## Appendix A The Derivation Process of Consumer Utility Function  
Referring to the linear demand function constructed by [PERSON] et al. [35], the utility of consumer purchasing from the merchants can be, respectively, expressed as follows. \[\left\{ \begin{array}{ll}U_{1}=\delta_{1}\lambda-\alpha p_{1}+\beta s_{1}\\ U_{2}=\delta_{2}\lambda-\alpha p_{2}+\beta s_{2}\end{array} \right. \tag{1}\]  
Rational consumers will only make a purchase when the utility is positive. When presented with multi-channel options, their selection process adheres to comparative utility optimization governed by max \(\{U_{1},U_{2},0\}\) behavioral patterns. The decision framework establishes parametric thresholds \((\lambda_{1},\lambda_{2},\lambda_{3})\) through product valuation analysis, corresponding to three pivotal operational scenarios \(U_{1}=0,U_{2}=0,\ U_{1}=U_{2}\). \[\left\{ \begin{array}{ll}\lambda_{1}&=\frac{\alpha p_{1}-\beta s_{1}}{\delta _{1}}\\ \lambda_{2}&=\frac{\alpha p_{2}-\beta s_{2}}{\delta_{2}}\\ \lambda_{3}&=\frac{\alpha(p_{1}-p_{2})-\beta(s_{1}-s_{2})}{\delta_{1}-\delta _{2}}\end{array} \right. \tag{2}\]  
The disparities existing between the parametric thresholds can be designated as follows. \[\left\{ \begin{array}{ll}\Delta\lambda^{a}=\lambda_{1}-\lambda_{2}=\frac{ \alpha(\delta_{2}p_{1}-\delta_{1}p_{2})-\beta(\delta_{2}s_{1}-\delta_{1}s_{2} )}{\delta_{2}\delta_{2}}\\ \Delta\lambda^{b}=\lambda_{3}-\lambda_{1}=\frac{\alpha(\delta_{2}p_{1}-\delta _{1}p_{2})-\beta(\delta_{2}s_{1}-\delta_{1}s_{2})}{\delta_{1}(\delta_{1}-\delta _{2})}\end{array} \right. \tag{3}\]When \(\lambda_{1}\geq\lambda_{2}\), so \(\Delta\lambda^{a}\geq 0\); thus, \(\alpha(\delta_{2}p_{1}-\delta_{1}p_{2})-\beta(\delta_{2}s_{1}-\delta_{1}s_{2})\geq 0\). Combining \(\delta_{1}>\delta_{2}\) can obtain \(\Delta\lambda^{b}\geq 0\) and, finally, we can deduce that \(\lambda_{3}\geq\lambda_{1}\geq\lambda_{2}\). When \(\lambda_{1}\leq\lambda_{2}\), so \(\Delta\lambda^{a}\leq 0\); thus, \(\alpha(\delta_{2}p_{1}-\delta_{1}p_{2})-\beta(\delta_{2}s_{1}-\delta_{1}s_{2})\leq 0\). Combining \(\delta_{1}>\delta_{2}\) can obtain \(\Delta\lambda^{b}\leq 0\) and, finally, we can deduce that \(\lambda_{3}\leq\lambda_{1}\leq\lambda_{2}\). According to the magnitude of the utility, consumers' purchasing decisions can be divided into four situations:  
1. When \(U_{i}<0\), which implies that \(\lambda<\lambda_{i}\), the consumers will not make a purchase from either Merchant 1 or Merchant 2, as the utility derived from the purchase would be negative;
2. When \(U_{1}\geq 0\) and \(U_{1}\geq U_{2}\), which indicates that \(\lambda\geq\lambda_{1}\) and \(\lambda>\lambda_{3}\), the consumers are more inclined to purchase from Merchant 1, as this offers a higher utility, while Merchant 2 would not receive any purchases;
3. When \(U_{2}\geq 0\) and \(U_{1}<U_{2}\), which implies that \(\lambda\geq\lambda_{2}\) and \(\lambda<\lambda_{3}\), the consumers are more likely to purchase from Merchant 2, as this provides a higher utility, while Merchant 1 would not receive any purchases;
4. When \(U_{1}=U_{2}\geq 0\), which means that \(\lambda=\lambda_{3}\) and \(\lambda\geq\max\{\lambda_{1},\lambda_{2}\}\), the consumers are indifferent between purchasing from Merchant 1 or Merchant 2, as the utilities are the same. Consequently, given different product valuations, the demand for merchants across various price ranges can be articulated as three distinct scenarios (Figure 11). Given varying product valuations, the demand for merchants across different price ranges can be outlined in the following three scenarios:  
1. In case \(\lambda_{1}\leq\lambda_{2}\), it follows that \(\lambda_{3}\leq\lambda_{1}\); consumers do not make purchases from Merchant 2, resulting in a demand of \(D_{2}=0\). When \(\lambda\in[\lambda_{1},1]\), consumers purchase from Merchant 1, with the demand expressed as \(D_{1}=1-\lambda_{1}\); when \(\lambda\in[0,\lambda_{1}]\), no purchase behavior occurs. As a result, the price range satisfies the following conditions.