Hyperbolicity conditions for non-cooperative differential games

The Nash equilibrium is a classical and fundamental type of solution in non-cooperative game theory, both in static and dynamic, or differential, problems. In the latter we focus our attention to the study of existence of Nash equilibrium strategies in feedback form. By optimal control theory this problem can be connected to the study of a system of partial differential equation, with terminal time conditions, satisfied by the value function. By PDE theory we can find conditions under which the Cauchy problem is “well-posed”. A basic requirement is that the system should be hyperbolic. We observe that in generic dimension, even just for two players, the hyperbolicity condition generally fails. We can recover some sort of weak hyperbolicity considering a semi-cooperative approach. Lastly, in one space dimension, using the theory on conservation laws, we can state reasonable conditions for which the system is strictly hyperbolic.