Oligopoly Game Research Articles

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.

Distributed-Observer-Based Nash Equilibrium Seeking Algorithm for Quadratic Games With Nonlinear Dynamics

In this article, a class of distributed quadratic games is considered over an undirected graph. The issue on the communication topology restriction is introduced and the players’ dynamics are with nonlinear dynamics and unknown time-varying perturbation. A distributed Nash equilibrium seeking algorithm is proposed based on a high-gain observer method, and the convergence is analyzed by the Lyapunov stability theory. It is shown that each player estimates the rival players’ states, and the errors between the players’ states and the Nash equilibrium are ultimately bounded by a small bound. Moreover, the presented algorithm is free of chattering phenomena because it is designed using the hyperbolic tangent function instead of the signum function to dominant the perturbation. The effectiveness of the proposed algorithm is validated via a simulation of the oligopoly game in which five firms produce the same products in a duopoly market structure.

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 Differential Oligopoly Game with Hyperbolic Demand, Cubic Cost and Capital Accumulation

Dynamic Differential Oligopoly Game with Hyperbolic Demand, Cubic Cost and Capital Accumulation

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.

Adaptive rules for discrete-time Cournot games of high competition level markets

For the Cournot-like oligopoly games of n firms, where the Cournot adjustment fails to converge, we propose adjustment processes originating from the family of the Moving Averages. In markets of linear demand, where firms have private and linear on quantities cost functions, these adaptive rules turn the games into discrete-time linear systems with delays. With an out of the box proof, we determine the least number of delays (m) that ensures the game of n players converges to its equilibrium. The Simple Moving Average rule (fixed number of delays) and the Cumulative Moving Average rule (constantly increasing number of delays), which is also known as “fictitious play”, are the main rules considered. Along with a hybrid rule, result of their combination, they are all studied for their convergent properties and compared in a bench-marking framework to indicate the different trajectories and identify their suitability in applications.

A limit theorem for the core of Betrand oligopoly games with externalities

This paper formulates the concept of Betrand oligopoly partition games. Furthermore, it is demonstrated that such well-defined games have positive externalities. More importantly, the core is proved to approximate the Equal Division solution in infinite markets.

Global Dynamics and Synchronization in a Duopoly Game with Bounded Rationality and Consumer Surplus

Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.

Capacity investment and green R&D in a dynamic oligopoly under the potential shift in environmental damage

We develop a differential oligopoly game to investigate firms’ capacity investment and green R&D efforts in the presence of the potential shift in environmental damage and under the spillover effect of R&D activities among firms. We find that when both the probability of potential shift in environmental damage and the efficacy of R&D activities are high, the spillover effect will discourage the R&D effort but encourage the capacity investment. Otherwise, the spillover effect will encourage the R&D effort but discourage the capacity investment. Moreover, the potential shift in environmental damage can significantly impact the capacity and green R&D decisions as well as the Pigouvian tax, especially in the case of a large number of firms, a high profitability of the product, a high level of interest rate, and a high level of R&D spillover among firms.

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.

Dynamic efficiencies of the 1997 Boeing‐McDonnell Douglas merger

AbstractWe evaluate the welfare effects of the 1997 Boeing‐McDonnell Douglas merger in the medium‐sized, wide‐body aircraft industry. We find that the merger led to lower prices. To explain the price drop, we develop a dynamic oligopoly game with learning‐by‐doing. We quantify the welfare effects of the merger by incorporating both increased market power and merger efficiencies from accelerated learning‐by‐doing. Our dynamic analysis indicates that net consumer surplus increased by as much as $5.14 billion, whereas a static model ignoring efficiencies of learning‐by‐doing predicts a $0.92 billion loss.

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.

Equivalence results when firms compete in prices/quantities and innovation effort

I consider the classical oligopoly game in which firms choose prices or quantities at the same time each chooses an innovation effort that directly establishes its (constant) marginal cost of production. I show how this game can be viewed as completely equivalent to a standard oligopoly game in which firms simply choose prices or quantities given a strictly concave cost of producing output. Hence, the effect of changing parameters unrelated to technology (e.g., the number of competing firms, the size of the market, or the number of merging firms) can be studied with no loss in a setting in which firms cannot innovate and produce using an increasing marginal returns technology. Exactly the same equivalence persists when performing welfare analysis, so the comparison between welfare benchmarks and equilibrium outcomes is the same as when firms choose prices or quantities given a strictly concave cost of producing output.

Optimal taxation in a common resource oligopoly game

In this paper, we study the optimal tax policy in a differential oligopoly game where the competing firms share the access to a productive renewable resource. We show that, in a Feedback Nash Equilibrium of the game, a linear Markov tax, imposed on the output, and specified as an affine function of the available resource stock, leads the competing firms to produce the socially optimal quantities over time, thus overcoming the dynamic interplay between the tragedy of the commons and the firms’ market power. The optimal tax turns out to be independent from the resource stock in a monopoly, and it cannot be defined in a duopoly.

A Differential Oligopoly Game Under Price Competition

A dynamic actual demand function was used to portray a market environment for fast moving consumer goods and to establish oligopoly differential games under price competition. We confirmed the stable point in n-player price competition as the saddle point of differential games, and acquired the optimal price and demand at equilibrium. Analysis on optimal price and demand shows that, to obtain more profits, a manufacturer should control costs, rapidly occupy the market, and improve product uniqueness.

Stackelberg versus Cournot: A differential game approach

We propose an infinite-horizon differential oligopoly game where, at each point in time, m Stackelberg leaders and n−m Stackelberg followers exploit a common-pool renewable resource and sell their harvest in the marketplace at a price that depends on total harvest. We derive a feedback-generalized-Stackelberg–Nash–Cournot equilibrium (a generalization of the feedback Stackelberg equilibrium), nesting feedback Stackelberg and feedback Cournot as special cases. We proceed with a comparison between the feedback Stackelberg and the feedback Cournot equilibria, and find a number of interesting results in contrast with “static” oligopoly theory. As to the relative efficiency of the two equilibria, we show that the Cournot equilibrium can be more efficient than the Stackelberg equilibrium. This holds true in the short-run, at the steady-state, and in terms of discounted welfare.

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.

A dynamic differential oligopoly game with sticky price and advertising

PurposeThe purpose of this study is to analyze comparatively the properties of open-loop and closed-loop equilibria in a dynamic oligopoly model with price dynamics and reflexive behavior of market agents.Design/methodology/approachTo consider dynamic competitive markets, the authors focus on a differential game theory in oligopolistic structures, using analytical models to illustrate how advertising effort, good differentiation and price stickiness interact simultaneously in the open-loop and the closed-loop Nash equilibria. The comparative assessment of these equilibria obtains some significant results.FindingsAn optimization model that enriches the continuous time is presented. Under the open-loop and the closed-loop, Nash equilibrium showed an increase in the total output, advertising in price stickiness and promotional efficiency, while there was a decrease in product differentiation and advertising promotional efficiency. However, the open-loop equilibrium levels are larger than the closed-loop equilibrium. Under the closed-loop information, the long-run equilibrium was faster than the opened-loop in a dynamic oligopoly. The graphical illustration was used to present the behavior of the model parameters.Practical implicationsThis study helps managers to choose an appropriate price and advertising adjustment to maximize profit. The obtained results may help firms to make the smart decision and may provide managers the valuable tool for making decisions in the competitive market environments.Originality/valueThis is a first attempt to analyze a dynamic oligopoly in the differentiated market environment. It considers a joint action of the output and advertising in shaping the closed-loop and the open-loop equilibria with N competitors in a dynamic competitive setting.

Technology choice in an evolutionary oligopoly game

In this paper, we propose and analyze a two-stage oligopoly game in which firms first simultaneously choose production technologies and in the second stage simultaneously choose production quantities. After characterizing the Nash equilibrium of the game, we cast our static model in a dynamic setting exploring the stability properties of the market equilibrium in two different cases: (i) exogenously distributed technologies and Cournot adjustments and (ii) endogenously distributed technologies in an infinite population game with Cournot–Nash equilibrium outputs. The main aim of the paper is that of extending the results about Cournot oligopoly stability in an evolutionary setting of heterogeneous decreasing returns-to-scale technologies. We show how the interplay between production decisions and R&D decisions can generate endogenous market fluctuations leading to complex dynamic phenomena.

On the first mover advantage in Stackelberg quantity games

In economic models, “sales equals production” is typically treated as an identity and not as an equilibrium outcome. This distinction, however, matters when production is sequential because of off-equilibrium path behavior. This paper shows that the first mover advantage in the standard Stackelberg oligopoly game in quantities may be reduced when “sales equals production” is no longer treated as an identity. Moving first does not per se produce a strategic advantage. It is only first moves that are sufficiently costly that produce this advantage. With costless production, the advantage disappears completely and the Cournot–Nash outcome is obtained.