Model assumptions
Hypothesis 1. Blockchain technology can improve the safety, synergy and efficiency of online carhailing, and further enhance the efficiency of urban transportation systems. According to the innovation maturity theory proposed by Jay Paap and Ralph Katz^{45}, we set technical efficiency as \(P=T{C}_{3}\), where \(T\) is the blockchain technology maturity, and \({C}_{3}\) is the R&D investment for the platform to introduce blockchain technology.
Hypothesis 2. The basic revenue of the platform is R. When the platform chooses active regulation, it pays operating cost \({C}_{2}\) and technology R&D cost \({C}_{3}\) and generates social welfare \(S\); when the platform chooses negative regulation, it causes social welfare losses \(L\). In the case of the introduction of blockchain, the platform’s negative regulation pays high additional cost \(d\) (the cost of data tampering and information fabrication, loss of reputation, etc.).
Hypothesis 3. When the government chooses active governance, it pays total governance cost \(C\) consisting of fixed governance cost and marginal governance cost. Blockchain technology reduces the risk of information asymmetry by reducing the probability of information distortion, thereby effectively reducing the cost of governance. Therefore, blockchain technology has a certain degree of substitution with governance. The higher blockchain technology maturity is, the lower the marginal governance cost is. We set the total governance cost as \(C={C}_{1}+(1T)/k\), where \({C}_{1}\) is the fixed governance cost and \(k\) is a constant representing the governance capacity.
Hypothesis 4. When the government chooses active governance, it encourages the online carhailing industry to introduce blockchain for innovative development by increasing investment, providing financial subsidies, and introducing relevant policies. These inputs are set as government innovation input \(F\). In addition, the government imposes fine \(f\) on the platform that chooses negative regulation.
On this basis, we derive the payoff matrix of the platform and the government under blockchain technology, presented in Table 4.
Evolutionary stability analysis
According to Table 4, when the platform chooses active regulation, the expected revenue \({E}_{x1}\) is as follows:
$${E}_{x1}=y\left(R+P+F{C}_{2}{C}_{3}\right)+\left(1y\right)\left(R+P{C}_{2}{C}_{3}\right)=yF+R+P{C}_{2}{C}_{3}$$
(7)
When the platform chooses active regulation, the expected revenue \({E}_{x2}\) is as follows:
$${E}_{x2}=y\left(R+Ffd\right)+\left(1y\right)\left(Rd\right)=yFyf+Rd$$
(8)
The average expected revenue \(\overline{{E }_{x}}\) of the platform is as follows:
$$\overline{{E }_{x}}=x{E}_{x1}+\left(1x\right){E}_{x2}=xPx{C}_{2}x{C}_{3}+y\alpha Fyf+Rd+xyf+xd$$
(9)
Similarly, the average expected revenue \(\overline{{E }_{y}}\) of the government is as follows:
$${E}_{y1}=x\left(SFC\right)+\left(1x\right)\left(fFLC\right)=xS+fFLCxf+xL$$
(10)
The replicated dynamic equations of the platform and the government obtained by calculation are as follows:
$$\left\{\begin{array}{c}F\left(x\right)=\frac{{d}_{x}}{{d}_{t}}=x\left({E}_{x1}\overline{{E }_{x}}\right)=x\left(1x\right)\left(P+yf+d{C}_{2}{C}_{3}\right)\\ F\left(y\right)=\frac{{d}_{y}}{{d}_{t}}=y\left({E}_{y1}\overline{{E }_{y}}\right)=y\left(1y\right)\left(fxfCF\right)\end{array}\right.$$
(11)
We calculate the replicated dynamic equations of the platform and the government when \(F\left(x\right)=0\) and \(F\left(y\right)=0\). Then, the five local equilibrium points of the platform and the government strategic choices can be calculated as \({M}_{1}\left(0 , 0\right)\), \({M}_{2}\left(0 , 1\right)\), \({M}_{3}\left(1 , 0\right)\), \({M}_{4}\left(1 , 1\right)\), and \({M}_{5}\left({x}_{0} , {y}_{0}\right)\), where \({x}_{0}=\frac{fCF}{f}\) and \({y}_{0}=\frac{{C}_{2}+{C}_{3}dT{C}_{3}}{f}\).
Platform evolutionary stability analysis
The choice of evolutionary stability strategy usually depends on the initial state of both players. When \(y={y}_{0}\), \(x\) is stable at any level of \(0\le x\le 1\), as shown in Fig. 1a. When \(y\ne {y}_{0}\), \({x}^{*}=0\) and \({x}^{*}=1\) are stable. When \({x}^{*}\) satisfies \({F}^{^{\prime}}\left(x\right)<0\), \({x}^{*}\) is an evolutionarily stable strategy. Therefore, when \(y<{y}_{0}\), \(x=0\) is stable. When the governance is too low, the platform chooses negative regulation, as shown in Fig. 1b. When \(y>{y}_{0}\), \(x=1\) is stable, meaning that when governance reaches a certain level, active regulation is the optimal strategy for the platform in the game system, as shown in Fig. 1c.
Government’s evolutionary stability analysis
When \(x={x}_{0}\), \(y\) is stable at any level of \(0\le y\le 1\), as shown in Fig. 2a. When \(x\ne {x}_{0}\), \({y}^{*}=0\) and \({y}^{*}=1\) are stable. When \({y}^{*}\) satisfies \({F}^{^{\prime}}\left(y\right)<0\), \({y}^{*}\) is an evolutionarily stable strategy. Therefore, when \(x<{x}_{0}\), \(y=1\) is stable. Additionally, when the probability that the platform chooses active regulation is too low, active governance is the optimal strategy of the government in the game system, as shown in Fig. 2b. When \(x>{x}_{0}\) and \(y=0\) is stable. When the probability that the platform chooses active regulation reaches a certain level, the government eventually chooses negative governance, as shown in Fig. 2c.
Evolutionary path analysis
According to the Jacobian matrix solution method, we calculate the Jacobian matrix of the game system as follows:
$$J=\left[\begin{array}{cc}(12x)(T{C}_{3}+yf+d{C}_{2}{C}_{3})& x(1x)f\\ y(1y)f& (12y)(fxfCF)\end{array}\right]$$
(18)
Then, we calculate the determinant \((detJ)\) and trace \((trJ)\) of the Jacobian matrix, and the calculation results are shown in Table 5.
When the Jacobian matrix meets \(detJ>0\) and \(trJ<0\), the equilibrium point of the replicated dynamic equation is the ESS point. According to Table 5, the game system has a saddle point \(\left({x}_{0},{y}_{0}\right)\). The evolutionary stability result of the system changes with the different value ranges of \({x}_{0}\) and \({y}_{0}\). Among them, it is obviously not true when \({x}_{0}>1\). Therefore, we discuss the values of \({x}_{0}\) and \({y}_{0}\) in different situations.

(1)
When \({x}_{0}<0\) and \({y}_{0}<0\), \(fCF<0\) and \({C}_{2}+{C}_{3}dT{C}_{3}<0\) so that \(\left(1 , 0\right)\) is the ESS point of the system.

(2)
When \({x}_{0}<0\) and \(0<{y}_{0}<1\), \(fCF<0\) and \(0<{C}_{2}+{C}_{3}dT{C}_{3}<f\),so that \(\left(0 , 0\right)\) is the ESS point of the system.

(3)
When \({x}_{0}<0\) and \({y}_{0}>1\), \(fCF<0\) and \({C}_{2}+{C}_{3}dT{C}_{3}>f\) so that \(\left(0 , 0\right)\) is the ESS point of the system.

(4)
When \(0<{x}_{0}<1\) and \({y}_{0}<0\), \(fCF>0\) and \({C}_{2}+{C}_{3}dT{C}_{3}<0\) so that \(\left(1 , 0\right)\) is the ESS point of the system.

(5)
When \(0<{x}_{0}<1\) and \(0<{y}_{0}<1\), then \(fCF>0\) and \(0<{C}_{2}+{C}_{3}dT{C}_{3}<f\) and there is no stable point in the system.

(6)
When \(0<{x}_{0}<1\) and \({y}_{0}>1\), \(fCF>0\) and \({C}_{2}+{C}_{3}dT{C}_{3}>f\) so that \(\left(0 , 1\right)\) is the ESS point of the system.
Based on the above discussion, the ESS points and the parameter conditions of the game system are shown in Table 6. The corresponding evolution phase diagrams for the six situations are shown in Fig. 3.
Key parameter analysis

(1)
Key parameters related to the platform
In states I and IV, the platform chooses negative regulation and active regulation, respectively. It is required to meet \(\frac{\partial F(x)}{\partial x}<0\) so that the two states tend to be stable, and the conditions are \(T{C}_{3}+d{C}_{2}{C}_{3}<0\) and \(T{C}_{3}+d{C}_{2}{C}_{3}>0\), respectively. Among them, the platform technology R&D cost \({C}_{3}\) and the additional cost of passive regulation \(d\) are important parameters that affect the platform. When the technology R&D cost is small and the additional cost is large, the expected revenue of the platform choosing active regulation is less than 0. For additional cost \(d\), when the blockchain technology maturity is low, the platform chooses negative regulation to lose less, and the technical efficiency obtained by applying the blockchain is not enough to offset the governance cost of active regulation \(({C}_{2}+{C}_{3})\). Thus, the probability of the platform choosing active regulation is low. For technical efficiency \(P=T{C}_{3}\), since blockchain technology maturity \(T\) is an uncontrollable parameter, to change the platform’s strategy in the game system to an ideal state, the platform should not only make reasonable use of the government’s innovation subsidies but also improve technology R&D efficiency, thereby reducing the technology R&D cost \({C}_{3}\).

(2)
Key parameters related to the government
In states I and II, the government chooses negative governance and active governance, respectively. It is required to meet \(\frac{\partial F(y)}{\partial y}<0\) so that the two states tend to be stable, and the conditions are \(fCF<0\) and \(fCF>0\), respectively. For total governance cost \(C={C}_{1}+(1T)/k\), since blockchain technology maturity \(T\) is an uncontrollable parameter for the government, the key parameters for the establishment of \(fCF>0\) are the government innovation input \(F\) and punishment intensity \(f\). When punishment intensity is small and innovation input is large, the expected revenue of the government choosing active governance is less than 0. When the platform chooses negative regulation, it brings high governance costs to the government, resulting in the net revenue of the government choosing that the active governance is less than 0, meaning that the punishment intensive is less than the sum of innovation input and governance cost. Driven by interests, the government tends to choose negative governance. At the same time, according to the conditions, government innovation input is not the larger the better. Therefore, to change the government’s strategy in the game system to an ideal state, the government should not only increase the punishment intensity for the platform’s negative regulatory behaviours, but also provide reasonable subsidies for the platform’s application of blockchain to increase the willingness of the platform to apply blockchain technology without affecting the interests of the government, and finally achieve the optimal state of government governance.
Numerical experiment
Based on the above analysis, the evolutionary stability results of the system depend on the initial conditions and changes in the relevant parameters. To more intuitively reflect on the evolutionary path of the game between the platform and the government and the influence of the parameter values on evolutionary stability results, this paper uses the Python programming language to carry out numerical experiments. The parameter initialisation values are shown in Table 7, and the system evolution diagrams are shown in Figs. 4, 5, 6, 7, 8, 9, 10 and 11. The horizontal axis in the figures represents the evolution time \(t\) of the system, and the vertical axis represents the proportion \(x(t)\) of the platform choosing active regulation and the proportion \(y(t)\) of the government choosing active governance.
The impact of innovation input and punishment intensity
Technology R&D cost (\({C}_{3}\)). To make the system evolve to state IV, the condition \({C}_{2}+{C}_{3}dT{C}_{3}<0\) should be satisfied. We set the technology development cost \({C}_{3}\) as 2 and 8 and obtain the evolution curves of \(x(t)\) and \(y(t)\) following the change of \({C}_{3}\), as shown in Figs. 4 and 5. The figures show that when the technology R&D cost is relatively large, as \({C}_{3}=8\), the platform tends to choose negative regulation, and when the technology R&D cost is reduced to \({C}_{3}=2\), the platform turns to active regulation.
Additional cost (\(d\)). We set the additional cost \(d\) as 2 and 8 and obtain the evolution curves of \(x(t)\) and \(y(t)\) following the change in d, as shown in Figs. 6 and 7. The figures show that when the additional cost is relatively small, as \(d=2\), the platform tends to choose negative regulation, and when the additional cost increases to \(d=8\), the platform turns to active regulation.
The impact of the technology R&D cost and the additional cost
Government innovation input (\(F\)). To make the system evolve to state \(\mathrm{II}\), the conditions \(fCF>0\), \({C}_{2}+{C}_{3}dT{C}_{3}>f\) should be satisfied. We set the government innovation input \(F\) as 6 and 2, and obtain the evolution curves of \(x(t)\) and \(y(t)\) following the change of \(F\), as shown in Figs. 8 and 9. The figures show that when innovation input is relatively large, as \(F=6\), the government tends to choose negative governance, and when innovation input is reduced to \(F=2\), the government turns to active governance.
Punishment intensity (\(f\)). We set the punishment intensity \(f\) as 6 and 2 and obtain the evolution curves of \(x(t)\) and \(y(t)\) following the change of f, as shown in Figs. 10 and 11. The figures show that when punishment intensity is relatively small, as \(f=2\), the government tends to choose negative governance, and when punishment intensity increases to \(f=6\), the government turns to active governance.
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