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

Abstract

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.

On Stackelberg Leader with Min-Max Followers to Solve Fuzzy Continuous Static Games

“…Khalifa and Kumar [21] investigated cooperative continuous static games in a crisp environment, defining and determining the first-kind stability set corresponding to the solution without differentiability. Under uncertainty environment, some recent researchers for cooperative games have been introduced (Mallozzi and Messalli [22], Bigdeli and Hassanpour [23], Elshafei [24], Zaichenko [25], Zidan et al [26], Donahue et al [27], Ganzfried [28], Zhou et al [29], Krishankumar et al [30,31], Tukhtasinov et al [32], Sivagami et al [33], Zhao et al [34], Megahed [35], Romanuke [36], and Khalifa et al [37]).…”

On Stability of Continuous Cooperative Static Games with Possibilistic Parameters in the Objective Functions

“…In this section, the proposed approach has been compared with some existing literatures [23][24][25][26][27] in terms of their characteristic features to illustrate the advantages of the suggested approach. e results for this analysis are summarized in Table 1.…”

“…In contrast, all other existing models fail to deal with it. Other than that the method proposed in [24,26,27] also derived the e cient solution for the problem along with the proposed one, but all these existing methods have considered the uncertainties with fuzzy variables; however, in the proposed method, a possibility variable has been used to address the uncertainties. e proposed method suggested the interactive approach based on the decision-maker preferences.…”

“…Khalifa and Kumar [23] studied the cooperative continuous static games in a crisp environment, defined them, and found the firstkind stability set corresponding to the solution without differentiability. Several researchers [24][25][26][27][28][29] have recently enriched the theory of cooperative games by considering the degree of uncertainties in the analysis.…”