Linear-quadratic mean field Stackelberg games: Master equations and decentralized feedback strategies


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This paper studies linear-quadratic Stackelberg games with a major player (leader) and \(N\) minor players (followers). To design decentralized strategies in the \(N+1\) player model, we construct a mean field limit model consisting of the leader and a representative follower and use dynamic programming to derive two master equations. We analyze quadratic solutions to the master equations and characterize existence and uniqueness by a pair of Riccati ordinary differential equations. The master equation-based solution is time consistent and provides decentralized feedback strategies in finite populations. As in feedback solutions of standard two-player dynamic Stackelberg games, the leader's equilibrium strategy in the mean field model does not have global optimality in minimizing its cost, and this feature makes the equilibrium analysis much more intricate than in mean field games (Huang, 2010). To characterize the performance of the decentralized strategies, we extend a procedure of Ekeland and Lazrak (2006) introduced for time inconsistent optimal control, so that the game of \(N+1\) players is interpreted as being played by a stream of short-lived agents. Subsequently, the set of decentralized strategies is shown to be an \(\varepsilon_N\)-Stackelberg equilibrium, where \(\varepsilon_N=o(1)\).

, 36 pages

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