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.

中文翻译：

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基于Pontryagin极大原理的连续更新的可转移效用合作微分对策。

我们考虑一类利用庞特里亚金极大值原理不断更新的合作差分博弈。假定在每个时刻，玩家具有或使用关于在固定持续时间的封闭时间间隔内定义的游戏结构的信息。随着时间的流逝，有关游戏结构的信息将会更新。本文的主题是构造具有连续更新的这类差分游戏的参与者的合作策略，他们的合作轨迹，特征函数和合作解决方案，尤其是以庞特里亚金的最大原理为最优条件。为了证明这种方法的新颖性，我们建议比较合作策略，轨迹，特征函数，以及经典（初始）差分游戏和具有连续更新的差分游戏的相应Shapley值。我们的方法为冲突控制流程提供了更深入的建模方法。在一个特定的示例中，我们证明了玩家在游戏开始时的行为勇敢并不断进行更新，因为他们缺乏整个游戏的信息，并且“本质上是时间不一致的”。相反，在初始模型中，参与者更加谨慎，这意味着他们一开始不敢排放过多的污染。我们证明了玩家在游戏开始时的行为勇敢并不断进行更新，因为他们缺乏整个游戏的信息，并且“本质上是时间不一致的”。相比之下，在初始模型中，参与者更加谨慎，这意味着他们一开始不敢排放过多的污染。我们证明了玩家在游戏开始时的行为勇敢并不断进行更新，因为他们缺乏整个游戏的信息，并且“本质上是时间不一致的”。相反，在初始模型中，参与者更加谨慎，这意味着他们一开始不敢排放过多的污染。