Quantum-Inspired Pricing Model Reveals Chaos Hiding in Dual-Channel Markets

When a manufacturer sells its products both through its own direct online channel and through a traditional retailer, the two prices do not evolve independently. They are locked in a strategic dance, each move prompting a countermove, each adjustment shaping what customers come to expect. A new study published in Quantum Information Processing by Jiayu Shen of Nanjing University of Industry Technology, Lian Shi of Anhui University of Finance and Economics, and Kai Zhu of Jiangsu University of Technology shows that this dance can descend into genuine chaos, and, more strikingly, that the chaos can be tamed with carefully designed feedback controllers. The work, published on 8 October 2026 as volume 25, article 335 of the journal, merges quantum game theory, behavioral economics, and nonlinear dynamics into a single mathematical framework for one of the most common competitive structures in modern commerce.

The starting point of the model is a dual-channel supply chain, a structure that has become ubiquitous in the era of direct-to-consumer sales. A manufacturer operates a direct channel, selling to consumers itself, while simultaneously supplying a retailer who competes for the same customers under a wholesale contract. Economists have long studied such arrangements with static game-theoretic tools, typically computing a Stackelberg equilibrium in which the manufacturer, as the leader, sets a wholesale price and the two channels then respond. Shen and colleagues preserve this static Stackelberg benchmark to fix the contractual wholesale price, but they then depart from tradition in two important directions that transform a well-behaved equilibrium problem into a rich nonlinear dynamical system.

The first departure is the quantum-inspired coupling of prices. Rather than treating the direct-channel price and the retail price as separate decision variables adjusted by independent learning rules, the authors introduce a strategic coupling mapping with an intensity parameter gamma, which generates the observable prices from an underlying entangled structure. The idea descends from the quantum game theory literature that began with the landmark 1999 work of Eisert, Wilkens, and Lewenstein in Physical Review Letters, where strategies were encoded in quantum states and entanglement was shown to alter equilibrium outcomes dramatically. In the pricing context, the coupling parameter plays an analogous role: it measures how strongly the two channels’ pricing decisions are interwoven, so that a change in one channel’s price is not merely a competitive signal but part of a jointly generated strategic state. As gamma varies, the effective landscape of the pricing game changes, and with it the stability of any long-run price configuration.

The second departure brings consumer psychology into the dynamics. Shoppers do not respond only to the price they see today; they carry internal reference prices, benchmarks formed from past experience against which current offers are judged as gains or losses. This reference-price effect, documented empirically since Winer’s 1986 brand-choice model and formalized in work by Hardie, Johnson, and Fader and by Briesch and colleagues, is captured here through exponential smoothing: consumers maintain channel-specific reference prices that are updated each period as a weighted average of the previous reference and the current observed price. The memory embedded in these reference prices means that demand today depends on the entire history of pricing, not just the present moment. Combined with multiplicative bounded-rationality learning, in which each channel adjusts its price in proportion to the marginal profitability of a price change rather than solving for the exact optimum, the model becomes a four-dimensional nonlinear discrete-time dynamical system with its own independently computed dynamic fixed point, distinct from the static Stackelberg benchmark.

With the machinery in place, the authors subject the system to a full local stability analysis via the Jacobian matrix, the standard tool for determining whether small perturbations around a fixed point decay or grow. The results trace a familiar but compelling narrative in nonlinear dynamics. For moderate values of the coupling intensity and learning speeds, the system converges to its dynamic fixed point: prices settle into a stable configuration and the market behaves as classical theory would hope. But as the strategic coupling intensifies, the fixed point loses stability through a flip bifurcation, the canonical first step on the period-doubling route to chaos. Prices begin to oscillate between two values, then four, then eight, and eventually the oscillations become aperiodic and irregular, wandering across a strange attractor. The authors also document multi-band attractors, in which the chaotic trajectory alternates among disjoint bands, and irregular bounded oscillations that never settle yet never diverge, a signature of complex but constrained market behavior.

What makes these findings more than a mathematical curiosity is their economic interpretation. Period-doubling in a pricing model corresponds to a market that cycles between promotional and premium pricing with increasing unpredictability, precisely the pattern that plagues firms locked in price wars across competing channels. The reference-price memory acts as an amplifier of this instability: because consumers anchor on past prices, aggressive discounting today depresses the reference price and thus the perceived value of future full-price sales, feeding back into the channels’ next adjustments. The quantum-inspired coupling, meanwhile, changes the geometry of the strategic interaction and shifts the thresholds at which stability breaks down. Sensitivity analysis around the contractual wholesale price shows that the qualitative flip-bifurcation mechanism is preserved regardless of where the wholesale benchmark sits, suggesting that the instability is an intrinsic feature of the coupled learning process rather than an artifact of a particular contract.

The second half of the paper confronts the practical question: once chaos has taken hold, can it be controlled? Chaos control has a distinguished pedigree, beginning with the celebrated Ott-Grebogi-Yorke method of 1990 and Pyragas’s delayed-feedback scheme of 1992, both of which demonstrated that chaotic systems can be stabilized using small, carefully timed interventions. Shen and colleagues compare two controllers adapted to the pricing game. The first is a local state-feedback controller, which measures the current deviation of prices from the dynamic fixed point and applies a corrective adjustment proportional to that deviation. The second is a delayed-feedback controller in the Pyragas tradition, which instead compares present prices with their values one period earlier and acts to eliminate the difference, requiring no knowledge of where the fixed point actually lies, an attractive property when the target is only implicitly defined.

Both controllers succeed. In numerical experiments conducted in MATLAB, with code available from the corresponding author upon reasonable request and no external datasets used, each scheme drives the chaotic baseline regime back to the dynamic fixed point. The evidence is presented in the language of control theory: tracking errors, the gap between actual and target prices, fall below a prescribed threshold; the control inputs themselves decay toward zero, indicating that the interventions become vanishingly small once order is restored; and the spectral radii of the controlled system, which measure how rapidly perturbations shrink, drop below one, the mathematical criterion for asymptotic stability. In economic terms, either controller can convert a market trapped in erratic price oscillations into one that converges smoothly to a coherent pricing pattern, and the comparison offers firms or regulators a menu of interventions with different information requirements.

The study situates itself within a rapidly growing literature on quantum game models of economic competition. Recent years have seen quantum treatments of Cournot and Bertrand duopolies, triopolies with heterogeneous players, public-enterprise competition, and mixed duopolies with nonlinear demand, published largely in Quantum Information Processing by groups including Shi and Xu, Zhang and colleagues, Zhu and Zhou, Deng and colleagues, and the Wei group, alongside the present authors’ own 2025 model of dual channels under channel cooperation and service. What distinguishes the new contribution is the combination of three ingredients, quantum-inspired coupling, dual-channel structure with a contractual wholesale benchmark, and consumer reference-price memory, in a single dynamical framework, together with a systematic chaos-control analysis. Earlier dual-channel work, from Chiang, Chhajed, and Hess’s strategic analysis of direct marketing to Cai’s channel-selection models and Chen and Zhou’s omni-channel complexity study, had explored either reference effects or nonlinear dynamics, but rarely both, and none had introduced quantum coupling into this setting.

The implications reach beyond the mathematics. For managers of dual-channel firms, the results offer a caution: aggressive price-adjustment speeds and tightly coupled strategic responses, each individually rational, can jointly push a market past a stability threshold into behavior that no equilibrium analysis predicts. For researchers, the paper demonstrates that quantum game theory is maturing from an abstract formalism into a modeling toolkit capable of capturing strategic interdependence in ways that classical games cannot, and that behavioral memory and nonlinear dynamics belong in the same conversation. The authors acknowledge no external funding beyond support from the Key Project in Natural Science Research of the Anhui Provincial Department of Education, and they declare no conflicts of interest. Whether real pricing data will confirm the bifurcation thresholds their model identifies remains an open question, but the framework now exists to ask it, and the answer could reshape how firms think about the fine line between healthy competition and chaotic price warfare.

Subject of Research: Nonlinear dynamics and chaos control in a quantum-coupled dual-channel pricing game with consumer reference-price memory

Article Title: Quantum-coupled dual-channel pricing dynamics with reference-price memory

Article References: Shen, J., Shi, L., & Zhu, K. (2026). Quantum-coupled dual-channel pricing dynamics with reference-price memory. Quantum Information Processing, 25(10), Article 335. https://doi.org/10.1007/s11128-026-05360-5

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05360-5

Keywords: quantum game theory, dual-channel supply chain, reference price, chaos control, bifurcation, nonlinear dynamics, bounded rationality, Stackelberg game, delayed feedback, price war, flip bifurcation, Quantum Information Processing