Dual expressions of decomposed subspaces of finite games

The vector space structure of non-cooperative games is investigated. For its 5 well known subspaces: pure potential games, non-strategic games, pure harmonic games, potential games, and harmonic games, dual expressions are discussed systematically in this paper. They are: (1) geometric expression, which provides the bases for each subspaces; (2) algebraic expression, which provides algebraic equation(s) for games in the subspaces to be satisfied. Some additional geometric relationship among some subspaces are also provided. Based on the geometric and/or algebraic structures the Nash equilibriums of each subspaces are also explored.