  In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics
and quadratic utility 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 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 game is analyzed 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 first case, the gradients are computed exactly
using the model whereas they are estimated using Monte-Carlo simulations in the
second case. Numerical experiments show the convergence of the two players'
controls as well as the utility function when the two algorithms are used in
different scenarios.
