Generalized Nash Equilibrium Model Research Articles

Supply chain network equilibrium with outsourcing for fresh agricultural products under stochastic demands

Abstract Accepted by: M. Zied Babai This paper examines a supply chain network focusing on the distribution of fresh agricultural products, incorporating outsourcing and stochastic demands considerations. Initially, we establish a generalized Nash equilibrium model with stochastic demands among fresh produce firms. Subsequently, we transform this model into a mixed complementarity system using the Karush–Kuhn–Tucker conditions. Utilizing the Fischer–Burmeister function, we further convert the mixed complementarity system into a set of nonlinear equations, amenable to solution via GAMS software. We conduct numerical experiments and perform sensitivity analysis on key parameters. Our findings suggest that higher subsidy rates incentivize firms towards production outsourcing, particularly benefiting low-income farmers, thereby potentially increasing profits. Moreover, market fluctuations play a pivotal role in ensuring the stability and profitability of firms, with moderate fluctuations presenting opportunities for fresh enterprises to adapt, innovate and capitalize on evolving market conditions. These results offer valuable insights for effective management of fresh produce supply chains.

A Domain Decomposition Method for Solution of a PDE-Constrained Generalized Nash Equilibrium Model of Biofilm Community Metabolism

A Domain Decomposition Method for Solution of a PDE-Constrained Generalized Nash Equilibrium Model of Biofilm Community Metabolism

Optimizing the Growing Dual Credit Requirements for Automobile Manufacturers in China’s Dual Credit Policy

Dual credit policy (DCP) is a market-based mechanism introduced by the Chinese government to promote the new energy vehicle (NEV) industry and improve energy savings in China. To offer sufficient impetus for the NEV industry while providing sufficient transitional buffer time for automobile manufacturers (AMs), the government needs to scientifically design how to gradually increase its dual credit requirement for AMs year by year. To assist the multi-year DCP design, this paper proposes a generalized Nash equilibrium model to predict AMs’ short-term decisions (i.e., vehicle production and credit trading) and long-term decisions (i.e., investment in production capacity expansion and research and development) under any DCP, considering the interactions among AMs’ decisions, vehicle prices, and credit price. Based on the equilibrium model, we then develop a bi-level programming problem to optimize the multi-year DCP. With numerical experiments, we show that implementing the optimal DCP can effectively enhance the market share of NEVs. In the context of the optimal multi-year DCP, the credit requirements set by the government should maintain a relatively low threshold during the initial years, but rise rapidly after that. Such optimal DCP offers AMs sufficient transition time while compelling a quick shift in their developmental strategies.

Radio and Computing Resource Allocation in Co-Located Edge Computing: A Generalized Nash Equilibrium Model

Mobile Network Operators (MNO) can reduce their Capital and Operational Expenditure (CAPEX) and (OPEX) with the help of tower sharing approach by utilizing the physical infrastructure equipped by a third party tower provider to expand their network coverage. Moreover, Computing Resource Providers (CRP) are also setting up their micro-datacenters at tower stations to provide the Multi-access Edge Computing (MEC) services by cooperating with tower providers. Since both the communication and computing services contribute to the task offloading in MEC, the resource allocation has become a challenging problem. In this paper, we formulate the joint uplink, downlink, and computing resources allocation problem in which the objectives of both MNOs and CRP are to minimize their OPEX. The task offloading is modeled as a network of queues where the end-to-end latency is calculated based on the performance of the queue network. Then, the formulated problem is transformed into a Generalized Nash Equilibrium Problem (GNEP) to capture the conflicting interests in the resource allocation among MNOs and CRP. To solve the formulated GNEP efficiently, two decentralized algorithms are proposed by introducing the penalty parameters to the coupling constraints. In addition, the convergence and performance of the algorithms on different parameters are analyzed.

Investigating How the Equilibria of the Electricity Market are Affected by Modeling the Strategic Behavior of Consumers

The paper proposes a formulation for a generalized Nash equilibrium model which incorporates the strategic biddings of some consumers capable of providing reserve and balancing services in the day-ahead and balancing markets, respectively. The strategic bidding in the electricity market is modelled using a bilevel optimization programming, where the electricity cost of consumers is minimized at the Upper Level (UL), and the energy and reserve costs in the market clearing process are co-optimized at the lower level (LL). By means of the strong duality theorem, the original bilevel model is transformed into an equivalent single-level Mathematical Problem with Equilibrium Constraints (MPEC). The joint problem of all MPECs, one per consumer, constitutes an Equilibrium Problem with Equilibrium Constraints (EPEC). The resulting EPEC is finally formulated as an auxiliary Mixed-Integer Linear Programming (MILP) problem. To this end, an exact linearization technique and Fortuny-Amat transformation are adopted to substitute the nonlinear terms and the complementarity conditions. In addition, a parametrization technique is used to replace the dual variable associated with the strong duality equation which appears in the EPEC problem. The diagonalization method is also adopted in this part as an ex-post analysis to verify the obtained solutions of the resulted MILP. Finally, a 3-bus illustrative example and the IEEE RTS 24-Bus System and 118-Bus System are considered to investigate the performance of the proposed approach.

A multicountry, multicommodity stochastic game theory network model of competition for medical supplies inspired by the Covid-19 pandemic

In this paper, we construct the first stochastic Generalized Nash Equilibrium model for the study of competition among countries for limited supplies of medical items (PPEs, ventilators, etc.) in the disaster preparedness and response phases in the Covid-19 pandemic. The government of each country is faced with a two-stage stochastic optimization problem in which the first stage is prior to the pandemic declaration and the second stage is post the pandemic declaration. We provide the theoretical constructs, a qualitative analysis, and an algorithm, accompanied by convergence results. Both illustrative examples are presented as well as algorithmically solved numerical examples, inspired by the need for N95 masks and ventilators. The results reveal that, in addition to the preparedness of countries before the pandemic declaration, their ability to adapt to the conditions in different scenarios has a significant impact on their overall success in the management of the pandemic crisis. The framework can capture competition for other medical supplies, including Covid-19 vaccines and possible treatments, with modifications to handle perishability.

Supply chain network competition among blood service organizations: a Generalized Nash Equilibrium framework

In this paper we present a Generalized Nash Equilibrium model of supply chain network competition among blood service organizations which compete not only for blood donors but also for business from hospitals and medical centers. The model incorporates not only link capacities and associated arc multipliers to capture perishability, but also bounds on the number of donors in regions as well as lower and upper bounds on the demands at the demand points in order to ensure needed amounts for surgeries, treatments, etc., while reducing wastage. The concept of a variational equilibrium is utilized to transform the problem into a variational inequality problem, and alternative formulations are given. A Lagrange analysis yields economic insights. The proposed algorithmic procedure is then applied to a series of numerical examples in order to illustrate the impacts of disruptions in the form of a reduction on the number of donors as well as that of decreases in capacities of critical links such as testing and processing on RBC prices, demands, net revenues of the blood service organizations, and their overall utilities.

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

AbstractIn this paper, we propose a new cybersecurity investment supply chain game theory model, assuming that the demands for the product are known and fixed and, hence, the conservation law of each demand market is fulfilled. The model is a generalized Nash equilibrium model with nonlinear budget constraints for which we define the variational equilibrium, which provides us with a variational inequality formulation. We construct an equivalent formulation, enabling the analysis of the influence of the conservation laws and the importance of the associated Lagrange multipliers. We find that the marginal expected transaction utility of each retailer depends on this Lagrange multiplier and its sign. Finally, numerical examples with reported equilibrium product flows, cybersecurity investment levels, and Lagrange multipliers, along with individual firm vulnerability and network vulnerability, illustrate the obtained results.

Investigating the effect of Arctic sea routes on the global maritime container transport system via a generalized Nash equilibrium model

This paper proposes a generalized Nash equilibrium model to investigate if Arctic routes can be used as a “relief valve” for current intercontinental sea routes. This model is presented as a Stacke...

A game-theoretic model of car ownership and household time allocation

The explosive growth of private cars in China and other developing countries has attracted a great deal of renewed research interest in car ownership. This paper investigates households’ car ownership decision-making process from the perspective of household time allocation. Applying the game-theoretic approach to capturing household members’ interactive decision-making mechanism, we propose a two-stage model that links household members’ short-term time allocation decisions to long-term car ownership decisions. The first stage models the bargaining of household members (e.g., husband and wife) over the car ownership decision, taking into consideration of government policies for regulating car ownership; and the second stage is a generalized Nash equilibrium model for activity-travel pattern analysis incorporating individuals’ interactions concerning activity participation. The existence and uniqueness of the generalized Nash equilibrium solution is examined, and a heuristic procedure that combines backwards induction and method of exhaustion is adopted to solve the two-stage game. The proposed model is applied to an empirical case study in Beijing, which demonstrates the applicability of the model in predicting car ownership and examining interactions between car ownership and household time allocation. The empirical model is applied to assess the impacts of plate-number-based vehicle usage rationing policies on car ownership and time allocation to travel and daily activities. Results show that the model can be applied to evaluate the car ownership impacts of car usage rationing policies.

Benefits of coordinating congestion management in electricity transmission networks: Theory and application to Germany

This article analyzes the coordination of congestion management in the electricity grid and identifies the benefits from closer cooperation among Transmission System Operators. Mimicking the German situation with four Transmission System Operators in charge of relieving grid congestion, in particular by redispatch of power plants, we set up a model with shared transmission network constraints. Through different valuations of these constraints we consider cases of coordination. Based on a Generalized Nash Equilibrium model, we suggest an intuitive approach to introducing coordination. An application to German data provides evidence that more coordination is beneficial, providing channels through which redispatch volumes and specific costs are influenced. We discuss implications of our results for security of supply and network expansion.

Sample average approximation method for a class of stochastic generalized Nash equilibrium problems

We consider a class of stochastic generalized Nash equilibrium problems (SGNEPs). Under some mild conditions, we reformulate the SGNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce an interior point method to solve the SAA problems. Comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic generalized Nash equilibrium model in river basin pollution game.

Liquidity risks on power exchanges: a generalized Nash equilibrium model

The extreme volatility of electricity prices makes their financial derivatives important instruments for asset managers. Even if the volume of derivative contracts traded on Power Exchanges has been growing since the inception of the restructuring of the sector, electricity remains considerably less liquid than other commodity markets. This paper assesses the effect of limited liquidity in power exchanges using an equilibrium model where agents cannot hedge up to their desired level. Mathematically, the problem is formulated as a two stage stochastic Generalized Nash equilibrium with possibly multiple equilibria. Computing a large panel of solutions, we show how the risk premium and players profits are affected by illiquidity. We also dhow that the illiquidity in the FTR market affects the trades in the electricity futures market.

The Benefit of Coordinating Congestion Management in Germany

The management of congestion within the German electricity transmission network has become more important during the last years. This emerging relevance is caused by the increase of renewable generation and the partial phaseout of nuclear power plants. Both developments yield a change in the transmission flow pattern and thus the need for congestion management. Currently, four German transmission system operators (TSOs) are in charge of managing congestion using curative methods, particularly re-dispatch of power plants. However, the existence of four TSOs within Germany induces the question whether coordination between them in managing national congestion would be beneficial. To address this issue, we apply a generalized Nash equilibrium model to analyze different degrees of coordination, covering the German electricity market with a detailed representation of the generation and network structure. Our results indicate that the costs of congestion management decrease in a rising degree of coordination as TSOs take into account congestion in other operators' zones. Total costs are highest in case each TSO is solely responsible for its own zone, and lowest if one integrated entity is in charge of mitigating congestion. We conclude that, in a setup with multiple TSOs, inducing coordination, for instance through a common market, has the potential of lowering the overall costs of congestion management.

A Generalized Nash Equilibrium Model of Market Coupling in the European Power System

“Market Coupling” is currently seen as the most advanced market design in the restructuring of the European electricity market. Market Coupling, by construction, introduces what is generally referred to as an incomplete market: it leaves several constraints out of the market and hence avoids pricing them. This may or may not have important consequences in practice depending on the case on hand. Quasi-Variational Inequality problems and the associated Generalized Nash Equilibrium can be used for representing incomplete markets. Recent papers propose methods for finding a set of solutions of Quasi-Variational Inequality problems. We apply one of these methods to a subproblem of market coupling namely the coordination of counter-trading. This problem is an illustration of a more general question encountered, for instance, in hierarchical planning in production management. We first discuss the economic interpretation of the Quasi-Variational Inequality problem. We then apply the algorithmic approach to a set of stylized case studies in order to illustrate the impact of different organizations of counter-trading. The paper emphasizes the structuring of the problem. A companion paper considers the full problem of Market Coupling and counter-trading and presents a more extensive numerical analysis.

The generalized Nash equilibrium model for oligopolistic transit market with elastic demand

This paper presents a bilevel transit fare equilibrium model for a deregulated transit system. In the upper-level problem, the transit competition is portrayed as an n-player, non-cooperative game by changing the fare structure of each of a set of transit lines separately so as to maximize the profit of each transit operator within the oligopolistic market. We show that there exists a generalized Nash game between transit operators, which can be formulated as a quasi-variational inequality problem. In the lower-level problem, the passengers’ response to the equilibrium fare structure of the transit operators is represented by the stochastic user equilibrium transit assignment model with elastic OD demand. As a result, the bilevel transit fare equilibrium problem is presented in the Stackelberg form and solved by a heuristic solution algorithm based on a sensitivity analysis approach. A numerical example is given to illustrate the competition mechanism on the transit network and some useful findings are presented on competitive operations.