3.4. Model Analysis

Assuming that the expected benefit of the e-commerce company when choosing the nondifferential pricing strategy is U₁₁, the expected benefit of the e-commerce company when choosing the differential pricing strategy is U₁₂, and the average expected benefit of the e-commerce company is U1¯, which are defined as follows:

According to the Malthusian dynamic equation, the replication dynamic equation of the e-commerce company is obtained as follows:

The first partial derivative of F (x) for x is as follows:

Based on the stability theorem of differential equations, the e-commerce company implements the strategy of nondifferential pricing in the stable state must meet the conditions: F(x) = 0, and F_x′(x) < 0.

When r > r₀, the stable strategy of the e-commerce company is nondifferential pricing; when r < r₀, the stable strategy of the e-commerce company is differential pricing; when r = r₀, the e-commerce company cannot determine the stable strategy. Where the threshold is as follows:

Assume H(r)=P_n − P_d − yΔP+y(M+I_e)[r+(1 − r)α]+(1 − y)rI_e+zF, when y[M − α(M+I_e)] > 0, (∂H/∂r) > 0, then H (r) is considered to be an increasing function of r. When r > r₀, H (r) > 0, F(x)|_x=1=0, and F_x′(x)|_x=1 < 0, so x = 1 has stability; When r < r₀, H (r) < 0, F(x)|_x=0=0, and F_x′(x)|_x=0 < 0, so x = 0 has stability; when r = r₀, H (r) = 0, F(x)=0, and F_x′(x)=0, so x is stable at all levels in the range of 0 to 1, that is, the company's strategy does not change over time, regardless of the proportion of company choosing to price differentially.

Proposition 1 states that the increase of the proportion of the government strict supervision to e-commerce company will change the stable strategy of e-commerce company from differential pricing to nondifferential pricing; Similarly, the decline of the proportion of the government strict supervision to e-commerce company will change the stable strategy of e-commerce company from nondifferential pricing to differential pricing. Therefore, the government's strict supervision for e-commerce company is essential, and the government should take measures to improve strict supervision for the e-commerce company.

Based on Proposition 1, the phase diagram of the strategy evolution of e-commerce company is shown in Figure 2.

Phase diagram of strategy evolution of e-commerce company.

Inference 1: with the increase of the value of P_n, M, I_e, F, and α, the e-commerce company is more inclined to implement the nondifferential pricing strategy, when other parameters remain unchanged. Similarly, with the increase of the value of P_d and ∆P, the e-commerce company is more inclined to implement the differential pricing strategy. It shows that the proportion of e-commerce company implementing nondifferential pricing strategy is directly proportional to the benefits of nondifferential pricing, the fines imposed by the government and platform on e-commerce company for differential pricing and the probability of consumers' discovery, and inversely proportional to the benefits of e-commerce company implementing differential pricing strategy.

Since r₀=(P_d+yΔP − P_n − αy(M+I_e) − zF/(1 − α)yM+(1 − αy)I_e), the volume of V_x1 in Figure 2 represents the proportion of nondifferential pricing by the e-commerce company, and the corresponding volume of V_x0 represents the proportion of differential pricing by the e-commerce company. When the value of P_n, M, I_e, F, and α gradually increases, the value of r₀ will gradually decrease, and the volume of V_x1 will increase at this time, indicating that the proportion of e-commerce company to implement nondifferential pricing increases; When the value of P_d and ∆P gradually increases, the value of r₀ will gradually increase, and the volume of V_x1 will decrease at this time, indicating that the proportion of e-commerce company to implement nondifferential pricing decreases.

Assuming that the expected benefit of the consumer when choosing loyalty strategy to e-commerce company is U₂₁, the expected benefit of the consumer when choosing disloyalty strategy to e-commerce company is U₂₂, and the average expected benefit of the consumer is U2¯, which are defined as follows:

According to the Malthusian dynamic equation, the replication dynamic equation of consumer is obtained as follows:

The first partial derivative of F (y) for y is as follows:

Based on the stability theorem of differential equations, consumer implements the strategy of loyalty in the stable state must meet the conditions: F(y) = 0, and F_y′(y) < 0.

When x > x₀, the stable strategy of the consumer is loyalty; when x < x₀, the stable strategy of the consumer is disloyalty; when x = x₀, the consumer cannot determine the stable strategy. Where the threshold is as follows:

Assume H(x)=U_l − U_d − (1 − x)ΔP+(1 − x)[M+(1 − r)αM] − (1 − x)(1 − r)C_c+(1 − x)z(1 − β)F, when M − ΔP+(1 − r)(αM − C_c)+z(1 − β)F > 0, (∂H/∂x) > 0, H (x) is considered to be an increasing function of x. When x > x₀, H (x) > 0, F(y)|_y=1=0, and F_y′(y)|_y=1 < 0, so y = 1 has stability; When x < x₀, H (x) < 0, F(y)|_y=0=0, and F_y′(y)|_y=0 < 0, so y = 0 has stability; When x = x₀, H (x) = 0, F(y)=0, and F_y′(y)=0, so y is stable at all levels in the range of 0 to 1, that is, the consumer's strategy does not change over time, regardless of the proportion of consumer choosing to be loyal.

Proposition 2 states that the increase of the proportion of nondifferential pricing of e-commerce company will change the stable strategy of consumer from disloyalty to loyalty; Similarly, the decline of the proportion of nondifferential pricing of e-commerce company will change the stable strategy of consumer from loyalty to disloyalty. Therefore, e-commerce company should reduce the degree of difference in pricing for consumers and try to retain consumers.

Based on Proposition 2, the phase diagram of the strategy evolution of consumer is shown in Figure 3.

Phase diagram of strategy evolution of consumer.

Inference 2: with the increase of the value of U_l, M, F, α, and β, the consumer is more inclined to be loyalty strategy to the e-commerce company, when other parameters remain unchanged. Similarly, with the increase of the value of U_d, ∆P, and C_c, the consumer is more inclined to be disloyalty strategy to the e-commerce company. It shows that the proportion of consumer being loyalty strategy to e-commerce company is directly proportional to the utility obtained by the loyal consumer from purchasing goods, the fines imposed by the government and e-commerce platform for differential pricing of e-commerce company, and the probability of consumers' discovery, and inversely proportional to the utility obtained by the disloyal consumer in purchasing goods, the additional benefit obtained by the e-commerce company in implementing differential pricing, the proportion of fines imposed by the platform to the e-commerce company and the cost of consumer complaints.

Since x₀=1 − (U_l − U_d/ΔP+(1 − r)C_c − [1+(1 − r)α]M − z(1 − β)F), the volume of V_y1 in Figure 3 represents the proportion of loyalty to e-commerce company by the consumer, and the corresponding volume of V_y0 represents the proportion of disloyalty to e-commerce company by the consumer. When the value of U_l, M, I_e, F, and α gradually increases, the value of x₀ will gradually decrease, and the volume of V_y1 will increase at this time, indicating that the proportion of loyalty to e-commerce company by the consumer increases; When the value of U_d, ∆P, β and C_c gradually increase, the value of x₀ will gradually increase, and the volume of V_y1 will decrease at this time, indicating that the proportion of loyalty to e-commerce company by consumer decreases.

Assuming that the expected benefit of the e-commerce platform when choosing the information supervision strategy is U₃₁, the expected benefit of the e-commerce platform when choosing the information nonsupervision strategy is U₃₂, and the average expected benefit of the e-commerce platform is U3¯, which are defined as follows:

According to the Malthusian dynamic equation, the replication dynamic equation of e-commerce platform is obtained as follows:

The first partial derivative of F (z) for z is as follows:

Based on the stability theorem of differential equations, e-commerce platform implements the strategy of information supervision in the stable state must meet the conditions: F(z) = 0, and F_z′(z) <0.

When y > y₀, the e-commerce platform will choose information supervision as the stable strategy; when y < y₀, the e-commerce platform will choose information nonsupervision as the stable strategy; when y = y₀, the e-commerce platform cannot determine the stable strategy. Where the threshold is as follows:

Assume H(y)=F − C_p+yT_p − xF − (1 − x)rI_p − (1 − x)y(1 − r)I_p, when T_p − (1 − x)(1 − r)I_p > 0, (∂H/∂x) >0, H (y) is considered to be an increasing function of y. When y > y₀, H (y) > 0, F(z)|_z=1=0, and F_z′(z)|_z=1 < 0, so z = 1 has stability; When y < y₀, H (y) < 0, F(z)|_z=0=0, and F_z′(z)|_z=0 < 0, so z = 0 has stability; When z = z₀, H (y) = 0, F(z)=0, and F_z′(z)=0, so z is stable at all levels in the range of 0 to 1, that is, the e-commerce platform's strategy does not change over time, regardless of the proportion of e-commerce platform choosing information supervision.

Proposition 3 states that the increase of the proportion of consumer loyalty will change the stable strategy of e-commerce platform from information nonsupervision to information supervision. Similarly, the decline of the proportion of consumer loyalty will change the stable strategy of e-commerce platform from information supervision to information nonsupervision. Therefore, if the consumer can be loyal to the e-commerce company in the platform, the platform will also actively supervise its subordinate company.

Based on Proposition 3, the phase diagram of the strategy evolution of the e-commerce platform is shown in Figure 4.

Phase diagram of strategy evolution of e-commerce platform.

Inference 3: with the increase of the value of F, β, and T_p, the e-commerce platform is more inclined to implement the information supervision strategy, when other parameters remain unchanged. Similarly, with the increase of the value of C_p and I_p, the e-commerce platform is more inclined to implement the information nonsupervision strategy. It shows that the proportion of e-commerce platform implementing information supervision strategy is directly proportional to the fines imposed by the platform for differential pricing of e-commerce company, the proportion of fines imposed by the e-commerce platform for differential pricing of e-commerce company, and the reputation value brought by the loyal consumer to the platform, and inversely proportional to the cost of the platform's information supervision on e-commerce company and the fines by government imposed on the platform for nonsupervision of e-commerce company information resulting in differential pricing.

Since y₀=(C_p+(1 − x)rI_p − (1 − x)βF/T_p − (1 − x)(1 − r)I_p), the volume of V_z1 in Figure 4 represents the proportion of information supervision of e-commerce company by the platform, and the corresponding volume of V_z0 represents the proportion of information nonsupervision by the platform. When the value of F, β, and T_p gradually increase, the value of y₀ will gradually decrease, and the volume of V_z1 will increase at this time, indicating that the proportion of e-commerce platform to implement information supervision increases; When the value of C_p and I_p gradually increases, the value of y₀ will gradually increase, and the volume of V_z1 will decrease at this time, indicating that the proportion of e-commerce platform to implement information supervision decreases.

Assuming that the expected benefit of government regulatory department when government implementing the strategy of strictly supervising is U₄₁, the expected benefit of government regulatory department when government implementing the strategy of loosely supervising is U₄₂, and the average expected benefit of the government regulatory department is U4¯, which are defined as follows:

According to the Malthusian dynamic equation, the replication dynamic equation of the government regulatory department is obtained as follows:

The first partial derivative of F (r) for r is as follows:

Based on the stability theorem of differential equations, government regulatory department implements the strategy of strictly supervising in the stable state must meet the conditions: F(r) = 0, and F_r′(r) < 0.

When z > z₀, the government regulatory department will choose strict supervision as the stable strategy; when z < z₀, the stable strategy of the government regulatory department will choose loose supervision as the stable strategy; when z = z₀, the government regulatory department cannot determine the stable strategy. Where the threshold is as follows:

Assume H(z)=−C_g+xR+(1 − x)I_e+(1 − x)(1 − z)I_p+(1 − x)N − (1 − x)yα[I_e+(1 − z)I_p], when (∂H/∂x) < 0, H (z) is considered to be an increasing function of z. When z < z₀, H (z) > 0, F(r)|_r=1=0, and F_r′(r)|_r=1 < 0, so r = 1 has stability; When z > z₀, H (z) < 0, F(r)|_r=0=0, and F_r′(r)|_r=0 < 0, so r = 0 has stability; When z = z₀, H (z) = 0, F(r)=0, and F_r′(r)=0, so z is stable at all levels in the range of 0 to 1, that is, the government regulatory department's strategy does not change over time, regardless of the proportion of government regulatory department choosing to strict supervision.

Proposition 4 states that the decline of the proportion of information supervision of e-commerce company by e-commerce platform will change the stable strategy of government regulatory department from loose supervision to strict supervision; Similarly, the increase of the proportion of information supervision of e-commerce company by e-commerce platform will change the stable strategy of government regulatory department from strictly supervising to loosely supervising. Therefore, the government's strict supervision on e-commerce company is the necessary measure under the unfavorable conditions of the e-commerce platform's information supervision on e-commerce company.

Based on Proposition 4, the phase diagram of strategy evolution of the government regulatory department is shown in Figure 5.

Phase diagram of strategy evolution of government regulatory department.

Inference 4: With the increase of the value of R, I_e, I_p, and N, the government regulatory department is more inclined to implement the strict supervision strategy, when other parameters remain unchanged. Similarly, with the increase of the value of C_g and α, the government is more inclined to implement the loose supervision strategy. It shows that the proportion of government regulatory department implementing strict supervision strategy is directly proportional to the social benefits obtained, the fines punished by the government on e-commerce company and platform, and the social reputation loss caused by differential pricing under the government's loose supervision, and inversely proportional to the cost for the government to strictly supervise and the proportion of consumer discovering differential pricing.

Since z₀=1 − (C_g − xR − (1 − x)[(1 − αy)I_e+N]/(1 − x)(1 − αy)I_p), the volume of V_r1 in Figure 5 represents the proportion of strictly supervised by the government, and the corresponding volume of V_r0 represents the proportion of loosely supervised by government. When the value of R, I_e, I_p, and N gradually increases, the value of z₀ will gradually increase, and the volume of V_r1 will increase at this time, indicating that the proportion of strict supervision by government regulatory department increases; When the value of C_g and α gradually increase, the value of z₀ will gradually decrease, and the volume of V_r1 will decrease at this time, indicating that the proportion of strict supervision by government regulatory department increases decreases.