An algorithm for computing explicit expressions for orthogonal projections onto finite-game subspaces

The space of finite games can be decomposed into three orthogonal subspaces, which are the subspaces of pure potential games, nonstrategic games, and pure harmonic games as shown in a paper by Candogan et al. [2]. This decomposition provides a systematic characterization for the space of finite games. Explicit expressions for the orthogonal projections onto the subspaces are helpful in analyzing general properties of finite games in the subspaces and the relationships of finite games in different subspaces. In the work by Candogan et al., for the two-player case, explicit expressions for the orthogonal projections onto the subspaces are given. In the current paper, we give an algorithm for computing explicit expressions for the n-player case by developing our framework in the semitensor product of matrices and the group inverses of matrices. Specifically, using the algorithm, once we know the number of players, no matter whether we know their number of strategies or their payoff functions, we can obtain explicit expressions for the orthogonal projections. These projections can then be used to analyse the dynamical behaviors of games belonging to these subspaces.