*:*

(53)

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

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

� � 2 1ð Þ � *ω*<sup>1</sup>

� � 2 1ð Þ � *ω*<sup>2</sup>

� �

� �

!

!

*Carbon Capture*

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

**Proposition 8.** *The limit of expectation E K t* ð Þ ð Þ *and variance D K t* ð Þ ð Þ *in the Nash non-cooperative game feedback equilibrium must satisfy*

$$E(K(t)) = \frac{\hat{\mu}\_1 + \hat{\mu}\_2}{\xi} + e^{-\xi t} \left(\hat{K}\_0 - \frac{\hat{\mu}\_1 + \hat{\mu}\_2}{\xi}\right), \quad \lim\_{t \to \infty} E(K(t)) = \frac{\hat{\mu}\_1 + \hat{\mu}\_2}{\xi}.\tag{63}$$

$$D(K(t)) = \frac{\rho^2 \left[ (\ddot{\mu}\_1 + \ddot{\mu}\_2) - 2 \left( \ddot{\mu}\_1 + \ddot{\mu}\_2 - \xi \ddot{K}\_0 \right) e^{-\xi t} + \left( \ddot{\mu}\_1 + \ddot{\mu}\_2 - 2 \xi \ddot{K}\_0 \right) e^{-2\xi t} \right]}{2\xi^2} \tag{64}$$

$$\lim\_{t \to \infty} D(E(K(t))) = \frac{\rho^2(\tilde{\mu}\_1 + \tilde{\mu}\_2)}{2\xi^2}. \tag{65}$$

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

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

*<sup>c</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

*<sup>c</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

*βR*

Hence, the solution of the HJB equation is an unary function with *K*, *Vc* ¼ *aK* þ *b*, where *a* and *b* are constant that need to be solved. This implies that

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

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>a</sup> <sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þþ *<sup>a</sup>*ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup>

Substituting the results of Eqs. (72) and (73), into *Vc* ¼ *aK* þ *b*, will obtain the

**Proposition 10.** The limit of expectation and variance in cooperative game

*ξ*

*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>ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> � *<sup>ξ</sup>K*<sup>0</sup> *<sup>e</sup>*�*ξ<sup>t</sup>* <sup>þ</sup> ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>2</sup> � <sup>2</sup>*ξK*<sup>0</sup> *<sup>e</sup>*�2*ξ<sup>t</sup>*

lim*<sup>t</sup>*!<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

*Proof.* Poof of Proposition 10 is like the derivation of Propositions 6 and 8.

ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup>* and *<sup>μ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*2þ*β*<sup>2</sup> ð Þð Þþ *<sup>ρ</sup>*þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*1þ*ϑ*<sup>2</sup>

, 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>

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

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

� 1 2

� *<sup>∂</sup>Vc*ð Þ *<sup>K</sup> <sup>∂</sup><sup>K</sup> <sup>ξ</sup><sup>K</sup>*

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

*βR*

*LR*

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

*LF*

<sup>þ</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þ *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

*<sup>β</sup><sup>F</sup> <sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

*<sup>∂</sup><sup>K</sup>* ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup>

*<sup>b</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þ� *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þ ð Þþ *<sup>a</sup>*ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup>

**5.1 The limit of expectation and variance**

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

where *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>α</sup>*1þ*β*<sup>1</sup> ð Þð Þþ *<sup>ρ</sup>*þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*1þ*ϑ*<sup>2</sup>

feedback equilibrium satisfy

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

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

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

> � 1 2

> > *<sup>∂</sup>Vc*ð Þ *<sup>K</sup>*

þ

results.

**75**

Substituting the results of Eqs. (69) and (70), obtain

*<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þ *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

*βF* 

*βF* <sup>2</sup>

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

*α*<sup>1</sup> þ *β*<sup>1</sup> ð Þþ *V*<sup>0</sup>

*<sup>c</sup>* , *L<sup>F</sup> c* as:

*<sup>c</sup>*ð Þ *K* ð Þ *ϑ*<sup>1</sup> þ *ϑ*<sup>2</sup>

(71)

(73)

*<sup>ξ</sup> :* (74)

*βR* <sup>2</sup>

> *βR*

*βF* <sup>&</sup>gt; <sup>0</sup>*:*

*<sup>β</sup><sup>R</sup>* , (69)

*<sup>β</sup><sup>F</sup> :* (70)

*<sup>β</sup><sup>R</sup> <sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þþ *<sup>V</sup>*<sup>0</sup>

ð Þ *<sup>ρ</sup>* <sup>þ</sup> *<sup>ξ</sup> :* (72)

<sup>2</sup>*ξ*<sup>2</sup> (75)

ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup> :*

<sup>2</sup>*ξ*<sup>2</sup> , (76)

$$\begin{array}{l} \textbf{where } \widetilde{\mu}\_{1} = \frac{\theta \beta^{\mathrm{F}}[a\_{1}(\rho\_{1} + \xi) + (\Gamma + \delta)] + \theta \beta^{\mathrm{R}}[a\_{2}(\rho\_{1} + \xi) + (\Gamma + \delta)]}{\beta^{\mathrm{R}}\beta^{\mathrm{F}}(\rho\_{1} + \xi)} \; \mathsf{and} \; \mathsf{C} \\\widetilde{\mu}\_{2} = \frac{(1 - \theta)\beta^{\mathrm{F}}[\beta\_{1}(\rho\_{2} + \xi) + (\Gamma + \delta)] + (1 - \theta)\beta^{\mathrm{R}}[\beta\_{2}(\rho\_{2} + \xi) + (\Gamma + \delta)]}{\beta^{\mathrm{R}}\beta^{\mathrm{F}}(\rho\_{2} + \xi)} \; . \end{array} $$

Poof of Proposition 8 is like the derivation of Proposition 6.

#### **5. Cooperative game**

Under cooperative game paradigm, Player I and Player II will choose to collaborate/share their hybrid-enabling technology development knowledge while sharing the payoff function in order to maximize their total payoffs. As a result, hybridenabling technology can be improved through this effort as well.

**Proposition 9.** *If above conditions are satisfied, then the feedback cooperative equilibria are defined as*

$$L\_{\varepsilon}^{R} = \frac{(a\_1 + \beta\_1)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^{\mathbb{R}}}, \quad L\_{\varepsilon}^{F} = \frac{(a\_2 + \beta\_2)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^{\mathbb{F}}},\tag{66}$$

and the optimal cooperative payoff function under hybrid-enabling technology on renewable sources and on fossil fuel, respectively. *Vc*ð Þ¼ *<sup>K</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup> <sup>K</sup>* <sup>þ</sup> *<sup>b</sup>:*

where *b*, is given in the proof.

*Proof.* The objective function (optimal sharing payoff function) satisfies the following equation.

$$J(K\_0) = \max\_{\left\{L\_c^F, L\_c^F\right\} 0} E\left\{ \int\_0^\infty e^{-\rho t} \left[ \left( a\_1 L\_c^R(t) + a\_2 L\_c^F + \rho\_1 \tilde{L}\_c^R + \rho\_2 \tilde{L}\_c^F + (\Gamma + \delta)K \right) \right. \right. \right\}$$

Then the optimal revenue sharing function satisfies the following HJB equation

$$\begin{split} \rho V\_{\epsilon}(K) &= \max\_{\left\{L\_{\epsilon}^{R}, L\_{\epsilon}^{F}\right\} \ge 0} \left\{ \left[a\_{1}L\_{\epsilon}^{R}(t) + a\_{2}L\_{\epsilon}^{F} + \rho\_{1}\tilde{L}\_{\epsilon}^{R} + \rho\_{2}\tilde{L}\_{\epsilon}^{F} + (\Gamma + \delta)K\right] \right\} \\ &\quad - \frac{1}{2}\rho^{R}\left(L\_{\epsilon}^{R}\right)^{2} - \frac{1}{2}\rho^{F}\left(L\_{\epsilon}^{F}\right)^{2} \\ &\quad + \frac{\partial V\_{\epsilon}(K)}{\partial K} \left[\theta\_{1}\left(L\_{\epsilon}^{R}, L\_{\epsilon}^{F}\right) + \theta\_{2}\left(\tilde{L}\_{\epsilon}^{R}, \tilde{L}\_{\epsilon}^{F}\right) - \xi K\right] + \frac{1}{2}\frac{\partial^{2}V\_{\epsilon}(K)}{\partial K^{2}}\rho^{2}(K) \right\} \end{split} \tag{68}$$

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

Via the first order conditions, now obtain the optimal values *LR <sup>c</sup>* , *L<sup>F</sup> c* as:

$$L\_c^{\mathbb{R}} = \frac{(\alpha\_1 + \beta\_1) + V\_c'(K)(\theta\_1 + \theta\_2)}{\beta^{\mathbb{R}}},\tag{69}$$

$$L\_{\mathfrak{c}}^{F} = \frac{(a\_2 + \beta\_2) + V\_{\mathfrak{c}}'(K)(\mathfrak{d}\_1 + \mathfrak{d}\_2)}{\beta^F}. \tag{70}$$

Substituting the results of Eqs. (69) and (70), obtain

$$\begin{split} \rho V\_{c}(K) &= \max\_{\{\tau\_{e}^{\theta}, L\_{e}^{\theta}\}} \left\{ (a\_{1} + \beta\_{1}) \left( \frac{(a\_{1} + \beta\_{1}) + V\_{c}(K)(\theta\_{1} + \theta\_{2})}{\rho^{\mathbf{g}}} \right) \\ &+ (a\_{2} + \beta\_{2}) \left( \frac{(a\_{2} + \beta\_{2}) + V\_{c}(K)(\theta\_{1} + \theta\_{2})}{\rho^{\mathbf{g}}} \right) - \frac{1}{2} \rho^{\mathbf{g}} \left( \frac{(a\_{1} + \beta\_{1}) + V\_{c}(K)(\theta\_{1} + \theta\_{2})}{\rho^{\mathbf{g}}} \right)^{2} \\ &- \frac{1}{2} \rho^{\mathbf{g}} \left( \frac{(a\_{2} + \beta\_{2}) + V\_{c}(K)(\theta\_{1} + \theta\_{2})}{\rho^{\mathbf{g}}} \right)^{2} - \frac{\partial V\_{c}(K)}{\partial K} \xi K \\ &+ \frac{\partial V\_{c}(K)}{\partial K} \left[ (\theta\_{1} + \theta\_{2}) \left( \frac{(a\_{1} + \beta\_{1}) + V\_{c}(K)(\theta\_{1} + \theta\_{2})}{\rho^{\mathbf{g}}} \right) \right] \end{split} \tag{71}$$

Hence, the solution of the HJB equation is an unary function with *K*, *Vc* ¼ *aK* þ *b*, where *a* and *b* are constant that need to be solved. This implies that

$$\overline{a} = \frac{(\Gamma + \delta)}{(\rho + \xi)}.\tag{72}$$

$$\overline{b} = \left( (a\_1 + \beta\_1) - \frac{((a\_1 + \beta\_1) + \overline{a}(\theta\_1 + \theta\_2))}{2} + (\theta\_1 + \theta\_2)\overline{a} \right) \left( \frac{(a\_1 + \beta\_1) + \overline{a}(\theta\_1 + \theta\_2)}{\beta^{\mathbb{R}}} \right)$$

$$1 + \left( (a\_2 + \beta\_2) - \frac{((a\_2 + \beta\_2) + \overline{a}(\theta\_1 + \theta\_2))}{2} + (\theta\_1 + \theta\_2)\overline{a} \right) \left( \frac{(a\_2 + \beta\_2) + \overline{a}(\theta\_1 + \theta\_2)}{\beta^F} \right) > 0. \tag{73}$$

Substituting the results of Eqs. (72) and (73), into *Vc* ¼ *aK* þ *b*, will obtain the results.

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

**Proposition 10.** The limit of expectation and variance in cooperative game feedback equilibrium satisfy

$$E(K(t)) = \frac{\overline{\mu}\_1 + \overline{\mu}\_2}{\xi} + e^{-\xi t} \left(\tilde{K}\_0 - \frac{\overline{\mu}\_1 + \overline{\mu}\_2}{\xi}\right), \qquad \lim\_{t \to \infty} E(K(t)) = \frac{\overline{\mu}\_1 + \overline{\mu}\_2}{\xi}.\tag{74}$$

$$D(K(t)) = \frac{\rho^2 \left[ (\overline{\mu}\_1 + \overline{\mu}\_2) - 2(\overline{\mu}\_1 + \overline{\mu}\_2 - \xi K\_0)e^{-\xi t} + (\overline{\mu}\_1 + \overline{\mu}\_2 - 2\xi K\_0)e^{-2\xi t} \right]}{2\xi^2} \tag{75}$$

$$\lim\_{t \to \infty} D(E(K(t))) = \frac{\rho^2(\overline{\mu}\_1 + \overline{\mu}\_2)}{2\xi^2},\tag{76}$$

where *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>α</sup>*1þ*β*<sup>1</sup> ð Þð Þþ *<sup>ρ</sup>*þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*1þ*ϑ*<sup>2</sup> ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup>* and *<sup>μ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*2þ*β*<sup>2</sup> ð Þð Þþ *<sup>ρ</sup>*þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*1þ*ϑ*<sup>2</sup> ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup> : Proof.* Poof of Proposition 10 is like the derivation of Propositions 6 and 8.

**4.1 The limit of expectation and variance**

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

*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>

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

*Carbon Capture*

**5. Cooperative game**

*libria are defined as*

following equation.

*J K*ð Þ¼ <sup>0</sup> max *LR <sup>c</sup>* , *<sup>L</sup><sup>R</sup>* f g*<sup>c</sup>* <sup>0</sup>

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

> � 1 2 *β<sup>R</sup> LR c* � �<sup>2</sup>

> > *<sup>∂</sup>Vc*ð Þ *<sup>K</sup>*

*<sup>∂</sup><sup>K</sup> <sup>ϑ</sup>*<sup>1</sup> *<sup>L</sup><sup>R</sup>*

þ

**74**

*<sup>c</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þð Þþ *<sup>ρ</sup>* <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup>

where *b*, is given in the proof.

*E* ð<sup>∞</sup> 0 *e*

*LR*

*non-cooperative game feedback equilibrium must satisfy*

**Proposition 8.** *The limit of expectation E K t* ð Þ ð Þ *and variance D K t* ð Þ ð Þ *in the Nash*

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

*<sup>β</sup>RβF*ð Þ *<sup>ρ</sup>*1þ*<sup>ξ</sup> and*

*<sup>β</sup>RβF*ð Þ *<sup>ρ</sup>*2þ*<sup>ξ</sup> :*

<sup>2</sup>*ξ*<sup>2</sup> (64)

<sup>2</sup>*ξ*<sup>2</sup> *:* (65)

*<sup>c</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þð Þþ *<sup>ρ</sup>* <sup>þ</sup> *<sup>ξ</sup>* ð Þ <sup>Γ</sup> <sup>þ</sup> *<sup>δ</sup>* ð Þ *<sup>ϑ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup>

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

ð Þ *<sup>ρ</sup>*þ*<sup>ξ</sup> <sup>K</sup>* <sup>þ</sup> *<sup>b</sup>:*

*<sup>c</sup>* þ ð Þ Γ þ *δ K*Þ

ð Þ *K* � (66)

(67)

(68)

*<sup>ξ</sup> :* (63)

*ξ* � �

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

where *<sup>μ</sup>*~<sup>1</sup> <sup>¼</sup> *θβF*½*α*1ð Þþ *<sup>ρ</sup>*1þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>* �þ*θβR*½ � *<sup>α</sup>*2ð Þþ *<sup>ρ</sup>*1þ*<sup>ξ</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>*

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

Under cooperative game paradigm, Player I and Player II will choose to collaborate/share their hybrid-enabling technology development knowledge while sharing the payoff function in order to maximize their total payoffs. As a result, hybrid-

**Proposition 9.** *If above conditions are satisfied, then the feedback cooperative equi-*

and the optimal cooperative payoff function under hybrid-enabling technology

*Proof.* The objective function (optimal sharing payoff function) satisfies the

*<sup>c</sup>* ðÞþ*<sup>t</sup> <sup>α</sup>*2*L<sup>F</sup>*

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

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

h i

Then the optimal revenue sharing function satisfies the following HJB equation

nh i

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

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

� *ξK*

þ 1 2 *∂*2 *Vc*ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

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

*<sup>c</sup>* þ ð Þ Γ þ *δ K*Þ

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

Poof of Proposition 8 is like the derivation of Proposition 6.

enabling technology can be improved through this effort as well.

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

on renewable sources and on fossil fuel, respectively. *Vc*ð Þ¼ *<sup>K</sup>* ð Þ <sup>Γ</sup>þ*<sup>δ</sup>*

�*ρ<sup>t</sup> α*1*L<sup>R</sup>*

*<sup>c</sup>* ðÞþ*<sup>t</sup> <sup>α</sup>*2*LF*

� h�

*α*1*LR*

� 1 2 *β<sup>F</sup> LF c* � �<sup>2</sup>

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

## **6. Comparative analysis of equilibrium results**

**Proposition 11.** *The outcome of the game depends on the parameters of the game and the type of the equilibrium one considers.*

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

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

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

and

*extra cost, such that V*ð Þ II

*Proof.* When 0 ≤*θ* ≤ <sup>2</sup>

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

and

**77**

<sup>Δ</sup>*V*ð Þ II ð Þ¼ *<sup>K</sup> <sup>V</sup>*ð Þ II

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

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

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

**Proposition 12.** *For any K* ≥ *0, under the condition that Player II pay an extra cost*

*for sharing hybrid-enabling technology. Then the optimal sharing payoff of hybridenabling technology of Player I reaches higher than the optimal sharing payoff under the*

*Similarly, the optimal sharing payoff of hybrid-enabling technology of Player II reaches higher than the optimal sharing payoff under the condition that Player II do not provide*

*condition that player II does not provide extra cost. This implies that V*ð Þ<sup>I</sup>

*<sup>N</sup>* ð Þ *K :*

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

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

ð Þ *ρ*<sup>1</sup> þ *ξ*

In this chapter a complete study of an energy market by considering a Bertrand duopoly game with two power plants using endogenous hybrid-enabling technology was presented. Numerous game paradigms were articulated and defined including Stackelberg, Nash non-cooperative and cooperative games as well as their relevant equilibria via a feedback control strategy. Mathematically, the necessary conditions under which a power plant will move from taking part in a non-cooperative Nash game to participate as a leader in a Stackelberg game was derived. In doing so, this model allowed us to quantify the optimal level of subsidy for sharing the hybrid-enabling technology. We then adopted the concept of limit expectation and variance of the improvement degree to identify the influence of random factors of external environment and limitations of the decision maker. It is found that for a given level of payoff distribution the Stackelberg equilibria with technological enhancements, the knowledge sharing paradigm dominates the Nash equilibria. In both Stackelberg and Nash games, optimal technological enhancements for power plants were found to be proportional to the government subsidy, but the variance improvement degree of the Stackelberg game differed to the results of the

Nash non-cooperative game due to the influence of random factors.

3, establish that

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

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

*b*<sup>1</sup> >0,

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

*b*<sup>2</sup> >0*:*

<sup>¼</sup> *<sup>b</sup>*<sup>1</sup> � ^

<sup>¼</sup> *<sup>b</sup>*<sup>2</sup> � ^

**7. Concluding remarks**

*β*~*R*

*β*~*F*

ð Þ *ρ*<sup>2</sup> þ *ξ*

ð Þ *ρ*<sup>2</sup> þ *ξ*

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

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

ð Þ *ρ*<sup>1</sup> þ *ξ*

*<sup>K</sup>* <sup>þ</sup> ^ *b*1

>0, (83)

>0*:* (84)

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

ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup> <sup>K</sup>* <sup>þ</sup> ^

*<sup>N</sup>* ð Þ *K .*

(85)

*b*2

(86)

*Proof.* (i) Player I, will participate in a Stackelberg game to share more hybridenabling technology under the condition that Player II pay much more extra cost for hybrid-enabling technology

$$\frac{a\_1(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi)+\theta\_1^R(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^R(\rho\_2+\xi)(\rho\_1+\xi)}\qquad(77)$$

$$-\frac{(1-\theta)(\beta\_1(\rho\_2+\xi)+(\Gamma+\delta))\theta\_2^R}{\tilde{\rho}^R(\rho\_2+\xi)}>0,$$

and

$$\frac{a\_2(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi)+\theta\_1^F(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^F(\rho\_2+\xi)(\rho\_1+\xi)}\qquad(78)$$

$$-\frac{(1-\theta)(\beta\_2(\rho\_2+\xi)+(\Gamma+\delta))\theta\_2^F}{\hat{\rho}^F(\rho\_2+\xi)} > 0.$$

(ii) Player I will prefer to participate in a cooperative game over a noncooperative game with Player II under the condition such that

$$\frac{(a\_1 + \rho\_1)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^{\mathbb{R}}} - \frac{\theta[a\_1(\rho\_1 + \xi) + (\Gamma + \delta)]}{\rho^{\mathbb{R}}(\rho\_1 + \xi)} > 0,\tag{79}$$

and

$$\frac{(a\_2 + \rho\_2)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^F} - \frac{\theta[a\_2(\rho\_1 + \xi) + (\Gamma + \delta)]}{\rho^F(\rho\_1 + \xi)} > 0. \tag{80}$$

(iii) The total payoff for Player I under a Stackelberg game exceeds the total payoff of Nash non-cooperative game with Player II under the condition such that

$$\frac{a\_1(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi)+\theta\_1^R(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^R(\rho\_2+\xi)(\rho\_1+\xi)}\qquad\text{(81)}$$

$$-\frac{(a\_1+\beta\_1)(\rho+\xi)+(\Gamma+\delta)(\theta\_1+\theta\_2)}{(\rho+\xi)\rho^R} > 0,$$

and

$$\frac{a\_2(2-\theta)(\rho\_2+\xi)(\rho\_1+\xi)+\theta\_1^F(\Gamma+\delta)((2-2\theta)(\rho\_1+\xi)+\theta(\rho\_2+\xi))}{2\rho^F(\rho\_2+\xi)(\rho\_1+\xi)}\qquad\text{(82)}$$

$$-\frac{(a\_2+\beta\_2)(\rho+\xi)+(\Gamma+\delta)(\theta\_1+\theta\_2)}{(\rho+\xi)\rho^F} > 0.$$

(iv)*.* Player II will prefer to participate in a cooperative game over a noncooperative game with Player I under the condition such that

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

$$\frac{(a\_1 + \rho\_1)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^{\mathbb{R}}} - \frac{(\mathbb{1} - \theta)[\beta\_1(\rho\_2 + \xi) + (\Gamma + \delta)]}{\tilde{\beta}^{\mathbb{R}}(\rho\_2 + \xi)} > 0,\tag{83}$$

and

**6. Comparative analysis of equilibrium results**

*the type of the equilibrium one considers.*

*<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>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>*

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

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

hybrid-enabling technology

and

*Carbon Capture*

and

and

**76**

**Proposition 11.** *The outcome of the game depends on the parameters of the game and*

*Proof.* (i) Player I, will participate in a Stackelberg game to share more hybridenabling technology under the condition that Player II pay much more extra cost for

� 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 <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>*

(ii) Player I will prefer to participate in a cooperative game over a noncooperative game with Player II under the condition such that

ð Þ *ρ*<sup>2</sup> þ *ξ*

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

(iii) The total payoff for Player I under a Stackelberg game exceeds the total payoff of Nash non-cooperative game with Player II under the condition such that

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

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

(iv)*.* Player II will prefer to participate in a cooperative game over a noncooperative game with Player I under the condition such that

*β*~*F*

ð Þ *ρ*<sup>2</sup> þ *ξ*

*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>*βR*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>* (77)

<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>* (78)

2

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

*<sup>β</sup><sup>F</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* <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>* (81)

<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>* (82)

ð Þ *<sup>ρ</sup>* <sup>þ</sup> *<sup>ξ</sup> <sup>β</sup><sup>R</sup>* <sup>&</sup>gt;0,

ð Þ *<sup>ρ</sup>* <sup>þ</sup> *<sup>ξ</sup> <sup>β</sup><sup>F</sup>* <sup>&</sup>gt;0*:*

>0*:*

> 0, (79)

>0*:* (80)

2

>0,

$$\frac{(a\_2 + \rho\_2)(\rho + \xi) + (\Gamma + \delta)(\theta\_1 + \theta\_2)}{(\rho + \xi)\rho^F} - \frac{(\mathbf{1} - \theta)[\beta\_2(\rho\_2 + \xi) + (\Gamma + \delta)]}{\tilde{\rho}^F(\rho\_2 + \xi)} > 0. \tag{84}$$

**Proposition 12.** *For any K* ≥ *0, under the condition that Player II pay an extra cost for sharing hybrid-enabling technology. Then the optimal sharing payoff of hybridenabling technology of Player I reaches higher than the optimal sharing payoff under the condition that player II does not provide extra cost. This implies that V*ð Þ<sup>I</sup> *<sup>S</sup>* ð Þ *<sup>K</sup>* <sup>≥</sup>*V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K . Similarly, the optimal sharing payoff of hybrid-enabling technology of Player II reaches higher than the optimal sharing payoff under the condition that Player II do not provide extra cost, such that V*ð Þ II *<sup>S</sup>* ð Þ *<sup>K</sup>* <sup>≥</sup>*V*ð Þ II *<sup>N</sup>* ð Þ *K :*

*Proof.* When 0 ≤*θ* ≤ <sup>2</sup> 3, establish that

$$\begin{split} \Delta V^{(\text{l})}(K) &= V\_S^{(\text{l})}(K) - V\_N^{(\text{l})}(K) = \frac{\theta(\Gamma + \delta)}{(\rho\_1 + \xi)} K + b\_1 - \frac{\theta(\Gamma + \delta)}{(\rho\_1 + \xi)} K + \hat{b}\_1 \\\\ &= b\_1 - \hat{b}\_1 > 0, \end{split} \tag{85}$$

and

$$
\Delta V^{(\text{II})}(K) = V\_S^{(\text{II})}(K) - V\_N^{(\text{II})}(K) = \frac{(1-\theta)\left(\Gamma + \delta\_\circ\right)}{(\rho\_1 + \xi)} K + b\_2 - \frac{(1-\theta)(\Gamma + \delta)}{(\rho\_1 + \xi)} K + \hat{b}\_2 \tag{86}
$$

$$
= b\_2 - \hat{b}\_2 > 0. \tag{86}
$$

### **7. Concluding remarks**

In this chapter a complete study of an energy market by considering a Bertrand duopoly game with two power plants using endogenous hybrid-enabling technology was presented. Numerous game paradigms were articulated and defined including Stackelberg, Nash non-cooperative and cooperative games as well as their relevant equilibria via a feedback control strategy. Mathematically, the necessary conditions under which a power plant will move from taking part in a non-cooperative Nash game to participate as a leader in a Stackelberg game was derived. In doing so, this model allowed us to quantify the optimal level of subsidy for sharing the hybrid-enabling technology. We then adopted the concept of limit expectation and variance of the improvement degree to identify the influence of random factors of external environment and limitations of the decision maker. It is found that for a given level of payoff distribution the Stackelberg equilibria with technological enhancements, the knowledge sharing paradigm dominates the Nash equilibria. In both Stackelberg and Nash games, optimal technological enhancements for power plants were found to be proportional to the government subsidy, but the variance improvement degree of the Stackelberg game differed to the results of the Nash non-cooperative game due to the influence of random factors.

Furthermore, we have shown that due to optimal price reaction functions being upward sloping, the subsidy level plays a decisive role on the payoff function of power plant II as the leader in a Stackelberg game. This model shows that cost reducing R&D investments with efficient hybrid-enabling technology innovation/s strengthens one's competitive bargaining position via the level of subsidy for Power Plant I to become a follower in the Stackelberg game. By analyzing this stochastic differential game model, we capture the government subsidy incentive as well as the subsidy that the leader (Power Plant II) pays the follower (Power Plant I) to share hybrid-enabling technology.

*L*~*R*

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

*L*~*F*

*b*1, *V*ð Þ II

ð Þ *<sup>ρ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>* , ^

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

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

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

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

*V*ð Þ<sup>I</sup>

*<sup>N</sup>* <sup>¼</sup> *<sup>a</sup>*^1*<sup>K</sup>* <sup>þ</sup> ^

where

and

**79**

*<sup>N</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>N</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>b</sup>*<sup>1</sup> <sup>¼</sup> <sup>Φ</sup>^ <sup>1</sup> *ρ*1

1

1

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

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

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

*β*~*R* !

! *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

1

1

2

2

Substituting *a*1&*a*<sup>2</sup> into Eqs. (A.2), (A.3), (A.5) and (A.6) and simplifying, obtain the optima effort level of technological improvements. By substituting the above results into Eqs. (A.1) and (A.4) we obtain the optimal sharing payoff

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

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

<sup>2</sup> <sup>þ</sup> *<sup>ϑ</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>*

� � *<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>

� �

� �

� �

� �

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

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

functions, for power plant (I) and power plant (II).

*<sup>N</sup>* <sup>¼</sup> *<sup>a</sup>*^2*<sup>K</sup>* <sup>þ</sup> ^

*<sup>N</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 2 *<sup>β</sup>*~*<sup>R</sup>* <sup>¼</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>*

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

*<sup>N</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 2 *<sup>β</sup>*~*<sup>F</sup>* <sup>¼</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>*

Hence, the solution of the HJB equation is an unary function with *K*, such that

, *<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>*

*b*2. Hence, finally *a*^1, ^

2

2 *<sup>β</sup>*~*<sup>F</sup> :* (A.6)

*b*<sup>2</sup> as:

*<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>Φ</sup>^ <sup>2</sup> *ρ*2

*b*1, *a*^2, and^

ð Þ *<sup>ρ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>* , ^

1

1

<sup>2</sup> *a*<sup>1</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*~*R* !

*β*~*F* !

*β*~*F* !

2

2

>0*:*

*βR* � �

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

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

*<sup>β</sup>*~*<sup>R</sup>* , (A.5)

, (A.7)

2

> 0*:*

(A.8)

(A.9)

The proposed quantitative framework could assist policymakers when determining the appropriate R&D incentives for the development of hybrid-enabling technology within the energy market to achieve desired short and long-term environmental objectives with respect to budget limitations and environmental considerations.
