Open access peer-reviewed chapter - ONLINE FIRST

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

By Ryle S. Perera

Submitted: July 13th 2020Reviewed: September 14th 2020Published: October 21st 2020

DOI: 10.5772/intechopen.94016

Downloaded: 58

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. [12] 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. [13] 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 [14], 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 [15], 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, [16] 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.

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 public-private 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 [31], the Cournot equilibria in an oligopolistic electricity market subject to a linear demand function is examined. In [18], 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 [10], the equilibrium strategies in random-demand procurement auctions in the electricity market is obtained and presented a method for explicit calculation of the bid strategies is presented. [19] 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 [4, 6, 22, 29, 34], the role of government as a leading player who intervenes in competitive electricity markets to promote environmental protection is evaluated. In [26], the role of government, when managing environmental sustainability in a complete electricity market in a Stackelberg game paradigm is examined. In [27], 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 F(e.g., natural gas, petroleum or coal) and renewable sources 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 pijijFR, (where pijis the price of energy i, given that the opponent player’s 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 Πij,ijFRsubject 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 Kt, 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:

  1. 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 impacts of market power on prices.

  2. 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 identify the influence of random factors.

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

  4. 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 duopoly.

  5. we reveal that for a given level of payoff distribution the Stackelberg equilibria under endogenous hybrid-technology innovation and the sharing paradigm dominate the Nash equilibria.

  6. 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 results of the Nash game.

  7. 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 (endogenous) capacities was used.

  8. our model shows that robust cost-reducing R&D investments with effective hybrid-enabling technology innovation strengthens an innovator’s competitive position and the Stackelberg structure emerges as an equilibrium outcome, allowing each power plant to optimally use energy sources to produce electricity while maximizing their payoffs.

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 Fand renewable sources Rto 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 [11, 20, 21, 28, 34] to analyzed environmental issues. Especially [7] 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.

2. Model setup

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 [17] 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 GHG-emission levels to the maximum accepted quantitative level, η¯F, 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 one-shot non-zero-sum game, represents the benchmark game of the population. If pijpjiis 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 [32].

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

Dij=aijβijpij+τi+γijpji+τj,i,jFRE1

where Dijis the demand function for the power plants employing the energy source iFRagainst the power plant which use the energy source jFR.τiis the tariff imposed by government subject to the power source i. For example government impose a tariff-rate quota (TRQs) τF, for fossil fuels (F) and a feed-in-tariff (FITs) τR, for renewable sources (R). pijis 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>0and γij>0, are independent constants that captures the demand sensitivity of a power plant subject to its own price βijand 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.

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

Πij=pijCiviDijFi=pijCiviaijβijpij+τi+γijpji+τjFi,E2

where i,jFRand Ci>0is 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>0is the initial setup cost of the power plants when using the energy source i). We also assume that pijCivi>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 underτFτR, is given aspij=Λij+Ci+vi.

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

Πijpij=aij2βijpijCivi+γijpjiCjvj+γijτj+Cj+vjβijτi+Ci+vi=0.E3

Defining Λij=pijCiviand using Λijand Λji, rewrite the first order conditions as:

2βijΛijγijΛji=aij+γijτj+Cj+vjβijτi+Ci+vi,2βjiΛjiγjiΛij=aji+γjiτi+Ci+viβjiτj+Cj+vj.E4

Simultaneously solving Eq. (4), obtain

Λij=2βjiaij+γijaji+βjiγijτj+Cj+vj+γijγji2βjiβijτi+Ci+vi4βjiβijγijγjiE5

such that βjiβijγijγji4.pijand pjiare obtained via pij=Λij+Ci+viand are the optimum prices if the profit functions are concave on pijand on pji. Then via the second order conditions, obtain the maximum point in the set as: 2Πijpij2=2βij<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 underτFτR,can be obtained as

Dij=βijΛij=βijθij+ωijτi+χijτj,Πij=βijΛij2Fi=βijθij+ωijτi+χijτj2Fi,E6

where

θij=2βjiaij+γijaji+βjiγijCj+vj+γijγji2βjiβijCi+vi4βjiβijγijγji,E7
ωij=γijγji2βjiβij4βjiβijγijγjiandχij=βjiγij4βjiβijγijγjiE8

Proof. Obtain the results by substituting Λijfrom 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=pji=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.

Power Plant II
Production StrategyFossil FuelRenewable Sources
Power Plant IFossil FuelΠFFΠFFΠFRΠRF
Renewable SourcesΠRFΠFRΠRRΠRR

Table 1.

Bi matrix for two power plants by different energy sources.

Note 1. Power and the payoff are measured on a utility scale consistent with the power plant’s preference ranking. Furthermore, [8, 35, 36] have applied symmetric two-person bi-matrix game in random contest to study evolutionary stable games.

Then via equation Πij=βijΛij2Fi=βijθij+ωijτi+χijτj2Fi, in Proposition 1, and the payoff matrix of the power plant I is given by:

A=a11a12a21a22=ΠF,FΠF,RΠR,FΠR,R=βF,FΛF,F2FFβF,RΛF,R2FFβR,FΛR,F2FRβR,RΛR,R2FR.E9

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

A=a11a21a12a22=ΠF,FΠR,FΠF,RΠR,R=βF,FΛF,F2FFβR,FΛR,F2FRβF,RΛF,R2FFβR,RΛR,R2FR.E10

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

ΠR,RΠF,RΠF,FΠR,FΠFR+ΠR,RΠR,RΠF,RΠF,FΠF,RΠR,F+ΠR,R.E11

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

  1. The probability that player I choosing row 1 is x and the probability that player I choosing row 2 is 1-x.

  2. The probability that player II choosing row 1 is y and the probability that player II choosing row 2 is 1-y.

Then the value of the game for Player I is

v1xy=xyΠF,F+x1yΠF,R+1xyΠR,F+1x1yΠR,R
=ΠF,FΠF,RΠR,F+ΠR,Ry+ΠF,RΠR,Rx+ΠR,FΠR,Ry+ΠR,R,E12

and the value of the game for Player II is

v2xy=xyΠF,F+x1yΠR,F+1xyΠF,R+1x1yΠR,R
=ΠF,FΠR,FΠF,R+ΠR,Rx+ΠF,RΠR,Ry+ΠR,FRΠR,Rx+ΠR,R.E13

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

v1=ΠF,FΠF,RΠR,F+ΠR,Ry+ΠF,RΠR,R=0.E14

Otherwise Player I can change x slightly and do better.

Similarly, for 0<y<1,

v2=ΠF,FΠR,FΠFR+ΠR,Rx+ΠF,RΠR,R=0.E15

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

ΠR,RΠF,RΠF,FΠR,FΠFR+ΠR,RΠR,RΠR,FΠF,FΠF,RΠR,F+ΠR,R.E16

Remark 1. Since the power plants plays a symmetric two person bimatrix game G, having two pure strategies ΠF,FΠR,F,ΠR,RΠFR, 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, [24, 25].

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 hybrid-enabling 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 [1, 2, 5]. 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 LRtdenotes the R&D effort level of technological improvements on renewable sources at time t, and LFtdenotes the R&D effort level of technological improvements on fossil fuel at time t, of Player I. L˜Rtdenotes the R&D effort level of technological improvements on renewable sources at time t, and L˜Ftdenotes 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 hybrid-enabling technology (Player I) and inferior hybrid-enabling technology (Player II) is denoted as CItand CIIt,which are the quadratic functions of the effect level of Player I and Player II at time t, respectively. Consider

CILRtLFtt=12βRtLRt2+βFtLFt2,E17

and

CIIL˜RtL˜Ftt=12β˜RtL˜Rt2+β˜FtL˜Ft2,E18

where 0<βRtβFtβ˜Rtβ˜Ft1and lower the βRtβFtβ˜Rtβ˜Ft, more effective is the technological development.

Let Ktdenote 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):

dKt=ϑ1tLRtLFt+ϑ2tL˜RtL˜FtξKtdt+φKdWtK0=K0>0.E19

ξ01, is the attenuation coefficient of hybrid-enabling technology. Let ϑ1t=ϑ1Rt+ϑ1Ftand ϑ2t=ϑ2Rt+ϑ2Ftdenote the influence of the effort level of hybrid-enabling technology sharing on collaboration innovation between Player I and Player II, at time t. Wtis a standard Brownian motion and φKtrandom interference factor on hybrid-enabling technology.

Let Πtdenotes the total profit under the hybrid-enabling technology system at time t. Let α1tα2tand β1tβ2tdenote 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:

Πt=α1tLRt+α2tLFt+β1tL˜Rt+β2tL˜Ft+Γ+δKt,E20

where

α1t=ΠR,RtΠF,FtΠR,FtΠF,Rt+ΠR,Rt,E21
α2t=ΠF,RtΠF,FtΠR,FtΠF,Rt+ΠR,Rt,Γ=ΓI+ΓII,δ=δI+δII,andE22
β1t=ΠR,RtΠF,FtΠR,FtΠF,Rt+ΠR,Rt,E23
β2t=ΠR,FtΠF,FtΠR,FtΠF,Rt+ΠR,Rt.E24

Γis the influence of the hybrid-enabling technology innovation on total revenue δ01; δis 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 Dijpijpji>0, given by Eq. (4), is twice continuously differentiable and

Dijpijpjipij=βij<0,andDijpijpjipji=γij>0pijpjiPI×PII.E25

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). [9] 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 Πij, given by Eq. (4):

Dijpijpjipji+pijCivi2DijpijpjipjipijEKt>0,E26

where Eis 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, [23]. This is a very intuitive and general condition, though clearly not a universal one. It is satisfied in particular if 2Dijpijpjipjipij>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 θtis the payoff distribution coefficient of player I at time t and θt01. 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 ωt=ω1tω2t, 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

JIK0=maxLSRLSF0E0eρ1tθtα1tLRt+α2tLFt+β1tL˜Rt+β2tL˜Ft+Γ+δKt12βRt1ω1LRt212βFt1ω2LFt2dt,E27

and

JIIK0=maxL˜SRL˜SFωt0E0eρ2t1θtα1tLRt+α2tLFt+β1tL˜Rt+β2tL˜Ft+Γ+δKt12β˜RtLRt212β˜FtL˜Ft212ω1βRtLRt212ω2βFtLFt2dt,E28

where ρ1and ρ2are the discount rates of Player I and Player II, respectively. In this feedback control strategy LSRt0, LSFt0,L S R (t) ≥ 0, L˜SRt0and L˜SFt0,are the control variables and ωt=ω1tω2t01. Kt>0is 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, [30].

3. A Stackelberg game under heterogeneous technology

Theory of strong Stackelberg reasoning is an improved version of an earlier theory [3], which provides an explanation of coordination in all dyadic (two-player) 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 counter-strategies 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 [33] and used to rationalize their solution of strictly competitive games.

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:

LSR=α12θρ2+ξρ1+ξ+ϑ1RΓ+δ22θρ1+ξ+θρ2+ξ2βRρ2+ξρ1+ξ,E29
LSF=α22θρ2+ξρ1+ξ+ϑ1FΓ+δ22θρ1+ξ+θρ2+ξ2βFρ2+ξρ1+ξ,E30
L˜SR=1θβ1ρ2+ξ+Γ+δϑ2Rβ˜Rρ2+ξ,E31
L˜SF=1θβ2ρ2+ξ+Γ+δϑ2Fβ˜Fρ2+ξ.E32

where LSR,LSFare the optimal effort level of hybrid-enabling technological improvements shared on renewable sources and fossil fuel at time t by Player I, respectively. L˜SR,L˜SFare 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

ω1=α123θ+ϑ1R2a2a1α12θ+ϑ1R2a2+a1,0θ230.otherwiseE33

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

ω2=α223θ+ϑ1F2a2a1α22θ+ϑ1F2a2+a1,0θ230.otherwiseE34

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

VSIK=θΓ+δ)ρ1+ξK+b1,VSIIK=1θΓ+δρ2+ξK+b2,E35

where a1, a2, b1 and b2 are given in the proof.

Proof. We define the optimal revenue functions for Player I and Player II under hybrid-enabling technology as VSIKand VSIIK, respectively, which are continuously differentiable. Applying HJB equation to VSIK, for Player I, we obtain

ρ1VSIK=maxLSRLSF0θα1LSRt+α2LSF+β1L˜SR+β2L˜SF+Γ+δK
12βR1ω1LSR212βF1ω2LSF2
+VSIKKϑ1LSRLSF+ϑ2L˜SRL˜SFξK+122VSIKK2φ2K.E36

Via the first order conditions, we obtain the optimal values LSRLSFfor Player I as:

LSR=θα1+VSIKϑ1RβR1ω1,E37
LSF=θα2+VSIKϑ1FβF1ω2,E38

where VSIKKVSIK.The optimal sharing revenue function, VSIIK, for Player II and the associated HJB equation is

ρ2VSIIK=maxL˜SRL˜SF01θα1LSR+α2LSF+β1L˜SR+β2L˜SF+Γ+δK
12β˜RLSRt212β˜FL˜SF212ω1βRLSR212ω2βFLSF2
+VSIIKKϑ1LSRLSF+ϑ2L˜SRL˜SFξK+122VSIIKK2φ2K.E39

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

ρ2VSIIK=maxL˜SRL˜SF01θα1θα1+VSIKϑ1RβR1ω1+α2θα2+VSIKϑ1FβF1ω2+β1L˜SR+β2L˜SF+Γ+δK12β˜RL˜SRt212β˜FL˜SF212ω1βRθα1+VSIKϑ1RβR1ω1212ω2βFθα2+VSIKϑ1FβF1ω22+VSIIKKϑ1Rθα1+VSIKϑ1RβR1ω1+ϑ1Fθα2+VSIKϑ1FβF1ω2+ϑ2L˜SRL˜SFξK+122VSIIKK2φ2K.E40

Via the first order conditions of (Eq. (40)), we obtain the optimal values L˜RL˜Ffor Player II as:

L˜SR=1θβ1+VSIIKϑ2Rβ˜R,E41
L˜SF=1θβ2+VSIIKϑ2Fβ˜F.E42

And the optimal value for ω1ω2

ω1=α123θ+ϑ1R2VSIIKVSIKα12θ+ϑ1R2VSIIK+VSIK,E43

and

ω2=α223θ+ϑ1F2VSIIKVSIKα22θ+ϑ1F2VSIIK+VSIK.E44

Hence, the solution of the HJB equation is an unary function with K (K as the independent variable), we define VSI=a1K+b1and VSII=a2K+b2, where a1,b1,a2,andb2are constants that need to be solved. Simplifying (Eq. (39)), obtain:

ρ1VSIK=θα1θα1+a1ϑ1RβR1ω1+α2θα2+a1ϑ1FβF1ω2+β11θβ1+a2ϑ2Rβ˜RE45
+β21θβ2+a2ϑ2Fβ˜F+Γ+δK12βR1ω1θα1+a1ϑ1RβR1ω1212βF1ω2θα2+a1ϑ1FβF1ω22ξKa1+ϑ1Rθα1+a1ϑ1RβR1ω1+ϑ1Fθα2+a1ϑ1FβF1ω2a1+ϑ2R1θβ1+a2ϑ2Rβ˜R+ϑ2F1θβ2+a2ϑ2Fβ˜Fa1,E46

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

ρ2VSIIK=1θα1θα1+a1ϑ1RβR1ω1+α2θα2+a1ϑ1FβF1ω2+β11θβ1+a2ϑ2Rβ˜R+β21θβ2+a2ϑ2Fβ˜F+Γ+δK12β˜R1θβ1+a2ϑ2Rβ˜R212β˜F1θβ2+a2ϑ2Fβ˜F212ω1βRθα1+a1ϑ1RβR1ω1212ω2βFθα2+a1ϑ1FβF1ω22+ϑ1Rθα1+a1ϑ1RβR1ω1+ϑ1Fθα2+a1ϑ1FβF1ω2a2ξKa2+ϑ2R1θβ1+a2ϑ2Rβ˜R+ϑ2F1θβ2+a2ϑ2Fβ˜Fa2.E47

This implies that,

a1=θΓ+δρ1+ξ,b1=Φ1ρ1,a2=1θΓ+δρ2+ξ,b2=Φ2ρ2,E48

where

Φ1=α1θθα1+a1ϑ1R2+ϑ1Ra1θα1+a1ϑ1RβR1ω1+α2θθα2+a1ϑ1F2+ϑ1Fa1θα2+a1ϑ1FβF1ω2+β1θ+ϑ2Ra11θβ1+a2ϑ2Rβ˜R+β2θ+ϑ2Fa21θβ2+a2ϑ2Fβ˜F>0,E49

and

Φ2=1θα1ω1θα1+a1ϑ1R21ω1+ϑ1Ra2θα1+a1ϑ1RβR1ω1+1θα2ω2θα2+a1ϑ1F21ω2+ϑ1Fa2θα2+a1ϑ1FβF1ω2+1θβ11θβ1+a2ϑ2R2+ϑ2Ra21θβ1+a2ϑ2Rβ˜R+1θβ21θβ2+a2ϑ2F2+ϑ2Fa21θβ2+a2ϑ2Fβ˜F>0.E50

Substituting the results of a1and a2into 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 hybrid-enabling 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.

dKt=μ1+μ2ξKtdt+φKdWtK0=K0>0,E51

where

μ1=ϑ1α12θρ2+ξρ1+ξ+ϑ1RΓ+δ22θρ1+ξ+θρ2+ξ2βRρ2+ξρ1+ξ
+α22θρ2+ξρ1+ξ+ϑ1FΓ+δ22θρ1+ξ+θρ2+ξ2βFρ2+ξρ1+ξ,E52

and

μ2=ϑ21θβ1ρ2+ξ+Γ+δϑ2Rβ˜Rρ2+ξ+1θβ2ρ2+ξ+Γ+δϑ2Fβ˜Fρ2+ξ.E53

Proposition 6. The limit of expectationEKt, and varianceDKtin the Stackelberg game feedback equilibrium must satisfy

EKt=μ1+μ2ξ+eξtK˜0μ1+μ2ξ,limtEKt=μ1+μ2ξ.E54
DKt=φ2μ1+μ22μ1+μ2ξK˜0eξt+μ1+μ22ξK˜0e2ξt2ξ2E55
limtDEKt=φ2μ1+μ22ξ2,E56

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

dKt2=2μ1+μ2+φ2K2ξK2dt+2φKKdWtK02=K˜02>0.E57

Then EKtand EKt2can be defined as:

dEKt=μ1+μ2ξKtdtK0=K0>0.E58
dEKt2=2μ1+μ2+φ2KEK2ξEK2dtK02=K02>0,E59

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 hybrid-enabling technology sharing concept to maximize their profits.

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

LNR=θα1ρ1+ξ+Γ+δβRρ1+ξ,LNF=θα2ρ1+ξ+Γ+δβFρ1+ξ.E60
L˜NR=1θβ1ρ2+ξ+Γ+δβ˜Rρ2+ξ,L˜NF=1θβ2ρ2+ξ+Γ+δβ˜Fρ2+ξ,E61

where LNR,LNFare the optimal level of hybrid-enabling technological advantage on renewable sources and on fossil fuel at time t for Player I, respectively. L˜NR,L˜NFare 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

VNIK=θΓ+δ)ρ1+ξK+b̂1,VNIIK=1θΓ+δρ2+ξK+b̂2,E62

where b̂1and b̂2are given in the proof.

Proof. See Appendix A.

4.1 The limit of expectation and variance

Proposition 8. The limit of expectationEKtand varianceDKtin the Nash non-cooperative game feedback equilibrium must satisfy

EKt=μ̂1+μ̂2ξ+eξtK̂0μ̂1+μ̂2ξ,limtEKt=μ̂1+μ̂2ξ.E63
DKt=φ2μ˜1+μ˜22μ˜1+μ˜2ξK˜0eξt+μ˜1+μ˜22ξK˜0e2ξt2ξ2E64
limtDEKt=φ2μ˜1+μ˜22ξ2.E65

where μ˜1=θβFα1ρ1+ξ+Γ+δ+θβRα2ρ1+ξ+Γ+δβRβFρ1+ξandμ˜2=1θβFβ1ρ2+ξ+Γ+δ+1θβRβ2ρ2+ξ+Γ+δβRβFρ2+ξ.

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, hybrid-enabling technology can be improved through this effort as well.

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

LcR=α1+β1ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβR,LcF=α2+β2ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβF,E66

and the optimal cooperative payoff function under hybrid-enabling technology on renewable sources and on fossil fuel, respectively. VcK=Γ+δρ+ξK+b¯.

where b¯,is given in the proof.

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

JK0=maxLcRLcR0E0eρtα1LcRt+α2LcF+β1L˜cR+β2L˜cF+Γ+δKE67

Then the optimal revenue sharing function satisfies the following HJB equation

ρVcK=maxLcRLcF0α1LcRt+α2LcF+β1L˜cR+β2L˜cF+Γ+δK
12βRLcR212βFLcF2
+VcKKϑ1LcRLcF+ϑ2L˜cRL˜cFξK+122VcKK2φ2KE68

Via the first order conditions, now obtain the optimal values LcRLcFas:

LcR=α1+β1+VcKϑ1+ϑ2βR,E69
LcF=α2+β2+VcKϑ1+ϑ2βF.E70

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

ρVcK=maxLcRLcF0α1+β1α1+β1+VcKϑ1+ϑ2βR+α2+β2α2+β2+VcKϑ1+ϑ2βF12βRα1+β1+VcKϑ1+ϑ2βR212βFα2+β2+VcKϑ1+ϑ2βF2VcKKξK+VcKKϑ1+ϑ2α1+β1+VcKϑ1+ϑ2βRE71

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

a¯=Γ+δρ+ξ.E72
b¯=α1+β1α1+β1+a¯ϑ1+ϑ22+ϑ1+ϑ2a¯α1+β1+a¯ϑ1+ϑ2βR
+α2+β2α2+β2+a¯ϑ1+ϑ22+ϑ1+ϑ2a¯α2+β2+a¯ϑ1+ϑ2βF>0.E73

Substituting the results of Eqs. (72) and (73), into Vc=a¯K+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

EKt=μ¯1+μ¯2ξ+eξtK˜0μ¯1+μ¯2ξ,limtEKt=μ¯1+μ¯2ξ.E74
DKt=φ2μ¯1+μ¯22μ¯1+μ¯2ξK0eξt+μ¯1+μ¯22ξK0e2ξt2ξ2E75
limtDEKt=φ2μ¯1+μ¯22ξ2,E76

where μ¯1=α1+β1ρ+ξ+Γ+δϑ1+ϑ2ρ+ξand μ¯2=α2+β2ρ+ξ+Γ+δϑ1+ϑ2ρ+ξ.

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

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.

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

α12θρ2+ξρ1+ξ+ϑ1RΓ+δ22θρ1+ξ+θρ2+ξ2βRρ2+ξρ1+ξE77
1θβ1ρ2+ξ+Γ+δϑ2Rβ˜Rρ2+ξ>0,

and

α22θρ2+ξρ1+ξ+ϑ1FΓ+δ22θρ1+ξ+θρ2+ξ2βFρ2+ξρ1+ξE78
1θβ2ρ2+ξ+Γ+δϑ2Fβ˜Fρ2+ξ>0.

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

α1+β1ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβRθα1ρ1+ξ+Γ+δβRρ1+ξ>0,E79

and

α2+β2ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβFθα2ρ1+ξ+Γ+δβFρ1+ξ>0.E80

(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

α12θρ2+ξρ1+ξ+ϑ1RΓ+δ22θρ1+ξ+θρ2+ξ2βRρ2+ξρ1+ξE81
α1+β1ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβR>0,

and

α22θρ2+ξρ1+ξ+ϑ1FΓ+δ22θρ1+ξ+θρ2+ξ2βFρ2+ξρ1+ξE82
α2+β2ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβF>0.

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

α1+β1ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβR1θβ1ρ2+ξ+Γ+δβ˜Rρ2+ξ>0,E83

and

α2+β2ρ+ξ+Γ+δϑ1+ϑ2ρ+ξβF1θβ2ρ2+ξ+Γ+δβ˜Fρ2+ξ>0.E84

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 hybrid-enabling technology of Player I reaches higher than the optimal sharing payoff under the condition that player II does not provide extra cost. This implies thatVSIKVNIK. 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 thatVSIIKVNIIK.

Proof. When 0θ23, establish that

ΔVIK=VSIKVNIK=θΓ+δρ1+ξK+b1θΓ+δρ1+ξK+b̂1=b1b̂1>0,E85

and

ΔVIIK=VSIIKVNIIK=1θΓ+δ)ρ1+ξK+b21θΓ+δρ1+ξK+b̂2=b2b̂2>0.E86

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.

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.

Proof. The optimal profit function for power plant I satisfies the following HJB equation such that VNIK:

ρ1VNIK=maxLNRLNF0θα1LNRt+α2LNF+β1L˜NR+β2L˜NF+Γ+δK12βRLNR212βFLNF2+VNIKKϑ1LNRLNF+ϑ2L˜NRL˜NFξK+122VNIKK2φ2KEA.1

Via the first order conditions, first obtain the optimal values LNRLIRfor power plant I as:

LNR=θα1+VNIKϑ1RβR=θα1+a1ϑ1RβR,EA.2
LNF=θα2+VNIKϑ1FβF=θα2+a1ϑ1FβFEA.3

where VNIKKVNIK.The HJB for power plant (II), using VNIIKK=VNIIK,then obtain

ρ2VNIIK=maxL˜NRL˜NF1θα1LNRt+α2LNF+β1L˜NR+β2L˜NF+Γ+δK12β˜RL˜NR212β˜FL˜NF2+VNIKKϑ1LNRLNF+ϑ2L˜NRL˜NFξK+122VNIKK2φ2K.EA.4

Substituting Eqs. (A.2) and (A.3) results to Eq. (A.4) and via the first order conditions, we obtain the optimal values L˜NRL˜NFfor power plant II as:

L˜NR=1θβ1+VNIIKϑ2Rβ˜R=1θβ1+a2ϑ2Rβ˜R,EA.5
L˜NF=1θβ2+VNIIKϑ2Fβ˜F=1θβ2+a2ϑ2Fβ˜F.EA.6

Hence, the solution of the HJB equation is an unary function with K, such that VNI=â1K+b̂1, VNII=â2K+b̂2. Hence, finally â1,b̂1,â2,andb̂2as:

â1=θΓ+δρ1+ξ,b̂1=Φ̂1ρ1,â2=1θΓ+δρ2+ξ,b̂2=Φ̂2ρ2,EA.7

where

Φ̂1=α1θθα1+a1ϑ1R2+ϑ1Ra1θα1+a1ϑ1RβR+α2θθα2+a1ϑ1F2+ϑ1Fa1θα2+a1ϑ1FβF1ω2+β1θ+ϑ2Ra11θβ1+a2ϑ2Rβ˜R+β2θ+ϑ2Fa11θβ2+a2ϑ2Fβ˜F>0.EA.8

and

Φ̂2=1θα1+ϑ1Ra2θα1+a1ϑ1RβR1ω1+1θα2+ϑ1Fa2θα2+a1ϑ1FβF1ω2+1θβ11θβ1+a2ϑ2R2+ϑ2Ra21θβ1+a2ϑ2Rβ˜R+1θβ21θβ2+a2ϑ2F2+ϑ2Fa21θβ2+a2ϑ2Fβ˜F>0.EA.9

Substituting a1&a2into 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 functions, for power plant (I) and power plant (II).

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Ryle S. Perera (October 21st 2020). Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources [Online First], IntechOpen, DOI: 10.5772/intechopen.94016. Available from:

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