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Affine minimax variational inequalities and matrix two-person games

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Abstract

The concept of minimax variational inequality was proposed by Huy and Yen (Acta Math Vietnam 36, 265–281, 2011). This paper establishes some properties of monotone affine minimax variational inequalities and gives sufficient conditions for their solution stability. Then, by transforming a two-person zero sum game in matrix form (Barron in Game Theory. An Introduction, 2nd edn, Wiley, New Jersey, 2013) to a monotone affine minimax variational inequality, we prove that the saddle point set in mixed strategies of the matrix game is a nonempty compact polyhedral convex set and it is locally upper Lipschitz everywhere when the game matrix is perturbed. The rate of convergence of the extragradient method of Korpelevich applied to the matrix game is also discussed.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2018.308 and the Grant MOST 105-2221-E-039-009-MY3 (Taiwan). The authors would like to thank Prof. Nguyen Dong Yen for useful discussions on the subject. The careful readings and insightful comments of the two anonymous referees are gratefully acknowledged.

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  1. Department of Mathematics, Hanoi Pedagogical University 2, Vinh Phuc, Vietnam

    Duong Thi Kim Huyen

  2. Center for General Education, China Medical University, 40402, Taichung, Taiwan

    Jen-Chih Yao

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  1. Duong Thi Kim Huyen
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  2. Jen-Chih Yao
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Correspondence to Jen-Chih Yao.

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Huyen, D.T.K., Yao, JC. Affine minimax variational inequalities and matrix two-person games. J. Fixed Point Theory Appl. 23, 22 (2021). https://doi.org/10.1007/s11784-021-00851-7

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