Quantum Cournot Duopoly Model Reveals Chaos When Entanglement Rises

A new study in Quantum Information Processing presents a quantum version of the classic Cournot duopoly game, extending the Li–Du–Massar (LDM) quantization scheme to a four-dimensional system that describes competition between two substitute products. The model, built by Nengfa Wang and Xingjing Zhang, merges quantum game theory with nonlinear dynamics to explore how firms with bounded rationality adjust their output decisions over time.

The researchers introduce squeezing parameters alpha and beta into the LDM framework, using them to characterize the coordination of product strategies in the two-product setting. Firms are assumed to be boundedly rational, so the model adopts a discrete-time dynamic system in which outputs are adjusted step by step rather than instantaneously reaching equilibrium.

To assess whether the market settles down or spirals out of control, the authors analyze the stability of the equilibrium using Jacobian eigenvalues and the Jury criterion. These standard tools of nonlinear dynamics allow them to map the conditions under which the quantum equilibrium remains stable and the conditions under which it breaks down.

Numerical simulations reveal the complex behaviors the system can exhibit, including bifurcations, high-dimensional singular attractors, and well-defined stability regions. The simulations show how the interplay between quantum features and the speed of firms’ adjustments shapes the long-run behavior of the market.

Two variables emerge as decisive for stability: entanglement and the adjustment speed. The equilibrium remains stable when either the adjustment speed or the entanglement is low. When either becomes high, the system undergoes bifurcation and eventually descends into chaos, meaning output decisions no longer converge to a predictable equilibrium.

Entanglement plays a particularly consequential role. As entanglement increases, the stable region of the parameter space is compressed and market uncertainty rises. In other words, stronger quantum correlations between the players’ strategic choices make the duopoly more fragile, shrinking the range of adjustment speeds for which orderly dynamics survive.

The paper also conducts a comparative analysis with the bounded-rational continuous-variable quantum game model of Yang and Gong, positioning the new four-dimensional framework against earlier approaches to quantized economic competition.

The work illustrates how tools from quantum information, such as squeezing and entanglement, can be combined with the mathematics of nonlinear dynamical systems to model strategic economic interaction. By identifying the thresholds at which stability gives way to bifurcation and chaos, the study offers a structured picture of when a quantum duopoly with substitutable products behaves predictably and when it does not, based on the model’s Jacobian analysis and numerical simulations.

Subject of Research: A quantum Cournot duopoly game model with product substitutability and its nonlinear stability dynamics

Article Title: Dynamic analysis of an LDM quantum Cournot duopoly game with product substitutability

Article References: Wang, N., & Zhang, X. (2026). Dynamic analysis of an LDM quantum Cournot duopoly game with product substitutability. Quantum Information Processing, 25(11), Article 350. https://doi.org/10.1007/s11128-026-05377-w

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05377-w

Keywords: quantum game theory, Cournot duopoly, LDM quantization, entanglement, bifurcation, chaos, stability analysis, bounded rationality, nonlinear dynamics, product substitutability, Dynamic, analysis