Duopoly Game Research Articles

On complex dynamics of Cournot-Bertrand game with asymmetric market information

A Cournot-Bertrand duopoly game that is characterized as bounded rational firms is introduced by a discrete dynamical map. The second firm in the game is characterized by knowing some information about the next time production of its opponent. The game’s equilibrium points are calculated and their conditions which ensuring stability are obtained for the boundary points. Due to the complex form of Nash point its stability loss is analyzed under varying some of the game’s parameters. The numerical simulation of Nash equilibrium point gives rise to periodic and chaotic attractors. Using some parameters’ values the structure of basins of attraction for some attracting set that changes that structure from simple to complex is determined. We also calculate the critical curves of the map’s game and show that it is noninvertible.

Common Ownership and Corporate Social Responsibility

Abstract This paper studies the effect of common ownership on corporate social responsibility (CSR). We find that common ownership is positively associated with a firm’s CSR score. The effect is stronger for firms in more competitive industries. We propose a two-stage duopoly game in which CSR serves as a commitment device to expand output aggressively to understand the empirical results. (JEL G30, D21, D22, L13, L21, L22) Received December 10, 2019; editorial decision September 9, 2020 by Editor: Gregor Matvos. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Dynamics of a heterogeneous quantum Cournot duopoly with adjusting players and quadratic costs

Based on bounded rationality and adaptive expectation, this paper sets up a dynamic quantum Cournot duopoly game model by introducing quadratic costs instead of linear costs and analyzes the stability of quantum equilibria, the impact of quantum entanglement on stability region and dynamic behavior of the system. The result shows that the stability region is affected by cost coefficients difference and quantum entanglements as key parameters between models with quadratic and linear costs and is especially found to be smaller than those with linear costs. Chaotic dynamics is confirmed to depend mainly on the adjustment speeds of bounded rational player, and the quantum entanglements can accelerate or delay the occurrence of the system’s chaotic behavior depending on cost coefficient differences. Numerical simulations demonstrate the complexity features of the dynamic system, stress similarities and differences between models with quadratic and linear costs and give new evidences in comparison with linear costs.

Quantum Stackelberg duopoly game with isoelastic demand function

We apply Frąckiewicz's scheme to investigate a quantum Stackelberg duopoly game with isoelastic demand function and unequal marginal costs. We analyze the existence conditions of quantum equilibrium, and the influences of relative marginal cost (m) and degree of quantum entanglement (γ) on the optimal profits and the first-mover advantage in theory. We demonstrate the theoretical analysis by numerical simulations. The results show that an increase of m will increase one player's profit, but will decrease the other player's profit for any fixed γ. An increase of γ will increase both players' profits for any fixed m. An increase of γ will enhance the first-mover advantage for 0<m<1, but will suppress it for 1<m<2.

Pricing and strategy selection in a closed-loop supply chain under demand and return rate uncertainty

Closed-loop supply chain (CLSC) decision-making involves many uncertainties, which makes the decision-making process more complex and diversified. This study considered a two-stage CLSC consisting of an original manufacturer and a third-party recycler. Without any government policy support, considering the effects of market demand, product return rate, and consumer perceived value, a CLSC decision model based on market demand with a [0,1] distribution was established. The model analyzes three situations—a manufacturer monopoly, the Cournot duopoly game, and the Stackelberg competition game—and solves them. The optimal values of decision variables such as optimal pricing, market demand, and all parties’ profits in the CLSC are obtained, and a strict mathematical proof is given. Through the model-solving process, the effects of product return rate and consumer perceived value on decision variables are analyzed; then, the profit allocation between the original manufacturer and the third-party recycler under different cooperation modes is analyzed. In addition, the four combinations of competition and cooperation are analyzed based on game theory. The Nash equilibrium solution and Pareto optimal solution of the four modes are analyzed by drawing a bimatrix Nash equilibrium table. The results indicate that the cooperation–cooperation mode is difficult to produce automatically, and government policy guidance and support are often needed to achieve Pareto optimality. Finally, a numerical example is given to validate the proposed model. In this way, the proposed model provides reliable theoretical support for the decision-making of both sides in a CLSC.

A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit

The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions.

Investigating the article processing charge of journals in the gold open access market: A game theory approach

AbstractAs a promising solution for enhancing knowledge communication as well as alleviating financial pressure of institutional libraries, gold open access (gold OA) has attracted wide attention all over the world. But it is a hard work to fully disclose how the equilibrium article processing charge is established in the gold OA market. To deal with this challenge, this paper firstly formulates the competition among journals in this market as a three‐stage Hotelling duopoly game, which is able to reveal its dynamic and competitive features affected by academic reputation, publication delays and article processing charges of journals. Then backward induction is applied to derive the market equilibria. Finally, an empirical study with 1,346 journals is conducted to verify the credibility of the theoretical solutions. The results show that gold OA journal with relatively higher academic reputation and shorter publication delay could charge higher article processing charge. And the publication delays of gold OA journals tend to cluster together in a limited time interval.

Nonzero-Sum Stochastic Differential Reinsurance Games with Jump–Diffusion Processes

In this paper, a nonzero-sum stochastic differential reinsurance game is studied. A model including controls for the market share (advertising), investment, and reinsurance policies is considered. A jump–diffusion process is used to represent insurance claims. Necessary conditions that would lead to the Nash equilibrium can be found in the duopoly game we consider. Cases with and without controls on the reinsurance are discussed separately. Closed-form solutions for optimal strategies are derived by applying the Hamilton–Jacobi–Bellman equation. Numerical examples are given to validate the correctness of our results.

Common Ownership and Corporate Social Responsibility

This paper studies the effect of common ownership on corporate social responsibility (CSR). We find that common ownership is positively associated with a firm's CSR score. The effect is stronger for firms in more competitive industries. We propose a two-stage duopoly game in which CSR serves as a commitment device to expand output aggressively to understand the empirical results.

The chaotic dynamics of a quantum Cournot duopoly game with bounded rationality

This paper analyzes the chaotic dynamics of a quantum Cournot duopoly game with bounded rational players by applying quantum game theory. We investigate the impact of quantum entanglement on the stability of the quantum Nash equilibrium points and chaotic dynamics behaviors of the system. The result shows that the stability region decreases with the quantum entanglement increasing. The adjustment speeds of bounded rational players can lead to chaotic behaviors, and quantum entanglement accelerates the bifurcation and chaos of the system. Numerical simulations demonstrate the chaotic features via stability region, bifurcation, largest Lyapunov exponent, strange attractors, sensitivity to initial conditions and fractal dimensions.

Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game

The Bischi–Naimzada game is a market competition between two firms with the objective of maximizing profits under limited information. In this article, we study a more generalized and realistic situation that takes into account the sales constraints. we generalize the economic model suggested by Bischi–Naimzada by introducing and studying the maximization of profits based on sales constraints. Our motivation in this paper is the studying of profit and sales constraints maximization and their influences on the game’s dynamics. The local stability of the equilibrium points of the proposed model is discussed. It examines how the dynamics of the proposed two-dimensional competition game model focusing on changes in both the speed of the adjustment and the sales constraint parameters. The map describing the game is proven to be noninvertible and yields many multi-stable, complex dynamics and the coexistence chaotic attractors may arise. The global behavior of the map is achieved by studying the critical curves. The numerical simulations demonstrate the coexistence of two attractors and complex structures of the attraction basins. Several examples are discussed in order to confirm all the analytical results obtained.

Complex Dynamical Behaviors of a Mixed Duopoly Game Based on Intellectual Property Rights Protection

Based on the oligopoly game theory and the intellectual property rights protection policy, we investigate the complex dynamical behaviors of a mixed duopoly game with quadratic cost. In the new system, a few parameters are improved by considering intellectual property rights protection and the stability conditions of the Nash equilibrium point are discussed in detail. A set of the two-dimensional bifurcation diagrams is demonstrated by using numerical modeling, and these diagrams show abundant complex dynamical behaviors, such as coexistence of attractors, different bifurcation, and fractal structures. These dynamical properties can present the long-run effects of strengthening intellectual property protection.

A Dynamic Duopoly Cournot Model with R&amp;D Efforts and Its Dynamic Behavior Analysis

In this paper, a dynamic two-stage Cournot duopoly game with R&amp;D efforts is built. Then, the local stability of the equilibrium points are discussed, and the stability condition of the Nash equilibrium point is also deduced through Jury criterion. The complex dynamical behaviors of the built model are investigated by numerical simulations. We found that the unique route to chaos is flip bifurcation, and the increase of adjusting speed will cause the system to lose stability and produce more complex dynamic behavior. In addition, we also found the phenomenon of multistability in the given model. Several kinds of coexistence of attractors are shown. In particular, we found that boundary attractors can coexist with internal attractors, which also aggravates the complexity of the system. At last, the chaotic state in the built system has been successfully controlled.

Lifetime Employment and Stackelberg Mixed Duopoly Games with a Foreign Labour-Managed Competitor

This article considers two Stackelberg games in which a state-owned firm competes against a foreign labour-managed firm. The first game is as follows. In the first stage, the state-owned firm decides whether to offer lifetime employment as a strategic commitment device. In the second stage, the foreign labour-managed firm decides whether to offer lifetime employment as a strategic commitment device. In the third stage, both firms simultaneously and independently choose actual outputs. The structure of the second game is nearly identical and differs only in the order in which the firms decide on the offer of lifetime employment in the first two stages. This study presents the equilibrium outcomes of these two Stackelberg games. JEL: C72, D21, F23, L30

A discrete fractional-order Cournot duopoly game

This paper focuses on associating the Cournot duopoly problem with long-memory effects. First of all, we generate a discrete fractional-order Cournot duopoly game by introducing the Caputo fractional-order difference calculus to the classical duopoly theory. In the fractional-order game, participants can make their decisions by taking full advantage of their historical information. Then we discuss both Nash equilibria and local stability of the game by employing the linear approximation theory. At last, we numerically validate the main results by using bifurcation diagrams, phase portraits, the largest Lyapunov exponent, and the 0–1 test algorithms.

A Bertrand Duopoly Game with Long-Memory Effects

Reconsidering the Bertrand duopoly game based on the concept of long short-term memory, we construct a fractional-order Bertrand duopoly game by extending the integer-order game to its corresponding fractional-order form. We build such a Bertrand duopoly game, in which both players can make their decisions with long-memory effects. Then, we investigate its Nash equilibria, local stability, and numerical solutions. Using the bifurcation diagram, the phase portrait, time series, and the 0-1 test for chaos, we numerically validate these results and illustrate its complex phenomena, such as bifurcation and chaos.

Equilibrium existence in the Hotelling model with convex production costs

We revisit the spatial duopoly model à la Hotelling (1929), to show that, provided the parameter scaling marginal cost is sufficiently high, quadratic production costs guarantee equilibrium existence in presence of linear transportation costs and a uniform distribution, with minimum product differentiation and no undercutting. We also discuss the conditions under which partial coverage arises. Finally, we extend the duopoly game to generalise the condition for the existence of the pure-strategy equilibrium to a class of convex production cost functions.

The Influences of Asymmetric Market Information on the Dynamics of Duopoly Game

We investigate the complex dynamic characteristics of a duopoly game whose players adopt a gradient-based mechanism to update their outputs and one of them possesses in some way certain information about his/her opponent. We show that knowing such asymmetric information does not give any advantages but affects the stability of the game’s equilibrium points. Theoretically, we prove that the equilibrium points can be destabilized through Neimark-Sacker followed by flip bifurcation. Numerically, we prove that the map describing the game is noninvertible and gives rise to several stable attractors (multistability). Furthermore, the dynamics of the map give different shapes of quite complicated attraction basins of periodic cycles.

The dynamics of a quantum Bertrand duopoly with differentiated products and heterogeneous expectations

Based on the bounded rationality and naïve expectation, the paper sets up a dynamic model of quantum Bertrand duopoly game with differentiated products by applying quantum game theory. We analyze the effect of quantum entanglement on stability of the quantum equilibrium points and complex dynamic behaviors. The result shows that the quantum entanglement can enhance the stability of the system. The adjustment speeds of bounded rational player can lead to chaotic characteristics of the system, the quantum entanglement can control the system’s chaos. Numerical simulations demonstrate the chaotic features via bifurcation, largest Lyapunov exponent, strange attractors, fractal dimensions and sensitivity to initial conditions.

Effective ambient charges on non-point source pollution in a two-stage Bertrand duopoly

ABSTRACT This paper analyses the effectiveness of the ambient charges for controlling emissions of non-point source pollutions. To this end, we construct a two-stage Bertrand duopoly game, in which optimal abatement technologies are chosen first and then the optimal prices as well as the optimal productions are determined. It is shown that the ambient charge is always effective at the second stage. Since the effect could be ambiguous at the first stage, this paper sheds light on the conditions under which the ambient charge becomes effective.