Optimal training for adversarial games

We consider adversarial games solved by a continuous version of the simultaneous gradient descent method, whose associated differential system is induced by a Hamiltonian function. In this case the solution obtained by training does never converge to the Nash equilibrium, but it might be closest to it at some special time instance. We analyse this optimal training time in two distinct situations: the hyperbolic and elliptic types of equilibria, covering the case of quadratic Hamiltonians. The case of more general Hamiltonian functions can be treated similarly after they are replaced by their quadratic approximations.