Shi Yin and Baizhou Li * A stochastic di erential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment

Abstract: Considering the fact that the development of low carbon economy calls for the low carbon technology sharing between interested enterprises, this paper study a stochastic di erential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises. In the paper, we consider the random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker. In the model, superior enterprises and inferior enterprises are separated entities, and they play Stacklberg master-slave game, Nash non-cooperative game, and cooperative game, respectively. We discuss the feedback equilibrium strategies of superior enterprises and inferior enterprises, and it is found that some random interference factors in sharing system can make the variance of improvement degree of low carbon technology level in the cooperation game higher than the variance in the Stackelberg game, and the result of Stackelberg game is similar to the result of Nash game. Additionally, a government subsidy incentive and a special subsidy that inferior enterprises give to superior enterprises are proposed.


Introduction
Global environment is an indivisible whole ecosystem. There seems to be rather compelling evidence that environmental pollution, resource depletion and global warming are issues that we seriously need to be concerned about today. Against this background, the development of low carbon technology has become an important support for global social and economic power. Responding to the development, low carbon technological innovation is playing a vital role in development of low carbon technology. How to achieve the low carbon technological innovation in enterprises is not only an important factor a ecting regional development of low carbon economy, but also the decisive factor for enterprises to acquire sustainable competitiveness and adapt to the competitive environment of future market. However, the implementation of low carbon technological innovation requires greater cost, and enterprises are faced with a great deal of pressure on capital investment. Therefore, low carbon technology sharing has become a vital role in the development of low carbon technology. Promotion of low carbon technology sharing calls for cooperation between interested enterprises. In this paper, we present a stochastic di erential game of low carbon technology sharing in innovation system of superior enterprises and inferior enterprises under uncertain environment. Our objective is to nd the optimal strategy of low carbon technology sharing and explore the key factors and mechanism of low carbon technology sharing.
Game theory has been used as an e ective tool to study knowledge, information and technology sharing. For example, Koessler [1] provided a simple Bayesian game model for the study of knowledge sharing; the study shows that their equilibrium is always a sequential equilibrium of the associated extensive form game with communication. In 2006, Cress and Martin [2] extended the model of Koessler to study knowledge sharing and rewards based on a game-theoretical perspective. It has been found that rewarding contributions with a cost-compensating bonus can be an e ective solution at the group level. Furthermore, Bandyopadhyay and Pathak [3] modelled a game of the knowledge interaction between two teams in two separate rms; it has been found that when the degree of complementarity of knowledge is higher enough, better payo s can be achieved if the top management enforces cooperation between the employees. Wu et al. [4] established a evolutionary game model of information sharing in network organization to analyze its dynamic evolutionary procedure. Their study showed that the key factors that a ect the system's evolution, cooperation pro t, initial cost of the cooperation, are obtained and researched. Ou et al. [5] modelled a game theory model to analyze the impact of important factors for low carbon international technology transfer. Their study showed that reduction of the control fees and taxes and increases of domestic subsidies all e ectively promote transfers. Xu and Xu [6] used prospect theory into evolutionary game theory to construct a perceived bene t matrix to explore the internal mechanism of low carbon technology innovation di usion under environmental regulation; theoretical study and numerical simulation showed that increasing subsidy factor, carbon tax rate and regulatory e ort can all induce enterprises to adopt low carbon technology innovation, and carbon tax rate has the strongest sensitivity. Gong and Xue [7] studied a game model of cooperative innovation between ICT low carbon developers and industrial enterprises. The authors considered that the sharing proportion played a key role in the cooperation, and the preferential tax policy of the government can coordinate the con ict in their cooperation, and the government incentive and regulatory penalty can promote both sides to improve input in both sharing arrangement modes.
Form the analysis of above studies, it is a mainstream trend that many scholars use game theory to study the sharing of low carbon technology. These studies have laid the method foundation for this article but we nd that most of the game models established in the literature are based on the static framework. In fact, with the rapid development of science, technology and information, the frequency and speed of low carbon technology upgrading have also improved dramatically. It means that the dynamic behavior of decision maker should be considered in the study of low carbon technology sharing in the same spatio-temporal region. In addition, Gao and Zhong [8] used di erential game approach to study the dynamic strategies for information sharing. Their study showed that the superior enterprises bene ts most when both rms fully cooperate, but the inferior enterprises enjoys the highest integral pro t when both rms only cooperate in information sharing and the lowest integral pro t. Meanwhile, low carbon technology related research have been widely studied recently due to their potential applications. Zhao et al. [9] derive the optimal solutions of the Nash equilibrium without cost sharing contract and the Stackelberg equilibrium with the integrator as the leader who partially shares the cost of the e orts of the supplier. Their study showed that cost sharing contract is an e ective coordination mechanism. Yu and Shi [10] used a stochastic di erential game model to study knowledge sharing between enterprise and university. However, stochastic di erential game model is seldom used in low carbon technology sharing and most of studies do not consider some random interference factors in sharing system. In fact, the process of decision making is often subject to various random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker [11]. The random interference factors can lead to a great uncertainty in equilibrium results because they are di cult to capture by the decision makers [12]. In this paper, we study the low carbon technology sharing in innovation system of superior enterprises and inferior enterprises under uncertain environment in the case of stochastic intervention.
The structure of this paper is organized as follows. In Section 2, stochastic di erential game formulation is provided. In Section 3, we resolve models of Stacklberg master-slave game. In Section 4, we resolve models of Nash non-cooperative game. Section 5 is devoted to models of cooperative game, and comparative analysis of equilibrium results are presented in Section 6. Section 7 summarizes the paper.

Stochastic di erential game formulation
For the sake of simplicity, enterprises of low carbon technology sharing can be divided into two interest groups: superior enterprises and inferior enterprises, which store, respectively, large quantities of low carbon technologies and heterogeneous resources of low carbon technologies. In the paper, we study a low carbon technology innovation system that consists of a single superior enterprise C and a single inferior enterprise E. In order to clarify the above problem, we further assume that decision makers are completely rational, full information, and aim to maximize their return.
Let L C (t) denote the e ort level of superior enterprises at time t, and let L E (t) denote the e ort level of inferior enterprises in the sharing process of low carbon technology. For further consideration, the sharing cost of low carbon technology can be denoted by C C (t) and C E (t) which are the quadratic functions of the e ort level of superior enterprises and inferior enterprises at time t, respectively. Consider where c C (t) and c E (t) are the cost coe cients of superior enterprises and inferior enterprises at time t, respectively. Let K(t) denote the technology level of low carbon in collaborative innovation system of superior enterprises and inferior enterprises at time t. In the sharing process of low carbon technology, the collaborative innovation between superior enterprises and inferior enterprises can improve the technology level of low carbon. Let σ C (t) and η E (t) denote the in uence of the e ort level of low carbon technology sharing on collaborative innovation between superior enterprises and inferior enterprises, respectively, at time t, namely, innovation capability coe cient of low carbon technology. The dynamics of technology level of low carbon are governed by the stochastic di erential equation Hence where δ is the attenuation coe cient of low carbon technology, δ ∈ ( , ]; z(t) and εK(t) are the standard Wiener process and random interference factors of superior enterprises and inferior enterprises at time t, respectively. Let π(t) denote the total payo of low carbon in collaborative innovation system at time t. Let α(t) and β(t) denote the in uence of the e ort level of low carbon technology sharing on the total income of superior enterprises and inferior enterprises, respectively, at time t, namely, the marginal return coe cient of low carbon technology. Total payo function can be expressed as where γ is the in uence of the technology innovation of low carbon on total revenue, namely, innovation in uence coe cient of low carbon technology, γ ∈ ( , ]; λ is the government subsidy coe cient of low carbon technology based on increments of low carbon technology level in collaborative innovation, λ ∈ ( , ]. We further assume that the total revenue is allocated between two participants, and θ(t) is the payo distribution coe cient of superior enterprises at time t, θ(t)∈ [ , ]. Although inferior enterprises have heterogeneous resources of low carbon technologies, superior enterprises store large quantities of low carbon technologies. Many practical low carbon technologies can be acquired by inferior enterprises in the sharing process of low carbon technology. Therefore, inferior enterprises need to pay much more extra sharing cost of low carbon technology. Let ω(t) denote the subsidy of low carbon technology, which inferior enterprises give to superior enterprises. The objective function of superior enterprises and inferior enterprises satisfy the following partial di erential equations where ρ is the discount rate of low carbon technology of superior enterprises and inferior enterprises, ρ ∈ ( , ].
There are three control variables, , and a state variable K(t)≥ in the sharing model of low carbon technology. Feedback control has been used more and more widely in analysis of information and economic systems [13]. Moreover, feedback control strategy has better control e ect, compared with open-loop control strategy. Therefore, we use feedback control strategy to analyze sharing model of low carbon technology.

Resolving models of Stacklberg master-slave game
In the sharing process of low carbon technology between superior enterprises and inferior enterprises, inferior enterprises can acquire many practical low carbon technologies from superior enterprises, and then inferior enterprises need to pay much more extra sharing cost of low carbon technology. In order to promote the technology sharing of low carbon, the inferior enterprises (the leaders) determine an optimal sharing e ort level and an optimal subsidy of low carbon technology sharing, and then the superior enterprises (the followers) choose their optimal sharing e ort level according to the optimal sharing e ort level and subsidy. This leads to a Stackelberg equilibrium.

. Stacklberg master-slave solutions
where L S C and L S E are the optimal e ort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively, where V S C (K) and V S E (K) are the optimal sharing payo function of low carbon technology of superior enterprises and inferior enterprises respectively, Proof. In order to obtain the Stacklberg equilibrium, there exists a optimal sharing revenue function of low carbon technology, V S C (K) and V S E (K), which is a continuous di erentiable function. First, we use backward induction to solve optimal control problem. The optimal sharing revenue function, V C (K), satis es the following Hamilton-Jacobi-Bellman equation For solving formula (10), using extreme conditions and searching for the optimal value of L C by setting the rst partial derivative equal to zero, we can get Second, the optimal sharing revenue function, V E (K), satis es the following Hamilton-Jacobi-Bellman equation Substituting the result of (11) into (12), we can obtain Performing the indicated maximization in (13) and searching for the optimal value of L E and ω by setting the rst partial derivative equal to zero, we can get Substituting the results of (11), (14a) and (14b) into (10) and (12), we can get The solution of the HJB equation is a unary function with K as independent variable. As [11], we have where a , b , a and b are the constants to be solved. Setting the rst partial derivative to formula (17), we can get Substituting the results of (17) and (18) into (15) and (16), we can get Substituting the results of a and a into (11), (14a) and (14b), we can further get Substituting the results of (17) and (18) into (7), we can get Hence, the optimal total payo of low carbon technology sharing can be expressed as follows Equations (21)-(22) indicate that, under model of Stacklberg game, the e ort level of superior enterprises and inferior enterprises is proportional to the government subsidy of low carbon technological innovation and the innovation capability of low carbon technology; the e ort level of superior enterprises and inferior enterprises is inversely proportional to the sharing cost and the discount rate of low carbon technology; the sharing payo of low carbon technology is proportional to the marginal return of low carbon technology.

. The limit of expectation and variance
From Proposition 3.1, the payo of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, under Stacklberg game equilibrium, it is necessary to study the limit of expectation and variance. Substituting the results of (8a) and (8b) into (4), we can get .
Proposition 3.2. The limit of expectation and variance in Stackelberg game feedback equilibrium satisfy Proof.
Lemma 3.3 (see [14). ] Itô's lemma is an identity used in Itô calculus to nd the di erential of a time-dependent function of a stochastic process. If f (x) is quadratic continuous di erentiate, t ∈ ∀ satisfy the following Itô equation where B (t) is the Brownian motion. According to formula (24), using Itô equation, we can get We can derive the expectation value for both sides of (24) and (27), and then E (K (t)) and E (K (t)) satisfy the following set of non-homogeneous linear di erential equations Solving the above non-homogeneous linear di erential equation leads to .

Resolving models of Nash non-cooperative game
Under Nash non-cooperative game, superior enterprises and inferior enterprises will simultaneously and independently choose their optimal e ort levels of low carbon technology sharing based on maximization of their pro ts.

. Nash non-cooperative game solutions
where L N C and L N E are the optimal e ort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively, where V N C (K) and V N E (K) are the optimal sharing payo functions of low carbon technology of superior enterprises and inferior enterprises, Proof. According to su cient conditions for static feedback equalization, there exists an optimal sharing revenue function of low carbon technology, which is a continuous di erentiable function. The optimal sharing revenue function satis es the following Hamilton-Jacobi-Bellman equation In order to maximize their pro ts, the inferior enterprises are so rational that they cannot accept the optimal subsidy of low carbon technology sharing, ω=0. For solving formula (32a) and (32b), using extreme conditions and searching for the optimal value of L C by setting the rst partial derivative equal to zero, we can get Substituting the results of (33a) and (33b) into (32a) and (32b), we can obtain The solution of the HJB equation is a unary function with K as independent variable. As [11], we have where a , b , a and b are the constants to be solved. Substituting the result of (36) into (34) and (35), we can get Using the K ≥ to (37) and (38), parameter values of the optimal value function can be expressed as follows Substituting the results of a , b , a and b into (33a), (33b) and (36), we can get the optimal e ort level of low carbon technology sharing and the optimal sharing payo function of low carbon technology of superior enterprises and inferior enterprises, respectively.

. The limit of expectation and variance
From Proposition 4.1, the payo of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, under Nash equilibrium, it is necessary to study the limit of expectation and variance. Substituting the results of (30a) and (30b) into (4), we can get .

Proposition 4.2. The limit of expectation and variance in Nash non-cooperative game feedback equilibrium satisfy
Proof. The proof of Proposition 4.2 is similarly to Proposition 3.2, so we do not repeat it here.

Resolving models of cooperative game
Under cooperative game, superior enterprises and inferior enterprises will choose their optimal e ort levels and sharing payo function of low carbon technology sharing based on maximization of their total payo . Thus, low carbon technology level can be further improved through cooperation between superior enterprises and inferior enterprises.

. Cooperative game solutions
Proposition 5.1. If the above conditions are satis ed, the feedback cooperative game equilibria are where L C C and L C E are the optimal e ort level of low carbon technology sharing of superior enterprises and inferior enterprises, respectively, where V C (K) is the optimal sharing payo function of low carbon technology of superior enterprises and inferior Proof. Under cooperative game, the sharing revenue function satis es the following equation In order to obtain the cooperative equilibrium state in this case, we assume that sharing revenue function of low carbon technology is a continuous di erentiable function. The optimal sharing revenue function satis es the following Hamilton-Jacobi-Bellman equation For solving formula (46), using extreme conditions and searching for the optimal value of L C by setting the rst partial derivative equal to zero, we can get Substituting the results of (47a) and (47b) into (46), we can obtain The solution of the HJB equation is a unary function with K as independent variable. As [11], we have where a and b are the constants to be solved. Substituting the result of (49) into (48), we can get Using the K ≥ to (50), parameter values of the optimal value function can be expressed as follows Substituting the results of a and b into (47a), (47b) and (49), we can get the optimal e ort level of low carbon technology sharing and the optimal sharing payo function of low carbon technology of superior enterprises and inferior enterprises, respectively.

. The limit of expectation and variance
From Proposition 5.1, the payo of superior enterprises and inferior enterprises is related to the improvement degree of low carbon technical level, whose possible values are numerical outcomes of a random phenomenon by various random interference factors. Therefore, it is necessary to study the limit of expectation and variance. Substituting the results of (43a) and (43b) into (4), we can get where τ = σ[α(ρ+δ)+σ(γ+λ)] .
Proposition 5.2. The limit of expectation and variance in cooperative game feedback equilibrium satisfy Proof. The proof of Proposition 5.2 is similarly to Proposition 3.2, so we do not repeat it here.

Comparative analysis of equilibrium results
According to the ≤ θ ≤ , we can get L C Proof. From Proposition 3.1 and Proposition 4.1, we can get According to the ≤ θ ≤ , we can further get V S Proposition 6.2 indicates that, under the condition that inferior enterprises give a subsidy to superior enterprises, the subsidy of low carbon technology is an incentive mechanism which can promote low carbon technology sharing between superior enterprises and inferior enterprises. Superior enterprises and inferior enterprises can share more low carbon technologies through this mechanism. Proposition 6.3. Under cooperative game, the total payo exceeds the total payo of Stacklberg master-slave game, and the total payo of Stacklberg master-slave game exceeds the total payo of Nash non-cooperative game in collaborative innovation system. That is to say, there exist V C Proof. According to Proposition 3.1, Proposition 4.1 and Proposition 5.1, we can get According to the ≤ θ ≤ , we can get V C (K) ≥ V S (K) and Proof. According to Proposition 3.2, Proposition 4.2 and Proposition 5.2, we can get Similarly, we can get (62a) The rst derivative of − e −δt + e − δt function of t is greater than 0 for t ∈ ( , ∞). When t → , we have − e −δt + e − δt = , and then we can get D K (t) − D (K (t)) > . Similarly, we can get Proposition 6.4 indicates that enterprises can create and bring new low carbon technologies better than in case of the Stackelberg master slave game. However, some random interference factors in sharing system can make the variance of the improvement degree of cooperation game higher than the variance of the Stackelberg master slave game. That is to say, enterprises need to bear more risk to achieve higher payo in sharing system under the cooperative game. Similarly, the result of Stackelberg game is similar to the result of Nash game. Therefore, di erent game modes are chosen by enterprises with di erent risk preferences. Cooperative game may be chosen by some enterprises with high risk preference, while Stackelberg game may be chosen by enterprises with moderate risk preference. The risk averse entity may choose Nash non-cooperative game.

Conclusions
In this paper, we have shown a stochastic di erential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment. In our model, we use the limit of expectation and variance of the improvement degree to identify the in uence of random factors. According to Hamilton-Jacobi-Bellman equation, we get the optimal e ort level of low carbon technology sharing, the subsidy of low carbon technology, the optimal sharing payo and the total payo of low carbon in collaborative innovation system of superior enterprises and inferior enterprises, respectively in the above game models. By comparing and analyzing of equilibrium results, we have shown that the e ort level of superior enterprises and inferior enterprises is proportional to the government subsidy of low carbon technological innovation and the innovation capability of low carbon technology; the e ort level of superior enterprises and inferior enterprises is inversely proportional to the sharing cost and the discount rate of low carbon technology; the sharing payo of low carbon technology is proportional to the marginal return of low carbon technology. Moreover, we have shown that some random interference factors in sharing system can make the variance of the improvement degree of cooperation game higher than the variance of the Stackelberg master slave game. Similarly, the result of Stackelberg game is similar to the result of Nash game. By analyzing this stochastic di erential game models, we have also provided a government subsidy incentive and a subsidy that inferior enterprises give to superior enterprises.