Full Length Paper Series A

Mathematical Programming

, Volume 141, Issue 1, pp 193-215

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Computing pure Nash and strong equilibria in bottleneck congestion games

  • Tobias HarksAffiliated withSchool of Business and Economics, Maastricht University Email author 
  • , Martin HoeferAffiliated withDepartment of Computer Science, RWTH Aachen University
  • , Max KlimmAffiliated withDepartment of Mathematics, TU Berlin
  • , Alexander SkopalikAffiliated withTU Dortmund


Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria—a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly NP-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.


Bottleneck congestion games Computation of strong equilibria Improvement dynamics

Mathematics Subject Classification

91A10 Noncooperative games 91A46 Combinatorial games 91-08 Computational methods 90B18 Communication networks