Title
Mixed Leader-Follower Game With Constraints: A Lagrange Multiplier Method
Abstract
We discuss an open-loop backward Stackelberg differential game involving a single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation for which the terminal- instead of the initial condition is specified a priori; the decisions of the leader consist of a static terminal-perturbation and a dynamic linear-quadratic control.
In addition, the terminal control is subject to an expectation constraint. For the information pattern, the leader announces both terminal and open loop dynamic decisions at the initial time while taking into account the best response of the follower.
Then, two interrelated optimization problems are sequentially solved by the follower (a backward linear-quadratic problem) and the leader (a mixed terminal-perturbation and backward-forward LQ problem). Our open-loop Stackelberg equilibrium is represented by some coupled backward-forward stochastic differential equations (BFSDEs) with mixed initial-terminal conditions.