On Decomposed Subspaces of Finite Games

This note provides the detailed description of the decomposed subspaces of finite games. First, the basis of potential games and the basis of non-strategic games are revealed. Then the bases of pure potential and pure harmonic subspaces are also obtained. These bases provide an explicit formula for the decomposition, and are convenient for investigating the properties of the corresponding subspaces. As an application, we consider the dynamics of networked evolutionary games (NEGs). Three problems are considered: 1) the dynamic equivalence of evolutionary games; 2) the dynamics of near potential games; and 3) the decomposition of NEGs.