Probabilistic linguistic evolutionary game with risk perception in applications to carbon emission reduction decision making

Carbon emission reduction, an effective way to facilitate carbon neutrality, has gained increasing attention in government policy and scientific research. However, the establishment of a sustainable carbon emission reduction market is a complex game between governments and enterprises. In addition, it is difficult to obtain precise evaluations of the political and environmental factors in most cases. Irrational enterprises with a profit-seeking nature bring challenges to the strategy selection. To bridge this gap, we propose a probabilistic linguistic evolutionary game to model strategic behavior in carbon emission reduction assistant decision making. First, we introduce a probabilistic linguistic payoff matrix to describe the uncertain payoffs of players. A new distance measure for the probabilistic variables is also proposed to construct the prospect payoff matrix in the prospect theory framework. Then, the evolutionary dynamics and the probabilistic linguistic evolutionary stability of the proposed methods are analyzed. A comprehensive case study for carbon emission reduction with comparisons is presented for validation.


Introduction
With the rapid development of the social economy, more greenhouse gases are being emitted into the atmosphere. Concentration of greenhouse gases results in adverse impacts on climate change, which, in turn, greatly threatens human survival [1]. To alleviate the unpredictable weather caused by climate change, most of countries have joined carbon neutrality initiatives and taken joint actions to balance carbon emissions and greenhouse gas removal activity. China, as one of the world's largest economies, announced a carbon neutrality goal in 2020. Carbon Xiang Wang xiangwangnu@163.com 1 Liaoning, China 2 Department of Marketing and E-Business, Nanjing University, Nanjing 210023, China 3 School of Management Science and Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China 4 College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China emission reduction (CER). An efficient way to narrow the carbon emission gap and facilitate carbon neutrality is not always easy. Governmental regulations on manufacturers and markets will not always motivate all enterprises to join in the carbon emission reduction initiatives due to the potential cost of low-carbon production. Moreover, government accountability and the level of compensation in the CER make it hard to develop the optimal strategy. Modeling participants' decision-making behaviors and responses in strategic games have become a hot spot in CER research. The game-theoretic model has become prominent in recent years [2]. Game theory (GT) aims to mathematically study the competition and cooperation among game players and has enjoyed great success in modeling and understanding economic competitions [3], social behavior in competitive situations [4], politicalstrategic decisions making [5], and so on. Mainstream GT research in the CER utilizes the cooperative game model or leader-follower game model to simulate the game behaviors of the government and enterprises. The players in these models are postulated to be totally rational and the game information is assumed to be complete. The complexity and dynamic environment make it harder for the player to play an ideal and rational game. Evolutionary game (EG) theory [6], inspired by the revolution in biology, was proposed to solve such problems. The equilibrium in the EG does not depend on strict constraints of superrational game players and shared common knowledge. Instead, the system equilibrium is dynamically achieved through a balanced process of trial and error. The EG can model the bounded rational decision-making behavior in the group and accordingly predict the trend of individual strategies in the game process. The capability of mistake-tolerance and suitability for EG is superior to that of traditional gametheoretic models. As a result, it has gained great attention in CER research [7].
The main research points of EG in CER concentrate on three aspects. (a) The green technology innovation game facilitates the low-carbon production of manufacturers. Generally, an EG game with the government and firms [8] or with consumers included [9] is established. The game results indicate that carbon taxes, subsidies, and government regulations exert different influences on the achievement of carbon neutrality. (b) The green supply chain supports carbon emission reduction. Carbon policies, including carbon caps, carbon taxes, and cap and trade, are preset, based on which a bilateral or trilateral EG model is devised to analyze enterprises' behaviors under mixed carbon policies and the critical factors that encourage lowcarbon emissions [10][11][12]. (c) The EG-based simulation model. The emphasis of this area focuses not on the mathematical design of the evolutionary game system but on scenarios with possible policy interventions. Generally, each individual in the EG is modeled as an intelligent agent that follows specific evolutionary rules. Policy interventions, such as carbon taxes, asymmetric penalties, emission reduction costs, and revenues are combined into different scenarios for the simulated system. Then, the intervention intensity of the carbon emission-related strategies and the system behaviors are comprehensively analyzed. Representative works can be found into [13,14].
Because player game behaviors are essentially the decision-making activities, cognitive uncertainty will widely exist in the whole game process. Especially regarding political or environmental factors, precise evaluation results are difficult to achieve. The fuzzy cooperative games mainly use fuzzy set theory [15] to handle uncertain and fuzzy knowledge. Researchers recently focus on fuzzy bilateral zero-sum games and fuzzy cooperative games. Then, fuzzy zero-sum games, games with fuzzy goals, and fuzzy mixed strategies are studied [16]. The improved forms based on extensions of fuzzy set have also been proposed. Xu et al. [17] proposed an intuitionistic fuzzy zero-sum game from the multi-attribute decision-making perspective. Xue et al. [18] defined a hesitant fuzzy matrix game and developed the bi-objective nonlinear programming models. Yang [19] integrated the triangular intuitionistic hesitant fuzzy set into the zero-sum matrix game and provided a matrix game-based decision making model. Zou [20] presented a cooperative game with fuzzy payoffs based on the Shapely function. Combinations of the cooperative fuzzy games with convex games, exact games, and extendable games have also been comprehensively analyzed in terms of the properties and their interrelations [21]. A fuzzy clustering cooperative game data envelopment analysis method [22] was also developed, in which the decision making units are first clustered and then treated as players in a cooperative game. Yu et al. [23] generalized the fuzzy cooperative games with a fuzzy coalition structure. For the evolutionary game, Gu et al. [24] first proposed evolutionary dynamics with fuzzy payoffs and introduced the fuzzy Moran process to update the strategy selection. Other works use fuzzy set theory and evolutionary game theory to solve economic problems in uncertain environments, such as the evolutionary game and fuzzy system-based simulation model used to analyze employees' safety behaviors in enterprises [25], the joint evolutionary game and fuzzy rule-based method to analyze the strategic behaviors of manufacturers in different market factors, and the economic benefit analysis method integrating the evolutionary game and fuzzy logic in green buildings [26].
One of the main challenges of strategic decision making in an uncertain environment is the quantification of qualitative information [27]. Decision-makers are more inclined to express their preferences with linguistic variables in uncertain environments. Describing both the cognitive and statistical uncertainty of the semantic information simultaneously directly influences the decision-making quality. To handle this problem, Pang et al. [28] proposed the probabilistic linguistic term set (PLTS).
The PLTS can not only describe the hesitancy in the judgments of decision-makers for multiple linguistic terms but also provide preference degrees for such hesitancy. Gou et al. [29] defined the probabilistic double hierarchy linguistic term set to improve the accuracy of PLTS. Wang [30] studied the partial and total orders of the PLTS and then proved their feasibility with practical cases. To estimate the incomplete information in PLTSs, Xiao et al. [31] introduced the knowledge-matching degree and proposed a consensus-reaching process for PLTSs. To solve the uncertainty in the probability information, Wu et al. [32] develop PLTS with interval probabilities based on belief and plausibility measures. In addition, the dual linguistic term set [33] based on the cloud model was also proposed, which allows decision-makers to express probabilistic information with linguistic variables. A review of the recent developments and applications for PLTS can be found in [34].
Accordingly, the PLTS has been applied to game theory due to its effectiveness in representing uncertain evaluations. Mi [35] utilized the probabilistic linguistic term set to solve the two-person and zero-sum matrix game and established the mathematical model between game players. Li [36] proposed a decision support system based on the probabilistic linguistic matrix game. Other related work uses linguistic variables to represent the payoff in the game while ignoring the statistical uncertainty, such as the hesitant fuzzy linguistic matrix game, in which players express their payoffs with a set of possible linguistic values [37], the proportional linguistic matrix game, where the payoff is represented by continuous linguistic scale domains [38], and the matrix game with 2-tuple linguistic payoffs [39]. However, most of the existing studies mainly utilize PLTS to solve the matrix game; the evolutionary game process in the fuzzy linguistic environment has not been fully investigated.
In addition, existing analysis methods for evolutionary games are based on expected utility theory, and then the payoff matrix is objective: the subjective preferences of players are not considered. Especially when handling economic decision making problems, the potential risk of loss may result in different decision-making behavior, for instance, risk-averse attitudes [40]. Considering that the players in the evolutionary game are bounded rational, the players will behave with different preferences for gains and losses. Then the decision making of the strategies will also be influenced. How the uncertain environment affects the game pattern and the subsequent risk attitudes that shape the game equilibrium should be also investigated.
Motivated by the above analyses, we aim to propose a probabilistic linguistic evolutionary game model that can quantify the uncertainty and incompleteness of subjective evaluations in games. In addition, the integration with PLTS makes it efficient in handling linguistic information with epistemic and statistical uncertainty. Since uncertainty often results in risk attitudes in the decision-making process, prospect theory (PT) is introduced to describe the risk preferences of players. The players' probabilistic linguistic payoffs are transformed into a prospect payoff matrix and then integrated into the evolutionary dynamics. Mutations of the strategies and the equilibrium of the game are also discussed. Furthermore, the proposed model is utilized to analyze CER problems with both risk and uncertainty factors considered.
The remainder of this paper is organized as follows. The requisite background knowledge is presented in Section 2. Section 3 introduces the probabilistic linguistic evolutionary game (PLEG) model. A new distance measure for the probabilistic linguistic variables is defined in preparation for quantifying the players' risk preferences. The strategy update rules and stable equilibrium solution of the PLEG are also discussed. A simulated case study and comprehensive analyses of the proposed methods are presented in Section 4. Section 5 concludes the paper.

Preliminary
The proposed methods are defined in the probabilistic linguistic term set (PLTS) framework. The related definitions and operations are briefly reviewed.
The PLTS is defined based on the linguistic term set (LTS). An LTS S is described by S = {s α |α = 1, 2, ..., τ }, where s α is the linguistic variables and τ is the linguistic term scale [41]. The LTS is an ordered set with s j > s k for any j > k. The negation operation on the LTS is defined as neg(s α ) = s β satisfying α + β = τ . Then the PLTS defined on the finite LTS S is expressed as is the probability assigned to the linguistic term L (k) and #L(p) is the cardinality of the PLTS [28]. The term L (k) (p (k) ) can be treated as the basic element of the PLTS.
The addition and multiplication operation laws of PLTSs are defined as follows [28]: Let L 1 (p) and L 2 (p) be two PLTSs; then, it holds that and The subtraction of PLTSs is defined as follows [42]: where ( > 0) is a sufficiently small value. For the convenience of comparing PLTSs, different comparison methods are proposed for practical applications. The existing methods can be roughly classified into two types: those based on the statistical characteristics of the PLTS and those based on the orders of the PLTS. Pang et al. [28] proposed a score function and deviation degree based on the expectation and variance to compare PLTSs. Lin et al. [43] introduced the concentration degree and proposed a novel score function to compare PLTSs. Wang et al. [44] proposed methods for comparing PLTSs by defining the partial and total orders. Since the order of the PLTS is not the critical concern of this paper, we utilize the statistical characteristics-based ranking method defined in [28]. The score function of a PLTS describes its average information, denoted as μ(L(p)) = s lineα . The deviation degree describes the variation in the PLTS and takes the The general comparison rules for PLTSs can be defined as follows. For two PLTSs L 1 (p) and L 2 (p), it holds that and σ (L 1 (p)) = σ (L 2 (p)).

Probabilistic linguistic evolutionary dynamics
Evolutionary game theory overcomes the deficiencies of traditional game theory, which mainly depends on complete rationality and information. The equilibrium of the game is achieved through dynamic learning or imitation strategies. It is more robust when confronted with disturbance and noise. To make the evolutionary game capable of handling uncertain games, especially in an uncertain linguistic environment, and broaden its application, we propose the probabilistic linguistic evolutionary game, as follows (Table 1).

The probabilistic linguistic evolutionary game
Definition 1 (probabilistic linguistic game) A game is a probabilistic linguistic evolutionary game if all the players' payoffs are represented by the probabilistic linguistic term set and an evolutionary equilibrium exists in the game.
The physical structure of a probabilistic linguistic evolutionary game G can be described by where N indicates the player set, is the strategy space of the players in N, and is the probabilistic fuzzy payoff set related to the strategies in . For simplicity and without loss of generality, we provide a game with two players and each player with binary strategies. Let N = {A, B} and the strategy space corresponding to N is is a binary pair noted as = {( SA i , SB i )}. In most strategic games, the gains brought by the actions of players are difficult to quantify precisely. The outcomes of some tactics can only be estimated at The payoff matrix in the probabilistic linguistic evolutionary game D(L 1 , L 2 ) The probabilistic linguistic Bhattacharyya distance between L 1 and L 2

E(A), E(B) The expectations of strategies A and B L(p)
A probabilistic linguistic term set U The payoff of the government's strategies V The payoff of the enterprises' strategies F The subsidy of the government to the enterprises H The penalty for passive enterprises The strategy space of the players The probabilistic fuzzy payoff of players' strategies v(x) The prospect value function s α The linguistic variables x, y The probability of choosing strategies A and B μ(L(p)) The score function of a probabilistic linguistic term set σ (L(p)) The deviation degree of a probabilistic linguistic term set λ The loss-aversion coefficient in prospect theory ψ(χ) The dominance degree of the payoffs between actions and the expectations θ Impact of the subsidy γ Impact of the penalty the qualitative level. The PLTS provides a flexible tool to evaluate such payoffs. In the probabilistic linguistic environment, the players' payoff matrix M = (M A , M B ) is denoted by: where each j i represents a probabilistic linguistic evaluation of the actions taken in the strategy space.
Evolutionary game theory postulates that game players are irrational. As a result, the gains and losses of different strategies will influence the decision-making results. In other words, the risk-aversion behaviors of the players should also be fully considered in the game process. To achieve this, we introduce the prospect payoff matrix in the PLEG. The natural logarithm based value function [45] is utilized in view of its good discriminating ability. The value function describing the utility of different possible outcomes is denoted by: The logarithm-based value function can implement a wide range of risk-taking outcomes and explain more loss-aversion behaviors. As a result, it can provide more discriminative risk results in practical applications.
To calculate the risk prospect values of different strategies in the PLEG, we first define the probabilistic linguistic Bhattacharyya distance measure and then develop the dominance degrees of the strategies. The Bhattacharyya distance is a stochastic metric that describes the similarity of two samples based on their probability distributions. As a result, it has been widely used in stochastic model updating [46] and uncertainty characterization [47]. Existing distance measures of PLTS are mainly based on Euclidean distance, and the probabilistic information is partly handled as the importance degrees [34]. In other words, the probability distribution provided by the PLTS is not fully considered. Given that the PLTS provides linguistic variables over a specific distribution, we introduce this uncertainty quantification metric and propose the probabilistic linguistic Bhattacharyya distance.
Let L 1 (p) and L 2 (p) be two PLTSs, then the probabilistic linguistic Bhattacharyya distance is defined as where σ L 1 , σ L 2 denote the deviations of PLTSs L 1 (p) and L 2 (p), respectively; μ L 1 and μ L 2 represent the corresponding score functions of L 1 (p) and L 2 (p), respectively. It can easily be proven that the Bhattacharyya distance defined in (7) However, the proposed distance does not strictly object to the triangle inequality.
We define the dominance degree of the payoff between the selected action and the average expectation for each player to calculate their prospect values. Let X = [x 1 − x] T represent the probability of choosing strategies A 1 and A 2 for player A. It holds that 0 ≤ x ≤ 1. Similarly, the probability of choosing strategies B 1 and B 2 can be expressed as Y = [y 1 − y] T with 0 ≤ y ≤ 1. Then, the overall payoff expectations of the two players yield: and Then, the dominance degrees of each strategy can be calculated by where χ ∈ A i , B j (i, j = 1, 2) and E(χ) are the probabilistic linguistic expectations defined in (8) and (9). By substituting (10) in (6), we can obtain the prospect values of different actions for each player. Then, the prospect payoff matrix can be denoted by represents the dominance degree of each strategy.
The adapted matrix in (11) transforms the original payoffs into prospect values, and the risk preference of players is then integrated. The different risk attitudes toward the potential gains and losses of different decision strategies can be tuned by the parameters of the value function in (6).
Equilibrium in the EG is achieved dynamically in the evolutionary process through trial and error. Such stable equilibrium is called the evolutionarily stable equilibrium (ESE) and the corresponding strategies are named evolutionarily stable strategies (ESS) [48]. Similarly, the probabilistic linguistic evolutionarily stable strategy (PLESS) of the probabilistic linguistic evolutionary game (PLEG) can be formulated as follows.

The update rules of the probabilistic fuzzy EG
Generally, most of the update rules in the EG are based on the imitation rule [49]. The players prefer to copy a strategy that leads to a higher payoff in the previous game round. The representative imitation-based update rules include the unconditional imitation rule [50], and the replicator rule [51]. The unconditional rule depicts the evolution process from a macro perspective. It assumes that the players choose the strategy of themselves and the limited neighbors unconditionally. This process is easy to implement and calculate mathematically. However, the deficiency is obvious. This updated rule is deterministic and prone to noise and uncertainty. In contrast, the replicator imitation rule is a micro-modeling method. Players' following strategies are stochastically selected with conditional probability related to the gains of previous strategies. The imitation process is stochastic and more in accord with the practical situation.
As a result, we develop the replicator dynamics of the probabilistic fuzzy EG based on the replicator rule. For the 2 × 2 PLEG, each player takes two possible strategies. The transition probability of a specific strategy for players in the evolutionary process is proportional to the payoff differences between the current and the average. The expectations of different strategies for players can be obtained from (8) and (9), denoted as E(A i ) = E(A|X i = 1) and E(B i ) = E(B|Y i = 1). Then the update rules for the strategies can be defined as follows. anḋ x is the transition probability corresponding to strategy A 1 and y is the transition probability corresponding to strategy B 1 . Accordingly, E(A| x=1 ) and E(B| y=1 ) represent the payoff expectations of strategies A 1 and B 1 .
The replication of a strategy will be intensified if its payoff is higher than the average. As a result, the lowaverage strategy will be substituted with the higher one in the game process.
By solving the replicator dynamics constituted by (13) and (14), we can obtain the equilibrium points of the PLEG. Stability analysis of the equilibrium points can be conducted via Lyapunov methods. Let J be the Jacobian matrix of the replicator dynamics. If all the eigenvalues of J at the equilibrium points are negative, then the system is in a stable state. The equilibrium point is called the stable point in Fig. 1. Accordingly, the solution of the system is the PLESS. If all the eigenvalues are positive, then the system is unstable regardless of its initial state. If both negative and positive eigenvalues exist, then the system converges along the negative eigenvalue direction but diverges along the positive eigenvalue direction. The equilibrium point is named a saddle point. Additionally, the system is unstable.
Based on the above methods, we can provide a PLEGbased CER strategy decision-making analysis method, which is depicted in Fig. 2. First, the initial CER project scheme is evaluated by the PLTS data manipulation method; then, the probabilistic linguistic payoff matrix of the players is obtained. Based on the proposed distance measure and the prospect value function, the original payoff matrix is transformed into the prospect matrix. Once fed into the PLEG dynamics, we can calculate the PLESS of the dynamic system and the evolutionary path related to the initial solution. By analyzing the evolution phase and

Case study
Carbon neutrality in China needs ongoing effort to be achieved. Carbon reduction is a difficult task for both governments and enterprises. It involves the adjustment and upgrading of the whole manufacturing structure. The government must formulate a series of supportive policies. However, whether the enterprises answer the call of CER is another concern. Policymakers should consider all the influencing factors in CER and the enterprises' response and then provide the optimal solution. To validate the proposed methods and provide assistant decision support for policymakers on the CER, we conduct a CER game between the government and enterprises.
In a bilateral PLEG with the government and enterprises as participants (depicted in Table 2), suppose that both players have two pure strategies, namely, support the policy or oppose the policy. Let U 11 and U 21 in Table 2 be the payoffs of the government's strategies under different attitudes toward enterprises, while U 12 and U 22 represent the payoffs with the counteractant enterprises in response to the government strategies, respectively. Similarly, V 11 and V 21 indicate enterprises' gains in joining the carbon emission when the government advances the carbon emission policy and opposes such policy, respectively. V 12 and V 22 represent the enterprises' payoffs with disapproval attitudes under different government policies. The term F represents the subsidy of the government to the enterprises that respond to the carbon emission policy positively. In contrast, H describes the penalty for the passive enterprises. Parameters θ and γ depict the impact of the subsidy and penalty on the government's payoffs in the game, respectively. It holds that 0 ≤ θ, γ ≤ 1.
To illustrate the proposed PLEG model, we conduct different simulated cases in the following.

Typical situations for the carbon emission game
In the CER proposal, both the government and enterprise can freely choose to cooperate (support carbon emission) or not. Considering the benefit of a green living environment and the responsibility of the government, CER will become a common consensus in the long term. However, enterprises will inevitably refuse to upgrade their industrial structures and manufacturing techniques due to their profit-seeking nature. The government plays a key role in achieving carbon neutrality, especially through economic subsidies and penalty policies. We present four typical case studies for different practical situations. The payoffs of the government and the enterprises in the game are represented by PLTS due to the uncertainty in the evaluations. Detailed cases are listed in Tables 4, 5, 6, 7 in Appendix A. For calculation convenience, the initial probabilities of the cooperation strategy for the government and the enterprises are set to 0.6 and 0.2 in the given cases, respectively. This implies that the government is more willing to initially choose to support strategies than enterprises. The evolutionary path and PLESSs for the four cases are illustrated in Figs. 3 and 4. Case 1 and Case 2 describe the evolutionary path to the PLESS with both players eventually choosing cooperation. The differences lie in how the PLESS is achieved and how the risk-aversion influences the evolutionary probabilities of their strategic decision making.
In Case 1, the determinant and trace values of the Jacobian for the PLEG dynamic system are Det 1 (0.6,0.2) = 0.025 and T r 1 (0.6,0.2) = −0.446, respectively. The initial strategy probabilities lie in the stable equilibrium state and the PLEG system will eventually converge to PLESS point (1, 1) (Fig. 3a). The evolving probabilities of supporting the CER strategy for Case 1 are illustrated in Fig. 4a. The government shows high enthusiasm for advancing the CER policy. The government strives to achieve CER in the game by maintaining the support strategy (Fig. 4a). However, the enterprises wait and try to stay out of the carbon market in the initial phase. Reflected in the evolutionary probabilities, the probability of choosing the support strategy for enterprises decreases in the early stage. The main reason for this behavior is that the profit in the CER market is relatively low and the prospects are ambiguous at first. With the continuous upsurge of promoting the CER market from the government, the enterprises' payoffs for the support strategy increase. Accordingly, the decision-making probability of the support strategy also increases as the game proceeds.
In Case 2, the probability of the support strategy for the government slowly increases because the prospect values of choosing the support strategy are always higher than those  2) = 0.799 at the initial state, which indicates that the initial phase is unstable. Accordingly, P LEG 2 will evolve to the PLESS point (1,1). This is also the sole PLESS solution for P LEG 2 (Fig. 3b). As a result, the probabilities of cooperation strategies for both players increase over time in Fig. 4b and finally reach the PLESS, that is, both selecting cooperation and reaching a win-win situation. The increase rate of the enterprises is higher than that of the government in that the introduced PT manipulates the risk-aversion attitudes of the enterprises and motivates the strategy with higher payoffs and lower risks.
In Case 3, the situation is slightly different. The government's gains of leveraging the CER policy are unattractive compared with the situations in Case 1 and Case 2. The payoffs of the government for the support strategy are lower not only than those in Case 1 but also for selecting the opposite strategy. The subsidy takes over much of the government's revenue in propelling CER, while the income from the penalty for uncooperative enterprises is disproportionate. All these factors hinder the enhancement of the supportive policy of the government. As a result, the probability of choosing the support strategy decreases rapidly over time in Fig. 4c. Enterprises, on the contrary, are more willing to join the carbon emission market initially due to the remarkable subsidy for low-carbon production. However, as policy support and financial support from the government weaken over the game, the profit for supporting CER also decreases for enterprises. The probability of the support strategy inevitably decreases, which is consistent with the government's decision-making results (Fig. 4c). In addition, the equilibrium analysis of this dynamic system indicates that the single PLESS is at point (0, 0). The players in the game both choose noncooperation (Fig. 3c), which coincides with the above analysis. The CER will fail in this situation.
In Case 4, the phase of the dynamic system implies that there is no stable equilibrium state, not to mention the PLESS. The initial state results in a saddle point (Fig. 3d). Consequently, the players' probabilities in Fig. 4d represent a cyclical fluctuation pattern. The variation in the probability of the enterprises' strategy lags behind that of the government. The main reason for this phenomenon lies in the subsidy to the enterprises for supporting the CER. Due to the maximum profit goal of the enterprises, their decision-making behavior will follow a strategy that brings high payoffs, namely, the support strategy in this scenario. Once the subsidy policy fades away with the decreasing intention of supporting the carbon market, the enterprises will also withdraw. As the government's payoff decreases with the low probability of cooperation but the enterprise remains on the verge of the cooperation strategy, the government will still try to revert to the cooperation strategy that brings high returns. Such periodic game behavior will repeat over time based on each player's expected returns for the cooperation strategy. The lagged circulated evolution also reflects the leader-follower relationships between the government and the enterprises.

Sensitivity analysis
One of the main goals of the proposed methods is to analyze the critical factors facilitating the CER initiative. In the following, we will investigate how the predefined factors and the risk-aversion attitudes influence the PLEG equilibrium. Then, constructive suggestions and managerial advice are provided.
As mentioned above, the decision-making pattern of enterprises follows governmental policy and actions. In other words, the government's attitudes toward the CER strategy may directly influence enterprises' game behaviors. To demonstrate this and analyze the sensitivity of the enterprises' game strategy, we take Case 1 as an example. The initial probabilities of the government for support strategy x are set from 0.2 to 1, while the others remain the same. The responses of the enterprise in terms of cooperation probability p B 1 to the government are illustrated in Fig. 5.
When the government tends to choose the opposed strategy (such as when x ≤ 0.5), the enterprises make the same choice. When the government behaves with a relatively high desire for the support strategy (x = 0.6), the support strategy of the enterprises shows a hysteretic increase with the government, as analyzed in Fig. 4a. When the support strategy of the government is sufficiently high, the hesitation by the enterprises disappears, and the desire for choosing the support strategy increases rapidly. As a result, the government plays a critical role in influencing the game results, and proactive actions and policies will motivate enterprises to participate in CER.
Another sensitive factor in the CER game is the subsidy, which is an attractive element that stimulates the growth of the CER scale in enterprises. To investigate how the subsidy changes the game behavior of the players, we utilize Case 3 to conduct a comprehensive analysis.
The PLEG related to Case 3 illustrates noncooperation game results. We propose a series of modified subsidy schemes for Case 3 from low to high levels. Because heavily subsidized policies are usually accompanied by supervisory and punitive measures, the penalty parameter γ in Table 2 is regulated proportionally to the subsidy scale. Details of the adjusted schemes are listed in Table 3. The probabilities of the cooperation strategy for the government and the enterprises under the given schemes are illustrated in Fig. 6.
Scheme 1 and Scheme 2 in Table 3 correspond to low levels of subsidies. The P LEG 3 still fails to bring the essential changes in terms of PLESS. Both the government and the enterprises ultimately refuse to choose the cooperation strategy. The differences lie in the duration of the deterioration process. The increased subsidy (F 2 ) makes the players insist on the support strategy for a longer time, especially for the enterprises. However, such a stimulus is temporary and unstable. The PLESSs in Scheme 1 and Scheme 2 are both (0, 0), and the players eventually choose to withdraw from the CER. When the subsidies are relatively higher (for instance, Scheme 3 and   1). Namely, the cooperation strategy prevails in the game results. The high subsidies also make the participants reach equilibrium more quickly; this is more noticeable for enterprises. Even though the tendency by cooperation in the government increases slowly initially, the willingness to select the support strategy by the enterprises still intensively increases (Scheme 3). As a result, the subsidy policy and supportive attitudes of the government are significant in the CER game and will directly affect the implementation of the CER initiative.
A distinctive feature of the proposed methods is the risk-aversion behavior in the game. The sensitivity of the PLEG to different risk preferences is the next focus. The risk-aversion behavior makes the government more circumspect in taking the support strategy in the game (Figs. 7a and 8a). In contrast, the enterprises show a more aggressive attitude toward joining the CER initiative (Figs. 7b and 8b). In Case 1, the PLESS of the game comes to the cooperation strategy. The subsidy of the government persists and the main beneficiaries are the enterprises. The large value of λ makes the enterprises more sensitive to the penalty risk in the game. The profitseeking nature impels enterprises to make quicker choices in the game. However, this game behavior pattern reverses in Case 3. The PLESS of Case 3 indicates that both players choose to oppose the CER. The low level of gains for the support strategy makes the government take a quicker response to the risk of the support strategy. The probability evidently decreases. Compared with that of the government, the strong risk-aversion effect makes enterprises persistent in the support strategy. It takes more time to shift to the opposed strategy with the increasing strength of riskaversion for enterprises (Fig. 8b). In conclusion, the riskaversion effect will contribute to facilitating cooperative strategies for players and restraining the uncooperative game results in the PLESS. This is also consistent with the practical decision making behavior in the market.

Comparisons
To investigate the risk attitudes of the players and illustrate the advantages of the proposed methods, we conduct a comparative analysis of the proposed methods. The prospect value matrix in (11) is detached from the proposed method for comparison. Additionally, Case 1 and Case 3 are selected in the comparative study. The game results under different methods are depicted in Fig. 9.
When ignoring the risk attitudes of players and the potential risk loss, the game results represent remarkable changes. Both the government and the enterprises take  prudent actions in the game. Specifically, the government adjusts its strategy from opposition to cooperation. This is mainly caused by the introduced risk-aversion effect. The achievement of the cooperation equilibrium in the enterprise takes more time, and the cautious wait-and-see attitude is observed in Fig. 9a. The swift equilibrium of cooperation in the PLEG disappears once the prospect value is introduced. A similar pattern can be observed in Case 3. The risk attitudes thoroughly change the PLESS from cooperation to noncooperation. The government selects the opposed strategy initially because the gain of supporting the CER is less attractive from the prospect value perspective. Unsurprisingly, enterprises show a hysteretic response to the government's action. The probability of cooperation strategy decreases promptly only when the enterprises perceive the economic losses caused by the increasing probability of the opposed strategy in the government.
In addition, we compare our method with the evolutionary game and classical matrix game to illustrate its advantages.
We take Case 1 as the comparative example. First, we transform the PLTS payoffs into crisp forms by means of a score function. Next, we use the evolutionary game model and the matrix game model to analyze the equilibrium of Case 1. The detailed results of the two methods are illustrated in Fig. 10.
The enterprise will choose to cooperate in the game. In contrast, the government cooperation probability increases slowly at first and finally decreases to zero. The ESS of Case 1 with the EG method is (0, 1), which indicates that only the enterprise will choose to cooperate in the game. Compared with the results in Fig. 4a, the riskaversion attitudes disappear for enterprises. This leads to the aggressive cooperation strategy by enterprises. Accordingly, the government tends to decrease cooperation intention in the game and obtain its optimal profit.
For the matrix game, the Nash equilibrium is (0, 1), in accordance with the results of the EG method. Accordingly, the game strategies of the government and the enterprise are similar to those in the EG situation.
The reason behind such differences is caused by two main aspects. One is that the PLTS in the proposed model provides more granular information in the game. The players can make more deliberate decisions in an uncertain environment. The other is that our methods take the risk preference into account. The risk-aversion decision-making behaviors will also result in different strategy preferences. All these factors make the proposed method more reliable in practical applications.

Conclusions
The evolutionary game models the strategic behaviors and optimal decision-making of limited-rational players through self-reinforcement learning in a dynamic process. As a result, it is applicable to many complex strategic game problems. The research of evolutionary games in uncertain environments has barely begun, and the existing works mainly try to use the fuzzy membership function to handle cognitive uncertainty. In the carbon reduction policy game, it is more convenient to obtain semantic descriptions rather than precise and crisp values. To bridge this gap, we develop PLEG for the carbon emission reduction game. The PLTS provides a more flexible tool for the player to express their gain and loss. It can quantify the uncertainty and incompleteness of subjective evaluations in the games. Besides, the integration with PLTS makes it efficient in handling linguistic information with epistemic and statistical uncertainties. One of the most remarkable postulates in the evolutionary game is the bounded-rational players in the decision making process. However, most of existing evolutionary games still manipulate the payoff of players with the expected utility. As a result, we introduce prospect theory and construct the prospect payoff matrix in PLEG. The notable feature of the prospect payoff matrix is that it is more sensitive to risk-aversion behavior. This is beneficial to both the government and the enterprises in carbon emission reduction development. The sensitivity analysis in the case study also demonstrates the role of risk attitudes in the game equilibrium and strategic decisionmaking. The evolutionary dynamics of the PLEG are based on the stochastic process. We utilize the replicator dynamics with the principles of maximizing group payoffs in the evolution process. The comparative analysis under different carbon emission reduction scenarios verifies the validity and advantage of the proposed methods.
The proposed methods provide a flexible tool to handle games that involve uncertainty, especially in the linguistic environment. The PLTS can compute with words and quantify the linguistic preference of players. Meanwhile, our methods allow for risk attitudes in the decision-making behaviors of players and integrate them into the game pattern. As a result, the proposed methods are more competitive and reliable in practical applications. Moreover, the proposed methods use only replicator dynamics to describe the game behaviors of players. The stochastic process is not fully considered. The other stochastic evolutionary dynamics in the uncertain environment should be further investigated and compared in future research. In addition, the integration of risk decision-making models into the payoff matrix is also an interesting research direction.