A Simplest Differential Game on a Plane with Four Participants

Abstract

Introduction. The article presents a simplest differential game with four participants. The players move on a plane and can do simple movements. The game under considering comes down to a cooperative differential game. The dynamic stability of such optimality principles as the S-kernel and Shapley vector is shown. Materials and Methods. The standard procedures of the cooperative game theory are applied to the analysis and decision of a cooperative differential game. The conditional and optimum trajectories, along which the players move, are found using the Pontryagin’s maximum principle. When constructing the characteristic function, the minimax approach is used. Results. The optimum strategy of the players, conditional and optimum trajectories of their movements at various ways of formation of coalitions are written out explicitly. The characteristic function is constructed according to the accepted max-min principle; the S-kernel and Shapley vector are considered as a decision. The components of the Shapley vector are written out explicitly; the fact that the Shapley vector is an element of the S-kernel and nonemptiness of the S-kernel, when the players are moving along an optimum trajectory, are shown. Using the results of the static cooperative game theory for researching differential games, we face the problems, which are connected with specifics of the differential equations of the movement. As a priority, the problem of the dynamic stability of the optimality principles under consideration is identified. In the work, the dynamic stability of the Shapley vector and S-kernel is shown. Discussion and Conclusion. The results of the research show that the analysis of the dynamic stability of the optimality principles considered is relevant.