Market Equilibrium Problem Research Articles

Existence results for a class of quasi-variational inequalities and applications to a noncooperative model

The aim of the paper is to investigate the existence of solutions for the class of strongly pseudomonotone quasi-variational inequalities. The weak Mosco continuity of multifunctions has a key role to reach this aim. As an application, the existence of equilibrium distributions for a dynamic oligopolistic market equilibrium problem with adaptive set of feasible solutions is obtained.

Notes on random optimal control equilibrium problem via stochastic inverse variational inequalities

The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

The new full-Newton step interior-point algorithm for the Fisher market equilibrium problems based on a kernel function

In this paper, we propose a full-Newton step interior-point method for the weighted linear complementarity problem (wLCP) model of the Fisher market equilibrium problem. The weighted complementarity problem (wCP) is a generalization of the complementarity problem (CP) with the nonnegative weight vector, where the zero on the right-hand side is replaced by the nonnegative weight vector. The importance of a wCP lies in the fact that many equilibrium problems in science, engineering, and economics can be reformulated as wCPs, which lead to the development of highly efficient algorithms. We extend a full-Newton step interior-point method (IPM) to the wLCP model of the Fisher problem. New search directions are obtained by using a kernel function in the scaled Newton system. The algorithm takes only full-Newton steps, thus, avoiding the calculation of the step size which is computationally advantageous. Under the suitable assumptions, the algorithm is shown to have global convergence and polynomial complexity. Some numerical results indicate the efficiency of the algorithm. To the best of our knowledge, this is the first full-Newton step IPM for Fisher problems, which uses the kernel function to obtain new search directions.

Nonconvex equilibrium models for energy markets: exploiting price information to determine the existence of an equilibrium

Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g. switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems.

Bi-level optimization based two-stage market clearing model considering guaranteed accommodation of renewable energy generation

The existing electricity market mechanisms designed to promote the consumption of renewable energy generation complicate network participation in market transactions owing to an unfair market competition environment, where the low cost renewable energy generation is not reflected in the high bidding price of high cost conventional energy generation. This study addresses this issue by proposing a bi-level optimization based two-stage market clearing model that considers the bidding strategies of market players, and guarantees the accommodation of renewable energy generation. The first stage implements a dual-market clearing mechanism that includes a unified market for trading the power generations of both renewable energy and conventional energy units, and a subsidy market reserved exclusively for conventional generation units. A re-adjustment clearing mechanism is then proposed in the second stage to accommodate the power generation of remaining renewable energy units after first stage energy allocations. Each stage of the proposed model is further described as a bi-level market equilibrium problem and is solved using a co-evolutionary algorithm. Finally, numerical results involving an improved IEEE 39-bus system demonstrate that the proposed two-stage model meets the basic requirements of incentive compatibility and individual rationality. It can facilitate the rational allocation of resources, promote the economical operation of electric power grids, and enhance social welfare.

A random time-dependent noncooperative equilibrium problem

The paper deals with the random time-dependent oligopolistic market equilibrium problem. For such a problem the firms’ point of view has been analyzed in Barbagallo and Guarino Lo Bianco (Optim. Lett. 14: 2479–2493, 2020) while here the policymaker’s point of view is studied. The random dynamic optimal control equilibrium conditions are expressed by means of an inverse stochastic time-dependent variational inequality which is proved to be equivalent to a stochastic time-dependent variational inequality. Some existence and well-posedness results for optimal regulatory taxes are obtained. Moreover a numerical scheme to compute the solution to the stochastic time-dependent variational inequality is presented. Finally an example is discussed.

Impact of connected and autonomous vehicle technology on market penetration and route choices

Before having a massive deployment of fully connected and autonomous vehicles (CAVs), CAVs with different automation levels and human-driven vehicles (HVs) will coexist on roads for a long time. To quantify the impacts of CAV technology on vehicle market penetration and travelers’ route choices in the future from the perspective of transportation planning, this paper investigates a two-sided market equilibrium problem, which considers the vehicle market on one side and the road traffic equilibrium market on the other side. The two markets interactively affect each other through market penetration, information quality, and spatial distribution of congestion, which are all endogenously determined in our model. In the vehicle market, two-stage decision-making is considered to describe various vehicle choices users may face in the future. In the first stage, users between each origin–destination (OD) pair choose a vehicle type between HV and CAV. In the second stage, CAV users further choose a vehicle automaton level. To account for the similarity of CAVs with different automation levels, we use a nested logit (NL) model to capture the two-stage decision-making. A vehicle choice with a higher market penetration provides higher-quality information, which affects the traffic equilibrium market. For the traffic equilibrium, route choices of users with different information quality are described by a multinomial logit (MNL) model. The spatial distribution of congestion determined by the traffic equilibrium also affects vehicle choices in the vehicle market. Specifically, users with higher quality information are more likely to choose routes with the lowest travel time. Consequently, a vehicle choice with a lower expected travel time attracts more users. The two-sided market equilibrium is formulated as a combined NL-MNL model so as to solve the two interactive market equilibria simultaneously. Then, we explore the properties of the equilibrium state. Sufficient and necessary conditions for the path flow pattern and demand pattern at equilibrium are derived, respectively. Based on the properties of the equilibrium state, we derive an equivalent variational inequality (VI) for the combined NL-MNL model. A new approach is provided to deriving two sufficient conditions, either of which guarantees the uniqueness of the VI solution. To solve the proposed problem efficiently, we develop a path-based modified self-regulated averaging (PMSRA) algorithm embedded with a modified K-shortest path method. Finally, numerical experiments are conducted to analyze the effects of CAV technology and demonstrate algorithm efficiency. Sensitivity analysis of parameters in the algorithm is also performed. Our results show that the market penetration of CAVs at the early stage of introduction is low due to the high purchase cost. With the development of CAV technology and mass production, fully automated CAVs may gradually dominate the market, while partially automated CAVs tend to be squeezed out of the market. In addition, our results reveal that the travel time saving from CAV technology and high-quality information is more pronounced for long trips and congested networks.

Long-run market equilibria in coupled energy sectors: A study of uniqueness

We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions determined at repeated spot markets. We show that in our multi-sector model, multiplicity of equilibria may occur, even if all assumptions hold that would be sufficient for uniqueness in a single-sector model. We then contribute to the literature by deriving sufficient conditions for the uniqueness of short- and long-run market equilibrium in coupled markets of multiple energy sectors. We illustrate via simple examples that these conditions are indeed required to guarantee uniqueness in general. The uniqueness result is an important step to be able to incorporate the proposed market equilibrium problem in more complex computational multilevel equilibrium models, in which uniqueness of lower levels is a prerequisite for obtaining meaningful solutions. Our analysis also paves the way to understand and analyze more complex sector coupling models in the future.

A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem

<p style='text-indent:20px;'>The weighted complementarity problem (wCP) can be applied to a large variety of equilibrium problems in science, economics and engineering. Since formulating an equilibrium problem as a wCP may lead to highly efficient algorithms for its numerical solution, wCP is a nontrivial generalization of the complementarity problem. In this paper we consider a special weighted linear complementarity problem (wLCP), which is the more general optimization of the Fisher market equilibrium problem. A full-modified-Newton infeasible interior-point method (IIPM) for the special wLCP is proposed. The algorithm reformulates the central path of the perturbed problem as an equivalent system of equations, and uses only full-Newton steps at each iteration, so-called a feasibility step (i.e., a full-modified-Newton step) and several usual centering steps. The polynomial complexity of the algorithm is as good as the best known iteration bound for these types of IIPMs in linear optimization.</p>

Strategic bidding in price coupled regions

With the emerging deregulated electricity markets, a part of the electricity trading takes place in day-ahead markets where producers and retailers place bids in order to maximize their profit. We present a price-maker model for strategic bidding from the perspective of a producer in Price Coupled Regions (PCR) considering a capacitated transmission network between local day-ahead markets. The aim for the bidder is to establish a production plan and set its bids taking into consideration the reaction of the market. We consider the problem as deterministic, that is, the bids of the competitors are known in advance. We are facing a bilevel optimization problem where the first level is a Unit Commitment problem, modeled as a Mixed Integer Linear Program (MILP), and the second level models a market equilibrium problem through a Linear Program. The problem is first reformulated as a single level problem. Properties of the optimal spot prices are studied to obtain an extended formulation that is linearized and tightened using new valid inequalities. Several properties of the spot prices allow to reduce significantly the number of binary variables. Two novel heuristics are proposed, the first applicable in PCR, the second for general formulations with Special Ordered Sets (SOS) of type 1. Our computational experiments highlights the risk of a loss for the bidder if some aspects usually not considered in the literature, such as Price Coupled Regions, or an accurate UC problem, are not taken into account. They also show that the reformulation techniques, combined with new valid inequalities, allow to solve much larger instances than the current state-of-the-art. Finally, our experiments also show that the proposed heuristics deliver very high quality solutions in a short computation time.

A peer-to-peer energy trading market embedded with residential shared energy storage units

The high penetration of distributed energy resources with local renewable energy consumption facilitates the emergence of peer-to-peer (P2P) energy trading, where a peer can share excessive energy with local peers. P2P energy trading is expected to be a promising business model in the future transactive energy market. Influenced by the sharing economy principle, shared energy storage(SES) is investigated to reduce the costly initial investment and improve the utilization rate of storage devices. Given the context, this paper proposes an equilibrium model of a P2P energy trading market, considering the deployment of shared energy storage in the residential consumer-side. The market equilibrium problem consists of two interwoven games. Considering the benefit conflicts between energy sellers and buyers in P2P energy trading, the P2P transaction between peers can be described as a non-cooperative game. Considering the limited SES capacity resources, the competition among SES users can be viewed as a generalized Nash equilibrium problem. These two games in the P2P market interact with each other and can be summarized as a holistic equilibrium problem, in which the P2P transaction price is endogenously determined and storage capacity is allocated. The market equilibrium problem can be solved based on Karush-Kuhn-Tucker (KKT) optimality conditions and linearization techniques, finally cast as a mixed-integer linear program. Numerical simulation demonstrates that P2P energy trading benefits all participants and the introduction of shared energy storage can further reduce energy costs.

Numéraire problems and market adjustments

AbstractFirstly, this paper makes a distinction between two notions of numéraire in order to clarify the problem of market equilibrium which depends on the choice of a standard of value. This problem has recently re‐emerged, although it smoldered before and during the “Two Cambridge Controversies” on capital theory in the Seventies. Secondly, it presents a collection of equilibrium/disequilibrium analysis which exhibits numéraire dependence. Thirdly, it takes a neo‐Walrasian model of non‐monetary general equilibrium as a benchmark to argue that a prominent role of numéraire should be attributed to fiat money in the analysis of market adjustment. Such a redirection of focus would avoid some typical cases of numéraire dependence that characterize a frictionless barter economy.

Gamma -Robust electricity market equilibrium models with transmission and generation investments

We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Gamma -robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Gamma -robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.

Long-run market equilibria in coupled energy sectors: A study of uniqueness

We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions determined at repeated spot markets. We show that in our multi-sector model, multiplicity of equilibria may occur, even if all assumptions hold that would be sufficient for uniqueness in a single-sector model. We then contribute to the literature by deriving sufficient conditions for the uniqueness of short- and long-run market equilibrium in coupled markets of multiple energy sectors. We illustrate via simple examples that these conditions are indeed required to guarantee uniqueness in general. The uniqueness result is an important step to be able to incorporate the proposed market equilibrium problem in more complex computational multilevel equilibrium models, in which uniqueness of lower levels is a prerequisite for obtaining meaningful solutions. Our analysis also paves the way to understand and analyze more complex sector coupling models in the future.

Variational Inequality Type Formulations of General Market Equilibrium Problems with Local Information

We suggest a new approach to creation of general market equilibrium models involving economic agents with local and partial knowledge about the system and under different restrictions. The market equilibrium problem is then formulated as a quasi-variational inequality that enables us to establish existence results for the model in different settings. We also describe dynamic processes, which fall into information exchange schemes of the proposed market model. In particular, we propose an iterative solution method for quasi-variational inequalities, which is based on evaluations of the proper market information only in a neighborhood of the current market state without knowledge of the whole feasible set and prove its convergence.

Stochastic variational formulation for a general random time-dependent economic equilibrium problem

In the paper, in a Hilbert space setting, a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses is studied and the random time-dependent Cournot–Nash equilibrium principle by means of a stochastic variational inequality is characterized. Then, some existence results to such problem are established and the stochastic continuity of the equilibrium solution is proved. Moreover a simple numerical example illustrates the theoretical results.

Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium problems

Our contribution in this paper, we introduce and analyze two new hybrid algorithms by combining Mann iteration and inertial method for solving split fixed point problems of demicontractive mappings and equilibrium problems in a real Hilbert space. By using a new technique of choosing step size, our algorithms do not need any prior information on the operator norm. In fact, an inertial type algorithm was proposed in order to accelerate its convergence rate. We then prove weak and strong convergence of proposed methods under some control conditions. Moreover, some numerical experiments for image restoration problems and oligopolistic market equilibrium problems are also provided for supporting our main results.

Limit vector variational inequalities and market equilibrium problems

In a finite-dimensional setting we investigate the solvability of a general vector variational inequality via the convergence of solutions of suitable approximating vector variational inequalities defined with more regular data. The theoretical results obtained in a very general framework are successfully applied to the study of a vector market equilibrium problem where instead of exact values of the cost mapping, feasible set and order cone, only approximation sequences of these data are available.

On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium

The general tensor variational inequalities, recently introduced in Barbagallo et al. (J Nonconvex Anal 19:711–729, 2018), are very useful in order to analyze economic equilibrium models. For this reason, the study of existence and regularity results for such inequalities has an important rule to the light of applications. To this aim, we start to consider some existence and uniqueness theorems for tensor variational inequalities. Then, we investigate on the approximation of solutions to tensor variational inequalities by using suitable perturbed tensor variational inequalities. We establish the convergence of solutions to the regularized tensor variational inequalities to a solution of the original tensor variational inequality making use of the set convergence in Kuratowski’s sense. After that, we focus our attention on some stability results. At last, we apply the theoretical results to examine a general oligopolistic market equilibrium problem.

Economic Benefits of Integrating Solar-Powered Heat Pumps Into a CHP System

As a promising distributed energy resource, solar-powered heat pumps (SPHPs) can achieve significant system benefits. On the one hand, the accommodation of solar generation can be highly improved on the demand side. On the other hand, SPHPs can efficiently provide heat energy for consumers, which reduces the operational costs of central heat supply from combined heat and power (CHP) units. In this paper, a planning framework is developed to evaluate the economic benefits of integrating SPHPs into a CHP system. The framework is formulated as a bilevel market equilibrium model. On the upper level, given the locational marginal prices (LMPs) and district heating prices (DHPs), the consumers at different buses will strategically invest in SPHPs. Thus, the net electric and heating loads vary with the investment in the SPHPs. On the lower level, LMPs are updated by direct current optimal power flow, and DHPs are updated by a cost-plus pricing method. Because of the nonlinearity of this market equilibrium problem, an algorithm is proposed to iteratively solve the bilevel model, and a criterion is designed to achieve a desired solution. Case studies based on the IEEE 14-bus and 118-bus systems demonstrate that by investing in SPHPs on the demand side, the operational costs of CHP systems can be effectively reduced in both the electric and heating sectors.