Multilevel multi-leader multiple-follower games with nonseparable objectives and shared constraints

Multi-leader multi-follower (MLMF) games are hierarchical games in which a collection of players in the upper-level, called leaders, compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the players in the lower-level, called followers. MLMF games serve as an important model in game theory to address compromises among multiple interacting decision units within a hierarchical system where multiple decision makers are involved at each level of the hierarchy. Such problems arise in a variety of contexts in economics, engineering, operations research and other fields and are of great importance in strategic decision making. In this paper, MLMF games with multiple hierarchical levels are considered. A reformulation of some class of multilevel-MLMF games into multilevel single-leader single-follower games is proposed, and equivalence between the original problem and the reformulated one is established. Using this equivalent reformulation, a solution procedure is proposed for such games. The proposed solution approach can effectively solve some class of multilevel-MLMF games whose objective functions at each level have non-separable terms where the shared constraints at each level are polyhedral. Our results improve previous works of Kulkarni and Shanbhag (IEEE Trans Autom Control 60(12):3379–3384, 2015) and that of Kassa and Kassa (J Glob Optim 68(4):729–747, 2017).