Cooperative Stochastic Differential Games Research Articles

Pareto efficiency of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps

This paper investigates a wide class of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps, which can model uncertainties of internal mode transition and characterize occasional sudden changes in the games, respectively. Firstly, the necessary conditions which guarantee the existence of Pareto solutions are obtained by utilizing the Lagrange multiplier method and the stochastic maximum principle with Markov jumps and Poisson jumps. Then the sufficient conditions which guarantee the existence of Pareto efficient strategies are derived. Secondly, the well-posedness of cooperative stochastic differential games (CSDG) with Markov jumps and Poisson jumps in infinite horizon is established when the solution of generalized algebraic Riccati equation (GARE) exists. Furthermore, we can obtain Pareto solutions by introducing the coupled algebraic Lyapunov equations (ALEs). Finally, the numerical examples verify the theoretical results.

Pareto efficiency of finite-horizon mean-field cooperative stochastic differential games with poisson jumps

This paper investigates finite-horizon mean-field cooperative stochastic differential games (CSDG) with Poisson jumps. The game model can describe the case where the game has external disturbances and suffers sudden changes. First, for the fixed initial state, the necessary conditions which guarantee the existence of Pareto solutions can be obtained by utilizing the Lagrange multiplier method and the mean-field type stochastic maximum principle (MF-SMP) with Poisson jumps. Then, the sufficient conditions which guarantee the existence of Pareto efficient strategies are derived. Next, for any initial states, the well-posedness of finite-horizon mean-field CSDG with Poisson jumps is established under the assumption that the solution of generalized algebraic Riccati equations (GAREs) exists. Further, all Pareto solutions can be obtained by constructing an algebraic Lyapunov equations (ALEs). Finally, a numerical example given illustrates our results.

Pareto efficiency in the infinite horizon mean-field type cooperative stochastic differential game

This paper discusses the linear quadratic (LQ) mean-field type stochastic differential game with non-homogeneous terms in infinite horizon. Employing the equivalent description of Pareto efficiency, necessary conditions for the existence of Pareto solutions are presented under an assumption on the Lagrange multiplier set. Two conditions are introduced to guarantee that the assumption is established. Further, sufficient conditions for a control to be Pareto efficient are put forward in terms of the necessary conditions, the convexity condition on the homogeneous weighted sum cost functional and a transversality condition. The characterization of Pareto solutions is also studied for the homogeneous case. If the system is MF-L2-stabilizable, then the solvability of the related generalized algebraic Riccati equations (GAREs) provides a sufficient condition under which the cost functionals are convex and all Pareto efficient strategies can be obtained by the weighted sum minimization method. In addition, by introducing two algebraic Lyapunov equations (ALEs), we derive all Pareto solutions.

Stochastic differential reinsurance games in diffusion approximation models

In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

Fitted Finite Volume Method of Three Transboundary Pollution in Three Gorges Reservoir Area of Chongqing City with Emission Permits Trading by Cooperative Stochastic Differential Game

Fitted Finite Volume Method of Three Transboundary Pollution in Three Gorges Reservoir Area of Chongqing City with Emission Permits Trading by Cooperative Stochastic Differential Game

An Empirical Study for Transboundary Pollution of Three Gorges Reservoir Area with Emission Permits Trading

In this paper, we discuss a cooperative stochastic differential game for the transboundary industrial pollution problems of Three Gorges Reservoir Area. Base on the stochastic optimal control theory, we derive the Hamilton–Jacobi–Bellman equations for the cooperative games. Furthermore we solve the Hamilton–Jacobi–Bellman equations by using a fitted finite volume method. Finally, an empirical study base on the datum of Three Gorges Reservoir Area is given to demonstrate the efficiency and usefulness of the numerical method.

A Cooperative Stochastic Differential Game of Transboundary Industrial Pollution between Two Asymmetric Nations

Considering the fact that transboundary pollution control calls for the cooperation between interested parties, this paper studies a cooperative stochastic differential game of transboundary industrial pollution between two asymmetric nations in infinite-horizon level. In this paper, we model two ways of transboundary pollution: one is an accumulative global pollutant with an uncertain evolutionary dynamic and the other is a regional nonaccumulative pollutant. In our model, firms and governments are separated entities and they play a Stackelberg game, while the governments of the two nations can cooperate in pollution reduction. We discuss the feedback Nash equilibrium strategies of governments and industrial firms, and it is found that the governments being cooperative in transboundary pollution control will set a higher pollution tax rate and make more pollution abatement effort than when they are noncooperative. Additionally, a payment distribution mechanism that supports the subgame consistent solution is proposed.

Optimal coordination strategy of dynamic supply chain based on cooperative stochastic differential game model under uncertain conditions

The optimal coordination strategy is achieved when the dynamic supply chain profit is maximized. The profit of dynamical supply chain is decided by the corresponding effort level of each node enterprise. This paper presents a cooperative stochastic differential game model to research the optimal coordination strategy of a dynamic supply chain under uncertain conditions and studies how to coordinate the effort level of node enterprise to maximize supply chain profit. For this purpose, a cooperative stochastic differential game model of the enterprises’ effort level is constructed, and the optimal solution of the model is solved by converting it to the solution of the equivalent stochastic partial differential equation. As a result, the optimal strategy of a local node enterprise is achieved. Considering the gap between the independent decision of each node enterprise and the optimal decision of the supply chain system, the coordination mechanisms of the dynamic supply chain, including the temporary static allocation mechanism and the compensation mechanism, are given to narrow the gap to achieve the optimal coordination strategy under uncertain conditions.

Cooperative economic growth

We comparatively study optimal economic growth in a simple endogenous growth model and under two different games, i.e., dynamic sequential game and cooperative stochastic differential game, between a representative household and a typical self-interested politician. Sequential equilibrium solution is derived by applying Backward Induction Principle and corresponding optimal economic growth rate is endogenously determined. Moreover, cooperative equilibrium solution is established with group rationality, individual rationality and sub-game consistency requirements fulfilled, and it is further confirmed that the representative household will save more, and the self-interested politician will tax less, thereby leading to much faster economic growth, when compared to those of the sequential equilibrium solution.

Governance Mechanism for Global Greenhouse Gas Emissions: A Stochastic Differential Game Approach

Today developed and developing countries have to admit the fact that global warming is affecting the earth, but the fundamental problem of how to divide up necessary greenhouse gas reductions between developed and developing countries remains. In this paper, we propose cooperative and noncooperative stochastic differential game models to describe greenhouse gas emissions decision makings of developed and developing countries, calculate their feedback Nash equilibrium and the Pareto optimal solution, characterize parameter spaces that developed and developing countries can cooperate, design cooperative conditions under which participants buy the cooperative payoff, and distribute the cooperative payoff with Nash bargaining solution. Lastly, numerical simulations are employed to illustrate the above results.

Agent behaviors and coordinative mechanism

PurposeThe purpose of this paper is to improve the behaviors coordination mechanism, to maintain the system's long time‐scale and stable competitive capability, when the agents in the system focus on cooperating with each other.Design/methodology/approachEffort level for every agent, whose dynamics can be described as a stochastic partial differential equation, and the incentive of effort as the control of the corresponding agent, are introduced to describe agents' behavior abstracted. The cooperative stochastic differential game model is constructed: first, the optimal resolve trajectory mapping with profit maximization of the system are obtained, then the transitory imputation coupled with effort initial state of the system by introducing dynamic Shapley value imputation method. Based on the results obtained, the profit distribution strategies and the equilibration incentive compensation mechanism are given, due to the evolution law of the payoff and the state variable.FindingsIt is concluded that: the transitory compensation to agent for efforts and incentive, which can be changed with the system state at current and in history and in future changed, would guarantee the realization of the Shapley value imputation throughout the game horizon.Originality/valueIn this paper, the interactivity between agents in the system is considered first. The dynamical Shapley imputation mechanism and the transitory compensatory mechanism are provided to make the imputation more stable and feasible.

Comparative Studies on Cooperative Stochastic Differential Game and Dynamic Sequential Game of Economic Maturity

In the paper, we are encouraged to investigate the effect of game structure imposed on the minimum-time needed to economic maturity in a dynamic macroeconomic model. Indeed, we have established a basic framework for the comparative study of the cooperative stochastic differential game and dynamic sequential game of economic maturity. Moreover, in a simple stochastic growth model, closed-form solution of the minimum-time needed to economic maturity has been derived with the explicit condition, under which it is confirmed that cooperation between the representative household and the self-interested politician will definitely lead us to much faster economic maturity than that of sequential action, supplied, too. Finally, our model supports the comparative study of the minimum-time needed to economic maturity under different political-institution constraints.

A stochastic differential game theory modeling and analyzing for Multi-Local-Worlds graph directed

Separating the management system into several Local-Worlds, the multi-players cooperation game and non-cooperation game in a graph are constructed to describe the interaction between agents in a certain Local-World and between different Local-Worlds respectively. Furthermore, the corresponding analytic processes that consist of two different game theory models are specified, then the profits, the optimal decision of agent, in this system with cooperative stochastic differential game model between Agents and non-cooperative stochastic differential game model between Super-Agents considered simultaneously, are given.

International Cooperation on Pollution Control Between Asymmetric Countries

The purpose of this study was to investigate international cooperation on pollution control between asymmetric countries under uncertainty. We considered any two countries that are asymmetric with respect to their technological level and the amount of environmental damage they incur. Production processes result in the emission of pollutants that are added to the existing pollution stock common to both countries. The stock of pollution evolves stochastically, according to the geometric Brownian motion. The governments of the two countries set their pollutant emission levels at the Markov perfect equilibrium to maximize their expected net present value of social welfare. In addition, they set their pollutant emission levels at cooperative equilibrium to maximize the sum of their expected net present value of social welfare.In cooperative stochastic differential games, a credible cooperative agreement must be subgame consistent. Subgame consistency ensures extension of an optimal policy to a later starting time, and any possible state brought about by prior optimal behaviour of the governments remains optimal.We considered a cooperative game in which the governments of two countries agree to maximize and divide the sum of their expected net present value of social welfare in a way that shares the gain from their cooperation proportional to the relative sizes of their expected non-cooperative net present value of social welfare at every instant of time. In conclusion, the country with a higher technological level or that incurs lower environmental damage obtains a larger instantaneous payoff at the subgame consistent solution.JEL Classification: F18, L13, Q58

Stochastic Differential Game of Transboundary Pollution and International Environmental Policy

The purpose of this paper is to investigate a stochastic differential game of transboundary pollution and environmental policy. We considered two asymmetric countries. The production process results in the emission of pollutants that are added to the existing stock common to both countries. The pollution accumulation dynamics is affected by an additive shock, the mean and variance are known. In both countries, the government controls emission levels. Uncertainty is incorporated into the model through a shock. The stock of pollution evolves stochastically according to geometric Brownian motion. In order to maximize the expected net present value of social welfare, the two governments should set the emission levels. We formulate stochastic differential games that need to be solved by the two governments and derive unique Markov-perfect Nash and cooperative equilibrium solutions. Subsequently, we propose a mechanism to derive the payoff distribution procedures of subgame consistent solutions in cooperative stochastic differential games for the two governments, and are able to derive a subgame consistent solution based on the Nash bargaining axioms. In conclusion, we show how the governments agree to maximize the sum of their expected payoffs and divide the total cooperative payoff so that the Nash bargaining outcome is maintained at every instant of time. Moreover, we show conditions guaranteeing individual rationality.JFL Classification: F18, L13, Q58

A cooperative stochastic differential game of transboundary industrial pollution

Though cooperation in environmental control holds out the best promise of effective actions, limited success has been observed because existing multinational joint initiatives fail to satisfy the property of subgame consistency. A cooperative solution is subgame consistent if the solution optimality principle is maintained in any subgame which starts at a later time with any feasible state brought about by prior optimal behaviors. This paper presents a cooperative stochastic differential game of transboundary industrial pollution with two novel features. The first feature is that industrial production creates short-term local impacts and long-term global impacts on the environment. Secondly, a subgame consistent cooperative solution is derived in this stochastic differential game together with a payment distribution mechanism that supports the subgame consistent solution. This is the first time that pollution management is analyzed in a cooperative stochastic differential game framework under these novel features.

SOLUTION MECHANISMS FOR COOPERATIVE STOCHASTIC DIFFERENTIAL GAMES

Cooperative stochastic differential games constitute a highly complex form of decision making under uncertainty. In particular, interactions between strategic behaviors, dynamic evolution, stochastic elements and solution agreement have to be considered simultaneously. This complexity leads to great difficulties in the derivation of dynamically stable solutions. Despite urgent calls for cooperation in the global economy, the lack of formal analyses has precluded rigorous analysis of this problem. In this paper, mechanisms for the derivation of dynamically stable solutions to cooperative stochastic differential games are presented. Games with transferable payoffs and those with nontransferable payoffs are considered. Numerical illustrations are also provided.

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.

TECHNICAL NOTE: NONTRANSFERABLE INDIVIDUAL PAYOFFS IN COOPERATIVE STOCHASTIC DIFFERENTIAL GAMES

This note develops a mechanism for the derivation of individual player's payoff functions in cooperative stochastic differential games with nontransferable payoffs. An illustrative example is provided.

Subgame Consistent Cooperative Solutions in Stochastic Differential Games

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that: (i) the extension of the solution policy to a later starting time and to any possible state brought about by the prior optimal behavior of the players would remain optimal; (ii) all players do not have incentive to deviate from the initial plan. In this paper, we develop a mechanism for the derivation of the payoff distribution procedures of subgame consistent solutions in stochastic differential games with transferable payoffs. The payoff distribution procedure of the subgame consistent solution can be identified analytically under different optimality principles. Demonstration of the use of the technique for specific optimality principles is shown with an explicitly solvable game. For the first time, analytically tractable solutions of cooperative stochastic differential games with subgame consistency are derived.