Nash Cournot Research Articles

Nash–Cournot Equilibrium and Its Impact on Network Transmission Congestion

This paper evaluates the impact of congestion on transmission lines when the operation cost is minimized using economic dispatch (ED), comparing the results obtained with the Nash–Cournot Equilibrium (NCE). A methodology is developed for the optimal power flow solution through the NCE, considering the network topology (upper and lower generation limits, upper and lower limits of the transmission lines, and power balance) for a nine-node system without and considering two bilateral power transactions. The results show that the operation cost is higher when the NCE is implemented than ED. However, the problem of congestion in the transmission lines is reduced due to the equilibrium obtained in the power dispatch against minimizing the operation cost in the dispatch; the transmission lines with the most significant participation tend to become congested when additional bilateral transactions occur. Finally, the above is verified by obtaining the mean and median of the transmission line percentages used in the two simulations.

A note on the uniqueness of Nash–Cournot equilibria in an oligopolistic energy market with renewable generation and demand uncertainty

This work explores the uniqueness of Nash–Cournot equilibria in wholesale electricity markets with both thermal and renewable generation in the presence of demand uncertainty. Criteria guaranteeing a unique equilibrium are derived for an energy market where each firm owns a share of renewable generation. This work expands upon the uniqueness results proven by Lagerlöf (2006) to incorporate the presence of renewable generation into the market in the manner of Acemoglu et al. (2017). A numerical case study is used to illustrate the potential impacts of increasing renewable generation on equilibrium uniqueness and directions for future research are discussed.

Stochastic Optimal Strategies and Management of Electric Vehicles and Microgrids

This study combines the Nash–Cournot competition model and the stochastic optimization model to examine the impact of electric vehicle (EV) quantity fluctuations on microgrid operations, aiming to optimize energy usage in a competitive electricity market. Integrating distributed energy resources and bidirectional charging, microgrids offer a novel approach for energy optimization, aiding in renewable energy generation, peak demand management, and emission reduction. Empirical evidence highlights benefits in Taiwan’s electricity market and net-zero emissions target by 2050, with a case study demonstrating enhanced local renewable energy generation due to EVs and microgrid integration. As the number of EVs increases, electricity sales from microgrids decline, but electricity purchases remain stable. The degree of electricity liberalization also influences the supply and demand dynamics of the electricity market. Microgrids selling electricity only to the main grid increases total power consumption by 65.55 million MWh, reducing the market share of the state-owned utility (Taipower). Conversely, allowing retailers to purchase from microgrids increases total consumption by 30.87 million MWh with a slight market share decrease for Taipower. This study contributes to providing an adaptable and flexible general model for future studies to modify and expand based on different scenarios and variables to shape energy and environmental policies.

On the credibility of threats to avoid the deployment of solar geoengineering

AbstractWe analyze how geoengineering in the form of solar radiation management (SRM), associated with the potential of high collateral damages, affects the governance architecture of climate agreements. We investigate under which conditions signatories to a climate agreement can avoid the deployment of SRM and implement a climate agreement on mitigation. We show that a climate agreement with all countries can be stable with the threat to deploy SRM in case a country free-rides. The threat is deterrent if collateral damages are perceived to be sufficiently high (lower threshold), but only credible if those damages are not too high (upper threshold). SRM deployment is the only threat available to signatories if they choose mitigation levels simultaneously with non-signatories (Nash–Cournot scenario). However, if signatories choose mitigation levels before non-signatories (Stackelberg scenario), an additional punishment option arises. Then if collateral damages are sufficiently large, signatories can reduce their mitigation levels and impose a heavier burden on non-signatories that would find it profitable to avoid the deployment of SRM. We show that our results are robust in two analytical frameworks frequently employed in the game-theoretic analysis of international environmental agreements.

Tseng-type splitting projection algorithms for equilibrium problems in Hilbert spaces

In this work, based on the subgradient-type projection method considered by Santos and Scheimberg (Optimization, 2017), we propose some Tseng-type splitting algorithms for solving equilibrium problems whose bifunction is decomposed into the sum of two bifunctions in Hilbert spaces. We establish the weak and strong convergence of the proposed algorithm under standard assumptions. Numerical results in Nash–Cournot oligopolistic electricity market equilibrium model and comparisons with other related methods demonstrate the effectiveness of the proposed algorithm.

A timestamp-based projected gradient play for distributed Nash equilibrium seeking in monotone games

In this paper, a timestamp-based projected gradient play is proposed for Nash equilibrium seeking over communication networks for strongly monotone games. Its difference from the well-known consensus-based Nash equilibrium seeking method is that each player’s local estimate of other players’ actions is updated through a timestamp-based broadcasting protocol instead of typical consensus protocols. We show that the proposed timestamp-based projected gradient play is indeed a projected gradient play with time delay, which can be further transformed into a disturbed full-information projected gradient play. Hence, we prove its convergence to the Nash equilibrium in the case of diminishing step-size and to the ϵ-approximation Nash equilibrium in the case of fixed step-size. Both convergence results are established on jointly strongly connected networks. The major advantage of the proposed approach is that it is intrinsically flexible to communication delay, disturbance, and asynchronization. Furthermore, its convergence rate is empirically comparable to the full-information projected gradient play with the same step-sizes. Illustrative numerical simulations are presented for distributed Nash equilibrium seeking in both quadratic and non-quadratic Nash–Cournot games over different types of communication networks.

Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games

The goal in this paper is to approximate the Price of Stability (PoS) in stochastic Nash games using stochastic approximation (SA) schemes. PoS is amongst the most popular metrics in game theory and provides an avenue for estimating the efficiency of Nash games. In particular, knowing the value of PoS can help with designing efficient networked systems, including transportation networks and power market mechanisms. Motivated by the absence of efficient methods for computing the PoS, first we consider stochastic optimization problems with a nonsmooth and merely convex objective function and a merely monotone stochastic variational inequality (SVI) constraint. This problem appears in the numerator of the PoS ratio. We develop a randomized block-coordinate stochastic extra-(sub)gradient method where we employ a novel iterative penalization scheme to account for the mapping of the SVI in each of the two gradient updates of the algorithm. We obtain an iteration complexity of the order ϵ − 4 that appears to be best known result for this class of constrained stochastic optimization problems, where ϵ denotes an arbitrary bound on suitably defined infeasibility and suboptimality metrics. Second, we develop an SA-based scheme for approximating the PoS and derive lower and upper bounds on the approximation error. To validate the theoretical findings, we provide preliminary simulation results on a networked stochastic Nash Cournot competition.

Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms

This paper proposes a distributed algorithm for games with shared coupling constraints based on the variational approach and the proximal-point algorithm. The paper demonstrates the effectiveness of the proximal-point algorithm in distributed computing of generalized Nash equilibrium (GNE) problems using local data and communication with neighbors in any networked game. The algorithm achieves the goal of reflecting local decisions in the Nash–Cournot game under partial-decision information while maintaining the distributed nature and convergence of the algorithm.

Behavior of prosumers in Smart Grid: A comparison of Net Energy Metering and Billing Schemes, and Game Theory-based Local Electricity Market

Net energy metering (NEM) and net energy billing (NEB) are the most commonly used self-consumption schemes for prosumers in countries around the world. The concept of the local electricity market (LEM) is emerging as the potential successor of the self-consumption schemes, with several already implemented pilot projects. The comparative study of prosumers’ behavior in terms of optimal economic self-dispatching under NEM, NEB, and game-theory-based LEM energy policies is performed in this paper. The non-cooperative game-theoretic methodology is used for the calculation of Nash–Cournot equilibrium which is adopted as an LEM clearing mechanism. The prosumers’ daily energy patterns (generation, consumption, and storage) and payoffs are analyzed. Diversities of prosumers’ assets (distributed energy resources (DERs) and storage devices) and load profiles are considered. The distribution utility’s role is considered in terms of its own payoff for supplying the necessary energy to ensure the power balance equation. The numerical analysis is performed using the model of the benchmark test distribution network IEEE 33-node. The obtained results are discussed in terms of the correlation between prosumers’ behavior and the adopted energy policy. The strengths and weaknesses of the most common energy policies are highlighted from the prosumers’ perspective. The obtained results confirm that the transition from NEM to NEB self-consumption scheme could provide a better payoff to prosumers with dispatchable DERs under variable electricity price for excess energy produced. The residential prosumers equipped with SDs or dispatchable DERs were able to achieve the best payoffs in the case of Game Theory-based LEM.

Strategic Investment in Power and Heat Markets: A Nash–Cournot Equilibrium Model

The current efforts to address the expansion planning of power and heat production facilities can lead to suboptimal results from a market perspective because these efforts often implicitly assume that power and heat systems are planned by a single entity. The present study addresses this issue by establishing a Nash–Cournot equilibrium model, where the producers determine the quantity of power and heat that they produce independently and simultaneously, and thereby collectively affect the price. The model considers three types of producers, including power-only, combined power and heat, and heat-only producers, where each producer determines its own expansion investment strategies independently with the aim of maximizing its own profit. An investment equilibrium is obtained by reformulating the equilibrium model as a mixed complementarity problem. Numerical results carried out on two test systems illustrate the impact of transmission congestion and supply-side market power on the investment equilibria.

Nash–Cournot power market model with a high penetration of prosumers: A distributionally robust optimization approach

The penetration of renewables on the demand side has recently attracted a large amount of attention. Its dual identity has an important impact on the competition between power producers and market dispatching. This study analyzed the participation of prosumers in the Nash–Cournot model for power market clearing. On this basis, we considered the uncertainty of renewable energy production, under which we used the distributionally robust optimization (DRO) model to produce a market equilibrium. Moreover, the Column-and-Constraint Generation algorithm was employed to solve the model. As a result, the impact of prosumers penetration and their production uncertainty on market equilibria was quantified and analyzed. Numerical simulations on an IEEE-30 bus system investigated the impact of the uncertainty of renewable energy output, the penetration level of wind power, and the demand level on the market equilibria. Test results demonstrate that the social welfare obtained from the proposed DRO model is 6.89% higher than that obtained from traditional robust optimization models.

A Stackelberg–Nash–Cournot Equilibrium in a Pollution Reduction Scheme

This paper presents a Stochastic Stackelberg–Nash–Cournot Equilibrium model with continuous market demand distribution to examine the effectiveness of ambient charges as an effective policy measure for reducing nonpoint source pollution in a hybrid scheme. To do so, we consider the supply side of an energy market with hybrid technology that competes in an oligopoly market setting. Within such a setting, each power plant or firm uses a mix of fossil fuels (F) and renewable energy sources (R) to generate power at any given time. The demand for electricity is not realized at the time when the firm (leader) makes the decision. The competition between the two energy sources available to leader is assumed to be of Nash–Cournot equilibria, implying that they use one energy source to generate electricity, whilst holding the other energy source as a constant when the followers reactions are known. Based on the assumption that the demand function is affine and power plants cost functions are quadratic, we obtain the Stackelberg–Nash–Cournot equilibrium. Hence, our analysis provides an interesting insight into the effectiveness of using ambient charges, within the context of a Stochastic Stackelberg–Nash–Cournot competition, as an environmental economic policy measure when included within a robust hybrid scheme. From an economical point of view, this allows pollutants to develop specific control technologies by undertaking research and development (R&D) measures or production processes to maintain emissions standards in a hybrid scheme. From a policy implementations point of view, the environmental authority can use the pollution abatement technology ratio to set ambient charges and industry specific pollutant quantitative limits subject to technological variations.

Bi-level pricing and inventory strategies for perishable products in a competitive supply chain

This paper aims to develop a new bi-level game model for joint pricing and inventory decisions in a competitive supply chain consisting of a dominant manufacturer, who produces single perishable product from deteriorating raw materials, and two follower retailers who face nonlinear price-dependent demand and operate under Cournot assumptions. Three levels of warehousing including raw material warehouse, final product warehouse, and retail warehouses with exponential deterioration rates are considered to explore the joint impact of deterioration rate and price elasticity on the equilibrium inventory decisions. A Stackelberg–Nash–Cournot model is developed to seek the equilibrium prices, quantities, and replenishment cycles and is solved through an exact methodology. A numerical example is presented to validate the proposed model and comprehensive sensitivity analyses are carried out to measure the impact of the model’s key parameters including the deterioration rate in the producer’s and the retailers’ warehouses, the retail and competitor price elasticity, and the market scale on the equilibrium.

Distributed Nash equilibrium seeking of aggregative games under networked attacks

AbstractThis paper investigates the distributed Nash equilibrium seeking problem of aggregative games in the presence of networked attacks. A switched algorithm with the fast estimate dynamics is proposed to explore the effect of attacks on seeking a Nash equilibrium. To guarantee the convergence of this algorithm, such a switched strategy is modeled as a hybrid system by using the hybrid automaton and time‐ratio conditions. Then, by constructing an appropriate Lyapunov function, we can prove that the hybrid dynamical system is exponentially stable and the Nash equilibrium can be achieved exponentially though the communication network is affected by attacks. Finally, a Nash–Cournot game is provided to verify the main results.

Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications

Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash–Cournot oligopolistic equilibrium, and more are presented.

Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications

The objective of this study is to introduce two new extragradient methods to solve equilibrium problems involving two distinct classes of bifunction. Based on the pseudomonotonicity hypothesis and the specific Lipschitz-type cost bifunction condition, we have shown a weak convergence theorem for the first proposed method. The bifunction is strongly pseudomonotone in the second method, but the step-size rule does not depend on the strongly pseudomonotone constant and Lipschitz-type constants. In contrast, the first convergence result, we prove a strong convergence theorem with the use of a particular class of variable step-size rule sequences. To confirm the validity of the proposed convergence results, we examine two well-known Nash–Cournot equilibrium models for the numerical experiment and show that the competitive advantage of our proposed methods is based on time of execution and number of iterations.

Strictly and [formula omitted]-robust counterparts of electricity market models: Perfect competition and Nash–Cournot equilibria

This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classic welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash–Cournot models. Thus, an analog of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.

Modeling Post-Liberalized European Gas Market Concentration—A Game Theory Perspective

The question of whether the liberalization of the gas industry has led to less concentrated markets has attracted much interest among the scientific community. Classical mathematical regression tools, statistical tests, and optimization equilibrium problems, more precisely non-linear complementarity problems, were used to model European gas markets and their effect on prices. In this research, the parametric and nonparametric game theory methods are employed to study the effect of the market concentration on gas prices. The parametric method takes into account the classical Cournot equilibrium test, with assumptions on cost and demand functions. However, the non-parametric method does not make any prior assumptions, a factor that allows greater freedom in modeling. The results of the parametric method demonstrate that the gas suppliers’ behavior in Austria and The Netherlands gas markets follows the Nash–Cournot equilibrium, where companies act rationally to maximize their payoffs. The non-parametric approach validates the fact that suppliers in both markets follow the same behavior even though one market is more liquid than the other. Interestingly, our findings also suggest that some of the gas suppliers maximize their ‘utility function’ not by only relying on profit, but also on some type of non-profit objective, and possibly collusive behavior.

An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems

The paper introduces an inertial extragradient subgradient method with self-adaptive step sizes for solving equilibrium problems in real Hilbert spaces. Weak convergence of the proposed method is obtained under the condition that the bifunction is pseudomonotone and Lipchitz continuous. Linear convergence is also given when the bifunction is strongly pseudomonotone and Lipchitz continuous. Numerical implementations and comparisons with other related inertial methods are given using test problems including a real-world application to Nash–Cournot oligopolistic electricity market equilibrium model.

Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications

A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance.