**Abstract**

In this chapter, we quantify an optimal level of subsidy for the sharing of hybrid-enabling technology innovation in an energy market while examining its Bertrand-Nash equilibrium. We formulate this as a Stochastic Differential Game (SDG) and analyze the stability of the Stuckenberg, Nash and cooperative equilibria via a feedback control strategy. We then adopt limit expectation and variance of the improvement degree to identify the influence of the external environment on the decision maker. We show that the game depends on its parameters and the equilibria chosen. Ultimately, our use of short-run price competition characterized by strategic supplies for renewable and fossil resources provides a more robust model than that presented by Bertrand-Edgworth with endogenous capacity. As a result, we highlight that R&D investments in hybrid-enabling technology can ensure immediate reliability and affordability within energy production and implementation of policy instruments.

**Keywords:** Bertrand duopoly game, cooperative game, hybrid-enabling technology, Nash non-cooperative game, Stackelberg game, stochastic differential game

## **1. Introduction**

In recent years, many researchers have developed models to discuss the importance of lowering carbon emissions and its potential impact on society by examining economic growth, international trade, and health benefits. Khan et al. [1] examined the relationship between green logistics indices, economic, environmental, and social factors through the perspective of Asian emerging economies. By adopting a Fully Modified OLS (FMOLS) Model and Dynamic OLS (DOLS) they claimed that logistics operations, particularly the efficiency of customs clearance processes, quality of logistics services and trade and transport-related infrastructure are positively and significantly correlated with per capita income, manufacturing value added and trade openness, whereas greater logistics operations are negatively associated with social and environmental problems including, climate change, global warming, carbon emissions, and poisoning atmosphere. Khan et al. [2] examined the potential relationship between public health expenditures, logistics

performance indices, renewable energy, and ecological sustainability in members of the Association of Southeast Asian Nations by applying the structural equation modeling approach. They showed that the use of renewable energy in logistics operations will improve environmental and economic performance to reduce emissions, whereas environmental performance is negatively correlated with public health expenditures, indicating that greater environmental sustainability can improve human health and economic growth. In [3], economic growth and environmental sustainability in the South Asian Association for Regional Cooperation using the data from the South Asian Association for Regional Cooperation (SAARC) member countries from 2005 to 2017 was examined. Adopting the panel autoregressive distributed lag technique to examine the hypotheses, they find that environmental sustainability is strongly and positively associated with national scale-level green practices, including renewable energy, regulatory pressure, eco-friendly policies, and the sustainable use of natural resources. In [4], the consumption of renewable energy with international trade and environmental quality in Nordic countries from 2001 to 2018 is investigated. Their findings concluded that renewable energy is strongly and positively associated with international trade in Nordic countries. Furthermore, [5] adopted multi-criteria-decision-making techniques to examine barriers in the sustainable supply chain management (SSCM) when firms are facing heavy pressure to adopt green practices in their supply chain (SC) operations to achieve better socio-environmental sustainability.

method for explicit calculation of the bid strategies is presented. [9] proposed a Nash bargaining game model to examine how governments can determine the taxes and subsidies in a competitive electricity market whilst achieving their environmental objectives. In [10–14], the role of government as a leading player who intervenes in competitive electricity markets to promote environmental protection is evaluated. In [15], the role of government, when managing environmental sustainability in a complete electricity market in a Stackelberg game paradigm is examined. In [16], a more robust trans-boundary industrial pollution reduction strategy for global emission collaborations is presented. The dynamics of each country's quantity of pollution is modeled as a Brownian motion with Jumps to capture the systematic jumps caused by surprise effects arising from policy uncertainties within the economy. However, a crucial limitation within many of these environmental policy models, is that technological change is incorporated as an exogenous variable and does not consider the role of endogenous hybrid-enabling technology or other technological breakthroughs, hence limiting the dynamics of

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

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

We quantify an optimal level of subsidy for the sharing of hybrid-enabling technology in an energy market under a Bertrand-duopoly game. We formulate a Stochastic Differential Game (SDG) to analyze the stability of the Stackelberg, Nash and cooperative equilibria via a feedback control strategy. We then adopt limit expectation and variance of the improvement degree to identify the influence of external environment limitations on the decision maker. We show that the game depends on its parameters and the equilibria chosen. We consider an electricity market composed of power plants I and II, with each one having the choice between fossil fuels ð Þ *F* (e.g., natural gas, petroleum or coal) and renewable sources ð Þ *R* (e.g., biomass, solar, wind, wave, geothermal or hydroelectric). Such hybrid power plants play a crucial ameliorating role in managing the long-standing problem of climate change and ensure immediate reliability and affordability of energy production,

In this model, we consider a Bertrand duopoly game for two power plants under

energy *j*) is determined as a Nash equilibrium of the game where each player wants to optimize his/her demand. We then search for the Nash equilibria, and the optimal proportions that maximizes the ð Þ Π *ij*, *ij*∈ f g *F*, *R* subject to the source type of energy that has been used. Once all these parameters have been fixed, the game becomes dynamic due to the evolution of a hybrid-enabling technology level *K t*ð Þ, prompted by Research and Developments (R&D) measures undertaken by each power plant. Hence, each player must fix a time-dependent effect level associated with this hybrid-enabling technology. In doing so, this study makes the following

, (where *pij* is the price of energy *<sup>i</sup>*, given that the opponent player'<sup>s</sup>

endogenous hybrid-enabling technology. In the first stage the matrix of prices

contributions to existing game theory/energy economics literature:

impacts of market power on prices.

identify the influence of random factors.

i. stochastic endogenous hybrid-enabling technology innovation is

introduced into a two-player stochastic differential game with random interference factors, which capture uncertain external environment factors and the internal limitations within the shared hybrid-enabling technology decision process. In doing so, we provide a framework to quantify the

ii. we show that both power plants invest in R&D measures and that the limit of expectation and variance of the improvement degree can be applied to

whilst reducing Greenhouse Gas (GHG) emissions.

these models.

*pij*

**59**

*ij*∈f g *F*,*R*

Around the world governments, businesses and individuals have committed to reducing carbon emission. As a result, the energy economy is highly exposed to these processes. As industries push for renewable energies, technology will need to step in to ensure reliability of the power supply. Therefore, there remains a need for exploiting the role of hybrid technology, its dynamics, limitations on the reduction of pollution levels and policy implementation within the wider carbon emissions debate. This is due to the vital role hybrid technology plays in energy production processes and the ability for the energy system to offer a better energy security. Development of such lower carbon emission policies has potential benefits to the environment and ecological sustainability to those economies. However, within many of these environmental policy models, technology is incorporated as an exogenous variable and limited attention is given to endogenous technology, other technological breakthroughs, potential government subsidies or collaborative innovations to integrate low carbon technology in environmental economics. Such interventions will promote the renewable energy sector to use natural resources and undertake publicprivate partnership investments to minimize dependence on fossil fuel derived energy.

To investigate the effects of hybrid-enabling technology when producing energy to meet consumption demand, we assume energy producing firms follow the Bertrand game paradigm. In the presence of government subsidy for the development and sustainability of renewable energy, tax on pollution created by energy producing firms will motivate them to undertake Research & Development (R&D) measures to improve hybrid enabling technologies to further reduce the level of carbon pollution. As a result, from an economic point of view it is an interesting question to examine the Bertrand-Nash equilibrium under such a dynamic environment. This chapter examines this concept via a Stochastic Differential game paradigm.

Many researchers have applied game theory to study carbon reduction behavior in electricity markets. In [6], the Cournot equilibria in an oligopolistic electricity market subject to a linear demand function is examined. In [7], the power suppliers bidding behavior is evaluated under the supply function equilibrium (SFE) paradigm, where the market power of an independent system operator (ISO) is modeled as a bi-level multi-objective problem. In [8], the equilibrium strategies in randomdemand procurement auctions in the electricity market is obtained and presented a

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

method for explicit calculation of the bid strategies is presented. [9] proposed a Nash bargaining game model to examine how governments can determine the taxes and subsidies in a competitive electricity market whilst achieving their environmental objectives. In [10–14], the role of government as a leading player who intervenes in competitive electricity markets to promote environmental protection is evaluated. In [15], the role of government, when managing environmental sustainability in a complete electricity market in a Stackelberg game paradigm is examined. In [16], a more robust trans-boundary industrial pollution reduction strategy for global emission collaborations is presented. The dynamics of each country's quantity of pollution is modeled as a Brownian motion with Jumps to capture the systematic jumps caused by surprise effects arising from policy uncertainties within the economy. However, a crucial limitation within many of these environmental policy models, is that technological change is incorporated as an exogenous variable and does not consider the role of endogenous hybrid-enabling technology or other technological breakthroughs, hence limiting the dynamics of these models.

We quantify an optimal level of subsidy for the sharing of hybrid-enabling technology in an energy market under a Bertrand-duopoly game. We formulate a Stochastic Differential Game (SDG) to analyze the stability of the Stackelberg, Nash and cooperative equilibria via a feedback control strategy. We then adopt limit expectation and variance of the improvement degree to identify the influence of external environment limitations on the decision maker. We show that the game depends on its parameters and the equilibria chosen. We consider an electricity market composed of power plants I and II, with each one having the choice between fossil fuels ð Þ *F* (e.g., natural gas, petroleum or coal) and renewable sources ð Þ *R* (e.g., biomass, solar, wind, wave, geothermal or hydroelectric). Such hybrid power plants play a crucial ameliorating role in managing the long-standing problem of climate change and ensure immediate reliability and affordability of energy production, whilst reducing Greenhouse Gas (GHG) emissions.

In this model, we consider a Bertrand duopoly game for two power plants under endogenous hybrid-enabling technology. In the first stage the matrix of prices *pij ij*∈f g *F*,*R* , (where *pij* is the price of energy *<sup>i</sup>*, given that the opponent player'<sup>s</sup> energy *j*) is determined as a Nash equilibrium of the game where each player wants to optimize his/her demand. We then search for the Nash equilibria, and the optimal proportions that maximizes the ð Þ Π *ij*, *ij*∈ f g *F*, *R* subject to the source type of energy that has been used. Once all these parameters have been fixed, the game becomes dynamic due to the evolution of a hybrid-enabling technology level *K t*ð Þ, prompted by Research and Developments (R&D) measures undertaken by each power plant. Hence, each player must fix a time-dependent effect level associated with this hybrid-enabling technology. In doing so, this study makes the following contributions to existing game theory/energy economics literature:


performance indices, renewable energy, and ecological sustainability in members of the Association of Southeast Asian Nations by applying the structural equation modeling approach. They showed that the use of renewable energy in logistics operations will improve environmental and economic performance to reduce emissions, whereas environmental performance is negatively correlated with public health expenditures, indicating that greater environmental sustainability can improve human health and economic growth. In [3], economic growth and environmental sustainability in the South Asian Association for Regional Cooperation using the data from the South Asian Association for Regional Cooperation (SAARC)

member countries from 2005 to 2017 was examined. Adopting the panel

*Carbon Capture*

(SC) operations to achieve better socio-environmental sustainability.

to meet consumption demand, we assume energy producing firms follow the

examines this concept via a Stochastic Differential game paradigm.

**58**

Bertrand game paradigm. In the presence of government subsidy for the development and sustainability of renewable energy, tax on pollution created by energy producing firms will motivate them to undertake Research & Development (R&D) measures to improve hybrid enabling technologies to further reduce the level of carbon pollution. As a result, from an economic point of view it is an interesting question to examine the Bertrand-Nash equilibrium under such a dynamic environment. This chapter

Many researchers have applied game theory to study carbon reduction behavior in electricity markets. In [6], the Cournot equilibria in an oligopolistic electricity market subject to a linear demand function is examined. In [7], the power suppliers bidding behavior is evaluated under the supply function equilibrium (SFE) paradigm, where the market power of an independent system operator (ISO) is modeled as a bi-level multi-objective problem. In [8], the equilibrium strategies in randomdemand procurement auctions in the electricity market is obtained and presented a

autoregressive distributed lag technique to examine the hypotheses, they find that environmental sustainability is strongly and positively associated with national scale-level green practices, including renewable energy, regulatory pressure, eco-friendly policies, and the sustainable use of natural resources. In [4], the consumption of renewable energy with international trade and environmental quality in Nordic countries from 2001 to 2018 is investigated. Their findings concluded that renewable energy is strongly and positively associated with international trade in Nordic countries. Furthermore, [5] adopted multi-criteria-decision-making techniques to examine barriers in the sustainable supply chain management (SSCM) when firms are facing heavy pressure to adopt green practices in their supply chain

Around the world governments, businesses and individuals have committed to reducing carbon emission. As a result, the energy economy is highly exposed to these processes. As industries push for renewable energies, technology will need to step in to ensure reliability of the power supply. Therefore, there remains a need for exploiting the role of hybrid technology, its dynamics, limitations on the reduction of pollution levels and policy implementation within the wider carbon emissions debate. This is due to the vital role hybrid technology plays in energy production processes and the ability for the energy system to offer a better energy security. Development of such lower carbon emission policies has potential benefits to the environment and ecological sustainability to those economies. However, within many of these environmental policy models, technology is incorporated as an exogenous variable and limited attention is given to endogenous technology, other technological breakthroughs, potential government subsidies or collaborative innovations to integrate low carbon technology in environmental economics. Such interventions will promote the renewable energy sector to use natural resources and undertake publicprivate partnership investments to minimize dependence on fossil fuel derived energy. To investigate the effects of hybrid-enabling technology when producing energy

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

implementing a cooperative game we examine feedback equilibria and the limit of expectation and variance under hybrid-enabling technology. In Section 6, comparative analysis of equilibrium results are described. Section 7 concludes the study.

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

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

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

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

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

<sup>þ</sup> *<sup>γ</sup>ij pji* <sup>þ</sup> *<sup>τ</sup> <sup>j</sup>*

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

, *<sup>i</sup>*, *<sup>j</sup>*∈f g *<sup>F</sup>*, *<sup>R</sup>* (1)

Appendix at the end of the chapter contains proofs.

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

unilateral incentive to change his or her behavior).

electricity prices and the associated cost functions.

*Dij* ¼ *aij* � *βij pij* þ *τ<sup>i</sup>*

**2. Model setup**

quadratic as in [23].

**61**


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 spending is costly, and the presumption is that R&D spending is somehow connected to increased innovation, revenue growth and profits.

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 Hamilton-Jacobi-Bellman-Isaacs equations.

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 implementing a cooperative game we examine feedback equilibria and the limit of expectation and variance under hybrid-enabling technology. In Section 6, comparative analysis of equilibrium results are described. Section 7 concludes the study. Appendix at the end of the chapter contains proofs.
