Cournot Oligopoly Game Research Articles

Correction to: Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

Correction to: Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

A game-theoretic approach for pollution control initiatives

There are numerous regional, national, and international efforts to establish mechanisms to curb pollution and emissions. In this paper, we mainly focus on emissions trading systems. Using a static Cournot oligopoly game, we investigate a multi-product multi-pollutant market in which several supply chains compete in a non-cooperative manner in their product markets. This study characterizes conditions under which an efficient solution procedure converges to the unique Nash equilibrium of the oligopoly. Meanwhile, its partners establish a cooperative triopoly game within each supply chain in a nonsuperadditive characteristic function form. Drawing on the cooperative game theory literature, we provide a closed-form solution for a rational distribution of joint rewards between the supply chain’s partners.

Solving the Cournot Oligopoly Game under Incomplete Information with the Possibility of Cooperative Interaction

The balance of commodity markets is an extremely important and pressing challenge. There is an active development of research aimed at solving such problem. However, well-known theoretical models do not always adequately reflect the desire of agents to make such individual decisions and search for such compromises to ensure the greatest profit and bring the market to equilibrium.
 The article considers the solution of the Cournot oligopoly game under incomplete information for the case of linear demand and linear costs. The approaches of the reflexive games theory and collective behavior theory are the basis for the construction of the decision-making process by agents. The policy of optimal responses of agents to the expected output volumes of the environment under traditional assumptions about the independence of agents may not lead to equilibrium. Therefore, the article considers and proposes non-traditional solutions that allow agents to cooperate. It is shown that it is enough to jointly regulate the process only at initial stage. After that, it should come to equilibrium if, in the future, agents independently adjust their own output, counting on the maximum expected profit. The necessary assertions and their mathematical proofs are given.

Modelling Cournot Games as Multi-agent Multi-armed Bandits

We investigate the use of a multi-agent multi-armed bandit (MA-MAB) setting for modeling repeated Cournot oligopoly games, where the firms acting as agents choose from the set of arms representing production quantity (a discrete value). Agents interact with separate and independent bandit problems. In this formulation, each agent makes sequential choices among arms to maximize its own reward. Agents do not have any information about the environment; they can only see their own rewards after taking an action. However, the market demand is a stationary function of the total industry output, and random entry or exit from the market is not allowed. Given these assumptions, we found that an ε-greedy approach offers a more viable learning mechanism than other traditional MAB approaches, as it does not require any additional knowledge of the system to operate. We also propose two novel approaches that take advantage of the ordered action space: ε-greedy+HL and ε-greedy+EL. These new approaches help firms to focus on more profitable actions by eliminating less profitable choices and hence are designed to optimize the exploration. We use computer simulations to study the emergence of various equilibria in the outcomes and do the empirical analysis of joint cumulative regrets.

Oligopoly dynamics with isoelastic demand: The joint effects of market saturation and strategic delegation

In the framework of a Cournot oligopoly game with isoelastic demand, we examine the simultaneous presence of both market saturation and strategic delegation. Although these two (realistic) aspects have already been considered in the literature each on its own, we aim at deepening their joint interactions when matched together in oligopolistic competition. In addition, we admit the possibility that delegation activities actuated by firms to weaken or even exclude competitors from the market may cease if undertaken by successful players, which thus regain their pure profit maximizing behavior. In this context, a limited market saturation level (positively) influences the effectiveness of delegation strategies and, at the same time, can sustain equilibrium configurations for the winning (monopolistic) firm even under the isoelastic market structure. Through local stability analysis, we show how the combination of strategic delegation with market saturation contributes to determine the equilibrium number of active players and the local asymptotic stability of the (economically relevant) equilibrium. Moreover, non-equilibrium dynamics reveal the presence of periodic cycles along which a firm is active while its competitors alternatively exits and enters the market. We show why these interesting scenarios are due to the joint interplay between strategic delegation and market saturation.

Multi-Layer Cournot-Congestion Model

Geopolitical instability, climate change and black swan events disrupt the trade and logistics of resources around the globe. Events such as the unforeseen closure of the Suez Canal or the cessation of trade between some players due to wars or embargos are some examples of this. The problem of predicting local price change under modification of an underlying transport network unites elements of game theory, network theory and transport theory. The micro-economic Cournot oligopoly game involves modelling economic actors as rational players attempting to maximise profit. Under fixed transport conditions, analytical results can be found on the equilibria. Similarly, the transport layer can be analytically solved using techniques for selfish routing. Where trade and transport layers are linked together there is inter-layer feedback wherein players attempt to maximise their utility. We looked at the nature of the approach towards a new equilibrium under this instantaneous network change. In this respect our findings indicate that players benefit significantly from taking advantage of the non-simultaneous responses to the market rather than from moving to a new equilibrium utility. We found that when uncoupled, both the upper and lower layers have concave utility curves meaning there exist unique and stable equilibria in both cases. This leads to the multilayer model having non-unique stable equilibria for which general solutions exist.

A comprehensive framework for optimal planning of competing energy hubs based on the game theory

The concept of energy hub has been used recently to study multi-carrier energy systems. A model has been proposed in this paper for planning multi-hub energy system considering the competition between the hubs. The energy hubs are interconnected by a power grid. Hubs supply the demands for heat and electricity using various technologies and access to several energy carriers. To this aim, load zones were used to incorporate demand profiles for both heat and electricity in the different seasons of the year. Supplying the demand most cost-effectively is the objective function of each hub. These hubs select the optimum strategy separately in a competitive space. The power grid is owned by Regional Transmission Operator (RTO) which is operated based on DC Power Flow. Moreover, elastic electricity demand is also addressed for the market in each region. The proposed model formulate the problem as a Cournot Oligopoly game using Karush–Kuhn–Tucker (KKT) conditions to consider the objective function of the hubs as well as that of RTO. Once solved, yearly capacity-building is determined as well as the flow of energy carriers for each technology, regional electricity prices and the amounts of electricity transmission between regions for each load zone and each hub. The proposed model has been applied to a test system and applicability of the proposed model was verified. Finally, the results of competitive modeling were compared with those of the centralized modeling.

Calmness of a Perturbed Cournot Oligopoly Game with Nonsmooth Cost Functions

This article deals with the calmness of a solution map for a Cournot Oligopoly Game with non-smooth cost functions. The fact that the cost functions are not supposed to be differentiable allows to consider cases where some firms have different units of production, with different marginal costs. In order to obtain results concerning calmness, we use a new technique based on an outer coderivative and on a mathematical induction on the number of players. It is concluded that the methodology used for the proofs can be replicated, in order to study the metric subregularity and calmness of multifunctions in a more general way.

Distributed cooperative control algorithm for optimal power sharing for AC microgrids using Cournot game theory

Research on the optimal power allocation of large-scale distributed generator (DG) units based on user power generation to access microgrids (MGs) in a multi-agent system framework has recently become the focus of modern grid and energy concerns. In this paper, according to the Cournot oligopoly game, the Nash equilibrium point between the power generation company and power generation user of the MG operating in island mode is obtained. According to the obtained Nash equilibrium point, the optimal ratio of power generated by the power generation company and by the power generation user in the model is calculated. At the same time, to achieve the maximum benefit and stable operation of the MG, a distributed cooperative control algorithm based on consensus theory is proposed. This control algorithm can cause each DG to generate power according to the total consumption load. The optimal power generation ratio distribution based on the Nash equilibrium point eliminates the steady-state frequency deviation of each DG in the MG, thereby ensuring the user’s power quality. The simulation results show that the control algorithm can achieve the above research goals.

Cournot oligopoly game-based local energy trading considering renewable energy uncertainty costs

Abstract Facilitated by advanced information and communication technologies (ICTs), local energy trading develops rapidly, playing an important role in the energy supply chain. Thus, it is essential to develop local trading models and strategies that can benefit participants, not only stimulating local balancing but also promoting renewable penetration. This paper proposes a new local energy trading decision-making model for suppliers by using the Cournot Oligopoly game, considering the uncertainty costs of renewable energy. Four types of representative energy providers are modelled, traditional thermal generation, wind power, photovoltaic (PV) power, and electricity storage. The revenue of these technologies is extensively formulated according to their operation cost, investment cost, and income from selling energy. The uncertainty cost of renewable generation is integrated into the trading, modelled as a penalty for potential energy shortage that is derived from output probability distribution function (PDF). This trading model is formulated as a non-cooperative Cournot oligopoly game to enable energy suppliers to maximize their profits through local trading considering price. The response of the customer to energy price variations, i.e. demand elasticity, is also included in the model. A unique Nash equilibrium (NE) and optimum strategies are derived by the proposed Optimal-Generation-Plan (OGP) Algorithm. As demonstrated in a typical local market, the proposed approach can effectively model and resolve multiple suppliers’ competition in local energy trading. It can work as a vehicle to facilitate the trading between various generation technologies and customers, realising local balancing and benefiting all market participants with enhanced revenue and reduced energy bills.

Aggregative games and oligopoly theory: short‐run and long‐run analysis

AbstractWe compile an IO toolkit for aggregative games with positive and normative comparative statics results for asymmetric oligopoly in the short and long run. We characterize the class of aggregative Bertrand and Cournot oligopoly games, and the subset for which the aggregate is a summary statistic for consumer welfare. We close the model with a monopolistically competitive fringe for long‐run analysis. Remarkably, we show strong neutrality properties in the long run across a wide range of market structures. The results elucidate aggregative games as a unifying principle in the literature on merger analysis, privatization, Stackelberg leadership, and cost shocks.

Reinforcement Learning in a Cournot Oligopoly Model

This paper analyzes the learning behavior of firms in a repeated Cournot oligopoly game. Literature shows the degree of information and cognitive capacity of learning firms is a key factor that determines long run outcome of an oligopoly market. In particular, when firms possess the knowledge of market demand and are capable of computing the optimal production quantity given the output of other firms, the resulting market outcome is the Nash equilibrium. On the other hand, imitation that assumes low behavioral sophistication of firms generally favors higher output and converges to the Walrasian equilibrium. In this paper, a reinforcement learning algorithm with low cognitive requirement is adopted to model firms’ learning behavior. Reinforcement learning firms observe past production choices and fine tune them to improve profits. Analytical result shows that the Nash equilibrium is the only fixed point of the reinforcement learning process. Convergence to the Nash equilibrium is observed in computational simulations. When firms are allowed to imitate the most profitable competitor, all states between the Nash equilibrium and the Walrasian equilibrium can be reached. Furthermore, the long run outcome shifts towards the Nash equilibrium as the length of firms’ memory increases.

On a Cournot Game with Bounded Rational Players, Convex, Log-Linear Demand and Consumer Surplus

Based on the Cournot oligopoly game and the nonlinear dynamics theory, we study the behavior of semi-public enterprises by considering corporate social responsibility into their objectives. The model that is established is a dynamical Cournot-type duopoly model with bounded rationality containing the consumer surplus. We suppose quadratic cost function and a convex, log-linear demand function. The game is modeled with a system of two difference equations. Existence and stability of equilibriums of this system are studied. More complex chaotic and unpredictable trajectories are resulted studying this discrete dynamical system. The complex dynamics of the system are demonstrated numerically via computing Lyapunov numbers, sensitivity dependence on initial conditions, and bifurcation diagrams.
 Keywords: Cournot duopoly game; Discrete dynamical system; Homogeneous expectations; Stability; Chaotic Behavior; Consumer Surplus.

Dynamic Cournot oligopoly game based on general isoelastic demand

This paper explores a nonlinear Cournot oligopoly with n firms displaying general isoelastic demand. The marginal profits-based gradient rule and the expectation rule Local Monopolistic Approximation were employed in two Cournot oligopoly games. Nash equilibrium stability analysis is carried out on each of the two games to throw light on the effects of demand elasticity and other parameters on the dynamics of the game. Our results show that the influence of demand elasticity on stability depends on firms’ expectation rules.

On the coalitional stability of monopoly power in differentiated Bertrand and Cournot oligopolies

In this article, we revisit the classic comparison between Bertrand and Cournot competition in the presence of a cartel of firms that faces outsiders acting individually. This competition setting enables to deal with both non-cooperative and cooperative oligopoly games. We concentrate on industries consisting of symmetrically differentiated products where firms operate at a constant and identical marginal cost. First, while the standard Bertrand–Cournot rankings still hold for Nash equilibrium prices, we show that the results may be altered for Nash equilibrium quantities and profits. Second, we define cooperative Bertrand and Cournot oligopoly games with transferable utility on the basis of their non-cooperative foundation. We establish that the core of a cooperative Cournot oligopoly game is strictly included in the core of a cooperative Bertrand oligopoly game when the number of firms is lower or equal to 25. Moreover, we focus on the aggregate-monotonic core, a subset of the core, that has the advantage to select point solutions satisfying both core selection and aggregate monotonicity properties. We succeed in comparing the aggregate-monotonic cores between Bertrand and Cournot competition regardless of the number of firms. Finally, we study a class of three-firm oligopolies with asymmetric costs in which the core inclusion property mentioned above still holds. We also provide numerical examples to illustrate the difficulty to generalize this result to an arbitrary number of firms because of negative equilibrium quantities.

Differentiated Cournot duopoly game with fractional-order and its discretization

PurposeFractional calculus provides powerful tool to build more realistic and accurate mathematical models in economic field. This paper aims to explore a proposed fractional-order differentiated Cournot duopoly game and its discretized game.Design/methodology/approachConditions for existence and uniqueness of the proposed game’s solution are derived. The existence of Nash equilibrium point and its local and global stability are obtained. Furthermore, local stability analysis of the discretized game is investigated. The effects of fractional-order on game’s dynamics are examined, along with other parameters of the game, via the 2D bifurcation diagrams in planes of system’s parameters are acquired.FindingsTheoretical and numerical simulation results demonstrate rich variety of interesting dynamical behaviors such as period-doubling and Neimark–Sacker bifurcations, attractors’ crises in addition to chaotic attractors. The results demonstrated that the stability Nash equilibrium point of the game can be lost by period doubling or Neimark–Sacker bifurcations.Practical implicationsOligopoly games are pivotal in the mathematical modeling of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also macroeconomic analysis of business cycles, innovation and growth.Originality/valueAlthough the Cournot game and its variants have attracted great interest among mathematicians and economists since the time of its proposition till present, memory effects in continuous-time and discrete-time Cournot duopoly game have not been addressed yet. To the best of author’s knowledge, this can be considered as the first attempt to investigate this problem of fractional-order differentiated Cournot duopoly game. In addition, studying more realistic models of Cournot oligopoly games plays a pivotal role in the mathematical investigation and better understanding of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also in macroeconomic analysis of business cycles, innovation and growth.

DEREGULATION CONTROL BY MERGERS AND ACQUISITIONS: A GAME THEORETIC ANALYSIS OF THE CHINESE AIRLINE INDUSTRY

The major challenges of deregulation are lax market entry, sudden surge in new market entrants, and the intense price wars that ensue, thus causing major losses for any industry. This paper investigates whether deregulation can be structured through a controlled Mergers and Acquisitions (M&As) process by means of government intervention, and how this promotes the performance of the players in the industry. We study this in the context of the Chinese aviation industry as an ideal microcosm of our problem statement. This is because China’s civil aviation industry has witnessed many of the above challenges since its deregulation and economic reforms in 1979, which saw the beginning of a transformation from a fully state-owned machinery to a rent-seeking private sector. The post controlled deregulation process through M&As led to three dominant carriers: Air China Limited (AC), China Southern Airlines (CS), and China Eastern Airlines (CE). Using a 3-player non-cooperative perfect information Cournot oligopoly game model, the strategic efficacy of the government intervention to consolidate the industry based on operating expenses, air passenger revenue, and profit data are investigated respectively. All three airlines are better off after the exercise, with the industry facing a more sustainable growth by the intervention.

The persistence of social strategies under increasing competitive pressure

In this paper we study the long run outcome if firms are repeatedly matched to play a N-firm Cournot oligopoly game and can select their type of business objective. Firms can either be profit-maximizing or socially concerned, also pursuing consumer welfare. We use an indirect evolutionary approach to distinguish between the endogenous, profit-maximizing choice of the firms’ business objectives and their corresponding market behavior. Our main result is that the prospects of long run survival of firms pursuing social strategies strongly depends on the number of firms in the industry. If a social strategy entails higher marginal production costs, it is best suited if the number of firms in the industry is not too large. In particular, we demonstrate that the relation between long survival and the level of competitiveness of the industry is inverse U-shaped. We confirm this result by using the concept of coalition stability and further show that if a high number of firms adopt a social strategy, welfare might be lower than with just profit-maximizing firms.

Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products

This paper proposes a Cournot game organized by three competing firms adopting bounded rationality. According to the marginal profit in the past time step, each firm tries to update its production using local knowledge. In this game, a firm’s preference is represented by a utility function that is derived from a constant elasticity of substitution (CES) production function. The game is modeled by a 3-dimensional discrete dynamical system. The equilibria of the system are numerically studied to detect their complex characteristics due to difficulty to get an explicit form for those equilibria. For the proposed utility function, some cases with different value parameters are considered. Numerical simulations are used to provide an experimental evidence for the complex behavior of the evolution of the system. The obtained results show that the system loses its stability due to different types of bifurcations.

The impact of cost uncertainty on Cournot oligopoly games

In this paper, Cournot oligopoly model is examined with an uncertain quadratic cost function. A random quadratic cost function is introduced in this model. The model consists of three competed firms on which each firm in the model wants to maximize its expected profit and also wants to minimize its uncertainty by minimizing the cost. To handle this multi-objective optimization problem, the expectation and worst-case approaches are adopted used to study uncertainty. The existence and uniqueness of the equilibrium are obtained. The asymptotic behavior of the equilibrium point is also investigated. Complete stability and bifurcation analysis are provided. The obtained theoretical results are verified by numerical simulation.