Multi-parametric approach for multilevel multi-leader-multi-follower games using equivalent reformulations

Multilevel multi-leader multi-follower games address compromises among multiple interacting decision agents within a hierarchical system in which multiple followers are involved at each lower-level unit and more than one decision maker (multiple leaders) are involved in the upper-level. The leaders' decisions are affected not only by reactions of the followers but also by various relationships among the leaders themselves. In general, multiple-leaders multiple-followers (MLMF) game serve as an important modeling tool in game theory with many applications in economics, engineering, operations research and other fields. In this paper, we have reformulated a multilevel-MLMF game into an equivalent multilevel single-leader multi-follower (SLMF) game by introducing a suppositional (or dummy) leader, and hence the multiple leaders in the original problem become followers in the second level. If the resulting multilevel-SLMF game consists of separable terms and parameterized common terms across all the followers, then the problem is further transformed into equivalent multilevel programs having a single leader and single follower at each level of the hierarchy. The proposed solution approach can solve multilevel multi-leader multi-follower problems whose objective values at all levels have common but having different positive weights of non-separable terms. This result improves the work of Kulkarni and Shanbhag (2015).