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

To understand why this is more than wordplay, it helps to start with the physics. Quantum energy teleportation, first proposed by Masahiro Hotta in 2008, is a protocol that allows energy to be extracted at a distant location by exploiting entanglement in the ground state of a many-body quantum system. In an ordinary quantum system, the ground state is the lowest-energy configuration, and a local region of it can appear as a so-called passive state, one from which no work can be extracted by local operations alone. Yet the region is not isolated: it is entangled with its surroundings. If a distant party performs a measurement on their part of the system and communicates the result over a classical channel, the recipient can apply a conditional operation tailored to that outcome. The measurement injects energy locally at the sender’s site, and the conditional operation at the receiver’s site then unlocks energy that was previously inaccessible, effectively delivering usable energy without any physical carrier traveling between the two points.

The protocol respects all the usual physical constraints. No energy travels faster than light, because the classical communication of the measurement outcome is essential, and the sender must pay an energy cost that, as Ikeda proves rigorously in the paper, is always at least as large as the total energy delivered to the receivers. The proof relies on the fact that the ground state minimizes the total energy, so any manipulation that raises the energy of the sender’s region must account for the energy extracted elsewhere. What makes the protocol economically interesting is precisely this structure: a supplier pays a cost, consumers receive a benefit, and the accounting between them is governed by the laws of quantum mechanics rather than by contract law. Ikeda demonstrated the protocol experimentally on IBM superconducting quantum processors, using pairs of qubits on the ibm_strasbourg, ibm_sherbrooke and ibm_kyiv devices, with modern error mitigation techniques such as matrix-free measurement mitigation and zero-noise extrapolation to extract clean signals from noisy hardware.

With this physical substrate in place, the paper builds economic machinery on top of it. The central move is to treat teleported energy as a commodity and the entangled ground state as a shared resource, then to ask what happens to classical game-theoretic models when players can additionally choose quantum strategies. In classical game theory, following John von Neumann, Oskar Morgenstern and John Nash, a game is defined by players, strategies and payoffs, and an equilibrium is a set of strategies from which no player benefits by deviating alone. Ikeda extends this framework by allowing strategies to be quantum operations, including measurements and conditional unitary gates, which means the space of possible actions is vastly larger than in the classical setting. New equilibria appear that have no classical counterpart, and in some cases the quantum strategies allow players to escape dilemmas, such as the free-rider problem, that are unavoidable when strategies are restricted to classical choices.

The paper works through a remarkable range of standard industrial organization models. Ikeda constructs quantum versions of the Bertrand and Cournot duopoly models, the two canonical frameworks for describing competition between firms. In Cournot competition, firms choose quantities and the market price adjusts; in Bertrand competition, firms choose prices and consumers buy from the cheapest supplier. When firms can share or trade entangled resources and deploy quantum strategies, the resulting equilibria differ from the classical ones, offering unconventional ways to divide the market or to extract surplus. The analysis extends to monopoly, to Hotelling-style location games in which firms choose where to position themselves along a market, to contestable markets where the threat of entry disciplines incumbents, and to perfect competition. In each case the quantum layer does not simply reproduce the classical results with extra steps; it introduces genuinely new strategic possibilities rooted in the correlation structure of entangled states.

Perhaps the most provocative applications concern public goods and incentive problems. Moral hazard, the situation in which an agent’s effort cannot be perfectly observed by a principal, has been a cornerstone of contract theory since Bengt Holmström’s 1979 work on observability. Free riding on public goods, analyzed classically by Paul Samuelson and others, similarly arises because individual contributions are hard to verify or reward. Ikeda shows that quantum energy teleportation offers a lens on these problems: because the measurement and feedback operations in the protocol are verifiable parts of a physical process, and because entanglement creates correlations that cannot be replicated classically, agents can be given incentives that depend on quantum information. The paper unveils strategies that transcend classical limits, suggesting that in a future quantum economy, the design of contracts and contribution mechanisms might exploit resources that classical mechanism design cannot even describe.

The study also reaches into matching theory, the branch of economics founded on the Gale-Shapley algorithm that underlies everything from school choice to organ exchange. Ikeda incorporates entangled resources into matching models and proves that stable and optimal allocations can be achieved in settings where the participants’ payoffs are tied to quantum states. In his formalism, a matching corresponds to a configuration whose total payoff is encoded in a Hamiltonian, and a stable matching is the ground state of that Hamiltonian, with penalty terms ensuring that no participant has an incentive to block the outcome. The proofs in the paper’s appendix show that the ground state satisfies individual rationality, meaning no participant prefers to opt out, which is exactly the stability condition that market designers care about. This connection between quantum many-body physics and market stability is one of the paper’s most elegant technical contributions.

None of this is pure abstraction. Ikeda’s earlier work demonstrated quantum energy teleportation on real superconducting hardware in 2023, and the new paper includes a detailed appendix on the experimental methods, showing how mid-circuit measurements and conditional operations are implemented, how noise is mitigated, and how the latest Eagle-generation processors run the protocol ten to one hundred times faster than the devices used two years earlier. The programming code for the simulations and experiments is publicly available under an MIT license on the author’s GitHub repository. Related results by other groups, including experimental activation of strongly passive states and proposals for quantum key distribution based on teleportation protocols, indicate a rapidly maturing research area in which the physics of entanglement and the engineering of quantum processors advance together.

The intellectual lineage of the paper is also worth noting. The idea that information is physical traces back through Maxwell’s demon, Leo Szilard’s analysis of measurement and entropy, and the work of Rolf Landauer and Charles Bennett. Quantum money, proposed by Stephen Wiesner in the 1980s and developed by later researchers, showed that quantum states could serve as unforgeable economic objects. Ikeda’s own 2022 paper with Shumpei Aoki on quantum games and quantum economic behavior laid the groundwork for the present synthesis. What is new here is the scale of the ambition: rather than treating one economic object quantum mechanically, the paper rebuilds an entire toolkit of microeconomics, from auctions and mechanism design to repeated games and cooperative solution concepts like the Shapley value, on quantum foundations.

Caveats remain, as they must in any nascent field. The models are theoretical, the demonstrations are small-scale, and no one knows whether entangled energy markets will ever exist outside the laboratory. Yet the paper makes a serious case that the question is worth asking. If quantum networks eventually distribute entanglement as routinely as fiber optics distribute bandwidth, then the economics of that distribution, including who pays for measurements, who profits from teleported energy, and how stable allocations are reached, will need a theory. Quantum Information Economics, as sketched in this work, is an early attempt to provide one, and it suggests that the strategic landscape of a quantum-enabled economy may be richer, and stranger, than anything classical game theory has imagined.

Subject of Research: Quantum energy teleportation as a commodity in quantum game theory and economic models

Article Title: Quantum games and economics through teleportation

Article References: Ikeda, K. (2026). Quantum games and economics through teleportation. Quantum Information Processing, 25(10), Article 331. https://doi.org/10.1007/s11128-026-05341-8

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05341-8

Keywords: quantum energy teleportation, quantum game theory, Quantum Information Economics, entanglement, Bertrand competition, Cournot duopoly, moral hazard, public goods, matching theory, market design, superconducting qubits, quantum networks