Open-Loop Strategies in Nonzero-Sum Differential Game with Multilevel Hierarchy
The paper is concerned with the construction of open-loop strategies for the n-person nonzero-sum differential game with multilevel hierarchy. The dynamics of the first player (leader) is defined by its own position and control. The player of further levels of hierarchy knows the position and control of the players of the upper hierarchical levels. At the same time the dynamics and payoff functional of the player do not depend on the position and control of lower hierarchical levels.
We solve this problem with the help of consequent solutions of optimal control problems for each player. Using the solution of Hamilton—Jacobi equation and the results of optimal control theory we construct the open-loop controls of the players. The specifics of this problem is the construction of solution for the Hamilton—Jacobi equation with the Hamiltonian discontinuous w.r.t. phase variable. In this case we use the notion of multivalued solution proposed by Subbotin. We show that the open-loop controls provide a Nash equilibrium in the differential game with multilevel hierarchy and the set of payoffs of the players is described by the multivalued solution of the corresponding Hamilton—Jacobi equation.
KeywordsDiscontinuous Hamilton—Jacobi equations Nonzero-sum differential game Nash equilibrium
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