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

The stakes are both environmental and economic. Decommissioned crystalline silicon modules contain lead, cadmium and tin, substances linked to brain disorders, cancers and damage to the lungs, liver, kidneys and prostate if they leach from improper dumps. Yet the same panels are also a treasure chest of recoverable material. IRENA estimates that recycling end-of-life PV panels could unlock roughly 450 million dollars in recoverable value by 2030 and 15 billion dollars by 2050. The problem, the study’s authors Shikha Tiwari, Eshwar Dayal and Lakshay argue, is that existing waste management systems simply fail to capture this stream, because of economic and logistical constraints. Solar installations range from sprawling utility-scale parks to scattered urban rooftops, and that geographic fragmentation drives up the cost of collecting heavy, fragile freight from thousands of dispersed points.

The core of the new work is a bi-level optimization model, a mathematical structure designed for situations where two decision-makers with conflicting goals interact. In this formulation, the collection centre operator acts as the leader, seeking to minimise total cost, which includes fixed facility costs, vehicle routing expenses and the incentives paid out. The end users of solar panels act as followers, rationally choosing whether to hand their retired panels to the formal recycling system and, if so, whether to request a home pickup or deliver the panels themselves to a drop-off point. The lower level of the model maximises the incentives users receive, while the upper level minimises the operator’s total cost. The authors frame this as a Stackelberg game, a classic game-theoretic construct in which the leader anticipates the follower’s best response before committing to its own decision.

What makes the framework distinctive is that it stitches together, in a single model, four elements that previous studies treated largely in isolation: forecasting of future waste volumes, optimisation of facility locations, vehicle routing, and financial incentives. The authors’ comparative analysis of the literature shows that most prior works incorporated at most two of these dimensions. Some optimised facility placement but ignored routing; others modelled incentives but lacked any mathematical backbone for reverse logistics. By contrast, the new model makes incentives endogenous, meaning the payment levels themselves are decision variables that respond to waste volumes and costs, rather than fixed assumptions plugged in from outside.

The forecasting component is built on the Weibull distribution, a statistical function widely used to model failure rates over time. Rather than assuming every panel dies at exactly year 25, the Weibull approach produces a probabilistic retirement curve that rises during operational degradation and accelerates near end of life, which the authors argue is far closer to real-world behaviour than a fixed-lifespan model in which all panels retire simultaneously. The study runs two scenarios: a regular-loss baseline with a shape factor of about 2.49, and an early-loss pessimistic scenario with a shape factor of 5.38, reflecting premature failures from harsh weather or handling damage. Future installed capacity is projected using segmented linear growth rates, declining from 7.8 percent in 2021-2025 to 3.5 percent by 2036-2040, a deliberate choice to avoid the overestimation that exponential extrapolation produces over a 20-year horizon.

That overestimation problem is vividly demonstrated in the results. When an exponential growth assumption with a 14 percent compound annual growth rate is applied to Jaipur district, the model projects more than 2,300 megawatts of installed capacity in a single district by 2040, and a cumulative capacity overestimate of more than 100 percent. Such inflated numbers would lead planners to open unnecessary collection centres, oversize their fleets and set incentive levels for waste that will never materialise. The segmented linear approach, by contrast, produces projections that the authors consider implementable, and it feeds district-level waste estimates for Jaipur from 2021 to 2040, a spatial resolution fine enough to support actual facility location and routing decisions, unlike the national or state-level estimates that dominate the existing literature.

For the case study, Jaipur, the district with the highest installed solar capacity in India’s top solar state, was divided into 95 equal square grids, with each grid centroid serving as a waste generation site. Thirty-seven authorised e-waste collection centres provided the pool of candidate facilities. The routing side of the model is a capacitated vehicle routing problem, solved with Miller-Tucker-Zemlin subtour elimination constraints to ensure that every collection route is a single continuous loop returning to its depot, rather than a set of disconnected fragments. Haversine distances between grid centroids and centres ground the routing calculations in real geography. The model also enforces a profitability rule: revenue from selling collected panels to recyclers must exceed total operating cost by at least 10 percent, a threshold reflecting administrative overhead, price volatility in recovered materials and the need to fund infrastructure investment.

The results reveal an elegant adaptive behaviour. Under regular loss, the model keeps just two collection centres open until 2035, then adds one per year, reaching six by 2039. Under early loss, where cumulative waste over 2021-2030 is roughly 31 times higher than in the regular scenario, the network scales to as many as 14 centres. Crucially, the model remains profitable in every year of both scenarios, adjusting centre counts and incentive levels without any structural redesign. The incentive dynamics are particularly telling: at low waste volumes, pickup incentives sit near their upper limit because door-to-door collection is cheap when there is little to collect, but as volumes surge, the model widens the gap between drop-off and pickup payments, effectively shifting the logistical burden onto end users and encouraging them to deliver panels themselves. Incentive values eventually stabilise at around 2,500 rupees per kilowatt for pickup and 3,500 for drop-off.

The study also derives four key performance indicators that translate the mathematics into measurable outcomes: cost per tonne collected, average haul distance per tonne, waste recovery rate, and carbon dioxide emissions per tonne. The recovery rate holds steady at about 84 percent across all years and scenarios, while emissions per tonne fall dramatically as waste density rises, dropping from over 35 kilograms of CO2 per tonne in the sparse early years to under 2 kilograms by 2039 in the regular-loss case. Sensitivity analysis shows that minimum collection rates above 70 percent yield a healthy profit-to-cost ratio, that collection centre capacities of 40,000 kilograms or more stabilise the network at two facilities, and that vehicle capacities of 4,000 kilograms or more minimise cost. Computationally, the framework scales gracefully: even at 132 grid cells, generating over four million variables, the solver converged in under 42 seconds with optimality gaps below 2.5 percent.

The implications reach well beyond Jaipur. The authors recommend that policymakers provide subsidies or extended producer responsibility funding during the low-volume window of 2026 to 2030, explicitly incentivise self-drop-off since full home pickup is fiscally unviable, and use the model’s quantified costs to set collection targets that are both ambitious and financially realistic. For operators, the message is that flat-rate incentives are inefficient; payments should be dynamically adjusted, centres should open in stages, and the pickup-drop-off gap should widen during waste surges. The framework is designed to be replicated across regions and time frames, and with modifications could extend to other e-waste streams. Future work, the authors note, could incorporate stochastic user behaviour, the informal recycling sector as a competitor, and real-time GIS integration. But the central insight stands: the coming wave of solar waste is not merely a liability to be buried, but a feedstock whose recovery can be engineered, optimised and made to pay for itself.

Subject of Research: Bi-level optimization modeling for sustainable end-of-life solar photovoltaic waste management and reverse logistics within a circular economy framework

Article Title: Development of a bi-level optimization model for sustainable Solar PV Waste management within a circular economy framework

Article References: Tiwari, S., Dayal, E., & Lakshay (2026). Development of a bi-level optimization model for sustainable Solar PV Waste management within a circular economy framework. Cleaner Engineering and Technology, 35, Article 101327. https://doi.org/10.1016/j.clet.2026.101327

Image Credits: AI Generated

DOI: 10.1016/j.clet.2026.101327

Keywords: solar PV waste, bi-level optimization, circular economy, reverse logistics, Weibull distribution, vehicle routing problem, Stackelberg game, recycling incentives, extended producer responsibility, Jaipur case study, e-waste management, renewable energy