Feedback Equilibrium Research Articles

Time-consistent consumption, investment, and proportional reinsurance in market models with Markovian regime switching

This paper outlines an analysis of equilibrium strategies within a game-theoretic framework addressing discounting stochastic scenarios involving consumption, investment, and reinsurance problems. The controlled state process follows a multi-dimensional linear stochastic differential equation influenced by Brownian motion and Poison jump process under a Markovian regime-switching environment. The objective functional encompasses both running and terminal costs explicitly linked to general discount functions, introducing time inconsistency in the model. Open-loop Nash equilibrium controls are detailed, supported by necessary and sufficient equilibrium conditions and a verification outcome. Furthermore, a state feedback equilibrium strategy is attained through a specific partial differential–difference equation. The study delves into investment consumption and equilibrium, reinsurance/new business strategies, specifically examining power and logarithmic utility functions in select cases. To validate the theoretical findings, a numerical example is presented, demonstrating their efficacy.

Dynamic order allocation in a duopoly hybrid workforce of competition: A machine learning approach

We develop a continuous-time stochastic differential game model that aims to capture market demand and stochastic cross-network effects, and we seek to find equilibrium order allocation strategies between the firm and the platform. By solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation system, we obtain the feedback equilibrium. For a simple scenario, we derive the analytical solution which indicates that the equilibrium expenditures depend only on the marginal market thickness, and that market thickness is consistently advantageous for the value function. For the complex scenario, we propose a machine learning approach based on the Deep Galerkin Method to solve high-dimensional nonlinear HJB systems, and we demonstrate its good convergence properties. Based on reliable parameter values, our simulation results show that: (1) For higher market thickness, value functions exhibit greater sensitivity to changes in cross-network effects. (2) For a given cross-network effect, the equilibrium acquisition and retention expenditures display significant sensitivity to market thickness and time, respectively. To show the interaction between the two platforms, we present the following two results. One is that for the platform adopting a high pricing strategy, its acquisition expenditures exceed those of its competitor, while retention expenditures are the opposite. The other is that as the cross-network effect of the platform increases, its maximum profit initially rises and then declines, and the time of occurrence of the maximum profit monotonically decreases. In contrast, the competitor’s maximum profit initially declines and then rises, and the time of occurrence of the maximum profit monotonically increases.

Measures for the Importance of Information Exchange in Linear Quadratic Differential Games Over Networks

The objective of this paper is to study and characterize the role and the importance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information</i> in achieving a feedback (Nash) equilibrium strategy in linear quadratic (LQ) differential games whenever the underlying players are distributed over a (physical or logic) network. It is assumed that each player should achieve a desired goal, quantified by an individual cost functional, in a competitive dynamic environment - captured by an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> - by relying only on the information and data exchanged with other players according to a prescribed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network:</i> the objective of the paper is to establish the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">value</i> of such an information exchange pattern towards achieving a more favorable (social) equilibrium. Interestingly, it is not assumed that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network</i> coincide. Moreover, since the ability of achieving a certain Nash equilibrium strategy may be lost even by removing a single communication link in the network - thus partially limiting the use of the metrics discussed above - in the second part of the paper we also consider the value of the information in forming approximate Nash equilibrium strategies, namely the so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -Nash equilibria. Finally, the newly defined metrics are corroborated - together with a few suggested constructive results to characterize such values - by means of numerical examples.

Mixed leadership stochastic differential game in feedback information pattern with applications

This paper is devoted to a mixed leadership stochastic differential game on a finite horizon in feedback information mode, where the control variables enter into the diffusion term of the state equation. A verification theorem for the feedback Stackelberg–Nash equilibrium is obtained by using a system of coupled and fully nonlinear parabolic partial differential equations. A necessary condition for the feedback Stackelberg–Nash equilibrium is obtained as well. Theoretic results are applied to deal with a dynamic innovation and pricing decision problem where the buyer acts as the leader in the pricing decisions and the dynamic model is stochastic. Via the solutions of coupled Riccati equations, feedback equilibrium strategies of innovation and pricing are explicitly expressed. Not only the local existence and uniqueness of the solutions of the coupled Riccati equations, but also some sufficient conditions for the existence of their solutions, are obtained. Some numerical simulation results are given to discuss the effects of model parameters on the feedback equilibrium strategies.

Co‐op advertising in randomly fluctuating markets

We consider a dynamic Stackelberg game between a manufacturer and a retailer facing a randomly changing market environment (boom–bust cycle) modeled as a Markov process over time. The manufacturer announces the national advertising effort and a cooperative (co‐op) advertising subsidy, and the retailer sets the local advertising effort. We derive the optimal feedback equilibrium advertising strategies and show that the channel members are forward‐looking in their advertising decisions. In the single‐crisis case, both the necessary and sufficient conditions for a positive subsidy reveal that the subsidy decision hinges on the relative profit margins before and after the crisis. Our comparative statics analysis indicates that the presence of national advertising enriches the members' advertising behaviors significantly. Furthermore, the pre‐crisis (post‐crisis) aggregate advertising contribution of the supply chain generally decreases (increases) in the damage rate and the crisis likelihood, as in the single‐firm case. But the individual advertising behaviors of the members may exhibit richer structures, depending on the comparative advantages of the members in each state. In addition, our numerical studies demonstrate that channel members can suffer sizable losses when they adopt myopic advertising strategies over their regime‐switching counterparts. Finally, we extend our analysis to incorporate retail and brand competition and show that the channel members' advertising behaviors are reasonably consistent with those without competition.

Comparative Study on Low-Carbon Strategy and Government Subsidy Model of Pharmaceutical Supply Chain

Despite the growing urgency to curb carbon emissions worldwide, the healthcare industry, particularly the pharmaceutical industry, has received little attention from the sustainability community in terms of its contribution to the global carbon footprint. This paper constructs a differential game model of the secondary pharmaceutical supply chain consisting of pharmaceutical enterprises and medical institutions in the context of centralized drug procurement policy, considering the effects of health insurance reimbursement and consumers’ low-carbon preferences, and compares and analyzes the feedback equilibrium strategies of low-carbon inputs and marketing efforts, supply chain profits, and social welfare levels under four government subsidy models and further discusses them with arithmetic examples. The results illustrated that government subsidies have a significant impact on the low-carbon investment of pharmaceutical enterprises and the low-carbon marketing of medical institutions; subsidies for pharmaceutical enterprises can significantly increase the low-carbon investment and profit level of pharmaceutical enterprises; subsidies for medical institutions can effectively promote the implementation of the “zero-rate” policy and the realization of the emission reduction target under the centralization policy of medical institutions, increase the market demand for low-carbon drugs, and thus gain higher profits; the dual-subsidy model of the government brings higher social welfare than the single-subsidy model, and under a reasonable subsidy ratio, the profit and social welfare of the whole supply chain can be maximized.

Carbon emission reduction decisions in construction supply chain based on differential game with government subsidies

In the context of a low-carbon economy, carbon emission reduction (CER) has become an environmental issue of world concern. As a key industry sector for CER, buildings' CER behavior in the low-carbon building supply chain is affected by the policies of various national governments. This paper fills this research gap by studying and analyzing the CER decision of the construction supply chain under government subsidies with the help of differential game theory and MATLAB numerical analysis software. In this paper, the differential game model with the centralized decision-making (CDM), decentralized decision-making (DDM) and the introduction of cost-sharing contract (CSC) for CER in the construction supply chain under the government subsidies model with developers and contractors as stakeholders in the construction industry are constructed with the help of differential game theory, taking the amount of CER in construction as the state variable, and the optimal feedback equilibrium strategies of both parties in the 3 decision-situations are derived and compared, as well as the optimal trajectory of CER in construction over time. The study found that after the introduction of CSC, the total profit of the construction supply chain under DDM reaches the level of CDM. Complemented by government subsidies, the building CER and the market demand for low-carbon buildings reach optimal values for both parties under DDM.

Conditional LQ time-inconsistent Markov-switching stochastic optimal control problem for diffusion with jumps

The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.

A robust time‐inconsistent linear‐quadratic problem

AbstractThis paper is concerned with a deterministic time‐inconsistent linear‐quadratic optimal control with model uncertainty, where nonexponential time discounting appears in the objective functional. If the objective functional is viewed as an output/observation on the lifetime horizon, the disturbance attenuation of this time‐inconsistent linear‐quadratic problem is modelled and solved under the philosophy of minimax optimization, and resorts to a time‐inconsistent zero‐sum linear‐quadratic dynamic game with the controller being a minimizing player and the disturbance being an imaginary maximizing player. The definition of robust mixed equilibrium solution is introduced, whose existence is fully characterized. An example is constructed to show that the mixed equilibrium solution exists for all the initial pairs even though neither the open‐loop equilibrium control nor the feedback equilibrium strategy exists for some initial pairs. Furthermore, for a time‐consistent version of the zero‐sum linear‐quadratic game, another example is presented to indicate the case that the existence of open‐loop time‐consistent equilibrium does not imply the existence of open‐loop precommitted equilibrium.

Dynamic innovation and pricing decisions in a supply-Chain

This paper studies dynamic innovation and pricing decisions in a two-echelon supply chain. We model a distribution channel where a seller sells a product to an independent buyer who ultimately sells it to the customers. We refer to innovation as efforts made on the product quality improvement, or on process improvement. Both the players can put innovation efforts over time which in turn may enhance the goodwill of the product in market. The product demand increases with goodwill and decreases with the retail price. The innovation efforts can also impact the unit processing cost of the product at the upstream firm’s end positively or negatively. We model the problem as a Stackelberg differential game in which the seller first announces its wholesale price and innovation efforts over time and the buyer responds by deciding the retail price and its innovation efforts over time. We obtain feedback equilibrium strategies for a central decision maker in centralized channel, and for both the players in a decentralized channel. We also obtain several useful managerial insights using analytical as well as numerical means.

Mean-variance equilibrium asset-liability management strategy with cointegrated assets

This paper investigates asset-liability management problems in a continuous-time economy. When the financial market consists of cointegrated risky assets, institutional investors attempt to make profit from the cointegration feature on the one hand, while on the other hand they need to maintain a stable surplus level, that is, the company’s wealth less its liability. Challenges occur when the liability is random and cannot be fully financed or hedged through the financial market. For mean–variance investors, an additional concern is the rational time-consistency issue, which ensures that a decision made in the future will not be restricted by the current surplus level. By putting all these factors together, this paper derives a closed-form feedback equilibrium control for time-consistent mean–variance asset-liability management problems with cointegrated risky assets. The solution is built upon the Hamilton–Jacobi–Bellman framework addressing time inconsistency. doi: 10.1017/S1446181120000164

Pricing policies of a dynamic green supply chain with strategies of retail service

PurposeThe paper investigates firms' optimal pricing policies and green strategies in a dynamic green supply chain with consideration of different retail service strategies. The purpose of the paper is to address the following research questions: (1) What are the optimal pricing policies and green strategies of the dynamic decentralized supply chain with the competitive or supportive retail service? (2) How does the dynamic consumer's perception of green product affect these equilibrium solutions?Design/methodology/approachThe paper establishes the dynamic game models and then derives a firm's instantaneous and steady-state feedback equilibrium solutions in three scenarios as follows: (1) the integrated supply chain; (2) the decentralized supply chain with competitive retail service and (3) the decentralized supply chain with supportive retail service. Finally, we conduct numerical analyses to compare the firm's instantaneous and steady-state equilibrium solutions and profit in the three scenarios.FindingsThe theoretical and numerical analysis results suggest that the supportive retail service is less inefficient than the competitive retail service in the decentralized supply chain and that the types of retail service have no influence on the green strategy. Moreover, a firm's myopia leads to lowering the greenness degree, retail service level and severe price competition, resulting in economic losses. Consumers’ initial perception of greenness degree determines whether the retailer should adopt the skimming pricing strategy or penetration pricing strategy. Furthermore, only when consumers’ perception of greenness degree is higher than a threshold, will the manufacturer produce green product with positive greenness degree.Originality/valueThis is one of few studies on the effect of different types of retail service on horizontal competition in green supply chain. The extension of the static study by adopting differential game approaches provides researchers with a deeper understanding of the application of retail service in green supply chain.

Research on Ecological Compensation in the Yellow River Basin Based on the Perspective of Game Theory

The Yellow River Basin is an important ecological function zone in China, and it plays an important role in the national economic and social development pattern. However, the over-utilization of water resources has made the ecological and environmental problems in the river basin more and more serious. For this reason, the coordinated governance of the areas along the Yellow River basin has become very important. Based on the static game model, this paper conducts a comprehensive analysis of the coordinated governance of the Yellow River basin in the upstream and downstream areas, constructs a static game model of ecological compensation between the Shanxi and Henan governments under coordinated governance, and studies the feedback equilibrium strategies of the Shanxi and Henan governments. This article provides theoretical basis and suggestions for collaborative governance to achieve sustainable development. The game results show that: reducing the cost of protecting the ecological environment of the upstream basin, increasing the benefits of protecting the ecological environment, and increasing the amount of ecological compensation can promote the implementation of the ecological compensation policy between the upstream and downstream in the basin.

Dynamic game in agriculture and industry cross-sectoral water pollution governance in developing countries

Agriculture in developing countries is a vulnerable sector and is subsidized. Environmental restrictions on agriculture sector are minimal. The excessive use of chemical fertilizers and pesticides has caused serious non-point source water pollution. Although previous studies have focused on agricultural water pollution, the cooperation between agricultural and industrial sectors in water pollution mitigation has not been thoroughly discussed. In this context, this paper introduces a cross-sectoral water pollution dynamic model. We studied the feedback (subgame perfect) equilibrium of the non-cooperative game between two sectors. To discern the impact of environmental policy, we compared the equilibrium solutions of two scenarios that differ in terms of environmental policy for agricultural sector mitigation. The results show that when agricultural and industrial sectors work together to mitigate water pollution, the accumulated common pollutants level decreases; when sectors consider mitigation measures in their objectives, they tend to emit more pollutants; the damage produced by both sectors is lower when both sectors participate in mitigation. Our policy recommendation is that cross-sectoral water pollution control is essential and developing countries should include both agricultural and industrial sectors in environmental governance to maximize the coordination of cross-sectoral environmental policies.

Groundwater extraction among overlapping generations: a differential game approach

Groundwater is a common resource that has been wasted for years. Today, we pay the consequences of such inappropriate exploitation and we are aware that it is necessary to realize policies in order to guarantee the use of this resource for future generations. In fact, the irrational exploitation of water by agents, nevertheless it is a renewable resource, may cause its exhaustion. In our paper, we develop a differential game to determine the efficient extraction of groundwater resource among overlapping generations. We consider intragenerational as well as intergenerational competition between extractors that exploit the resource in different time intervals, and so the horizons of the players in the game are asynchronous. Feedback equilibria have been computed in order to determine the optimal extraction rate of “young” and “old” agents that coexist in the economy. The effects of the withdrawal by several generations are numerically and graphically analyzed in order to obtain results on the efficiency of the groundwater resource.

MEAN–VARIANCE EQUILIBRIUM ASSET-LIABILITY MANAGEMENT STRATEGY WITH COINTEGRATED ASSETS

AbstractThis paper investigates asset-liability management problems in a continuous-time economy. When the financial market consists of cointegrated risky assets, institutional investors attempt to make profit from the cointegration feature on the one hand, while on the other hand they need to maintain a stable surplus level, that is, the company’s wealth less its liability. Challenges occur when the liability is random and cannot be fully financed or hedged through the financial market. For mean–variance investors, an additional concern is the rational time-consistency issue, which ensures that a decision made in the future will not be restricted by the current surplus level. By putting all these factors together, this paper derives a closed-form feedback equilibrium control for time-consistent mean–variance asset-liability management problems with cointegrated risky assets. The solution is built upon the Hamilton–Jacobi–Bellman framework addressing time inconsistency.

The Shrinking of Beijing and the Rising of Xiong’an: Optimize Population Migration in terms of Transport Service

The population of Beijing has already come to its loading capacity. The China central government plans to build an ideal city named Xiong’an nearby Beijing. The city is expected to work as a carrying hub for noncapital functions of Beijing. The central government does not rush to build before a deliberated urban planning is accomplished. For sustainable development, a difficulty faced by urban planners is that the maximum number of people can be migrated from Beijing to Xiong’an with constraint on level of transport service. This paper developed a specialized bilevel programming model where the upper level is to ensure a predetermined transport service level regarding to population migration, while the lower level is feedback equilibrium between trip generation and traffic assignment. To be more specific, trip is generated by the gravity model, and traffic is assigned by the user equilibrium model. It is well known that the bilevel programming problem is tough and challenging. A try-and-error algorithm is designed for the upper-level model, and a method of successive average (MSA) is developed for the lower-level model. The effectiveness of the model and algorithm is validated by an experimental study using the current transport network between Beijing and Xiong’an. It shows that the methods can be very useful to identify the maximum population migration subject to level of transport service.

Mean Field Stackelberg Games: State Feedback Equilibrium

We study mean field Stackelberg games between a major player (the leader) and a large population of minor players (the followers). By treating the mean field as part of the dynamics of the major player and a representative minor player, we Markovianize the decision problems and employ dynamic programming to determine the equilibrium strategy in a state feedback form. We show that for linear quadratic (LQ) models, the feedback equilibrium strategy is time consistent. We further give the explicit solution in a discrete-time LQ model.

Optimal Strategies of Product Price, Quality, and Corporate Environmental Responsibility.

With the awakening of environmental consciousness, more and more firms desire to go “green” by shifting their focus of corporate social responsibility (CSR) from charitable contributions to environmental actions called corporate environmental responsibility (CER). We develop a monopoly differential game to depict optimal corporate strategies of product price, quality, and CER. Using the Hamilton–Jacobi–Bellman (HJB) equation, we analyze optimal feedback equilibrium strategies for pricing and investing in both quality and CER with/without government subsidies. Numerical simulations show that government subsidy can improve CER and profit.

Equilibrium Solutions of Multiperiod Mean-Variance Portfolio Selection

This is a companion paper of [Mixed equilibrium solution of time-inconsistent stochastic linear-quadratic problem, SIAM J. Control Optim., vol. 57, no. 1, 533-569, 2019], where general theory has been established to characterize the open-loop equilibrium control, feedback equilibrium strategy and mixed equilibrium solution for a time-inconsistent stochastic linear-quadratic problem. This note is, on the one hand, to test the developed theory of that paper and on the other hand to push the solvability of multiperiod mean-variance portfolio selection. A nondegenerate assumption, which is popular in the existing literature about multiperiod mean-variance portfolio selection, has been removed in this note; and neat conditions have been obtained to characterize the existence of equilibrium solutions.