Solving Quadratic Multi-Leader-Follower Games by Smoothing the Follower's Best Response

TL;DR

This paper introduces a smoothing approach to solve quadratic multi-leader-follower games with nonsmooth best response functions, proving existence, uniqueness, and convergence of solutions, and compares computational methods.

Contribution

It develops a novel smoothing technique to reformulate nonsmooth Nash equilibrium problems as smooth ones, enabling efficient solution methods and theoretical guarantees.

Findings

Existence and uniqueness of solutions for all smoothing parameters.

Accumulation points of Nash equilibria satisfy s-stationarity.

Numerical comparison shows efficiency of proposed algorithms.

Abstract

We derive Nash equilibria for a class of quadratic multi-leader-follower games using the nonsmooth best response function. To overcome the challenge of nonsmoothness, we pursue a smoothing approach resulting in a reformulation as a smooth Nash equilibrium problem. The existence and uniqueness of solutions are proven for all smoothing parameters. Accumulation points of Nash equilibria exist for a decreasing sequence of these smoothing parameters and we show that these candidates fulfill the conditions of s-stationarity and are Nash equilibria to the multi-leader-follower game. Finally, we propose an update on the leader variables for efficient computation and numerically compare nonsmooth Newton and subgradient methods.