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Europhysics Letters (EPL)

Quantum Cournot duopoly game with general isoelastic demand and asymmetric production costs

Xinli Zhang, Xiaohan Zhang, Hou Wen

Published May 30, 2022• 7 citations
Publisher

Abstract

Abstract This letter explores the quantum version of a Cournot duopoly game with general isoelastic demand and asymmetric production costs by applying Li-Du-Massar's minimal quantization rules, and it especially analyzes the existence region of quantum equilibrium, and the influences of quantum entanglement , difference in marginal costs ( k ) and elasticity of demand on the optimal profits of both firms. The results show that the existence region decreases with γ and k increasing. A larger elasticity of demand can destroy the profits of both firms. If positive γ and k are more favourable to the profits of two firms. If the first firm's profit increases with γ increasing for fixed k , but decreases with k increasing for fixed γ . The second firm's profit increases with k increasing for any fixed γ . As to the influences of γ on the second firm's profit, when k is less than a critical value, it increases with γ increasing, otherwise it decreases with γ increasing for fixed k .

DOI
10.1209/0295-5075/ac747d

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Abstract This letter explores the quantum version of a Cournot duopoly game with general isoelastic demand and asymmetric production costs by applying Li-Du-Massar's minimal quantization rules, and it especially analyzes the existence region of quantum equilibrium, and the influences of quantum entanglement , difference in marginal costs ( k ) and elasticity of demand on the optimal profits of both firms. The results show that the existence region decreases with γ and k increasing. A larger elasticity of demand can destroy the profits of both firms. If positive γ and k are more favourable to the profits of two firms. If the first firm's profit increases with γ increasing for fixed k , but decreases with k increasing for fixed γ . The second firm's profit increases with k increasing for any fixed γ . As to the influences of γ on the second firm's profit, when k is less than a critical value, it increases with γ increasing, otherwise it decreases with γ increasing for fixed k .
The development of quantum information theory based on quantum entangled states is advancing rapidly and has promoted the emergence of quantum game theory. Quantum game theory is quantum information processing applied to classical games, which solves problems that classical games cannot solve. This paper explores the quantum Cournot-Bertrand game model of two firms in a duopoly market with bounded rationality and product substitutability. The first and second firms choose production output and sales price as their decision variables, respectively. We first define the quantum Cournot-Bertrand dynamical system and study the existence and locally stable conditions of equilibrium points. Then we explore bifurcation conditions and bifurcation types. Finally, we do the bifurcation analysis and explore the impacts of the model parameters as the potential market size of products, the two firms' linear marginal cost, the players' speed of strategy adjustment, the product substitution index, and the quantum entanglement degree on the Nash equilibrium stability and bifurcation using numerical analysis and simulation, and provide economic explanations for the results.
Based upon a modification of Li et al. 's "minimal" quantization rules ( Phys. Lett. A 306 (2002) 73), we investigate the quantum version of the Cournot and Bertrand oligopoly. In the Cournot oligopoly, the profit of each of the N firms at the Nash equilibrium point rises monotonically with the measure of the quantum entanglement. Only at maximal entanglement, however, does the Nash equilibrium point coincide with the Pareto optimal point. In the Bertrand case, the Bertrand Paradox remains for finite entanglement ( i.e. , the perfectly competitive stage is reached for any N ⩾2), whereas with maximal entanglement each of the N firms will still have a non-zero shared profit. Hence, the Bertrand Paradox is completely resolved. Furthermore, a perfectly competitive market is reached asymptotically for N → ∞ in both the Cournot and Bertrand oligopoly.
On the basis of the bounded rationality and adaptive expectations, we investigate the dynamics of quantum Cournot duopoly game with heterogeneous players by using the Li et al. scheme. Further, we discuss the effect of quantum entanglement on local stability of the system. The results show that: (i) The special form of quantum Cournot duopoly game is the classical Cournot duopoly game, and the quantum Cournot duopoly game with naïve expectation is also a special form of quantum Cournot duopoly game with adaptive expectation. (ii) The increase or decrease of the stable region is dependent on the degree of quantum entanglement. It is related to the difference of relative marginal cost. (iii) The numerical simulation reveals the complexity characteristics of quantum dynamic Cournot game system through the analysis of stability region, bifurcation, strange attractor and sensitivity to initial conditions, and shows the difference from the classical dynamic Cournot model.
Abstract Drawing inspiration from the Frackiewicz and Li-Du-Massar quantum schemes, we provide some multiplayer generalizations of their techniques. Initially, we introduce a multiplayer version of the Frackiewicz quantum scheme, and give quantum Cournot oligopoly games with both three and n players. Our findings indicate that quantum equilibrium profits decrease monotonically with increasing entanglement levels of environments participated by oneself, but increase monotonically with the entanglement levels of other players. Subsequently, we present a multiplayer extension of the Li-Du-Massar quantum scheme, and analyze quantum Cournot oligopoly games with three and n players. We observe that quantum equilibrium profit exhibits a positive correlation with entanglement levels. Furthermore, the quantum equilibrium quantity and profit exhibit a negative correlation with the number of players for both multiplayer quantum schemes.
The aim of this paper is to investigate Cournot-type competition in the quantum domain with the use of the Li-Du-Massar scheme for continuous-variable quantum games. We derive a formula which, in a simple way, determines a unique Nash equilibrium. The result concerns a large class of Cournot duopoly problems including the competition, where the demand and cost functions are not necessary linear. Further, we show that the Nash equilibrium converges to a Pareto-optimal strategy profile as the quantum correlation increases. In addition to illustrating how the formula works, we provide the readers with two examples.