Optimization frameworks and sensitivity analysis of Stackelberg mean-field games

TL;DR

This paper introduces a new analytical framework for Stackelberg mean-field games with one leader and many followers, enabling sensitivity and robustness analysis under model uncertainty.

Contribution

It establishes an equivalence between the Stackelberg mean-field game and a minimax optimization problem, facilitating analytical and numerical analysis of Nash equilibria.

Findings

The game value is sensitive to model perturbations, showing non-vanishing sub-optimality.

A relaxed, more pessimistic approach improves near-optimal solutions under model uncertainty.

The framework allows for robustness analysis of the leader's strategy in uncertain environments.

Abstract

This paper proposes and studies a class of discrete-time finite-time-horizon Stackelberg mean-field games, with one leader and an infinite number of identical and indistinguishable followers. In this game, the objective of the leader is to maximize her reward considering the worst-case cost over all possible $\epsilon$-Nash equilibria among followers. A new analytical paradigm is established by showing the equivalence between this Stackelberg mean-field game and a minimax optimization problem. This optimization framework facilitates studying both analytically and numerically the set of Nash equilibria for the game; and leads to the sensitivity and the robustness analysis of the game value. In particular, when there is model uncertainty, the game value for the leader suffers non-vanishing sub-optimality as the perturbed model converges to the true model. In order to obtain a near-optimal solution, the leader needs to be more pessimistic with anticipation of model errors and adopts a relaxed version of the original Stackelberg game.