Non-cooperative and semi-cooperative differential games

In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.