  In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics
and quadratic cost are studied under infinite-horizon discounted utility
function. ZSMFTG are a class of games in which two decision makers whose
utilities sum to zero, compete to influence a large population of
indistinguishable agents. In particular, the case in which the transition and
utility functions depend on the state, the action of the controllers, and the
mean of the state and the actions, is investigated. The optimality conditions
of the game are analysed for both open-loop and closed-loop controls, and
explicit expressions for the Nash equilibrium strategies are derived. Moreover,
two policy optimization methods that rely on policy gradient are proposed for
both model-based and sample-based frameworks. In the model-based case, the
gradients are computed exactly using the model, whereas they are estimated
using Monte-Carlo simulations in the sample-based case. Numerical experiments
are conducted to show the convergence of the utility function as well as the
two players' controls.
