Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Evolutionary Quantum Cournot duopoly game in the purview of the modified EWL scheme

A. V. S. Kameshwari, S. Balakrishnan

Published Jan 6, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1109/comsnets63942.2025.10885664

Related papers

Evolutionary stability and payoff equivalence in an Eisert-Wilkens-Lewenstein quantum game

Azhar Iqbal, Derek Abbott

An evolutionarily stable strategy (ESS) is a strategy that, once adopted by a population, cannot be invaded by any sufficiently rare alternative (mutant) strategy. We analyse evolutionary stability in an Eisert--Wilkens--Lewenstein (EWL) quantization of a symmetric $2\times2$ coordination game, whose classical mixed equilibrium $p^*=1/2$ is not an ESS. In the restricted two-parameter EWL strategy space, the corresponding quantum strategy is a strict symmetric Nash equilibrium and an ESS for every entanglement strength $γ>0$. In the full $SU(2)$ strategy space it remains a Nash equilibrium, but in the unreduced game, in which every unitary is a distinct strategy, exact payoff duplicates preclude strictness and ESS for every pure strategy. For the agreement-payoff class, the payoff factors through a Bloch map; identifying payoff-equivalent unitaries gives the reduced space $U(1)\backslash SU(2)\cong S^2$, where strictness and ESS are restored for $γ>0$. An arbitrarily small perturbation outside this class that preserves $p^*=1/2$ yields a profitable deviation, so the reduced-space result is not generic.

quant-pharXiv:2607.21103•Quantum Information Processing•Jul 24, 2026•0 citations

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

A. V. S. Kameshwari, S. Balakrishnan

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.

Physical review. E•Jan 5, 2024•10 citations

Interplay of noise, memory and entangling operator in quantum Stackelberg-Bertrand duopoly game

A. V. S. Kameshwari, S. Balakrishnan

Abstract In this work, we make an attempt to understand how noise, memory and entangling operators collectively decide the profit of the firms Here we have studied the quantum version of Stackelberg-Bertrand duopoly game using modified EWL scheme in both correlated and uncorrelated noise channels. When this game is analyzed for amplitude damping channels following interesting results are obtained: Firstly, decoherence in channel 2 effects the profit function of the firms more than that of channel 1. Secondly, in the case of correlated noise memory prevents the death of entanglement at maximum noise. Finally, the profit function of the firms depends upon noise, memory, strategies, and entangling operator in any given game setting.

Physica Scripta•Apr 11, 2023•5 citations

Dynamics of a quantum Cournot-Bertrand duopoly model based on bounded rationality and product substitutability

Yu-Chung Chang

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.

Quantum Information Processing•Jul 9, 2025•6 citations

Chaotic dynamics of a quantum Cournot duopoly game with heterogeneous expectations

Yuhuai Zhang, Jing Yuan, Yili Zhang

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.

International Journal of Quantum Information•Feb 3, 2023•4 citations

Evolutionary Stability in Quantum Games

Azhar Iqbal, Taksu Cheon

In evolutionary game theory an Evolutionarily Stable Strategy (ESS) is a refinement of the Nash equilibrium concept that is sometimes also recognized as evolutionary stability. It is a game-theoretic model, well known to mathematical biologists, that was found quite useful in the understanding of evolutionary dynamics of a population. This chapter presents an analysis of evolutionary stability in the emerging field of quantum games.

arXiv:0706.1413•IMPERIAL COLLEGE PRESS eBooks•Sep 1, 2008•7 citations
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.
An evolutionarily stable strategy (ESS) is a strategy that, once adopted by a population, cannot be invaded by any sufficiently rare alternative (mutant) strategy. We analyse evolutionary stability in an Eisert--Wilkens--Lewenstein (EWL) quantization of a symmetric 2×22\times22×2 coordination game, whose classical mixed equilibrium p∗=1/2p^*=1/2p∗=1/2 is not an ESS. In the restricted two-parameter EWL strategy space, the corresponding quantum strategy is a strict symmetric Nash equilibrium and an ESS for every entanglement strength γ>0γ>0γ>0. In the full SU(2)SU(2)SU(2) strategy space it remains a Nash equilibrium, but in the unreduced game, in which every unitary is a distinct strategy, exact payoff duplicates preclude strictness and ESS for every pure strategy. For the agreement-payoff class, the payoff factors through a Bloch map; identifying payoff-equivalent unitaries gives the reduced space U(1)\SU(2)≅S2U(1)\backslash SU(2)\cong S^2U(1)\SU(2)≅S2, where strictness and ESS are restored for γ>0γ>0γ>0. An arbitrarily small perturbation outside this class that preserves p∗=1/2p^*=1/2p∗=1/2 yields a profitable deviation, so the reduced-space result is not generic.
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 In this work, we make an attempt to understand how noise, memory and entangling operators collectively decide the profit of the firms Here we have studied the quantum version of Stackelberg-Bertrand duopoly game using modified EWL scheme in both correlated and uncorrelated noise channels. When this game is analyzed for amplitude damping channels following interesting results are obtained: Firstly, decoherence in channel 2 effects the profit function of the firms more than that of channel 1. Secondly, in the case of correlated noise memory prevents the death of entanglement at maximum noise. Finally, the profit function of the firms depends upon noise, memory, strategies, and entangling operator in any given game setting.
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.
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.