Class Of Cooperative Differential Games Research Articles

Cooperative differential games with continuous updating using Hamilton–Jacobi–Bellman equation

This paper examines a class of cooperative differential games with continuous updating. Here it is assumed that at each time instant players have or use information about the game structure defined for a closed time interval with fixed duration. The current time continuously evolves with the updating interval. The main problems considered in a cooperative setting with continuous updating is how to define players' cooperative behaviour, how to construct a cooperative trajectory, how to define the characteristic function and how to arrive at a cooperative solution. This paper also addresses the properties of the solution and presents some techniques to fix the process by which a cooperative solution is constructed. Theoretical results are demonstrated on a differential game model of non-renewable resource extraction, initial and continuous updating versions are also considered. Comparison of cooperative strategies, trajectories, characteristic functions and corresponding Shapley values is presented.

Strong Time-Consistent Solution for Cooperative Differential Games with Network Structure

One class of cooperative differential games on networks is considered. It is assumed that interaction on the network is possible not only between neighboring players, but also between players connected by paths. Various cooperative optimality principles and their properties for such games are investigated. The construction of the characteristic function is proposed, taking into account the network structure of the game and the ability of players to cut off connections. The conditions under which a strong time-consistent subcore is not empty are studied. The formula for explicit calculation of the Shapley value is derived. The results are illustrated by the example of one differential marketing game.

Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle

We consider a class of cooperative differential games with continuous updating making use of the Pontryagin maximum principle. It is assumed that at each moment, players have or use information about the game structure defined in a closed time interval of a fixed duration. Over time, information about the game structure will be updated. The subject of the current paper is to construct players’ cooperative strategies, their cooperative trajectory, the characteristic function, and the cooperative solution for this class of differential games with continuous updating, particularly by using Pontryagin’s maximum principle as the optimality conditions. In order to demonstrate this method’s novelty, we propose to compare cooperative strategies, trajectories, characteristic functions, and corresponding Shapley values for a classic (initial) differential game and a differential game with continuous updating. Our approach provides a means of more profound modeling of conflict controlled processes. In a particular example, we demonstrate that players’ behavior is braver at the beginning of the game with continuous updating because they lack the information for the whole game, and they are “intrinsically time-inconsistent”. In contrast, in the initial model, the players are more cautious, which implies they dare not emit too much pollution at first.

Cooperative differential game model with continuous updating

The paper considers and describes the class of cooperative differential games with continuous updating. Such a class of differential games is new, at the moment only the classnoncooperative game models with continuous updating have been studied. This paper describes the process of constructing cooperative strategies, cooperative trajectory, characteristicfunction and cooperative solution with continuous updating. Cooperative case of limited resource extraction game model with continuous updating is considered. Optimal strategies,characteristic function and cooperative solution are constructed. The Shapley vector is used as a cooperative solution. The numerical simulation results are demonstrated in the Matlabenvironment.

On class of non-transferable utility cooperative differential games with continuous updating

This paper considers and describes the class of cooperative differential games with the non-transferable utility and continuous updating. It is the first detailed paper about the application of continuous updating approach to the non-transferable utility differential games. The process of how to construct Pareto optimal strategy with continuous updating and Pareto trajectory is described. Another important contribution is that the property of subgame consistency is adopted for the class of games with continuous updating. The resource extraction game model is used as an example. The Pareto optimal strategies and corresponding trajectory are constructed, and the set of Pareto optimal strategies satisfying the subgame consistency property is presented. The results of numerical simulation are demonstrated in the Matlab environment, and the conclusion is drawn.

DYNAMICALLY CONSISTENT SOLUTION FOR A POLLUTION MANAGEMENT GAME IN COLLABORATIVE ABATEMENT WITH UNCERTAIN FUTURE PAYOFFS

This paper presents dynamically stable solutions to a class of cooperative differential games of pollution management in collaborative abatement with uncertain future payoffs. Collaborative abatement leads to a decrease in cost due to reduction in duplicated efforts in the process of joint development and brings about an enhancement in the effectiveness of abatement activities via the sharing of knowledge from individual nation's research. It is believed by many researchers to be the key to effective pollution reduction. Uncertainties in future economic performance are prevalent in fast developing countries like China, Brazil and India. This type of uncertainties often hinders the reaching of cooperative agreements in joint pollution control initiatives. In dynamic cooperative games, a credible cooperative agreement has to be dynamically consistent. For dynamic consistency to hold, the agreed upon optimality principle must remain in effect at any instant of time throughout the game along the optimal state trajectory contingent upon the realization of specific random events. In this analysis, dynamically consistent cooperative solutions and analytically tractable payoff distribution procedures contingent upon specific random events are derived. This approach widens the application of cooperative differential game theory to environmental problems where future payoffs are not known with certainty.

Dynamically stable principles of optimality in cooperative differential games on quick action operation

The class of cooperative differential games on quick-action operation is determined, and the principles of optimality in it and questions of existence of solutions are considered. A superadditive characteristic set which is the analogue of a characteristic function in a cooperative game is constructed. The-concept of sharing and of sharing predominance is introduced. The principle of dynamic stability is defined in a cooperative game on quick-action operation. The theorem on the existence of a dynamically stable c-nucleus is proved. The application of the proposed treatment of games of group pursuit on quick-action operation is considered.