Decompositions of finite games: From weighted inner product to standard inner product

To investigate the topological structure of finite games, various decomposions of finite games have been proposed. The inner product of vectors plays a key role in the decomposition of finite games. This paper considers the effect of different inner products on the orthogonal decomposition of finite games. We found that only when the compatible condition is satisfied, a common decomposition can be induced by the standard inner product and the weighted inner product simultaneously. To explain the result, we studied the existing decompositions, including potential based decomposition, zero-sum based decomposition, and normalization based decomposition. For zero-sum based decomposition and normalization based decomposition, we redefine their subspaces in a linear algebraic framework, which shows their physical meanings clearly. Bases of subspaces in these two decompositions are constructed.