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Gauss–Seidel Method for Multi-leader–follower Games

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Abstract

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.

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Notes

  1. There are several stationarity concepts in MPECs, such as strong (S-) stationarity, Bouligand (B-) stationarity and Clarke (C-) stationarity. Among those stationarity concepts, S-stationarity and B-stationarity are considered strong, in particular.

  2. We found that there are some typos in Corollary 3.5 of [16]. Hence we corrected them.

  3. The authors of [13] have confirmed that there were typos in the matrices \(H_{\mathrm {I}}\) and M shown there. We use the correct data in our numerical experiments and obtained almost the same results as those of Table 1 in [13].

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Acknowledgements

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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  1. Department of Systems and Mathematical Sciences, Graduate School of Science and Engineering, Nanzan University, Nagoya, Japan

    Atsushi Hori & Masao Fukushima

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  1. Atsushi Hori
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  2. Masao Fukushima
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Correspondence to Masao Fukushima.

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Hori, A., Fukushima, M. Gauss–Seidel Method for Multi-leader–follower Games. J Optim Theory Appl 180, 651–670 (2019). https://doi.org/10.1007/s10957-018-1391-5

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