Search Results (68)

Rare earths are a strategically vital mineral resource on a global scale; the security of their supply chain and their recycling face severe challenges amid dual pressures of resource scarcity and environmental protection. This study focuses on the reverse supply chain for rare earth permanent magnet materials—specifically those based on praseodymium and neodymium—constructing a two-level game model comprising a rare earth oligopoly (Stackelberg leader) and two recyclers (Cournot followers). It systematically analyzes the dual decision-making behavior of recyclers between “recycling and selling” and “in-house remanufacturing”, and examines the impact of two policy instruments—government subsidies and environmental taxes—on the supply chain equilibrium. The study employs reverse induction to solve the game equilibrium and combines this with numerical simulation methods to compare the differing effects of the two policies on key indicators such as product output, market share, profit distribution, waste recovery volume, and recycling rates. The results indicate that there is strategic coordination and resource competition in recyclers’ decisions regarding recycling and remanufacturing, causing them to assume dual roles as both “suppliers” and “competitors” within the supply chain. Subsidy policies significantly incentivize recycling and remanufacturing activities, thereby increasing the recycling rate, but exert a slight squeeze on the profits of rare earth conglomerates. Environmental tax policies effectively curb primary mining and promote resource circulation, but may have a negative impact on the remanufacturing industry. Currently, value within the closed-loop rare earth supply chain is highly concentrated among upstream oligopolistic enterprises, whilst the recycling segment suffers from insufficient economic incentives and significant policy dependency. This study provides theoretical support and decision-making references for the government to optimize policy combinations and promote the high-quality development of the rare earth recycling industry. Full article

This paper investigates how quantum entanglement and heterogeneous expectations jointly affect the stability, complexity, and controllability of a Cournot triopoly with isoelastic demand. Based on the Li–Du–Massar quantization scheme, we construct a discrete-time quantum Cournot triopoly in which three firms adopt different updating mechanisms: boundedly rational adjustment, naïve and adaptive expectations. The quantum boundary equilibrium and the unique interior quantum Nash equilibrium are derived explicitly. By linearizing the resulting three-dimensional nonlinear map and applying the Jury criterion, we obtain analytical local stability conditions for the interior equilibrium. The results show that increasing the entanglement level reduces the admissible range of the adjustment speed, thereby shrinking the stability domain and making the market dynamics more prone to bifurcation and chaos. Numerical simulations further reveal a typical transition from stable convergence to flip bifurcation, period-doubling cascades, chaotic attractors, and sensitive dependence on initial conditions. Finally, a control parameter is introduced to rescale the effective adjustment speed of the boundedly rational firm. This mechanism preserves the equilibrium set while restoring convergence to a stable fixed point once the control intensity exceeds a critical threshold. The findings highlight the joint role of entanglement, expectation heterogeneity, and nonlinear demand in shaping complex quantum oligopoly dynamics. Full article

Driven by the rapid growth of the low-altitude economy, drone logistics is emerging as a critical component of modern smart logistics systems. This study aims to examine how heterogeneous logistics service providers (LSPs) select among technological innovation, inter-firm cooperation, and hybrid strategies, as well as how these strategic choices affect ESG performance. We develop a two-stage duopoly Cournot game model that accounts for asymmetric logistics capabilities and consumers’ service-quality sensitivity, and compare the three strategic arrangements against a benchmark scenario without innovation or cooperation. Results show that a capability-driven Matthew effect already exists in the benchmark market. Technological innovation may further widen the performance gap between firms, yet it generates the highest social welfare by improving service quality and preserving market competition. Pure cooperation enhances coordination efficiency and environmental performance, but may reduce consumer surplus by weakening competition. The hybrid strategy generally delivers the highest system profit and robust environmental performance, while its advantages depend on market parameters and require sound benefit-sharing governance mechanisms. This study contributes to sustainable drone logistics research by integrating strategic interaction, firm heterogeneity and ESG outcomes into a unified framework, and provides targeted managerial and policy implications for innovation support, alliance governance and competition regulation. Full article

Within closed-loop supply chains (CLSCs), limited attention has been given to firms’ production and abatement decisions involving carbon permit transfer between an original equipment manufacturer (OEM) and an independent remanufacturer (IR) under a cap-and-trade policy (CTP). Several questions remain unresolved: How does CTP reshape production and abatement decisions through carbon permit transfer under binding emission constraints? Can such regulation reallocate abatement responsibilities between firms and generate environmental and economic benefits? To address these questions, this paper develops a constrained Cournot game model capturing competitive interactions between an OEM and an IR under regulation. The results show that CTP reallocates abatement responsibilities toward the firm with a lower abatement difficulty, inducing full abatement as a corner solution. When remanufactured products exhibit a high low-carbon level, a moderate increase in cap stringency promotes remanufacturing output and market share. By contrast, once full abatement is reached, stricter regulation expands output. Moreover, when remanufacturing features substantial cost savings and a high low-carbon level, CTP consistently improves social welfare. This study provides insights into how cap-and-trade policies shape production and abatement decisions in CLSCs. Full article

This paper develops a practical computational framework for the Bayesian Cournot model with bilateral incomplete cost information, where each player is uncertain about the opponent’s marginal cost, drawn from a continuous compact interval $[c_{*}, c^{*}]$ with $0<c_{*}<c^{*}<\infty$. The infinite dimensionality of the functional strategy spaces (mappings from types to production quantities) renders analytical closed-form solutions infeasible in this continuous-type setting. To overcome this challenge, we restrict the strategy spaces to finite-dimensional differentiable sub-manifolds—specifically, one-parameter families of oscillatory functions (cosine, sine, and mixed forms). After suitable affine Q-rescaling to map the oscillatory range into the production interval $[0, Q]$, and with parameter ranges satisfying $\alpha , \beta >(\pi /2)/c_{*}$, these curves ensure near-exhaustivity: the joint production map $(\alpha , \beta )\mapsto (x_{\alpha }\left(s\right), y_{\beta }\left(t\right))$ covers ${[0, Q]}^2$ densely for every fixed cost pair $(s, t)$, thereby recovering (up to density and closure) the full ex-post payoff space. We introduce the ex-post payoff mapping $\Phi (s, t, x, y)=(e_s(x, y)\left(t\right), f_t(x, y)\left(s\right))$, which collects every realizable payoff pair once nature draws the types and players select their strategies. The image of $\Phi$ defines the general payoff space of the game, and its non-dominated points constitute the general ex-post Pareto frontier—all efficient realized outcomes across type-strategy realizations, without dependence on private probability measures over types. Using multi-objective genetic algorithms, we numerically approximate this frontier (and selected collusive compromises) within the restricted but representative sub-manifolds. The resulting frontiers are computationally accessible, robust to parameter variations, and validated through hypervolume convergence, sensitivity analysis, and comparisons with NSGA-II, PSO and scalarization methods. The findings are significant because they provide decision-makers in oligopolistic markets (e.g., electric vehicles) with viable, implementable production policies that explore efficient trade-offs under genuine cost uncertainty, without requiring explicit forecasts of the opponent’s type distribution—a limitation of traditional expected-utility approaches. By focusing on ex-post efficiency, the method reveals belief-independent compromise solutions that may guide tacit coordination or collusive outcomes in real-world strategic settings. Full article

This study examines the optimal design of low-carbon policies for governments, firms, and consumers within a unified analytical framework. We develop a three-stage game-theoretic duopoly model with differentiated products, green R&D, and knowledge spillovers to analyze the effects and implications of low-carbon policies in a polluting industry. The analysis encompasses both Cournot and Bertrand competition under commitment and non-commitment regimes, as well as non-cooperative and cooperative R&D structures. Specifically, we (i) quantify the impacts of low-carbon policies on R&D, emissions, profits, and welfare across alternative competition modes, policy-timing regimes, and R&D organizations; (ii) examine the roles of key policy parameters across all scenarios; and (iii) provide an integrated and intuitive interpretation of the underlying economic mechanisms. Full article

For the first time, this paper introduces Nabla fractional calculus into the distributed Nash equilibrium (NE) seeking problem of aggregative games (AGs) with partial decision information in undirected communication networks, and proposes two novel fractional-order distributed algorithms. In the considered setting, each agent can access to only local information and collaboratively estimates the global aggregate through communication with its neighbors. Both algorithms adopt a backward-difference scheme followed by an implicit fractional-order gradient descent step. One updates local aggregate estimates via fractional-order dynamic tracking and the other uses fractional-order average dynamic consensus protocols. Under standard assumptions, convergence of both algorithms to the NE is rigorously proved using nabla fractional-order Lyapunov stability theory, achieving a Mittag-Leffler convergence rate. The feasibility of the developed schemes is verified via numerical experiments applied to a Nash-Cournot game and the coordination control of flexible robotic arms. Full article

This paper studies an efficient method for solving stochastic optimization problems formulated as stochastic variational inequalities with a quasi-monotone operator, where the cost function extends the classical monotone and pseudomonotone operators. Our proposed method iterates an adaptive stepsize that adjusts automatically without linesearch and includes a momentum term to accelerate the convergence. Each iteration requires only a single projection onto the feasible set, ensuring low computational complexity. Under standard assumptions, the algorithm achieves almost sure convergence and a proven convergence rate. Furthermore, numerical experiments demonstrate its superior performance, accuracy, stability, and efficiency compared with existing stochastic approximation schemes. We also apply the method to problems such as stochastic network bandwidth allocation, stochastic complementarity problems, and the networked stochastic Nash–Cournot game, showing its strength and practical usefulness. The obtained result is an extension of existing works in the literature. Full article

The dual-channel structure resulted from manufacturer encroachment could alter the incentives of downstream retailer to ex ante communicate demand forecast. And different types of channel competition need to be investigated in this dual-channel information sharing scenario. This paper aims to investigate retailer’s ex ante imperfect demand information sharing strategy given that upstream manufacturer has set up direct sales channel (manufacturer encroachment). The imperfect information sharing means the demand information shared is uncertain and has some error relative to the real-world demand condition. It examines two types of channel competition: quantity competition and price competition. Additionally, this study discusses the encroaching manufacturer’s incentives for adjusting channel substitution. The paper adopts a stylized game theoretic model to describe interactions between retailer and the encroaching manufacturer. Contrary to conventional wisdom, the paper shows that under manufacturer encroachment, it is always possible for ex ante demand information sharing. Specifically, in the Cournot competition scenario where retailer channel and the encroaching manufacturer direct channel compete in quantity, the encroaching manufacturer could encourage demand information communication through side payment. Furthermore, in the Bertrand competition scenario, retailer may voluntarily share demand information. In addition, in either quantity or price competition, the encroaching manufacturer has incentives to adjust channel substitution for profit maximization. Full article

This paper pioneers the introduction of nabla fractional calculus into distributed Nash equilibrium (NE) seeking for non-cooperative games (NGs), proposing several novel discrete-time fractional-order algorithms. We first develop a gradient play-based algorithm under perfect information and subsequently extend it to partial-information settings. Two types of communication network topologies among agents, namely connected undirected graphs and strongly connected unbalanced directed graphs, are explicitly considered. When the pseudo-gradient mapping of the NG is Lipschitz continuous and strongly monotone, the proposed algorithms are proven to achieve asymptotic convergence to the NE with at least a Mittag–Leffler convergence rate. Both the step size and the fractional order act as tunable parameters that jointly influence the convergence performance. Numerical experiments on potential games and Nash–Cournot games demonstrate the effectiveness of the proposed algorithms. Full article

This article concerns a Cournot duopoly game with homogeneous expectations. The cost functions of the two players are assumed to be asymmetric to capture possible asymmetries in firms’ technologies or firms’ input costs. Large values of the speed of adjustment of the players destabilize the Nash Equilibrium (N.E.) and cause the appearance of a chaotic trajectory in the Discrete Dynamical System (D.D.S.). The scope of this article is to control the chaotic dynamics that appear outside the stability field, assuming asymmetric cost functions of the two players. Specifically, one player uses linear costs, while the other uses nonlinear costs (quadratic or cubic). The cubic cost functions are widely used in the Economic Dispatch Problem. The delayed feedback control method is applied by introducing a new control parameter at the D.D.S. It is shown that larger values of the control parameter keep the N.E. locally asymptotically stable even for higher values of the speed of adjustment. Full article

A Cournot triopoly is a type of oligopoly market involving three firms that produce and sell homogeneous or similar products without cooperating with one another. In Cournot models, firms’ decisions about production levels play a crucial role in determining overall market output. Compared to duopoly models, oligopolies with more than two firms have received relatively less attention in the literature. Nevertheless, triopoly models are more reflective of real-world market conditions, even though analyzing their dynamics remains a complex challenge. A reaction–diffusion system of PDEs generalizing a nonlinear triopoly model describing a master–slave Cournot game is introduced. The effect of diffusion on the stability of Nash equilibrium is investigated. Self-diffusion alone cannot induce Turing pattern formation. In fact, linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. The conditions for the onset of cross-diffusion-driven instability are obtained via linear stability analysis, and the formation of several Turing patterns is investigated through numerical simulations. Full article

This study investigates dynamic behaviors within a competition Cournot duopoly framework incorporating consumer surplus, and social welfare through the bounded rationality method. The distinctive aspect of the competition game is the incorporation of discrete difference equations into the players’ optimization problems. Both rivals seek to achieve optimal quantity outcomes by maximizing their respective objective functions. The first firm seeks to enhance the average between consumer surplus and its profit, while the second firm focuses on its profit optimization with a social welfare component. The game map features four fixed points, with one being the Nash equilibrium point at the intersection of marginal objective functions. Our analysis explores equilibrium stability, dynamic complexities, basins of attraction, and the emergence of chaos through double routes via flip bifurcation and Neimark-Sacker bifurcations. We observe that increased adjustment speeds can destabilize the system, leading to a richness of dynamic complexity. Full article

The current generation of integrated power systems is represented by the Adaptive Power and Thermal Management System (APTMS). The coupled performance between the APTMS and the aircraft engine significantly increases the difficulty of energy management and optimization. This article establishes an energy-coupled Amesim model of the APTMS and the aircraft engine to analyze performance conflicts. Energy optimization based on the Stackelberg game model is established, with the aircraft engine as the leader and the APTMS as the follower. The Adaptive Chaotic Particle Swarm Optimization (ACPSO) algorithm is introduced to search for the game equilibrium solution. Simulation results indicate that this energy management strategy can achieve equilibrium and alleviate performance conflict. In flight, the optimal strategy depends on thrust–fuel flow characteristics and cooling power demand. Finally, compared with the multi-objective optimization algorithm MOPSO and the non-cooperative Cournot game model, the advantages of this energy management system based on the Stackelberg game are verified. Full article

We have built and investigated analytically and numerically a differential game model of Cournot oligopoly with consideration of pollution, network structure, and continuous updating. Up to this time, games with network structure and continuous updating were considered separately. We analyzed time consistency for a cooperative solution of the game. For a specific example, we built a non-empty subgame perfect subcore. We considered stochastic versions of the proposed model and received results similar to the deterministic case. Full article