Skew-symmetric games and symmetric-based decomposition of finite games

Lei Wang; Xinyun Liu; Ting Li; Jiandong Zhu; Lei Wang; Xinyun Liu; Ting Li; Jiandong Zhu

doi:10.3934/mmc.2022024

Mathematical Modelling and Control

2022, Volume 2, Issue 4: 257-267. doi: 10.3934/mmc.2022024

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Skew-symmetric games and symmetric-based decomposition of finite games

1.
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
2.
School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China

Received: 02 September 2022 Revised: 02 November 2022 Accepted: 18 December 2022 Published: 27 December 2022

In this paper, skew-symmetric games and a symmetric-based decomposition of finite games are investigated. First, necessary and sufficient conditions for testing skew-symmetric games are obtained by the semi-tensor product method based on adjacent transpositions. By using the obtained conditions for skew-symmetric games, a basis of the skew-symmetric game subspace is constructed. Then, the discriminant equations for a skew-symmetric game with the minimum number are derived. Furthermore, based on the basis of the skew-symmetric game subspace and that of the symmetric game subspace, a basis of the asymmetric game subspace is constructed, which completely solves the problem of symmetric-based decomposition of finite games. Finally, an illustrative example is provided to validate the obtained theoretical results.
- skew-symmetric games,
- asymmetric games,
- decomposition of finite games,
- semi-tensor product of matrices,
- adjacent transpositions
Citation: Lei Wang, Xinyun Liu, Ting Li, Jiandong Zhu. Skew-symmetric games and symmetric-based decomposition of finite games[J]. Mathematical Modelling and Control, 2022, 2(4): 257-267. doi: 10.3934/mmc.2022024

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Abstract

In this paper, skew-symmetric games and a symmetric-based decomposition of finite games are investigated. First, necessary and sufficient conditions for testing skew-symmetric games are obtained by the semi-tensor product method based on adjacent transpositions. By using the obtained conditions for skew-symmetric games, a basis of the skew-symmetric game subspace is constructed. Then, the discriminant equations for a skew-symmetric game with the minimum number are derived. Furthermore, based on the basis of the skew-symmetric game subspace and that of the symmetric game subspace, a basis of the asymmetric game subspace is constructed, which completely solves the problem of symmetric-based decomposition of finite games. Finally, an illustrative example is provided to validate the obtained theoretical results.

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