  Various decomposition 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 find that only when the compatible condition
is satisfied, a common decomposition can be induced by the standard inner
product and the weighted inner product. To explain the result, we studied the
existing decompositions, including potential based decomposition, zero-sum
based decomposition, and symmetry based decomposition.
