Matrix Expressions of Symmetric and Skew-symmetric Games

A generalization of swap matrix, called a permutation matrix, is introduced in this paper. Based on permutation matrices, we further investigate the matrix expressions of symmetric games and skew-symmetric games. First, the constructing algorithm of permutation matrices is provided, and some interesting properties are proposed. Then some necessary and sufficient algebraic conditions to verify ordinary symmetric games and skew-symmetric games are presented. Finally, we prove that ordinary symmetry is equivalent to the combination of weak symmetry and anonymity. Some illustrative examples are given to validate the main results.