Solving quadratic multi-leader-follower games by smoothing the follower's best response

ABSTRACT

We analyse the existence of 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 fulfil 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.

Keywords:

• Multi-leader-follower games
• Nash equilibria
• game theory
• equilibrium problems with equilibrium constraints

Acknowledgments

Special thanks to Michael Ferris and Olivier Huber for fruitful discussions on model extensions. This work was supported by the DFG under Grant STE2063/2-1.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the DFG under Grant STE2063/2-1.

Notes on contributors

Michael Herty

Michael Herty is professor for applied mathematics at RWTH Aachen University and author of numerous research articles.

Sonja Steffensen

Sonja Steffensen is researcher in applied mathematics at RWTH Aachen University and expert for bilevel optimization as well as for optimal control.

Anna Thünen

Anna Thünen is PhD candidate at RWTH Aachen University and interested in bilevel games with finitely and infinitely many followers.