Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 651–670 | Cite as

Gauss–Seidel Method for Multi-leader–follower Games

  • Atsushi Hori
  • Masao FukushimaEmail author


The multi-leader–follower game has many applications such as the bilevel structured market in which two or more enterprises, called leaders, have initiatives, and the other firms, called followers, observe the leaders’ decisions and then decide their own strategies. A special case of the game is the Stackelberg model, or the single-leader–follower game, which has been studied for many years. The Stackelberg game may be reformulated as a mathematical program with equilibrium constraints, which has also been studied extensively in recent years. On the other hand, the multi-leader–follower game may be formulated as an equilibrium problem with equilibrium constraints, in which each leader’s problem is an mathematical program with equilibrium constraints. However, finding an equilibrium point of an equilibrium problem with equilibrium constraints is much more difficult than solving a single mathematical program with equilibrium constraints, because each leader’s problem contains those variables which are common to other players’ problems. Moreover, the constraints of each leader’s problem depend on the other rival leaders’ strategies. In this paper, we propose a Gauss–Seidel type algorithm with a penalty technique for solving an equilibrium problem with equilibrium constraints associated with the multi-leader–follower game, and then suggest a refinement procedure to obtain more accurate solutions. We discuss convergence of the algorithm and report some numerical results to illustrate the behavior of the algorithm.


Multi-leader–follower game Equilibrium problem with equilibrium constraints S-stationary B-stationary 

Mathematics Subject Classification

91A06 91A10 90C33 



The authors are grateful to an anonymous referee for careful reading of the manuscript and helpful comments. This work was supported in part by Grant-in-Aid for Scientific Research (C) (26330029) from Japan Society for the Promotion of Science.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Systems and Mathematical Sciences, Graduate School of Science and EngineeringNanzan UniversityNagoyaJapan

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