The Cournot duopoly is one of the oldest workhorses of economic theory. In it, two firms simultaneously choose how much of a good to produce, and the market price adjusts according to total output. Each firm wants to maximize its own profit, and under the right conditions the two settle into a Nash equilibrium, a pair of production levels from which neither has an incentive to deviate. But real firms do not respond instantaneously. Market data take time to gather, production plans take time to implement, and decisions made today reflect information that is already stale. Economists have long known that such delays can destabilize otherwise well-behaved markets, sending quantities into cycles or even chaos. What Guerrini and Ragni show is that when the game is played with quantum strategies, the mathematics of delay acquires a remarkably clean structure.

Their model builds on a line of research that began in 1999, when David Meyer showed that games can admit quantum strategies and that these can outperform any classical play. Soon after, Jens Eisert, Martin Wilkens and Maciej Lewenstein formulated a general framework for quantum games in which players manipulate entangled qubits rather than choosing plain numbers. In the Cournot setting, later developed by Hongjun Li, Jiangfeng Du and Serge Massar in a continuous-variable formulation and by Paweł Frąckiewicz in an alternative scheme, the firms’ strategic choices become quantum operators acting on a shared entangled state. The degree of entanglement, together with the specific protocol used to translate quantum outputs into production quantities, shapes the payoff landscape that each firm navigates.

Guerrini and Ragni’s first contribution is a unified mathematical representation that encompasses both the Li–Du–Massar protocol and the Frąckiewicz protocol within a single framework. This unification matters because previous studies treated each protocol separately, making it hard to tell which features of a quantum duopoly’s dynamics are artifacts of a particular formalism and which are genuine properties of the underlying economics. By writing the two schemes in a common language, the authors can cleanly separate the effects of the quantum protocol from the effects of cost asymmetry between the firms, and they can ask which of these actually determines whether the market stays calm or breaks into oscillation.

The model assumes isoelastic demand, meaning the price elasticity of demand is constant, a standard and analytically convenient specification, and adds delayed marginal-profit adjustment: each firm continuously nudges its output in the direction of its marginal profit, but computed from quantities observed a fixed time earlier. The authors establish the mathematical foundations first, proving that the interior quantum Nash equilibrium exists, remains positive, and stays bounded for all time, so the model cannot blow up or produce nonsensical negative outputs. They then derive the exact characteristic equation of the linearized system, the transcendental equation whose roots govern whether small perturbations around equilibrium decay or grow.

The central result is expressed through a single number, m, defined as the ratio of the second firm’s marginal cost to the first firm’s. If m lies between 3 minus 2 times the square root of 2 and 3 plus 2 times the square root of 2, roughly between 0.172 and 5.828, the equilibrium is stable for every finite delay whatsoever. No matter how stale the information becomes, the market always returns to equilibrium. This is a delay-independent stability region, and its existence is a genuinely quantum feature of the model’s structure. Outside that window, when one firm’s costs are more than about 5.8 times or less than about 0.17 times the other’s, the picture changes dramatically: there exists a finite critical delay beyond which stability is lost.

At that critical delay, the authors show, a single conjugate pair of characteristic roots crosses the imaginary axis transversally, the signature of a Hopf bifurcation. In plain terms, the equilibrium loses stability not through an explosion but through the birth of sustained oscillations: output levels begin to cycle around the Nash equilibrium with a growing amplitude until nonlinear effects take over. Numerical simulations reported in the paper confirm the analytical prediction, displaying precisely the oscillatory dynamics expected once the delay exceeds the computed threshold. A root-continuation analysis, tracking the characteristic roots as parameters vary, further corroborates the exact boundary.

Perhaps the most conceptually important finding is what does and does not matter for the stability threshold. Whether a finite critical delay exists at all depends only on the relative marginal costs of the two firms, a purely economic quantity that no choice of quantum protocol can alter. What the entanglement between the players’ qubits and the choice between the Li–Du–Massar and Frąckiewicz schemes do control is the numerical location of that threshold, that is, how much delay the market can tolerate before oscillations set in. In other words, quantum structure tunes the market’s resilience, but the economic fundamentals decide whether resilience is unlimited or finite. This protocol-invariant boundary gives the study its title and offers a rare example of a result in quantum game theory that is robust across competing formalizations.

The work also connects two previously separate literatures. On one side, economists including Tönu Puu, Fabio Tramontana, Luca Gori, and Angelo Matsumoto and Ferenc Szidarovszky have extensively mapped the delayed and often chaotic dynamics of classical Cournot models with isoelastic demand, identifying Hopf bifurcations and stability switches in a variety of specifications. On the other side, a rapidly growing quantum-game community, including recent studies of memory, heterogeneous players, relative profit maximization, and asymmetric information in quantum Cournot and Bertrand settings, has documented complex dynamics in discrete-time quantum duopolies and triopolies. Guerrini and Ragni bring the delay-differential machinery of the first literature to bear on the second, providing the kind of exact characteristic-equation analysis that discrete-time studies often approximate.

As with all theoretical work, the model is a stylized rather than a literal description of markets; no one is suggesting that firms literally share entangled qubits when setting production quotas. But quantum game theory serves as a powerful generalization of classical strategic reasoning, and results like this one clarify which economic phenomena are robust to how the strategic interaction is formalized. The finding that cost asymmetry alone dictates whether delayed information can destabilize a quantum market, while entanglement and protocol merely shift the danger zone, offers a compact and testable intuition: the more lopsided the competition, the less tolerance the market has for stale information, and once the imbalance crosses the algebraic window bounded by 3 plus or minus 2 times the square root of 2, every quantum formulation inherits the same fragility. For a field still debating the right way to quantize a game, a result that survives the change of protocol is a small landmark, and a reminder that sometimes the deepest structure in a quantum model is the classical economics hiding underneath.

Subject of Research: Stability and delayed-information dynamics of quantum entangled Cournot duopoly games

Article Title: Delayed information and quantum entanglement in Cournot duopoly: a protocol-invariant boundary for oscillatory instability

Article References: Guerrini, L., & Ragni, S. (2026). Delayed information and quantum entanglement in Cournot duopoly: a protocol-invariant boundary for oscillatory instability. Quantum Information Processing, 25(10), Article 333. https://doi.org/10.1007/s11128-026-05359-y

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05359-y

Keywords: quantum game theory, Cournot duopoly, entanglement, delayed information, Nash equilibrium, Hopf bifurcation, isoelastic demand, stability switching, cost asymmetry, delay differential equations, oscillatory instability, Quantum Information Processing