Polynomial representation for orthogonal projections onto subspaces of finite games

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The space of finite games can be decomposed into three orthogonal subspaces [5], which are the subspaces of pure potential games, nonstrategic games and pure harmonic games. The orthogonal projections onto these subspaces are represented as the Moore-Penrose inverses of the corresponding linear operators (i.e., matrices) [5]. Although the representation is compact and nice, no analytic method is given to calculate Moore- Penrose inverses of these linear operators. Hence using their results, one cannot verify whether a finite game belongs to one of these subspaces. In this paper, jumping over calculating Moore-Penrose inverses of these linear operators directly, via using group inverses, in the framework of the semitensor product of matrices, we give explicit polynomial representation for these orthogonal projections and for potential functions of potential games. Using our results, one not only can determine whether a finite game belongs to one of these subspaces, but also can find an arbitrary finite game belonging to one of them. Besides, we give formal definitions for these types of games by using their payoff functions. Based on these results, more properties of finite games are revealed.