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### On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong

1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
• 收稿日期:2020-02-22 修回日期:2021-04-21 出版日期:2022-08-25 发布日期:2022-08-02
• 通讯作者: LI Haitao,Email:haitaoli09@gmail.com
• 作者简介:LIU Aixin,Email:aixinliu2020@163.com;LI Ping,Email:sdnulip@sdnu.edu.cn;YANG Xinrong,Email:xinrongyang2019@163.com
• 基金资助:
The paper was supported by the National Natural Science Foundation of China under Grant No. 62073202, and the Young Experts of Taishan Scholar Project under Grant No. tsqn201909076.

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong. On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games[J]. 系统科学与复杂性, 2022, 35(4): 1415-1428.

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong. On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1415-1428.

### On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games

LIU Aixin, LI Haitao, LI Ping, YANG Xinrong

1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
• Received:2020-02-22 Revised:2021-04-21 Online:2022-08-25 Published:2022-08-02
• Contact: LI Haitao,Email:haitaoli09@gmail.com
• Supported by:
The paper was supported by the National Natural Science Foundation of China under Grant No. 62073202, and the Young Experts of Taishan Scholar Project under Grant No. tsqn201909076.
This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games (FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third, by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.
This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games (FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third, by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.
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