Bertrand Duopoly Research Articles

Emission taxes and environmental R&D risk choices in a duopoly market

This study investigates how an emission tax rate affects risk choices in environmental research and development (ER&D) in Cournot and Bertrand duopolies. Firms choose the variance of their ER&D projects and compete in a differentiated product market. We find that the degree of risk increases in the emission tax rate and decreases in the degree of product differentiation. We further find that firms take considerable welfare risks when the emission tax rate is high relative to marginal environmental damage. Moreover, we find that firms choose riskier projects under Bertrand competition than under Cournot competition. This study helps firm managers choose appropriate ER&D projects in their competitive environments and helps policymakers formulate reasonable environmental tax policies.

Information sharing in a Cournot–Bertrand duopoly

Studies on information sharing in oligopolies focus on either Cournot or Bertrand markets. We consider a Cournot–Bertrand market where owners provide strategic managerial incentives and can share the details of their compensation contracts. We find that the Cournot firm punishes its manager for sales, whereas the Bertrand firm rewards sales. Both firms share contract information if the firms' products are sufficiently differentiated. However, if product differentiation is low, then either the Cournot firm or the Bertrand firm keeps the contract information private. Mandating information sharing can lead to an increase in consumer and social welfare but harms firms' profits.

Efficiency of Cournot and Bertrand Equilibria with Non-cooperative and Cooperative R&D in Presence of Product Differentiation

Strategic cooperative and non-cooperative R&D decisions are analysed in the presence of R&D spillover and product differentiation. Using a two-stage game it is shown that in both Cournot and Bertrand duopolies, R&D and output levels are larger and prices are lower under cooperative R&D agreements compared to non-cooperative R&D. For complementary and independent goods, these results are valid for any degree of R&D spillover and may hold for substitute goods even for sufficiently small R&D spillover. In terms of social welfare, cooperative R&D agreements dominate non-cooperation for both Cournot and Bertrand duopolies. As for relative efficiency of Cournot and Bertrand equilibria, it is shown that the traditional efficiency ranking may be reversed and Cournot equilibrium may dominate Bertrand equilibrium if the degree of product differentiation is low and the products are reasonably close substitutes for high degree of R&D spillover provided that R&D productivity is high. This result is stronger for cooperative R&D decision. JEL Codes: D21, D43, L13

To abate, or not to abate? A strategic approach on green production in Cournot and Bertrand duopolies

This research analyses firms' strategic choice of adopting an abatement technology in an environment with pollution externalities when the government levies an emission tax to incentivise firms to undertake emission-reducing actions. A set of different Nash equilibria – ranging from dirty to green production – arises in both quantity-setting and price-setting duopolies. Results show that if societal awareness towards a clean environment is relatively low (resp. high) and the index measuring the relative cost of abatement is relatively high (resp. low), the strategic interaction between two independent, competing, and selfish firms playing the abatement game leads them not to abate (resp. to abate) as the Pareto efficient outcome: no conflict exists between self-interest and mutual benefit to not undertake (resp. to undertake) emissions-reducing actions. Multiple Nash equilibria or a “green” prisoner's dilemma can also emerge in pure strategies. When the choice of adopting a green technology is an anti-prisoner's dilemma (deadlock), the society is better off, as social welfare under abatement is always larger than under no abatement. These findings suggest that living in a sustainable environment encourages the improvement of public education systems for the achievement of an eco-responsible attitude and the development of clean technologies through ad hoc R&D. • The article analyses the firms' abatement game in a duopoly. • There are pollution externalities and a welfare-maximising emission tax. • A set of different Nash equilibria, ranging from dirty to green production, arises. • Results hold both in quantity-setting and price-setting strategic contexts. • Sustainability requires ad hoc R&D and an eco-responsible attitude.

Dynamics of a Bertrand Duopoly Game of the Greek Oil Market and Application of the d-Backtest Method

This work presents the complexity that emerges in a Bertrand duopoly between two companies in the Greek oil market, one of which is semi-public and the other is private. The game uses linear demand functions for differentiated products from the existing literature and asymmetric cost functions that arose after approaches using the published financial reports of the two oil companies (Hellenic Petroleum and Motor Oil). The game is based on the assumption of homogeneous players who are characterized by bounded rationality and follow an adjustment mechanism. The players’ decisions for each time period are expressed by two difference equations. A dynamical analysis of the game’s discrete dynamical system is made by finding the equilibrium positions and studying their stability. Numerical simulations include bifurcation diagrams and strange attractors. Lyapunov numbers’ graphs and sensitivity analysis in initial conditions prove the algebraic results and reveal the complexity and chaotic behavior of the system focusing on the two parameters k1 and k2 (speed of adjustment for each player). The d-Backtest method is applied through which an attempt is made to control the chaos that appears outside the stability space in order to return to the locally asymptotically stable Nash equilibrium for the system.

Complexity of a Bertrand duopoly game with homogeneous expectations, quadratic cost functions and chaos control

Complexity of a Bertrand duopoly game with homogeneous expectations, quadratic cost functions and chaos control

General equilibrium of Bertrand game: A spatial computational approach

<abstract> In this paper, the competitive equilibrium of the Bertrand game is discussed with bounded rationality and the utility function. When the parameters changing and the number of firms increasing, the competitive equilibrium valve of Bertrand duopoly game is gradually being blurred. When the number of competitors is more than four, it is very difficult to derive the value of the equilibrium points. How to find the general competitive equilibrium points for Bertrand game, which is studied from spatial agglomeration with mean value theorem. A Bertrand duopoly game is proposed based on a demand function. Celestial bodies motion as method is introduced to handle the number and stability of competitive equilibrium points, and the stable points is symmetry. The results are supported by numerical computation and simulations. </abstract>

The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference

In this paper, a Cournot model and its dual Bertrand model where firms have comprehensive preferences are developed. The preference was proposed by Bowles (2004) based on the results of Rabin (1993) and Lebine (1998). Comparable static method is used to illustrate the impact of parameters on equilibrium. Three scenarios are classified by preference parameter: completely cooperative, hostile to each other and general Cournot or Bertrand model. In each situation, Cournot model and its dual model present different chaos phenomenon which have various and abundant strange attractors, while the shapes of the stability regions are similar in the same situation. In the static setting, the local stability analysis of equilibrium gives the bounded region of quantity or price. The one-dimensional Logistic mapping is applied to studying the dynamics of the models on invariant axes. The critical curves classify different regions according to the number of preimages (q1, q2) or (p1, p2). Besides, simulations give more intuitive results: the cycle attractor, chaotic attractor and the basin of attraction with “holes” are presented. The article also provides new findings that under the assumption of comprehensive preference, the Cournot model with substitutes (complements) and the Bertrand model with complements (substitutes) are still duality models in the dynamic settings.

Asymmetric Information on Price Can Affect Bertrand Duopoly Players with the Gradient-Based Mechanism

We study a Bertrand duopoly game in which firms adopt a gradient-based mechanism to update their prices. In this competition, one of the firms knows somehow the price adopted by the other firm next time step. Such asymmetric information of the market price possessed by one firm gives interesting results about its stability in the market. Under such information, we use the bounded rationality mechanism to build the model describing the game at hand. We calculate the equilibrium points of the game and study their stabilities. Using different sets of parameter values, we show that the interior equilibrium point can be destabilized through flip and Neimark–Sacker bifurcations. We compare the region of stability of the proposed model with a classical Bertrand model without asymmetric information. The results show that the proposed game’s map is noninvertible with type Z 0 − Z 2 or Z 1 − Z 3 , while the classical model is of type Z 0 − Z 2 only. This explains the quite complicated basins of attraction given for the proposed map.

Quantum Stackelberg–Bertrand duopoly

We apply Li et al.’s “minimal” quantization rules (Phys. Lett. A 306, 73, 2002) to investigate the quantum version of the Stackelberg–Bertrand duopoly, especially how the quantum entanglement affects the second-mover advantage in the Stackelberg–Bertrand duopoly. It is found that positive quantum entanglement is more favourable to the profit of the leader and destroys the second-mover advantage.

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.

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.

Spatial simulation of the quantum Bertrand duopoly game

The quantum Bertrand duopoly game is studied in this work via spatial numerical simulation. It is found that the implemented simulation converges to the Pareto optimal solution if the prices are strongly entangled, even if such a solution is not in Nash equilibrium.

Environmental Technological Choice in a Cournot-Bertrand Model

This paper investigates the abatement technology choice in Cournot-Bertrand, as well as in pure Cournot and Bertrand duopolies. It is argued that, under certain conditions, the Bertrand firm will choose a “dirtier” technology compared to its Cournot rival. Furthermore, when competing in quantities, firms choose greener technologies than when competing in prices. This is because higher abatement rates make a firm choosing quantity relatively more aggressive than a firm choosing price, irrespective of the strategic variable of its rival. Moreover, it is shown that the introduction of abatement technologies will increase total output in the market and decrease total emissions.

Maximum‐revenue tariffs versus free trade

AbstractWelfare with the maximum‐revenue tariff is compared to free‐trade welfare under Cournot duopoly with differentiated products; under Bertrand duopoly with differentiated products; and under perfect competition in the case of a large country able to affect its terms of trade. Under Cournot duopoly and Bertrand duopoly, assuming linear demands and constant marginal costs, welfare with the maximum‐revenue tariff is always higher than free‐trade welfare. Under perfect competition, assuming linear demand and supply, welfare with the maximum‐revenue tariff will be higher than free‐trade welfare if the country has sufficient market power.

Competitive Bundling in a Bertrand Duopoly

In competitive industries, some firms bundle their products, whereas others unbundle them; still other firms occupy a niche position and offer only a subset of products. No general theory has been advanced to explain this variety of bundling strategies. We characterize the strategies of two symmetric firms competing (in a Bertrand fashion) with regard to two homogeneous components. We model their decisions as a two-stage non-cooperative game. Firms in the first stage select their product offerings, which may include any single-component product and/or the bundle; in the second stage, firms simultaneously set their products' prices. We show that three types of equilibria always emerge in equilibrium: (i) Differentiated duopoly, in which one firm offers the bundle and the other firm offers only a single-component product; (ii) Monopoly, in which one firm offers the full set of products and the other firm stays out of the business; (iii) Perfect competition, in which both firms offer both single-component products (possibly along with the bundle) and compete on price, driving their profits down to zero. Hence, bundling can be anti-competitive (if it is used to defend a monopolistic position) or hyper-competitive (if it leads to head-to-head competition). In case of a differentiated duopoly, the bundler has a profit advantage|which remains moderate when customer valuations are negatively correlated and highly heterogeneous|but might grow out of bound otherwise. Therefore, bundling does not necessarily hurt customer welfare or competition, but under the wrong set of circumstances, it could be a terrible competitive weapon.

Quantum Games with Unawareness with Duopoly Problems in View

Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common knowledge among the players. Our purpose is to determine reasonable game outcomes when players’ perceptions about what game is actually played are limited. To this end, we consider a collection consisting of the classical and quantum games that specifies how each player views the game and how each player views the other players’ perceptions of the game. We show that a slight change in how the players perceive the game may considerably affect the result of the game and, in the case of maximally correlated strategies, may vary from the inefficient Nash equilibrium outcome in the classical Cournot duopoly to the Pareto optimal outcome. We complete our work by investigating in the same way the Bertrand duopoly model.

Analysis of a dynamics duopoly game with two content providers

Abstract This paper explores the idea of Bertrand duopoly in a context of bounded rationality, where two content providers (CP) endeavor to maximize their own profit. Each CP varies his price and fixes the level of his credibility of content. We suppose that one CP adjusts his price strategy according to a kind of smoothed information which average his marginal profit in the previous period and the one in a delayed period, while another CP makes its price strategy as the best response, which is a function the opponent action in the current period. Through a detailed analysis, we calculate explicitly the equilibrium points of the dynamical system, and we show the stability of its Nash equilibrium under some conditions. Numerical results reveal that while varying the model parameters (speed of adjustment and side payment), complex dynamic behaviors would occur, such as chaos. In addition, an appropriate method of chaos controlling is applied to force the system to go back to its stabilization behavior.

Evolutionary selection against iteratively weakly dominated strategies

This paper provides sufficient conditions under which regular payoff monotonic evolutionary dynamics (a class of imitative dynamics that includes the replicator dynamic) select against strategies that do not survive a sequence of iterated elimination of weakly dominated strategies. We apply these conditions to Bertrand duopolies and first-price auctions. Our conditions also imply evolutionary selection against iteratively strictly dominated strategies.