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

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