### Analysis of the stability of each participant’s strategy

In an evolving game, three players dynamically adjust their strategies by adjusting the values ​​of (X), (y) and (z), and optimize their strategies through learning and imitation. Scalable stable strategy (ESS) points should be robust to minor perturbations according to the stability theorem of differential equations and properties of ESS36. In order to clarify the strategic choice of each participant, the dynamic evolution trend of three actors is analyzed respectively, therefore, the critical condition for decision-making is identified.

#### Analysis of the stability of the strategy of private companies

The probability that private firms choose strategy P1 is (X). According to eq. (4), the first derivative of (F(x)) is as follows:

$$frac{dF(x)}{{dx}} = (1 – 2x)left[ {u_{C} (omega L_{P1} – L_{P1} ) + pi (y)pi (z)u_{C} (F + E_{p} ) + pi (y)pi (1 – z)u_{C} (S_{P} )} right]$$

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When (pi (y) = pi (y)* = frac{{u_{C} (L_{P1} – omega L_{P1} )}}{{pi (z)u_{C} ( F + E_{p} ) + pi (1 – z)u_{C} (S_{P} )}})so (frac{dF(x)}{{dx}} equiv 0). It indicates that all private company strategies are at a steady state. Otherwise, when (pi (y) ne pi (y)*)if on condition that (pi (y) the results are as follows: (frac{dF(x)}{{dx}}|_{x = 0} , (frac{dF(x)}{{dx}}|_{x = 1} > 0). In this case, it indicates that (x = 0) is ESS and private companies tend to choose the P2 strategy. If on condition that (pi (y) > pi (y)*)so, (frac{dF(x)}{{dx}}|_{x = 0} > 0), (frac{dF(x)}{{dx}}|_{x = 1} . In this case, it implies that (x = 1) is ESS and private companies prefer to choose the P1 strategy. Therefore, the probability response function (X) that private companies choose the hardworking behavior strategy P1 is as follows:

$$x = left{ {begin{array}{*{20}l} {0,} hfill & {if;pi (y) frac{{u_{C} (L_{P1} – omega L_{P1} )}}{{pi (z)u_{C} (F + E_{p} ) + pi (1 – z) u_ {C} (S_{P} )}}} hfill end{array} } right.$$

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The dynamic phase diagram of the private firm is therefore shown in Fig. 9. As noted above, the evolving steady state of private firms is closely tied to the strategic choice of citizens and government. Along with the increase in the probability of citizen participation in regulation, the probability of private firms adopting the effort behavior strategy P1 increases and gradually converges to 1. Further analysis of the model shows that, to induce private companies to choose the P1 strategy, increase the perceived value of government sanction (u_{C} (F)) and loss of reputation (u_{C} (S_{P} )) is an effective method.

#### Citizen Strategy Stability Analysis

The probability that citizens choose strategy C1 is (y). According to eq. (5), the first derivative of (F(y)) is as follows:

$$frac{dF(y)}{{dy}} = (1 – 2y)left[ { – u_{C} (C_{C1} ) + pi (1 – x)pi (z)u_{I} (E_{G} + E_{p} )} right]$$

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When (pi (z) = pi (z)* = frac{{u_{C} (C_{C1} )}}{{pi (1 – x)u_{I} (E_{G} + E_{p} )}})so (frac{dF(y)}{{dy}} equiv 0). This implies that all citizen strategies are at a steady state. Otherwise, when (pi (z) ne pi (z)*)if on condition that (pi (z) the results are as follows: (frac{dF(y)}{{dy}}|_{y = 0} , (frac{dF(y)}{{dy}}|_{y = 1} > 0). In this case, it means that (y = 0) is ESS and citizens prefer to choose strategy C2. If on condition that (pi (z) > pi (z)*)so, (frac{dF(y)}{{dy}}|_{y = 0} > 0), (frac{dF(y)}{{dy}}|_{y = 1} . In this case, it implies that (y = 1) is ESS and citizens are inclined to choose strategy C1. Therefore, the probability response function (y) citizens choose to impose the framework strategy C1 is as follows:

$$y = left{ {begin{array}{*{20}l} 0 hfill & {if;pi (z) frac{{u_{C} (C_{C1} )}}{{pi (1 – x) u_{I} (E_{G} + E_{p} )}}} hfill end{ array} } right.$$

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The dynamic phase diagram of the citizen is thus represented in Fig. 10. The evolving balance of citizens is closely linked to the strategic choice of private companies and government. With the increase in the probability of strict government supervision, the probability of the C1 strategy that citizens participate in the supervision increases and converges to 1. Moreover, it is an effective way to increase the perceived value of the remuneration (u_{I} (E_{G} )) and (u_{I} (E_{P} )) motivate the public to participate in monitoring.

#### Analysis of the stability of the government’s strategy

The probability that the government chooses strategy G1 is (z). According to eq. (6), the first derivative of (F(z)) is as follows:

$$frac{dF(z)}{{dz}} = (1 – 2z)left[ {u_{I} (G_{G1} – G_{G2} ) + pi (y)left[ { – u_{C} (C_{G} ) + pi (1 – x)left[ {u_{I} (F) + u_{C} (S_{G} – E_{G} )} right]} right]} right]$$

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When (pi (x) = pi (x)* = frac{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – C_{G} – E_{G} )} right] + u_{I} (G_{G1} – G_{G2} )}}{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – E_{G} )} right]}})so (frac{dF(z)}{{dz}} equiv 0). This implies that all government strategies are at a steady state. Otherwise, when (pi (x) ne pi (x)*)if on condition that (pi (x) the results are as follows: (frac{dF(z)}{{dz}}|_{z = 0} > 0), (frac{dF(z)}{{dz}}|_{z = 1} . In this case, it means that (z = 1) is the SSE and the government tends to choose the G1 strategy to implement strict regulation. If on condition that (pi (x) > pi (x)*)so, (frac{dF(z)}{{dz}}|_{z = 0} , (frac{dF(z)}{{dz}}|_{z = 1} > 0). In this case, it implies that (z = 0) is ESS and the government is inclined to choose the G2 strategy. Therefore, the probability response function (z) that the government selects the strict surveillance strategy G1 is as follows:

$$z = left{ {begin{array}{*{20}l} 0 hfill & {if;pi (x) > frac{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – C_{G} – E_{G} )} right] + u_{I} (G_{G1} – G_{G2} )}}{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – E_{G} )} right]}}} hfill {[0,1]} hfill & {if;pi (x) = frac{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – C_{G} – E_{G} )} right] + u_{I} (G_{G1} – G_{G2} )}}{{pi (y)left[ {u_{I} (F) + u_{C} (S_{G} – E_{G} )} right]}}} hfill 1 hfill & {if;pi (x) (12) The dynamic phase diagram of the government is thus represented in Fig. 11. The evolving equilibrium state of government is closely linked to the strategic choices of private companies and citizens. Along with the increase in the probability of private firms choosing the P2 strategy, the probability of the strict regulation strategy G1 increases and converges to 1. Further analysis based on the response function reveals that the increase in perceived benefits under strict supervision can promote positive regulatory behavior. Picture 11 ### Tripartite Scalable Game System Strategy Stability Analysis In combination with Eqs. (4), (5) and (6), an evolutionary tripartite dynamic system is obtained as follows:$$left{ {begin{array}{*{20}l} {F(x) = frac{dx}{{dt}} = x(1 – x)(V_{P1} – V_{ P2} ) = 0} hfill {F(y) = frac{dy}{{dt}} = y(1 – y)(V_{C1} – V_{C2} ) = 0} hfill {F(z) = frac{dz}{{dt}} = z(1 – z)(V_{G1} – V_{G2} ) = 0} hfill end{array} } right .$$(13) By solving the equations, 8 pure strategy equilibrium points are obtained, as (P_{1} (0,0,0)), (P_{2} (1,0,0)), (P_{3} (0,1,0)), (P_{4} (0,0,1)), (P_{5} (1,1,0)), (P_{6} (1,0,1)), (P_{7} (0,1,1)), (P_{8} (1,1,1)). According to Lyapunov’s stability theory, the equilibrium of the replicated dynamical system is stable only if the eigenvalues ​​of the Jacobian have negative real parts33. Indeed, when the eigenvalues ​​of the Jacobian matrix corresponding to an equilibrium point are negative, the equilibrium point is a local stable point. As long as at least one eigenvalue of the Jacobian matrix is ​​positive, the point is an unstable point. The Jacobian matrix of system (13) can be obtained as follows:$$J = left[ begin{gathered} J_{11} J_{12} J_{13} hfill J_{21} J_{22} J_{23} hfill J_{31} J_{32} J_{33} hfill end{gathered} right]$$(14)$$left{ {begin{array}{*{20}l} {J_{11} = frac{partial F(x)}{{partial x}} = (1 – 2x)left[ {u_{L} (omega L_{P1} – L_{P1} ) + pi (y)pi (z)u_{L} (F + E_{p} ) + pi (y)pi (1 – z)u_{L} (S_{P} )} right]} hfill {J_{12} = frac{partial F(x)}{{partial y}} = x(1 – x)left[ {pi (z)u_{L} (F + E_{p} ) + pi (1 – z)u_{L} (S_{P} )} right]} hfill {J_{13} = frac{partial F(x)}{{partial z}} = x(1 – x)left[ {pi (y)u_{L} (F + E_{p} ) – pi (y)u_{L} (S_{P} )} right]} hfill {J_{21} = frac{partial F(y)}{{partial x}} = y(1 – y)left[ { – pi (z)u_{G} (E_{G} + E_{p} )} right]} hfill {J_{22} = frac{partial F(y)}{{partial y}} = (1 – 2y)left[ { – u_{L} (C_{C1} ) + pi (1 – x)pi (z)u_{G} (E_{G} + E_{p} )} right]} hfill {J_{23} = frac{partial F(y)}{{partial z}} = y(1 – y)left[ {pi (1 – x)u_{G} (E_{G} + E_{p} )} right]} hfill {J_{31} = frac{partial F(z)}{{partial x}} = z(1 – z)left[ { – pi (y)left[ {u_{G} (F) + u_{L} (S_{G} – E_{G} )} right]} right]} hfill {J_{32} = frac{partial F(z)}{{partial y}} = z(1 – z)left[ { – u_{L} (C_{G} ) + pi (1 – x)left[ {u_{G} (F) + u_{L} (S_{G} – E_{G} )} right]} right]} hfill {J_{33} = frac{partial F(z)}{{partial z}} = (1 – 2z)left[ {u_{G} (G_{G1} – G_{G2} ) + pi (y)left[ { – u_{L} (C_{G} ) + pi (1 – x)left[ {u_{G} (F) + u_{L} (S_{G} – E_{G} )} right]} right]} right]} hfill end{array} } right.

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As shown in Table 3, since the cost of public guardianship is greater than 0, i.e. (u_{C} (C_{C1} ) > 0)then we find that the eigenvalue (lambda_{{{ 2}}} > 0) of (P_{3} (0,1,0)), (P_{5} (1,1,0)) and (P_{8} (1,1,1)). It is obvious that the benefits obtained under tight government control are greater than those under loose control. Due to (u_{I} (G_{G1} – G_{G2} ) > 0)therefore the eigenvalue (lambda_{{{ 3}}} > 0) in (P_{1} (0,0,0)) and (P_{2} (1,0,0)). As for the eigenvalue (lambda_{{{ 2}}}) of (P_{4} (0,0,1)), the reward and compensation for reporting opportunistic behavior outweighs the cost. We can then find that (u_{I} (E_{G} + E_{p} ) > u_{C} (C_{C1} ))So, (P_{4} (0,0,1)) is not a point of equilibrium. For (u_{C} (L_{P1} – omega L_{P1} ) > 0)so the eigenvalue (lambda_{{{ 1}}}) of (P_{6} (1,0,1)) is greater than 0. Based on the actual situation, the penalties for opportunistic behavior in private enterprises outweigh the benefits, i.e. (u_{C} (F + E_{p} ) > u_{C} (L_{P1} – omega L_{P1} )). Then eigenvalue (lambda_{{{ 1}}} > 0) of (P_{7} (0,1,1)). Therefore, there is no stable equilibrium point in the tripartite game model depending on the reality background.

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