Optimal leader functions for the reverse Stackelberg game: Splines and basis functions

In order to deal with the control of large-scale infrastructures, a multi-level approach may be required in which several groups of decision makers have different objectives. A game formulation can help to structure such a control task. The reverse Stackelberg game has a hierarchical structure in which the follower player acts subsequent to the leader´s disclosure of her leader function, which maps the follower decision space into the leader decision space. The problem of finding a leader function such that the leader´s objective function is optimized, given an optimal response with respect to the follower´s objective function, is in general a difficult problem. So far, the set of optimal affine leader functions has been delineated. However, for the more general class of nonlinear leader functions, no structured solution approach exists yet. In this paper, we consider several nonlinear structures for a leader function based on basis functions as well as based on interpolating splines and we show how these approaches can be adopted to find an optimal leader function.