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Physical review. E

Two-qubit entangling operators as chaos control in a discrete dynamic Cournot duopoly game

A. V. S. Kameshwari, S. Balakrishnan

Published Jan 5, 2024• 10 citations
Publisher

Abstract

The current trend in economics research is to incorporate quantum mechanical concepts to increase the security of business models. This interdisciplinary field of study represents real-world market dynamics more closely than do its classical counterparts. In this paper, we shed light on the significance of the two-qubit entangling operators in controlling chaos. We introduce a modified Eisert-Wilkens-Lewenstein scheme in a nonlinear Cournot duopoly game with complete and incomplete information. By doing so, the following interesting results are obtained: To begin, monopoly in a duopoly game can be avoided with the use of special perfect entanglers. Also, the stability analysis shows that there exists a class of entangling operators which can stabilize an unstable system and vice versa. Second, numerical analysis highlights the two-qubit entangling operators which can stabilize a chaotic system or at least delay chaos. Finally, we show that with an appropriate choice of initial state and speed of adjustments, entangling operators can decrease the sensitivity of the system. In short, while we know the importance of entangling operators in quantum game theory, in this paper we indicate the significance of operators in the context of a chaotic system.

DOI
10.1103/physreve.109.014207

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The current trend in economics research is to incorporate quantum mechanical concepts to increase the security of business models. This interdisciplinary field of study represents real-world market dynamics more closely than do its classical counterparts. In this paper, we shed light on the significance of the two-qubit entangling operators in controlling chaos. We introduce a modified Eisert-Wilkens-Lewenstein scheme in a nonlinear Cournot duopoly game with complete and incomplete information. By doing so, the following interesting results are obtained: To begin, monopoly in a duopoly game can be avoided with the use of special perfect entanglers. Also, the stability analysis shows that there exists a class of entangling operators which can stabilize an unstable system and vice versa. Second, numerical analysis highlights the two-qubit entangling operators which can stabilize a chaotic system or at least delay chaos. Finally, we show that with an appropriate choice of initial state and speed of adjustments, entangling operators can decrease the sensitivity of the system. In short, while we know the importance of entangling operators in quantum game theory, in this paper we indicate the significance of operators in the context of a chaotic system.
Abstract Recent economic research has focused on applying quantum mechanical models to understand complex markets. Real markets are usually complex with nonlinear inverse demand functions and dynamic participants. Furthermore, when these market games are quantized, players interact via quantum channels that are prone to decoherence. In this work, owing to the complexity of real markets, we introduce noise and memory in a dynamic quantum Stackelberg duopoly game for the first time. We shed light on the significance of two-qubit entangling operators, particularly special perfect entanglers (SPEs), in controlling chaos. The following noteworthy observations are made: First, although entangling operators cannot prevent monopolies in the market, they can shift leader dominance to follower dominance. Second, numerical analysis highlights the significance of the B-gate in a correlated noisy channel as a chaos control. We also highlight the significance of the controlled-NOT (CNOT), and double-CNOT (DCNOT) gates in controlling chaos in noisy, memoryless channel 2. Finally, we show that the CNOT and DCNOT gates are more sensitive than the B-gate. These novel features of two-qubit entangling operators in correlated, noisy and dynamic market games shed new light on the role of SPEs in quantum dynamic game settings.
It is well-known that the evolutionary dynamics is widely studied in classical game settings. In this work, for the first time, the notion of evolutionary dynamics is studied in the context of quantum Cournot duopoly game using a modified EWL scheme. Unlike the well-known Eisert-Wilkens-Lewenstein scheme and the Marinatto-Weber scheme, the modified EWL scheme admits a wide range of two-qubit entangling operators which are essential to understand game theory from the perspective of quantum operators. By applying this scheme and by using replicator dynamics, it is observed that these exists only one evolutionary stable strategy (ESS). Further, the identified stable strategy plays a crucial role in replacing the strategy-profit maximization (played by majority of the population) with the mutant strategy-revenue maximization. Also, these results highlight the insignificance of the entangling operators in deciding the ESS.
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 In the Cournot duopoly market, each firm is not completely rational, one firm's possession of information is not symmetrical, and their Nash equilibria are also usually inferior to Pareto optimality. The intersection of quantum game theory, nonlinear dynamics and asymmetric information may be an effective way to solve this problem. In this paper, we set up a multi-period quantum Cournot duopoly game model with bounded rationality based on asymmetric information, investigate the effects of quantum entanglement and asymmetric information on the local stability of quantum Nash equilibrium and complex dynamic behavior of the system. The result shows that an increase in the quantum entanglement decreases the quantum Nash equilibrium local stability and accelerates the occurrence of the system’s chaotic behavior, while the asymmetric information stabilizes the quantum Nash equilibrium and suppresses the chaotic behavior.The contradictive impact on the local stability of quantum Nash equilibrium creates the interesting properties of the game. Numerical simulations demonstrate these complex dynamical features via bifurcation, strange attractors, fractal dimensions and sensitivity to initial conditions.
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.