On Systematic Computation of Optimal Nonlinear Solutions for the Reverse Stackelberg Game

In control of large-scale intelligent infrastructures, multilevel optimization can serve as a useful framework to deal with overall complex problems, especially in networks that already exhibit a natural hierarchy of decision makers with different objectives. In particular, solution methods from the hierarchical game theory can be adopted. Here, we focus on solving problems that belong to the specific class of reverse Stackelberg games. In this game, a follower player acts subsequent to the leader´s revelation of her so-called leader function, which maps the leader decision space to the follower decision space. In general, the problem of finding a leader function such that the leader´s objective function is optimized while taking into account a follower decision that is optimal for the follower, is difficult to solve. We provide a structured solution approach for the class of nonlinear leader functions and make a comparison with the evolutionary approaches proposed in the literature. In particular, a continuous multilevel optimization approach and a gridding approach are proposed to compute an optimal leader function based on basis functions. Also, leader functions derived by interpolation are discussed. All approaches are illustrated and compared in a worked example, in which the required computation times and deviation from the desired solution are considered.