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

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.