To grasp the research question’s essence, simplify complex issues, and mirror real-life behaviors, the following assumptions are made for the game model:

1) Rational Economic Agents: All players (government, enterprises, consumers) act to maximize their own interests, with the government seeking maximum social welfare.

2) Perfect Information Dynamic Game: All parties have full information about each other’s strategies and payoffs through IT.

3) Sequential Game Order: The game proceeds in the order of government, enterprise, and consumer. The government sets policies, enterprises decide on low-carbon supply chain strategies, and consumers make purchasing decisions.

4) Simplified Cost and Profit: The model focuses only on the costs and benefits related to implementing low-carbon supply chains, setting other costs and profits to zero to highlight key factors.

To support the development of low-carbon supply chains in cross-border e-commerce, the government can provide special funds to help enterprises implement low-carbon supply chains, reduce their implementation costs, enhance carbon trading systems, and subsidize products purchased through low-carbon supply chains. This lowers consumers’ purchase costs. The government can also use fiscal and tax measures, such as tax breaks for enterprises adopting low-carbon supply chains and taxes on products from traditional supply chains, to narrow the cost gap between the two.

In this model, the government’s two strategies are named green regulation $a_{g1}$ and general regulation $a_{g2}$, with proportion $p_g$ and $1-p_g$, forming the government’s strategy space $A_g=\left\{a_{g1},a_{g2}\right\}$. Enterprises can choose low-carbon supply chain $a_{m1}$ or traditional supply chain $a_{m2}$, with proportion $p_m$ and $1-p_m$, making the enterprise’s strategy space $A_m=\left\{a_{m1},a_{m2}\right\}$. Consumers can purchase through low-carbon $a_{c1}$ or traditional $a_{c2}$ supply chains, with proportion $p_c$ and $1-p_c$, giving the consumer’s strategy space $A_c=\left\{a_{c1},a_{c2}\right\}$. The game tree below illustrates the game process among the government (G), enterprise (M), and consumer (C) ( Figure 1 ).

Other parameter settings for the above model are shown in Table 1 .

Based on the tripartite game tree model, the payoff matrices for the government, enterprises, and consumers are presented in Table 2 .

Note: $u_g\left(a_{gj}\right)$, $u_m\left(a_{mj}\right)$, and $u_c\left(a_{cj}\right)$ represent the payoff functions for the government, enterprises, and consumers, respectively, under different action combinations.

As shown in Table 2 , the payoff functions for each party under different strategies are presented in Table 3 .

When the expected payoffs of all strategies are equal, a mixed-strategy Nash equilibrium is achieved, as players’ payoffs become strategy-invariant. Specifically, when $u_{g1}=u_{g2}$, $u_{m1}=u_{m2}$, and $u_{c1}=u_{c2}$, the mixed-strategy Nash equilibrium for the government, enterprises, and consumers in the low-carbon supply chain implementation game of cross-border e-commerce can be derived as follows:

$\{\begin{array}{l}{p}_g^{*}=\frac{B+p_c{R}_1-\left(1-p_c\right)R_2-\left(C_1-C_2\right)}{B-\beta T}\\ p_m^{*}=\frac{U_2-\left(1-p_g\right)F}{U_1+U_2+\left(1-\beta \right)p_{g}T-\left(1-p_g\right)F}\\ p_c^{*}=\frac{T+\left(1-p_m\right)F-p_{m}B}{\left(1-p_m\right)F}\\ p_g^{*},p_m^{*},p_c^{*}\in \left[0,1\right]\end{array}$ (1)

Differentiating $p_c^{*}$ with respect to $T$ in Equation (1) yields:

$\frac{\partial p_c^{*}}{\partial T}=\frac{1}{\left(1-p_m\right)F}$ (2)

Equation (2) indicates that $p_c^{*}$ increases monotonically with $T$. Thus, the government’s green regulation strategy $a_{g1}$ encourages consumers to purchase via low-carbon supply chains, as the additional consumer benefits from increased government investment $T$ raise the probability $p_c^{*}$.

Differentiating $p_c^{*}$ with respect to $B$ in Equation (1) yields:

$\frac{\partial p_c^{*}}{\partial B}=\frac{-p_m}{\left(1-p_m\right)F}$ (3)

Equation (3) shows that $p_c^{*}$ decreases monotonically with $B$. Consequently, the government’s general regulatory strategy $a_{g2}$ based on fiscal incentives does not positively influence consumers. Instead, as tax breaks and benefits $B$ for enterprises rise, consumers are more inclined to purchase through traditional supply channels.

Differentiating $p_c^{*}$ with respect to $F$ in equation (1) yields:

$\frac{\partial p_c^{*}}{\partial F}=\frac{p_{m}B-T}{\left(1-p_m\right)F^2}$ (4)

Equation (4) reveals no strict monotonic relationship between $p_c^{*}$ and $F$. The impact of $F$ on $p_c^{*}$ largely hinges on the interplay between government’s effective investment $T$ under green regulation $a_{g1}$ and the benefits $B$ offered to enterprises under general regulation $a_{g2}$. If $p_{m}B-T>0$, $p_c^{*}$ increases with $F$; if $p_{m}B-T<0$, $p_c^{*}$ decreases with $F$. This suggests that with an imperfect incentive mechanism and insufficient government investment, increasing taxes on cross-border e-commerce products for consumers does not encourage them to switch to low-carbon supply chain channels.

In summary, the government can incentivize consumers to switch from traditional channels to cross-border e-commerce low-carbon supply chain channels by improving tax policies (such as setting reasonable tax breaks for enterprises) and increasing effective investment in low-carbon supply chains (such as developing low-carbon technologies, improving relevant infrastructure, promoting carbon trading and carbon credit mechanisms, and subsidizing price differentials caused by green barriers). Further analysis indicates that $\left|\partial p_c^{*}/\partial T\right|>\left|\partial p_c^{*}/\partial B\right|$, which demonstrates that $T$ has a more powerful influence on consumers’ choices.

Differentiating $p_m^{*}$ with respect to $F$ yields:

$\frac{\partial p_m^{*}}{\partial F}=\frac{-\left(1-p_g\right)\left(U_1+\left(1-\beta \right)p_{g}T\right)}{{\left(U_1+U_2+p_{g}T-\beta p_{g}T+p_{g}F-F\right)}^2}$ (5)

This indicates $p_m^{*}$ is a decreasing function of $F$. Thus, the government’s general regulatory strategy $a_{g2}$ based on fiscal incentives does not positively impact enterprises. Instead, as the carbon tax $F$ on cross-border e-commerce products increases, enterprises are more inclined to implement traditional supply chains.

Differentiating $p_m^{*}$ with respect to $T$ yields:

$\frac{\partial p_m^{*}}{\partial T}=\frac{\left(1-\beta \right)\left(U_2+p_{g}F-F\right)}{{\left(U_1+U_2+p_{g}T-\beta p_{g}T+p_{g}F-F\right)}^2}$ (6)

Since $U_2>F$ and $U_2+p_{g}F-F>0$, $p_m^{*}$ is an increasing function of $T$. This means increasing the government’s investment $T$ in low-carbon supply chains under the green regulation strategy $a_{g1}$ promotes enterprise adoption of low-carbon strategies $a_{m1}$.

Differentiating $p_m^{*}$ with respect to $\beta$ yields:

$\frac{\partial p_m^{*}}{\partial \beta }=\frac{p_{g}T\left(U_2+p_{g}F-F\right)}{{\left(U_1+U_2+p_{g}T-\beta p_{g}T+p_{g}F-F\right)}^2}$ (7)

This shows $p_m^{*}$ is an increasing function of $\beta$. By raising the proportion $\beta$ of enterprise profits from government investment, the government can boost the probability $p_m^{*}$ of enterprises implementing the low-carbon supply chain strategy $a_{m1}$.

In conclusion, government measures to increase enterprise benefits from implementing low-carbon supply chains, such as investing in low-carbon technologies and carbon trading mechanisms, can strongly motivate enterprises to adopt low-carbon supply chains.

Differentiating $p_g^{*}$ with respect to $T$ gives:

$\frac{\partial p_g^{*}}{\partial T}=\frac{b\left(B-C_1+C_2+p_c{R}_1-R_2+p_c{R}_2\right)}{{\left(B-\beta T\right)}^2}$ (8)

Since $p_g^{*}>0$, $B-\beta T>0$, $B-C_1+C_2+p_c{R}_1-R_2+p_c{R}_2>0$. This shows $p_g^{*}$ is an increasing function of $T$. In other words, the more funds $T$ the government allocates to low-carbon supply chains in cross-border e-commerce, the higher the probability $p_g^{*}$ that it will adopt the green regulation strategy $a_{g1}$.

Differentiating $p_g^{*}$ with respect to $\beta$ gives:

$\frac{\partial p_g^{*}}{\partial \beta }=\frac{\left(B-C_1+C_2+p_c{R}_1-R_2+p_c{R}_2\right)T}{{\left(B-\beta T\right)}^2}$ (9)

This indicates $p_g^{*}$ is an increasing function of $\beta$. Thus, raising the proportion $\beta$ of enterprise profits from government investment promotes the adoption of the green regulation strategy $a_{g1}$.

In summary, similar to the enterprise equilibrium analysis, when the government increases investment in low-carbon supply chains, enterprises gain more benefits and are more motivated to implement low-carbon strategies. In turn, the government, aiming for greater social welfare, becomes more proactive in investing in R&D, construction, and promotion, thus advancing the development of low-carbon supply chains in cross-border e-commerce.

This study focuses on cross-border e-commerce, and its findings may not generalize to other trade sectors (e.g., traditional manufacturing or service industries). Future work should extend the tripartite game model to broader supply chain contexts to enhance applicability.