The variables employed and their meanings within the tripartite evolutionary game model are shown in Table 1.

Assumption 1

We envisage a scenario in which the government, renewable power producers, and coal-fired power producers are the participants in strategic interaction within the model. Operating with economic rationality, this triad engages in an evolutionary game that unfolds under conditions of incomplete information symmetry. Through a process of continuous trial and error, equilibrium is ultimately achieved. Notably, the validity period of green certificates is 1 year, and speculative behavior is nonexistent.

Assumption 2

In the near term, the challenges surrounding investing in renewable power persist. The government pursues two distinct strategic paths: (vigorous enforcement of the mandatory quota system, accompanied by penalties for coal-fired power producers failing to meet quotas; application of non-mandatory quotas). Renewable power producers, in contrast, choose from the following strategies: (certification and sale of green certificates; continuance without certification, thereby forgoing fixed grid pricing subsidies). Finally, coal-fired power producers choose from the following strategies: (procurement of green certificates to fulfill government-imposed quota objectives; acceptance of governmental penalties if abstaining from purchasing green certificates).

Assumption 3

The tariff on renewable power under the FIT mechanism aligns with the sum of coal-fired power pricing and subsidies. Under the government’s pursuit of mandatory quota fulfillment, returns encompass the attainment of predetermined objectives and utility benefits (U) arising from public approval. The corresponding costs encompass administrative expenditures (C). When renewable power producers opt to sell green certificates, their earnings amount to a δ Q 0 + b Q 0 δ . Conversely, the choice to abstain from selling these certificates leads to earnings manifesting as subsidies ( ϖ Q 0 δ α ). A decision by coal-fired power producers to purchase green certificates yields − a δ Q 0 + b Q 0 δ , while forgoing the acquisition results in − F Q 0 δ .

Assumption 4

In the scenario in which renewable power producers certify and trade green certificates, earnings accrue in the absence of any outstanding financial obligations. Moreover, in the event that renewable power producers decline certification and continue under fixed grid pricing, the likelihood of eventually securing surplus subsidies beyond market prices is denoted as α .

Assumption 5

Within this framework, the probability that renewable power producers opt to certify and sell green certificates stands at x   0   <   x   <   1 , while the likelihood of refraining from certificate certification stands at 1 − x . For coal-fired power producers, the choice to purchase green certificates bears a probability of y   0   <   y   <   1 , whereas abstention from such procurement corresponds to 1 − y . Moreover, the government selects a mandatory or non-mandatory quota system with probability z   0   <   z   <   1 or 1 − z , respectively.

In this study, it is assumed that in the green certificate market, the supply function of green certificates is represented by P = a Q 0 + b and the demand function is given by Q = Q 0 δ . Building upon the aforementioned foundation and contingent upon the government’s selection of either a mandatory or a non-mandatory quota strategy, the strategic choices of renewable power producers, coal-fired power producers, and the government can be comprehensively depicted through the construction of distinct game payoff matrices (refer to Table 2 for the specific matrices pertaining to different scenarios). π , κ , and τ represent the revenues of renewable power producers, coal-fired power producers, and the government, respectively, under various strategic choices. The revenue equations under different strategic choices are as follows: π 1 , κ 1 , τ 1 = a δ Q 0 + b Q 0 δ , − a δ Q 0 + b Q 0 δ , U − C (1) π 2 , κ 2 , τ 2 = a δ Q 0 + b Q 0 δ , − F Q 0 δ , U − C (2) π 3 , κ 3 , τ 3 = ω ¯ Q 0 δ α , − a δ Q 0 + b Q 0 δ , U − C (3) π 4 , κ 4 , τ 4 = ω ¯ Q 0 δ α , − F Q 0 δ , U − C (4) π 5 , κ 5 , τ 5 = a δ Q 0 + b Q 0 δ , − a δ Q 0 + b Q 0 δ , 0 (5) π 6 , κ 6 , τ 6 = a δ Q 0 + b Q 0 δ , 0,0 (6) π 7 , κ 7 , τ 7 = ω ¯ Q 0 δ α , − a δ Q 0 + b Q 0 δ , 0 (7) π 8 , κ 8 , τ 8 = ω ¯ Q 0 δ α , 0,0 (8)

The interactions among the government, renewable power producers, and coal-fired power producers represent an evolutionary game involving distinct cohorts, thus constituting an asymmetric game.

The replicator dynamics equation for renewable power producers is as follows: L = F x = d x d t = x π s − π ¯ (9)

The expected utilities of the strategies of “opting for certification and selling green certificates” and “not certifying green certificates” for renewable power producers are denoted as π s , π s ¯ , and population utility π ¯ , respectively. π s = y z π 1 + 1 − y z π 2 + y 1 − z π 5 + 1 − y 1 − z π 6 (10) π s ¯ = y z π 3 + 1 − y z π 4 + y 1 − z π 7 + 1 − y 1 − z π 8 (11) π ¯ = x π s + 1 − x π s ¯ (12)

Substituting Eqs 10–12 into Eq. 9 yields: L = F x = d x d t = x π s − π ¯ = x 1 − x y z π 1 − π 3 − π 5 − π 7 + π 5 − π 7 y + π 2 − π 4 − π 6 − π 8 z 1 − y + π 6 − π 8 1 − y . (13)

The replicator dynamics equation for coal-fired power producers is as follows: H = F y = d y d t = y κ g − κ ¯ (14)

The utility expectations of coal-fired power producers choosing between “purchasing green certificates” and “not purchasing green certificates” are denoted as κ g , κ g ¯ , and population utility κ ¯ , respectively. κ g = x z κ 1 + z 1 − x κ 3 + 1 − z x κ 5 + 1 − x 1 − z κ 7 (15) κ g ¯ = x z κ 2 + z 1 − x κ 4 + 1 − z x κ 6 + 1 − x 1 − z κ 8 (16) κ ¯ = y κ g + 1 − y κ g ¯ (17)

Substituting Eqs 15–17 into Eq. 14 yields: H = F y = d y d t = y κ g − κ ¯ = y 1 − y x z κ 1 − κ 2 − κ 5 − κ 6 + κ 5 − κ 6 x + κ 3 − κ 4 − κ 7 − κ 8 z 1 − x + κ 7 − κ 8 1 − x (18)

The replicator dynamics equation for the government is as follows: G = F z = d z d t = y τ q − τ ¯ (19)

The government’s preference for selecting between a “mandatory quota system” and a “non-mandatory quota system” yields the expected utilities of τ q , τ q ¯ , and the population utility τ ¯ , respectively. τ q = x y τ 1 + x 1 − y τ 2 + 1 − x y τ 3 + 1 − x 1 − y τ 4 (20) τ q ¯ = xy τ 5 + x 1 − y τ 6 + 1 − x y τ 7 + 1 − x 1 − y τ 8 (21) τ ¯ = z τ g + 1 − z τ g ¯ (22)

Substituting Eqs 20–22 into Eq. 19 yields: Z = F z = d z d t = z τ q − τ ¯ = z 1 − z x y τ 1 − τ 5 − τ 3 − τ 7 + τ 3 − τ 7 y + τ 2 − τ 6 − τ 4 − τ 8 x 1 − y + τ 4 − τ 8 1 − y (23)

When the learning rates of F x = d x / d t = 0 , F y = d y / d t = 0 , and F z = d z / d t = 0 , which represent the three entities of the game, converge to zero, the evolutionary game reaches a state of local equilibrium. The partial equilibrium points are H 1 0,0,0 , H 2 1,0,0 , H 3 0,1,0 , H 4 0,0,1 , H 5 1,1,0 , H 6 1,0,1 , H 7 0,1,1 , and H 8 1,1,1 . This also holds when the system of Eq. 24 below is satisfied. y z π 1 − π 3 − π 5 − π 7 + π 5 − π 7 y + π 2 − π 4 − π 6 − π 8 z 1 − y + π 6 − π 8 1 − y = 0 x z κ 1 − κ 2 − κ 5 − κ 6 + κ 5 − κ 6 x + κ 3 − κ 4 − κ 7 − κ 8 z 1 − x + κ 7 − κ 8 1 − x = 0 x y τ 1 − τ 5 − τ 3 − τ 7 + τ 3 − τ 7 y + τ 2 − τ 6 − τ 4 − τ 8 x 1 − y + τ 4 − τ 8 1 − y = 0 (24)

Solving the system of Eq. 24 yields potential local equilibrium points within the evolutionary game, denoted as x * , y * , z * .

In the realm of multiagent evolutionary game theory, as expounded upon by Ritzberger ¹ in 1996, a strategy combination is asymptotically stable in the dynamic replicator system of the game if and only if it constitutes a strict Nash equilibrium—the states of asymptotic stability and ESS combinations are mutually necessary and sufficient. Therefore, within the tripartite evolutionary game under consideration in the present study, the evolutionarily stable strategy combinations manifest in H 1 0,0,0 , H 2 1,0,0 , H 3 0,1,0 , H 4 0,0,1 , H 5 1,1,0 , H 6 1,0,1 , H 7 0,1,1 and H 8 1,1,1 . Drawing inspiration from the analysis of multiagent evolutionary games concerning environmental regulations by Xv et al. (2017) as well as the work of Yang et al. (2020) based on the Lyapunov stability theory, and employing the eigenvalues of the Jacobian matrix as a determinant of the asymptotic stability of the local equilibrium points within a tripartite evolutionary game to corroborate ESSs (Liu and Dong, 2022), the Jacobian matrix in our tripartite evolutionary game can be deduced. J = J 11 J 12 J 13 J 21 J 22 J 23 J 31 J 32 J 33 = ∂ L ∂ x ∂ L ∂ y ∂ L ∂ z ∂ H ∂ x ∂ H ∂ y ∂ H ∂ z ∂ G ∂ x ∂ G ∂ y ∂ G ∂ z (25)

The partial derivatives of the dynamic equations for the tripartite evolutionary game involving renewable power producers, coal-fired power producers, and the government are obtained from Eq. 25 as follows: ∂ L ∂ x = 1 − 2 x y z π 1 − π 3 − π 5 − π 7 + π 5 − π 7 y + π 2 − π 4 − π 6 − π 8 z 1 − y + π 6 − π 8 1 − y (26) ∂ L ∂ y = x − x 2 z π 1 − π 3 − π 5 − π 7 + π 5 − π 7 − π 2 − π 4 − π 6 − π 8 z − π 6 − π 8 (27) ∂ L ∂ z = x − x 2 y [ π 1 − π 3 − π 5 − π 7 ] + [ π 2 − π 4 − π 6 − π 8 ] 1 − y (28) ∂ H ∂ x = y 1 − y z κ 1 − κ 2 − κ 5 − κ 6 + κ 5 − κ 6 − κ 3 − κ 4 − κ 7 − κ 8 z − κ 7 − κ 8 (29) ∂ H ∂ y = 1 − 2 y x z κ 1 − κ 2 − κ 5 − κ 6 + κ 5 − κ 6 x + κ 3 − κ 4 − κ 7 − κ 8 z 1 − x + κ 7 − κ 8 1 − x (30) ∂ H ∂ z = y 1 − y x κ 1 − κ 2 − κ 5 − κ 6 + κ 3 − κ 4 − κ 7 − κ 8 1 − x (31) ∂ G ∂ x = z 1 − z y τ 1 − τ 5 − τ 3 − τ 7 + τ 2 − τ 6 − τ 4 − τ 8 1 − y (32) ∂ G ∂ y = z 1 − z x τ 1 − τ 5 − τ 3 − τ 7 + τ 3 − τ 7 − τ 2 − τ 6 − τ 4 − τ 8 x − τ 4 − τ 8 (33) ∂ G ∂ z = 1 − 2 z x y τ 1 − τ 5 − τ 3 − τ 7 + τ 3 − τ 7 y + τ 2 − τ 6 − τ 4 − τ 8 x 1 − y + τ 4 − τ 8 1 − y (34)

Pursuant to the Lyapunov stability criteria, when all eigenvalues of the Jacobian matrix λ are negative, the local equilibrium point achieves gradual stability and is designated as an attractor; if the Jacobian matrix bears a single negative eigenvalue λ, the local equilibrium point becomes unstable and is classified as a source; and in cases where the eigenvalues of the Jacobian matrix consist of one negative and two positive or one positive and two negative values, the local equilibrium point becomes unstable, akin to a saddle point. Therefore, through differentiation of Eq. 24 with respect to the renewable power producers, coal-fired power producers, and the government acting in the tripartite evolutionary game, we derive the eigenvalues of the Jacobian matrix at the local equilibrium points H 1 0,0,0 , H 2 1,0,0 , H 3 0,1,0 , H 4 0,0,1 , H 5 1,1,0 , H 6 1,0,1 , H 7 0,1,1 , H 8 1,1,1 , and H * x * , y * , z * .

The simulated strategies of the government, renewable power producers, and coal-fired power producers in Scenario 1, where the market price of green certificates surpasses the subsidies for renewable power generation and the penalty for failing to meet mandatory quotas outweighs the market price of green certificates, are depicted in Figure 1.

In Scenario 1, these three parties progressively converge toward unity, ultimately establishing a tripartite evolutionary equilibrium at point H 8 1,1,1 . The government opts to enforce mandatory quotas, renewable power producers choose to authenticate and trade green certificates, and coal-fired power producers decide to purchase green certificates, thereby resulting in a successful green certificate transaction.

The simulated strategies of the government, renewable power producers, and coal-fired power producers in Scenario 2, where the market price of green certificates falls below the subsidies for renewable power generation and the penalty for failing to meet mandatory quotas surpasses the market price of green certificates, are illustrated in Figure 2.

In Scenario 2, the government and coal-fired power producers gradually converge toward unity, while renewable power producers converge toward zero and opt out of green certificate transactions. The tripartite evolutionary equilibrium stabilizes at point H 7 0,1,1 , with the government enforcing mandatory quotas, coal-fired power producers purchasing green certificates, and renewable power producers refusing to authenticate green certificates, thereby resulting in an unsuccessful green certificate transaction.

The simulated strategic dynamics among the government, renewable power producers, and coal-fired power producers in Scenario 3, wherein the price of green certificates falls below the subsidies for renewable power generation and the penalty for not meeting mandatory quotas is less than the market price of green certificates, are presented in Figure 3.

In Scenario 3, the government converges to a value of one, while renewable and coal-fired power producers gradually approach a value of zero, exhibiting resistance to engage in green certificate transactions. The tripartite evolutionary equilibrium stabilizes at point H 4 0,0,1 , wherein the government opts to enforce mandatory quotas, coal-fired power producers choose to acquire green certificates, and renewable power producers opt not to validate green certificates, ultimately leading to the collapse of green certificate transactions.

The strategic dynamics of the government, renewable power producers, and coal-fired power producers in Scenario 4, where the market price of green certificates exceeds the subsidies for renewable power generation and the penalty for not meeting mandatory quotas is less than the market price of green certificates, are illustrated in Figure 4.

In Scenario 4, the government and renewable power producers converge to a value of one, while coal-fired power producers converge to a value of zero, refraining from purchasing green certificates. The tripartite evolutionary equilibrium stabilizes at point AAAAA, with the government choosing to enforce mandatory quotas, renewable power producers opting to validate and trade green certificates, and coal-fired power producers opting not to purchase green certificates, ultimately causing the collapse of green certificate transactions.

By artificially simulating and evaluating the equilibrium strategies of the government, renewable power producers, and coal-fired power producers in the four aforementioned scenarios concerning mandatory and non-mandatory quotas, the conclusions drawn from the tripartite evolutionary game analysis are verified. Furthermore, to ensure that green certificate transactions become a shared strategy among power producers, the government must meticulously formulate the primary parameters of RPS and TGC, thereby averting Scenarios 2 to 4 and realizing the circumstances illustrated in Scenario 1. In the following section, we delve into a detailed discussion in the context of Scenario 1 concerning the impact of subsidy, quota, and penalty intensity parameter settings on green certificate transactions in the presence of mandatory quotas.

We proceed to investigate the impact of different subsidy withdrawal patterns on green certificate transactions under Scenario 1 in the evolutionary game among stakeholders, where the ultimate equilibrium of H 8 1,1,1 is achieved and remains stable. Four distinct subsidy withdrawal approaches have been devised. The first involves uniform withdrawal of subsidies; the second features gradual-then-rapid withdrawal; the third involves swift-then-gradual withdrawal; and the fourth employs stepwise subsidy withdrawal. Simulations and emulations of the effects of these divergent subsidy withdrawal strategies on green certificate transactions are depicted in Figure 5.

From Figure 5, it can be seen that under the first approach, coal-fired power producers reach equilibrium stability at Period 14, while renewable power producers attain equilibrium stability at Period 19. Notably, coal-fired power and renewable power producers do not synchronously adopt green certificate trading strategies, thereby hindering the achievement of green certificate transactions. Under the second approach (i.e., gradual-then-rapid subsidy withdrawal), both coal-fired and renewable power producers stabilize at equilibrium almost simultaneously by Period 20, thus facilitating the attainment of green certificate transactions, albeit with a lengthier equilibrium stabilization period. Under the fourth approach of stepwise subsidy withdrawal, both coal-fired and renewable power producers achieve equilibrium stability by Period 20. The third approach, which involves swift-then-gradual subsidy withdrawal, leads both coal-fired and renewable power producers to simultaneously achieve equilibrium stability by Period 15, thus facilitating the swift and efficient realization of green certificate transactions.

Consequently, in China’s transition from FIT to RPS–TGC regulation of renewable power pricing, opting for the third subsidy withdrawal approach (i.e., swift-then-gradual withdrawal) would enable the rapid realization of this transition.

In this section, a specific comparative analysis is conducted to assess the equilibrium stability tendencies of renewable and coal-fired power producers in the context of an effective RPS–TGC framework (namely, Scenario 1), when the compulsory quota of δ is 10%, 20%, or 30%. Subsequently, the impact of quota specifications on green certificate transactions is investigated. The simulation and emulation of the effects of three distinct quota ratios on green certificate transactions are illustrated in Figure 6. Points plotted with the • symbol represent a quota of 10%; those plotted with the o symbol illustrate a ratio of 20%; and those plotted with the * symbol represent a ratio of 30%.

When δ is set to 10%, as depicted in Figure 6, it becomes evident that the formation of a stable equilibrium strategy among the three entities requires a considerable amount of time. Given the reduced demand for green certificates within the TGC market due to the lower quota, the price of these certificates declines correspondingly. This price fluctuation creates an incentive for coal-fired power producers to engage in purchases, thereby elevating the probability of their doing so. At the same time, the fluctuating green certificate price diminishes the incentive for renewable energy producers to trade these certificates, which in turn reduces the probability of their engaging in market transactions. Given the steeper slope of the curve for coal-fired power producers, the curves representing the two parties intersect before converging on a common stable equilibrium strategy. Before the intersection point is reached, the probability of coal-fired power producers purchasing green certificates is lower than that of renewable power producers selling them. This situation reverses after the intersection point, thereby prolonging the completion time for green certificate transactions.

With δ set to 30%, the formation of a stable equilibrium strategy among the three entities also requires a substantial period of time. Furthermore, the time taken by renewable power producers to reach equilibrium stability is notably shorter than that required by coal-fired power producers, which impedes the green certificate transaction process. A higher δ value leads to an increased demand for green certificates in the TGC market, which causes the price of certificates to surge. This price fluctuation exerts a suppressive effect on coal-fired power producers’ incentives to purchase green certificates, thereby resulting in a decrease in the probability of their engaging in transactions. At the same time, the price signal enhances the incentive for renewable power producers to trade green certificates, which has the opposite effect. The steeper slope of the curve for renewable power producers leads to an intersection being reached before both parties attain a common stable equilibrium strategy. Prior to the intersection point being reached, the probability of coal-fired power producers purchasing green certificates surpasses that of renewable power producers selling them. This trend reverses after the intersection point is reached, thus extending the time required to finalize green certificate transactions.

Finally, in the scenario with a quota of 20%, both renewable and coal-fired power producers achieve strategic convergence and sustain a simultaneous ascent to equilibrium stability by the ninth period. This favorable outcome enhances the likelihood of a three-party green certificate transaction occurring. In summary, in contrast to the baseline scenario, setting δ either too low or too high impedes consensus formation and thus the success of green certificate transactions, which hinders the transition from FIT to RPS–TGC renewable energy pricing regulation.

We proceed to engage in a specific comparative analysis under the assumption of an efficacious RPS–TGC scenario in order to scrutinize the effects of three distinct penalty levels (namely, F = 1.3P, F = 1.5P, and F = 2P) on the equilibrium stability tendencies of renewable power producers, coal-fired power producers, and the governing body, with a specific focus on the implications for green certificate transactions. The ramifications of these divergent penalty levels for green certificate transactions are simulated and presented in Figure 7. Points plotted with the * symbol correspond to the F = 1.3P curve, which represents a low-penalty scenario; those plotted with the • symbol signify the F = 1.5P curve, which represents a moderate-penalty scenario; and those plotted with the o symbol denote the F = 2P curve, which represents a high-penalty scenario.

Upon analyzing Figure 7, it becomes evident that in the context of a low-penalty scenario, the establishment of a stable equilibrium strategy requires a substantial period of time. As the penalty factor (F) diminishes, the likelihood of coal-fired power producers purchasing green certificates decreases, while the probability of their incurring penalties from the governing body increases. Consequently, the rate of adjustment in the trading strategy of renewable energy producers with respect to green certificates also diminishes, but synchronizing the behaviors of these two parties remains challenging. This mutual lack of synchronization results in flatter curves for both parties, thus delaying the convergence to a mutually shared trading strategy. In the high-penalty scenario (F = 2P), coal-fired power producers experience relatively rapid evolution toward equilibrium stability in their strategic behavior, but this pace significantly outstrips that of renewable power producers, thereby leading to pronounced asynchrony. With an increase in the penalty factor (F), the propensity of coal-fired power producers to accept penalties from the governing body decreases, thus leading to a rapid escalation in the rate of adjustment in their trading strategy and resulting in steeper curves. Conversely, the rate of adjustment in the trading strategy of renewable power producers notably lags behind that of coal-fired power producers, as can be seen by the evident time delay. This asynchronous adjustment undermines the formation of a mutually shared trading strategy and postpones the achievement of a successful green certificate transaction.

In the context of a moderate-penalty scenario, both renewable and coal-fired power producers promptly synchronize their strategic behaviors and jointly converge toward an equilibrium stability strategy. This synchronization greatly contributes to the realization of a shared strategy selection within the dynamic framework of the tripartite game and facilitates green certificate transactions. In contrast with the moderate-penalty scenario, the adoption of excessively low or high penalties hinders the inclination of the agents in the tripartite game toward the shared strategic choice of engaging in green certificate transactions. This in turn impedes the transition from FIT pricing to RPS–TGC renewable energy pricing regulation.