A Regularized Nikaido-Isoda Function Approach to Multi-Leader-Follower Games

Abstract

A multi-leader--follower game (MLFG) is a hierarchical noncooperative game in which leaders compete at the upper level while taking into account the followers' best responses at the lower level. A typical approach to solving the MLFG reformulates it as an equilibrium problem with equilibrium constraints (EPECs) by replacing the lower-level game with its KKT conditions. Another approach, when each follower's response is unique, is to reformulate the MLFG as a Nash equilibrium problem by substituting these response functions into each leader's problem. However, both reformulations may lack scalability since higher-order derivatives may be required when solving the resulting problems. In this paper, we propose a new reformulation of the MLFG by exploiting a regularized Nikaido--Isoda function and approximating the MLFG by a single-level differentiable Nash equilibrium problem with a penalty parameter. The proposed reformulation neither requires derivative information on the followers' game nor assumes convexity of each follower's problem; hence, it can handle a broader class of MLFGs. Under global subanalyticity, we analyze the mathematical relationship between equilibria of the original MLFG and the proposed reformulation.