# Open-Loop Strategies in Nonzero-Sum Differential Game with Multilevel Hierarchy

## Abstract

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.

## Keywords

Discontinuous Hamilton—Jacobi equations Nonzero-sum differential game Nash equilibrium## References

- 1.Pan, L., Yong, J.: A differential game with multi-level of hierarchy. J. Math. Anal. Appl.
**161**(2), 522–544 (1991)MathSciNetCrossRefGoogle Scholar - 2.Xu, H., Mizukami, K.: Linear feedback closed-loop Stackelberg strategies for descriptor systems with multilevel hierarchy. J. Optim. Theory Appl.
**88**(1), 209–231 (1996)MathSciNetCrossRefGoogle Scholar - 3.Case, J.H.: Toward a theory of many player differential games. SIAM J. Control
**7**(2), 179–197 (1969)MathSciNetCrossRefGoogle Scholar - 4.Friedman, A.: Differential Games. Wiley-Interscience, Hoboken (1971)zbMATHGoogle Scholar
- 5.Starr, A.W., Ho, Y.C.: Non-zero Sum differential games. J. Optim. Theory Appl.
**3**(3), 184–206 (1969)CrossRefGoogle Scholar - 6.Kolpakova, E.A.: On the solution of a system of Hamilton-Jacobi equations of special form. Trudy Inst. Mat. i Mech. UrO RAN
**23**(1), 158–170 (2017)MathSciNetCrossRefGoogle Scholar - 7.Bressan, A., Shen, W.: Semi-cooperative strategies for differential games. Int. J. Game Theory
**32**, 1–33 (2004)MathSciNetCrossRefGoogle Scholar - 8.Cardaliaguet, P., Plaskacz, S.: Existence and uniqueness of a Nash equilibrium feedback for a simple nonzeo-sun differential game. Int. J. Game Theory
**32**, 33–71 (2003)CrossRefGoogle Scholar - 9.Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York-London (1972)zbMATHGoogle Scholar
- 10.Subbotin, A.I.: Generalized Solution of the first Order PDEs. The dynamical optimization Perspectives. Birkhauser, Boston-Berlin (1995)CrossRefGoogle Scholar
- 11.Subbotina, N.N.: The method of characteristics for Hamilton-Jacobi equations and applications to dynamical optimization. J. Math. Sci.
**135**(3), 2955–3091 (2006)MathSciNetCrossRefGoogle Scholar - 12.Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1999)zbMATHGoogle Scholar
- 13.Lakhtin, A.S., Subbotin, A.I.: The minimax and viscosity solutions in discontinuous partial differential equations of the first order. Dokl. Akad. Nauk
**359**(4), 452–455 (1998)MathSciNetzbMATHGoogle Scholar - 14.Kolpakova, E.A.: Solution for a System of Hamilton—Jacobi equations of special type and a link with nash equilibria. In: Frontiers of Dynamic Games, Static and Dynamic Game Theory: Foundations and Applications, pp. 53–66. Birkhauser, Basel (2018)Google Scholar