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Fractional calculus provides powerful tool to build more realistic and accurate mathematical models in economic field. This paper aims to explore a proposed fractional-order differentiated Cournot duopoly game and its discretized game.

Conditions for existence and uniqueness of the proposed game’s solution are derived. The existence of Nash equilibrium point and its local and global stability are obtained. Furthermore, local stability analysis of the discretized game is investigated. The effects of fractional-order on game’s dynamics are examined, along with other parameters of the game, via the 2D bifurcation diagrams in planes of system’s parameters are acquired.

Theoretical and numerical simulation results demonstrate rich variety of interesting dynamical behaviors such as period-doubling and Neimark–Sacker bifurcations, attractors’ crises in addition to chaotic attractors. The results demonstrated that the stability Nash equilibrium point of the game can be lost by period doubling or Neimark–Sacker bifurcations.

Oligopoly games are pivotal in the mathematical modeling of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also macroeconomic analysis of business cycles, innovation and growth.

Although the Cournot game and its variants have attracted great interest among mathematicians and economists since the time of its proposition till present, memory effects in continuous-time and discrete-time Cournot duopoly game have not been addressed yet. To the best of author’s knowledge, this can be considered as the first attempt to investigate this problem of fractional-order differentiated Cournot duopoly game. In addition, studying more realistic models of Cournot oligopoly games plays a pivotal role in the mathematical investigation and better understanding of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also in macroeconomic analysis of business cycles, innovation and growth.

This paper considers the brand awareness and anchor influence on consumers' live-streaming purchases, and explores the existence of “free-riding” behavior, the comparison of brand promotion effect and active live-streaming effect and the optimal strategic combination between the brand and the anchor. The authors investigate the evolutionary stabilization strategies of the bounded rational brand and anchor, and explore the conditions for the realization of the optimal strategy. Management suggestions for the development of live streaming commerce can be provided in this paper.

Two significant models are used in this paper. The Stackelberg model is used to study the “free-riding” behavior, the comparison of brand promotion effect and active live-streaming effect and the optimal strategic combination between the brand and the anchor. Using evolutionary game theory to get the evolutionary stable equilibrium strategies and analyze the binary equilibrium strategy of the bounded rational brand and anchor. In addition, relevant simulation analysis is conducted using realistic data to verify the conclusions and for further analysis, making the conclusions of the paper have realistic significance.

The study shows that “free-riding” behavior exists and the positive effect of brand promotion is greater than that of active live-streaming. The brand and the anchor take active actions as the optimal strategy. As the sensitivity coefficient of consumers to live-streaming effort and the sensitivity coefficient of consumers to brand promotion change, various evolutionary stabilization strategies will appear. When the two sensitivity coefficients are below a certain threshold, the game sides will reach the optimal strategic combination to obtain the maximum benefits. When they rise above this threshold, it is counterproductive instead. The system achieves the optimal strategic combination when the difference factor between effort cost and promotion cost must be higher than a certain value, but when it takes the smallest possible value, the game sides tend to take active actions. This study can provide management suggestions for the sustainable development of the live-streaming model.

This paper shows that under certain conditions, the brand and the anchor can evolve into the optimal strategy to maximize the profits of both parties, which has certain practical significance for the prosperous development of live streaming commerce. In future research, the authors will consider the regulatory role of the government and construct a more realistic game model to provide constructive suggestions for the sustainable prosperity of live streaming commerce. Meanwhile, there are also games between multiple brands and multiple anchors, as well as games among brands-anchors-the live streaming platforms, and the authors will conduct more in-depth research in the future.

So far, the co-impact of anchor influence and brand awareness has not been considered simultaneously in published articles. This paper provides theoretical guidance for the behavioral choices of the brand and the anchor under the live streaming commerce, which is conducive to the prosperous development of live streaming commerce.

The purpose of this paper is to test the invulnerability of the guarantee network at the equilibrium point.

This paper introduces a tractable guarantee network model that captures the invulnerability of the network in terms of cascade-based attack. Furthermore, the equilibrium points are introduced for banks to determine loan origination.

The proposed approach not only develops equilibrium analysis as an extended perspective in the guarantee network, but also applies cascading failure method to construct the guarantee network. The equilibrium points are examined by simulating experiment. The invulnerability of the guarantee network is quantified by the survival of firms in the simulating progress.

There is less study in equilibrium analysis of the guarantee network. Additionally, cascading failure model is expressed in the presented approach. Moreover, agent-based model can be extended in generating the guarantee network in the future study.

The approach of this paper presents a framework to analyze the equilibrium of the guarantee network. For this, the systemic risk of the whole guarantee network and each node's contribution are measured to predict the probability of default on cascading failure. Focusing on cascade failure process based on equilibrium point, the invulnerability of the guarantee network can be quantified.

Emotions, understood as evolving mental states, are pivotal in shaping individuals“' decision-making, especially in ambiguous information evaluation, probability estimation of events, and causality analysis. Public–private partnership (PPP) projects represent a confluence of “economic–environmental–social” dimensions, wherein stakeholder behavior follows the sequential progression of “cognition–emotion–action.” Consequently, comprehending the effects of emotional shifts on stakeholder's decision-making processes is vital to fostering the sustainability of PPP projects.

The paper utilizes rank-dependent expected utility and evolutionary game theory to systematically examine the influence of emotional factors on stakeholders' behavior and decision-making processes within PPP projects. The paper integrates three emotional state functions—optimism, pessimism and rationality—into the PPP framework, highlighting the intricate interactions among the government, private sector, surrounding public and the media. Furthermore, the paper amalgamates the evolutionary pathways of environmental rights incidents with the media's role. Through equilibrium analysis and numerical simulation, the paper delves into the diverse interplay of emotions across different phases of the environmental rights incident, assessing the impact of these emotions on the evolutionary game's equilibrium results.

Emotions significantly influence the microlevel decisions of PPP stakeholders, adapting continually based on event dynamics and media influences. When the private sector demonstrates optimism and the surrounding public leans toward rationality or pessimism, the likelihood of the private sector engaging in speculative behavior escalates, while the surrounding public refrains from adopting a supervisory strategy. Conversely, when the private sector is pessimistic and the public is optimistic, the system fails to evolve a stable strategy. However, when government regulation intensifies, the private sector opts for a nonspeculative strategy, and the surrounding public adopts a supervisory strategy. Under these conditions, the system attains a relatively optimal state of equilibrium.

The paper develops a game model to examine the evolutionary dynamics between the surrounding public and private sectors concerning environmental rights protection in waste incineration PPP projects. It illuminates the nature of the conflicting interests among project participants, delves into the impact of emotional factors on their decision-making processes and offers crucial perspectives for the governance of such partnerships. Furthermore, this paper provides substantive recommendations for emotional oversight to enhance governance efficacy.

The paper aims to elucidate effective strategies for promoting the adoption of green technology innovation within the private sector, thereby enhancing the value of public–private partnership (PPP) projects during the operational phase.

Utilizing prospect theory, the paper considers the government and the public as external driving forces. It establishes a tripartite evolutionary game model composed of government regulators, the private sector and the public. The paper uses numerical simulations to explore the evolutionary stable equilibrium strategies and the determinants influencing each stakeholder.

The paper demonstrates that government intervention and public participation substantially promote green technology innovation within the private sector. Major influencing factors encompass the intensity of pollution taxation, governmental information disclosure and public attention. However, an optimal threshold exists for environmental publicity and innovation subsidies, as excessive levels might inhibit technological innovation. Furthermore, within government intervention strategies, compensating the public for their participation costs is essential to circumvent the public's “free-rider” tendencies and encourage active public collaboration in PPP project innovation.

By constructing a tripartite evolutionary game model, the paper comprehensively examines the roles of government intervention and public participation in promoting green technology innovation within the private sector, offering fresh perspectives and strategies for the operational phase of PPP projects.

This paper aims to study the oscillation phenomenon before chaos as well as its mechanism of occurrence in the energy-saving and emission-reduction system.

The system dynamics analysis, phase portrait analyses, equilibrium point analysis and bifurcation curve were applied to this paper.

First, the authors find an oscillation phenomenon previous to chaos. Second, on the one hand, the existence of two unstable saddles is the reason for the occurrence of oscillation phenomenon. On the other hand, the increasing of carbon emissions can arouse oscillation phenomenon.

This paper finds an oscillation phenomenon previous to chaos in the energy-saving and emission-reduction system. The mechanism of occurrence of oscillation phenomenon is studied. The existence of two unstable saddles is the reason for the occurrence of such oscillation phenomenon. The oscillation is related with fold bifurcation. The study also provides a theoretical basis for the further study of chaos control.

The purpose of this study is to explore the coopetition relationships between platform owners and complementors in complementary product markets. Drawing on the coopetition theory, the authors examined the evolutionary trends of the coopetition relationships between platform owners and complementors and explore the main influence factors.

The authors used Lotka–Volterra model to analyze the coopetition relationship between platform owners and complementors, including the evolutionary trends as well as the results. Considering the feasibility of sample data collection, simulation is used to verify the effects of different factors on the evolution of coopetition relationships.

The results show that there are four possible results of the competition in the complementary products market. That comprises “winner-take-all for platform owners,” “winner-take-all for complementors,” “stable competitive coexistence” and “unstable competitive coexistence,” where “stable competitive coexistence” is the optimal evolutionary state. Moreover, the results of competitive evolution are determined by innovation subjects’ interaction parameters. However, the natural growth rate, the initial market benefits of the two innovators and the overall benefits of the complementary product markets influence the time to reach a steady state.

The study provides new insights into the entry of platform owners into complementary markets, and the findings highlight the fact that in complementary product markets, platform owners and complementors should seek “competitive coexistence” rather than “winner-takes-all.” Moreover, the authors also enrich the coopetition theory by revealing the core factors that influence the evolution of coopetition relationships, which further enhance the analysis of the evolutionary process of coopetition relationships.

The purpose of this paper is to survey literature on macroeconomic nonlinear dynamics.

The paper identifies five influential types of models where the possible generation of endogenous cycles and chaotic motion arises. First, the frameworks that make use of the one‐hump logistic type equation; second, the models inspired on the growth literature of the 1940s; third, intertemporal utility maximization problems with increasing returns; fourth, models that can be represented as piecewise dynamic maps; and, fifth, bounded rationality – heterogeneous expectations setups.

The attention will be mainly focused on the theme of business cycles; an interpretation of the deterministic real business cycle model with increasing returns is proposed and a graphical analysis of the underlying system shows that strange attractors are observable for specific sets of parameter values.

The study of endogenous cycles in macroeconomic literature has important implications for policy: if fluctuations are due to deterministic reasons this may imply that by manipulating policy parameters governments may be able to change the qualitative nature of the economy's dynamics.

The paper gives a comprehensive view of nonlinear dynamics in macroeconomics. It shows that various relevant subjects might be addressed in this kind of models, e.g. economic growth, asset pricing, business cycles, consumption decisions, among others.

The purpose of this paper is to employ an emerging phenomenon in China concerning collaborative carbon emission reduction (CCER) to investigate: first, the coordination of suppliers and manufacturers within supply chains to reduce carbon emissions, and second, the role of governmental policy in facilitating this process.

This paper draws upon evolutionary game theory to develop an evolutionary game model for CCER for suppliers and manufacturers within supply chains. This includes a detailed analysis of the evolutionary direction and process in different areas, both with, and in the absence of, governmental subsidies.

The results demonstrate that CCER is path dependent and that its evolutionary process is influenced by the following four factors: the initial status within supply chains; the cost; the additional benefit; and the investment risk related to CCER. The research also reveals that the reward provided by manufacturers is rational over the long term, due to the excessive cost of incentives potentially preventing the implementation of CCER.

This study represents the first attempt to investigate CCER within supply chains through the application of an evolutionary game-theoretic model. The investigation of multiple factors in the model will deepen understanding of the collaborative role required for the carbon emission reduction.

China's agricultural production trusteeship is characterized by the organic link between household operation of small-scale peasant economy and agricultural socialized services, which releases agricultural development vitality and promotes agricultural modernization. As one of the agricultural production trusteeship modes, the whole process trusteeship is suitable for the actual situation of China's aging population and labor force transfer. This paper aims to construct an evolutionary game model containing multistakeholder to explore the behavior decisions through numerical simulation and to provide useful suggestions for the formation of a positive and stable trusteeship relationship and the sound development of the whole process trusteeship of agricultural production.

The paper incorporates village committees, service organizations and farmers into the same research framework, selects “guarantee + dividends” as the income distribution method and applies the evolutionary game method to analyze behavioral choices and evolutionary paths of stakeholders. By constructing the expectation function, establishing the replicator dynamic equations and analyzing the tripartite evolutionary stability strategy, the authors explore the factors that affect the stakeholders’ strategy choice and determine asymptotically stable points and stability conditions.

(1) There is a game relationship among village committees, farmers and service organizations in the whole process trusteeship of agricultural production, asymptotically stable points (0,0,1) and (1,1,1) are obtained through calculation. (2) The proportion of stakeholders' strategy choice, the weight of the whole process trusteeship of agricultural production in the performance appraisal of the village committee, the village committee's supervision cost, the village committee's reputation effect and the penalty for false dividends of the service organization will affect the speed at which the curve representing the tripartite relationship approaches two asymptotically stable points.

The countermeasures proposed in the paper have excellent reference value. (1) For areas that have realized the project: Village committees can solve the trusteeship problems exposed in the initial areas and improve farmers' satisfaction with the project. (2) For areas that have not realized the project: Those regions will receive more experience references and enhance their confidence in this project. The limitation of the paper is that it takes the main grain-producing areas in only the three northeastern provinces of China as the research object. The next research object will be extended to the whole country.

This paper propose strategies for realizing the orderly operation of the whole process trusteeship of agricultural production: first, increase the proportion of stakeholders' strategy choices; second, reduce the village committee's supervision cost; third, increase the weight of the whole process trusteeship of agricultural production in the performance appraisal of the village committee; fourth, improve the village committee's reputation effect; fifth, increase the penalty for false dividends of the service organization.

Agricultural production trusteeship is in its initial stage in China. The interest relationships between stakeholders are not yet clear. The paper innovatively applies the evolutionary game method to the research field of the whole process trusteeship of agricultural production. According to conditions in China, based on ensuring the guaranteed income, the paper introduces the dividend income variable and establishes a tripartite game model of village committees, service organization and farmers. The paper provides suggestions for the orderly and healthy development of China's agricultural production trusteeship and provides experience for the operation of other modes of agricultural production trusteeship.