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Generalized Multilinear Games and Vertical Tensor Complementarity Problems

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Abstract

This paper generalizes the multilinear game where the payoff tensor of each player is fixed to the generalized multilinear game where the payoff tensor of each player is selected from a nonempty set of tensors. We prove the existence of \(\varepsilon \)-Nash equilibria for generalized multilinear games under the assumption that all involved sets of tensors are bounded, and the existence of Nash equilibria for generalized multilinear games under the assumption that all involved sets of tensors are compact. In particular, when all involved sets of tensors are finite, we show that finding a Nash equilibrium point for the generalized multilinear game is equivalent to solving a vertical tensor complementarity problem, and establish a one-to-one correspondence between the Nash equilibrium point of the game and the solution of the vertical tensor complementarity problem.

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Acknowledgements

The authors wish to express the gratitude to the associate editor and referees for their valuable comments and advice. The authors were partially supported by the National Natural Science Foundation of China (Grant Nos. 12171357 and 12371309)

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300350, People’s Republic of China

    Qingyang Jia, Zheng-Hai Huang & Yong Wang

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  1. Qingyang Jia
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  2. Zheng-Hai Huang
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  3. Yong Wang
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Correspondence to Zheng-Hai Huang.

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Communicated by Francesco Zirilli.

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Jia, Q., Huang, ZH. & Wang, Y. Generalized Multilinear Games and Vertical Tensor Complementarity Problems. J Optim Theory Appl 200, 602–633 (2024). https://doi.org/10.1007/s10957-023-02360-8

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