The paper introduces an analytical framework for decomposing finite strategic games within the multi-potential game(MPG) paradigm. Specifically, it partitions the finite non-cooperative game space into multi-potential subspace and multi-harmonic subspace, uncovering the common interests and conflicts of player subsets. Additionally, this approach provides a systematic methodology for calculating potential functions within MPGs. Within the MPG framework, a finite non-cooperative game characterized by its minimal potential index qG can be considered as a qG-person non-cooperative game. Furthermore, the Nash equilibrium of this qG-person non-cooperative game is examined. This analysis substantially enhances our comprehension of Nash equilibrium in finite non-cooperative games in terms of group welfare. Finally, an illustrative example is presented to demonstrate the theoretical results.