Evolutionary Game Analysis of Government and Enterprises Carbon-Reduction based on Public Willingness

7 This paper explores the evolutionary game of government and enterprises carbon-reduction with 8 public willingness constraints. On the basis of the features of government and enterprises in 9 energy saving and emission reduction system, the novel evolutionary game model is constructed. 10 The effects of behavioral strategy and willingness constraint strength are visualized by system 11 dynamics theory. With the aid of these visual indicators, the varying dynamic evolution path under 12 different situations is put forward. The economic interpretation of evolutionary stable strategies is 13 discussed. The results show that, public willingness can promote government-enterprise to achieve 14 the optimal state (action, carbon-reduction) spontaneously. The initial willingness can speed up the 15 convergence rate of these two players’ behaviors. The residents’ willingness further restrains the 16 behaviors of government and enterprises, which can eliminate the possibility of adopting passive 17 strategies and reduce the lag of strategies for both parties.


Introduction 20
Climate warming is a common challenge for all human beings. The extreme weather, 21 ecological imbalance and other problems are profoundly affecting the process of human 22 civilization. The mainstream believes that greenhouse gas emissions from human activities are the 23 main cause of global warming (Rosa and Dietz 2012). There was a long and arduous negotiation 24 between international communities in order to control greenhouse gas emissions. It has 25 successively reached such landmark international conventions as the United Nations Framework 26 Convention on climate change, the Kyoto Protocol and the Paris Agreement. In the development 27 and implementation of these conventions, people come to realize that energy saving and emission 28 reduction (ESER) is the only way to deal with global warming (Jiao et al. 2021). The smooth 29 implementation of ESER cannot be separated from the correct leadership of government. As a 30 matter of fact, no country is immune to the complex and multiple climate problems in the world. 31 Consequently it is very important to formulate ESER policies according to national conditions. 32 As the main source of carbon emissions, enterprises are the key actors in ESER (Yu et al. 33 2019). However, in reality, the number of enterprise that takes the initiative to take measures to 34 reduce carbon emissions is relatively small. The main reasons for this phenomenon lie in 35 government's inaction, enterprises' pursuit of short-term profit maximization, and weak public 36 environmental awareness (Liu 2012). Both enterprise emission reduction and government 37 regulation have to pay appropriate costs ). As the main driving force of ESER, 38 government should actively supervise and guide enterprises' emission reduction behavior (Peters 39 et al. 2010). Rational enterprises will identify policy orientation and decide whether to carry out 40 ESER according to the principle of profit maximization. Conversely, enterprise behavior can also 41 affect government decision-making (Price et al. 2010). ESER process will be promoted in the 42 game between government and enterprises. Due to the mutual influence and restriction between 43 government and enterprises (Zhu and Dou, 2007), the research on these two subjects is 44 exceptionally important. 45 Stackelberg game, Bayesian game, non cooperative game and other classical game theories 46 (Wang et al. 2017； Zu et al. 2018； Zhao et al. 2015 are widely used in the study of the mechanism 47 between government and enterprises. Classical game theory requires a basic hypothesis of 48 'complete rationality', and in terms of the degree of rationality, the requirement of complete 49 rationality is higher than that of "rational economic man hypothesis" in neoclassical economics 50 (Kreps, 1990). However, in real life, complete rationality is only an ideal state, and reasoning 51 errors in economic decision-making are inevitable. Factors such as insufficient consideration, 52 information cost, excitement and experience will lead to inaccurate and irrational decision-making 53 of government and enterprises (Nelson, 2009). Therefore, evolutionary game based on bounded 54 rationality has more practical significance. 55 Existing research based on evolutionary game model shows that, appropriate low-carbon 56 subsidies, reasonable carbon taxes (Zhao et  development of low-carbon markets. Then the production enterprises will be forced to adopt 64 low-carbon strategies. The application of two-population evolutionary game (Mahmoudi and 65 reputation. At this time, the comprehensive income of enterprises is recorded as . At the same 110 time, it can not be ignored that enterprises need to pay the corresponding cost to take emission 111 reduction measures, which is recorded as . On the contrary, if the enterprise chooses NR, it will 112 not be able to obtain relevant additional income, and the comprehensive income of the enterprise 113 is recorded as . And is necessarily greater than . 114 (3) For the government, the behavior of enterprises in carbon emission reduction will 115 inevitably affect the overall social welfare. Enterprises take CR measures, which is conducive to 116 the improvement of the ecological environment and low-carbon awareness, will inevitably 117 enhance social benefits, and social benefits will eventually turn into government achievements. 118 Therefore, when enterprises adopt CR strategy, the government's social benefits are recorded as 119 . When enterprises adopt NR strategy, the government's social benefits are recorded as . And 120 is necessarily greater than . 121 (4) The government's 'AC' includes positive and negative regulation of enterprises. Positive 122 regulation means that when enterprises take effective measures to reduce emissions, the 123 government gives subsidies in time to improve the enthusiasm of CR. Such subsidies are recorded 124 as S. Negative regulation means that the government takes decisive punishment measures to 125 weaken the motivation of enterprises to adopt negative strategies when enterprises are negative in 126 CR. Such punishment are recorded as F. In addition, it is worth noting that when the government 127 adopts AC strategy, it must pay human and material resources to supervise the enterprises, and it 128 also needs to formulate written reward and punishment policies. Therefore, the government's AC 129 cost is recorded as . 130 (5) There is no extra cost for the government and enterprises when they do not take measures, 131 and it is assumed that the above parameters are greater than zero. 132 The payoff matrix of government and enterprises is shown in Table 1 and table 2 It is assumed that the probability of CR is y and 1-y for NR (0≤y≤1); the probability of AC is 138 x and 1-x for NA (0≤x≤1). 139 The expected return of AC is recorded as , and the expected return of NA is recorded as 140 . According to the payoff matrix, the expected return of government's choice for AC and NA 141 are respectively = y( − − S) + (1 − y)( − + F) , = + (1 − ) . The 142 average expected return of government is = + (1 − ) . According to the Malthusian 143 equation (Friedman, 1991), the growth rate of the proportion of government groups choosing 'AC' 144 strategy over time is / . And / is proportional to the difference of the expected return 145 and average expected return of 'AC' strategy. From this, we can get the dynamic evolution of the 146 probability of government's AC strategy over time, that is, the dynamic equation of government's 147 behavior is : 148 (1) 149 The expected return of CR is recorded as , and the expected return of NR is recorded as 150 . According to the payoff matrix, the expected return of enterprises' choice for CR and NR are 151 average expected return of enterprises is = + (1 − ) . The replication dynamic 153 equation of enterprise behavior is as follows: 154 According to Eq. (1) and Eq. (2), the replication dynamic equations of government and 156 enterprises can form a two-dimensional dynamic system: 157 Friedman (1998) believes that the stability of local equilibrium points can be obtained by 165 analyzing the Jacobian matrix. The Jacobian matrix of the above dynamic system is expressed by 166 J as: 167 According to Friedman, the evolutionary stable strategy of system only exists when the 170 equilibrium point satisfies both the determinant greater than zero and the trace less than zero 171 (Det(J) > 0, Tr(J) < 0). According to the Jacobian matrix, the values of a、b、c、d at the five 172 equilibrium points are calculated, as shown in Table 3. 173 Table 3 174 Values at the local equilibrium point 175

177
Based on Table 3, it can be found that under the existing conditions, point (1, 1) cannot 178 become the evolutionary stable point. This means that government and enterprises could not reach 179 the optimal state (AC, CR). Next, we will consider whether the introduction of public willingness 180 can improve this situation. 181 With economic development and the rapid improvement of people's living standards, people's 182 requirements for environmental quality improvement are getting higher and higher. This also puts 183 forward brand-new requirements for the two important subjects (government and enterprises) in 184 the emission reduction work. This article believes that public willingness mainly contains three 185 levels of connotation: First, the impact of residents' willingness on enterprises. If enterprises do 186 not adopt CR strategy, casusing excessive carbon emissions, the public will form negative opinion 187 on enterprises, resulting in the loss of enterprise reputation and the decline of stock price 188 (enterprises with large carbon emissions are generally listed companies). And in the long run, it 189 will affect the overall performance of enterprises. The second is the influence of residents' 190 willingness on the government. If government does not effectively supervise and restrict 191 enterprises that exceed carbon emissions, residents will question government's ruling ability, 192 reduce the "loyal investment" to the government, and affect government's credibility. Finally, what 193 is more special is that the willingness of enterprises also has a certain impact on the government. 194 Although enterprises are a powerful group relative to the public, they are both managed objects 195 relative to the government. Therefore, enterprises can be regarded as a part of the "public". The 196 development of enterprises is inseparable from the support of relevant policy and subsidies. If 197 enterprises respond to the call of government to actively ESER, but the government is indifferent 198 and does not provide assistance to it, enterprises will also have opinions on government's inaction 199 and protest to the government in various ways. 200 In this situation, the payoff matrix of government and enterprises will change accordingly. 201 Assuming that government has no action to give enterprises corresponding carbon-reduction 202 subsidies, enterprises will appeal and lobby to the government. At this time, the coordination cost 203 and possible compensation loss of government are recorded as .
It can also be regarded as 204 the constraint of enterprises' willingness on the government. If enterprises do not reduce emissions, 205 regardless of government's action or inaction, the negative evaluation of residents will bring brand 206 and reputation losses to enterprises, which is recorded as .
can also be regarded as the 207 constraint of residents' willingness on enterprises. If both government and enterprises do not make 208 a difference in carbon-reduction, not only do enterprises lose their brand and reputation, but the 209 credibility of government is also affected. The loss of government is recorded as .
can also 210 be regarded as the constraint of residents' willingness on the government. (Assume all parameters 211 are greater than zero). 212 The new payoff matrix of government and enterprises is shown in Table 4 and table 5. 213

Table 4 214
The government revenues under different enterprise behaviors (public willingness). 215

Table 5 216
The enterprise profits under different government behaviors (public willingness). 217 After the introduction of public willingness, the two-dimensional dynamic system of 218 government -enterprises game is as follows: 219 Let F( )= 0 and F( ) = 0, we can get five local equilibrium points, which are 221 With the introduction of public willingness, the new Jacobian matrix is represented by J as: 224

Proof 235
According to the new Jacobian matrix J , the values of 、 、 、 at five equilibrium 236 points are calculated. As shown in Table 6. 237 Table 6 238 Values at the local equilibrium point (public willingness) 239 Equilibrium point

241
Due to the uncertainty of parameters, there are twelve cases of stability at equilibrium point. 242 By analyzing the symbols of determinant and trace in Jacobian matrix under different cases 255 (Table 7), the stability of each equilibrium point can be obtained. As shown in Table 7, the 256 evolutionary stable point of case ① ③ ⑥ is (0, 0). The evolutionary stable point of case ② ⑧ 257 is (0, 1). The evolution stable point of case ⑨⑫ is (1, 0). The evolution stable point of case ⑤ 258 ⑩ ⑪ is (1, 1). Specifically, there are two evolutionary stable points (0, 0) and (1, 1) in case ④ 259 and no evolutionary stable point in case ⑦. Then Proposition 1 can be proved. 260 (1) "+" means greater than zero; "-" Represents less than zero; "N" stands for numerical 267 uncertainty. 268 (2) Notes on the stability of equilibrium points. Taking case① as an example, according to the 269 principle of evolutionary stability, the determinant (Det (J)) of point (0,0) is greater than 0, and the 270 trace (Tr (J)) is less than 0, which means that (0,0) is the evolutionary stable point of the system. 271 (3) The determinant of point (0,1) and (1,0) is less than 0, and the trace is uncertain, which 272 means that these two points are the saddle points of the system. 273 It is a meaningless point. 282 According to Proposition 1 and its Proof, we can draw the following significant conclusions: 283 − . Under this circumstance, enterprises choose NR when government does not act, and 285 government chooses NA when enterprises do not reduce emissions. Residents' willingness has 286 weak constraints on enterprises ( ) and the government ( ), so both sides lack strong 287 constraints. In the long-term evolution process, there is an opportunistic tendency on both players 288 of the game, and they have the motivation to choose passive behaviors. Therefore, the evolution 289 equilibrium point of system must have (0,0), and there are strategies (NA, NR) between 290 government and enterprises. At this time, the state of carbon-reduction is most passive, as shown 291 in case ①③④⑥ of Table 7. 292 Under this circumstance, residents' willingness has strong constraints on enterprises. So regardless 295 of government's action or inaction, enterprises can gain more profits by choosing CR. From < 296 + , it can be seen that the constraint of enterprises' willingness on the government is not 297 strong enough, so government will choose NA when enterprises reduce emissions. At this time, 298 the evolutionary stable strategy is (NA, CR), and the carbon-reduction reaches a suboptimal state, 299 as shown in case ②⑧ of Table 7. Under such circumstance, due to the constraint of residents' willingness on 303 enterprises ( ) is not enough, so regardless of government's action or inaction, NR is the best 304 choice for enterprises. According to − > − , we can see the constraint of residents' 305 willingness on the government ( ) is strong. So government will take active action after 306 confirming the behavior of enterprises. At this time, the evolutionary stable strategy is (AC, NR), 307 and the carbon-reduction also reaches the suboptimal state, as shown in case ⑨⑫ of Table 7. Under such circumstance, government's choice of AC is more favorable when enterprises reduce 312 emissions, and the best choice for enterprises when government acts is CR, which shows that both 313 government and enterprises have incentives to take active actions to promote ESER. Obviously, 314 the strengthening of public willingness has effectively promoted the government and enterprises to 315 take positive actions. There must be an evolutionary stable strategy (AC, CR), at which time 316 carbon-reduction can reach an optimal state, as shown in case ④⑤⑩⑪ of Table 7. 317 In particular, after joining the two conditions of − + < 0 and − − + < 318 0, government and enterprises may have negative behavior patterns. Although there are constraints 319 of public willingness, there will be opportunistic tendency between the government and 320 enterprises. Therefore, after a long-term evolution, (0,0)will also be a stable point of system 321 evolution, as shown in case ④ of Table 7.  、 and are not assigned temporarily, they 341 will be changed according to the actual situation in various cases. Based on the parameter 342 assignment and the actual situation, the situation of − + < − − will not occur, 343 so this paper does not give a specific analysis of case ③⑥⑨⑫. 344

The cases of government-enterprise cannot achieve the optimal path spontaneously 345
According to Proposition 1 and its proof, only when enterprise willingness is strong enough to 346 restrain the government, the evolutionary stable point of system exists (1, 1), then government and 347 enterprises can spontaneously move towards the optimal path of ESER. The following are 348 situations in which the enterprise willingness has a weak constraint on the government ( < + 349 ), and the two parties cannot spontaneously move towards the optimal path of ESER. No matter what the initial willingness of both parties is, they will eventually converge to the 362 strategy (NA, NR). Due to the constraints of enterprise willingness and residents' willingness on 363 government are relatively weak, NA is the best choice for government whether enterprises reduce 364 emissions or not. The strategy of enterprises will eventually tend to NR with the determination of 365 government behavior. 366 Fig. 2 and Fig. 3 respectively depict the dynamic evolution path of government and enterprise 367 behavior strategies over time in case①. The vertical axis of the two graphs respectively represents 368 the probability of AC and probability of CR. Horizontal axis represents time t. The data in this 369 study are simulated data, so the unit of t is not specifically set (such as year, month, etc.), but 370 refers to a general unit of time, which is used to examine the convergence speed of government 20 and enterprise behavior. In order to better compare and analyze the impact of different initial 372 willingness on the evolution of system, this paper sets the initial willingness of enterprises to 373 reduce emissions and the initial willingness of governments to act as low, medium and high levels, 374 namely (0.2,0.5,0.8). As shown in Fig. 2, The rays from the three endpoints of = 0.8, = 375 0.5, = 0.2 indicate the evolution path of government behavioral strategy when government's 376 initial willingness is high, medium, and low. There are three rays of red, green and blue from each 377 endpoint. They respectively show the evolution path of government behavioral strategy under the 378 fixed government's initial willingness and different enterprises' initial willingness (y = 0.2, y = 379 0.5, y = 0.8).

383
As shown in Fig. 2, government's behavior probability will eventually converge to zero. 384 Beyond that, the greater the enterprises' initial willingness to reduce emissions, the faster the 385 government's behavioral probability converges toward zero. Reflected in the graph, the time 386 for the blue, green, and red rays to converge to 0 is 0.75, 1.1, and 1.35, respectively. Since the cost 387 of government action ( + ) is large when enterprises reduce emissions, if government 21 observes that enterprises have a strong willingness to reduce emissions during the evolutionary 389 game, it will speed up the adjustment of its own behavior. More specifically, when the initial 390 willingness of enterprises is low (e.g., y = 0.2), the convergence rate of x will accelerate as 391 government's initial willingness increases. Shown graphically, the ray from = 0.8 is the fastest 392 of three red rays. The situation is similar for = 0.5. Due to the weak constraint of residents' 393 willingness on the government ( ), government will urgently tend to the favorable strategy of 394 NA when the willingness of enterprises is low. If the initial willingness of enterprises is high ( = 395 0.8), government will not adjust its own behavior quickly because of the relatively strong 396 constraint of enterprise willingness ( ). The convergence rate of government's action probability 397 will only accelerate with the decrease of its initial willingness.

400
As shown in Fig. 3, enterprises' behavior probability will eventually converge to zero. 401 Beyond that, the convergence rate of will be accelerated with the reduction of enterprise's 402 initial willingness. Reflected graphically, when government's initial willingness is certain (e.g., 403 = 0.2), the ray starting from = 0.2 converges the fastest among the three red rays. On the other 404 hand, the convergence rate of y will slow down with enhancement of government's initial 405 willingness, and will also rise briefly when the government's initial willingness is strong. 406 Reflected in the graph, the time for red, green and blue rays to converge to 0 is 0.43, 0.54 and 0.66 407 respectively. Moreover, the blue and green rays will rise briefly at t ∈ [0,0.25] and then fall. The 408 main reason is that the profit of enterprises is greatly influenced by government behavior. When 409 government's initial willingness is strong, enterprises will increase the probability of emission 410 reduction, and then decrease with government's inaction. The evolutionary stable point of the system is (0, 1), and the evolutionary stable strategy is (NA, 424 CR). In addition, the behavior patterns of two parties are similar to Case ⑧. In terms of parameter 425 conditions, case ⑧ enhances the constraint of residents' willingness on the government ( ) 426 compared with case ②. However, government can avoid losses caused by resident's blame 427 because enterprises must choose ER. Therefore, in the long-term evolution, government will also 428 choose NA (Due to space limitations, the evolution graph and corresponding analysis of the above 429 two cases are omitted).

The cases of achieving the optimal path spontaneously 443
In the following cases, the willingness of enterprises has a strong constraint on the 444 government( > + ). Both government and enterprises can spontaneously move towards the 445 optimal path of ESER. The enhancement of residents' willingness to restrain government and 446 enterprises( 、 ) can further improve the situation of carbon-reduction. 1). In this case, government will take active actions when enterprises reduce emissions, and 456 enterprises will choose emission reduction when government acts. Conversely, vice versa. So 457 there are two evolutionary stable strategies in the system, namely (AC, CR) and (NA, NR). This 458 situation can partly tend to the optimal state of carbon-reduction. So the following article will 459 examine whether we can increase situations of (AC, CR) through the change of parameters, so as 460 to optimize the path of ESER. 461  Fig. 6, it can be found that when government's initial 465 willingness is high ( = 0.8) or low ( = 0.2), the enterprises' initial willingness has no 466 significant impact on government's behavior. When government's initial willingness is moderate 467 ( = 0.5), government behavior will be influenced by the initial willingness of enterprises. Only 468 when the initial willingness of enterprises is low ( = 0.2), government's action probability 469 will tend to 0, otherwise it will tend to 1. Shown graphically, all three rays starting from = 0.8 470 converge to 1 and the convergence speed is approximately the same. All three rays starting from 471 = 0.2 converge to 0 and the speed is about the same. Starting from = 0.5, the blue and green ray 472 converge to 1, and the red ray converge to 0. In this case, the constraint of residents' willingness 473 on the government ( ) are relatively weak. Although enterprise willingness ( ) has a strong 474 constraint on the government, government can fully bear the losses caused by irrational strategies. 475 Due to the existence of behavioral inertia, when the initial willingness of government is relatively 476 certain ( = 0.2 or = 0.8), it will not change its own strategy. When government's initial 477 willingness is uncertain ( = 0.5), it will determine its optimal strategy based on the behavior of 26

enterprises. 479
On the other hand, government's initial willingness has a positive correlation with the 480 convergence rate of its behavior probability. As shown in the graph, when enterprises' initial 481 willingness is determined (such as = 0.5), the convergence time of two green rays from = 482 0.8 and = 0.5 to 1 is 0.9 and 1.15 respectively. Obviously, the improvement of government's 483 initial willingness accelerates the convergence of to 1. Given = 0.2, the convergence time of 484 two red rays from = 0.5 and = 0.2 to 0 is 0.95 and 0.7 respectively. It shows that the 485 decrease of government's initial willingness accelerates the convergence of to 0.

488
As shown in Fig.7, we can know that government's initial willingness has a great impact on 489 the behavior of enterprises. When the initial willingness of government is high ( = 0.8) or low ( 490 = 0.2), the behavior probability of enterprises will eventually tend to 1 or 0. When government's 491 initial willingness is moderate ( =0.5), enterprises will make a choice based on their initial 492 willingness. Only when enterprises' initial willingness is low ( =0.2), will tend to 0, 493 otherwise it will converge to 1. Graphically, all three blue rays converge to 1, and all three red rays 494 converge to 0. Only one green ray starting from y = 0.2 tend to 0, while the other two green rays 27 converge to 1. Due to the weak constraint of residents' willingness ( ) on enterprises, the best 496 choice for enterprises is NR when government does not act. However, when government has a 497 strong willingness of action, it is difficult for enterprises to bear the double losses caused by 498 government's fine ( ) and residents' willingness ( ). So enterprises will inevitably choose CR. 499 When government's initial willingness is moderate, enterprises will determine the ultimate 500 strategy according to their initial willingness. 501 On the other hand, government's initial willingness will also affect the convergence rate of 502 enterprise behavior probability. Reflected in the graph, the convergence time of blue ray from 503 y=0.5 to 1 is 0.23, while that of green ray from y=0.5 is 0.42. It shows that the enhancement of 504 government's initial willingness will accelerate the convergence rate of y to 1. The convergence 505 time of red ray from y = 0.2 to 0 is 0.26, while that of green ray from y = 0.2 is 0.45. It shows that 506 the decrease of government's initial willingness will accelerate the convergence rate of y to 0. found that with the enhancement of residents' constraint on enterprises, situations of (AC, CR) 520 have increased. The state of carbon-reduction has been optimized to some extent. 521 If , , and increase at the same time, they are = 103, = 20, and = 12. 522 The parameter conditions are also consistent with case ④. 523 , and are improved together. Compared to Fig. 8 and 9, situations of (AC, CR) have 527 increased more in Fig. 10. The state of carbon-reduction has been greatly optimized. It shows that 528 a joint strengthen of resident and enterprise willingness is more conducive to the achievement of 529 the optimal path. 530 According to the above analysis, the increase of , , and is conducive to the 531 optimization of carbon-reduction status. However, the continuous improvement of and 532 will change the case ④ into other cases. So we will try to keep and unchanged ( = 10, 533 = 5), continuously strengthen the willingness of enterprises ( ). Then observe whether the 534 behavior of government and enterprises will change significantly. Through a large number of 535 parameter simulation and simulation, it is found that with the continuous improvement of 536 enterprise willingness, there are fewer and fewer situations of (NA, NR). Until is greater than 537 or equal to 412, the situation of (NA, NR) disappears.

542
As shown in Fig. 11 and Fig. 12, when is 411, the situation will converge to (NA, NR) 543 only when the initial behavior probability of two players is (0.1, 0.1). When is increased to 544 412, government and enterprises will converge to (AC, CR) under any initial behavior probability. 545 Under this circumstances, the result of two equilibrium strategies does not match the case④. The 546 main reason is that the willingness of enterprises is too strong. If government does not give 547 enterprises subsidies for reducing emissions, government will be unable to bear the huge losses 548 both parties is, they will eventually converge to the strategy (AC, CR). 559  Fig. 15 respectively depict the dynamic evolution path of government and 562 enterprises over time in case ⑤. As shown in Fig. 14, government's behavior probability x will 563 eventually converge to 1. Beyond that, the convergence rate of x is positively related to 564 government's initial willingness. Reflected in the graph, when enterprises' initial willingness is 565 given (e.g., = 0.2), the convergence time of three red rays from x = 0.8, x = 0.5 and x = 0.2 is 566 0.86, 1.08 and 1.26 respectively. On the other hand, the initial willingness of enterprises has no 567 significant impact on government's behavior probability. Shown graphically, when government's 568 initial willingness is given (e.g., = 0.8), the convergence time of red, green and blue rays is 569 approximately the same. Because the constraint of enterprises' willingness ( ) is relatively strong, 570 government is unwilling to bear the loss of credibility and coordination costs brought by 571 enterprises' harsh criticism. At the same time, the loss of government action will not be too much 572 (− + ) when enterprises do not reduce emissions. Therefore, no matter what enterprises' initial 573 willingness is, government's behavioral strategy will not fluctuate greatly.

576
As shown in Fig. 15, enterprises' behavior probability y will eventually converge to 1. Beyond 577 that, the convergence rate of y is positively related to enterprises' initial willingness. Expressed 578 graphically, when government's initial willingness is given (e.g., = 0.2), the ray starting from y 579 = 0.8 converges at the fastest rate among the three red rays. On the other hand, enhancement of 580 government's initial willingness is conducive to increasing the convergence rate of y. Shown 581 graphically, the time for blue, green and red rays to converge to 1 is 0.075, 0.095 and 0.135 582 respectively. In this case, residents' willingness has a strong constraint on enterprises. And if 583 enterprises do not reduce emissions while government acts, they will also bear the fine from 584 government. Therefore, once enterprises observe that government has a strong willingness to act, 585 they will enhance their emission reduction efforts to avoid being punished by government. 586 Comparing Fig. 14 and Fig. 15, it can be found that the convergence rate of y is much faster 587 than that of . Due to the strong constraint of residents' willingness on enterprises( ), whether 588 government acts or does not act, the best choice for enterprises is CR. At the same time, due to the 589 weak constraint of residents' willingness on the government ( ) and the strong constraint of 590 enterprise willingness ( ), government behavior is mainly affected by enterprises. Government 591 will ultimately choose AC according to the behavior of enterprises. So there is a lag in the 592 government's behavioral strategy.   Under the incentive of maximizing their own interests, both government and enterprises will take 616 active actions to promote carbon-reduction. Compared with case ⑤ and case ⑩, case ⑪ avoids 617 possible losses caused by the lag strategy of government or enterprises. The path of ESER is 618 further optimized. 619

Policy recommendations 620
Government and enterprises could not reach the optimal path of ESER through spontaneous 621 behavior for lack of necessary constraints. In the formulation and implementation of 622 carbon-reduction policies, the impact of public willingness must be fully considered. A series of 623 measures must be taken to strengthen the constraints of public willingness on the government and 624 enterprises.
Since the constraint of enterprise willingness directly determines whether the optimal path can 626 be realized, the enhancement of this constraint is of paramount importance. The shift from section 627 4.1 to 4.2 mainly depends on the enhancement of enterprise willingness. Regular symposiums for 628 business people and the establishment of government-enterprise information exchange platforms 629 are helpful in conveying the wishes of enterprises. Enterprises should pay attention to the 630 maintenance of their own rights and interests in ESER. They can exert pressure on the government 631 through enterprise groups or industry associations, which can force government to take action to 632 improve their consistency in the work of carbon-reduction. For example, the increasing 633 willingness of enterprises in case ④ can eliminate the passive behavior of government, and 634