A}(3v-2c(1+{r}_{F})-2v{r}_{F})-2v)}{2a{k}_{A}(1+{r}_{F})}$. When $\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c<{p}_{p}^{Ns}\le \frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$, the solution process is similar to case (1.1), more specifically, (1.2.1) when ${k}_{A}\le \frac{2(1-{r}_{F})}{3-2{r}_{F}}$, the globally optimal decision is

$${p}_{p}^{Ns*}=\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right].$$

Furthermore,

$${\Pi }_{p}^{Ns}\left(\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)=\frac{v({k}_{A}(3v-2c(1+{r}_{F})-2v{r}_{F})-2v)}{2a{k}_{A}(1+{r}_{F})}=\frac{vH\left({k}_{A}\right)}{2a{k}_{A}(1+{r}_{F})}.$$

If ${k}_{A}\le \frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $H\left(0\right)<0$ and $H\left(\frac{2(1-{r}_{F})}{3-2{r}_{F}}\right)=\frac{2c(2c+3v{r}_{F}-2(c+v){r}_{F}^{2})}{-3+2{r}_{F}}<0$, thus ${p}_{p}^{As*}=\frac{v}{{k}_{A}}$. (1.2.2) When ${k}_{A}>\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{{\partial }^{2}{\Pi }_{p}^{Ns}\left({p}_{p}^{Ns}\right)}{\partial {\left({p}_{p}^{Ns}\right)}^{2}}>0$, the globally optimal decision is likewise

$$\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right].$$

$H\left(1\right)=-c(2c-v+2(c+v){r}_{F})$, in general, ${r}_{F}\approx 0.05<0.5$, $H\left(1\right)$ is an increasing function of $v$. Moreover, if $v\le \frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, $H\left(1\right)\le 0$, and thus $H\left({k}_{A}\right)<0$ holds constantly,

$$\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right]=\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c.$$

If $\frac{2c(1+{r}_{F})}{1-2{r}_{F}}<v<a\left(1+{r}_{F}\right)$, $H\left(1\right)>0$, there exists a threshold $\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}$ such that $H\left(\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\right)=0$. And in the case that $\frac{2(1-{r}_{F})}{3-2{r}_{F}}<{k}_{A}<\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}$, $H\left({k}_{A}\right)\le 0$, therefore

$$\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right]=\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c.$$

If $\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1$, $H\left({k}_{A}\right)>0$,

$$\begin{aligned}&\mathrm{argmax}\left[{\Pi }_{p}^{Ns}\left(\frac{v}{{k}_{A}}\right),{\Pi }_{p}^{Ns}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)\right]\\ &\quad =\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c. \end{aligned}$$

Since $\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}$ is a decreasing function of $v$, if $v\le \frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, we have $\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\notin \left(\mathrm{0,1}\right)$.

In summary, the optimal charge for the platform over the entire defined domain is the following result.

(1.a) When $B\le \mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right)$, or both $B>\frac{cv}{a(1+{r}_{F})}$ and $v\le \frac{2c(1+{r}_{F})}{1-2{r}_{F}}$ are valid at the same time, ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}$, ${p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$; ${\Pi }_{p}^{Ns*}={CS}^{Ns*}=0$; ${\Pi }_{s}^{Ns*}=\left(1+{r}_{F}\right)B$; ${SW}^{Ns*}=\left(1+{r}_{F}\right)B$.

(1.b) When $\mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right)<B\le \frac{cv}{a(1+{r}_{F})}$, the optimal strategies of PSSC, the equilibrium market demand and the equilibrium profits or utility of each subject are shown in Eqs. (B7)–(B13).

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c,& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B7)

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B8)

$${D}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B}{c}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B9)

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}},& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B10)

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}},& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B11)

$$\begin{aligned}&{CS}^{Ns*}\\ &\;=\left\{\begin{array}{ll}0 , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{1}{a}{\int }_{0}^{\frac{v-{k}_{A}{p}_{s}^{Ns*}}{\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}}\left[v+{k}_{A}\left(1+{r}_{F}\right){x}_{i}-{k}_{A}{p}_{s}^{Ns*}-\left(1+{r}_{F}\right){x}_{i}\right]d{x}_{i} & \\ \quad =\frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}},& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.\end{aligned}$$

(B12)

$${SW}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B13)

(1.c) When $B>\frac{cv}{a(1+{r}_{F})}$ and $v>\frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, the equilibrium results are shown in Eqs. (B14)–(B20).

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B14)

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ {v}^{+},& \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B15)

$${D}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{v}{a\left(1+{r}_{F}\right)}, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B16)

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{v\left({k}_{A}\left(3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}\right)-2v\right)}{2a{k}_{A}\left(1+{r}_{F}\right)}, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B17)

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B , &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{a{k}_{A}\left(1+{r}_{F}\right)}+(1+{r}_{F})B,& \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B18)

$${CS}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)}, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B19)

$${SW}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B ,& 0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{v\left(v\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)-2c\left(1+{r}_{F}\right)\right)}{2a\left(1+{r}_{F}\right)}+(1+{r}_{F})B,& \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

(B20)

The results of Proposition 1 can be obtained by further sorting.□

1.2.2 Proof of Proposition 2

The maximized profit for the seller is.

$$\left\{\begin{array}{ll}\mathrm{max }&{\Pi }_{s}^{As}\left({p}_{s}^{As}\right)=\left({p}_{s}^{As}-{p}_{p}^{As}\right)D-\left(1+{r}_{s}\right)\left(cD-B\right)\\ s.t. &cD\ge B\end{array}\right..$$

(B21)

First, the financial constraint is not considered. According to the piecewise demand function, we consider the cases of ${p}_{s}^{As}\le v$, $v<{p}_{s}^{As}\le \frac{v}{{k}_{A}}$, respectively. When${p}_{s}^{As}\le v$, the seller’s pricing decision is${p}_{s}^{As}=v$. When$v<{p}_{s}^{As}\le \frac{v}{{k}_{A}}$, given the platform charge, the seller’s price response function is

$${p}_{s}^{As}\left({p}_{p}^{As}\right)=\mathrm{max}\left[\mathrm{min}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2},\frac{v}{{k}_{A}}\right),v\right]=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &{p}_{p}^{As}>\frac{v}{{k}_{A}}-\left(1+{r}_{s}\right)c\\ \frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2},& \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c<{p}_{p}^{As}\le \frac{v}{{k}_{A}}-\left(1+{r}_{s}\right)c\\ v , &{p}_{p}^{As}\le \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c\end{array}\right..$$

Second, consider the capital constraint. When ${p}_{s}^{As}=v$, the constraint is satisfied. When ${p}_{s}^{As}>v$, the constraint can be reduced to ${p}_{s}^{As}\le \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}$.

Thus, the seller’s response function after considering the financial constraint is as follows.

(i) When $B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\ge v$. The seller’s price response function can be further expressed as

$$\begin{aligned}&{p}_{s}^{As}\left({p}_{p}^{As}\right)\\&\;=\left\{\begin{array}{ll}argmax\left({\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right),{\Pi }_{s}^{As}\left({v}^{-}\right)\right), &{p}_{p}^{As}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}\\ argmax\left({\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right),{\Pi }_{s}^{As}\left({v}^{-}\right)\right), &\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}\\ argmax\left({\Pi }_{s}^{As}\left({v}^{+}\right),{\Pi }_{s}^{As}\left({v}^{-}\right)\right) , &{p}_{p}^{As}\le \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c\end{array}\right..\end{aligned}$$

(B22)

Next, we further simplify Eq. (B22). (a)

$${\Pi }_{s}^{As}\left({v}^{+}\right)=\left[v-{p}_{p}^{As}-\left(1+{r}_{s}\right)c\right]\frac{v}{a\left(1+{r}_{F}\right)}+\left(1+{r}_{s}\right)B\le {\Pi }_{s}^{As}\left({v}^{-}\right)=v-{p}_{p}^{As}-\left(1+{r}_{s}\right)c+\left(1+{r}_{s}\right)B,$$

that is $\mathrm{argmax}\left({\Pi }_{s}^{As}\left({v}^{+}\right),{\Pi }_{s}^{As}\left({v}^{-}\right)\right)={v}^{-}$. (b)

$${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}-{p}_{p}^{As}\right)\frac{B}{c},$$

$$\frac{\partial }{\partial {p}_{p}^{As}}\left({\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\right)=1-\frac{B}{c}>0,$$

and

$$\frac{\partial }{\partial B}\left({\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\right)=\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}-{p}_{p}^{As}\right)\frac{1}{c}<0,$$

thus if

$${p}_{p}^{As}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

$${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)>\frac{(1-{k}_{A})(aB(B-2c)(1+{r}_{F})+{c}^{2}v)}{{c}^{2}{k}_{A}}$$

holds, and further, if $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$,

$${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)>\frac{\left(1-{k}_{A}\right)\left(aB\left(B-2c\right)\left(1+{r}_{F}\right)+{c}^{2}v\right)}{{c}^{2}{k}_{A}}>0,$$

while $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, there exists a point ${p}_{p}^{As{\ddag} }$ which satisfies

$${p}_{p}^{As{\ddag} }=\frac{B(aB(1+{r}_{F})(1-{k}_{A})-cv)+{k}_{A}{c}^{2}(v-(c-B)(1+{r}_{s}))}{(c-B)c{k}_{A}}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

and for

$$\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}<{p}_{p}^{As}\le {p}_{p}^{As{\ddag} },$$

there is

$${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\le {\Pi }_{s}^{As}\left({v}^{-}\right),$$

for ${p}_{p}^{As}>{p}_{p}^{As{\ddag} }$, ${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)>{\Pi }_{s}^{As}\left({v}^{-}\right)$ is obtained. (c)

$${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)=\frac{{\left(v-{k}_{A}\left(\left(1+{r}_{s}\right)c+{p}_{p}^{As}\right)\right)}^{2}}{4a\left(1-{k}_{A}\right){k}_{A}\left(1+{r}_{F}\right)}+\left(1+{r}_{s}\right)B,$$

and

$$\frac{\partial }{\partial {p}_{p}^{As}}\left({\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\right)=1-\frac{v-{k}_{A}({p}_{p}^{As}+\left(1+{r}_{s}\right)c)}{2a(1-{k}_{A})(1+{r}_{F})},$$

$$\frac{\partial }{\partial {\left({p}_{p}^{As}\right)}^{2}}{\left({\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\right)}^{2}=\frac{{k}_{A}}{2a(1-{k}_{A})(1+{r}_{F})}>0,$$

therefore,

$$\frac{\partial }{\partial {p}_{p}^{As}}\left({\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\right)>1-\frac{v}{a\left(1+{r}_{F}\right)}>0,$$

that is ${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)$ is an increasing function of ${p}_{p}^{As}$. If ${p}_{p}^{As}=\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c$, we have

$${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)=-\frac{v\left(1-{k}_{A}\right)\left(a-v+a{r}_{F}\right)}{a{k}_{A}\left(1+{r}_{F}\right)}<0;$$

if

$${p}_{p}^{As}=\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

$${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)=\frac{\left(1-{k}_{A}\right)\left(aB\left(B-2c\right)\left(1+{r}_{F}\right)+{c}^{2}v\right)}{{c}^{2}{k}_{A}}.$$

If $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, $\frac{\left(1-{k}_{A}\right)\left(aB\left(B-2c\right)\left(1+{r}_{F}\right)+{c}^{2}v\right)}{{c}^{2}{k}_{A}}\ge 0$ holds, by the Zero Theorem, there exists a unique point

$${p}_{p}^{As{\dag}}=\frac{v-2a(1+{r}_{F})(1-{k}_{A})\left(1-\sqrt{1-\frac{v}{a(1+{r}_{F})}}\right)-{k}_{A}c(1+{r}_{s})}{{k}_{A}}$$

such that in case

$${p}_{p}^{As{\dag}}\le {p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

$${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)\ge 0$$

is got; in case $\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{s}\right)c<{p}_{p}^{As}<{p}_{p}^{As{\dag}}$, ${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)<0$. While $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, then $\frac{\left(1-{k}_{A}\right)\left(aB\left(B-2c\right)\left(1+{r}_{F}\right)+{c}^{2}v\right)}{{c}^{2}{k}_{A}}<0$, thus we have

$${\Pi }_{s}^{As}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2}\right)-{\Pi }_{s}^{As}\left({v}^{-}\right)<0.$$

From the above analysis we can obtain the following simplification of Eq. (B22).

When $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, the seller’s optimal price response function is

$${p}_{s}^{As}\left({p}_{p}^{As}\right)=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& {p}_{p}^{As}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}\\ \frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2},& {p}_{p}^{As{\dag}}<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}\\ {v}^{-} , &{p}_{p}^{As}\le {p}_{p}^{As{\dag}}\end{array}\right..$$

(B23)

When $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, the seller’s optimal price response function is

$${p}_{s}^{As}\left({p}_{p}^{As}\right)=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& {p}_{p}^{As}>{p}_{p}^{As{\ddag} }\\ {v}^{-} ,& {p}_{p}^{As}\le {p}_{p}^{As{\ddag} }\end{array}.\right.$$

(B24)

(ii) When $B>\frac{vc}{a\left(1+{r}_{F}\right)}$, $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}<v$, and thus we have

$$\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}<v\le \mathrm{max}\left[\mathrm{min}\left(\frac{v}{2{k}_{A}}+\frac{{p}_{p}^{As}+\left(1+{r}_{s}\right)c}{2},\frac{v}{{k}_{A}}\right),v\right],{p}_{s}^{As}\left({p}_{p}^{As}\right)={v}^{-}.$$

Based on the seller’s response function, we can examine the optimal charge. The profit function of the platform is

$${\Pi }_{p}^{As}\left({p}_{p}^{As}\right)={p}_{p}^{As}D+{\int }_{i\in S}\left[\mathrm{min}\left[{p}_{s}^{As},\left(1+{r}_{F}\right){x}_{i}\right]-\left(1+{r}_{F}\right){p}_{s}^{As}\right]f\left({x}_{i}\right)d{x}_{i}+\left({r}_{s}-{r}_{F}\right)\left(cD-B\right).$$

(B25)

Similar to Proposition 1, we first examine the optimal choice of the platform over different charging intervals, and then find its optimal charging decision over the entire defined domain.

(2.1) $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$. If

$${p}_{p}^{As}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

$${\Pi }_{p}^{As}\left({p}_{p}^{As}\right)=\frac{B(2c{k}_{A}{p}_{p}^{As}+2(1+{r}_{F})(aB\left(1+{r}_{F}\right)-cv)-aB{k}_{A}(1+3{r}_{F}+2{r}_{F}^{2}))}{2{c}^{2}{k}_{A}}.$$

In order make the seller participate in platform operation, the charge needs to satisfy the condition

$${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}-{p}_{p}^{As}\right)\frac{B}{c}\ge 0,$$

i.e., ${p}_{p}^{As}\le \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}$, and thus the charge is $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}$; if

$${p}_{p}^{As{\dag}}<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

$$\begin{aligned}&{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)=\frac{1}{8a{(1-{k}_{A})}^{2}{k}_{A}(1+{r}_{F})}\\ &\quad\left\{\begin{array}{l}{k}_{A}{\left(v-{k}_{A}\left(c\left(1+{r}_{s}\right)+{p}_{p}^{As}\right)\right)}^{2}+2\left(1-{k}_{A}\right)\left(-v+{k}_{A}\left(c\left(1+{r}_{s}\right)+{p}_{p}^{As}\right)\right)\\ \left[v\left(1+{r}_{F}\right)+{k}_{A}\left[-{p}_{p}^{As}\left(1-{r}_{F}\right)+c\left(1+{r}_{F}\right)\left(1+{r}_{s}\right)\right]\right]\\ +8(1-{k}_{A}){k}_{A}({r}_{F}-{r}_{s})\left[aB(1-{k}_{A})(1+{r}_{F})+\frac{1}{2}c\left(-v+{k}_{A}(c\left(1+{r}_{s}\right)+{p}_{p}^{As}\right)\right]\end{array}\right\}.\end{aligned}$$

The first-order and second-order derivatives of platform’s profit with respect to the charge are Eqs. (B26) and (B27), respectively.

$$\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {p}_{p}^{As}}=\frac{2v+{k}_{A}^{2}(c+3{p}_{p}^{As}+3c{r}_{s}-2{r}_{F}(2c+{p}_{p}^{As}+c{r}_{s}))+{k}_{A}(-3v-2{p}_{p}^{As}-2c{r}_{s}+2{r}_{F}(2c+{p}_{p}^{As}+c{r}_{s}))}{4a{(-1+{k}_{A})}^{2}(1+{r}_{F})},$$

(B26)

$$\frac{{\partial }^{2}{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {\left({p}_{p}^{As}\right)}^{2}}=\frac{{k}_{A}^{2}\left(3-2{r}_{F}\right)-2{k}_{A}(1-{r}_{F})}{4a{(1-{k}_{A})}^{2}(1+{r}_{F})}.$$

(B27)

(2.1.1) When ${k}_{A}<\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{{\partial }^{2}{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {\left({p}_{p}^{As}\right)}^{2}}<0$, and at the point

$${p}_{p}^{As \Delta }=\frac{2v+c{k}_{A}^{2}(1+3{r}_{s}-2{r}_{F}(2+{r}_{s}))+{k}_{A}(-3v-2c{r}_{s}+2c{r}_{F}(2+{r}_{s}))}{{k}_{A}(2-2{r}_{F}-{k}_{A}(3-2{r}_{F}))},$$

there is $\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {p}_{p}^{As}}=0$, besides,

$$\begin{aligned}&{p}_{p}^{As \Delta }-\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}\\&\quad=\frac{2\left(1-{k}_{A}\right)\left(2aB+cv{r}_{F}-2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(-3aB+{c}^{2}+2aB{r}_{F}\right)\right)}{c{k}_{A}\left(2-2{r}_{F}+{k}_{A}\left(-3+2{r}_{F}\right)\right)}\\&\quad\ge \frac{2\left(1-{k}_{A}\right)\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{{k}_{A}\left(2-2{r}_{F}+{k}_{A}\left(-3+2{r}_{F}\right)\right)}>0,\end{aligned}$$

and thus the optimal charge is $\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}$; (2.1.2) when ${k}_{A}=\frac{2(1-{r}_{F})}{3-2{r}_{F}}$,

$$\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {p}_{p}^{As}}=\frac{2c+3v{r}_{F}-2(c+v){r}_{F}^{2}}{2a(1+{r}_{F})}>0,$$

in the interval of

$${p}_{p}^{As{\dag}}<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}},$$

the optimal charge of the platform is likewise $\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}$; (2.1.3) when ${k}_{A}>\frac{2(1-{r}_{F})}{3-2{r}_{F}}$, $\frac{{\partial }^{2}{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)}{\partial {\left({p}_{p}^{As}\right)}^{2}}>0$, the optimal charge is

$$\mathrm{argmax}\left[{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right),{\Pi }_{p}^{As}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}(1+{r}_{s})}{c{k}_{A}}\right)\right].$$

If ${p}_{p}^{As}\le {p}_{p}^{As{\dag}}$,

$$\begin{aligned}{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)&={p}_{p}^{As}-\left(1+{r}_{F}\right)v+{\int }_{0}^{\frac{v}{1+{r}_{F}}}\left(1+{r}_{F}\right){x}_{i}f\left({x}_{i}\right)d{x}_{i}+{\int }_{\frac{v}{1+{r}_{F}}}^{a}vf\left({x}_{i}\right)d{x}_{i}\\&\quad+\left({r}_{s}-{r}_{F}\right)\left(c-B\right)={p}_{p}^{As}+\left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right),\end{aligned}$$

${\Pi }_{p}^{As}\left({p}_{p}^{As}\right)$ is an increasing function of ${p}_{p}^{As}$ and thus ${p}_{p}^{As}={p}_{p}^{As{\dag}}$.

In summary, the optimal charge of the platform over the entire definition domain is

$${p}_{p}^{As*}=\mathrm{argmax}\left[{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right),{\Pi }_{p}^{As}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}\right),{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right].$$

Since

$${\Pi }_{p}^{As}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}\right)=\frac{B(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}(3aB-2{c}^{2}+aB{r}_{F}-2aB{r}_{F}^{2}-2{c}^{2}{r}_{s}))}{2{c}^{2}{k}_{A}},$$

$${\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=\frac{B(aB{k}_{A}(1-{r}_{F}-2{r}_{F}^{2})-2{r}_{F}(cv-aB\left(1+{r}_{F}\right)))}{2{c}^{2}{k}_{A}},$$

we can easily get

$${\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}\right)=\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)+{k}_{A}{c}^{2}\left(1+{r}_{s}\right)\right)}{{c}^{2}{k}_{A}}>0.$$

While

$${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)=\left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a(1-{k}_{A})(1+{r}_{F})\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}(1+{r}_{s})}{{k}_{A}},$$

$$\begin{aligned}&{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\\&\quad=\frac{B(aB{k}_{A}(1-{r}_{F}-2{r}_{F}^{2})-2{r}_{F}(cv-aB\left(1+{r}_{F}\right)))}{2{c}^{2}{k}_{A}}-(B-c-v){r}_{F}+\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}\\&\qquad+B{r}_{s}-c{r}_{s}-\frac{v-2a(1-{k}_{A})(1+{r}_{F})\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}(1+{r}_{s})}{{k}_{A}},\end{aligned}$$

and

$$\frac{\partial }{\partial {k}_{A}}\left({\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right)=\frac{-a{B}^{2}{r}_{F}^{2}+{c}^{2}\left[v-2a\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)\right]+{r}_{F}\left[-a{B}^{2}+Bcv-2a{c}^{2}\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)\right]}{{c}^{2}{k}_{A}^{2}},$$

denote that

$${\mathcalligra{l}}\left(B\right)=-a{r}_{F}\left(1+{r}_{F}\right){B}^{2}+cv{r}_{F}B+{c}^{2}\left[v-2a\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)\right]-2a{c}^{2}{r}_{F}\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right),$$

then

$$\frac{\partial }{\partial {k}_{A}}\left({\Pi }_{p}^{As}\left(\frac{c\left(v-c{k}_{A}\right)-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right)$$

has the same sign as ${\mathcalligra{l}}\left(B\right)$. $\frac{\partial {\mathcalligra{l}}\left(B\right)}{\partial B}={r}_{F}\left(cv-2aB\left(1+{r}_{F}\right)\right)$, ${\mathcalligra{l}}\left(B\right)$ is a concave function of $B$. According to the discriminant of quadratic roots,

$${c}^{2}{v}^{2}{r}_{F}^{2}+4a{c}^{2}{r}_{F}\left(1+{r}_{F}\right)\left(v-2a\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)\right)\le 0,$$

and thus ${\mathcalligra{l}}\left(B\right)\le 0$ holds constantly, i.e., ${\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)$ is a decreasing function of ${k}_{A}$. Since

$$\begin{aligned}&\underset{{k}_{A}\to 0}{\mathrm{lim}}\left[{\Pi }_{p}^{As}\left(\frac{c\left(v-c{k}_{A}\right)-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right]\\&\quad=\underset{{k}_{A}\to 0}{\mathrm{lim}}\left[\frac{a{B}^{2}(1-{r}_{F}-2{r}_{F}^{2})}{2{c}^{2}}-\left(B-c-v\right){r}_{F}+\frac{{v}^{2}}{2a+2a{r}_{F}}+B{r}_{s}\right. \\ &\qquad\left. -c{r}_{s}-\frac{1}{{c}^{2}{k}_{A}}\left({\mathcalligra{l}}\left(B\right)-{k}_{A}\left(2a\left(1+{r}_{F}\right)-c(1+{r}_{s})\right)\right)\right]\\&\quad=+\infty \gg 0,\end{aligned}$$

there exists ${\tilde{k}}_{A}>0$ such that if

$${k}_{A}<\mathrm{min}\left({\tilde{k}}_{A},1\right), {\Pi }_{p}^{As}\left(\frac{c\left(v-c{k}_{A}\right)-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)>0,$$

and if

$$\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1,{\Pi }_{p}^{As}\left(\frac{c\left(v-c{k}_{A}\right)-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)<0.$$

Therefore, when $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, the equilibrium results are shown in Eqs. (B28)–(B34).

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {p}_{p}^{As{\dag}},& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B28)

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {v}^{-} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B29)

$${D}^{As*}=\left\{\begin{array}{ll}\frac{B}{c},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ 1 , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B30)

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} ,& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}\left(1+{r}_{s}\right)}{{k}_{A}}, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B31)

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ v-\frac{v}{{k}_{A}}+\frac{2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}}+B\left(1+{r}_{s}\right), &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B32)

$${CS}^{As*}=\left\{\begin{array}{ll}\frac{1}{a}{\int }_{0}^{\frac{v-{k}_{A}{p}_{s}^{As*}}{\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}}\left[v+{k}_{A}\left(1+{r}_{F}\right){x}_{i}-{k}_{A}{p}_{s}^{As*}-\left(1+{r}_{F}\right){x}_{i}\right]d{x}_{i}=\frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}} , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{1}{a}{\int }_{0}^{\frac{{p}_{s}^{As*}}{1+{r}_{F}}}\left[v+{k}_{A}\left(1+{r}_{F}\right){x}_{i}-{k}_{A}{p}_{s}^{As*}-\left(1+{r}_{F}\right){x}_{i}\right]d{x}_{i}+\frac{1}{a}{\int }_{\frac{{p}_{s}^{As*}}{1+{r}_{F}}}^{a}\left[v+\left(1+{r}_{F}\right){x}_{i}-{p}_{s}^{As*}-\left(1+{r}_{F}\right){x}_{i}\right]d{x}_{i}=\frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)},& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B33)

$${SW}^{As*}=\left\{\begin{array}{ll}\frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a\left(1+{r}_{F}\right)} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B34)

(2.2) $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$. If

$${p}_{p}^{As}>{p}_{p}^{As{\ddag} },{\Pi }_{p}^{As}\left({p}_{p}^{As}\right)=\frac{B(2c{k}_{A}{p}_{p}^{As}+2(1+{r}_{F})(aB\left(1+{r}_{F}\right)-cv)-aB{k}_{A}(1+3{r}_{F}+2{r}_{F}^{2}))}{2{c}^{2}{k}_{A}},$$

to make the seller participate in the platform operation, the platform charge needs to satisfy the condition that ${\Pi }_{s}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\ge 0$, i.e., ${p}_{p}^{As}\le \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}$, and thus the platform’s charge is $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}$. When ${p}_{p}^{As}\le {p}_{p}^{As{\ddag} }$, its charging strategy is ${p}_{p}^{As{\ddag} }$.

The optimal charge of the platform over the whole definition domain is

$$\mathrm{argmax}\left[{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right),{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right].$$

Where

$${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)=(B-c-v){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B(aB(1+{r}_{F})(1-{k}_{A})-cv)+{k}_{A}{c}^{2}(v-(c-B)(1+{r}_{s}))}{(c-B)c{k}_{A}},$$

$$\frac{\partial }{\partial {k}_{A}}\left({\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)\right)=\frac{B(aB\left(1+{r}_{F}\right)-cv)\left(c\left(1-{r}_{F}\right)+B{r}_{F}\right)}{(c-B){c}^{2}{k}_{A}^{2}}<0,$$

and

$$\underset{{k}_{A}\to 0}{\mathrm{lim}}\left[{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)\right]=+\infty \gg 0,$$

there exists a threshold ${\breve{k}}$ such that for

$${k}_{A}<\mathrm{min}\left({\breve{k}},1\right),{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)>0,$$

and for

$$\mathrm{min}\left({\breve{k}},1\right)\le {k}_{A}<1,{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)<0.$$

Therefore, when $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, the equilibrium results of each subject are shown in Eqs (B35)–(B41).

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {p}_{p}^{As{\ddag} } , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B35)

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {v}^{-} ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B36)

$${D}^{As*}=\left\{\begin{array}{ll}\frac{B}{c}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ 1, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

(B37)

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}\left(v-\left(c-B\right)\left(1+{r}_{s}\right)\right)}{\left(c-B\right)c{k}_{A}},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B38)

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{B\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)\left(1+{r}_{s}\right)\right]}{\left(c-B\right)c{k}_{A}},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B39)

$${CS}^{As*}=\left\{\begin{array}{ll}\frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)} ,& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B40)

$${SW}^{As*}=\left\{\begin{array}{ll}\frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a\left(1+{r}_{F}\right)} ,& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B41)

(2.3) $B>\frac{vc}{a\left(1+{r}_{F}\right)}$.

$${\Pi }_{p}^{As}\left({p}_{p}^{As}\right)={p}_{p}^{As}D+{\sum }_{i\in S}\left[\mathrm{min}\left[{p}_{s}^{As},\left(1+{r}_{F}\right){x}_{i}\right]-\left(1+{r}_{F}\right){p}_{s}^{As}\right]+\left({r}_{s}-{r}_{F}\right)\left(cD-B\right)={p}_{p}^{As}+(B-c-v){r}_{F}-\frac{{v}^{2}}{2a+2a{r}_{F}}-B{r}_{s}+c{r}_{s}.$$

The participation constraints of the seller is ${\Pi }_{s}^{As}\left({v}^{-}\right)=v-{p}_{p}^{As}-\left(1+{r}_{s}\right)\left(c-B\right)\ge 0$, that is ${p}_{p}^{As}\le v-\left(1+{r}_{s}\right)\left(c-B\right)$, and thus the optimal charge is ${p}_{p}^{As*}=v-\left(1+{r}_{s}\right)\left(c-B\right)$. The optimal strategy of the seller, the market equilibrium demand and the equilibrium profits or utility of each subject are ${p}_{s}^{As*}={v}^{-}$, ${D}^{As*}=1$, ${\Pi }_{p}^{As*}=B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}}{2a(1+{r}_{F})}$, ${\Pi }_{s}^{As*}=0$, ${CS}^{As*}=\frac{{v}^{2}(1-{k}_{A})}{2a(1+{r}_{F})}$, ${SW}^{As*}=B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a(1+{r}_{F})}$, respectively.

The results of Proposition 2 can be obtained by further sorting.□

1.2.3 Proof of Proposition 3

Single buyer’s credit: obviously there is $\frac{\partial {p}_{s}^{Ns*}}{\partial {k}_{A}}\le 0$; when

$$B\le \mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right),\frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}<0;$$

when

$$\mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right)<B\le \mathrm{max}\left(\mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right),\frac{cv}{2a(1+{r}_{F})}\right),\frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}<0$$

and the inequations

$$\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}>\frac{(2{k}_{A}-1)v}{{k}_{A}}>\frac{(2{k}_{A}-1)v}{{k}_{A}}-\left(1+{r}_{F}\right)c$$

holds. When

$$\mathrm{max}\left(\mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right),\frac{cv}{2a(1+{r}_{F})}\right)<B\le \frac{cv}{a(1+{r}_{F})},$$

we can get

$$\left\{\begin{array}{ll}\frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}<0, &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}>0, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

while $B>\frac{cv}{a(1+{r}_{F})}$, there is

$$\left\{\begin{array}{ll}\frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}<0 ,& 0<{k}_{A}\le \min\left(\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})},1\right)\\ \frac{\partial {p}_{p}^{Ns*}}{\partial {k}_{A}}>0 , &\min\left(\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})},1\right)<{k}_{A}<1\end{array}\right..$$

Dual credit: $\frac{\partial {p}_{s}^{As*}}{\partial {k}_{A}}\le 0$ obviously holds; when

$$B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})},\frac{\partial {p}_{p}^{As{\dag}}}{\partial {k}_{A}}=\frac{2a\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-v}{{k}_{A}^{2}}>0,$$

and thus we have

$$\left\{\begin{array}{ll}\frac{\partial {p}_{p}^{As*}}{\partial {k}_{A}}<0, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {p}_{p}^{As*}}{\partial {k}_{A}}>0, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.;$$

when

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)},\frac{\partial {p}_{p}^{As{\ddag} }}{\partial {k}_{A}}=\frac{B\left(cv-aB\left(1+{r}_{F}\right)\right)}{(c-B)c{k}_{A}^{2}}>0,$$

we get

$$\left\{\begin{array}{ll}\frac{\partial {p}_{p}^{As*}}{\partial {k}_{A}}<0,&0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {p}_{p}^{As*}}{\partial {k}_{A}}>0,&\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right..$$

By merging and organizing, we get the content of Proposition 3.□

1.2.4 Proof of Proposition 4

Single buyer’s credit: By collapsing, the equilibrium result of the platform and the seller in Proposition 1 can be further expressed as the following result.

When $\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1$ and $\frac{2c(1+{r}_{F})}{1-2{r}_{F}}<v<a\left(1+{r}_{F}\right)$, the equilibrium strategies are

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c, &\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c ,& B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.,$$

(B42)

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& \frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ {v}^{+} , &B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right..$$

(B43)

In other cases, we have ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}-\left(1+{r}_{F}\right)c$ and ${p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$.

Based on the above rearrangement results, we easily get that when conditions

$$\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)},\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1$$

and $\frac{2c(1+{r}_{F})}{1-2{r}_{F}}<v<a\left(1+{r}_{F}\right)$) are all met, $B$ has an effect on the equilibrium results and $\frac{\partial {p}_{p}^{Ns*}}{\partial B}<0$; $\frac{\partial {p}_{s}^{Ns*}}{\partial B}<0$.

Dual credit: $\frac{\partial {p}_{s}^{As*}}{\partial B}\le 0$ obviously holds. When $B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}$, we have

$$\left\{\begin{array}{ll}\frac{\partial {p}_{p}^{As*}}{\partial B}<0,& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {p}_{p}^{As*}}{\partial B}=0, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right..$$

$$\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}=-\frac{(1-{k}_{A})(aB(B-2c)\left(1+{r}_{F}\right)+{c}^{2}v)}{{(B-c)}^{2}c{k}_{A}},$$

if

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)},\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}>0,$$

and we can get ${p}_{p}^{As{\ddag} }={p}_{p}^{As{\dag}}$ at the point

$$B=c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)},$$

when $B=\frac{vc}{a\left(1+{r}_{F}\right)},$ ${p}_{p}^{As{\ddag} }=v-\left(1+{r}_{s}\right)c<v-\left(1+{r}_{s}\right)\left(c-B\right)$, and thus ${p}_{p}^{As{\dag}}\le {p}_{p}^{As{\ddag} }<v-\left(1+{r}_{s}\right)\left(c-B\right)$.

By combining and organizing, we get Proposition 4.□

1.2.5 Proof of Proposition 5

Single buyer’s credit: According to Proposition 1, when ${k}_{A}^{\varsigma }<{k}_{A}<1$, ${k}_{A}$ affects the equilibrium profits of the agents if

$$\mathrm{min}\left(\frac{cv}{a\left(1+{r}_{F}\right)},\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}\right)<B\le \frac{cv}{a(1+{r}_{F})},$$

and

$$\frac{\partial {\Pi }_{p}^{Ns*}}{\partial {k}_{A}}=\frac{B(aB\left(1-{r}_{F}^{2}\right)+cv{r}_{F})}{{c}^{2}{k}_{A}^{2}}>0;$$

$$\frac{\partial {\Pi }_{s}^{Ns*}}{\partial {k}_{A}}=-\frac{a{B}^{2}\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}^{2}}<0;$$

$$\frac{\partial {CS}^{Ns*}}{\partial {k}_{A}}=-\frac{a{B}^{2}\left(1+{r}_{F}\right)}{2{c}^{2}}<0;$$

$$\frac{\partial {SW}^{Ns*}}{\partial {k}_{A}}=\frac{B\left(2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)-aB{k}_{A}^{2}(1+{r}_{F})\right)}{2{c}^{2}{k}_{A}^{2}},$$

when

$$\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}\le \sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)}},$$

there is $\frac{\partial {SW}^{Ns*}}{\partial {k}_{A}}\ge 0$; when

$$\mathrm{max}\left(\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},\sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)}}\right)<{k}_{A}<1,$$

there is $\frac{\partial {SW}^{Ns*}}{\partial {k}_{A}}<0$. If $B>\frac{cv}{a(1+{r}_{F})}$, then we have

$$\frac{\partial {\Pi }_{p}^{Ns*}}{\partial {k}_{A}}=\frac{{v}^{2}}{a{k}_{A}^{2}(1+{r}_{F})}>0;$$

$$\frac{\partial {\Pi }_{s}^{Ns*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{a{k}_{A}^{2}(1+{r}_{F})}<0;$$

$$\frac{\partial {CS}^{Ns*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a(1+{r}_{F})}<0;$$

$$\frac{\partial {SW}^{Ns*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0.$$

Dual credit: If $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, we get

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}=\frac{B{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{{c}^{2}{k}_{A}^{2}}>0, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}=\frac{\partial {p}_{p}^{As*}}{\partial {k}_{A}}>0 , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

and

$${{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)|}_{0<{k}_{A}<\mathrm{min}\left({\tilde{k}}_{A},1\right)}<{{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)|}_{\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1}<{{\Pi }_{p}^{As*}\left({p}_{p}^{As{\dag}}\right)|}_{\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1},$$

thus$\frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}>0$. The first order derivative of the seller’s profit with respect to ${k}_{A}$ is then

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{s}^{As*}}{\partial {k}_{A}}=0 , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {\Pi }_{s}^{As*}}{\partial {k}_{A}}=-\frac{\partial {p}_{p}^{As{\dag}}}{\partial {k}_{A}}<0, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

moreover,

$$\frac{\partial {{\Pi }_{s}^{As*}|}_{\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1}}{\partial v}=\frac{(1-{k}_{A})\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}}>0,$$

$${{\Pi }_{s}^{As*}|}_{\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1}>B\left(1+{r}_{s}\right)\ge {{\Pi }_{s}^{As*}|}_{0<{k}_{A}<\mathrm{min}\left({\tilde{k}}_{A},1\right)}.$$

The first order derivative of consumer surplus with respect to ${k}_{A}$ is

$$\left\{\begin{array}{ll}\frac{\partial {CS}^{As*}}{\partial {k}_{A}}=-\frac{a{B}^{2}\left(1+{r}_{F}\right)}{2{c}^{2}}<0, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {CS}^{As*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0 , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

and $\mathrm{min}\left({\tilde{k}}_{A},1\right)$ is an discontinuous point and at this point

$${\frac{a{B}^{2}(1-{k}_{A})(1+{r}_{F})}{2{c}^{2}}|}_{{k}_{A}=\mathrm{min}\left({\tilde{k}}_{A},1\right)}<{\frac{{v}^{2}(1-{k}_{A})}{2a(1+{r}_{F})}|}_{{k}_{A}=\mathrm{min}\left({\tilde{k}}_{A},1\right)}.$$

The first order derivative of social welfare with respect to ${k}_{A}$ is

$$\left\{\begin{array}{ll}\frac{\partial {SW}^{As*}}{\partial {k}_{A}}=\frac{B\left(2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}^{2}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {SW}^{As*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0 ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

and $\frac{\partial {SW}^{As*}}{\partial {k}_{A}}\ge 0$ when $0<{k}_{A}\le \sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)}}$; when

$$\mathrm{min}\left(\sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)},}{\tilde{k}}_{A},1\right)<{k}_{A}<\mathrm{min}\left({\tilde{k}}_{A},1\right),\frac{\partial {SW}^{As*}}{\partial {k}_{A}}<0.$$

Due to${{\Pi }_{p}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{-}}={{\Pi }_{p}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{+}}$, ${{\Pi }_{s}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{-}}<{{\Pi }_{s}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{+}}$),

$${{CS}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{-}}<\frac{a(1-{\tilde{k}}_{A})(1+{r}_{F})}{2{c}^{2}}\frac{{c}^{2}{v}^{2}}{{\left(a\left(1+{r}_{F}\right)\right)}^{2}}=\frac{{v}^{2}(1-{\tilde{k}}_{A})}{2a(1+{r}_{F})}={{CS}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{+}},$$

thus ${{SW}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{-}}<{{SW}^{As*}|}_{{k}_{A}\to {{\tilde{k}}_{A}}^{+}}$, as long as

$${k}_{A}>\mathrm{ min}\left(\sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)},}{\tilde{k}}_{A},1\right),$$

we have $\frac{\partial {SW}^{As*}}{\partial {k}_{A}}<0$.

If $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, we have

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}=\frac{B{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{{c}^{2}{k}_{A}^{2}}>0,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}=\frac{\partial {p}_{p}^{As{\ddag} }}{\partial {k}_{A}}>0 , &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

and

$${{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)|}_{0<{k}_{A}<\mathrm{min}\left({\breve{k}},1\right)}<{{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)|}_{\mathrm{min}\left({\breve{k}},1\right)\le {k}_{A}<1}<{{\Pi }_{p}^{As*}\left({p}_{p}^{As{\ddag} }\right)|}_{\mathrm{min}\left({\breve{k}},1\right)\le {k}_{A}<1},$$

hence there is $\frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}>0$ throughout the domain of definition; for the seller, we have

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{s}^{As*}}{\partial {k}_{A}}=0 , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {\Pi }_{s}^{As*}}{\partial {k}_{A}}=-\frac{\partial {p}_{p}^{As{\ddag} }}{\partial {k}_{A}}<0, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

and similarly,

$$\left\{\begin{array}{ll}\frac{\partial {CS}^{As*}}{\partial {k}_{A}}=-\frac{a{B}^{2}\left(1+{r}_{F}\right)}{2{c}^{2}}<0, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {CS}^{As*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0 ,& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,\mathrm{min}\left({\breve{k}},1\right)$$

is also the discontinuous point, and the first-order derivative of social welfare with respect to ${k}_{A}$ is

$$\left\{\begin{array}{ll}\frac{\partial {SW}^{As*}}{\partial {k}_{A}}=\frac{B\left(2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}^{2}},& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {SW}^{As*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0 , &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

when $0<{k}_{A}\le \sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)}}$, $\frac{\partial {SW}^{As*}}{\partial {k}_{A}}\ge 0$; when

$${k}_{A}>\mathrm{min}\left(\sqrt{\frac{2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)}{aB\left(1+{r}_{F}\right)},}{\breve{k}},1\right),\frac{\partial {SW}^{As*}}{\partial {k}_{A}}<0.$$

If $B>\frac{vc}{a\left(1+{r}_{F}\right)}$, $\frac{\partial {\Pi }_{p}^{As*}}{\partial {k}_{A}}=\frac{\partial {\Pi }_{s}^{As*}}{\partial {k}_{A}}=0$ and $\frac{\partial {CS}^{As*}}{\partial {k}_{A}}=\frac{\partial {SW}^{As*}}{\partial {k}_{A}}=-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<0$.

By combining and organizing, we get Proposition 5.□

1.2.6 Proof of Proposition 6

Single buyer’s credit: According to Eqs. (B42) and (B43), we can obtain the equilibrium demand and the profits of each subject, and the results are as follows.

$${D}^{Ns*}=\left\{\begin{array}{ll}0 ,& B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{B}{c} , &\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{v}{a\left(1+{r}_{F}\right)} , &B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.$$

(B44)

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 ,& B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}},& \frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{v\left({k}_{A}\left(3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}\right)-2v\right)}{2a{k}_{A}\left(1+{r}_{F}\right)} , &B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.$$

(B45)

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B ,& B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}}, &\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{a{k}_{A}\left(1+{r}_{F}\right)}+\left(1+{r}_{F}\right)B, &B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.$$

(B46)

$$\begin{aligned}&{CS}^{Ns*}\\&\quad=\left\{\begin{array}{ll}0 , &B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{1}{a}{\int }_{0}^{\frac{v-{k}_{A}{p}_{s}^{Ns*}}{\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}}\left[v+{k}_{A}\left(1+{r}_{F}\right){x}_{i}-{k}_{A}{p}_{s}^{Ns*}-\left(1+{r}_{F}\right){x}_{i}\right]d{x}_{i},& B>\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}\end{array}\right.\\&\quad=\left\{\begin{array}{ll}0 ,& B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}}, &\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)} ,& B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.\end{aligned}$$

(B47)

$${SW}^{Ns*}=\left\{\begin{array}{ll}\left(1+{r}_{F}\right)B , &B\le \frac{2c(v{r}_{F}+c{k}_{A}(1+{r}_{F}))}{a(1+{r}_{F})({k}_{A}(3-2{r}_{F})-2(1-{r}_{F}))}\\ \frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}},& \frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ \frac{v\left(v\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)-2c\left(1+{r}_{F}\right)\right)}{2a\left(1+{r}_{F}\right)}+\left(1+{r}_{F}\right)B , &B>\frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.$$

(B48)

In the other cases, we have ${\Pi }_{p}^{Ns*}={CS}^{Ns*}=0$; ${\Pi }_{s}^{Ns*}={SW}^{Ns*}=\left(1+{r}_{F}\right)B$.

According to the above results, $B$ has an impact on platform profit as well as consumer surplus if and only if

$$\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)},$$

and

$$\frac{\partial {\Pi }_{p}^{Ns*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)-c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{{c}^{2}{k}_{A}}>0,$$

what’s more if

$$B=\frac{cv}{a\left(1+{r}_{F}\right)},\frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}}=\frac{v({k}_{A}(3v-2c(1+{r}_{F})-2v{r}_{F})-2v)}{2a{k}_{A}(1+{r}_{F})},$$

thus $\frac{\partial {\Pi }_{p}^{Ns*}}{\partial B}\ge 0$. The first-order derivatives of seller’s profit and consumer surplus with respect to $B$ are $\frac{\partial {\Pi }_{s}^{Ns*}}{\partial B}=\frac{(2aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A})(1+{r}_{F})}{{c}^{2}{k}_{A}}>0$; $\frac{\partial {CS}^{Ns*}}{\partial B}=\frac{aB\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}}>0$, and at the point $B=\frac{cv}{a\left(1+{r}_{F}\right)}$, $\frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}}=\frac{{v}^{2}\left(1-{k}_{A}\right)}{a{k}_{A}\left(1+{r}_{F}\right)}+(1+{r}_{F})B$; $\frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}}=\frac{{v}^{2}(1-{k}_{A})}{2a(1+{r}_{F})}$, therefore $\frac{\partial {\Pi }_{s}^{Ns*}}{\partial B}>0$, $\frac{\partial {CS}^{Ns*}}{\partial B}\ge 0$. The first-order derivative of social welfare with respect to $B$ is

$$\frac{\partial {SW}^{Ns*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}}>\frac{Ba\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)-c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{{c}^{2}{k}_{A}}>0.$$

When $B>\frac{cv}{a\left(1+{r}_{F}\right)}$, $B$ only has an effect on the seller’s profit as well as social welfare, and $\frac{\partial {\Pi }_{s}^{Ns*}}{\partial B}=\frac{\partial {SW}^{Ns*}}{\partial B}=1+{r}_{F}>0$. When $B=\frac{cv}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{p}^{Ns*}, {\Pi }_{s}^{Ns*}, {CS}^{Ns*}, {SW}^{Ns*}$ are all continuous functions, and thus for $B>\frac{2c\left(v{r}_{F}+c{k}_{A}\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)\left({k}_{A}\left(3-2{r}_{F}\right)-2\left(1-{r}_{F}\right)\right)}$, we have $\frac{\partial {\Pi }_{s}^{Ns*}}{\partial B}>0$ and $\frac{\partial {SW}^{Ns*}}{\partial B}>0$.

Dual credit: If $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, the first order derivative of the platform profit with respect to $B$ is

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{p}^{As*}}{\partial B}=\frac{{r}_{F}\left(2aB-cv+2aB{r}_{F}\right)+aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)}{{c}^{2}{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {\Pi }_{p}^{As*}}{\partial B}={r}_{F}-{r}_{s}<0 ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right..$$

When $0<{k}_{A}<\mathrm{min}\left({\tilde{k}}_{A},1\right)$, there exists $\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))}$, and $\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))}<c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}$, if $0<B<\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))}$, there is $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}<0$, if $B>\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))}$, $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}>0$ holds. And given $B$, ${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)>{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)$. Further, since ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)$ is a decreasing function of $B$, we can get

$${{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)|}_{B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}}<{{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)|}_{c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}}<{{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)|}_{c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}}.$$

$\frac{\partial {\Pi }_{s}^{As*}}{\partial B}\ge 0$ clearly holds and ${p}_{p}^{As{\ddag} }={p}_{p}^{As{\dag}}$ when $B=c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}$, and thus

$${{\Pi }_{s}^{As*}|}_{B={\left(c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)}^{-}}<{{\Pi }_{s}^{As*}|}_{B={\left(c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)}^{+}}.$$

Similarly, $\frac{\partial {CS}^{As*}}{\partial B}\ge 0$ holds; while the first-order derivative of social welfare with respect to $B$ is

$$\left\{\begin{array}{ll}\frac{\partial {SW}^{As*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \frac{\partial {SW}^{As*}}{\partial B}=1+{r}_{F}>0 ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

there exists

$$\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}\left(2-{k}_{A}-2{r}_{F}\right)+2{r}_{F})}<\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))},$$

$$\frac{\partial {SW}^{As*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}}<0$$

for $0<B<\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}\left(2-{k}_{A}-2{r}_{F}\right)+2{r}_{F})}$, and

$$\frac{\partial {SW}^{As*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}}>0$$

if $B>\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}\left(2-{k}_{A}-2{r}_{F}\right)+2{r}_{F})}$.

If $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, then we have

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{p}^{As*}}{\partial B}=\frac{{r}_{F}\left(2aB-cv+2aB{r}_{F}\right)+aB{k}_{A}(1-{r}_{F}-2{r}_{F}^{2})}{{c}^{2}{k}_{A}},&0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {\Pi }_{p}^{As*}}{\partial B}={r}_{F}-{r}_{s}-\frac{(1-{k}_{A})(aB(B-2c)\left(1+{r}_{F}\right)+{c}^{2}v)}{{(B-c)}^{2}c{k}_{A}},&\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

and within the range of $0<{k}_{A}<\mathrm{min}\left({\breve{k}},1\right)$, due to $B>c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}>\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}+2{r}_{F}(1-{k}_{A}))}$, we get $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}>0$. And in the range of $\mathrm{min}\left({\breve{k}},1\right)\le {k}_{A}<$ 1, the sign of $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}$ is related to $B$. Specifically, when

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B<c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)},$$

$\frac{\partial {\Pi }_{p}^{As*}}{\partial B}=\frac{{r}_{F}\left(2aB-cv+2aB{r}_{F}\right)+aB{k}_{A}(1-{r}_{F}-2{r}_{F}^{2})}{{c}^{2}{k}_{A}}<0$; while

$$B>c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)},$$

there is $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}>0$. Since ${p}_{p}^{As{\ddag} }<v-\left(1+{r}_{s}\right)\left(c-B\right)$, the seller’s price is ${v}^{-}$ and thus we have

$${{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)|}_{c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}}<{{\Pi }_{p}^{As}\left({p}_{p}^{As*}\right)|}_{c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}}=B-c+v+\left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}<{{\Pi }_{p}^{As}\left({p}_{p}^{As*}\right)|}_{B>\frac{vc}{a\left(1+{r}_{F}\right)}}.$$

The first-order derivative of the seller’s profit with respect to $B$ is

$$\left\{\begin{array}{ll}\frac{\partial {\Pi }_{s}^{As*}}{\partial B}=0 , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {\Pi }_{s}^{As*}}{\partial B}=-\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}+1+{r}_{s},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

there exists

$$c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)}.$$

Since $\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}>0$ and $1+{r}_{s}>{r}_{s}-{r}_{F}$, we get

$$c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)}>c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left({r}_{s}-{r}_{F}\right)\right)}.$$

If

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B<c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)},$$

there is $\frac{\partial {\Pi }_{s}^{As*}}{\partial B}=-\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}+1+{r}_{s}>0$; while

$$B>c-\frac{c\sqrt{\left(1-{k}_{A}\right)\left(a\left(1+{r}_{F}\right)-v\right)\left(a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)\right)}}{a\left(1+{r}_{F}\right)-{k}_{A}\left(a\left(1+{r}_{F}\right)-c\left(1+{r}_{s}\right)\right)},$$

$\frac{\partial {\Pi }_{s}^{As*}}{\partial B}=-\frac{\partial {p}_{p}^{As{\ddag} }}{\partial B}+1+{r}_{s}<0$ holds. The first-order derivative of consumer surplus with respect to $B$ is constant with $\frac{\partial {CS}^{As*}}{\partial B}\ge 0$; while the first-order derivative of social welfare with respect to $B$ is

$$\left\{\begin{array}{ll}\frac{\partial {SW}^{As*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}},& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{\partial {SW}^{As*}}{\partial B}=1+{r}_{F}>0 , &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

because

$$B>c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}>\frac{cv{r}_{F}}{a(1+{r}_{F})({k}_{A}\left(2-{k}_{A}-2{r}_{F}\right)+2{r}_{F})}$$

holds, we can get

$$\frac{\partial {SW}^{As*}}{\partial B}=\frac{Ba\left(1+{r}_{F}\right)\left(2{r}_{F}+{k}_{A}\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)\right)-cv{r}_{F}}{{c}^{2}{k}_{A}}>0,$$

and thus given ${k}_{A}$,$\frac{\partial {SW}^{As*}}{\partial B}>0$.

If $B>\frac{vc}{a\left(1+{r}_{F}\right)}$, we have $\frac{\partial {\Pi }_{p}^{As*}}{\partial B}>0;\frac{\partial {SW}^{As*}}{\partial B}>0;\frac{\partial {CS}^{As*}}{\partial B}\ge 0$.

By further simplifying and organizing, we get Proposition 6.□

1.2.7 Proof of Proposition 7

Before proceeding to the proof, we first compare the key demarcation points.

(3.1) $\frac{cv}{a\left(1+{r}_{F}\right)}$ and $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}$

$$\frac{cv}{a\left(1+{r}_{F}\right)}-\left(c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)=\frac{c\left(v+\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}-a\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)},$$

and the difference between them with respect to the second order derivative of $v$ is

$$\frac{{\partial }^{2}}{\partial {v}^{2}}\left(\frac{cv}{a\left(1+{r}_{F}\right)}-\left(c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}\right)\right)=-\frac{ac\left(1+{r}_{F}\right)}{4{\left(a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)\right)}^{3/2}}<0,$$

and at points $v=0=a\left(1+{r}_{F}\right)$, $\frac{cv}{a\left(1+{r}_{F}\right)}-\left(c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)=0$, and thus $\frac{cv}{a\left(1+{r}_{F}\right)}>\left(c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)$.

(3.2) $\frac{cv}{a\left(1+{r}_{F}\right)}$ and $\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}$

$$\frac{cv}{a\left(1+{r}_{F}\right)}-\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}=\frac{c\left(\left(1-2{r}_{F}\right)v-2c\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)},$$

in general ${r}_{F}\approx 0.05<0.5$, thus $\frac{c\left(\left(1-2{r}_{F}\right)v-2c\left(1+{r}_{F}\right)\right)}{a\left(1+{r}_{F}\right)}$ is an increasing function of $v$ and $\frac{cv}{a\left(1+{r}_{F}\right)}\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$ for $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, otherwise, $\frac{cv}{a\left(1+{r}_{F}\right)}>\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$.

(3.3) $\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}$ and $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$

$$\frac{{\partial }^{2}}{\partial {v}^{2}}\left(\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}-\left(c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}\right)\right)=-\frac{ac\left(1+{r}_{F}\right)}{4{\left(a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)\right)}^{3/2}}<0,$$

and if ${r}_{F}<0.25$, which is a reasonable assumption,

$$\frac{\partial }{\partial v}\left(\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}-\left(c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}\right)\right)<-\frac{c\left(1-4{r}_{F}\right)}{2a\left(1+{r}_{F}\right)}<0,$$

and at points

$$v={v}^{r{\dag}}=\frac{a{r}_{F}(3+4{r}_{F})-a-8c{r}_{F}(1+{r}_{F})+(1+{r}_{F})\sqrt{a\left(a-8a{r}_{F}\left(1-2{r}_{F}\right)+16c{r}_{F}\right)}}{8{r}_{F}^{2}},$$

we have $\frac{2({c}^{2}(1+{r}_{F})+cv{r}_{F})}{a(1+{r}_{F})}=\left(c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}\right)$, thus there is

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<\frac{cv}{a\left(1+{r}_{F}\right)}$$

if the range of $\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}<v\le \mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)$ is met; and if

$$\mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)<v\le a\left(1+{r}_{F}\right),\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<\frac{cv}{a\left(1+{r}_{F}\right)}$$

holds.

We summarize the results of the above analysis in the following Table 2.

Table 2 Comparison of key demarcation points

Then, we compare the equilibrium strategies.

(3.a) $B>\frac{vc}{a\left(1+{r}_{F}\right)}$. (3.a.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, the equilibrium strategies of PSSC in buyer’s credit model are ${p}_{p}^{Ns*}={p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$, respectively, while the equilibrium strategies of both parties in the dual credit model are ${p}_{p}^{As*}=v-\left(1+{r}_{s}\right)\left(c-B\right)$ and ${p}_{s}^{As*}={v}^{-}$. Clearly ${p}_{p}^{Ns*}>{p}_{p}^{As*}$; ${p}_{s}^{Ns*}\ge {p}_{s}^{As*}$. (3.a.2) When $v>\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, the equilibrium strategies in the buyer’s credit model are

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} ,& 0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}}, &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ {v}^{+},& \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right..$$

And both sides’ equilibrium strategies in dual credit model are then ${p}_{p}^{As*}=v-\left(1+{r}_{s}\right)\left(c-B\right)$ and ${p}_{s}^{As*}={v}^{-}$. Clearly ${p}_{s}^{Ns*}\ge {p}_{s}^{As*}$, and we focus on the comparison of platform charges. Since $\frac{v}{{k}_{A}}>v-\left(1+{r}_{s}\right)\left(c-B\right)$ and $\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c-\left(v-\left(1+{r}_{s}\right)\left(c-B\right)\right)$ is an increasing function of ${k}_{A}$, and at the point ${k}_{A}=\frac{v}{v-\left(1+{r}_{s}\right)B+c\left({r}_{s}-{r}_{F}\right)}$,$\frac{\left(2{k}_{A}-1\right)v}{{k}_{A}}-\left(1+{r}_{F}\right)c=v-\left(1+{r}_{s}\right)\left(c-B\right)$, the comparison of equilibrium charges in two credit models is.

$$\left\{\begin{array}{ll}{p}_{p}^{Ns*}>{p}_{p}^{As*}, &otherwise\\ {p}_{p}^{Ns*}\le {p}_{p}^{As*},& \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\le {k}_{A}\le \max\left(\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})},\mathrm{min}\left(\frac{v}{v-\left(1+{r}_{s}\right)B+c\left({r}_{s}-{r}_{F}\right)},1\right)\right)\end{array}\right..$$

(B49)

(3.a.1) and (3.a.2) are combined and collated to obtain comparison results as

$$\left\{\begin{array}{ll}{p}_{p}^{Ns*}>{p}_{p}^{As*},& otherwise\\ {p}_{p}^{Ns*}\le {p}_{p}^{As*},& {k}_{A}^{\varsigma }\le {k}_{A}\le \max\left({k}_{A}^{\varsigma },\mathrm{min}\left(\frac{v}{v-\left(1+{r}_{s}\right)B+c\left({r}_{s}-{r}_{F}\right)},1\right)\right)\end{array}\right..$$

(B50)

(3.b) $B\le \frac{vc}{a\left(1+{r}_{F}\right)}$. (3.b.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, according to Table 2, we can easily classify the equilibrium strategies in two credit models based on the size of seller’s capital. Specifically, if $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}; {p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$;

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {p}_{p}^{As{\dag}} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {v}^{-} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right..$$

Since $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}<\frac{v}{{k}_{A}}$,

$${p}_{p}^{As{\dag}}<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}<\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{F}\right)}{c{k}_{A}}<\frac{v}{{k}_{A}},$$

we can get ${p}_{p}^{Ns*}>{p}_{p}^{As*}$, while ${p}_{s}^{Ns*}>{p}_{s}^{As*}$ clearly holds. If $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}; {p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$;

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {p}_{p}^{As{\ddag} } ,&\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {v}^{-} , &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right..$$

$${p}_{p}^{As{\ddag} }-\left(\frac{v}{{k}_{A}}-\left(1+{r}_{s}\right)c\right)=\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}v-\left(c-B\right)cv}{\left(c-B\right)c{k}_{A}}=\frac{\left(1-{k}_{A}\right)\left(a{B}^{2}\left(1+{r}_{F}\right)-{c}^{2}v\right)}{\left(c-B\right)c{k}_{A}}<0,$$

thus ${p}_{p}^{As{\ddag} }<\frac{v}{{k}_{A}}-\left(1+{r}_{s}\right)c<\frac{v}{{k}_{A}}$, and ${p}_{p}^{Ns*}>{p}_{p}^{As*}$ holds, ${p}_{s}^{Ns*}>{p}_{s}^{As*}$ still holds constant.

(3.b.2) When $\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}<v\le \mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)$, if $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}; {p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$;

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {p}_{p}^{As{\dag}} ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {v}^{-} ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

The comparison results are the same as in case (3.b.1), i.e., ${p}_{p}^{Ns*}>{p}_{p}^{As*}$; ${p}_{s}^{Ns*}>{p}_{s}^{As*}$. If

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},$$

${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}; {p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$;

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {p}_{p}^{As{\ddag} } ,& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {v}^{-} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

and similarly there are ${p}_{p}^{Ns*}>{p}_{p}^{As*}$ and ${p}_{s}^{Ns*}>{p}_{s}^{As*}$. If $\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$,

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {p}_{p}^{As{\ddag} } ,& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ {v}^{-} ,&\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

obviously ${p}_{s}^{Ns*}>{p}_{s}^{As*}$, $\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}>\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c$, and

$${p}_{p}^{As{\ddag} }-\left[\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right]=\frac{\left[aB\left(1+{r}_{F}\right)(2c-B)-{c}^{2}v\right]\left(1-{k}_{A}\right)}{(c-B)c{k}_{A}}-\left({r}_{s}-{r}_{F}\right)c,$$

since $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$, $aB\left(1+{r}_{F}\right)\left(2c-B\right)-{c}^{2}v>0$ holds when

$$\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}<c+\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)},$$

i.e., ${p}_{p}^{As{\ddag} }-\left[\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right]$ is a decreasing function of ${k}_{A}$, and at the point ${k}_{A}=\frac{aB\left(2c-B\right)\left(1+{r}_{F}\right)-{c}^{2}v}{aB\left(2c-B\right)\left(1+{r}_{F}\right)-{c}^{2}v+{c}^{2}(c-B)\left({r}_{s}-{r}_{F}\right)}$, ${p}_{p}^{As{\ddag} }=\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c$, therefore when $\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$, the comparison results of the equilibrium charges are

$$ \begin{aligned} & \left\{ {\begin{array}{*{20}l} {p_{p}^{{Ns*}} > p_{p}^{{As*}} ,} \hfill & {\quad {\text{otherwise}}} \hfill \\ {p_{p}^{{Ns*}} \le p_{p}^{{As*}} ,} \hfill & {\quad \frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}} < k_{A} \le {\text{max}}\left( {\frac{{aB\left( {2c - B} \right)\left( {1 + r_{F} } \right) - c^{2} v}}{{aB\left( {2c - B} \right)\left( {1 + r_{F} } \right) - c^{2} v + c^{2} \left( {c - B} \right)\left( {r_{s} - r_{F} } \right)}},\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},{\text{min}}\left( {{\breve{k}} _{A} ,1} \right)} \right)} \hfill \\ \end{array} } \right. \\ & \quad = \left\{ {\begin{array}{*{20}l} {p_{p}^{{Ns*}} > p_{p}^{{As*}} ,} \hfill & {\quad {\text{otherwise}}} \hfill \\ {p_{p}^{{Ns*}} \le p_{p}^{{As*}} ,} \hfill & {\quad k_{A}^{\varsigma } < k_{A} \le {\text{max}}\left( {\frac{{aB\left( {2c - B} \right)\left( {1 + r_{F} } \right) - c^{2} v}}{{aB\left( {2c - B} \right)\left( {1 + r_{F} } \right) - c^{2} v + c^{2} \left( {c - B} \right)\left( {r_{s} - r_{F} } \right)}},k_{A}^{\varsigma } ,k_{A}^{\tau } } \right)} \hfill \\ \end{array} } \right.. \\ \end{aligned} $$

(B51)

(3.b.3) When $\mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)<v\le a\left(1+{r}_{F}\right)$, if $B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$, ${p}_{p}^{Ns*}=\frac{v}{{k}_{A}}; {p}_{s}^{Ns*}=\frac{v}{{k}_{A}}$;

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {p}_{p}^{As{\dag}} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {v}^{-} ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

there exists ${p}_{p}^{Ns*}>{p}_{p}^{As*}$; ${p}_{s}^{Ns*}>{p}_{s}^{As*}$. If

$$\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)},$$

$${p}_{p}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{Ns*}=\left\{\begin{array}{ll}\frac{v}{{k}_{A}} , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.;$$

$${p}_{p}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},&0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {p}_{p}^{As{\dag}},&\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.;$$

$${p}_{s}^{As*}=\left\{\begin{array}{ll}\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ {v}^{-} , &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

$$\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}-v=\frac{\left(1-{k}_{A}\right)\left[cv-a\left(1+{r}_{F}\right)B\right]}{c{k}_{A}}>0,$$

so we have ${p}_{s}^{Ns*}\ge {p}_{s}^{As*}$. Since

$${p}_{p}^{As{\dag}}<{p}_{p}^{As}\le \frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{s}\right)}{c{k}_{A}}<\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-{k}_{A}{c}^{2}\left(1+{r}_{F}\right)}{c{k}_{A}}<\frac{v}{{k}_{A}},$$

the comparison results are

$$\left\{\begin{array}{ll}{p}_{p}^{Ns*}>{p}_{p}^{As*},& otherwise\\ {p}_{p}^{Ns*}\le {p}_{p}^{As*}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\le {k}_{A}\le \min\left({\tilde{k}}_{A},1\right)\end{array}\right.=\left\{\begin{array}{ll}{p}_{p}^{Ns*}>{p}_{p}^{As*}, &otherwise\\ {p}_{p}^{Ns*}\le {p}_{p}^{As*}, &{k}_{A}^{\varsigma }\le {k}_{A}\le \max\left({k}_{A}^{\varsigma },{k}_{A}^{\tau }\right)\end{array}\right..$$

(B52)

If $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, the result of such case is shown in Eq. (B51).

Denote

$${k}_{A}^{\nu }=\left\{\begin{array}{ll}\max\left({k}_{A}^{\varsigma }{k}_{A}^{\tau }\right) , &\min\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)<B\le \max\left(\mathrm{min}\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right),c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)\\ \max\left(\frac{aB\left(2c-B\right)\left(1+{r}_{F}\right)-{c}^{2}v}{aB\left(2c-B\right)\left(1+{r}_{F}\right)-{c}^{2}v+{c}^{2}\left(c-B\right)\left({r}_{s}-{r}_{F}\right)},{k}_{A}^{\varsigma },{k}_{A}^{\tau }\right), &\max\left(\mathrm{min}\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right),c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\right)<B\le \frac{vc}{a\left(1+{r}_{F}\right)}\\ \max\left({k}_{A}^{\varsigma },\mathrm{min}\left(\frac{v}{v-\left(1+{r}_{s}\right)B+c\left({r}_{s}-{r}_{F}\right)},1\right)\right) , &B>\frac{vc}{a\left(1+{r}_{F}\right)}\end{array}\right.,$$

and summarizing the results of the above analysis, we can obtain Proposition 7.□

1.2.8 Proof of Proposition 8

(4.a) $B>\frac{vc}{a\left(1+{r}_{F}\right)}$. (4.a.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, ${\Pi }_{p}^{Ns*}=0$, ${\Pi }_{p}^{As*}=B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}}{2a(1+{r}_{F})}$, in the interval of

$$\frac{vc}{a\left(1+{r}_{F}\right)}<B<\mathrm{max}\left(\frac{{v}^{2}+2a(1+{r}_{F})(c-v+(c+v){r}_{F})}{2a{(1+{r}_{F})}^{2}},\frac{vc}{a\left(1+{r}_{F}\right)}\right),$$

${\Pi }_{p}^{As*}<0$, and thus ${\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}$, if

$$B>\mathrm{max}\left(\frac{{v}^{2}+2a(1+{r}_{F})(c-v+(c+v){r}_{F})}{2a{(1+{r}_{F})}^{2}},\frac{vc}{a\left(1+{r}_{F}\right)}\right),$$

we have ${\Pi }_{p}^{As*}>{\Pi }_{p}^{Ns*}$. (4.a.2) When $v>\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$,

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{v\left({k}_{A}\left(3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}\right)-2v\right)}{2a{k}_{A}\left(1+{r}_{F}\right)}, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

${\Pi }_{p}^{As*}=B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}}{2a(1+{r}_{F})}$.

$${\Pi }_{p}^{As*}>\frac{2av+2cv-2ac-{v}^{2}+2c\left(-2a+v\right){r}_{F}-2a(c+v){r}_{F}^{2}}{2a(1+{r}_{F})}=Z\left(v\right)$$

and $Z\left(v\right)$ is a concave function of $v$. $Z\left(\frac{2c(1+{r}_{F})}{1-2{r}_{F}}\right)=\frac{c(1+{r}_{F})(a\left(1-2{r}_{F}\right)-4c{r}_{F})}{a{(1-2{r}_{F})}^{2}}$, $Z\left(a\left(1+{r}_{F}\right)\right)=\frac{1}{2}a(1+{r}_{F})(1-2{r}_{F})$, if ${r}_{F}<0.5$, $Z\left(\frac{2c(1+{r}_{F})}{1-2{r}_{F}}\right)>0, Z\left(a\left(1+{r}_{F}\right)\right)>0$, thus $Z\left(v\right)>0$ holds, i.e., ${\Pi }_{p}^{As*}>0$, at which time $\frac{{v}^{2}+2a(1+{r}_{F})(c-v+(c+v){r}_{F})}{2a{(1+{r}_{F})}^{2}}<\frac{vc}{a\left(1+{r}_{F}\right)}$. Besides, $\frac{\partial {\Pi }_{p}^{Ns*}}{\partial {k}_{A}}>0$ and ${\Pi }_{p}^{Ns*}=0$ at ${k}_{A}=\frac{2v}{3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}}$, thus ${\Pi }_{p}^{Ns*}\ge 0$.

$${\Pi }_{p}^{As*}-\frac{v\left({k}_{A}\left(3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}\right)-2v\right)}{2a{k}_{A}\left(1+{r}_{F}\right)}>{\Pi }_{p}^{As*}-\frac{v\left(v-2c\left(1+{r}_{F}\right)-2v{r}_{F}\right)}{2a\left(1+{r}_{F}\right)}=P\left(v\right),$$

since $P\left(v\right)$ is a concave function of $v$, we can obtain

$$P\left(v\right)\ge \mathrm{max}\left(P\left(a\left(1+{r}_{F}\right)\right),P\left(\frac{2c(1+{r}_{F})}{1-2{r}_{F}}\right)\right)=\mathrm{max}\left(B(1+{r}_{F},{\Pi }_{p}^{As*}\right)>0,$$

so there is a inequation ${\Pi }_{p}^{As*}>{\Pi }_{p}^{Ns*}$.

Thus, the comparisons of platform profits in the two credit models are

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}, &\frac{vc}{a\left(1+{r}_{F}\right)}<B<max\left(\frac{{v}^{2}+2a(1+{r}_{F})(c-v+(c+v){r}_{F})}{2a{(1+{r}_{F})}^{2}},\frac{vc}{a\left(1+{r}_{F}\right)}\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*}, &otherwise\end{array}\right..$$

(B53)

(4.b) $B\le \frac{vc}{a\left(1+{r}_{F}\right)}$. (4.b.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, and further if $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, ${\Pi }_{p}^{Ns*}=0$;

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}\left(1+{r}_{s}\right)}{{k}_{A}},& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

$$\frac{\partial {\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)}{\partial {k}_{A}}=\frac{{r}_{F}B(cv-aB\left(1+{r}_{F}\right))}{{c}^{2}{k}_{A}^{2}}>0$$

and

$$\underset{{k}_{A}\to 0}{\mathrm{lim}}{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=-\infty \ll 0,$$

thus there exists ${\hat{k}}_{A}^{{\dag}}=\frac{2{r}_{F}(cv-aB\left(1+{r}_{F}\right)}{aB(1-{r}_{F}-2{r}_{F}^{2})}$ such that ${\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=0$, meanwhile, for ${k}_{A}<{\hat{k}}_{A}^{{\dag}}$, ${\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)<0$, and ${\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)>0$ for ${k}_{A}>{\hat{k}}_{A}^{{\dag}}$. On the other hand,

$${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)=\left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a(1-{k}_{A})(1+{r}_{F})\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}(1+{r}_{s})}{{k}_{A}},$$

$$\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)}{\partial {k}_{A}}=-\frac{v-2a(1+{r}_{F})+2a(1+{r}_{F})\sqrt{1-\frac{v}{a+a{r}_{F}}}}{{k}_{A}^{2}},$$

$$\frac{{\partial }^{2}{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)}{\partial {k}_{A}\partial v}=-\frac{1}{{k}_{A}^{2}}\left(1-\frac{1}{\sqrt{1-\frac{v}{a+a{r}_{F}}}}\right)>0,$$

thus,

$$\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)}{\partial {k}_{A}}>{\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)}{\partial {k}_{A}}|}_{v=0}=0,$$

i.e., ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)$ is an increasing function of ${k}_{A}$, and $\underset{{k}_{A}\to 0}{\mathrm{lim}}{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)=-\infty \ll 0$, so there exists

$${\hat{k}}_{A}=\frac{2a(1+{r}_{F})\left[v+2a(1+{r}_{F})\sqrt{1-\frac{v}{a(1+{r}_{F})}}-2a(1+{r}_{F})\right]}{-4{a}^{2}+2ac+{v}^{2}+4{a}^{2}\sqrt{1-\frac{v}{a(1+{r}_{F})}}+2a{r}_{F}^{2}\left[-B+c+v+2a\left(-1+\sqrt{1-\frac{v}{a(1+{r}_{F})}}\right)\right]+2aB{r}_{s}+2a{r}_{F}\left[-B+2c+v+4a\left(-1+\sqrt{1-\frac{v}{a(1+{r}_{F})}}\right)+B{r}_{s}\right]}>0$$

such that ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)=0$. And ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)<0$ for ${k}_{A}<{\hat{k}}_{A}$, while ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)>0$ if ${k}_{A}>{\hat{k}}_{A}$. Furthermore,

$$\frac{\partial {\hat{k}}_{A}}{\partial B}=\frac{2a(1+{r}_{F})\left[v+2a(1+{r}_{F})\sqrt{1-\frac{v}{a(1+{r}_{F})}}-2a(1+{r}_{F})\right]\left(2a{r}_{s}\left(1+{r}_{s}\right)-2a{r}_{F}\left(1+{r}_{F}\right)\right)}{{\left(-4{a}^{2}+2ac+{v}^{2}+4{a}^{2}\sqrt{1-\frac{v}{a(1+{r}_{F})}}+2a{r}_{F}^{2}(-B+c+v+2a(-1+\sqrt{1-\frac{v}{a(1+{r}_{F})}}))+2aB{r}_{s}+2a{r}_{F}(-B+2c+v+4a(-1+\sqrt{1-\frac{v}{a(1+{r}_{F})}})+B{r}_{s})\right)}^{2}},$$

and since $v+2a\left(1+{r}_{F}\right)\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}-2a\left(1+{r}_{F}\right)\le 0$, with the assumption ${r}_{s}>{r}_{F}$, $\frac{\partial {\hat{k}}_{A}}{\partial B}\le 0$ holds, i.e., ${\hat{k}}_{A}$ is a decreasing function of$B$.

According to the definition of ${\tilde{k}}_{A}$, ${\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)={\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)$ if ${k}_{A}={\tilde{k}}_{A}$, thus when

$${\left({\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right)|}_{{k}_{A}={\tilde{k}}_{A}}=0,$$

the equation ${\hat{k}}_{A}^{{\dag}}={\hat{k}}_{A}={\tilde{k}}_{A}$ is right; when

$${\left({\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right)|}_{{k}_{A}={\tilde{k}}_{A}}>0,$$

we can get ${\hat{k}}_{A}^{{\dag}}<{\hat{k}}_{A}<{\tilde{k}}_{A}$; while

$${\left({\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)-{\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)\right)|}_{{k}_{A}={\tilde{k}}_{A}}<0,$$

there exists ${\tilde{k}}_{A}<{\hat{k}}_{A}<{\hat{k}}_{A}^{{\dag}}$.

Based on the above analysis, we can obtain the following results.

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}, &0<{k}_{A}<\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*},& \min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B54)

If $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{p}^{Ns*}=0$;

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} ,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}\left(v-\left(c-B\right)\left(1+{r}_{s}\right)\right)}{\left(c-B\right)c{k}_{A}}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

$$\frac{\partial {\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)}{\partial {k}_{A}}=\frac{B(cv-aB\left(1+{r}_{F}\right))}{(c-B)c{k}_{A}^{2}}>0,$$

thus ${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)$ is an increasing function of ${k}_{A}$ and $\underset{{k}_{A}\to 0}{\mathrm{lim}}{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)=-\infty \ll 0$, so there exists

$${\check{k}}_{A}=\frac{2aB(1+{r}_{F})(aB-cv+aB{r}_{F})}{2{a}^{2}{B}^{2}+c\left(-B+c\right){v}^{2}-2a{c}^{2}\left(B-c+v\right)+2a\left(2a{B}^{2}+c\left({B}^{2}+2{c}^{2}-B\left(3c+v\right)\right)\right){r}_{F}+2a\left(a{B}^{2}+\left(B-c\right)c\left(B-c-v\right)\right){r}_{F}^{2}+2aBc(c-B){r}_{s}(1+{r}_{F})}>0$$

such that ${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)=0$, and ${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)\le 0$ for$0<{k}_{A}\le \mathrm{min}\left({\check{k}}_{A},1\right)$, while ${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)>0$ for $\mathrm{min}\left({\check{k}}_{A},1\right)<{k}_{A}<1$. And given $B$, by the definition of the optimal platform charge at this point, we have ${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)<{\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)$ and thus

$${\check{k}}_{A}<{\hat{k}}_{A}\left(c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}\right).$$

Then by the monotonicity of ${\hat{k}}_{A}$, we can get ${\check{k}}_{A}<{\hat{k}}_{A}$.

Therefore, similar to the previous process, we can obtain the following results.

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*},& 0<{k}_{A}<\min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*},& \min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\le {k}_{A}<1\end{array}\right.$$

(B55)

(4.b.2) $\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}<v\le \mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)$. The first case is$B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, which by the previous analysis yields

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*},& 0<{k}_{A}<\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*}, &\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\le {k}_{A}<1\end{array}\right..$$

Similarly, if

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},$$

${\Pi }_{p}^{Ns*}=0$;

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} ,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}\left(v-\left(c-B\right)\left(1+{r}_{s}\right)\right)}{\left(c-B\right)c{k}_{A}}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

the comparison result is $\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*},& 0<{k}_{A}<\min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*}, &\min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\le {k}_{A}<1\end{array}\right.$. While $\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$,

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}\left(v-\left(c-B\right)\left(1+{r}_{s}\right)\right)}{\left(c-B\right)c{k}_{A}},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

in the range of $\mathrm{min}\left({\breve{k}},1\right)\le {k}_{A}<1$, we have

$${\Pi }_{p}^{As}\left({p}_{p}^{As{\ddag} }\right)>{\Pi }_{p}^{As*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)>{\Pi }_{p}^{Ns*}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right)$$

by the definition of optimal charge.

Thus, we can obtain the following results.

$$ \left\{ {\begin{array}{*{20}c} {{\Pi }_{p}^{{As{*}}} < {\Pi }_{p}^{Ns*} , 0 < k_{A} < \min\left( {k_{A} ,\hat{k}_{A}^{\dag } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},1} \right)} \\ {{\Pi }_{p}^{{As{*}}} \ge {\Pi }_{p}^{Ns*} , \min\left( {k_{A} ,\hat{k}_{A}^{\dag } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},1} \right) \le k_{A} < 1} \\ \end{array} } \right. $$

(B56)

(4.b.3) $\mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)<v\le a\left(1+{r}_{F}\right)$. If $B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{p}^{Ns*}=0$;

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}\left(1+{r}_{s}\right)}{{k}_{A}},& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

thus we can obtain

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}, &0<{k}_{A}<\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*},& \min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right)\le {k}_{A}<1\end{array}\right..$$

If

$$\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)},$$

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})\right)}{2{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{v-2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)-c{k}_{A}\left(1+{r}_{s}\right)}{{k}_{A}}, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

when $\mathrm{min}\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1$, by the definition we get

$${\Pi }_{p}^{As}\left({p}_{p}^{As{\dag}}\right)>{\Pi }_{p}^{As}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)>{\Pi }_{p}^{Ns*}\left(\frac{cv-2aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)}{c{k}_{A}}-\left(1+{r}_{F}\right)c\right),$$

and the comparison of platform charges results in

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}, &0<{k}_{A}<\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*},& \min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},1\right)\le {k}_{A}<1\end{array}\right..$$

Finally, when $c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$,

$${\Pi }_{p}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(-2aB-2cv{r}_{F}+2aB{r}_{F}^{2}+{k}_{A}\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)\right)}{2{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{p}^{As*}=\left\{\begin{array}{ll}\frac{B\left(aB{k}_{A}\left(1-{r}_{F}-2{r}_{F}^{2}\right)-2{r}_{F}\left(cv-aB\left(1+{r}_{F}\right)\right)\right)}{2{c}^{2}{k}_{A}} ,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \left(B-c-v\right){r}_{F}-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}+{r}_{s}\left(c-B\right)+\frac{B\left(aB\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)-cv\right)+{k}_{A}{c}^{2}\left(v-\left(c-B\right)\left(1+{r}_{s}\right)\right)}{\left(c-B\right)c{k}_{A}}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

it is clear that there are

$$\left\{\begin{array}{ll}{\Pi }_{p}^{As*}<{\Pi }_{p}^{Ns*}, &0<{k}_{A}<\min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},1\right)\\ {\Pi }_{p}^{As*}\ge {\Pi }_{p}^{Ns*}, &\min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},1\right)\le {k}_{A}<1\end{array}\right..$$

Let

$${k}_{A}^{\xi }=\left\{\begin{array}{ll}\min\left({\hat{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right),& B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}\\ \min\left({\check{k}}_{A},{\hat{k}}_{A}^{{\dag}},1\right),& c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\end{array}\right.,$$

then the above results can be further organized as Proposition 8.□

1.2.9 Proof of Proposition 9

(5.a) $B>\frac{vc}{a\left(1+{r}_{F}\right)}$. (5.a.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$,${\Pi }_{s}^{Ns*}=\left(1+{r}_{F}\right)B>0={\Pi }_{s}^{As*}$. (5.a.2) When $v>\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$,

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}(1+{r}_{F})B , &0<{k}_{A}\le \frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{a{k}_{A}\left(1+{r}_{F}\right)}+(1+{r}_{F})B, &\frac{2v}{v(3-2{r}_{F})-2c(1+{r}_{F})}<{k}_{A}<1\end{array}\right.$$

, ${\Pi }_{s}^{As*}=0$, ${\Pi }_{s}^{Ns*}>{\Pi }_{s}^{As*}$ holds constantly.

(5.b) $B\le \frac{vc}{a\left(1+{r}_{F}\right)}$. (5.b.1) When $v\le \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, if $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$ holds, ${\Pi }_{s}^{Ns*}=(1+{r}_{F})B$ and

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ v-\frac{v}{{k}_{A}}+\frac{2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}}+B\left(1+{r}_{s}\right), &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.$$

,

$$\frac{\partial {\Pi }_{s}^{As*}\left({v}^{-}\right)}{\partial v}=\frac{(1-{k}_{A})\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}}>0,$$

thus ${\Pi }_{s}^{As*}\left({v}^{-}\right)>B\left(1+{r}_{s}\right)>{\Pi }_{s}^{Ns*}$. The comparisons of the seller’s profit in the two models are

$$ \left\{ {\begin{array}{*{20}l} {{\Pi }_{s}^{As*} \ge {\Pi }_{s}^{Ns*} , } \hfill & {\quad \min\left( {\tilde{k}_{A} ,1} \right) \le k_{A} < 1} \hfill \\ {{\Pi }_{s}^{As*} < {\Pi }_{s}^{Ns*} , } \hfill & {\quad 0 < k_{A} < \min\left( {\tilde{k}_{A} ,1} \right)} \hfill \\ \end{array} } \right.. $$

(B57)

If $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{s}^{Ns*}=(1+{r}_{F})B$ and

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0,&0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{B\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)(1+{r}_{s})\right]}{(c-B)c{k}_{A}},&\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

the seller’s profit comparison obviously results in

$$ \left\{ {\begin{array}{*{20}l} {\Pi _{s}^{{As*}} \ge \Pi _{s}^{{Ns*}} ,} \hfill & {\min \left( {{\breve{k}}_{A} ,1} \right) \le k_{A} < 1} \hfill \\ {\Pi _{s}^{{As*}} < \Pi _{s}^{{Ns*}} ,~} \hfill & {0 < k_{A} < \min \left( {{\breve{k}}_{A} ,1} \right)} \hfill \\ \end{array} } \right. $$

(5.b.2) When $\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}<v\le \mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)$, if the seller’s capital satisfy $B\le c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}$, then we have

$$\left\{\begin{array}{ll}{\Pi }_{s}^{As*}\ge {\Pi }_{s}^{Ns*}, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\\ {\Pi }_{s}^{As*}<{\Pi }_{s}^{Ns*}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\end{array}\right..$$

If

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$$

is satisfied, ${\Pi }_{s}^{Ns*}=(1+{r}_{F})B$ and

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{B\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)\left(1+{r}_{s}\right)\right]}{\left(c-B\right)c{k}_{A}},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

the comparison results are the same as in (5.b.1). And when the condition $\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$ holds,

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}(1+{r}_{F})B ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}},& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{B\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)\left(1+{r}_{s}\right)\right]}{\left(c-B\right)c{k}_{A}}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

since

$${\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=\frac{B\left\{\left(-aB\left(1-{k}_{A}\right)-{c}^{2}{k}_{A}\right)(1+{r}_{F})+\frac{c\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)(1+{r}_{s})\right]}{(c-B)}\right\}}{{c}^{2}{k}_{A}},$$

$$\frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)}{\partial {k}_{A}}=\frac{B(aB(c(2+{r}_{F})-B)-{c}^{2}v)}{(c-B){c}^{2}{k}_{A}^{2}}$$

and

$$c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}<\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right],$$

then in the range of

$$\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right]<B\le \frac{cv}{a\left(1+{r}_{F}\right)}<\frac{1}{2}c\left[2+{r}_{F}+\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right],$$

$$\frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)}{\partial {k}_{A}}>0.$$

Moreover, due to the fact that

$${\left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)|}_{{k}_{A}\to 0}=\frac{B({c}^{2}v-aB(c(2+{r}_{F})-B))}{(c-B){c}^{2}{k}_{A}}<0,$$

there exists a cut-off point ${k}_{A}^{\iota }\in \left(\mathrm{0,1}\right)$, for ${k}_{A}>{k}_{A}^{\iota }$, ${\Pi }_{s}^{As*}\left({v}^{-}\right)>{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)$. And if $B<\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right]$,

$$\frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)}{\partial {k}_{A}}<0,$$

$${\left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)|}_{{k}_{A}\to 1}>0,$$

thus ${\Pi }_{s}^{As*}\left({v}^{-}\right)>{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)$ holds.

Based on the above analysis, we can get the results of comparing seller’s profit in both models as follows: when

$$\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B<\mathrm{max}\left(\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right],\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}\right),$$

, we get

$$\left\{\begin{array}{ll}{\Pi }_{s}^{As*}\ge {\Pi }_{s}^{Ns*}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\\ {\Pi }_{s}^{As*}<{\Pi }_{s}^{Ns*}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\end{array}\right.;$$

when

$$\mathrm{max}\left(\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right],\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}\right)<B\le \frac{cv}{a\left(1+{r}_{F}\right)},$$

then we have

$$ \left\{ {\begin{array}{*{20}c} {{\Pi }_{s}^{As*} \ge {\Pi }_{s}^{Ns*} , \min\left( {{\breve{k}}_{A} ,1} \right) \le k_{A} < \max\left( {{\text{min}}\left( {{\breve{k}}_{A} ,1} \right),\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}}} \right) or \max\left( {{\text{min}}\left( {{\breve{k}}_{A} ,1} \right),\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},k_{A}^{\iota } } \right) \le k_{A} < 1} \\ {{\Pi }_{s}^{As*} < {\Pi }_{s}^{Ns*} , otherwise} \\ \end{array} } \right.. $$

(B58)

(5.b.3) When $\mathrm{max}\left({v}^{r{\dag}},\frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}\right)<v\le a\left(1+{r}_{F}\right)$, if $B\le \frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}$, ${\Pi }_{s}^{Ns*}=(1+{r}_{F})B$ and

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 , &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ v-\frac{v}{{k}_{A}}+\frac{2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}}+B\left(1+{r}_{s}\right),& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

thus the comparison results are

$$\left\{\begin{array}{ll}{\Pi }_{s}^{As*}\ge {\Pi }_{s}^{Ns*}, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\\ {\Pi }_{s}^{As*}<{\Pi }_{s}^{Ns*}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\end{array}\right..$$

If

$$\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)}<B\le c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)},$$

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}(1+{r}_{F})B ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}},& \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\\ v-\frac{v}{{k}_{A}}+\frac{2a\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)\left(1-\sqrt{1-\frac{v}{a\left(1+{r}_{F}\right)}}\right)}{{k}_{A}}+B\left(1+{r}_{s}\right), &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\end{array}\right.,$$

and since

$${\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=v+B\left(1+{r}_{s}\right)-\frac{v+\frac{B(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A})(1+{r}_{F})}{{c}^{2}}-2a(1-{k}_{A})(1+{r}_{F})\left(1-\sqrt{1-\frac{v}{a+a{r}_{F}}}\right)}{{k}_{A}},$$

and

$$\frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)\right)}{\partial {k}_{A}}=\frac{{c}^{2}v+a(1+{r}_{F})\left({B}^{2}-2{c}^{2}\left(1-\sqrt{1-\frac{v}{a+a{r}_{F}}}\right)\right)}{{c}^{2}{k}_{A}^{2}},$$

$$\frac{{\partial }^{2}\left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{v}{{k}_{A}}\right)\right)}{\partial {k}_{A}\partial B}=\frac{2aB(1+{r}_{F})}{{c}^{2}{k}_{A}^{2}}>0,$$

$$\frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{v}{{k}_{A}}\right)\right)}{\partial {k}_{A}}\le \frac{\partial \left({\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{v}{{k}_{A}}\right)\right)}{\partial {k}_{A}}{|}_{B=c-\frac{c\sqrt{a\left(1+{r}_{F}\right)\left(a-v+a{r}_{F}\right)}}{a\left(1+{r}_{F}\right)}}=0,$$

we can conclude ${\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)$ is a decreasing function of ${k}_{A}$. Since ${\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)=B\left({r}_{s}-{r}_{F}\right)>0$ for ${k}_{A}=1$, ${\Pi }_{s}^{As*}\left({v}^{-}\right)-{\Pi }_{s}^{Ns*}\left(\frac{cv-a\left(1+{r}_{F}\right)\left(1-{k}_{A}\right)B}{c{k}_{A}}\right)>0$ is obtained. The comparison results are

$$\left\{\begin{array}{ll}{\Pi }_{s}^{As*}\ge {\Pi }_{s}^{Ns*}, &\min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1\\ {\Pi }_{s}^{As*}<{\Pi }_{s}^{Ns*}, &0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right)\end{array}\right..$$

If $c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B\le \frac{vc}{a\left(1+{r}_{F}\right)}$,

$${\Pi }_{s}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left(aB\left(1-{k}_{A}\right)+{c}^{2}{k}_{A}\right)\left(1+{r}_{F}\right)}{{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right.,$$

$${\Pi }_{s}^{As*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{B\left[\left(cv-aB\left(1+{r}_{F}\right)\right)\left(1-{k}_{A}\right)+{k}_{A}c\left(c-B\right)\left(1+{r}_{s}\right)\right]}{\left(c-B\right)c{k}_{A}},& \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

and according to the analysis of (5.b.2), we can obtain the following result: when

$$c-\frac{c\sqrt{a(1+{r}_{F})(a-v+a{r}_{F})}}{a(1+{r}_{F})}<B<\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right],$$

there are

$$\left\{\begin{array}{ll}{\Pi }_{s}^{As*}\ge {\Pi }_{s}^{Ns*}, &\min\left({\breve{k}},1\right)\le {k}_{A}<1\\ {\Pi }_{s}^{As*}<{\Pi }_{s}^{Ns*}, &0<{k}_{A}<\min\left({\breve{k}},1\right)\end{array}\right.;$$

when $\frac{1}{2}c\left[2+{r}_{F}-\sqrt{\frac{{a(2+{r}_{F})}^{2}-4v}{a}}\right]<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$, the comparison results in Eq. (B58).

By rearranging all the above results, we get Proposition 9.□

1.2.10 Proof of Proposition 10

When $B\le \frac{cv}{a(1+{r}_{F})}$,

$${CS}^{Ns*}\le \left\{\begin{array}{ll}0 , &0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}\le 1\end{array}\right.,$$

$${CS}^{As*}=\left\{\begin{array}{ll}\frac{a{B}^{2}\left(1-{k}_{A}\right)\left(1+{r}_{F}\right)}{2{c}^{2}},& 0<{k}_{A}<\min\left({\tilde{k}}_{A},1\right) or 0<{k}_{A}<\min\left({\breve{k}},1\right)\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)} ,& \min\left({\tilde{k}}_{A},1\right)\le {k}_{A}<1 or \min\left({\breve{k}},1\right)\le {k}_{A}<1\end{array}\right.,$$

since $0<\frac{a{B}^{2}(1-{k}_{A})(1+{r}_{F})}{2{c}^{2}}\le \frac{{v}^{2}(1-{k}_{A})}{2a(1+{r}_{F})}$, ${CS}^{As*}>{CS}^{Ns*}$ holds constant.

When $B>\frac{vc}{a\left(1+{r}_{F}\right)}$,

$${CS}^{Ns*}=\left\{\begin{array}{ll}0 ,& 0<{k}_{A}\le \min\left(\frac{2v}{3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}},1\right)\\ \frac{{v}^{2}\left(1-{k}_{A}\right)}{2a\left(1+{r}_{F}\right)},& \min\left(\frac{2v}{3v-2c\left(1+{r}_{F}\right)-2v{r}_{F}},1\right)<{k}_{A}<1\end{array}\right.;$$

${CS}^{As*}=\frac{{v}^{2}(1-{k}_{A})}{2a(1+{r}_{F})}$, again with ${CS}^{As*}\ge {CS}^{Ns*}$.

In summary, ${CS}^{As*}>{CS}^{Ns*}$ holds constant.□

1.2.11 Proof of Proposition 11

(7.a) When $B>\frac{vc}{a\left(1+{r}_{F}\right)}$ and

$$\begin{aligned}&v>\frac{2c(1+{r}_{F})}{1-2{r}_{F}},\\&{SW}^{Ns*}=\left\{\begin{array}{ll}(1+{r}_{F})B , &0<{k}_{A}\le {k}_{A}^{\varsigma }\\ \frac{v\left(v\left(2\left(1-{r}_{F}\right)-{k}_{A}\right)-2c\left(1+{r}_{F}\right)\right)}{2a\left(1+{r}_{F}\right)}+(1+{r}_{F})B,& {k}_{A}^{\varsigma }<{k}_{A}<1\end{array}\right.,\\&{SW}^{As*}=B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a(1+{r}_{F})}.\end{aligned}$$

When ${k}_{A}^{\varsigma }<{k}_{A}<1$, ${SW}^{As*}-{SW}^{Ns*}$= $\frac{(a(1+{r}_{F})-v)(v\left(1-{r}_{F}\right)-c(1+{r}_{F}))}{a(1+{r}_{F})}>0$; while $0<{k}_{A}\le {k}_{A}^{\varsigma }$, ${SW}^{As*}-{SW}^{Ns*}=v\left(1-{r}_{F}\right)-c\left(1+{r}_{F}\right)-\frac{{v}^{2}{k}_{A}}{2a(1+{r}_{F})}$. Since $v>\frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, there exist $a\left(1+{r}_{F}\right)\ge \frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, i.e., $a\left(1-2{r}_{F}\right)\ge 2c$, ${SW}^{As*}-{SW}^{Ns*}=v\left(1-{r}_{F}\right)-c\left(1+{r}_{F}\right)-\frac{{v}^{2}{k}_{A}}{2a\left(1+{r}_{F}\right)}>v\left(1-{r}_{F}\right)-c\left(1+{r}_{F}\right)-\frac{{v}^{2}}{2a\left(1+{r}_{F}\right)}>0$.

(7.b) When $B\le \mathrm{min}\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)$, ${SW}^{Ns*}=(1+{r}_{F})B$ and

$${SW}^{As*}=\left\{\begin{array}{ll}\frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}},& 0<{k}_{A}<{k}_{A}^{\tau }\\ B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a\left(1+{r}_{F}\right)} , &{k}_{A}^{\tau }\le {k}_{A}<1\end{array}\right..$$

If $0<{k}_{A}<{k}_{A}^{\tau }$, ${SW}^{As*}-{SW}^{Ns*}=\frac{B\left[Ba\left(1+{r}_{F}\right)\left(-{k}_{A}^{2}+2{k}_{A}\left(1-{r}_{F}\right)+2{r}_{F}\right)-2{c}^{2}{k}_{A}(1+{r}_{F})-2cv{r}_{F}\right]}{2{c}^{2}{k}_{A}}$, and $-{k}_{A}^{2}+2{k}_{A}\left(1-{r}_{F}\right)+2{r}_{F}>0$. Let $M\left({k}_{A}\right)=Ba\left(1+{r}_{F}\right)\left(-{k}_{A}^{2}+2{k}_{A}\left(1-{r}_{F}\right)+2{r}_{F}\right)-2{c}^{2}{k}_{A}(1+{r}_{F})-2cv{r}_{F}$, $M\left({k}_{A}\right)$ is a concave function of ${k}_{A}$, and $M\left(0\right)=2{r}_{F}\left(Ba\left(1+{r}_{F}\right)-cv\right)<0$, $M\left(1\right)=Ba\left(1+{r}_{F}\right)-2{c}^{2}\left(1+{r}_{F}\right)-2cv{r}_{F}<2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)-2{c}^{2}\left(1+{r}_{F}\right)-2cv{r}_{F}=0$, therefore, there exist two points ${k}_{A}^{{\mathcalligra{o}}1},{k}_{A}^{{\mathcalligra{o}}2}$,$0<{k}_{A}^{{\mathcalligra{o}}1}<{k}_{A}^{{\mathcalligra{o}}2}<1$, such that $M\left({k}_{A}\right)=0$ and if $\mathrm{min}\left({k}_{A}^{\tau },{k}_{A}^{{\mathcalligra{o}}1}\right)<{k}_{A}<\mathrm{min}\left({k}_{A}^{\tau },{k}_{A}^{{\mathcalligra{o}}2}\right)$, $M\left({k}_{A}\right)>0$ holds. Furthermore if $v>\frac{2c(1+{r}_{F})}{1-2{r}_{F}}$, according to the analysis of (7.a), we can obtain the following results.

$$ \left\{ {\begin{array}{*{20}l} {SW^{As*} \le SW^{Ns*} , } \hfill & {\quad otherwise} \hfill \\ {SW^{As*} > SW^{Ns*} ,} \hfill & {\quad \min\left( {k_{A}^{\tau } ,k_{A}^{{\mathcalligra{o}}1} } \right) < k_{A} < \min\left( {k_{A}^{\tau } ,k_{A}^{{\mathcalligra{o}}2} } \right) or k_{A}^{\tau } \le k_{A} < 1} \hfill \\ \end{array} } \right. $$

(B59)

(7.c) When

$$\begin{aligned}&\mathrm{min}\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)<B\le \frac{cv}{a\left(1+{r}_{F}\right)},\\&{SW}^{Ns*}=\left\{\begin{array}{ll}(1+{r}_{F})B ,& 0<{k}_{A}\le \frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\\ \frac{B\left[2{r}_{F}(aB\left(1+{r}_{F}\right)-cv)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}}, &\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}<{k}_{A}<1\end{array}\right., \\&{SW}^{As*}=\left\{\begin{array}{ll}\frac{B\left[2{r}_{F}\left(aB\left(1+{r}_{F}\right)-cv\right)+2{k}_{A}aB\left(1-{r}_{F}^{2}\right)-aB{k}_{A}^{2}\left(1+{r}_{F}\right)\right]}{2{c}^{2}{k}_{A}}, &0<{k}_{A}<{k}_{A}^{\tau }\\ B-c+v+(B-c-v){r}_{F}-\frac{{v}^{2}{k}_{A}}{2a\left(1+{r}_{F}\right)} , &{k}_{A}^{\tau }\le {k}_{A}<1\end{array}\right.\end{aligned},$$

$$\mathrm{min}\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)<B\le \frac{cv}{a\left(1+{r}_{F}\right)}$$

is a non-empty set if and only if $v\ge \frac{2c\left(1+{r}_{F}\right)}{1-2{r}_{F}}$, and at this point there must be $v>\frac{2c(1+{r}_{F})}{2\left(1-{r}_{F}\right)-{k}_{A}}$. Further, when

$$0<{k}_{A}<\mathrm{min}\left({k}_{A}^{\tau },\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})}\right),$$

$${SW}^{As*}-{SW}^{Ns*}=\frac{B\left[Ba\left(1+{r}_{F}\right)\left(-{k}_{A}^{2}+2{k}_{A}\left(1-{r}_{F}\right)+2{r}_{F}\right)-2{c}^{2}{k}_{A}(1+{r}_{F})-2cv{r}_{F}\right]}{2{c}^{2}{k}_{A}},$$

$M\left(0\right)=2{r}_{F}\left(Ba\left(1+{r}_{F}\right)-cv\right)<0$ and $M\left(1\right)=Ba\left(1+{r}_{F}\right)-2{c}^{2}\left(1+{r}_{F}\right)-2cv{r}_{F}>0$. There exists the unique point ${k}_{A}^{{\mathcalligra{o}}1}$ such that $M\left({k}_{A}\right)=0$, and $M\left({k}_{A}\right)<0$ if

$$0<{k}_{A}<\mathrm{min}\left({k}_{A}^{\tau },\frac{2(aB+cv{r}_{F}-aB{r}_{F}^{2})}{(1+{r}_{F})(3aB-2{c}^{2}-2aB{r}_{F})},{k}_{A}^{{\mathcalligra{o}}1}\right),$$

and if

$${\text{min}}\left( {k_{A}^{\tau } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},k_{A}^{{\mathcalligra{o}}1} } \right) < k_{A} < {\text{min}}\left( {k_{A}^{\tau } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}}} \right),M\left( {k_{A} } \right) > 0.$$

According to the definition of the optimal solution of platform profit function, as well as the functions of seller’s profit and consumer surplus, we can get $SW^{As*} > SW^{Ns*}$ in the range of

$${\text{max}}\left( {k_{A}^{\tau } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}}} \right) \le k_{A} < 1.$$

Thus, the results of comparing social welfare in the two models are

$$ \left\{ {\begin{array}{*{20}l} {SW^{As*} \ge SW^{Ns*} ,} \hfill & {\quad otherwise} \hfill \\ {SW^{As*} < SW^{Ns*} ,} \hfill & {\quad 0 < k_{A} < \min\left( {k_{A}^{\tau } ,\frac{{2\left( {aB + cvr_{F} - aBr_{F}^{2} } \right)}}{{\left( {1 + r_{F} } \right)\left( {3aB - 2c^{2} - 2aBr_{F} } \right)}},k_{A}^{{\mathcalligra{o}}1} } \right)} \hfill \\ \end{array} } \right.. $$

(B60)

Let

$${k}_{A}^{\upsilon }=\left\{\begin{array}{l}{k}_{A}^{\tau } , B\le \min\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)\\ \min\left({k}_{A}^{\tau },\frac{2\left(aB+cv{r}_{F}-aB{r}_{F}^{2}\right)}{\left(1+{r}_{F}\right)\left(3aB-2{c}^{2}-2aB{r}_{F}\right)},{k}_{A}^{{\mathcalligra{o}}1}\right), \min\left(\frac{2\left({c}^{2}\left(1+{r}_{F}\right)+cv{r}_{F}\right)}{a\left(1+{r}_{F}\right)},\frac{vc}{a\left(1+{r}_{F}\right)}\right)<B\le \frac{cv}{a\left(1+{r}_{F}\right)}\\ 0 , B>\frac{vc}{a\left(1+{r}_{F}\right)}\end{array}\right.$$

, by further sorting and simplification, Proposition 11 is proved.□