Nash Cournot Research Articles

Joint implementation of tradable credit and road pricing in public-private partnership networks considering mixed equilibrium behaviors

This paper investigates joint road charging schemes in a public-private partnership (PPP) network by simultaneously taking into account Cournot-Nash (CN) players and user equilibrium (UE) players. Each joint scheme comprises a tradable credit plan for public roads and a regular tolling plan for private roads. We show that, under UE-CN mixed equilibrium, there exist anonymous nonnegative joint schemes that can support a system optimum link flow pattern. By using preemptive approach, we further design three bi-objective optimization models with hybrid implementation of tradable credit and road pricing. Numerical examples demonstrate that the proposed methods are effective in managing PPP networks.

Braess Paradox of traffic networks with mixed equilibrium behaviors

Under the user equilibrium (UE) behavior assumption, the Braess Paradox (BP) and its variations have been well investigated. However, users do not always follow the UE behavior. In reality, there are likely quiet a few non-collaborative Cournot–Nash (CN) players coexisting with UE players in the common traffic network. Users in a CN player are completely collaborative to minimize their total travel cost and users subordinating to different players are perfectly competitive. Considering both UE and CN players in the congested network, it remains unclear that under what conditions the BP will occur. In this paper, the BP occurrence conditions under the UE–CN mixed equilibrium are firstly investigated using the classical Braess network with linear link cost function. Then, the BP conditions are studied to the ordinary grid network with nonlinear link cost function. It is shown that the BP occurrence in the conventional Braess network depends upon the link travel time function parameters and the demand level of users controlled by UE/CN players, and the BP occurs in the grid network only for certain demand combinations of users under one UE player and two CN players.

Coalition formation feasibility and Nash–Cournot equilibrium problems in electricity markets: A Fuzzy ASA approach

A numerical method based on a stochastic global optimization paradigm is presented and applied to the calculation of Nash–Cournot equilibria in electricity markets. The proposed method uses and solves GNEP's (generalized Nash equilibrium problems) by means of the Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) algorithm. Concepts of cooperative game theory are used too, such as the bilateral Shapley value. This approach makes it possible to study the feasibility of coalition formation processes in several scenarios, and a case study based on the IEEE 30-bus system is used in order to present and discuss in detail the proposal. The main advantage of the new technique is to simplify and extend current methods, as explained along the text.

Strategic bidding for multiple price-maker hydroelectric producers

In a market comprised of multiple price-maker firms, the payoff each firm receives depends not only on one’s own actions but also on the actions of the other firms. This is the defining characteristic of a non-cooperative economic game. In this article, we ask: What is the revenue-maximizing production schedule for multiple price-maker hydroelectric producers competing in a deregulated, bid-based market? In every time stage, we seek a set of bids such that, given all other price-maker producers’ bids, no price-maker can improve (increase) their revenue by changing their bid; i.e., a pure-strategy Nash–Cournot equilibrium. From a theoretical game theory perspective, the analysis on the underlying non-cooperative game is lacking. Specifically, existing approaches are not able to detect when multiple equilibria exist and consider any equilibrium found optimal. In our approach, we create interpolations for each price-maker’s best response function using mixed-integer linear programming formulations within a dynamic programming framework. In the presence of multiple Nash equilibria, when one exists, our approach finds the equilibrium that is Pareto optimal. If a Pareto-optimal Nash equilibrium does not exist, we use a tailored bargaining algorithm to determine a unique solution. To illustrate some of the finer details of our method, we present three examples and a case study on an electricity market in Honduras.

A sustainable vegetable supply chain using plant factories in Taiwanese markets: A Nash–Cournot model

Sustainable plant factory systems are able to provide steady and high-quality plants to markets while using less labor, water, nutrition, and pesticides. A plant factory is a controlled environment for plant production systems with artificial light, temperature, humidity, carbon dioxide, water supply, and cultivation solution. This paper focuses on the entry and competition of a plant factory supply chain in vegetable markets, using a Nash–Cournot model to simulate this competition. The Lagrangian multiplier method is used to derive KKT optimality conditions for the model. Combining the optimality conditions yields a linear complementarity problem (LCP), which is solved by GAMS and PATH. A case study of the plant factory supply chain in nine Taiwanese vegetable markets is presented. The research simulates the impact of the location of plant factories, number of firms, and different market demands. The results show that total production and profits of the plant factory supply chain increase as transportation costs decrease. In addition, the producer surplus, consumer surplus, and total surplus of the plant factory supply chain in Taiwanese markets improve when factories are located close to the markets. A sensitivity analysis is conducted which shows the impact of market share and production cost on the plant factory supply chain. While the case study focuses on the Taiwanese agricultural commodity production, the methodology and analysis procedures have generalizability to similar plant production industry problems in other contexts.

Strategy Equilibrium in Stackelberg Model with Transmission Congestion in Electricity Market

Nash Cournot Equilibrium (NCE) has been widely used in a competitive electricity market to analyze generation firms’ strategic production quantities. Congestion on a transmission network may lead to a mixed strategy NCE. Mixed strategy is complicated to understand, difficult to compute, and hard to implement in practical market. However, Stackelberg model based equilibrium does not have any mixed strategy, even under congestion in a transmission line. A guide to understanding mixed strategy equilibrium is given by analyzing a cycling phenomenon in the players’ best choices. This paper connects the concept of leader-follower in Stackelberg model with relations between generation firms on both sides of the congested line. From the viewpoint of social welfare, the surplus analysis is presented for comparison between the NCE and the Stackelberg equilibrium (SE).

Non-renewable resource Stackelberg games

The market structure for many mineral industries can be described as oligopoly with potential for Stackelberg leadership. This paper derives and analyzes dynamically consistent extraction equilibria in a two-period discrete-time “Truly” Stackelberg (TS) model of non-renewable resource extraction, where firms move sequentially within each period and where both the leader and follower have market power. We show how the leader may be able to manipulate extraction patterns by exploiting resource constraints. Whether the leader wants to speed up its own production relative to the Cournot–Nash (CN) equilibrium depends on the shape of its iso-profit curve, which is affected by the two firms’ relative stock endowments and relative production costs. If the leader extracts faster, then the follower extracts slower, but in aggregate the industry extracts faster. Unlike static Stackelberg games, the follower does not necessarily have a second mover disadvantage.

Tradable credit schemes on networks with mixed equilibrium behaviors

This paper analyzes and designs tradable credit schemes on networks with two types of players, namely, a finite number of Cournot–Nash (CN) players and an infinite number of (infinitesimal) Wardrop-equilibrium (WE) players. We first show that there are nonnegative anonymous credit schemes that yield system optimum, when transaction costs are not considered. We then analyze how transaction costs would affect the trading and route-choice behaviors of both CN and WE players, and discuss the equilibrium conditions on the coupled credit market and transportation network in the presence of transaction costs. A variational inequality is formulated to describe the equilibrium and is subsequently applied to a numerical example to assess the impacts of transaction costs on a tradable credit system. As expected, transaction costs reduce the trading volume of credits and change their market price. They also change the way how players respond to credit charges in their route choices and cause efficiency losses to the credit schemes that are previously designed without considering transaction costs. With transaction costs, travel costs of WE players will likely increase while those of CN players may decrease due to their higher adaptability in routing strategies.

Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers

We derive sufficient conditions for the existence and uniqueness of the Stackelberg–Nash–Cournot equilibria for a supply chain problem with a single manufacturer and multiple asymmetric retailers and characterize the first and second order derivatives of the total equilibrium quantities. The Stackelberg manufacturer is assumed to supply a homogeneous product to all retailers with the retail price determined by a general nonlinear inverse demand function. We provide several extensions of our previous results [G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474] obtained for a similar supply chain with symmetric retailers.

On Nash–Cournot games with price caps

In this paper, we study an N-person Nash–Cournot game with a price cap. Under certain mild conditions, we show that this game can be formulated as a complementarity problem. Based on this formulation, we study various properties of the game, including equilibrium existence, computability, and a description of the equilibrium set when there are multiple equilibria.

Solving Discretely-Constrained Nash–Cournot Games with an Application to Power Markets

This paper provides a methodology to solve Nash–Cournot energy production games allowing some variables to be discrete. Normally, these games can be stated as mixed complementarity problems but only permit continuous variables in order to make use of each producer’s Karush–Kuhn–Tucker conditions. The proposed approach allows for more realistic modeling and a compromise between integrality and complementarity to avoid infeasible situations.

On price taking behavior in a nonrenewable resource cartel–fringe game

We consider a nonrenewable resource game with one cartel and a set of fringe members. We show that (i) the outcomes of the closed-loop and the open-loop nonrenewable resource game with the fringe members as price takers (the cartel–fringe game à la Salant, 1976) coincide and (ii) when the number of fringe firms becomes arbitrarily large, the equilibrium outcome of the closed-loop Nash game does not coincide with the equilibrium outcome of the closed-loop cartel–fringe game. Thus, the outcome of the cartel–fringe open-loop equilibrium can be supported as an outcome of a subgame-perfect equilibrium. However the interpretation of the cartel–fringe model, where from the outset the fringe is assumed to be price taker, as a limit case of an asymmetric oligopoly with the agents playing Nash–Cournot, does not extend to the case where firms can use closed-loop strategies.

A Generalized Nash–Cournot Model for the Northwestern European Natural Gas Markets with a Fuel Substitution Demand Function: The GaMMES Model

This article presents a dynamic Generalized Nash–Cournot model to describe the evolution of the natural gas markets. The major players along the gas chain are depicted including: producers, consumers, storage and pipeline operators, as well as intermediate local traders. Our economic structure description takes into account market power and the demand representation tries to capture the possible fuel substitution that can be made between the consumption of oil, coal, and natural gas in the overall fossil energy consumption. We also take into account long-term contracts in an endogenous way, which makes the model a Generalized Nash Equilibrium problem. We discuss some means to solve such problems. Our model has been applied to represent the European natural gas market and forecast, until 2030, after a calibration process, consumption, prices, production, and natural gas dependence. A comparison between our model, a more standard one that does not take into account energy substitution, and the European Commission natural gas forecasts is carried out to analyze our results. Finally, in order to illustrate the possible use of fuel substitution, we studied the evolution of the natural gas price as compared to the coal and oil prices.

Efficiency Loss of Mixed Equilibrium Behaviors with Polynomial Cost Functions

This paper investigates the efficiency loss of selfish routing simultaneously with user equilibrium (UE) player and Cournot-Nash (CN) players. The upper bound of the efficiency loss of UE-CN mixed equilibrium with polynomial cost functions is obtained by the scaling method and the nonlinear programming method, respectively. It is shown that the upper bound of efficiency loss obtained by the scaling method only depends on the polynomial degree of the link travel cost function and the upper bound of efficiency loss obtained by the nonlinear programming method depends on the number of the CN players besides the aforementioned factors. The numerical tests validate our analytical results. KEY WORDS: efficiency loss, UE-CN mixed equilibrium, variational inequality, selfish routing game

Impact of wind integration on electricity markets: a chance-constrained Nash Cournot model

SUMMARY Wind-powered electricity generation is growing at a high rate around the world, mainly driven by the associated environmental benefits. However, because of the uncertain nature of wind energy, large-scale integration of wind-powered generators into power systems introduces uncertainty of supply in the operation and planning of electric energy systems. A chance-constrained model is presented to study the strategic behavior of power suppliers (firms) with respect to the wind generation uncertainty in an electricity market. The linear complementarity problem of a Nash-Cournot competition is extended to account for the wind generation uncertainty. An iterative solution approach is introduced to solve the resultant joint chance-constrained programming problem. Five studies were conducted to show the impact of the amount, location and standard deviation of the wind generation and transmission line limits on the suppliers’ profit in the Nash-Cournot game for different confidence levels. The value of stochastic solution index is used to evaluate the suppliers’ loss of profit. The numerical results for a three-bus test system are given to verify the formulation and the solution approach. Copyright © 2011 John Wiley & Sons, Ltd.

A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers

A well-studied problem in the supply chain management literature considers a two-tier supply chain for a homogeneous product with a single manufacturer, multiple retailers and a general inverse demand function. The problem has been analyzed in the literature without a formal mathematical treatment of the existence/uniqueness of equilibria. Furthermore, the existence/uniqueness results derived for related models are not extendable to our model. The objective of this paper is to derive sufficient conditions for the existence/uniqueness of Stackelberg–Nash–Cournot equilibria for the two-tier problem.

Coordination of cooperative advertising models in a one-manufacturer two-retailer supply chain system

This paper considers cooperative advertising issues of a monopolistic manufacturer with competing duopolistic retailers. Four possible game structures (or power configurations), i.e., Stackelberg–Cournot, Stackelberg–Collusion, Nash–Cournot and Nash–Collusion, are discussed. Under each of four game structures, we develop a decision model for the three partners to design the optimal cooperative advertising policies. Through a comparison among the four models, we reveal how cooperative advertising policies and profits of all participants are affected by various competitive behaviors, and then determine whether the partners have any incentives to transit to a different structure. Also presented in the paper are a centralized decision model and a proposed cost-sharing contract, which is able to achieve perfect coordination of the considered channel, where the utility of risk preference is used to determine the fraction of local advertising costs shared by the manufacturer.

On the Characterization of Solution Sets of Smooth and Nonsmooth Convex Stochastic Nash Games

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous-strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness, may be made when player objectives are given by expectations. For instance, in deterministic regimes, a suitable coercivity condition associated with the gradient map is sufficient for the existence of a Nash equilibrium; in stochastic regimes, the application of this condition requires being able to analytically evaluate the expectation and its gradients. Our interest is in developing a framework that relies on the analysis of merely the integrands of the expectations; in the context of existence statement, we consider whether the satisfaction of a suitable coercivity condition in an almost-sure sense may lead to statements about the original stochastic Nash game. Notably, this condition also guarantees the existence of an equilibrium of the scenario-based Nash game. We consider a range of such statements for claiming the existence of stochastic Nash equilibria when payoff functions are both smooth and nonsmooth and when strategy sets are possibly coupled through a shared convex constraint. Notably the sufficiency conditions are less stringent, when one either imposes appropriate monotonicity requirements or requires that strategy sets be decoupled. Uniqueness, however, can be claimed by requiring that a strong monotonicity condition holds over a set of positive measure, rather than in an almost-sure sense. When strategy sets are coupled by shared convex expected-value constraints, a suitable regularity condition allows for claiming existence and uniqueness in the primal-dual space. We illustrate our approach by examining two extensions of stochastic Nash–Cournot games, of which the first allows for nonsmooth payoffs through the introduction of risk-measures, while the second allows for shared stochastic constraints.

Analysis of biomass co-firing systems in Taiwan power markets using linear complementarity models

Biomass co-firing systems in power plants generate electric power by the simultaneous combustion of biomass and fossil fuels. The co-firing process reduces investment costs by converting biomass energy into electricity in existing conventional power plants. Biomass co-firing significantly reduces carbon dioxide and sulfur dioxide emissions in power generation. To meet the increase in biomass demand, this paper has considered systematic energy crop production, which is expected to increase in the near future. Our aim is to analyze biomass co-firing systems in the Taiwanese electricity market. In this paper, we study two emerging biomass feedstocks: switchgrass and Miscanthus. We focus on the impact of energy crop co-firing on carbon dioxide and sulfur dioxide emissions for electricity generation. A Nash–Cournot competition model, which simulates potential biomass co-firing scenarios, is formulated for power markets. A case study conducted in the Taiwanese electricity market showed that biomass co-firing lowers total electricity demand and sale. Miscanthus is more economical than switchgrass in terms of the production cost and the land required to generate biopower for the same levels of biomass co-firing.

Nash–Cournot equilibrium of a deregulated electricity market using competitive coevolutionary algorithms

In this paper, a competitive coevolutionary approach is used to find out the Nash–Cournot equilibrium of an oligopolistic electricity market based on either centralized or bilateral structure, where the transmission constraints are considered. We consider a market game played by bounded evolutionary agents, adapting their strategies to maximize their profits in a competitive environment. Under the effect of the competition, the coevolved agents are led toward an equilibrium point where each agent population is invaded by an Evolutionary stable Strategy.