Entropy

Dynamics of a Quantum Common-Pool Resource Game with Homogeneous Players’ Expectations

Juan Grau-Climent, Luis García-Pérez, Ramón Alonso‐Sanz, Juan Carlos Losada

Published Nov 25, 2023• 2 citations

Abstract

In this work, we analyse a common-pool resource game with homogeneous players (both have boundedly rational expectations) and entanglement between players' strategies. The quantum model with homogeneous expectations is a differential approach to the game since, to the best of our knowledge, it has hardly been considered in previous works. The game is represented using a Cournot type payoff functions, limited to the maximum capacity of the resource. The behaviour of the dynamics is studied considering how the fixed points (particularly the Nash equilibrium) and the stability of the system vary depending on the different values of the parameters involved in the model. In the analysis of this game, it is especially relevant to consider the extent to which the resource is exploited, since the output of the players is highly affected by this issue. It is studied in which cases the resource can be overexploited, adjusting the parameters of the model to avoid this scenario when it is possible. The results are obtained from an analytical point of view and also graphically using bifurcation diagrams to show the behaviour of the dynamics.

DOI: 10.3390/e25121585

Related papers

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 ProcessingJul 9, 20256 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 InformationFeb 3, 20234 citations

Quantum Cournot model based on general entanglement operator

Katarzyna Bolonek-Lasoń

Abstract The properties of the Cournot model based on the most general entanglement operator containing quadratic expressions which is symmetric with respect to the exchange of players are considered. The degree of entanglement of games dependent on one and two squeezing parameters and their payoff values in Nash equilibrium are compared. The analysis showed that the relationship between the degree of entanglement of the initial state of the game and the payoff values in Nash equilibrium is ambiguous. The phase values included in the entanglement operator have a strong influence on the final outcome of the game. In a quantum duopoly based on the initial state of a game that depends on one squeezing parameter, the maximum possible payoff in Nash equilibrium cannot be reached when the value of the phase parameter is greater than zero, in contrast with a game that depends on two parameters.

arXiv:2406.16049Quantum Information ProcessingNov 5, 20242 citations

Asymmetric model of the dynamic quantum Cournot duopoly game with asymmetric information and heterogeneous players

Huaxi Chen, Wensheng Jia

Abstract Building on the existing symmetric quantization model of the dynamic Cournot duopoly game (CDG) with asymmetric information, we extend it to an asymmetric quantization model and study the stability of the quantum Bayesian Nash equilibrium (QBNE) under heterogeneous expectations. We analyze the influence of various parameters on the stability of QBNE, with a particular focus on the impact of the parameter α on system stability. The results show that when α < 1, under the same parameters, the quantum strategy of the asymmetric quantization model is more favorable for stabilizing the market. However, when α > 1, the quantum strategy of the symmetric quantization model is more conducive to stabilizing the market.

Chinese Physics BAug 29, 20251 citation

Complex dynamical analysis of the Frackiewicz quantum commons game with bounded rationality

Xingjing Zhang, Nengfa Wang, Kai Gu +1 more

Abstract In the classical commons game, shepherds’ complete rationality causes the Tragedy of the Commons (ToC). Based on bounded rationality, we develop a dynamic Frackiewicz quantum commons game model to explore quantum entanglement’s effects on grazing strategies and profits, analyze quantum equilibrium stability, and apply synchronization control. Our Contributions: (1) The stability of quantum equilibrium depends on the adjustment speed and the squeezing parameter: it remains stable at low adjustment speeds and when the squeezing parameter is close to its critical value (i.e., π 4 ), but exhibits bifurcation or chaos otherwise. (2) Enhanced entanglement expands the stability region, mitigates individual rationality-induced conflicts, and boosts collective profits. Within the Frackiewicz framework, the quantum model optimizes ToC outcomes: the number of sheep never exceeds the classical level, while profits rise with entanglement and never fall below the classical level. (3) Synchronization control ensures strategy evolution convergence to stabilize the system. Physically, entanglement strengthens strategy correlations, suppresses bounded rationality-induced chaos, and expands the stable parameter region, offering a quantum solution to ToC.

Physica ScriptaNov 27, 20251 citation

The Complexion of Multi-period Quantum Cournot Duopoly Game with Bounded Rationality Based on Asymmetric Information

Xinli Zhang, Jiahe Zhang

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.

Oct 5, 20220 citations