  The multi-leader--multi-follower game (MLMFG) involves two or more leaders
and followers and serves as a generalization of the Stackelberg game and the
single-leader--multi-follower game (SLMFG). Although MLMFG covers wide range of
real-world applications, its research is still sparse. Notably, fundamental
solution methods for this class of problems remain insufficiently established.
A prevailing approach is to recast the MLMFG as an equilibrium problem with
equilibrium constraints (EPEC) and solve it using a solver. Meanwhile,
interpreting the solution to the EPEC in the context of MLMFG may be complex
due to shared decision variables among all leaders, followers' strategies that
each leader can unilaterally change, but the variables are essentially
controlled by followers. To address this issue, we introduce a response
function of followers' noncooperative game that is a function with leaders'
strategies as a variable. Employing this approach allows the MLMFG to be solved
as a single-level differentiable variational inequality using a smoothing
scheme for the followers' response function. We also demonstrate that the
sequence of solutions to the smoothed variational inequality converges to a
stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the
smoothing method by numerical experiments and confirm its validity.
