**2. Model setup**

iii. mathematically, we show that the issue of the game depends on the parameters of the game and the type of equilibrium one considers.

duopoly.

*Carbon Capture*

iv. by applying the HJB equation we obtain the optimal effort level and the optimal level of subsidy for sharing hybrid-enabling technology via feedback equilibrium strategies whilst examining the Stackelberg equilibria, Nash equilibria and cooperative equilibria under Bertrand

v. we reveal that for a given level of payoff distribution the Stackelberg

vi. we show that in Stackelberg and Nash games, optimal hybrid-enabling technology innovation is proportional to the government subsidy, but the variance improvement degree of the Stackelberg game is different to the

vii. our characterization of the short run price competition by strategic supplies for renewable and fossil resources, provides a more robust model than that presented by Bertrand-Edgworth, in which price competition with fixed

effective hybrid-enabling technology innovation strengthens an innovator's

equilibrium outcome, allowing each power plant to optimally use energy

viii. our model shows that robust cost-reducing R&D investments with

sources to produce electricity while maximizing their payoffs.

spending is costly, and the presumption is that R&D spending is somehow

In recent years, researchers have incorporated the theory of SDGs, originated from [14, 17–20] to analyzed environmental issues. Especially [21] analyzed (two player) zero-sum stochastic differential games in a rigorous way, and proved that the upper and lower value functions of such games satisfy the dynamic programming principle whilst being the unique viscosity solutions of their associated

In Section 2, the proposed model and elements of evolutionary game theory are presented. In Section 3 by implementing the Stackelberg game we examine feedback Stackelberg equilibria, optimal level of subsidy for the shared hybrid-enabling technology from its counterpart and the limit of expectation and variance. In Section 4 by implementing a Nash game we examine feedback Nash equilibria and the limit of expectation and variance under hybrid-enabling technology. In Section 5 by

connected to increased innovation, revenue growth and profits.

Hamilton-Jacobi-Bellman-Isaacs equations.

**60**

competitive position and the Stackelberg structure emerges as an

Therefore, under a Stochastic Differential Game (SDG) paradigm with uncertainty, each power plant can optimally use energy sources to produce electricity while maximizing their payoffs. Each power plant is capable of using fossil fuels ð Þ *F* and renewable sources ð Þ *R* to produce electricity at any time. To maintain the generality of the proposed model, this model is not limited to a specific energy source. Hence, the terms }*F*} and }*R*}, are used throughout the paper. On the other hand the government encourages power plants to conform to a maximum accepted level of carbon emissions through strategies such as the imposition of tariffs on polluters as well as incentives for those who choose to undertake R&D measures to reduce their emission levels in order to maintain environmental sustainability. R&D

paradigm dominate the Nash equilibria.

results of the Nash game.

(endogenous) capacities was used.

equilibria under endogenous hybrid-technology innovation and the sharing

We propose that the production process of electricity leads to emissions and is proportional to the power industry's use of energy source. We assume that there are two power plants (Player I) and (Player II) in the energy market and each power plant is capable of using *fossil fuels* (*F*) and *renewable sources* (*R*) to generate power at any given time *t*. To reduce the level of Green House Gas (GHG)-emissions into the atmosphere (accordance with [22] Protocol), the government will set a maximum emission quantitative level, that is directly linked to the power industry's use of energy source *F*, when producing electricity. Government encourages the power industry to undertake necessary hybrid-enabling technology to reduce their GHGemission levels to the maximum accepted quantitative level, *η<sup>F</sup>*, and improve efficiency in renewables. We assume that the power plants change their strategies over time based on payoff comparisons based on hybrid-enabling technological advances. This contradicts with classical non cooperative game theory that analyzes how rational players will behave through static solution concepts such as the Nash equilibrium (NE) (i.e., a strategy choice for each player whereby no individual has a unilateral incentive to change his or her behavior).

Under the theory of evolutionary games, the production strategies in the *absence of any superior hybrid-enabling technological advances*, allows the power plants to play a symmetric two-person 2 � 2 bi-matrix game. Thus, for each power plant, we define the set Σ as its pure strategy given by the set of non-negative prices [0, ∞). According to the Bertrand game all firms setting the lowest price will split market demand equally (Hotelling type) and the profit can be calculated subject to the electricity prices and the associated cost functions.

Then each iteration of an evolutionary game, where two matched power plants in accordance with Bertrand paradigm compete with each market and play a oneshot non-zero-sum game, represents the benchmark game of the population. If *pij*, *pji* is the matrix of prices of power plants, respectively, then via Proposition 1 (given below), it will allow us to derive Nash equilibria of prices for these two matched power plants. On the demand side we assume that the preferences are quadratic as in [23].

We define the continuous demand function *Dij* , for each power plant as

$$D\_{\vec{\eta}} = \mathfrak{a}\_{\vec{\eta}} - \beta\_{\vec{\eta}} \left( p\_{\vec{\eta}} + \tau\_i \right) + \chi\_{\vec{\eta}} \left( p\_{ji} + \tau\_j \right), \quad i, j \in \{F, R\} \tag{1}$$

where *Dij* is the demand function for the power plants employing the energy source *i*∈ f g *F*, *R* against the power plant which use the energy source *j*∈f g *F*, *R : τ<sup>i</sup>* is the tariff imposed by government subject to the power source *i*. For example government impose a tariff-rate quota (TRQs) ð Þ *τ<sup>F</sup>* , for *fossil fuels* (*F*) and a feed-intariff (FITs) ð Þ *τ<sup>R</sup>* , for *renewable sources* (*R*). *pij* is the electricity price of the power plant that uses the energy source *i*, versus the power plant that employs the energy source *j*. *aij* >0, is the constant market base for the power plant that employs the energy source *i* versus the one which use the energy source *j*. The parameters *βij* >0 and *γij* > 0, are independent constants that captures the demand sensitivity of a

power plant subject to its own price *βij* and its rival's price *γij*. Eq. (1), concludes that the goods in the market are gross substitutes and that the demand function *Dij*, is increasing in the price of the rival firm *pji*.

such that *<sup>β</sup>jiβij* 6¼ *<sup>γ</sup>ijγji*

ð Þ *τF*, *τ<sup>R</sup>* , *can be obtained as*

where

*θij* ¼

(2) and simplifying.

**technology**

(utility) in **Table 1**.

its rival.

**Table 1.**

**63**

*D*<sup>∗</sup>

Π∗

*ij* <sup>¼</sup> *<sup>β</sup>ij*Λ<sup>∗</sup>

*ij* <sup>¼</sup> *<sup>β</sup>ij* <sup>Λ</sup><sup>∗</sup>

*ωij* ¼

*Proof.* Obtain the results by substituting Λ<sup>∗</sup>

*Bi matrix for two power plants by different energy sources.*

*ij* <sup>2</sup>

2*βjiaij* þ *γijaji* þ *βjiγij Cj* þ *v <sup>j</sup>*

*<sup>γ</sup>ijγji* � <sup>2</sup>*βjiβij*

<sup>4</sup> *: p*<sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

*ij* and *p*<sup>∗</sup>

second order conditions, obtain the maximum point in the set as: *<sup>∂</sup>*2Π*ij*

is negative confirming that the profit function is concave at this point.

*ij* <sup>¼</sup> *<sup>β</sup>ij <sup>θ</sup>ij* <sup>þ</sup> *<sup>ω</sup>ij<sup>τ</sup>* <sup>∗</sup>

optimum prices if the profit functions are concave on *pij* and on *pji*. Then via the

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

Since *βij* >0, implies that the second derivative of the profit function in equilibrium

**Proposition 2.** *At equilibrium prices the power plant's demand and profit under*

,

� *Fi* <sup>¼</sup> *<sup>β</sup>ij <sup>θ</sup>ij* <sup>þ</sup> *<sup>ω</sup>ij<sup>τ</sup>* <sup>∗</sup>

<sup>4</sup>*βjiβij* � *<sup>γ</sup>ijγji* and *<sup>χ</sup>ij* <sup>¼</sup> *<sup>β</sup>jiγij*

*Remark 1.* The only outcome where neither power plant has an incentive to deviate is when *pij* ¼ *pji* ¼ *ci*, which will be the Nash or Bertrand equilibrium for the game. The intuition behind this result is that power plants will keep *undercutting* the price of its rival until price equals marginal cost. In the long run price changes with marginal cost and industry production increases with demand and falls with marginal cost. One way for a power plant to avoid the Bertrand paradox and earn economic profit in a Bertrand setting is to have a competitive cost advantage over

**2.1 Production decisions of power plants with homogenous hybrid/enabling**

Restricting ourselves to a two matched symmetric two-person bi- matrix game in random contest in a one-population evolutionary game, we define the payoff

**Note 1.** Power and the payoff are measured on a utility scale consistent with the power plant's preference ranking. Furthermore, [24–26] have applied symmetric two-person bi-matrix game in random contest to study evolutionary stable games.

**Power Plant I** Fossil Fuel ð Þ Π*FF*, Π*FF* ð Þ Π*FR*, Π*RF*

Power Plant II Production Strategy Fossil Fuel Renewable Sources

Renewable Sources ð Þ Π*RF*, Π*FR* ð Þ Π*RR*, Π*RR*

*<sup>i</sup>* <sup>þ</sup> *<sup>χ</sup>ij<sup>τ</sup>* <sup>∗</sup> *j*

*ji* are obtained via *pij* ¼ Λ*ij* þ *Ci* þ *vi* and are the

*<sup>i</sup>* <sup>þ</sup> *<sup>χ</sup>ij<sup>τ</sup>* <sup>∗</sup> *j*

<sup>4</sup>*βjiβij* � *<sup>γ</sup>ijγji* , (7)

<sup>2</sup>

<sup>þ</sup> *<sup>γ</sup>ijγji* � <sup>2</sup>*βjiβij* ð Þ *Ci* <sup>þ</sup> *vi*

*∂p*<sup>2</sup> *ij*

� *Fi*,

<sup>4</sup>*βjiβij* � *<sup>γ</sup>ijγji* (8)

*ij* from Proposition 1 into Eqs. (1) and

¼ �2*βij* < 0*:*

(6)

The government's tariff policy for the power plants with respect to their source of energy for long-time periods are transparent, and this information is available to the public. Therefore, it is assumed that the competitive power plants follow the government's financial legislation, having the capability and technological skills to produce electricity from specific sources at any given time to meet energy demand. Then for each time-period the power plant will consider the tariff-rate quota or feed-in-tariff and adopt a pricing strategy for the selected energy source. Hence, we conclude that the production rate of the power plants is equal to the corresponding demand rates with a negligible internal consumption and waste rate.

To apply the Backward induction technique to investigate the equilibrium prices, demand, and profits, we define the profit function for each power plant as

$$\begin{split} \Pi\_{\vec{\eta}} &= \left( p\_{\vec{\eta}} - \mathbf{C}\_{i} - \boldsymbol{\nu}\_{i} \right) D\_{\vec{\eta}} - F\_{i} \\ &= \left( p\_{\vec{\eta}} - \mathbf{C}\_{i} - \boldsymbol{\nu}\_{i} \right) \left( a\_{\vec{\eta}} - \beta\_{\vec{\eta}} \left( p\_{\vec{\eta}} + \boldsymbol{\tau}\_{i} \right) + \boldsymbol{\gamma}\_{\vec{\eta}} \left( p\_{ji} + \boldsymbol{\tau}\_{j} \right) \right) - F\_{i}, \end{split} \tag{2}$$

where *i*, *j*∈ f g *F*, *R* and *Ci* >0 is the unit production cost of the power plant when using energy source *i*. *vi* >0, for any additional R&D unit cost for undertaking hybrid-enabling technology, for a power plant that rely on an energy source *i*, (*Fi* >0 is the initial setup cost of the power plants when using the energy source *i*). We also assume that *pij* � *Ci* � *vi* � � <sup>&</sup>gt;0*:* The firms' technologies are represented by their reduced cost functions. This assumes that all factor markets are perfectly competitive and – both here and in the models of imperfect competition in the output market – are not influenced by any strategic behavior of the firms in other markets. We will make alternative assumptions about those technologies. In the first assumption, pollution is proportional to output and firms do not have any further abatement technologies.

**Proposition 1.** *The equilibrium price for the power plants under* ð Þ *τF*, *τ<sup>R</sup> , is given as pij* ¼ Λ*ij* þ *Ci* þ *vi:*

*Proof.* Via the first order conditions of the profit function (Eq. (2)), obtain

$$\begin{split} \frac{\partial \Pi\_{\vec{\boldsymbol{\eta}}}}{\partial \mathbf{p}\_{\vec{\boldsymbol{\eta}}}} &= \mathbf{a}\_{\vec{\boldsymbol{\eta}}} - 2\beta\_{\vec{\boldsymbol{\eta}}} \Big( \mathbf{p}\_{\vec{\boldsymbol{\eta}}} - \mathbf{C}\_{\vec{\boldsymbol{\epsilon}}} - \boldsymbol{\nu}\_{\vec{\boldsymbol{\eta}}} \Big) + \boldsymbol{\gamma}\_{\vec{\boldsymbol{\eta}}} \Big( \mathbf{p}\_{\vec{\boldsymbol{\mu}}} - \mathbf{C}\_{\vec{\boldsymbol{\eta}}} - \boldsymbol{\nu}\_{\vec{\boldsymbol{\eta}}} \Big) + \boldsymbol{\gamma}\_{\vec{\boldsymbol{\eta}}} \Big( \boldsymbol{\tau}\_{\vec{\boldsymbol{\eta}}} + \mathbf{C}\_{\vec{\boldsymbol{\eta}}} + \boldsymbol{\nu}\_{\vec{\boldsymbol{\eta}}} \Big) \\ &- \beta\_{\vec{\boldsymbol{\eta}}} (\boldsymbol{\tau}\_{\vec{\boldsymbol{\eta}}} + \mathbf{C}\_{\vec{\boldsymbol{\eta}}} + \boldsymbol{\nu}\_{\vec{\boldsymbol{\eta}}}) = \mathbf{0}. \end{split} \tag{3}$$

Defining Λ*ij* ¼ *pij* � *Ci* � *vi* and using Λ*ij* and Λ*ji*, rewrite the first order conditions as:

$$\begin{cases} 2\beta\_{\vec{\eta}}\Lambda\_{\vec{\eta}} - \chi\_{\vec{\eta}}\Lambda\_{\vec{\mu}} &= a\_{\vec{\eta}} + \chi\_{\vec{\eta}}(\pi\_{j} + \mathbf{C}\_{j} + \boldsymbol{\nu}\_{j}) - \beta\_{\vec{\eta}}(\pi\_{i} + \mathbf{C}\_{i} + \boldsymbol{\nu}\_{i}), \\ 2\beta\_{\vec{\mu}}\Lambda\_{\vec{\mu}} - \chi\_{\vec{\mu}}\Lambda\_{\vec{\eta}} &= a\_{\vec{\mu}} + \chi\_{\vec{\mu}}(\pi\_{i} + \mathbf{C}\_{i} + \boldsymbol{\nu}\_{i}) - \beta\_{\vec{\mu}}(\pi\_{j} + \mathbf{C}\_{j} + \boldsymbol{\nu}\_{j}). \end{cases} \tag{4}$$

Simultaneously solving Eq. (4), obtain

$$\Lambda\_{\vec{\eta}} = \frac{2\beta\_{\vec{\mu}}a\_{\vec{\eta}} + \chi\_{\vec{\eta}}a\_{\vec{\mu}} + \beta\_{\vec{\mu}}\chi\_{\vec{\eta}}\left(\tau\_{\vec{\jmath}} + \left(\mathbf{C}\_{\vec{\jmath}} + \nu\_{\vec{\jmath}}\right)\right) + \left(\chi\_{\vec{\eta}}\chi\_{\vec{\mu}} - 2\beta\_{\vec{\mu}}\beta\_{\vec{\eta}}\right)\left(\tau\_{\vec{\imath}} + \left(\mathbf{C}\_{\vec{\imath}} + \nu\_{\vec{\imath}}\right)\right)}{\left(4\beta\_{\vec{\mu}}\beta\_{\vec{\eta}} - \chi\_{\vec{\eta}}\chi\_{\vec{\mu}}\right)} \tag{5}$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

such that *<sup>β</sup>jiβij* 6¼ *<sup>γ</sup>ijγji* <sup>4</sup> *: p*<sup>∗</sup> *ij* and *p*<sup>∗</sup> *ji* are obtained via *pij* ¼ Λ*ij* þ *Ci* þ *vi* and are the optimum prices if the profit functions are concave on *pij* and on *pji*. Then via the second order conditions, obtain the maximum point in the set as: *<sup>∂</sup>*2Π*ij ∂p*<sup>2</sup> *ij* ¼ �2*βij* < 0*:* Since *βij* >0, implies that the second derivative of the profit function in equilibrium is negative confirming that the profit function is concave at this point.

**Proposition 2.** *At equilibrium prices the power plant's demand and profit under* ð Þ *τF*, *τ<sup>R</sup>* , *can be obtained as*

$$\begin{aligned} D\_{\vec{\eta}}^{\*} &= \beta\_{\vec{\eta}} \Lambda\_{\vec{\eta}}^{\*} = \beta\_{\vec{\eta}} \Big( \theta\_{\vec{\eta}} + a \boldsymbol{\eta}\_{\vec{\eta}} \boldsymbol{\tau}\_{i}^{\*} + \boldsymbol{\chi}\_{\vec{\eta}} \boldsymbol{\tau}\_{j}^{\*} \Big), \\ \Pi\_{\vec{\eta}}^{\*} &= \beta\_{\vec{\eta}} \Big( \Lambda\_{\vec{\eta}}^{\*} \Big)^{2} - F\_{i} = \beta\_{\vec{\eta}} \Big( \theta\_{\vec{\eta}} + a \boldsymbol{\eta}\_{\vec{\eta}} \boldsymbol{\tau}\_{i}^{\*} + \boldsymbol{\chi}\_{\vec{\eta}} \boldsymbol{\tau}\_{j}^{\*} \Big)^{2} - F\_{i}, \end{aligned} \tag{6}$$

where

power plant subject to its own price *βij* and its rival's price *γij*. Eq. (1), concludes that the goods in the market are gross substitutes and that the demand function *Dij*, is

The government's tariff policy for the power plants with respect to their source of energy for long-time periods are transparent, and this information is available to the public. Therefore, it is assumed that the competitive power plants follow the government's financial legislation, having the capability and technological skills to produce electricity from specific sources at any given time to meet energy demand. Then for each time-period the power plant will consider the tariff-rate quota or feed-in-tariff and adopt a pricing strategy for the selected energy source. Hence, we conclude that the production rate of the power plants is equal to the corresponding

demand rates with a negligible internal consumption and waste rate.

*Dij* � *Fi*

To apply the Backward induction technique to investigate the equilibrium prices, demand, and profits, we define the profit function for each power plant as

*aij* � *βij pij* þ *τ<sup>i</sup>*

using energy source *i*. *vi* >0, for any additional R&D unit cost for undertaking hybrid-enabling technology, for a power plant that rely on an energy source *i*, (*Fi* >0 is the initial setup cost of the power plants when using the energy source *i*).

their reduced cost functions. This assumes that all factor markets are perfectly competitive and – both here and in the models of imperfect competition in the output market – are not influenced by any strategic behavior of the firms in other markets. We will make alternative assumptions about those technologies. In the first assumption, pollution is proportional to output and firms do not have any

� �

where *i*, *j*∈ f g *F*, *R* and *Ci* >0 is the unit production cost of the power plant when

**Proposition 1.** *The equilibrium price for the power plants under* ð Þ *τF*, *τ<sup>R</sup> , is given as*

þ *γij pji* � *Cj* � *v <sup>j</sup>* � �

Defining Λ*ij* ¼ *pij* � *Ci* � *vi* and using Λ*ij* and Λ*ji*, rewrite the first order condi-

� � � � <sup>þ</sup> *<sup>γ</sup>ijγji* � <sup>2</sup>*βjiβij*

4*βjiβij* � *γijγji*

2*βji*Λ*ji* � *γji*Λ*ij* ¼ *aji* þ *γji*ð Þ� *τ<sup>i</sup>* þ *Ci* þ *vi βji τ <sup>j</sup>* þ *Cj* þ *v <sup>j</sup>*

� � � *<sup>β</sup>ij*ð Þ *<sup>τ</sup><sup>i</sup>* <sup>þ</sup> *Ci* <sup>þ</sup> *vi* ,

� �

� � (5)

*Proof.* Via the first order conditions of the profit function (Eq. (2)), obtain

� � � �

þ *γij pji* þ *τ <sup>j</sup>*

>0*:* The firms' technologies are represented by

þ *γij τ <sup>j</sup>* þ *Cj* þ *v <sup>j</sup>* � �

� �*:*

ð Þ *τ<sup>i</sup>* þ ð Þ *Ci* þ *vi*

� *Fi*,

(2)

(3)

(4)

increasing in the price of the rival firm *pji*.

*Carbon Capture*

Π*ij* ¼ *pij* � *Ci* � *vi* � �

We also assume that *pij* � *Ci* � *vi*

further abatement technologies.

¼ *aij* � 2*βij pij* � *Ci* � *vi*

�*βij*ð Þ¼ *τ<sup>i</sup>* þ *Ci* þ *vi* 0*:*

Simultaneously solving Eq. (4), obtain

2*βjiaij* þ *γijaji* þ *βjiγij τ <sup>j</sup>* þ *Cj* þ *v <sup>j</sup>*

� �

2*βij*Λ*ij* � *γij*Λ*ji* ¼ *aij* þ *γij τ <sup>j</sup>* þ *Cj* þ *v <sup>j</sup>*

*pij* ¼ Λ*ij* þ *Ci* þ *vi:*

8 < :

*∂*Π*ij ∂pij*

tions as:

Λ*ij* ¼

**62**

¼ *pij* � *Ci* � *vi* � �

� �

$$\theta\_{ij} = \frac{\left(2\beta\_{ji}a\_{ij} + \chi\_{\vec{\eta}}a\_{\vec{\mu}i} + \beta\_{\vec{\mu}}\chi\_{\vec{\eta}}\left(\mathbf{C}\_{j} + \boldsymbol{\nu}\_{j}\right) + \left(\chi\_{\vec{\eta}}\chi\_{\vec{\mu}} - 2\beta\_{\vec{\mu}}\beta\_{\vec{\eta}}\right)\left(\mathbf{C}\_{i} + \boldsymbol{\nu}\_{i}\right)\right)}{\left(4\beta\_{ji}\beta\_{\vec{\mu}} - \chi\_{\vec{\eta}}\chi\_{\vec{\mu}}\right)},\tag{7}$$

$$\alpha\_{\vec{\eta}} = \frac{\left(\chi\_{\vec{\eta}}\chi\_{\vec{\mu}} - 2\beta\_{\vec{\mu}}\beta\_{\vec{\eta}}\right)}{\left(4\beta\_{\vec{\mu}}\beta\_{\vec{\eta}} - \chi\_{\vec{\eta}}\chi\_{\vec{\mu}}\right)} \text{ and } \chi\_{\vec{\eta}} = \frac{\beta\_{\vec{\mu}}\chi\_{\vec{\eta}}}{\left(4\beta\_{\vec{\mu}}\beta\_{\vec{\eta}} - \chi\_{\vec{\eta}}\chi\_{\vec{\mu}}\right)}\tag{8}$$

*Proof.* Obtain the results by substituting Λ<sup>∗</sup> *ij* from Proposition 1 into Eqs. (1) and (2) and simplifying.

*Remark 1.* The only outcome where neither power plant has an incentive to deviate is when *pij* ¼ *pji* ¼ *ci*, which will be the Nash or Bertrand equilibrium for the game. The intuition behind this result is that power plants will keep *undercutting* the price of its rival until price equals marginal cost. In the long run price changes with marginal cost and industry production increases with demand and falls with marginal cost. One way for a power plant to avoid the Bertrand paradox and earn economic profit in a Bertrand setting is to have a competitive cost advantage over its rival.

#### **2.1 Production decisions of power plants with homogenous hybrid/enabling technology**

Restricting ourselves to a two matched symmetric two-person bi- matrix game in random contest in a one-population evolutionary game, we define the payoff (utility) in **Table 1**.

**Note 1.** Power and the payoff are measured on a utility scale consistent with the power plant's preference ranking. Furthermore, [24–26] have applied symmetric two-person bi-matrix game in random contest to study evolutionary stable games.


**Table 1.**

*Bi matrix for two power plants by different energy sources.*

Then via equation Π<sup>∗</sup> *ij* <sup>¼</sup> *<sup>β</sup>ij* <sup>Λ</sup><sup>∗</sup> *ij* � �<sup>2</sup> � *Fi* ¼ *βij θij* þ *ωijτ<sup>i</sup>* þ *χijτ <sup>j</sup>* � �<sup>2</sup> � *Fi*, in Proposition 1, and the payoff matrix of the power plant I is given by:

$$A = \begin{bmatrix} \mathfrak{a}\_{11} & \mathfrak{a}\_{12} \\ \mathfrak{a}\_{21} & \mathfrak{a}\_{22} \end{bmatrix} = \begin{bmatrix} \Pi\_{F,F} & \Pi\_{F,R} \\ \Pi\_{R,F} & \Pi\_{R,R} \end{bmatrix} = \begin{bmatrix} \beta\_{F,F} \Lambda\_{F,F}^2 - F\_F & \beta\_{F,R} \Lambda\_{F,R}^2 - F\_F \\ \beta\_{R,F} \Lambda\_{R,F}^2 - F\_R & \beta\_{R,R} \Lambda\_{R,R}^2 - F\_R \end{bmatrix}. \tag{9}$$

*Remark 1.* Since the power plants plays a symmetric two person bimatrix game

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

*G,* having two pure strategies Π*F*,*<sup>F</sup>* 6¼ Π*R*,*F*, Π*R*,*<sup>R</sup>* 6¼ Π*FR*, imply that *G,* has an evolutionary stable strategy. Then the Nash equilibrium is an outcome in which the strategy chosen by each player is the best reply to the strategy chosen by the other. This best reply strategy yields the highest payoff to the player choosing it, given the

**2.2 Production decisions of power plants under endogenous hybrid/enabling**

Both players will undertake R&D measures on hybrid-enabling technology to ensure immediate reliability and affordability in energy production whilst reducing GHG-emissions. We assume that the strategic effects implemented by power plant I (Player I), has improved hybrid-enabling technology to generate energy and utilize energy sources in a much efficient way. This gives a superior advantage to power plant I overpower plant II (Player II) and both power plants are rational to maximize their profits. Although Power plant II has heterogeneous resources to hybridenabling technology, from a practical point of view it is logical for power plant I to share this technology with power plant II, because the price competition is typically characterized by a second-mover advantage. Many researchers have investigated the effects of these commitments in Cournot, Bertrand and Stackelberg setups. See [29–31]. Due to the government incentives, tariff-rate quota, feed-in-tariff and R&D incentive measures, the power companies will be competitive to improve their efficiency. Let *LR*ð Þ*<sup>t</sup>* denotes the *<sup>R</sup>*&*D effort level* of technological improvements on renewable sources at time *<sup>t</sup>*, and *<sup>L</sup><sup>F</sup>*ð Þ*<sup>t</sup>* denotes the *<sup>R</sup>*&*D effort level* of

ð Þ*t* denotes the

ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>F</sup>* ð Þ*t*

� �,

*K* <sup>p</sup> *dW t*ð Þ

(19)

technological improvements on fossil fuel at time *t*, of Player I. *L*~*<sup>R</sup>*

2 ~*β R* ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>R</sup>* ð Þ*t* � �<sup>2</sup>

ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>F</sup>* ð Þ*t*

Player I and Player II at time *t*, respectively. Consider

*<sup>C</sup>*<sup>I</sup> *LR*ð Þ*<sup>t</sup>* , *LF*ð Þ*<sup>t</sup>* , *<sup>t</sup>* � � <sup>¼</sup> <sup>1</sup>

ð Þ*t* , *t* � � <sup>¼</sup> <sup>1</sup>

more effective is the technological development.

*dK t*ðÞ¼ *<sup>ϑ</sup>*1ð Þ*<sup>t</sup> LR*ð Þ*<sup>t</sup>* , *LF*ð Þ*<sup>t</sup>* � � <sup>þ</sup> *<sup>ϑ</sup>*2ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>R</sup>*

*R*&*D effort level* of technological improvements on renewable sources at time *t*, and

ð Þ*t* denotes the *R*&*D effort level* of technological improvements on fossil fuel at time *t*, of Player II. For, further consideration, the sharing cost of advanced hybridenabling technology (Player I) and inferior hybrid-enabling technology (Player II) is denoted as *C*Ið Þ*t* and *C*IIð Þ*t* , which are the quadratic functions of the effect level of

<sup>2</sup> *<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup> LR*ð Þ*<sup>t</sup>* � �<sup>2</sup>

� � <sup>≤</sup>1 and lower the *<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>R</sup>*

Let *K t*ð Þ denote the evolution of the hybrid-enabling technology at time *t*, due to R&D collaborative innovation system of Player I and Player II at time *t*. The dynamics of hybrid-technology is governed by the stochastic differential equation (SDE):

� � � *<sup>ξ</sup>K t*ð Þ h i*dt* <sup>þ</sup> *<sup>φ</sup>* ffiffiffiffi

ð Þ*<sup>t</sup>* , *<sup>L</sup>*~*<sup>F</sup>* ð Þ*t*

þ ~*β F* ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>F</sup>* ð Þ*t* � �<sup>2</sup> � �, (18)

<sup>þ</sup> *<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup> <sup>L</sup><sup>F</sup>*ð Þ*<sup>t</sup>* � �<sup>2</sup> � �, (17)

strategy chosen by the co-player, [27, 28].

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

**technological advances**

*L*~*F*

and

8 < :

**65**

*<sup>C</sup>*II *<sup>L</sup>*~*<sup>R</sup>*

*K*ð Þ¼ 0 *K*<sup>0</sup> >0*:*

where 0 <sup>&</sup>lt; *<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>R</sup>*

ð Þ*<sup>t</sup>* , *<sup>L</sup>*~*<sup>F</sup>*

Obviously the bimatrix of the power plant II, is given by:

$$A = \begin{bmatrix} \mathfrak{a}\_{11} & \mathfrak{a}\_{21} \\ \mathfrak{a}\_{12} & \mathfrak{a}\_{22} \end{bmatrix} = \begin{bmatrix} \Pi\_{F,F} & \Pi\_{R,F} \\ \Pi\_{F,R} & \Pi\_{R,R} \end{bmatrix} = \begin{bmatrix} \beta\_{F,F} \Lambda\_{F,F}^2 - F\_F & \beta\_{R,F} \Lambda\_{R,F}^2 - F\_R \\ \beta\_{F,R} \Lambda\_{F,R}^2 - F\_F & \beta\_{R,R} \Lambda\_{R,R}^2 - F\_R \end{bmatrix}. \tag{10}$$

**Proposition 3.** *The Nash equilibrium for the Bi-matrix game G, is given as*

$$\left(\frac{\left(\Pi\_{\mathcal{R},\mathcal{R}}-\Pi\_{F,\mathcal{R}}\right)}{\left(\Pi\_{F,\mathcal{F}}-\Pi\_{\mathcal{R},\mathcal{F}}-\Pi\_{\mathcal{R}\mathcal{R}}+\Pi\_{\mathcal{R},\mathcal{R}}\right)},\frac{\left(\Pi\_{\mathcal{R},\mathcal{R}}-\Pi\_{F,\mathcal{R}}\right)}{\left(\Pi\_{F,\mathcal{F}}-\Pi\_{F,\mathcal{R}}-\Pi\_{\mathcal{R},\mathcal{F}}+\Pi\_{\mathcal{R},\mathcal{R}}\right)}\right).\tag{11}$$

*Proof.* Suppose players I and II use mixed strategies (*x*,1-*x*) and (*y*,1-*y*), respectively, where


Then the value of the game for Player I is

$$\upsilon\_{1}(\mathbf{x},\boldsymbol{\uprho}) = \mathbf{x}\boldsymbol{\uprho}(\boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{F}}) + \boldsymbol{\uprho}(\mathbf{1}-\boldsymbol{\uprho})(\boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{R}}) + (\mathbf{1}-\boldsymbol{\uprho})\boldsymbol{\uprho}(\boldsymbol{\Pi}\_{\boldsymbol{R},\boldsymbol{F}}) + (\mathbf{1}-\boldsymbol{\uprho})(\boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{R}})$$

$$= ( (\boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{F}} - \boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{R}} - \boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{F}} + \boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{R}})\boldsymbol{\uprho} + (\boldsymbol{\varPi}\_{\boldsymbol{F},\boldsymbol{R}} - \boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{R}})\boldsymbol{\uprho} + ((\boldsymbol{\varPi}\_{\boldsymbol{R},\boldsymbol{F}} - \boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{R}})\boldsymbol{\uprho} + \boldsymbol{\upPi}\_{\boldsymbol{R},\boldsymbol{R}}), \quad \text{(12)}$$

and the value of the game for Player II is

$$v\_2(\mathbf{x}, \boldsymbol{y}) = \mathbf{x}\boldsymbol{y}(\boldsymbol{\Pi}\_{F, \mathcal{F}}) + \boldsymbol{\varkappa}(\mathbf{1} - \boldsymbol{\jmath})(\boldsymbol{\Pi}\_{\mathcal{R}, F}) + (\mathbf{1} - \boldsymbol{\varkappa})\boldsymbol{\jmath}(\boldsymbol{\Pi}\_{F, \mathcal{R}}) + (\mathbf{1} - \boldsymbol{\varkappa})(\mathbf{1} - \boldsymbol{\jmath})(\boldsymbol{\Pi}\_{\mathcal{R}, \mathcal{R}})$$

$$= \left( (\boldsymbol{\Pi}\_{F, F} - \boldsymbol{\Pi}\_{\mathcal{R}, F} - \boldsymbol{\Pi}\_{F, \mathcal{R}} + \boldsymbol{\Pi}\_{\mathcal{R}, \mathcal{R}})\boldsymbol{\varkappa} + (\boldsymbol{\Pi}\_{F, \mathcal{R}} - \boldsymbol{\Pi}\_{\mathcal{R}, \mathcal{R}})\right)\boldsymbol{\jmath} + \left( (\boldsymbol{\Pi}\_{\mathcal{R}, F\mathcal{R}} - \boldsymbol{\Pi}\_{\mathcal{R}, \mathcal{R}})\boldsymbol{\varkappa} + \boldsymbol{\Pi}\_{\mathcal{R}, \mathcal{R}} \right). \tag{13}$$

Suppose (*X, Y*) yields a Nash equilibrium. Then for the given payoffs having 0<*x*<1 implies that

$$\boldsymbol{\nu}\_{1} = (\boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{F}} - \boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{R}} - \boldsymbol{\Pi}\_{\boldsymbol{R},\boldsymbol{F}} + \boldsymbol{\Pi}\_{\boldsymbol{R},\boldsymbol{R}}) \boldsymbol{y} + (\boldsymbol{\Pi}\_{\boldsymbol{F},\boldsymbol{R}} - \boldsymbol{\Pi}\_{\boldsymbol{R},\boldsymbol{R}}) = \mathbf{0}.\tag{14}$$

Otherwise Player I can change *x* slightly and do better. Similarly, for 0< *y*<1,

$$\boldsymbol{\nu}\_{2} = (\boldsymbol{\Pi}\_{F,F} - \boldsymbol{\Pi}\_{R,F} - \boldsymbol{\Pi}\_{FR} + \boldsymbol{\Pi}\_{R,R})\boldsymbol{\chi} + (\boldsymbol{\Pi}\_{F,R} - \boldsymbol{\Pi}\_{R,R}) = \mathbf{0}.\tag{15}$$

Otherwise Player II can change y slightly and do better. It follows that the unique Nash equilibrium (*x,y*), has

$$\left(\frac{\left(\Pi\_{\mathcal{R},\mathcal{R}}-\Pi\_{\mathcal{F},\mathcal{R}}\right)}{\left(\Pi\_{\mathcal{F},\mathcal{F}}-\Pi\_{\mathcal{R},\mathcal{F}}-\Pi\_{\mathcal{F}\mathcal{R}}+\Pi\_{\mathcal{R},\mathcal{R}}\right)},\frac{\left(\Pi\_{\mathcal{R},\mathcal{R}}-\Pi\_{\mathcal{R},\mathcal{F}}\right)}{\left(\Pi\_{\mathcal{F},\mathcal{F}}-\Pi\_{\mathcal{F},\mathcal{R}}-\Pi\_{\mathcal{R},\mathcal{F}}+\Pi\_{\mathcal{R},\mathcal{R}}\right)}\right).\tag{16}$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

*Remark 1.* Since the power plants plays a symmetric two person bimatrix game *G,* having two pure strategies Π*F*,*<sup>F</sup>* 6¼ Π*R*,*F*, Π*R*,*<sup>R</sup>* 6¼ Π*FR*, imply that *G,* has an evolutionary stable strategy. Then the Nash equilibrium is an outcome in which the strategy chosen by each player is the best reply to the strategy chosen by the other. This best reply strategy yields the highest payoff to the player choosing it, given the strategy chosen by the co-player, [27, 28].

### **2.2 Production decisions of power plants under endogenous hybrid/enabling technological advances**

Both players will undertake R&D measures on hybrid-enabling technology to ensure immediate reliability and affordability in energy production whilst reducing GHG-emissions. We assume that the strategic effects implemented by power plant I (Player I), has improved hybrid-enabling technology to generate energy and utilize energy sources in a much efficient way. This gives a superior advantage to power plant I overpower plant II (Player II) and both power plants are rational to maximize their profits. Although Power plant II has heterogeneous resources to hybridenabling technology, from a practical point of view it is logical for power plant I to share this technology with power plant II, because the price competition is typically characterized by a second-mover advantage. Many researchers have investigated the effects of these commitments in Cournot, Bertrand and Stackelberg setups. See [29–31]. Due to the government incentives, tariff-rate quota, feed-in-tariff and R&D incentive measures, the power companies will be competitive to improve their efficiency. Let *LR*ð Þ*<sup>t</sup>* denotes the *<sup>R</sup>*&*D effort level* of technological improvements on renewable sources at time *<sup>t</sup>*, and *<sup>L</sup><sup>F</sup>*ð Þ*<sup>t</sup>* denotes the *<sup>R</sup>*&*D effort level* of technological improvements on fossil fuel at time *t*, of Player I. *L*~*<sup>R</sup>* ð Þ*t* denotes the *R*&*D effort level* of technological improvements on renewable sources at time *t*, and

*L*~*F* ð Þ*t* denotes the *R*&*D effort level* of technological improvements on fossil fuel at time *t*, of Player II. For, further consideration, the sharing cost of advanced hybridenabling technology (Player I) and inferior hybrid-enabling technology (Player II) is denoted as *C*Ið Þ*t* and *C*IIð Þ*t* , which are the quadratic functions of the effect level of Player I and Player II at time *t*, respectively. Consider

$$\mathbf{C}\_{\rm I}(\boldsymbol{L}^{R}(\boldsymbol{t}), \boldsymbol{L}^{F}(\boldsymbol{t}), \boldsymbol{t}) = \frac{1}{2} \left( \boldsymbol{\rho}^{R}(\boldsymbol{t}) \left( \boldsymbol{L}^{R}(\boldsymbol{t}) \right)^{2} + \boldsymbol{\rho}^{F}(\boldsymbol{t}) \left( \boldsymbol{L}^{F}(\boldsymbol{t}) \right)^{2} \right), \tag{17}$$

and

Then via equation Π<sup>∗</sup>

*<sup>A</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>11</sup> *<sup>a</sup>*<sup>12</sup> *a*<sup>21</sup> *a*<sup>22</sup> � �

*Carbon Capture*

*<sup>A</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>11</sup> *<sup>a</sup>*<sup>21</sup> *a*<sup>12</sup> *a*<sup>22</sup> � �

respectively, where

0<*x*<1 implies that

**64**

Similarly, for 0< *y*<1,

unique Nash equilibrium (*x,y*), has

ð Þ Π*<sup>R</sup>*,*<sup>R</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* � Π*FR* þ Π*<sup>R</sup>*,*<sup>R</sup>*

*ij* <sup>¼</sup> *<sup>β</sup>ij* <sup>Λ</sup><sup>∗</sup>

<sup>¼</sup> <sup>Π</sup>*F*,*<sup>F</sup>* <sup>Π</sup>*F*,*<sup>R</sup>* Π*R*,*<sup>F</sup>* Π*R*,*<sup>R</sup>* � �

<sup>¼</sup> <sup>Π</sup>*F*,*<sup>F</sup>* <sup>Π</sup>*R*,*<sup>F</sup>* Π*F*,*<sup>R</sup>* Π*R*,*<sup>R</sup>* � �

ð Þ Π*<sup>R</sup>*,*<sup>R</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* � Π*FR* þ Π*<sup>R</sup>*,*<sup>R</sup>*

player I choosing row 2 is 1-*x*.

player II choosing row 2 is 1-*y*.

Then the value of the game for Player I is

and the value of the game for Player II is

Otherwise Player I can change *x* slightly and do better.

Obviously the bimatrix of the power plant II, is given by:

*ij* � �<sup>2</sup>

Proposition 1, and the payoff matrix of the power plant I is given by:

� *Fi* ¼ *βij θij* þ *ωijτ<sup>i</sup>* þ *χijτ <sup>j</sup>*

<sup>¼</sup> *<sup>β</sup>F*,*F*Λ<sup>2</sup>

<sup>¼</sup> *<sup>β</sup>F*,*F*Λ<sup>2</sup>

**Proposition 3.** *The Nash equilibrium for the Bi-matrix game G, is given as*

� �

*Proof.* Suppose players I and II use mixed strategies (*x*,1-*x*) and (*y*,1-*y*),

i. The probability that player I choosing row 1 is *x* and the probability that

ii. The probability that player II choosing row 1 is *y* and the probability that

*v*1ð Þ¼ *x*, *y xy*ð Þþ Π*<sup>F</sup>*,*<sup>F</sup> x*ð Þ 1 � *y* ð Þþ Π*<sup>F</sup>*,*<sup>R</sup>* ð Þ 1 � *x y*ð Þþ Π*<sup>R</sup>*,*<sup>F</sup>* ð Þ 1 � *x* ð Þ 1 � *y* ð Þ Π*<sup>R</sup>*,*<sup>R</sup>*

¼ ð Þ ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* þ Π*<sup>R</sup>*,*<sup>R</sup> y* þ ð Þ Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>R</sup> x* þ ð Þ ð Þ Π*<sup>R</sup>*,*<sup>F</sup>* � Π*<sup>R</sup>*,*<sup>R</sup> y* þ Π*<sup>R</sup>*,*<sup>R</sup>* , (12)

*v*2ð Þ¼ *x*, *y xy*ð Þþ Π*<sup>F</sup>*,*<sup>F</sup> x*ð Þ 1 � *y* ð Þþ Π*<sup>R</sup>*,*<sup>F</sup>* ð Þ 1 � *x y*ð Þþ Π*<sup>F</sup>*,*<sup>R</sup>* ð Þ 1 � *x* ð Þ 1 � *y* ð Þ Π*<sup>R</sup>*,*<sup>R</sup>*

¼ ð Þ ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* þ Π*<sup>R</sup>*,*<sup>R</sup> x* þ ð Þ Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>R</sup> y* þ ð Þ ð Þ Π*<sup>R</sup>*,*FR* � Π*<sup>R</sup>*,*<sup>R</sup> x* þ Π*<sup>R</sup>*,*<sup>R</sup> :* (13)

Suppose (*X, Y*) yields a Nash equilibrium. Then for the given payoffs having

Otherwise Player II can change y slightly and do better. It follows that the

� �

*v*<sup>1</sup> ¼ ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* þ Π*<sup>R</sup>*,*<sup>R</sup> y* þ ð Þ¼ Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>R</sup>* 0*:* (14)

*v*<sup>2</sup> ¼ ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* � Π*FR* þ Π*<sup>R</sup>*,*<sup>R</sup> x* þ ð Þ¼ Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>R</sup>* 0*:* (15)

, ð Þ <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>* � <sup>Π</sup>*<sup>R</sup>*,*<sup>F</sup>*

ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* þ Π*<sup>R</sup>*,*<sup>R</sup>*

*βF*,*R*Λ<sup>2</sup>

, ð Þ <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>* � <sup>Π</sup>*<sup>F</sup>*,*<sup>R</sup>*

ð Þ Π*<sup>F</sup>*,*<sup>F</sup>* � Π*<sup>F</sup>*,*<sup>R</sup>* � Π*<sup>R</sup>*,*<sup>F</sup>* þ Π*<sup>R</sup>*,*<sup>R</sup>*

*βR*,*F*Λ<sup>2</sup>

� �<sup>2</sup>

*<sup>F</sup>*,*<sup>F</sup>* � *FF <sup>β</sup>F*,*R*Λ<sup>2</sup>

" #

*<sup>R</sup>*,*<sup>F</sup>* � *FR <sup>β</sup>R*,*R*Λ<sup>2</sup>

*<sup>F</sup>*,*<sup>F</sup>* � *FF <sup>β</sup>R*,*F*Λ<sup>2</sup>

" #

*<sup>F</sup>*,*<sup>R</sup>* � *FF <sup>β</sup>R*,*R*Λ<sup>2</sup>

� *Fi*, in

*:* (9)

*:* (10)

*:* (11)

*:* (16)

*<sup>F</sup>*,*<sup>R</sup>* � *FF*

*<sup>R</sup>*,*<sup>R</sup>* � *FR*

*<sup>R</sup>*,*<sup>F</sup>* � *FR*

*<sup>R</sup>*,*<sup>R</sup>* � *FR*

$$\mathbf{C}\_{\rm II} \left( \bar{\boldsymbol{L}}^{R}(t), \bar{\boldsymbol{L}}^{F}(t), \boldsymbol{t} \right) = \frac{\mathbf{1}}{2} \left( \bar{\boldsymbol{\rho}}^{R}(t) \left( \bar{\boldsymbol{L}}^{R}(t) \right)^{2} + \bar{\boldsymbol{\rho}}^{F}(t) \left( \bar{\boldsymbol{L}}^{F}(t) \right)^{2} \right), \tag{18}$$

where 0 <sup>&</sup>lt; *<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>R</sup>* ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>F</sup>* ð Þ*t* � � <sup>≤</sup>1 and lower the *<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>R</sup>* ð Þ*<sup>t</sup>* , *<sup>β</sup>*~*<sup>F</sup>* ð Þ*t* � �, more effective is the technological development.

Let *K t*ð Þ denote the evolution of the hybrid-enabling technology at time *t*, due to R&D collaborative innovation system of Player I and Player II at time *t*. The dynamics of hybrid-technology is governed by the stochastic differential equation (SDE):

$$\begin{cases} dK(t) = \left[\theta\_1(t)\left(L^R(t), L^F(t)\right) + \theta\_2(t)\left(\tilde{L}^R(t), \tilde{L}^F(t)\right) - \xi K(t)\right]dt + \eta\sqrt{K}dW(t) \\\\ K(0) = K\_0 > 0. \end{cases} \tag{19}$$

*ξ*∈ð � 0, 1 , is the attenuation coefficient of hybrid-enabling technology. Let *<sup>ϑ</sup>*1ðÞ¼ *<sup>t</sup> <sup>ϑ</sup><sup>R</sup>* <sup>1</sup> ð Þþ*<sup>t</sup> <sup>ϑ</sup><sup>F</sup>* <sup>1</sup> ð Þ*<sup>t</sup>* � � and *<sup>ϑ</sup>*2ðÞ¼ *<sup>t</sup> <sup>ϑ</sup><sup>R</sup>* <sup>2</sup> ðÞþ*<sup>t</sup> <sup>ϑ</sup><sup>F</sup>* <sup>2</sup> ð Þ*<sup>t</sup>* � � denote the influence of the effort level of hybrid-enabling technology sharing on collaboration innovation between Player I and Player II, at time *<sup>t</sup>*. *W t*ð Þ is a standard Brownian motion and *<sup>φ</sup>* ffiffiffiffi *K* <sup>p</sup> ð Þ*<sup>t</sup>* � � random interference factor on hybrid-enabling technology.

where *E* is the expectations. The main implication of this is that it leads to reaction correspondences that are non-decreasing (in the sense that each selection is non-decreasing) but need not be single-valued or continuous. This has a very appealing and precise interpretation: The price elasticity of Power Plant *i*'s demand increases in the rival's price, [33]. This is a very intuitive and general condition,

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

higher price by a Power Plant's rival does not lower the responsiveness of the Power

We further assume that the total revenue is allocated between two players and

�*ρ*1*<sup>t</sup> <sup>θ</sup>*ð Þ*<sup>t</sup> <sup>α</sup>*1ð Þ*<sup>t</sup> LR*ðÞþ*<sup>t</sup> <sup>α</sup>*2ð Þ*<sup>t</sup> <sup>L</sup><sup>F</sup>*ðÞþ*<sup>t</sup> <sup>β</sup>*1ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>R</sup>*

� 1 2

> � 1 2 *β*~*F* ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>F</sup>* ð Þ*t* � �<sup>2</sup>

*<sup>β</sup><sup>F</sup>*ð Þ*<sup>t</sup>* ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> *LF*ð Þ*<sup>t</sup>* � �<sup>2</sup>

�*ρ*2*<sup>t</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup>*ð Þ*<sup>t</sup> <sup>α</sup>*1ð Þ*<sup>t</sup> LR*ðÞþ*<sup>t</sup> <sup>α</sup>*2ð Þ*<sup>t</sup> <sup>L</sup><sup>F</sup>*ðÞþ*<sup>t</sup> <sup>β</sup>*1ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>R</sup>*

*<sup>S</sup>* ð Þ*<sup>t</sup>* <sup>≥</sup> 0, L S R (t) <sup>≥</sup> 0, *<sup>L</sup>*~*<sup>R</sup>*

*<sup>∂</sup>pjipij*

ðÞþ*<sup>t</sup> <sup>β</sup>*2ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>F</sup>*

� *dt*� , ð Þ*t*

(27)

ð Þ*t*

(28)

*<sup>S</sup>* ð Þ*t* ≥ 0 and

> 0, if a

though clearly not a universal one. It is satisfied in particular if *<sup>∂</sup>*<sup>2</sup>*Dij pij*, *<sup>p</sup>* ð Þ*ji*

*θ*ð Þ*t* is the payoff distribution coefficient of player I at time *t* and *θ*ð Þ*t* ∈½ � 0, 1 . Although Player II has heterogeneous resources of hybrid-enabling technology, Player I can produce electricity more efficiently with lower GHG-emission, ensure immediate reliability and affordability in energy production. Then Player II, can acquire practical outcomes of this hybrid-enabling technological advances. To promote the hybrid-enabling technology, Player II (leader) determine an optimal sharing effort level and an optimal subsidy. Then Player I (follower) choose their optimal sharing effort level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium. Let *ω*ðÞ¼ *t* ð Þ *ω*1ð Þ*t* , *ω*2ð Þ*t* , denote the subsidy for hybrid-enabling technology, with Player II willing to pay to Payer I under collaboration. The objective functions of power plant I and power plant II

plant's demand to a change in own price.

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

satisfy the following partial differential equations

� h �

*E* ð<sup>∞</sup> 0 *e*

> � 1 2 *β*~*R*

*<sup>ω</sup>*2*β<sup>F</sup>*ð Þ*<sup>t</sup> LF*ð Þ*<sup>t</sup>* � �<sup>2</sup>

*<sup>β</sup><sup>R</sup>*ð Þ*<sup>t</sup>* ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> *<sup>L</sup><sup>R</sup>*ð Þ*<sup>t</sup>* � �<sup>2</sup>

� h �

ð Þ*<sup>t</sup> LR*ð Þ*<sup>t</sup>* � �<sup>2</sup>

� *dt*� ,

*<sup>S</sup>* ð Þ*<sup>t</sup>* <sup>≥</sup>0, *LF*

where *ρ*<sup>1</sup> and *ρ*<sup>2</sup> are the discount rates of Player I and Player II, respectively. In

*<sup>S</sup>* ð Þ*t* ≥0, are the control variables and *ω*ðÞ¼ *t* ð Þ *ω*1ð Þ*t* ,*ω*2ð Þ*t* ∈ð Þ 0, 1 . *K t*ð Þ>0 is the state variable. In feedback control process, it is assumed that players at every point in time have access to the current system and can make decisions accordingly to that state. Consequently, the players can respond to any disturbance in an optimal way. Hence, feedback strategies are robust for deviations and players can react to disturbances during the evolution of the game and adapt their actions

2

ðÞþ*<sup>t</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> K t*ð Þ�

� 1 2

*J*ð Þ<sup>I</sup> ð Þ *K*<sup>0</sup>

¼ max *LR <sup>S</sup>* , *<sup>L</sup><sup>F</sup>* f g*<sup>S</sup>* <sup>≥</sup><sup>0</sup>

and

<sup>þ</sup>*β*2ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>F</sup>*

�1 2

*L*~*F*

**67**

*E* ð<sup>∞</sup> 0 *e*

þð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> K t*ð ÞÞÞ � <sup>1</sup>

*J*ð Þ II ð Þ¼ *K*<sup>0</sup> max *L*~*R <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> <sup>S</sup>* ,*ω*ð Þ*t* � �≥<sup>0</sup>

*<sup>ω</sup>*1*β<sup>R</sup>*ð Þ*<sup>t</sup> LR*ð Þ*<sup>t</sup>* � �<sup>2</sup>

accordingly, [34].

this feedback control strategy *L<sup>R</sup>*

Let Πð Þ*t* denotes the total profit under the hybrid-enabling technology system at time *t*. Let ð Þ *α*1ð Þ*t* , *α*2ð Þ*t* and *β*1ð Þ*t* , *β*<sup>2</sup> ð Þ ð Þ*t* denote the influence of the effort level hybrid-enabling technology on the total profit of Player I and player II, respectively, at time *t*, namely, the marginal return coefficient of hybrid-enabling technology. Total profit function can be expressed as:

$$\Pi(t) = \left(a\_1(t)L^R(t) + a\_2(t)L^F(t)\right) + \left(\beta\_1(t)\tilde{L}^R(t) + \beta\_2(t)\tilde{L}^F(t)\right) + (\Gamma + \delta)K(t), \tag{20}$$

where

$$a\_1(t) \quad = \frac{\Pi\_{R,R}(t)}{\Pi\_{F,F}(t) - \Pi\_{R,F}(t) - \Pi\_{F,R}(t) + \Pi\_{R,R}(t)},\tag{21}$$

$$\begin{aligned} a\_2(t) &= \frac{-\Pi\_{F,\mathcal{R}}(t)}{\Pi\_{F,\mathcal{F}}(t) - \Pi\_{\mathcal{R},\mathcal{F}}(t) - \Pi\_{F,\mathcal{R}}(t) + \Pi\_{\mathcal{R},\mathcal{R}}(t)}, \\ \Gamma &= \Gamma\_{\mathcal{(I)}} + \Gamma\_{\langle\mathcal{II}\rangle}, \delta = \delta\_{\mathcal{(I)}} + \delta\_{\langle\mathcal{II}\rangle}, \text{and} \end{aligned} \tag{22}$$

$$\beta\_1(t) \quad = \frac{\Pi\_{R,R}(t)}{\Pi\_{F,F}(t) - \Pi\_{R,F}(t) - \Pi\_{F,R}(t) + \Pi\_{R,R}(t)}, \tag{23}$$

$$\beta\_2(t) \quad = \frac{-\Pi\_{R,F}(t)}{\Pi\_{F,F}(t) - \Pi\_{R,F}(t) - \Pi\_{F,R}(t) + \Pi\_{R,R}(t)}. \tag{24}$$

Γ is the influence of the hybrid-enabling technology innovation on total revenue *δ*∈ð � 0, 1 ; *δ* is the total government subsidy coefficient of hybrid-enabling technology based on increments of advances in hybrid-enabling technology.

**Proposition 4.** *At least one of the Power Plants has a second mover advantage.*

*Proof.* Demand function *Dij pij*, *pji* � �>0, given by Eq. (4), is twice continuously differentiable and

$$\frac{\partial D\_{\vec{\eta}}\left(p\_{\vec{\eta}}, p\_{\vec{\mu}}\right)}{\partial p\_{\vec{\eta}}} = -\beta\_{\vec{\eta}} < 0,\\
and\\
\frac{\partial D\_{\vec{\eta}}\left(p\_{\vec{\eta}}, p\_{\vec{\mu}}\right)}{\partial p\_{\vec{\mu}}} = \gamma\_{\vec{\eta}} > 0\\\forall \left(p\_{\vec{\eta}}, p\_{\vec{\mu}}\right) \in P\_{\mathcal{I}} \times P\_{\mathcal{II}}.\tag{25}$$

The first inequality says that each demand is downward sloping in own price, and the second that goods are substitutes (each demand increases with the price of the other good). [32] shows that in case of symmetric firms, there is a second-mover (first-mover) advantage for both players when each profit function is strictly concave in own action and strictly increasing (decreasing) in rival's action, and reaction curves are upward (downward) sloping.

Then a sufficient condition on the super-modularity of the profit function is obtained via the profit function Π*ij*, given by Eq. (4):

$$
\left[\frac{\partial D\_{\vec{\eta}}\left(p\_{\vec{\eta}}, p\_{\vec{\mu}}\right)}{\partial p\_{ji}} + \left(p\_{\vec{\eta}} - \mathbf{C}\_{i} - \nu\_{i}\right) \frac{\partial^{2} D\_{\vec{\eta}}\left(p\_{\vec{\eta}}, p\_{\vec{\mu}}\right)}{\partial p\_{ji} p\_{ij}}\right] \mathbf{E}(K(t)) > \mathbf{0},\tag{26}
$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

where *E* is the expectations. The main implication of this is that it leads to reaction correspondences that are non-decreasing (in the sense that each selection is non-decreasing) but need not be single-valued or continuous. This has a very appealing and precise interpretation: The price elasticity of Power Plant *i*'s demand increases in the rival's price, [33]. This is a very intuitive and general condition, though clearly not a universal one. It is satisfied in particular if *<sup>∂</sup>*<sup>2</sup>*Dij pij*, *<sup>p</sup>* ð Þ*ji <sup>∂</sup>pjipij* > 0, if a higher price by a Power Plant's rival does not lower the responsiveness of the Power plant's demand to a change in own price.

We further assume that the total revenue is allocated between two players and *θ*ð Þ*t* is the payoff distribution coefficient of player I at time *t* and *θ*ð Þ*t* ∈½ � 0, 1 . Although Player II has heterogeneous resources of hybrid-enabling technology, Player I can produce electricity more efficiently with lower GHG-emission, ensure immediate reliability and affordability in energy production. Then Player II, can acquire practical outcomes of this hybrid-enabling technological advances. To promote the hybrid-enabling technology, Player II (leader) determine an optimal sharing effort level and an optimal subsidy. Then Player I (follower) choose their optimal sharing effort level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium. Let *ω*ðÞ¼ *t* ð Þ *ω*1ð Þ*t* , *ω*2ð Þ*t* , denote the subsidy for hybrid-enabling technology, with Player II willing to pay to Payer I under collaboration. The objective functions of power plant I and power plant II satisfy the following partial differential equations

$$\begin{split} \mathcal{I}\_{(1)}(\mathbf{K}\_{0}) \\ = \max\_{\left\{L\_{S}^{R}, \boldsymbol{L}\_{S}^{F}\right\} \ge 0} & E\left\{ \int\_{0}^{\infty} e^{-\rho\_{1}t} \left[ \theta(t) \left( a\_{1}(t) \boldsymbol{L}^{R}(t) + a\_{2}(t) \boldsymbol{L}^{F}(t) + \boldsymbol{\beta}\_{1}(t) \boldsymbol{\bar{L}}^{R}(t) + \boldsymbol{\beta}\_{2}(t) \boldsymbol{\bar{L}}^{F}(t) \right) \right. \\ & \left. + (\boldsymbol{\Gamma} + \delta) \boldsymbol{K}(t) \right) \right) - \frac{1}{2} \boldsymbol{\beta}^{R}(t) \left( \boldsymbol{1} - \boldsymbol{o}\_{1} \right) \left( \boldsymbol{L}^{R}(t) \right)^{2} - \frac{1}{2} \boldsymbol{\beta}^{F}(t) \left( \boldsymbol{1} - \boldsymbol{o}\_{2} \right) \left( \boldsymbol{L}^{F}(t) \right)^{2} \Big] dt \Bigg{,} \end{split} \tag{27}$$

and

*ξ*∈ð � 0, 1 , is the attenuation coefficient of hybrid-enabling technology. Let

level of hybrid-enabling technology sharing on collaboration innovation between Player I and Player II, at time *<sup>t</sup>*. *W t*ð Þ is a standard Brownian motion and *<sup>φ</sup>* ffiffiffiffi

time *t*. Let ð Þ *α*1ð Þ*t* , *α*2ð Þ*t* and *β*1ð Þ*t* , *β*<sup>2</sup> ð Þ ð Þ*t* denote the influence of the effort level hybrid-enabling technology on the total profit of Player I and player II, respectively, at time *t*, namely, the marginal return coefficient of hybrid-enabling technology.

*<sup>α</sup>*2ðÞ ¼ *<sup>t</sup>* �Π*<sup>F</sup>*,*<sup>R</sup>*ð Þ*<sup>t</sup>*

*<sup>β</sup>*2ðÞ ¼ *<sup>t</sup>* �Π*<sup>R</sup>*,*<sup>F</sup>*ð Þ*<sup>t</sup>*

ogy based on increments of advances in hybrid-enabling technology.

� �

*<sup>∂</sup>Dij pij*, *pji* � �

*∂pji*

The first inequality says that each demand is downward sloping in own price, and the second that goods are substitutes (each demand increases with the price of the other good). [32] shows that in case of symmetric firms, there is a second-mover (first-mover) advantage for both players when each profit function is strictly concave in own action and strictly increasing (decreasing) in rival's action, and reaction

Then a sufficient condition on the super-modularity of the profit function is

*Dij pij*, *pji* � �

3

*<sup>∂</sup>pjipij*

<sup>2</sup> ðÞþ*<sup>t</sup> <sup>ϑ</sup><sup>F</sup>*

Let Πð Þ*t* denotes the total profit under the hybrid-enabling technology system at

ð Þþ*<sup>t</sup> <sup>β</sup>*2ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>F</sup>*

� �

Π*<sup>R</sup>*,*<sup>R</sup>*ð Þ*t*

<sup>Π</sup>*<sup>F</sup>*,*<sup>F</sup>*ð Þ�*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>F</sup>*,*<sup>R</sup>*ðÞþ*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>*ð Þ*<sup>t</sup>* ,

Π*<sup>R</sup>*,*<sup>R</sup>*ð Þ*t*

Γ is the influence of the hybrid-enabling technology innovation on total revenue *δ*∈ð � 0, 1 ; *δ* is the total government subsidy coefficient of hybrid-enabling technol-

**Proposition 4.** *At least one of the Power Plants has a second mover advantage.*

Γ ¼ Γð Þ<sup>I</sup> þ Γð Þ II , *δ* ¼ *δ*ð Þ<sup>I</sup> þ *δ*ð Þ II , and

<sup>2</sup> ð Þ*<sup>t</sup>* � � denote the influence of the effort

ð Þ*t*

<sup>Π</sup>*<sup>F</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>F</sup>*,*<sup>R</sup>*ðÞþ*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>*ð Þ*<sup>t</sup>* , (21)

<sup>Π</sup>*<sup>F</sup>*,*<sup>F</sup>*ð Þ�*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>F</sup>*,*<sup>R</sup>*ðÞþ*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>*ð Þ*<sup>t</sup>* , (23)

<sup>Π</sup>*<sup>F</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>F</sup>*ðÞ�*<sup>t</sup>* <sup>Π</sup>*<sup>F</sup>*,*<sup>R</sup>*ðÞþ*<sup>t</sup>* <sup>Π</sup>*<sup>R</sup>*,*<sup>R</sup>*ð Þ*<sup>t</sup> :* (24)

¼ *γij* >0∀ *pij*, *pji*

>0, given by Eq. (4), is twice continuously

∈*P*<sup>I</sup> � *P*II*:* (25)

5*EKt* ð Þ ð Þ >0, (26)

� �

*K* <sup>p</sup> ð Þ*<sup>t</sup>* � �

(22)

þ ð Þ Γ þ *δ K t*ð Þ, (20)

*<sup>ϑ</sup>*1ðÞ¼ *<sup>t</sup> <sup>ϑ</sup><sup>R</sup>*

*Carbon Capture*

where

<sup>1</sup> ð Þþ*<sup>t</sup> <sup>ϑ</sup><sup>F</sup>*

<sup>1</sup> ð Þ*<sup>t</sup>* � � and *<sup>ϑ</sup>*2ðÞ¼ *<sup>t</sup> <sup>ϑ</sup><sup>R</sup>*

Total profit function can be expressed as:

<sup>Π</sup>ðÞ¼ *<sup>t</sup> <sup>α</sup>*1ð Þ*<sup>t</sup> LR*ðÞþ*<sup>t</sup> <sup>α</sup>*2ð Þ*<sup>t</sup> LF*ð Þ*<sup>t</sup>* � � <sup>þ</sup> *<sup>β</sup>*1ð Þ*<sup>t</sup> <sup>L</sup>*~*<sup>R</sup>*

*α*1ðÞ ¼ *t*

*β*1ðÞ ¼ *t*

*Proof.* Demand function *Dij pij*, *pji*

curves are upward (downward) sloping.

*<sup>∂</sup>Dij pij*, *pji* � �

2 4

**66**

*∂pji*

obtained via the profit function Π*ij*, given by Eq. (4):

þ *pij* � *Ci* � *vi* � � *<sup>∂</sup>*<sup>2</sup>

¼ �*βij* <0, *and*

differentiable and

*<sup>∂</sup>Dij pij*, *pji* � �

*∂pij*

random interference factor on hybrid-enabling technology.

$$\begin{split} J\_{(\Pi)}(\mathcal{K}\_{0}) &= \max\_{\left\{ \begin{subarray}{c} \boldsymbol{L}\_{s}^{R}, \boldsymbol{F} \\ \left\{ \boldsymbol{L}\_{s}^{R}, \boldsymbol{L}\_{s}^{F}, \boldsymbol{\alpha}(t) \end{subarray} \right\} \geq 0 \end{split} \left\{ \begin{subarray}{c} \boldsymbol{\alpha} - \boldsymbol{\rho}\_{1} \boldsymbol{t} \\ \left\{ \boldsymbol{\alpha} - \boldsymbol{\theta}(t) \right\} \left( \boldsymbol{\alpha}\_{1}(t) \boldsymbol{L}^{R}(t) + \boldsymbol{\alpha}\_{2}(t) \boldsymbol{L}^{F}(t) + \boldsymbol{\beta}\_{1}(t) \boldsymbol{\tilde{L}}^{R}(t) \right) \right\} \\ + \boldsymbol{\rho}\_{2}(t) & \boldsymbol{\tilde{L}}^{F}(t) + (\boldsymbol{\Gamma} + \boldsymbol{\delta}) \boldsymbol{K}(t) \right\} - \frac{1}{2} \boldsymbol{\tilde{\rho}}^{R}(t) \left( \boldsymbol{L}^{R}(t) \right)^{2} - \frac{1}{2} \boldsymbol{\tilde{\rho}}^{F}(t) \left( \boldsymbol{\tilde{\boldsymbol{L}}}^{F}(t) \right)^{2} \\ - \frac{1}{2} \boldsymbol{\alpha}\_{1} \boldsymbol{\rho}^{R}(t) \left( \boldsymbol{L}^{R}(t) \right)^{2} - \frac{1}{2} \boldsymbol{\alpha}\_{2} \boldsymbol{\rho}^{F}(t) \left( \boldsymbol{L}^{F}(t) \right)^{2} \Big] \boldsymbol{d}t \right\}, \end{split} \tag{28}$$

where *ρ*<sup>1</sup> and *ρ*<sup>2</sup> are the discount rates of Player I and Player II, respectively. In this feedback control strategy *L<sup>R</sup> <sup>S</sup>* ð Þ*<sup>t</sup>* <sup>≥</sup>0, *LF <sup>S</sup>* ð Þ*<sup>t</sup>* <sup>≥</sup> 0, L S R (t) <sup>≥</sup> 0, *<sup>L</sup>*~*<sup>R</sup> <sup>S</sup>* ð Þ*t* ≥ 0 and *L*~*F <sup>S</sup>* ð Þ*t* ≥0, are the control variables and *ω*ðÞ¼ *t* ð Þ *ω*1ð Þ*t* ,*ω*2ð Þ*t* ∈ð Þ 0, 1 . *K t*ð Þ>0 is the state variable. In feedback control process, it is assumed that players at every point in time have access to the current system and can make decisions accordingly to that state. Consequently, the players can respond to any disturbance in an optimal way. Hence, feedback strategies are robust for deviations and players can react to disturbances during the evolution of the game and adapt their actions accordingly, [34].

## **3. A Stackelberg game under heterogeneous technology**

Theory of strong Stackelberg reasoning is an improved version of an earlier theory [35], which provides an explanation of coordination in all dyadic (twoplayer) common interest games. It provides an explanation of why players tend to choose strategies associated with a payoff-dominant Nash equilibrium. Its distinctive assumption is that players behave as though their co-players will anticipate any strategy choice and invariably choose a best reply to it. Stackelberg strategies resulting from this form of reasoning do not form Nash equilibria. The theory makes no predictions, because a non-equilibrium outcome is inherently unstable, leaving at least one player with a reason to choose differently and thereby achieve a better payoff. Strong Stackelberg reasoning is a simple theory, according to which players in dyadic games choose strategies that would maximize their own payoffs if their co-players could invariably anticipate their strategy choices and play counterstrategies that yield the maximum payoffs for themselves. The key assumption is relatively innocuous, first because game theory imposes no constraints on players' beliefs, apart from consistency requirements, and second because the theory does not assume that players necessarily believe that their strategies will be anticipated, merely that they behave as though that is the case, as a heuristic aid to choosing the best strategy. Strong Stackelberg reasoning is, in fact, merely a generalization of the minorant and majorant models introduced by [36] and used to rationalize their solution of strictly competitive games.

*Similarly, the optimal level of subsidy for sharing hybrid-enabling technology on fossil*

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

<sup>1</sup> ½ � 2*a*<sup>2</sup> � *a*<sup>1</sup>

, 0≤*θ* ≤

2 3

(34)

*K* þ *b*2, (35)

*<sup>S</sup>* ð Þ *K* , respectively, which are

*<sup>S</sup>* ð Þ *K* , for Player I, we

*<sup>S</sup>* þ ð Þ Γ þ *δ K*Þ

ð Þ *K* )

*<sup>S</sup>* , *L<sup>F</sup> S* � � for

, (37)

, (38)

*<sup>S</sup>* þ ð Þ Γ þ *δ K*

ð Þ *K* )

*:* (39)

*:* (36)

*<sup>S</sup>* ð Þ *K* , for

<sup>1</sup> ½ � 2*a*<sup>2</sup> þ *a*<sup>1</sup>

0*:* otherwise

*<sup>S</sup>* ð Þ¼ *<sup>K</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

*<sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup>*

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> *<sup>L</sup><sup>F</sup>*

1 2 *∂*2 *V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1

*<sup>S</sup>* ð Þ *<sup>K</sup> :* The optimal sharing revenue function, *<sup>V</sup>*0ð Þ II

*<sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup>*

nh i � �

*<sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup>*

1 2 *∂*2 *V*ð Þ II *<sup>S</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

*ω*2*β<sup>F</sup> L<sup>F</sup> S* � �<sup>2</sup>

� 1 2

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

*<sup>S</sup>* <sup>þ</sup> *<sup>α</sup>*2*LF*

*<sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � � � *<sup>ξ</sup><sup>K</sup>*

*ω*1*β<sup>R</sup> LR S* � �<sup>2</sup>

� 1 2

h i <sup>þ</sup>

nh i �

*<sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup>*

*S* � �<sup>2</sup>

*<sup>S</sup>* ð Þ *<sup>K</sup>* and *<sup>V</sup>*ð Þ II

ð Þ *ρ*<sup>2</sup> þ *ξ*

*The optimal sharing payoff functions under hybrid-enabling technology on renewable*

*Proof.* We define the optimal revenue functions for Player I and Player II

*<sup>S</sup>* ð Þþ*<sup>t</sup> <sup>α</sup>*2*LF*

� 1 2

*<sup>α</sup>*2ð Þþ <sup>2</sup> � <sup>3</sup>*<sup>θ</sup> <sup>ϑ</sup><sup>F</sup>*

*<sup>α</sup>*2ð Þþ <sup>2</sup> � *<sup>θ</sup> <sup>ϑ</sup><sup>F</sup>*

*sources and on fossil fuel for Player I and Player II are given below*

*<sup>K</sup>* <sup>þ</sup> *<sup>b</sup>*1, *<sup>V</sup>*ð Þ II

*θ α*1*LR*

h i <sup>þ</sup>

Via the first order conditions, we obtain the optimal values *LR*

*S* � �<sup>2</sup>

*<sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � � � *<sup>ξ</sup><sup>K</sup>*

*<sup>S</sup>* <sup>¼</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>S</sup>* <sup>¼</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> *<sup>L</sup><sup>R</sup>*

*LR*

*LF*

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*1*LR*

*fuel is given by:*

*V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ¼ *K*

*<sup>ρ</sup>*1*V*ð Þ<sup>I</sup>

þ *∂V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *K <sup>∂</sup><sup>K</sup> <sup>ϑ</sup>*<sup>1</sup> *LR*

Player I as:

where *<sup>∂</sup>V*ð Þ<sup>I</sup>

*<sup>ρ</sup>*2*V*ð Þ II

**69**

*<sup>S</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* � *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>S</sup>* ð Þ¼ *K* max *L*~*R <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � �≥<sup>0</sup>

> � 1 2 *β*~*<sup>R</sup> L<sup>R</sup> <sup>S</sup>* ð Þ*<sup>t</sup>* � �<sup>2</sup>

> > *∂V*ð Þ II *<sup>S</sup>* ð Þ *K <sup>∂</sup><sup>K</sup> <sup>ϑ</sup>*<sup>1</sup> *LR*

þ

Player II and the associated HJB equation is

� 1 2 *β*~*<sup>F</sup> L*~*<sup>F</sup> S* � �<sup>2</sup>

*<sup>S</sup>* , *L<sup>F</sup> S* � � <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup>*

obtain

*ω*<sup>2</sup> ¼

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

*θ* Γ þ *δ*<sup>Þ</sup> � � ð Þ *ρ*<sup>1</sup> þ *ξ*

under hybrid-enabling technology as *V*ð Þ<sup>I</sup>

*<sup>S</sup>* ð Þ¼ *K* max *LR <sup>S</sup>* , *<sup>L</sup><sup>F</sup>* f g*<sup>S</sup>* <sup>≥</sup><sup>0</sup>

> � 1 2

*<sup>S</sup>* , *LF S* � � <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup>*

8 >><

>>:

where *a*1, *a*2, *b*<sup>1</sup> and *b*<sup>2</sup> are given in the proof.

continuously differentiable. Applying HJB equation to *V*ð Þ<sup>I</sup>

To promote the sharing of hybrid-enabling technology, the Player II (the leader) determine an optimal sharing effort sharing level and an optimal subsidy scheme. Then the Player I (the follower) choose his/her optimal sharing level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium.

**Proposition 5.** *If above conditions are satisfied, the feedback Stackelberg leader (Player II)-follower (Player I) and equilibria is given as:*

$$L\_{\mathbb{S}}^{R} = \frac{a\_1(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi) + \theta\_1^{\mathbb{R}}(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\theta^{\mathbb{R}}(\rho\_2+\xi)(\rho\_1+\xi)},\tag{29}$$

$$L\_{\rm S}^{\rm F} = \frac{a\_{\rm 2}(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi) + \theta\_1^{\rm F}(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\theta^{\rm F}(\rho\_2+\xi)(\rho\_1+\xi)},\tag{30}$$

$$
\tilde{L}\_{\mathbb{S}}^{\mathbb{R}} = \frac{(\mathbb{1} - \theta)(\beta\_1(\rho\_2 + \xi) + (\Gamma + \delta))\theta\_2^{\mathbb{R}}}{\tilde{\rho}^{\mathbb{R}}(\rho\_2 + \xi)}, \tag{31}
$$

$$\tilde{L}\_{\rm S}^{F} = \frac{(\mathbf{1} - \boldsymbol{\theta})(\boldsymbol{\beta}\_2(\boldsymbol{\rho}\_2 + \boldsymbol{\xi}) + (\boldsymbol{\Gamma} + \boldsymbol{\delta}))\boldsymbol{\theta}\_2^F}{\tilde{\boldsymbol{\beta}}^F(\boldsymbol{\rho}\_2 + \boldsymbol{\xi})}. \tag{32}$$

where *LR <sup>S</sup>* , *LF <sup>S</sup>* are the optimal effort level of hybrid-enabling technological improvements shared on renewable sources and fossil fuel at time t by Player I, respectively. *L*~*<sup>R</sup> <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> <sup>S</sup>* are the optimal effort level of technological improvements shared on renewable sources and fossil fuel at time t by Player II, respectively.

*The optimal level of subsidy for sharing hybrid-enabling on renewable sources is given by*

$$a\_1 = \begin{cases} a\_1(2 - 3\theta) + \theta\_1^R[2a\_2 - a\_1] \\ \hline a\_1(2 - \theta) + \theta\_1^R[2a\_2 + a\_1] \\ 0. \end{cases}, \quad 0 \le \theta \le \frac{2}{3} \\ \tag{33}$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

*Similarly, the optimal level of subsidy for sharing hybrid-enabling technology on fossil fuel is given by:*

$$a\_2 = \begin{cases} \frac{a\_2(2-3\theta) + \theta\_1^F[2a\_2 - a\_1]}{a\_2(2-\theta) + \theta\_1^F[2a\_2 + a\_1]}, & 0 \le \theta \le \frac{2}{3} \\\\ 0. & \text{otherwise} \end{cases} \tag{34}$$

*The optimal sharing payoff functions under hybrid-enabling technology on renewable sources and on fossil fuel for Player I and Player II are given below*

$$V\_S^{(\text{I})}(K) = \frac{\theta(\Gamma + \delta)}{(\rho\_1 + \xi)} K + b\_1, \quad V\_S^{(\text{II})}(K) = \frac{(\mathbf{1} - \theta)(\Gamma + \delta)}{(\rho\_2 + \xi)} K + b\_2,\tag{35}$$

where *a*1, *a*2, *b*<sup>1</sup> and *b*<sup>2</sup> are given in the proof.

*Proof.* We define the optimal revenue functions for Player I and Player II under hybrid-enabling technology as *V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup>* and *<sup>V</sup>*ð Þ II *<sup>S</sup>* ð Þ *K* , respectively, which are continuously differentiable. Applying HJB equation to *V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *K* , for Player I, we obtain

$$\rho\_1 \mathbf{V}\_S^{(\text{I})}(K) = \max\_{\left\{ L\_S^R, \tilde{L}\_S^F \right\} \ge 0} \left\{ \left[ \theta \left( a\_1 \mathbf{L}\_S^R(t) + a\_2 \mathbf{L}\_S^F + \beta\_1 \tilde{\mathbf{L}}\_S^R + \beta\_2 \tilde{\mathbf{L}}\_S^F + (\Gamma + \delta) K \right) \right] \right.$$

$$- \frac{1}{2} \rho^R (\mathbf{1} - a\_1) \left( \mathbf{L}\_S^R \right)^2 - \frac{1}{2} \theta^F (\mathbf{1} - a\_2) \left( \mathbf{L}\_S^F \right)^2$$

$$+ \frac{\partial V\_S^{(\text{I})}(K)}{\partial \mathbf{K}} \left[ \theta\_1 \left( L\_S^R, L\_S^F \right) + \theta\_2 \left( \tilde{L}\_S^R, \tilde{L}\_S^F \right) - \xi K \right] + \frac{1}{2} \frac{\partial^2 V\_S^{(\text{I})}(K)}{\partial \mathbf{K}^2} \rho^2(K) \right]. \tag{36}$$

Via the first order conditions, we obtain the optimal values *LR <sup>S</sup>* , *L<sup>F</sup> S* � � for Player I as:

$$L\_{\mathbb{S}}^{\mathbb{R}} = \frac{\theta a\_1 + V\_{\mathbb{S}}'(\mathbb{K}) \theta\_1^{\mathbb{R}}}{\beta^{\mathbb{R}} (1 - a\_1)},\tag{37}$$

$$L\_S^F = \frac{\theta a\_2 + V\_S^{\prime(1)}(K)\theta\_1^F}{\beta^F(1 - a\_2)},\tag{38}$$

where *<sup>∂</sup>V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* � *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> :* The optimal sharing revenue function, *<sup>V</sup>*0ð Þ II *<sup>S</sup>* ð Þ *K* , for Player II and the associated HJB equation is

$$\rho\_2 V\_S^{(\text{II})}(K) = \max\_{\left\{\tilde{L}\_S^R, \tilde{L}\_S^F\right\} \ge 0} \left\{ \left[ (1-\theta) \left( a\_1 L\_S^R + a\_2 L\_S^F + \theta\_1 \tilde{L}\_S^R + \theta\_2 \tilde{L}\_S^F + (\Gamma + \delta) K \right) \right] \right. \\$$

$$- \frac{1}{2} \bar{\rho}^R \left( L\_S^R(\mathbf{t}) \right)^2 - \frac{1}{2} \bar{\rho}^F \left( \tilde{L}\_S^F \right)^2 - \frac{1}{2} a \alpha\_1 \rho^R \left( L\_S^R \right)^2 - \frac{1}{2} a \alpha \rho^F \left( L\_S^F \right)^2$$

$$+ \frac{\partial V\_S^{(\text{II})}(K)}{\partial K} \left[ \theta\_1 \left( L\_S^R, L\_S^F \right) + \theta\_2 \left( \tilde{L}\_S^R, \tilde{L}\_S^F \right) - \xi K \right] + \frac{1}{2} \frac{\partial^2 V\_S^{(\text{II})}(K)}{\partial K^2} \rho^2(K) \right\}. \tag{39}$$

**3. A Stackelberg game under heterogeneous technology**

solution of strictly competitive games.

*<sup>S</sup>* <sup>¼</sup> *<sup>α</sup>*1ð Þ <sup>2</sup> � *<sup>θ</sup>* ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> <sup>ϑ</sup><sup>R</sup>*

*<sup>S</sup>* <sup>¼</sup> *<sup>α</sup>*2ð Þ <sup>2</sup> � *<sup>θ</sup>* ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> <sup>ϑ</sup><sup>F</sup>*

*L*~*R*

*L*~*F*

*LR*

*Carbon Capture*

*LF*

where *LR*

respectively. *L*~*<sup>R</sup>*

*given by*

**68**

*<sup>S</sup>* , *LF*

*<sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup>*

*ω*<sup>1</sup> ¼

8 ><

>:

*(Player II)-follower (Player I) and equilibria is given as:*

Theory of strong Stackelberg reasoning is an improved version of an earlier theory [35], which provides an explanation of coordination in all dyadic (twoplayer) common interest games. It provides an explanation of why players tend to choose strategies associated with a payoff-dominant Nash equilibrium. Its distinctive assumption is that players behave as though their co-players will anticipate any strategy choice and invariably choose a best reply to it. Stackelberg strategies resulting from this form of reasoning do not form Nash equilibria. The theory makes no predictions, because a non-equilibrium outcome is inherently unstable, leaving at least one player with a reason to choose differently and thereby achieve a better payoff. Strong Stackelberg reasoning is a simple theory, according to which players in dyadic games choose strategies that would maximize their own payoffs if their co-players could invariably anticipate their strategy choices and play counterstrategies that yield the maximum payoffs for themselves. The key assumption is relatively innocuous, first because game theory imposes no constraints on players' beliefs, apart from consistency requirements, and second because the theory does not assume that players necessarily believe that their strategies will be anticipated, merely that they behave as though that is the case, as a heuristic aid to choosing the best strategy. Strong Stackelberg reasoning is, in fact, merely a generalization of the minorant and majorant models introduced by [36] and used to rationalize their

To promote the sharing of hybrid-enabling technology, the Player II (the leader) determine an optimal sharing effort sharing level and an optimal subsidy scheme. Then the Player I (the follower) choose his/her optimal sharing level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium. **Proposition 5.** *If above conditions are satisfied, the feedback Stackelberg leader*

*<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ *<sup>β</sup>*1ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> <sup>ϑ</sup><sup>R</sup>*

*<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ *<sup>β</sup>*2ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> <sup>ϑ</sup><sup>F</sup>*

*β*~*F*

improvements shared on renewable sources and fossil fuel at time t by Player I,

shared on renewable sources and fossil fuel at time t by Player II, respectively. *The optimal level of subsidy for sharing hybrid-enabling on renewable sources is*

*<sup>α</sup>*1ð Þþ <sup>2</sup> � <sup>3</sup>*<sup>θ</sup> <sup>ϑ</sup><sup>R</sup>*

*<sup>α</sup>*1ð Þþ <sup>2</sup> � *<sup>θ</sup> <sup>ϑ</sup><sup>R</sup>*

ð Þ *ρ*<sup>2</sup> þ *ξ*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*<sup>S</sup>* are the optimal effort level of hybrid-enabling technological

<sup>1</sup> ½ � 2*a*<sup>2</sup> � *a*<sup>1</sup>

<sup>1</sup> ½ � 2*a*<sup>2</sup> þ *a*<sup>1</sup>

0*:* otherwise

*<sup>S</sup>* are the optimal effort level of technological improvements

*β*~*R*

<sup>1</sup> ð Þ Γ þ *δ* ð Þ ð Þ 2 � 2*θ* ð Þþ *ρ*<sup>1</sup> þ *ξ θ ρ*ð Þ <sup>2</sup> þ *ξ* <sup>2</sup>*β<sup>R</sup>*ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* , (29)

<sup>1</sup> ð Þ Γ þ *δ* ð Þ ð Þ 2 � 2*θ* ð Þþ *ρ*<sup>1</sup> þ *ξ θ ρ*ð Þ <sup>2</sup> þ *ξ* <sup>2</sup>*β<sup>F</sup>*ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* , (30)

2

2

, 0≤*θ* ≤

2 3

, (31)

*:* (32)

(33)

*Carbon Capture*

Substituting the results of Eqs. (37) and (38) into Eq. (39), obtain

*<sup>ρ</sup>*2*V*ð Þ II *<sup>S</sup>* ð Þ¼ *K* max *L*~*R <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � �<sup>≥</sup> <sup>0</sup> ð Þ 1 � *θ <sup>α</sup>*<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1 � � *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> þ *<sup>α</sup>*<sup>2</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1 � � *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> 0 @ 2 4 8 < : <sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup> <sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup> <sup>S</sup>* þ ð Þ Γ þ *δ K* � � 1 2 *β*~*<sup>R</sup> L*~*<sup>R</sup> <sup>S</sup>* ð Þ*t* � �<sup>2</sup> � 1 2 *β*~*<sup>F</sup> L*~*<sup>F</sup> S* � �<sup>2</sup> � 1 2 *<sup>ω</sup>*1*β<sup>R</sup> θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1 *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> !<sup>2</sup> � 1 2 *<sup>ω</sup>*2*β<sup>F</sup> θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1 *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> !<sup>2</sup> 3 5 þ *∂V*ð Þ II *<sup>S</sup>* ð Þ *K ∂K ϑR* <sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1 h i *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> þ *ϑF* <sup>1</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1 h i *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup> <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � � � *<sup>ξ</sup><sup>K</sup>* 2 4 3 5 þ 1 2 *∂*2 *V*ð Þ II *<sup>S</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup> ð Þ *K* ) *:* (40)

Via the first order conditions of (Eq. (40)), we obtain the optimal values *L*~*R* , *<sup>L</sup>*~*<sup>F</sup>* � � for Player II as:

$$
\tilde{L}\_{\mathcal{S}}^{\mathbb{R}} = \frac{(1 - \theta)\beta\_1 + \boldsymbol{V}\_{\mathcal{S}}^{\prime(\text{II})}(K)\theta\_2^{\mathbb{R}}}{\tilde{\boldsymbol{\beta}}^{\mathbb{R}}},\tag{41}
$$

þ*β*<sup>2</sup>

� 1 2

þ *ϑR*

þ *ϑR*

*<sup>ρ</sup>*2*V*ð Þ II

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

*β*~*F* !

1

*<sup>β</sup>F*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

� � *<sup>β</sup>R*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

<sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

� �

and simplifying (Eq. (40)), obtain:

*<sup>S</sup>* ð Þ¼ *<sup>K</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

þ*β*<sup>2</sup>

� 1 2 ~*β*

þ *ϑR*

þ *ϑR*

This implies that,

where

**71**

*<sup>a</sup>*<sup>1</sup> <sup>¼</sup> *<sup>θ</sup>*ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

2

*θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

*<sup>β</sup>F*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � �<sup>2</sup>

2

þ *ϑF*

*<sup>β</sup>*~*<sup>R</sup>* <sup>þ</sup>

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

*β*~*F* !

*<sup>F</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

<sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* , *<sup>b</sup>*<sup>1</sup> <sup>¼</sup> <sup>Φ</sup><sup>1</sup>

� �

<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

~*β F* !<sup>2</sup>

1 � � *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

þ *ϑF*

*<sup>β</sup>*~*<sup>R</sup>* <sup>þ</sup>

2

*ρ*1

<sup>Φ</sup><sup>1</sup> <sup>¼</sup> *<sup>α</sup>*1*<sup>θ</sup>* � *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

<sup>þ</sup> *<sup>α</sup>*2*<sup>θ</sup>* � *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

<sup>2</sup> *a*<sup>1</sup>

<sup>2</sup> *a*<sup>2</sup>

<sup>þ</sup> *<sup>β</sup>*1*<sup>θ</sup>* <sup>þ</sup> *<sup>ϑ</sup><sup>R</sup>*

<sup>þ</sup> *<sup>β</sup>*2*<sup>θ</sup>* <sup>þ</sup> *<sup>ϑ</sup><sup>F</sup>*

" #

" #

1

<sup>1</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

*ϑF*

1 � � *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

þ ð Þ Γ þ *δ K*

� 1 2

<sup>1</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

*ϑF*

" #

2

2

" #

� � *<sup>β</sup>F*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

þð Þ Γ þ *δ K*Þ

� *ξKa*<sup>1</sup>

1

*a*1

<sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

� � *β*~*F*

<sup>þ</sup> *<sup>α</sup>*<sup>2</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

! � 1 2

1 � � *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

, *<sup>a</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

1

1

<sup>2</sup> <sup>þ</sup> *<sup>ϑ</sup><sup>F</sup>*

! *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

~*β R* !

*β*~*F* !

<sup>2</sup> <sup>þ</sup> *<sup>ϑ</sup><sup>R</sup>*

! *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

<sup>1</sup> *a*<sup>1</sup>

<sup>1</sup> *a*<sup>1</sup>

2

2

>0,

� �

� �

� � ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

� � ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

<sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

� � *β*~*F*

! � 1 2

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

*<sup>β</sup>R*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

2

1 � � *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

!

*<sup>ω</sup>*1*β<sup>R</sup> θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> � �<sup>2</sup>

*a*<sup>2</sup> � *ξKa*<sup>2</sup>

2

ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* , *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>Φ</sup><sup>2</sup>

*ρ*2

1

1

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> � �

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � �

*a*2*:*

*a*1,

þ *β*<sup>1</sup>

1

*<sup>β</sup>*~*<sup>R</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

*β*~*R* !<sup>2</sup>

> � 1 2

*θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>* 1

*<sup>β</sup>R*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> � �<sup>2</sup>

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

*β*~*R*

2

*<sup>ω</sup>*2*β<sup>F</sup> θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � �<sup>2</sup>

1

(47)

(49)

, (48)

2

(46)

$$
\tilde{L}\_{\mathcal{S}}^{F} = \frac{(1 - \theta)\beta\_2 + V\_{\mathcal{S}}^{\prime(\text{II})}(K)\theta\_2^F}{\tilde{\beta}^F}. \tag{42}
$$

And the optimal value for ð Þ *ω*1,*ω*<sup>2</sup>

$$\rho\_1 = \frac{a\_1(2 - 3\theta) + \theta\_1^R \left[2V\_S'^{(\text{II})}(K) - V\_S'^{(\text{I})}(K)\right]}{a\_1(2 - \theta) + \theta\_1^R \left[2V\_S'^{(\text{II})}(K) + V\_S'^{(\text{I})}(K)\right]},\tag{43}$$

and

$$\alpha\_{2} = \frac{a\_{2}(2 - 3\theta) + \theta\_{1}^{F} \left[2V\_{\mathcal{S}}^{\prime}\,^{(\text{II})}(K) - V\_{\mathcal{S}}^{\prime(\text{I})}(K)\right]}{a\_{2}(2 - \theta) + \theta\_{1}^{F} \left[2V\_{\mathcal{S}}^{\prime}\,^{(\text{II})}(K) + V\_{\mathcal{S}}^{\prime(\text{I})}(K)\right]}.\tag{44}$$

Hence, the solution of the HJB equation is an unary function with *K* (*K* as the independent variable), we define *V*ð Þ<sup>I</sup> *<sup>S</sup>* <sup>¼</sup> *<sup>a</sup>*1*<sup>K</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>1</sup> and *<sup>V</sup>*ð Þ II *<sup>S</sup>* ¼ *a*2*K* þ *b*2, where *a*1, *b*1, *a*2, and *b*<sup>2</sup> are constants that need to be solved. Simplifying (Eq. (39)), obtain:

$$\rho\_1 V\_S^{(1)}(K) = \theta \left( a\_1 \frac{\theta \alpha\_1 + a\_1 \theta\_1^R}{\int^{\mathbb{R}} (1 - \alpha\_1)} \right) + a\_2 \left( \frac{\theta \alpha\_2 + a\_1 \theta\_1^F}{\int^{\mathbb{F}} (1 - \alpha\_2)} \right) + \beta\_1 \left( \frac{(1 - \theta)\beta\_1 + a\_2 \theta\_2^R}{\int^{\mathbb{R}}} \right) \tag{45}$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

$$\begin{aligned} &+\beta\_2\left(\frac{(1-\theta)\beta\_2+a\_2\theta\_2^F}{\hat{\beta}^F}\right)+(\Gamma+\delta)K\right)-\frac{1}{2}\theta^R(1-\nu\_1)\left(\frac{\theta a\_1+a\_1\theta\_1^F}{\rho^R(1-\nu\_1)}\right)^2\\ &-\frac{1}{2}\theta^F(1-\nu\_2)\left(\frac{\theta a\_2+a\_1\theta\_1^F}{\rho^F(1-\nu\_2)}\right)^2-\xiKa\_1\\ &+\left[\frac{\theta\_1^R[\theta a\_1+a\_1\theta\_1^R]}{\rho^R(1-\nu\_1)}+\frac{\theta\_1^F[\theta a\_2+a\_1\theta\_1^F]}{\rho^F(1-\nu\_2)}\right]a\_1\\ &+\left[\frac{\theta\_2^R\left[(1-\theta)\beta\_1+a\_2\theta\_2^F\right]}{\hat{\beta}^R}+\frac{\theta\_2^F\left[(1-\theta)\beta\_2+a\_2\theta\_2^F\right]}{\hat{\beta}^F}\right]a\_1,\end{aligned} \tag{46}$$

and simplifying (Eq. (40)), obtain:

*<sup>ρ</sup>*2*V*ð Þ II *<sup>S</sup>* ð Þ¼ *<sup>K</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>* 1 � � *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>* 1 � � *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> þ *β*<sup>1</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>* 2 *β*~*R* ! þ*β*<sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>* 2 *β*~*F* ! þ ð Þ Γ þ *δ K* ! � 1 2 *<sup>β</sup>*~*<sup>R</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>* 2 *β*~*R* !<sup>2</sup> � 1 2 ~*β <sup>F</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>* 2 ~*β F* !<sup>2</sup> � 1 2 *<sup>ω</sup>*1*β<sup>R</sup> θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>* 1 *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> � �<sup>2</sup> � 1 2 *<sup>ω</sup>*2*β<sup>F</sup> θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>* 1 *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � �<sup>2</sup> þ *ϑR* <sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>* 1 � � *<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> þ *ϑF* <sup>1</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>* 1 � � *<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> " #*a*<sup>2</sup> � *ξKa*<sup>2</sup> þ *ϑR* <sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>* 2 � � *<sup>β</sup>*~*<sup>R</sup>* <sup>þ</sup> *ϑF* <sup>2</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>* 2 � � *β*~*F* " #*a*2*:* (47)

This implies that,

$$a\_1 = \frac{\theta(\Gamma + \delta)}{(\rho\_1 + \xi)}, \quad b\_1 = \frac{\Phi\_1}{\rho\_1}, \quad a\_2 = \frac{(\mathbb{1} - \theta)(\Gamma + \delta)}{(\rho\_2 + \xi)}, \quad b\_2 = \frac{\Phi\_2}{\rho\_2},\tag{48}$$

where

Substituting the results of Eqs. (37) and (38) into Eq. (39), obtain

� � 1 2 *β*~*<sup>R</sup> L*~*<sup>R</sup> <sup>S</sup>* ð Þ*t* � �<sup>2</sup>

*<sup>α</sup>*<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

� 1 2

Via the first order conditions of (Eq. (40)), we obtain the optimal values

*<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ II

*<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ II

<sup>1</sup> 2*V*<sup>0</sup> *S*

<sup>1</sup> 2*V*<sup>0</sup> *S*

> <sup>1</sup> 2*V*<sup>0</sup> *S*

<sup>1</sup> 2*V*<sup>0</sup> *S*

Hence, the solution of the HJB equation is an unary function with *K* (*K* as

where *a*1, *b*1, *a*2, and *b*<sup>2</sup> are constants that need to be solved. Simplifying (Eq. (39)),

*θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>* 1

!

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> � �

*β*

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1

þ

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> !<sup>2</sup>

h i

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

� 1 2 *β*~*<sup>F</sup> L*~*<sup>F</sup> S* � �<sup>2</sup>

<sup>1</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 2

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 2

ð Þ II ð Þ� *<sup>K</sup> VS*

ð Þ II ð Þþ *<sup>K</sup> VS*

ð Þ II ð Þ� *<sup>K</sup> VS*

ð Þ II ð Þþ *<sup>K</sup> VS*

*<sup>S</sup>* <sup>¼</sup> *<sup>a</sup>*1*<sup>K</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>1</sup> and *<sup>V</sup>*ð Þ II

þ *β*<sup>1</sup>

h i

h i

*<sup>ω</sup>*2*β<sup>F</sup> θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>α</sup>*<sup>2</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1

<sup>~</sup>*<sup>R</sup>* , (41)

*<sup>β</sup>*~*<sup>F</sup> :* (42)

0ð Þ<sup>I</sup> ð Þ *<sup>K</sup>*

0ð Þ<sup>I</sup> ð Þ *<sup>K</sup>* � � , (43)

0ð Þ<sup>I</sup> ð Þ *<sup>K</sup>*

0ð Þ<sup>I</sup> ð Þ *<sup>K</sup>* � � *:* (44)

*<sup>S</sup>* ¼ *a*2*K* þ *b*2,

2

(45)

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

*β*~*R*

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1

� �

*<sup>β</sup><sup>F</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

3 5

<sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup>*

*<sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � �

� *ξK*

(40)

3 5

� �

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

þ *ϑF*

ð Þ 1 � *θ*

*<sup>S</sup>* þ ð Þ Γ þ *δ K*

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> !<sup>2</sup>

> ð Þ *K* ) *:*

> > *L*~*R*

*L*~*F*

*<sup>α</sup>*1ð Þþ <sup>2</sup> � <sup>3</sup>*<sup>θ</sup> <sup>ϑ</sup><sup>R</sup>*

*<sup>α</sup>*1ð Þþ <sup>2</sup> � *<sup>θ</sup> <sup>ϑ</sup><sup>R</sup>*

*<sup>α</sup>*2ð Þþ <sup>2</sup> � <sup>3</sup>*<sup>θ</sup> <sup>ϑ</sup><sup>F</sup>*

*<sup>α</sup>*2ð Þþ <sup>2</sup> � *<sup>θ</sup> <sup>ϑ</sup><sup>F</sup>*

þ *α*<sup>2</sup>

*<sup>S</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1

<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

h i

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup>

2 4 8 < :

0 @

*<sup>ρ</sup>*2*V*ð Þ II

*Carbon Capture*

*<sup>S</sup>* ð Þ¼ *K* max *L*~*R <sup>S</sup>* , *<sup>L</sup>*~*<sup>F</sup> S* � �<sup>≥</sup> <sup>0</sup>

> � 1 2

> > þ *∂V*ð Þ II *<sup>S</sup>* ð Þ *K ∂K*

þ 1 2 *∂*2 *V*ð Þ II *<sup>S</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

for Player II as:

And the optimal value for ð Þ *ω*1,*ω*<sup>2</sup>

*ω*<sup>1</sup> ¼

*ω*<sup>2</sup> ¼

the independent variable), we define *V*ð Þ<sup>I</sup>

*θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>* 1

*<sup>β</sup><sup>R</sup>*ð Þ <sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> � �

*L*~*R* , *<sup>L</sup>*~*<sup>F</sup>* � �

and

obtain:

**70**

*<sup>ρ</sup>*1*V*ð Þ<sup>I</sup>

*<sup>S</sup>* ð Þ¼ *K θ α*<sup>1</sup>

<sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup>*

*<sup>S</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup>*

*<sup>ω</sup>*1*β<sup>R</sup> θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*ϑR*

2 4

$$\begin{split} \Phi\_{1} &= \left( a\_{1}\theta - \frac{\left(\theta a\_{1} + a\_{1}\theta\_{1}^{R}\right)}{2} + \theta\_{1}^{R}a\_{1} \right) \left( \frac{\theta a\_{1} + a\_{1}\theta\_{1}^{R}}{\beta^{R}(1 - a\_{1})} \right) \\ &+ \left( a\_{2}\theta - \frac{\left(\theta a\_{2} + a\_{1}\theta\_{1}^{F}\right)}{2} + \theta\_{1}^{F}a\_{1} \right) \left( \frac{\theta a\_{2} + a\_{1}\theta\_{1}^{F}}{\beta^{F}(1 - a\_{2})} \right) \\ &+ \left( \beta\_{1}\theta + \theta\_{2}^{R}a\_{1} \right) \left( \frac{(1 - \theta)\beta\_{1} + a\_{2}\theta\_{2}^{R}}{\tilde{\beta}^{R}} \right) \\ &+ \left( \beta\_{2}\theta + \theta\_{2}^{F}a\_{2} \right) \left( \frac{(1 - \theta)\beta\_{2} + a\_{2}\theta\_{2}^{F}}{\tilde{\beta}^{F}} \right) > 0, \end{split} \tag{49}$$

and

$$\begin{split} \Phi\_{2} &= \left( (1-\theta)a\_{1} - \frac{a\_{1}\left(\theta a\_{1} + a\_{1}\theta\_{1}^{\mathbb{R}}\right)}{2(1-\alpha\_{1})} + \theta\_{1}^{\mathbb{R}}a\_{2} \right) \frac{\left(\theta a\_{1} + a\_{1}\theta\_{1}^{\mathbb{R}}\right)}{\theta^{\mathbb{R}}(1-\alpha\_{1})} \\ &+ \left( (1-\theta)a\_{2} - \frac{a\_{2}\left(\theta a\_{2} + a\_{1}\theta\_{1}^{\mathbb{R}}\right)}{2(1-\alpha\_{2})} + \theta\_{1}^{\mathbb{R}}a\_{2} \right) \frac{\left(\theta a\_{2} + a\_{1}\theta\_{1}^{\mathbb{R}}\right)}{\theta^{\mathbb{R}}(1-\alpha\_{2})} \\ &+ \left( (1-\theta)\theta\_{1} - \frac{((1-\theta)\theta\_{1} + a\_{2}\theta\_{2}^{\mathbb{R}})}{2} + \theta\_{2}^{\mathbb{R}}a\_{2} \right) \left( \frac{(1-\theta)\theta\_{1} + a\_{2}\theta\_{2}^{\mathbb{R}}}{\tilde{\rho}^{\mathbb{R}}} \right) \\ &+ \left( (1-\theta)\theta\_{2} - \frac{((1-\theta)\theta\_{2} + a\_{2}\theta\_{2}^{\mathbb{R}})}{2} + \theta\_{2}^{\mathbb{R}}a\_{2} \right) \left( \frac{(1-\theta)\theta\_{2} + a\_{2}\theta\_{2}^{\mathbb{R}}}{\tilde{\rho}^{\mathbb{R}}} \right) > 0. \end{split} \tag{50} $$

*DKt* ð Þ¼ ð Þ *<sup>φ</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> ð Þ� <sup>2</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> � *<sup>ξ</sup>K*<sup>~</sup> <sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94016*

ð Þ *<sup>K</sup>*ð Þ <sup>0</sup> <sup>2</sup> <sup>¼</sup> *<sup>K</sup>*~<sup>2</sup>

**4. Nash non cooperative game**

*Nash equilibria will be:*

*LR*

*L*~*R*

where *LR*

*V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ¼ *K*

where ^

**73**

*<sup>N</sup>*, *LF*

*b*<sup>1</sup> and ^

*Proof.* See Appendix A.

(

results.

*Proof.* Applying Itô's lemma to (Eq. (51)), obtain:

<sup>0</sup> > 0*:*

Then *EKt* ð Þ ð Þ and *EKt* ð Þ ð Þ <sup>2</sup> can be defined as:

�

ð Þ *<sup>K</sup>*ð Þ <sup>0</sup> <sup>2</sup> <sup>¼</sup> *<sup>K</sup>*<sup>2</sup> <sup>0</sup> <sup>&</sup>gt;0, (

*<sup>N</sup>* <sup>¼</sup> *θ α*½ � <sup>1</sup>ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

ð Þ *ρ*<sup>2</sup> þ *ξ*

on renewable sources at time t for Player II, respectively.

*sources and on fossil fuel for Player I and Player II are given below*

*<sup>K</sup>* <sup>þ</sup> ^

*b*<sup>2</sup> are given in the proof.

*<sup>N</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ½ � *<sup>β</sup>*1ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> β*~*R*

> *θ* Γ þ *δ*<sup>Þ</sup> � � ð Þ *ρ*<sup>1</sup> þ *ξ*

� �*e*�*ξ<sup>t</sup>* <sup>þ</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> � <sup>2</sup>*ξK*<sup>~</sup> <sup>0</sup> � �*e*�2*ξ<sup>t</sup>* � �

lim*t*!<sup>∞</sup> *DEKt* ð Þ¼ ð Þ ð Þ *<sup>φ</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> ð Þ

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources*

*dE K t* ð Þ¼ ð Þ ½ � *μ*<sup>1</sup> þ *μ*<sup>2</sup> � *ξK t*ð Þ *dt*

*dE K t* ð Þ ð Þ <sup>2</sup> <sup>¼</sup> <sup>2</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>φ</sup>*<sup>2</sup> ½ � ð Þ*<sup>K</sup> E K*ð Þ� <sup>2</sup>*ξE K*<sup>2</sup> � � � � *dt*

Solving the above non-homogeneous linear differential equation, will obtain the

Under Nash-non-cooperative game setting, Player I and Player II simultaneously

**Proposition 7.** *If above conditions are satisfied, the feedback non-cooperative game*

*<sup>N</sup>* <sup>¼</sup> *θ α*½ � <sup>2</sup>ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

*<sup>N</sup>* are the optimal level of hybrid-enabling technological advantage

*<sup>N</sup>* ð Þ¼ *<sup>K</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*<sup>N</sup>* <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ½ � *<sup>β</sup>*2ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> β*~*F*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*<sup>β</sup><sup>F</sup>*ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> :* (60)

*<sup>K</sup>* <sup>þ</sup> ^

, (61)

*<sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N*

*b*2, (62)

and independently choose their optimal efforts levels of heterogeneous hybrid-

, *L*~*<sup>F</sup>*

on renewable sources and on fossil fuel at time t for Player I, respectively. *L*~*<sup>R</sup>*

*b*1, *V*ð Þ II

are the optimal level of hybrid-enabling technological advantage on fossil fuel and

*The optimal sharing payoff functions under hybrid-enabling technology on renewable*

enabling technology sharing concept to maximize their profits.

*<sup>β</sup><sup>R</sup>*ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* , *LF*

*dKt* ð Þ ð Þ <sup>2</sup> <sup>¼</sup> <sup>2</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>φ</sup>*<sup>2</sup> ð Þ*<sup>K</sup>* � <sup>2</sup>*ξK*<sup>2</sup> � �*dt* <sup>þ</sup> <sup>2</sup>*φ<sup>K</sup>* ffiffiffiffi

*K*ð Þ¼ 0 *K*<sup>0</sup> >0*:*

<sup>2</sup>*ξ*<sup>2</sup> (55)

*K* <sup>p</sup> *dW t*ð Þ

<sup>2</sup>*ξ*<sup>2</sup> , (56)

(57)

(58)

(59)

Substituting the results of *a*<sup>1</sup> and *a*<sup>2</sup> into Eqs. (37), (38), (41) and (42), and simplifying, we obtain the optimal effort level of hybrid-enabling technological improvements. By substituting optimal values given in Eqs. (48)–(50) into Eqs. (46) and (47) obtain the optimal sharing payoff functions under hybridenabling technology on renewable sources and fossil fuel for Player I and Player II.

#### **3.1 The limit of expectation and variance**

The payoff of Player I and Player II, under the Stackelberg game paradigm is related to the improvement degree of hybrid-enabling technology via Proposition 4. To analyze the limit of expectations and variance under Stackelberg game equilibrium rewrite (Eq. (19)) as follows.

$$\begin{cases} dK(t) = [\mu\_1 + \mu\_2 - \xi K(t)]dt + \varrho \sqrt{K} dW(t) \\ K(0) = K\_0 > 0, \end{cases} \tag{51}$$

where

$$\mu\_1 = \theta\_1 \left[ \frac{a\_1(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi) + \theta\_1^R(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^R(\rho\_2+\xi)(\rho\_1+\xi)} \right]$$

$$+ \frac{a\_2(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi) + \theta\_1^F(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^F(\rho\_2+\xi)(\rho\_1+\xi)} \Big],\tag{52}$$

and

$$\mu\_2 = \theta\_2 \left[ \frac{(1-\theta)(\beta\_1(\rho\_2+\xi)+(\Gamma+\delta))\theta\_2^{\mathbb{R}}}{\tilde{\boldsymbol{\beta}}^{\mathbb{R}}(\rho\_2+\xi)} + \frac{(1-\theta)(\beta\_2(\rho\_2+\xi)+(\Gamma+\delta))\theta\_2^{\mathbb{R}}}{\tilde{\boldsymbol{\beta}}^{\mathbb{F}}(\rho\_2+\xi)} \right]. \tag{53}$$

**Proposition 6.** *The limit of expectation E K t* ð Þ ð Þ *, and variance D K t* ð Þ ð Þ *in the Stackelberg game feedback equilibrium must satisfy*

$$E(K(t)) = \frac{\mu\_1 + \mu\_2}{\xi} + e^{-\xi t} \left(\tilde{K}\_0 - \frac{\mu\_1 + \mu\_2}{\xi}\right), \quad \lim\_{t \to \infty} E(K(t)) = \frac{\mu\_1 + \mu\_2}{\xi}.\tag{54}$$

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

$$D(K(t)) = \frac{\rho^2 \left[ (\mu\_1 + \mu\_2) - 2 \left( \mu\_1 + \mu\_2 - \xi \tilde{K}\_0 \right) e^{-\xi t} + \left( \mu\_1 + \mu\_2 - 2 \xi \tilde{K}\_0 \right) e^{-2\xi t} \right]}{2\xi^2} \tag{55}$$

$$\lim\_{t \to \infty} D(E(K(t))) = \frac{\rho^2(\mu\_1 + \mu\_2)}{2\xi^2},\tag{56}$$

*Proof.* Applying Itô's lemma to (Eq. (51)), obtain:

$$\begin{cases} d(K(t))^2 = \left[2(\mu\_1 + \mu\_2 + \rho^2)K - 2\xi K^2\right]dt + 2\rho K \sqrt{K}dW(t) \\\ (K(\mathbf{0}))^2 = \tilde{K}\_0^2 > \mathbf{0}. \end{cases} \tag{57}$$

Then *EKt* ð Þ ð Þ and *EKt* ð Þ ð Þ <sup>2</sup> can be defined as:

$$\begin{cases} dE(K(t)) = [\mu\_1 + \mu\_2 - \xi K(t)]dt \\ K(\mathbf{0}) = K\_0 > \mathbf{0}. \end{cases} \tag{58}$$

$$\begin{cases} dE(K(t))^2 = \left[ [2(\mu\_1 + \mu\_2 + \varrho^2)K]E(K) - 2\xi E(K^2) \right] dt\\ \left( K(\mathbf{0}) \right)^2 = K\_0^2 > \mathbf{0}, \end{cases} \tag{59}$$

Solving the above non-homogeneous linear differential equation, will obtain the results.

#### **4. Nash non cooperative game**

Under Nash-non-cooperative game setting, Player I and Player II simultaneously and independently choose their optimal efforts levels of heterogeneous hybridenabling technology sharing concept to maximize their profits.

**Proposition 7.** *If above conditions are satisfied, the feedback non-cooperative game Nash equilibria will be:*

$$L\_N^R = \frac{\theta[a\_1(\rho\_1 + \xi) + (\Gamma + \delta)]}{\rho^\mathbb{R}(\rho\_1 + \xi)},\\L\_N^F = \frac{\theta[a\_2(\rho\_1 + \xi) + (\Gamma + \delta)]}{\rho^\mathbb{F}(\rho\_1 + \xi)}.\tag{60}$$

$$\tilde{L}\_N^R = \frac{(\mathbf{1} - \boldsymbol{\theta})[\boldsymbol{\beta}\_1(\rho\_2 + \xi) + (\boldsymbol{\Gamma} + \boldsymbol{\delta})]}{\tilde{\boldsymbol{\beta}}^R(\rho\_2 + \xi)},\\ \tilde{L}\_N^F = \frac{(\mathbf{1} - \boldsymbol{\theta})[\boldsymbol{\beta}\_2(\rho\_2 + \xi) + (\boldsymbol{\Gamma} + \boldsymbol{\delta})]}{\tilde{\boldsymbol{\beta}}^F(\rho\_2 + \xi)},\tag{61}$$

where *LR <sup>N</sup>*, *LF <sup>N</sup>* are the optimal level of hybrid-enabling technological advantage on renewable sources and on fossil fuel at time t for Player I, respectively. *L*~*<sup>R</sup> <sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N* are the optimal level of hybrid-enabling technological advantage on fossil fuel and on renewable sources at time t for Player II, respectively.

*The optimal sharing payoff functions under hybrid-enabling technology on renewable sources and on fossil fuel for Player I and Player II are given below*

$$V\_N^{(\text{I})}(K) = \frac{\theta(\Gamma + \delta\_{\text{I}})}{(\rho\_1 + \xi)} K + \hat{b}\_1, \quad V\_N^{(\text{II})}(K) = \frac{(1 - \theta)(\Gamma + \delta)}{(\rho\_2 + \xi)} K + \hat{b}\_2,\tag{62}$$

where ^ *b*<sup>1</sup> and ^ *b*<sup>2</sup> are given in the proof. *Proof.* See Appendix A.

and

*Carbon Capture*

<sup>Φ</sup><sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*<sup>1</sup> � *<sup>ω</sup>*<sup>1</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> � ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> � ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

**3.1 The limit of expectation and variance**

(

þ*α*2ð Þ <sup>2</sup> � *<sup>θ</sup>* ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> <sup>ϑ</sup><sup>F</sup>*

rium rewrite (Eq. (19)) as follows.

where

and

**72**

*μ*<sup>2</sup> ¼ *ϑ*<sup>2</sup>

*μ*<sup>1</sup> ¼ *ϑ*<sup>1</sup>

�

"

*EKt* ð Þ¼ ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup>

1

1

! *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

! *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

<sup>þ</sup> *<sup>ϑ</sup><sup>R</sup>* <sup>1</sup> *a*<sup>2</sup>

<sup>þ</sup> *<sup>ϑ</sup><sup>F</sup>* <sup>1</sup> *a*<sup>2</sup>

2

2

Substituting the results of *a*<sup>1</sup> and *a*<sup>2</sup> into Eqs. (37), (38), (41) and (42), and simplifying, we obtain the optimal effort level of hybrid-enabling technological improvements. By substituting optimal values given in Eqs. (48)–(50) into Eqs. (46) and (47) obtain the optimal sharing payoff functions under hybridenabling technology on renewable sources and fossil fuel for Player I and Player II.

The payoff of Player I and Player II, under the Stackelberg game paradigm is related to the improvement degree of hybrid-enabling technology via Proposition 4. To analyze the limit of expectations and variance under Stackelberg game equilib-

*dK t*ðÞ¼ ½ � *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> � *<sup>ξ</sup>K t*ð Þ *dt* <sup>þ</sup> *<sup>φ</sup>* ffiffiffiffi

<sup>2</sup>*β<sup>F</sup>*ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>*

2

**Proposition 6.** *The limit of expectation E K t* ð Þ ð Þ *, and variance D K t* ð Þ ð Þ *in the*

*ξ* � �

�*ξ<sup>t</sup> <sup>K</sup>*<sup>~</sup> <sup>0</sup> � *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup>

<sup>2</sup>*β<sup>R</sup>*ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>*

<sup>1</sup> ð Þ Γ þ *δ* ð Þ ð Þ 2 � 2*θ* ð Þþ *ρ*<sup>1</sup> þ *ξ θ ρ*ð Þ <sup>2</sup> þ *ξ*

*K*ð Þ¼ 0 *K*<sup>0</sup> > 0,

*<sup>α</sup>*1ð Þ <sup>2</sup> � *<sup>θ</sup>* ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þþ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> <sup>ϑ</sup><sup>R</sup>*

ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ *<sup>β</sup>*1ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> <sup>ϑ</sup><sup>R</sup>*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*β*~*R*

*Stackelberg game feedback equilibrium must satisfy*

*<sup>ξ</sup>* <sup>þ</sup> *<sup>e</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>ϑ</sup><sup>F</sup>*

<sup>2</sup> *a*<sup>2</sup>

<sup>2</sup> *a*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>ϑ</sup><sup>R</sup>*

1

1

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>R</sup>*

*β*~*R* !

ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*ϑ<sup>F</sup>*

*β*~*F* !

> *K* <sup>p</sup> *dW t*ð Þ

<sup>1</sup> ð Þ Γ þ *δ* ð Þ ð Þ 2 � 2*θ* ð Þþ *ρ*<sup>1</sup> þ *ξ θ ρ*ð Þ <sup>2</sup> þ *ξ*

<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>θ</sup>* ð Þ *<sup>β</sup>*2ð Þþ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup> <sup>ϑ</sup><sup>F</sup>*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*β*~*F*

, lim*<sup>t</sup>*!<sup>∞</sup> *EKt* ð Þ¼ ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup>

2

2

> 0*:*

(50)

(51)

�

, (52)

2

*<sup>ξ</sup> :* (54)
