  A common strategy for solving an unconstrained two-player Nash equilibrium
problem with continuous variables is applying Newton's method to the system of
nonlinear equations obtained by the corresponding first-order necessary
optimality conditions. However, when taking into account the game dynamics, it
is not clear what is the goal of each player when considering that they are
taking their current decision following Newton's iterates. In this paper we
provide an interpretation for Newton's iterate in view of the game dynamics as
follows: instead of minimizing the quadratic approximation of their objective
function parameterized by the other player current decision (as a typical
Jacobi-type strategy), we show that the Newton iterate follows this approach
but with the objective function parameterized by a prediction of the other
player action, considering that they are following the same Newtonian strategy.
This interpretation allows us to present a new Newtonian algorithm where a
backtracking procedure is introduced in order to guarantee that the computed
Newtonian directions, for each player, are descent directions for their
corresponding parameterized functions. Thus, besides favoring global
convergence, our algorithm also favors true minimizers instead of maximizers or
saddle points, differently from the standard Newton method, which does not
consider the minimization structure of the problem in the non-convex case.
Thus, our method is more robust in comparison with other Jacobi-type strategies
or the pure Newtonian approach, which is corroborated by our illustrative
numerical experiments. We also present a proof of the well-definiteness of the
algorithm under some standard assumptions, together with a thorough analysis of
its convergence properties taking into account the game dynamics.
