Cournot Duopoly Game Research Articles

A Remanufacturing Duopoly Game Based on a Piecewise Nonlinear Map: Analysis and Investigations

Abstract A remanufacturing Cournot duopoly game is introduced based on a nonlinear utility function in this paper. What we mean by remanufacturing here is that the second firm in this game receives used products and remanufacture them and then sell them again in the market. The bounded rationality mechanism is used to form a piecewise system that describes this game in discrete time periods. This piecewise system depends on five parameters and is defined on two regions separated by a border curve. The fixed points of this system in each region are calculated and their stability is discussed. Numerical simulations for this system exhibit the occurrence of different types of multiple attractors. We also give examples of different stable periodic cycles and chaotic attractors that are separated by the border curve or passing through it.

Dynamical Study of Competition Cournot-like Duopoly Games Incorporating Fractional Order Derivatives and Seasonal Influences

Abstract Cournot’s game is one of the most distinguished and influential economic models. However, the classical integer order derivatives utilized in Cournot’s game lack the efficiency to simulate the significant memory characteristics observed in many economic systems. This work aims at introducing a dynamical study of a more realistic proposed competition Cournot-like duopoly game having fractional order derivatives. Sufficient conditions for existence and uniqueness of the new model’s solution are obtained. The existence and local stability analysis of Nash equilibrium points along with other equilibrium points are examined. Some aspects of global stability analysis are treated. More significantly, the effects of seasonal periodic perturbations of parameters values are also explored. The multiscale fuzzy entropy measurements for complexity are employed for this case. Numerical simulations are presented in order to verify the analytical results. It is observed that the time-varying parameters induce very complicated dynamics in perturbed Cournot duopoly game compared with the unperturbed game.

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.

Stability and Hopf Bifurcation of a Stochastic Cournot Duopoly Game in a Blockchain Cloud Services Market Driven by Brownian Motion

The blockchain technology has been rapidly popularizing and gaining a quick mileage in many domains owing to its innate nature, so there must be several tremendous market opportunities and uncertainties for the blockchain cloud service. We propose a Cournot duopoly game in a blockchain cloud services market for cyber-physical-social systems (CPSS). Introducing Brownian noise disturbance to the deterministic Cournot duopoly game, we construct a stochastic Cournot duopoly game in a blockchain cloud service market with uncertainty. We analyze the stochastic stability of the Cournot duopoly game by the Lyapunov exponent method, the Lyapunov function, and the singular boundary theory. Then we investigate the stochastic Hopf bifurcation of the Cournot duopoly game by the invariant measure theory. At last, we numerically validate these results by the probability density. These results of stability and Hopf bifurcation illustrate complex behaviors of the stochastic Cournot duopoly game in the blockchain cloud services market.

Strategic delegation under cost asymmetry revised

We revisit the Cournot duopoly game with strategic delegation and asymmetric costs of Delbono et al. (2016). In particular, the authors claim that a Prisoner Dilemma always arises. However, we show that, by setting properly the admissible parameter set, if the firms are sufficiently different, the efficient firm is better off when both firms delegate production. Therefore, in contrast with the traditional view, we claim that a Prisoner Dilemma is not an inevitable outcome in a strategic delegation game.

Simulation of the quantum Cournot duopoly game

This article studies the quantum Cournot duopoly game via the iterated confronting of a large number of players laying in a two-dimensional lattice. In every iteration, every player plays with his nearest partners and adopts the strategy of his best paid nearest mate. Variable degree of quantum entanglement is taken into consideration in the study.

Effect of information asymmetry in Cournot duopoly game with bounded rationality

We investigate the effect of information asymmetry on a dynamic Cournot duopoly game with bounded rationality. Concretely, we study how one player’s possession of information about the other player’s behavior in a duopoly affects the stability of the Cournot-Nash equilibrium. We theoretically and numerically show that the information stabilizes the Cournot-Nash equilibrium and suppresses chaotic behavior in the duopoly.

Price Discrimination in Dynamic Cournot Competition

This paper introduces a new Cournot duopoly game and gives an applied study for price discrimination in a market by dynamic methods. One of two oligopolies has two different prices for a homogeneous product, while the other charges one kind of price. It is found that there is only one stable equilibrium for the discrete dynamic system, and a corresponding stable condition is given. Using a discriminative price is not always beneficial to a firm in equilibrium. If both oligopolies carry out price discrimination, the market’s average price is lower than when only one oligopoly does it. The results are verified by numerical simulations.

Nonlinear dynamics of a quantum Cournot duopoly game with heterogeneous players

In this paper, a dynamic quantum Cournot duopoly game with heterogeneous players is proposed by applying the Li–Du–Massar scheme. We analyze the influence of the degree of quantum entanglement (DQE) on stability and dynamics behavior of the system. The result shows that: (i) The corresponding classical Cournot duopoly game is a special case of the dynamic quantum Cournot duopoly game. (ii) The stability region may increase or decrease as the DQE increases. It depends on the relative marginal cost difference. (iii) Numerical simulations show that the complicated dynamics behaviors of quantum Cournot game is different from that of corresponding classical model.

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.

Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation

In this paper, a dynamical two-stage Cournot duopoly game with R&D spillover effect and product differentiation is established. The stability of all the equilibrium points is studied using Jury criterion, and then the stability condition is given. The direction of flip bifurcation is given by using central manifold theorem and norm form theory. By numerical simulation, two routes to chaos are studied through 2-D bifurcation diagram, one of which is flip bifurcation and the other is Neimark-Sacker bifurcation at period-2 point. The system will lose its stability as the speed of adjustment increases. In addition, a common nonlinear phenomenon named intermittent chaos is observed in the built model. Two types of intermittent chaos are displayed in time series plots and phase diagrams, one of which is PM-III intermittent chaos and the other is induced by crisis. And then four types of coexistence of attractors are illustrated through basin of attraction, which are coexistence of periodic attractor and chaotic attractor, coexistence of multiple chaotic attractors, coexistence of periodic attractor, quasi-periodic attractor and chaotic attractor, and coexistence of multiple periodic attractors, respectively.

Equilibrium selection of a homogenous duopoly with extrapolative foresight

This paper conducts a further analysis concerning equilibrium selection in a widely-studied nonlinear Cournot duopoly game, where each firm is capable to forecast its rival's output in the near future through a straightforward extrapolative foresight technology. Compared with the prior related work in which just one firm can implement such technology, this newly developed adjustment mechanism is endowed with more rationality in the sense that there appear more intelligent firms. It turns out this more rational adjustment mechanism does lead to more convergence to asymmetric equilibrium behavior for a weak foresight ability. Our finding shows that some undesirable complex behavior such as fluctuations and divergent trajectories can be inhibited by increasing the number of such rational firms.

Nonlinear Phenomena in Cournot Duopoly Model

The economic world is very dynamic, and most phenomena appearing in this world are mutually interconnected. These connections may result in the emergence of nonlinear relationships among economic agents. Research discussions about different markets’ structures cannot be considered as finished yet. Even such a well-known concept as oligopoly can be described with different models applying diverse assumptions and using various values of parameters; for example, the Cournot duopoly game, Bertrand duopoly game or Stackelberg duopoly game can be and are used. These models usually assume linear functions and make analyses of the behavior of the two companies. The aim of this paper is to consider a nonlinear inverse demand function in the Cournot duopoly model. Supposing there is a sufficiently large proportion among the costs of the two companies, we can possibly detect nonlinear phenomena such as bifurcation of limit values of production or deterministic chaos. To prove a sensitive dependence on the initial condition, which accompanies deterministic chaos, the concept of Lyapunov exponents is used. We also point out the fact that even though some particular values of parameters are irrelevant for the above-mentioned nonlinear phenomena, it is worth being aware of their existence.

Metric Subregularity in Generalized Equations

In this article, we study the metric subregularity of generalized equations using a new tool of nonsmooth analysis. We obtain a sufficient condition for a generalized equation to be metrically subregular, which is not a necessary condition for metric regularity, using a subtle adjustment of the Mordukhovich coderivative. We apply these results to the study of the metric subregularity in a Cournot duopoly game.

Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production

In this paper, a two-stage oligopoly game of semi-collusion in production is analyzed and expounded, where at first stage all firms compete in R&D and at second stage all firms coordinate the production activities in order to make their joint profit maximized. Not only the local stability of equilibriums, but also the existence, stability and direction of flip bifurcation of the discrete nonlinear model are investigated by using the normal form method and the center manifold theory. Then, the validity of the theoretical analysis is justified through numerical simulation. We find that the model we built can exhibit very complex dynamical behaviors, but it cannot undergo the Neimark–Sacker bifurcation. Also the coexistence of attractors is found through numerical simulation, and their basins of attraction are simulated. At last of this paper, the chaotic motion of the proposed model is controlled by delayed feedback control method.

Emotions in a repeated Cournot duopoly game: A psychophysiological experiment.

The reason why cooperation occurs in repeated games has remained a puzzle. Earlier literature has maintained that reciprocal behavior that gives rise to cooperation can be entirely self-regarding. However, experimental evidence shows that reciprocal behavior is other-regarding in many one-shot games. This other-regarding behavior is believed to have an emotional foundation. We hypothesized that emotions play a role in reciprocal behavior in repeated games as well. We tested this hypothesis by measuring the psychophysiological correlates of emotions from pairs of subjects as they played a repeated Cournot duopoly game. The players, who were in different rooms and remained anonymous to each other, made adjustment decisions to their production quantities that determined their payoffs in each round. Autonomic nervous system arousal was activated when the payoffs of both players decreased in a round, whereas positive affect was expressed when the payoffs of both players increased in a round. The disgust expression was related to a player's own one-sided increase in the payoff. Anger was expressed occasionally but less frequently when the outcome was the player's ideal outcome. An upwards adjustment of the production quantity was observed when the other player did not cooperate. This had the effect of decreasing the payoffs of both players and this was also related to an increase in the level of arousal. Our results provide evidence on how emotions are present in reciprocal behavior in a repeated social dilemma game. The results challenge recent behavioral research that advocates self-regarded motivations of cooperation in repeated games.

Nonlinear Complex Dynamics of Carbon Emission Reduction Cournot Game with Bounded Rationality

Based on the hypothesis of participant’s bounded rationality, our study formulated a novel Cournot duopoly game model of carbon emission reduction and, subsequently, analyzed the dynamic adjustment mechanism of emission reduction for enterprises. The existence and stability of the equilibrium solution of game are further discussed by the nonlinear dynamics theory. Our findings revealed that the parameters have key significance on the dynamic properties of the system. However, when the adjustment speed gets too large, the system loses the original stability and vividly demonstrates complex chaos phenomenon. Higher market prices in carbon trading have an outstanding impact on the stability of the system, which easily leads to system instability. Our study further controlled the chaos behavior of the power system by the delay feedback control. The results of the numerical analysis depict that the unstable behavior of the dynamic system can be controlled efficiently and quickly, in the quest to restore back a stable and orderly market. Our novel method is proved to have provided decision makers with effective solution to market instability.

Supplemental Material for Emotions in a Repeated Cournot Duopoly Game: A Psychophysiological Experiment

Supplemental Material for Emotions in a Repeated Cournot Duopoly Game: A Psychophysiological Experiment

Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control

Many researchers have used quadratic utility function to study its influences on economic games with product differentiation. Such games include Cournot, Bertrand, and a mixed-type game called Cournot-Bertrand. Within this paper, a cubic utility function that is derived from a constant elasticity of substitution production function (CES) is introduced. This cubic function is more desirable than the quadratic one besides its amenability to efficiency analysis. Based on that utility a two-dimensional Cournot duopoly game with horizontal product differentiation is modeled using a discrete time scale. Two different types of games are studied in this paper. In the first game, firms are updating their output production using the traditional bounded rationality approach. In the second game, firms adopt Puu’s mechanism to update their productions. Puu’s mechanism does not require any information about the profit function; instead it needs both firms to know their production and their profits in the past time periods. In both scenarios, an explicit form for the Nash equilibrium point is obtained under certain conditions. The stability analysis of Nash point is considered. Furthermore, some numerical simulations are carried out to confirm the chaotic behavior of Nash equilibrium point. This analysis includes bifurcation, attractor, maximum Lyapunov exponent, and sensitivity to initial conditions.

Violence and law enforcement in markets for illegal goods

In this article, I try to establish optimal law enforcement efforts in markets for illegal goods taking into account both consumption and violence externalities. I model competition between firms as a Cournot duopoly game where they produce an illegal good and sabotage each other to gain a larger share of the market. I show that socially optimal law enforcement can result in any of the following corner solutions: letting firms produce freely, partially intervene punishing one firm more than the other, or fully intervene to eliminate them both. Which solution is optimal depends on the size of consumption and violence externalities; the direct costs of law enforcement and sabotage; the weight of profits in the welfare function of the authority; and how cautious is the authority avoiding violence externalities while enforcing the law.