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

Linear-Quadratic Zero-Sum Mean-Field Type Games: Optimality Conditions and Policy Optimization